62DD18B7d01

Lecture Notes on

Computational Geomechanics:

Inelastic Finite Elements for

Pressure Sensitive Materials

by:

Prof. Boris Jeremic; Boris Jeremi (University of California, Davis)

with significant contributions (chronologically, as noted in chapters) by:

Prof. Zhaohui Yang; (University of Alaska, Anchorage)

Dr. Zhao Cheng; (EarthMechanics Inc.)

Dr. Guanzhou Jie; (Wells Fargo Securities)

Prof. Kallol Sett; (University of Akron)

Prof. Mahdi Taiebat; باتيمهدی ط

(University of British Columbia)

Dr. Matthias Preisig (Ecole Polytechnique Federale de Lausanne)

Mr. Nima Tafazzoli; (University of California, Davis)

Version: April 29, 2010, 9:23

Copyright and Copyleft, Boris Jeremic and all contributors

http://sokocalo.engr.ucdavis.edu/~jeremic/

http://sokocalo.engr.ucdavis.edu/~jeremic/CG/_TeX-files_.tgz

http://sokocalo.engr.ucdavis.edu/~jeremic/

Computational Geomechanics: Lecture Notes 2

The use of the modeling and simulation system FEI (these lecture notes and accompanying modeling, computational and visual-

ization tools) for teaching, research and professional practice is strictly encouraged. Copyright and Copyleft are covered by GPL1 and

Woody’s (as in Guthrie) license (adapted by B.J.):

“This work is Copylefted and Copyrighted worldwide, by the Authors, for an indefinite period of time, and anybody caught using

it without our permission, will be mighty good friends of ourn, ’cause we don’t give a darn.

Read it.

Learn it.

Use it.

Hack it.

Debug it.

Yodel it.

Enjoy it.

We wrote it,

that’s all we wanted to do.”

Comments are much appreciated! Special thanks to (in chronological order): Miroslav Жivkovi

(Miroslav ZHivkovic), Dmitry J. Nicolsky, Andrzej Niemunis, Robbie Jaeger, Gi¸rgo Periklèou (Yiorgos

Perikleous)

1http://www.gnu.org/copyleft/gpl.html

Jeremic et al. Version: April 29, 2010, 9:23

Contents

1 Introduction (1996–2003–) 13

1.1 Specialization to Computational Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2 Tour of Computational Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.1 Special Equilibrium Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.2 Generalized Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.3 Sources of Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3 Verification and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3.1 Role of Verification and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.4 Verification and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.4.1 On Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.4.2 Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.4.3 Application Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Finite Element Formulation for Single Phase Material (dry) (1994–) 23

2.1 Formulation of the Continuum Mechanics Incremental Equations of Motion . . . . . . . . . . . . 23

2.2 Finite Element Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3 Isoparametric 8 – 20 Node Finite Element Definition . . . . . . . . . . . . . . . . . . . . . . . . 34

3 Small Deformation Elasto–Plasticity (1991–1994–2002–2006–2010–) 39

3.1 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2 Elasto–plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2.1 Constitutive Relations for Infinitesimal Plasticity . . . . . . . . . . . . . . . . . . . . . . . 42

3.2.2 On Integration Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2.3 Midpoint Rule Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2.4 Crossing the Yield Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.2.5 Singularities in the Yield Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.3 A Forward Euler (Explicit) Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.4 A Backward Euler (Implicit) Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.4.1 Single Vector Return Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.4.2 Backward Euler Algorithms: Starting Points . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.4.3 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3


3.4.4 Consistent Tangent Stiffness Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.4.5 Gradients to the Potential Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.5 Elastic–Plastic Material Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.5.1 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.5.2 Yield Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.5.3 Plastic Flow Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.5.4 Hardening–Softening Evolution Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.5.5 Tresca Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.5.6 von Mises Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.5.7 Drucker-Prager Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

3.5.8 Modified Cam-Clay Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

3.5.9 Dafalias-Manzari Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4 Probabilistic Elasto–Plasticity (2004–) 111

5 Stochastic Elastic–Plastic Finite Element Method (2006–) 113

6 Large Deformation Elasto–Plasticity (1996–2004–) 115

6.1 Continuum Mechanics Preliminaries: Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.1.1 Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.1.2 Deformation Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.1.3 Strain Tensors, Deformation Tensors and Stretch . . . . . . . . . . . . . . . . . . . . . . 118

6.1.4 Rate of Deformation Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.2 Constitutive Relations: Hyperelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.2.2 Isotropic Hyperelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.2.3 Volumetric–Isochoric Decomposition of Deformation . . . . . . . . . . . . . . . . . . . . 126

6.2.4 Simo–Serrin’s Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.2.5 Stress Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.2.6 Tangent Stiffness Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.2.7 Isotropic Hyperelastic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.3 Finite Deformation Hyperelasto–Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

6.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

6.3.2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

6.3.3 Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.3.4 Implicit Integration Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.3.5 Algorithmic Tangent Stiffness Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.4 Material and Geometric Non–Linear Finite Element Formulation . . . . . . . . . . . . . . . . . . 149

6.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

6.4.2 Equilibrium Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149



6.4.3 Formulation of Non–Linear Finite Element Equations . . . . . . . . . . . . . . . . . . . . 150

6.4.4 Computational Domain in Incremental Analysis . . . . . . . . . . . . . . . . . . . . . . . 151

7 Solution of Static Equilibrium Equations (1994–) 157

7.1 The Residual Force Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

7.2 Constraining the Residual Force Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

7.3 Load Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

7.4 Displacement Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

7.5 Generalized, Hyper–Spherical Arc-Length Control . . . . . . . . . . . . . . . . . . . . . . . . . . 160

7.5.1 Traversing Equilibrium Path in Positive Sense . . . . . . . . . . . . . . . . . . . . . . . . 163

7.5.2 Predictor step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

7.5.3 Automatic Increments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

7.5.4 Convergence Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

7.5.5 The Algorithm Progress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

8 Solution of Dynamic Equations of Motion (1989–2006–) 169

8.1 The Principle of Virtual Displacements in Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 169

8.2 Direct Integration Methods for the Equations of Dynamic Equilibrium . . . . . . . . . . . . . . . 169

8.2.1 Newmark Integrator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

8.2.2 HHT Integrator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

9 Finite Element Formulation for Porous Solid–Pore Fluid Systems (1999–2005–) 173

9.1 General form of u–p–U Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

9.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

9.1.2 Governing Equations of Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

9.1.3 Modified Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

9.2 Numerical Solution of the u–p–U Governing Equations . . . . . . . . . . . . . . . . . . . . . . . 178

9.2.1 Numerical Solution of solid part equilibrium equation . . . . . . . . . . . . . . . . . . . . 179

9.2.2 Numerical Solution of fluid part equilibrium equation . . . . . . . . . . . . . . . . . . . . 180

9.2.3 Numerical Solution of flow conservation equation . . . . . . . . . . . . . . . . . . . . . . 181

9.2.4 Matrix form of the governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

9.2.5 Choice of shape functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

9.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

9.3.1 Verification Example: Drilling of a well . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

9.3.2 Verification Example: The Case of a Spherical Cavity . . . . . . . . . . . . . . . . . . . . 190

9.3.3 Verification Example: Consolidation of a Soil Layer . . . . . . . . . . . . . . . . . . . . . 193

9.3.4 Verification Example: Line Injection of a fluid in a Reservoir . . . . . . . . . . . . . . . . 200

9.3.5 Verification: Shock Wave Propagation in Saturated Porous Medium . . . . . . . . . . . . 204

9.4 u-p Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

9.4.1 Governing Equations of Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204



9.4.2 Numerical Solutions of the Governing Equations . . . . . . . . . . . . . . . . . . . . . . . 205

10 Earthquake–Soil–Structure Interaction (2002–) 209

10.1 Dynamic Soil-Foundation-Structure Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

10.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

10.3 The Domain Reduction Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

10.3.1 Method Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

10.3.2 Method Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

10.4 Numerical Accuracy and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

10.4.1 Grid Spacing ∆h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

10.4.2 Time Step Length ∆t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

10.4.3 Nonlinear Material Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

10.5 Domain Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

10.6 Verification using one-dimensional Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . . 219

10.6.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

10.6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

10.7 Case History: Simple Structure on Nonlinear Soil . . . . . . . . . . . . . . . . . . . . . . . . . . 224

10.7.1 Simplified Models for Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

10.7.2 Full nonlinear 3d Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

11 Parallel Computing in Computational Geomechanics (1998–2000-2005–) 247

12 Practical Applications (1994–) 249

12.1 Consolidation of Clays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

12.2 Staged Construction Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

12.3 Seismic Wave Propagation in Soils (Ground Motions) . . . . . . . . . . . . . . . . . . . . . . . . 252

12.4 Static and Dynamic Behavior of Pile Foundations in Dry and Saturated Soils . . . . . . . . . . . 253

12.5 Static and Dynamics Behavior of Shallow Foundations . . . . . . . . . . . . . . . . . . . . . . . 254

A nDarray 265

A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

A.2 nDarray Programming Tool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

A.2.1 Introduction to the nDarray Programming Tool . . . . . . . . . . . . . . . . . . . . . . . 266

A.2.2 Abstraction Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

A.3 Finite Element Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

A.3.1 Stress, Strain and Elastoplastic State Classes . . . . . . . . . . . . . . . . . . . . . . . . 271

A.3.2 Material Model Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

A.3.3 Stiffness Matrix Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

A.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

A.4.1 Tensor Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

A.4.2 Fourth Order Isotropic Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274


http://cml00.engr.ucdavis.edu/~matthias/OpenSees/index.html#1dDRM


A.4.3 Elastic Isotropic Stiffness and Compliance Tensors . . . . . . . . . . . . . . . . . . . . . . 274

A.4.4 Second Derivative of θ Stress Invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

A.4.5 Application to Computations in Elastoplasticity . . . . . . . . . . . . . . . . . . . . . . . 276

A.4.6 Stiffness Matrix Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

A.5 Performance Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

A.6 Summary and Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

B Useful Formulae 279

B.1 Stress and Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

B.1.1 Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

B.1.2 Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

B.2 Derivatives of Stress Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

C Closed Form Gradients to the Potential Function 287

D Hyperelasticity: Detailed Derivations 295

D.1 Simo–Serrin’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

D.2 Derivation of ∂2volW/(∂CIJ ∂CKL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

D.3 Derivation of ∂2isoW/(∂CIJ ∂CKL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

D.4 Derivation of wA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

D.5 Derivation of YAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299




List of Figures

2.1 Motion of body in stationary Cartesian coordinate system . . . . . . . . . . . . . . . . . . . . . . 24

2.2 General three dimensional body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3 Isoparametric 8–20 node brick element in global and local coordinate systems . . . . . . . . . . . 35

3.1 The pictorial representation of integration algorithms in computational elasto–plasticity: General-

ized Midpoint schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2 Pictorial representation of the intersection point problem in computational elasto–plasticity . . . . 60

3.3 Pictorial representation of the corner point problem in computational elasto–plasticity . . . . . . . 62

3.4 The pictorial representation of the apex point problem in computational elasto–plasticity . . . . . 63

3.5 Influence regions in the meridian plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.6 Normalized iso-error maps of Von-Mises model with linear isotropic hardening. . . . . . . . . . . 72

3.7 Relative iso-error maps of Von-Mises model with linear isotropic hardening. . . . . . . . . . . . . 73

3.8 Normalized iso-error maps of Von-Mises model with Armstrong-Frederick kinematic hardening. . 73

3.9 Relative iso-error maps of Von-Mises model with Armstrong-Frederick kinematic hardening. . . . 74

3.10 Normalized iso-error maps of Drucker-Prager perfectly plastic model. . . . . . . . . . . . . . . . 74

3.11 Relative iso-error maps of Drucker-Prager perfectly plastic model. . . . . . . . . . . . . . . . . . 75

3.12 Normalized iso-error maps of Drucker-Prager model with Armstrong-Frederick kinematic hardening. 75

3.13 Relative iso-error maps of Drucker-Prager model with Armstrong-Frederick kinematic hardening. 76

3.14 Normalized iso-error maps of Dafalias-Manzari model with average elastic moduli. . . . . . . . . 76

3.15 Relative iso-error maps of Dafalias-Manzari model with average elastic moduli. . . . . . . . . . . 77

3.16 Normalized iso-error maps of Dafalias-Manzari model with constant elastic moduli. . . . . . . . . 77

3.17 Relative iso-error maps of Dafalias-Manzari model with constant elastic moduli. . . . . . . . . . 78

3.18 Typical convergence for Von-Mises model with linear isotropic hardening (tolerance value 1×10−7). 78

3.19 Typical convergence for Drucker-Prager model with Armstrong-Frederick kinematic hardening (tol-

erance value 1×10−7). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.20 Typical convergence for Dafalias-Manzari model (tolerance value 1×10−7). . . . . . . . . . . . . 80

3.21 Residual norms for typical iteration steps (tolerance value 1×10−6). . . . . . . . . . . . . . . . . 83

3.22 Yield surface patterns in the meridian plane for isotropic granular materials (from Lade (1988b)) . 91

3.23 Deviatoric trace of typical yield surface for pressure sensitive materials. . . . . . . . . . . . . . . . 91

9


3.24 Various types of evolution laws that control hardening and/or softening of elastic–plastic material

models: (a) Isotropic (scalar) controlling equivalent friction angle and isotropic yield stress. (b)

Rotational kinematic hardening (second order tensor) controlling pivoting around fixed point (usu-

ally stress origin) of the yield surface. (c) Translational kinematic hardening (second order tensor)

controlling translation of the yield surface. (d) Distortional (fourth order tensor) controlling the

shape of the yield surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.25 Schematic illustration of the yield, critical, dilatancy, and bounding surfaces in the π-plane of

deviatoric stress ratio space (after Dafalias and Manzari 2004). . . . . . . . . . . . . . . . . . . 106

6.1 Displacement, stretch and rotation of material vector dXI to new position dxi. . . . . . . . . . . 116

6.2 Illustration of the equation Fij = RikUkj = vikRkj . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.3 Relative velocity dvi of particle Q at point q relative to particle P at point p. . . . . . . . . . . . 121

6.4 Volumetric isochoric decomposition of deformation. . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.5 Multiplicative decomposition of deformation gradient: schematics. . . . . . . . . . . . . . . . . . 135

6.6 Motion of body in stationary Cartesian coordinate system . . . . . . . . . . . . . . . . . . . . . . 150

7.1 Spherical arc-length method and notation for one DOF system. . . . . . . . . . . . . . . . . . . . 159

7.2 Influence of ψu and ψf on the constraint surface shape. . . . . . . . . . . . . . . . . . . . . . . . 160

7.3 Simple illustration of Bifurcation and Turning point. . . . . . . . . . . . . . . . . . . . . . . . . . 164

9.1 Fluid mechanics of Darcy’s flow (wi) versus real flow (Ui = wi/n). . . . . . . . . . . . . . . . . . 176

9.2 Shape functions used for coupled analysis, displacement u and pore pressure p formulation . . . . 184

9.3 Boundary Conditions for Drilling of a Borehole . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

9.4 The mesh generation for the study of borehole problem . . . . . . . . . . . . . . . . . . . . . . . 187

9.5 The comparison of radial solid displacement between analytical solution and experimental result

for drained behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188


for undrained behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

9.7 The comparison of radial solid displacement between two analytical solutions and expanded boundary190

9.8 The mesh generation for the study of spherical cavity . . . . . . . . . . . . . . . . . . . . . . . . 192


for drained behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193


for undrained behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

9.11 The comparison of radial solid displacement between analytical solution for undrained behavior

and experimental result for drained behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

9.12 Consolidation of a Soil Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

9.13 The comparison between analytical solution and experimental results for the normalized p during

the consolidation process against normalized depth z = z/h for various normalized t = cf t/h2 . . 199

9.14 The comparison between analytical solution and experimental results for the settlement . . . . . . 199



9.15 The mesh generation for the study of line injection problem . . . . . . . . . . . . . . . . . . . . . 201

9.16 The comparison between analytical solution and experimental result for pore pressure . . . . . . . 202

9.17 The comparison between analytical solution and experimental result for radial displacement . . . . 203

9.18 The comparison between analytical solution and experimental result for radial displacement . . . . 203

9.19 Compressional wave in both solid and fluid, comparison with closed form solution. . . . . . . . . . 205

10.1 Large physical domain with the source of load Pe(t) and the local feature (in this case a soil–

foundation–building system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

10.2 Simplified large physical domain with the source of load Pe(t) and without the local feature (in

this case a soil–foundation–building system. Instead of the local feature, the model is simplified

so that it is possible to analyze it and simulate the dynamic response as to consistently propagate

the dynamic forces Pe(t). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

10.3 DRM: Single layer of elements between Γ and Γe is used to create P eff . . . . . . . . . . . . . . 215

10.4 Resulting acceleration using Linear and Newton-Raphson algorithms . . . . . . . . . . . . . . . . 217

10.5 Absorbing Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

10.6 The analyzed models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

10.7 Using total motions to calculate P eff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

10.8 ue and ue obtained from free-field model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

10.9 As in Figure 10.8 but with absorbing boundary at the base . . . . . . . . . . . . . . . . . . . . . 223

10.10Acceleration time histories and Fourier amplitude spectra’s . . . . . . . . . . . . . . . . . . . . . 225

10.11Transfer Function between Surface and Base of Soil Layer . . . . . . . . . . . . . . . . . . . . . 228

10.12Displacement Time-Histories at surface of 1d Soil Column . . . . . . . . . . . . . . . . . . . . . 229

10.13Fourier Amplitudes at surface of 1d Soil Column . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

10.14Acceleration time history at lowest free node . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

10.15Transfer functions between acceleration at the soil surface and the base . . . . . . . . . . . . . . 232

10.16Difference between results of analysis with different time steps . . . . . . . . . . . . . . . . . . . 233

10.17Averaged differences between results of analysis with different time steps . . . . . . . . . . . . . 234

10.18Two-dimensional quasi-plane-strain model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

10.19The boundary conditions of the 2d model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

10.20Elastic homogeneous free-field model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

10.21The 2d SFSI-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

10.22Eigenmodes of SFSI-system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

10.23Fourier amplitude spectrum of input motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

10.24Parametric study of 15 linear structures with varying natural frequency. . . . . . . . . . . . . . . 237

10.25Parametric study of 15 nonlinear structures with varying natural frequency . . . . . . . . . . . . . 238

10.26Displacements in x-direction at the top of the nonlinear structures . . . . . . . . . . . . . . . . . 239

10.27Displacements in x-direction at the top of structures 1 and 2 . . . . . . . . . . . . . . . . . . . . 240

10.28Moments at the base of the linear column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

10.29Moments at the base of the nonlinear column . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

10.30Fourier amplitude spectra of moments at the base of nonlinear column . . . . . . . . . . . . . . . 242



10.31Average equivalent plastic strain at time t = 12 s . . . . . . . . . . . . . . . . . . . . . . . . . . 243

10.32Average equivalent plastic strain at time t = 14 s . . . . . . . . . . . . . . . . . . . . . . . . . . 243

10.33The full 3d-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

10.34Resulting displacements and moments for 3d SFSI model . . . . . . . . . . . . . . . . . . . . . . 245

12.1 Finite element for 1D consolidation analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

12.2 Finite element for 1D consolidation analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250


Chapter 1

Introduction (1996–2003–)

These Lecture Notes are in a perpetual state of change. They were influenced by some of my reading, in

particular by a number of books and lecture notes (Bathe and Wilson Bathe and Wilson (1976), Felippa Felippa

(1993), Lubliner Lubliner (1990), Crisfield Crisfield (1991), Chen and Han Chen and Han (1988), Zienkiewicz and

Taylor Zienkiewicz and Taylor (1991a,b), Malvern Malvern (1969), Dunica and Kolundzija Dunica and Kolundzija

(1986), Kojic Kojic (1997), Hjelmstad Hjelmstad (1997), Oberkampf et al. Oberkampf et al. (2002)).

The intent is to collect writings related to research done within my research group and make it available to

students in a number of graduate courses I teach, including Computational Geomechanics (ECI285), Nonlinear

Finite Elements for Elastic–Plastic Problems (ECI280A) and Dynamic Finite Elements (ECI280B). Each course

has its own set of lecture notes, however they are all part of a larger write–up, from which parts are used in

compiling notes for each of the courses. A number of research students have also contributed material presented

here, and their contributions are acknowledged and much appreciated.

1.1 Specialization to Computational Mechanics

In this section we start from general mechanics and specialize our interest toward the field of computational

mechanics...

Mechanics

Computational Mechanics

Statics and Dynamics

Linear and Nonlinear Analysis

Elastic and Inelastic Analysis

Discretization Methods

13


The Solution Morass

Smooth and Rough nonlinearities

1.2 Tour of Computational Mechanics

In this section we describe various examples of equilibrium path and set up basic terminology.

Equilibrium Path

1.2.1 Special Equilibrium Points

Critical Points

Limit Points

Bifurcation Points

Turning Points

Failure Points

1.2.2 Generalized Response

1.2.3 Sources of Nonlinearities

Tonti Diagrams

1.3 Verification and Validation

• How do we use experimental simulations to develop and improve models

• How much can (should) we trust model implementations (verification)

• How much can (should) we trust numerical simulations (validation)

Trusting Simulation Tools

• Verification: The process of determining that a model implementation accurately represents the developer’s

conceptual description and specification. Mathematics issue. Verification provides evidence that the model

is solved correctly.

• Validation: The process of determining the degree to which a model is accurate representation of the real

world from the perspective of the intended uses of the model. Physics issue. Validation provides evidence

that the correct model is solved.



Importance of V & V

• V & V procedures are the primary means of assessing accuracy in modeling and computational simulations

• V & V procedures are the tools with which we build confidence and credibility in modeling and computational

simulations

Maturity of Computational Simulations NRC committee (1986) identified stages of maturity in CFD

• Stage 1: Developing enabling technologies (scientific papers published)

• Stage 2: Demonstration of and Confidence in technologies and tools (capabilities and limitations of tech-

nology understood)

• Stage 3: Compilation of technologies and tools (capabilities and limitations of technology understood)

• Stage 4: Spreading of the effective use (changes the engineering process, value exceeds expectations)

• Stage 5: Mature capabilities (fully dependable, cost effective design applications)

1.3.1 Role of Verification and Validation

MathematicalModel

ComputerImplementationDiscrete Mathematics

Continuum Mathematics

Programming

Analysis

Code Verification

SimulationComputer

ValidationModel

Reality Model Discovery

and Building

Alternative V & V Definitions IEEE V & V definitions (1984):

• Verification: The process of determining whether the products of a given phase of the software development

cycle fulfill the requirements established during the previous phase

• Validation: The process of evaluating software at the end of the software development process to ensure

compliance with software requirements.



• Other organization have similar definitions:

– Software quality assurance community

– American Nuclear Society (safety analysis of commercial nuclear reactors)

– International Organization for Standardization (ISO)

Certification and Accreditation

• Certification: A written guarantee that a system or component complies with its specified requirements

and is acceptable for operational use (IEEE (1990)).

– Written guarantee can be issued by anyone (code developer, code user, independent code evaluator)

– Code certification is more formal than validation documentation

• Accreditation: The official certification that a model or simulation is acceptable for use for a specific

purpose (DOD/DMSO (1994))

– Only officially designated entities can accredit

– Normally appointed by the customers/users of the code or legal authority

– Appropriate for major liability or public safety applications

Independence of Computational Confidence Assessment

1. V&V conducted by the computational tool developer, No Independence

2. V&V conducted by a user from same organization

3. V&V conducted by a computational tool evaluator contracted by developer’s organization

4. V&V conducted by a computational tool evaluator contracted by the customer

5. V&V conducted by a computational tool evaluator contracted by the a legal authority High Independence

1.4 Verification and Validation

Real World

Benchmark PDE solutionBenchmark ODE solutionAnalytical solution

Complete System

Subsystem Cases

Benchmark Cases

Unit ProblemsHighly accurate solution

Experimental Data

Conceptual Model

Computational Model

Computational Solution

ValidationVerification



Verification: The process of determining that a model implementation accurately represents the developer’s

conceptual description and specification.

• Identify and remove errors in computer coding

– Numerical algorithm

verification

– Software quality

assurance practice

• Quantification of the

numerical errors in

computed solution

Benchmark PDE solutionBenchmark ODE solutionAnalytical solution

Highly accurate solution

Conceptual Model

Computational Model


Verification

Validation: The process of determining the degree to which a model is accurate representation of the real

world from the perspective of the intended uses of the model.

• Tactical goal:

Identification and minimization of

uncertainties and errors in

the computational model

• Strategic goal:

Increase confidence in

the quantitative predictive

capability of

the computational model

Real World

Complete System

Subsystem Cases

Benchmark Cases

Unit Problems

Experimental Data

Conceptual Model

Computational Model


Validation

1.4.1 On Validation

Goals of Validation Quantification of uncertainties and errors in the computational model and the experimental

measurements

• Goals on validation

– Tactical goal: Identification and minimization of uncertainties and errors in the computational model

– Strategic goal: Increase confidence in the quantitative predictive capability of the computational model

• Strategy is to reduce as much as possible the following:

– Computational model uncertainties and errors

– Random (precision) errors and bias (systematic) errors in the experiments

– Incomplete physical characterization of the experiment



Computational Model

Conceptual Model


Highly accurate solutionAnalytical solutionBenchmark ODE solutionBenchmark PDE solution

Comparison of agreement

Validation Procedure Uncertainty

• Aleatory uncertainty → inherent variation associated with the physical system of the environment (variation

in external excitation, material properties...). Also know known as irreducible uncertainty, variability and

stochastic uncertainty.

• Epistemic uncertainty → potential deficiency in any phase of the modeling process that is due to lack

of knowledge (poor understanding of mechanics...). Also known as reducible uncertainty, model form

uncertainty and subjective uncertainty

Types of Physical Experiments

• Traditional Experiments

– Improve the fundamental understanding of physics involved

– Improve the mathematical models for physical phenomena

– Assess component performance

• Validation Experiments

– Model validation experiments

– Designed and executed to quantitatively estimate mathematical model’s ability to simulate well defined

physical behavior

– The simulation tool (SimTool) (conceptual model, computational model, computational solution) is

the customer

Validation Experiments

• A validation experiment should be jointly designed and executed by experimentalist and computationalist

– Need for close working relationship from inception to documentation



– Elimination of typical competition between each

– Complete honesty concerning strengths and weaknesses of both experimental and computational sim-

ulations

• A validation Experiment should be designed to capture the relevant physics

– Measure all important modeling data in the experiment

– Characteristics and imperfections of the experimental facility should be included in the model

• A validation experiment should use any possible synergism between experiment and computational ap-

proaches

– Offset strength and weaknesses of computations and experiments

– Use high confidence simulations for simple physics to calibrate of improve the characterization of the

experimental facility

– Conduct experiments with a hierarchy of physics complexity to determine where the computational

simulation breaks (remember, SimTool is the customer!)

• Maintain independence between computational and experimental results

– Blind comparison, the computational simulations should be predictions

– Neither side is allowed to use fudge factors, parameters

• Validate experiments on unit level problems, hierarchy of experimental measurements should be made which

present an increasing range of computational difficulty

– Use of qualitative data (e.g. visualization) and quantitative data

– Computational data should be processed to match the experimental measurement techniques

• Experimental uncertainty analysis should be developed and employed

– Distinguish and quantify random and correlated bias errors

– Use symmetry arguments and statistical methods to identify correlated bias errors

– Make uncertainty estimates on input quantities needed by the SimTool

1.4.2 Prediction

• Prediction: use of computational model to foretell the state of a physical system under consideration under

conditions for which the computational model has not been validated

• Validation does not directly make a claim about the accuracy of a prediction

– Computational models are easily misused (unintentionally or intentionally)

– How closely related are the conditions of the prediction and specific cases in validation database

– How well is physics of the problem understood



Relation Between Validation and Prediction Quantification of confidence in a prediction:

• How do I quantify validation and its inference value in a predictions?

• How do I quantify verification and its inference value in a prediction?

• How far are individual experiments in my validation database from my physical system of interest?

1.4.3 Application Domain

System Parameter

Sys

tem

com

plex

ity

DomainValidation

DomainApplication

DomainApplication

System Parameter

Sys

tem

com

plex

ityDomain

Validation

System Parameter

Sys

tem

com

plex

ity

ApplicationDomain

DomainValidation

Inference

• Rarely applicable to engineering systems (certainly not for infrastructure objects like bridges, buildings, port

facilities, dams...)

• Even if the engineering system is small, environmental influences (generalized loads, conditions, wear and

tare) are hard to predict

• Human factors (take Mars rover example with a memory overflow, operator forgot to flush the memory...)

• Inference ⇒ Based on physics or statistics

• Validation domain is actually an aggregation of tests and thus might not be convex (bifurcation of behavior)

• Experimental facilities (e.g. NEES sites) provide for validation domain that are exclusively non–overlapping

with the application domain.

Importance of Models and Numerical Simulations

• Verified and Validated models can be used for assessing behavior of

– components or

– complete systems,

• with the understanding that the environmental influences cannot all be taken into the account prior to

operation

• but with a good model, their influence on system behavior can be assessed as need be (before or after the

event)



Prediction under Uncertainty

• Ever present uncertainty needs to be estimated for predictions

• Identify all relevant sources of uncertainty

• Create mathematical representation of individual sources

• Propagate representation of sources through modeling and simulation process (Probabilistic Elastic Plastic

Theory)

• Short Course on Verification and Validation in Computational Mechanics, by Dr. Willliam Oberkampf,

Sandia National Laboratories July 27th, 2003, Albuquerque, New Mexico.

• Material from Verification and Validation in Computational Mechanics web site http://www.usacm.org/vnvcsm/

at the USACM.

• William L. Oberkampf, Timothy G. Trucano, and Charles Hirsch. Verification, validation

and predictive capability in computational engineering and physics. In Proceedings of the Foundations for

Verification and Validation on the 21st Century Workshop, pages 1–74, Laurel, Maryland, October 22-23

2002. Johns Hopkins University / Applied Physics Laboratory.

• Patrick J. Roache. Verification and Validation in Computational Science and Engineering. Hermosa

publishers, 1998. ISBN 0-913478-08-3.

• Boris Jeremic, Gerik Scheuermann, Jan Frey, Zhaohui Yang, Bernd Hamman, Kenneth I. Joy and Hans

Haggen. Tensor Visualizations in Computational Geomechanics. International Journal for Numerical and

Analytical Methods in Geomechanics incorporating Mechanics of Cohesive–Frictional Materials, Vol 26.

Issue 10, pp 925-944, August 2002.




Chapter 2

Finite Element Formulation for Single

Phase Material (dry) (1994–)

2.1 Formulation of the Continuum Mechanics Incremental Equations

of Motion

Consider1 the motion of a general body in a stationary Cartesian coordinate system, as shown in Figure (2.1),

and assume that the body can experience large displacements, large strains, and nonlinear constitutive response.

The aim is to evaluate the equilibrium positions of the complete body at discrete time points 0,∆t, 2∆t, . . . ,

where ∆t is an increment in time. To develop the solution strategy, assume that the solutions for the static and

kinematic variables for all time steps from 0 to time t inclusive, have been obtained. Then the solution process

for the next required equilibrium position corresponding to time t+∆t is typical and would be applied repetitively

until a complete solution path has been found. Hence, in the analysis one follows all particles of the body in their

motion, from the original to the final configuration of the body. In so doing, we have adopted a Lagrangian ( or

material ) formulation of the problem.

In the Lagrangian incremental analysis approach we express the equilibrium of the body at time t+ ∆t using

the principle of virtual displacements. Using tensorial notation2 this principle requires that:

∫

t+∆tV

t+∆tτij δ t+∆teijt+∆tdV = t+∆tR (2.1)

where the t+∆tτij are Cartesian components of the Cauchy stress tensor, the t+∆teij are the Cartesian components

of an infinitesimal strain tensor, and the δ means ”variation in” i.e.:

δ t+∆teij = δ1

2

(∂ui

∂ t+∆txj+

∂uj∂ t+∆txi

)

=1

2

(∂δui

∂ t+∆txj+

∂δuj∂ t+∆txi

)

(2.2)

1detailed derivations and explanations are given in Bathe (1982)2Einstein’s summation rule is implied unless stated differently, all lower case indices (i, j, p, q, m, n, o, r, s, t, . . . ) can have values

of 1, 2, 3, and values for capital letter indices will be specified where need be.

23


t+ t∆

t+ t∆ ui

tui

t+ t∆u ix i0

ix0

ixt

Ω 0

Ωn

Ωn+1

1

22 20 t

2 3)P(

0

0V

t

t

AV

V

0

3330 tx x x

x x x

x x x,

, ,

t+ t

t+ t t+ t t+ t

t+ t

t+ t

t+ t

1tP( x , 2

tx , )t3

x

01P( x )

03x0

2x

0

Configuration

at time

t

Configuration

at time

Configuration

at time t+ t

∆

∆ ∆ ∆

∆

∆

∆

∆

, ,

,,

,

t+ t∆

i

i

ixx

x

1x,1xt,1x

t

A

A

i=1,2,3

++

+=

==

Figure 2.1: Motion of body in stationary Cartesian coordinate system

It should be noted that Cauchy stresses are ”body forces per unit area” in the configuration at time t+ ∆t, and

the infinitesimal strain components are also referred to this as yet unknown configuration. The right hand side of

equation (2.1), i.e. t+∆tR is the virtual work performed when the body is subjected to a virtual displacement at

time t+ ∆t:

t+∆tR =

∫

t+∆tV

(t+∆tfBi − ρut+∆t

i

)δut+∆ti

t+∆tdV +

∫

t+∆tS

t+∆tfSi δut+∆ti

t+∆tdS (2.3)

where t+∆tfBi and t+∆tfSi are the components of the externally applied body and surface force vectors, re-

spectively, and −ρuit+∆t is the inertial body force that is present if accelerations are present3, δui is the ith

component of the virtual displacement vector.

A fundamental difficulty in the general application of equation (2.1) is that the configuration of the body

at time t+ ∆t is unknown. The continuous change in the configuration of the body entails some important

consequences for the development of an incremental analysis procedure. For example, an important consideration

must be that the Cauchy stress at time t + ∆t cannot be obtained by simply adding to the Cauchy stresses at

time t a stress increment which is due only to the straining of the material. Namely, the calculation of the Cauchy

stress at time t+∆t must also take into account the rigid body rotation of the material, because the components

of the Cauchy stress tensor are not invariant with respect to the rigid body rotations4.

3This is based on D’Alembert’s principle.4However, that problem will not be addressed in this work since this work deals with Material–Nonlinear–Only analysis of solids,

thus excluding large displacement and large strain effects.



The fact that the configuration of the body changes continuously in a large deformation analysis is dealt with

in an elegant manner by using appropriate stress and strain measures and constitutive relations. When solving the

general problem5 one possible approach6 is given in Simo (1988). Previous discussion was oriented toward small

deformation, small displacement analysis leading to the use of Cauchy stress tensor τij and small strain tensor

eij .

In the following, we will briefly cover some other stress and strain measures particularly useful in large strain

and large displacement analysis.

The basic equation that we want to solve is relation (2.1), which expresses the equilibrium and compatibility

requirements of the general body considered in the configuration corresponding to time t+ ∆t. The constitutive

equations also enter (2.1) through the calculation of stresses. Since in general the body can undergo large

displacements and large strains, and the constitutive relations are nonlinear, the relation in (2.1) cannot be solved

directly. However, an approximate solution can be obtained by referring all variables to a previously calculated

known equilibrium configuration, and linearizing the resulting equations. This solution can then be improved by

iterations.

To develop the governing equations for the approximate solution obtained by linearization we recall that the

solutions for time 0,∆t, 2∆t, . . . , t have already been calculated and that we can employ the fact that the 2nd

Piola–Kirchhoff stress tensor is energy conjugate to the Green–Lagrange strain tensor:

∫

0V

t0Sij δ

t0ǫij

0dV =

∫

0V

(

0ρtρ

0txi,m

tτmn0txj,n

)(t0xk,i δ

ttekl

t0 xl,j

)0dV =

∫

0V

0ρtρ

tτmn δt0emn

0dV (2.4)

because:

t0xk,l

0txl,m = δkm

and since:

0ρ0dV = tρtdV

we have:

∫

0V

t0Sij δ

t0ǫij

0dV =

∫

0V

tτmn δttemn

tdV (2.5)

where 2nd Piola–Kirchhoff stress tensor is defined as:

t0Sij =

0ρtρ

0txi,m

tτmn0txj,n (2.6)

5That is, large displacements, large deformations and nonlinear constitutive relations.6This is still a ”hot” research topic!



where 0txj,n = ∂0xi

∂txmand

0ρtρ represents the ratio of the mass density at time 0 and time t, and the Green–Lagrange

strain is defined as:

t0ǫij =

1

2

(t0ui,j + t

0uj,i + t0uk,i

t0uk,j

)(2.7)

Then, by employing (2.5) we refer the stresses and strains to one of these known equilibrium configurations.

The choice lies between two formulations which have been termed total Lagrangian and updated Lagrangian

formulations.

In the total Lagrangian formulations, also termed Lagrangian formulation, all static and kinematic variables

are referred to the initial configuration at time 0, while in the updated Lagrangian formulation all static and

kinematic variables are referred to the configuration at time t. Both the total Lagrangian and updated Lagrangian

formulations include all kinematic nonlinear effects due to large displacement, large rotations and large strains,

but whether the large strain behavior is modeled appropriately depends on the constitutive relations specified.

The only advantage of using one formulation rather than the other lies in its greater numerical efficiency.

Using (2.5) in the total Lagrangian formulations we consider this basic equation:

∫

0V

t+∆t0 Sij δ

t+∆t0 ǫij

0dV = t+∆tR (2.8)

while in the updated Lagrangian formulations we consider:

∫

tV

t+∆tt Sij δ

t+∆tt ǫij

tdV = t+∆tR (2.9)

in which t+∆tR is the external virtual work as defined in (2.3). Approximate solution to the (2.8) and (2.9) can be

obtained by linearizing these relations. By comparing the total Lagrangian and updated Lagrangian formulations

we can observe that they quite analogous and that, in fact, the only theoretical difference between the two

formulations lies in the choice of different reference configurations for the kinematic and static variables. If in the

numerical solution the appropriate constitutive tensors are employed, identical results are obtained.

Coupling of large deformation, large displacement and material nonlinear analysis is still the topic of research

in the research community. Possible direction may be the use of both Lagrangian and Eulerian formulation

intermixed in one scheme.

2.2 Finite Element Discretization

Consider the equilibrium of a general three–dimensional body such as in Figure (2.2) (Bathe, 1996). The external

forces acting on a body are surface tractions fSi and body forces fBi . Displacements are ui and strain tensor7 is

eij and the stress tensor corresponding to strain tensor is τij .

Assume that the externally applied forces are given and that we want to solve for the resulting displacements,

strains and stresses. One possible approach to express the equilibrium of the body is to use the principle of virtual

7 small strain tensor as defined in equation: eij = 12

(ui,j + uj,i).



x2

x1

x3

1

1

2

2

3

3fB

f Bf B

fS

fS

fS

1r

r3

r2

Figure 2.2: General three dimensional body

displacements. This principle states that the equilibrium of the body requires that for any compatible, small

virtual displacements8 imposed onto the body, the total internal virtual work is equal to the total external virtual

work. This statement can be mathematically expressed using equation (2.10) for the body at time t + ∆t, and

since we are using the incremental approach let us drop the time dimension, so that all the equations are imposed

for the given increment9, at time t+ ∆t. The equation is now, using tensorial notation10:

∫

V

τij δeij dV =

∫

V

(fBi − ρui

)δui dV +

∫

S

fSi δui dS (2.10)

The internal work given on the left side of (2.10) is equal to the actual stresses τij going through the virtual

strains δeij that corresponds to the imposed virtual displacements. The external work is on the right side of (2.10)

and is equal to the actual (surface) forces fSi and (body) forces fBi −ρui going through the virtual displacements

δui.

It should be emphasized that the virtual strains used in (2.10) are those corresponding to the imposed body

and surface virtual displacements, and that these displacements can be any compatible set of displacements that

satisfy the geometric boundary conditions. The equation in (2.10) is an expression of equilibrium, and for different

virtual displacements, correspondingly different equations of equilibrium are obtained. However, equation (2.10)

also contains the compatibility and constitutive requirements if the principle is used in the appropriate manner.

8which satisfy the essential boundary conditions.9t + ∆t will be dropped from now one in this chapter.

10Einstein’s summation rule is implied unless stated differently, all lower case indices (i, j, p, q, m, n, o, r, s, t, . . . ) can have values

of 1, 2, 3, and values for capital letter indices will be specified where need be.



Namely, the displacements considered should be continuous and compatible and should satisfy the displacement

boundary conditions, and the stresses should be evaluated from the strains using appropriate constitutive relations.

Thus, the principle of virtual displacements embodies all requirements that need be fulfilled in the analysis of a

problem in solid and structural mechanics. The principle of virtual displacements can be directly related to the

principle that total potential Π of the system must be stationary.

In the finite element analysis we approximate the body in Figure (2.2) as an assemblage of discrete finite

elements with the elements being interconnected at nodal points on the element boundaries. The displacements

measured in a local coordinate system r1, r2 and r3 within each element are assumed to be a function of the

displacements at the N finite element nodal points:

u ≈ ua = HI uIa (2.11)

where I = 1, 2, 3, . . . , n and n is number of nodes in a specific element, a = 1, 2, 3 represents a number of

dimensions (can be 1 or 2 or 3), HI represents displacement interpolation vector, uIa is the tensor of global

generalized displacement components at all element nodes. The use of the term generalized displacements means

that both translations, rotations, or any other nodal unknown are modeled independently. Here specifically only

translational degrees of freedom are considered. The strain tensor is defined as:

eab =1

2(ua,b + ub,a) (2.12)

and the by using (2.11) we can define the approximate strain tensor:

eab ≈ eab =1

2(ua,b + ub,a) =

=1

2

(

(HI uIa),b + (HI uIb),a

)

=

=1

2((HI,b uIa) + (HI,a uIb)) (2.13)

The most general stress–strain relationship11 for an isotropic material is:

τab = Eabcd(ecd − e0cd

)+ τ0

ab (2.14)

where τab is the approximate Cauchy stress tensor, Eabcd is the constitutive tensor12, ecd is the infinitesimal

approximate strain tensor, e0cd is the infinitesimal initial strain tensor and τ0ab is the initial Cauchy stress tensor.

Using the assumption of the displacements within each finite element, as expressed in (2.11), we can now derive

equilibrium equations that corresponds to the nodal point displacements of the assemblage of finite elements. We

can rewrite (2.10) as a sum13 of integrations over the volume and areas of all finite elements:

⋃

m

∫

Vm

τab δeab dVm =

⋃

m

∫

Vm

(fBa − ρua

)δua dV

m +⋃

m

∫

Sm

fSa δuSa dSm (2.15)

11in terms of exact stress and strain fields but it holds for approximate fields as well.12This tensor can be elastic or elastoplastic constitutive tensor.13Or, more correctly as a union

S

m since we are integrating over the union of elements.



where m = 1, 2, 3, . . . , k and k is the number of elements. It is important to note that the integrations in (2.15)

are performed over the element volumes and surfaces, and that for convenience we may use different element

coordinate systems in the calculations. If we substitute equations (2.11), (2.12), (2.13) and (2.14) in (2.15) it

follows:

⋃

m

∫

Vm

(Eabcd

(ecd − e0cd

)+ τ0

ab

)δ

(1

2(HI,b uIa +HI,a uIb)

)

dV m =

⋃

m

∫

Vm

fBa δ (HI uIa) dVm −

⋃

m

∫

Vm

HJ üJa ρ δ (HI uIa) dVm +

⋃

m

∫

Sm

fSa δ (HI uIa) dSm (2.16)

or:

⋃

m

∫

Vm

(

Eabcd

((1

2(HJ,d uJc +HJ,c uJd)

)

− e0cd

)

+ τ0ab

)

δ

(1

2(HI,b uIa +HI,a uIb)

)

dV m =

=⋃

m

∫

Vm

fBa δ (HI uIa) dVm −

⋃

m

∫

Vm

HJ üJa ρ δ (HI uIa) dVm +

⋃

m

∫

Sm

fSa δ (HI uIa) dSm

(2.17)

We can observe that δ in the previous equations represents a virtual quantity but the rules for δ are quite similar to

regular differentiation so that δ can enter the brackets and ”virtualize” the nodal displacement14. It thus follows:

⋃

m

∫

Vm

(

Eabcd

((1


)

− e0cd

)

+ τ0ab

) (1

2(HI,b δuIa +HI,a δuIb)

)

dV m =

=⋃

m

∫

Vm

fBa (HIδuIa) dVm −

⋃

m

∫

Vm

HJ üJa ρ (HIδuIa) dVm +

⋃

m

∫

Sm

fSa (HIδuIa) dSm

(2.18)

Let us now work out some algebra in the left hand side of equation (2.18):

⋃

m

∫

Vm

(

Eabcd

((HJ,d uJc +HJ,c uJd)

2

)

− Eabcde0cd + τ0

ab

) ((HI,b δuIa +HI,a δuIb)

2

)

dV m =

=⋃

m

∫

Vm


⋃

m

∫

Vm

HJ üJa ρ HIδuIa dVm +

⋃

m

∫

Sm

fSa (HIδuIa) dSm

(2.19)

and further:

14since they are driving variables that define overall displacement field through interpolation functions



⋃

m

∫

Vm

((1


)

Eabcd

(1


))

dV m +

+⋃

m

∫

Vm

(

−Eabcd e0cd(

1


))

dV m +

+⋃

m

∫

Vm

(τ0ab

)(

1


)

dV m =

⋃

m

∫

Vm

fBa (HIδuIa) dVm

−⋃

m

∫

Vm

HJ üJa ρ HIδuIa dVm

+⋃

m

∫

Sm

fSa (HIδuIa) dSm (2.20)

Several things should be observed in the equation (2.20). Namely, the first three lines in the equation can be

simplified if one takes into account symmetries of Eijkl and τij . In the case of the elastic stiffness tensor Eijkl

major and both minor symmetries exist. In the case of the elastoplastic stiffness tensor, such symmetries exists if

a flow a rule is associated. If flow rule is non–associated, only minor symmetries exist while major symmetry is

destroyed15. As a matter of fact, both minor symmetries in Eijkl are the only symmetries we need, and the first

line of (2.20) can be rewritten as:

⋃

m

∫

Vm

((1

2(HJ,d uJc +HJ,c ujd)

)

Eabcd

(1


))

dV m =

=⋃

m

∫

Vm

(HJ,d uJc) Eabcd (HI,b δuIa) dVm =

=⋃

m

∫

Vm

(HI,b δuIa) Eabcd (HJ,d uJc) dVm (2.21)

Similar simplifications are possible in second and third line of equation (2.20). Namely, in the second line we can

use both minor symmetries of Eijkl so that:

⋃

m

∫

Vm

(

−Eabcd e0cd(

1


))

dV m =

=⋃

m

∫

Vm

(−Eabcd e0cd (HI,b δuIa)

)dV m (2.22)

and the third line can be simplified due to the symmetry in Cauchy stress tensor τij as:

15for more on stiffness tensor symmetries see sections (3.5.1, 3.3 and 3.4)



⋃

m

∫

Vm

(τ0ab

)(

1


)

dV m =

=⋃

m

∫

Vm

(τ0ab

)(HI,b δuIa) dV

m (2.23)

After these simplifications, equation (2.20) looks like this:

⋃

m

∫

Vm

(HI,b δuIa) Eabcd (HJ,d uJc) dVm +

+⋃

m

∫

Vm

(−Eabcd e0cd (HI,b δuIa)

)dV m +

⋃

m

∫

Vm

(τ0ab

)(HI,b δuIa) dV

m =

=⋃

m

∫

Vm


⋃

m

∫

Vm

HJ üJa ρ HIδuIa dVm +

⋃

m

∫

Sm

fSa (HIδuIa) dSm (2.24)

or if we leave the unknown nodal accelerations16 üJc and displacements uJc on the left hand side and move all

the known quantities on to the right hand side:

⋃

m

∫

Vm

HJ δac üJc ρ HIδuIa dVm +

⋃

m

∫

Vm

(HI,b δuIa) Eabcd (HJ,d uJc) dVm =

=⋃

m

∫

Vm

fBa (HIδuIa) dVm +

⋃

m

∫

Sm

fSa (HIδuIa) dSm +

+⋃

m

∫

Vm

(Eabcd e

0cd (HI,b δuIa)

)dV m −

⋃

m

∫

Vm

(τ0ab

)(HI,b δuIa) dV

m (2.25)

To obtain the equation for the unknown nodal generalized displacements from (2.25), we invoke the vir-

tual displacement theorem which states that virtual displacements are any, non zero, kinematically admissible

displacements. In that case we can factor out nodal virtual displacements δuIa so that equation (2.25) becomes:

[⋃

m

∫

Vm

HJ δac üJc ρ HI dVm +

⋃

m

∫

Vm

(HI,b) Eabcd (HJ,d uJc) dVm

]

δuIa =

=⋃

m

[∫

Vm

fBa HI dVm

]

δuIa +⋃

m

[∫

Sm

fSa HI dSm

]

δuIa +

+⋃

m

[∫

Vm

(Eabcd e

0cd HI,b

)dV m

]

δuIa −⋃

m

[∫

Vm

(τ0ab

)HI,b dV

m

]

δuIa (2.26)

and now we can just cancel δuIa on both sides:

16It is noted that üJc = δac üJa relationship was used here, where δac is the Kronecker delta.



⋃

m

∫

Vm

HJ δac ρ HI üJcdVm +

⋃

m

∫

Vm

(HI,b) Eabcd (HJ,d uJc) dVm =

=⋃

m

∫

Vm

fBa HI dVm +

⋃

m

∫

Sm

fSa HI dSm +

+⋃

m

∫

Vm

(Eabcd e

0cd HI,b

)dV m −

⋃

m

∫

Vm

(τ0ab

)HI,b dV

m (2.27)

One should also observe that in the first line of equation (2.27) generalized nodal accelerations üJc and generalized

nodal displacements uJc are unknowns that are not subjected to integration so they can be factored out of the

integral:

⋃

m

∫

Vm

HJ δac ρ HI dVm üJc

+⋃

m

∫

Vm

HI,b Eabcd HJ,d dVm uJc

=⋃

m

∫

Vm

fBa HI dVm +

⋃

m

∫

Sm

fSa HI dSm +

+⋃

m

∫

Vm

(Eabcd e

0cd HI,b

)dV m −

⋃

m

∫

Vm

(τ0ab

)HI,b dV

m (2.28)

We can now define several tensors from equation (2.28):

(m)MIacJ =

∫

Vm

HJ δac ρ HI dVm (2.29)

(m)KIacJ =

∫

Vm

HI,b Eabcd HJ,d dVm (2.30)

(m)FBIa =

∫

Vm

fBa HI dVm (2.31)

(m)FSIa =

∫

Sm

fSa HI dSm (2.32)



(m)Fe0mn

Ia =

∫

Vm

Eabcd e0cd HI,b dV

m (2.33)

(m)Fτ0

mn

Ia =

∫

Vm

τ0ab HI,b dV

m (2.34)

where (m)KIacJ is the element stiffness tensor, (m)FBIa is the tensor of element body forces, (m)FSIa is the tensor

of element surface forces, (m)Fe0mn

Ia is the tensor of element initial strain effects, (m)Fτ0

mn

Ia is the tensor of element

initial stress effects. Now equation (2.28) becomes:

⋃

(m)

(m)MIacJ üJc +⋃

(m)

(m)KIacJ uJc =⋃

m

(m)FBIa +⋃

m

(m)FSIa +⋃

m

(m)Fe0mn

Ia −⋃

m

(m)Fτ0

mn

Ia (2.35)

By summing17 all the relevant tensors, a well known equation is obtained:

MAacB üBc +KAacB uBc = FAa (2.36)

A,B = 1, 2, . . . ,# of nodes

a, c = 1, . . . ,# of dimensions (1, 2 or 3)

where:

MAacB =⋃

m

(m)MIacJ ; KAacB =⋃

m

(m)KIacJ (2.37)

are the system mass and stiffness tensors, respectively, üBc is the tensor of unknown nodal accelerations, and uBc

is the tensor of unknown generalized nodal displacements, while the load tensor is given as:

FAa =⋃

m

(m)FBIa +⋃

m

(m)FSIa +⋃

m

(m)Fe0mn

Ia −⋃

m

(m)Fτ0

mn

Ia (2.38)

After assembling the system of equations in (2.37) it is relatively easy to solve for the unknown displacements

uLc either for static or fully dynamic case. It is also very important to note that in all previous equations, omissions

of inertial force term (all terms with ρ) will yield static equilibrium equations. Description of solutions procedures

17Summation of the element volume integrals expresses the direct addition of the element tensors to obtain global, system tensors.

This method of direct addition is usually referred to as the direct stiffness method.



for static linear and nonlinear problems are described in some detail in chapter 7. In addition to that, solution

procedures for dynamic, linear and nonlinear problems are described in some detail in chapter 8.

A note on the final form of the tensors used is in order. In order to use readily available system of equation

solvers equation (2.37) will be rewritten in the following from:

MPQ üP +KPQ uP = FQ P,Q = 1, 2, . . . , (DOFperNode)N (2.39)

where MPQ is system mass matrix, KPQ is system stiffness matrix and FQ is the loading vector. Matrix form of

equation 2.37, presented as equation 2.39 is obtained flattening the system mass tensor MAacB, system stiffness

tensor KAacB , unknown acceleration tensor üBc, unknown displacement tensor uBc and the system loading tensor

FAa. Flattening from the fourth order mass/stiffness tensors to two dimensional mass/stiffness matrix is done by

simply performing appropriate (re–) numbering of nodal DOFs in each dimension. Similar approach is used for

unknown accelerations/displacements and for loadings.

Static Analysis: Internal and External Loads. Internal and external loading tensors can be defined as:

(fIa)int =⋃

(m)

(m)KIacJ uJc =⋃

m

∫

Vm

τab HI,b dVm (2.40)

(fIa)ext =⋃

m

(m)FBIa +⋃

m

(m)FSIa +⋃

m

(m)Fe0mn

Ia −⋃

m

(m)Fτ0

mn

Ia (2.41)

where (fIa)int is the internal force tensor and (fIa)ext is the external force tensor. Equilibrium is obtained when

residual:

rIa(uJc, λ) = (fIa (uJc))int − λ (fIa)ext (2.42)

is equal to zero, r(u, λ) = 0. The same equation in flattened form yields:

r(u, λ) = fint(u) − λfext = 0 (2.43)

2.3 Isoparametric 8 – 20 Node Finite Element Definition

The basic procedure in the isoparametric18 finite element formulation is to express the element coordinates and

element displacements in the form of interpolations using the local three dimensional19 coordinate system of the

18name isoparametric comes from the fact that both displacements and coordinates are defined in terms of nodal values. Super-

parametric and subparametric finite elements exists also.19in the case of element presented here, that is isoparametric 8 – 20 node finite element.



element. Considering the general 3D element, the coordinate interpolations, using indicial notation20 are:

xi = HA (rk) xAi (2.44)

where A = 1, 2, . . . , n and n is the total number of nodes associated with that specific element, xAi is the i-th

coordinate of node A, i = 1, 2, 3, k = 1, 2, 3 and HA are the interpolation functions defined in local coordinate

system of the element, with variables r1, r2 and r3 varying from −1 to +1.

The interpolation functions HA for the isoparametric 8–20 node are the so called serendipity interpolation

functions mainly because they were derived by inspection. For the finite element with nodes numbered as in

Figure (2.3) they are given21 in the following set of formulae:

r1

r2

r33

49

11

14

15

x

x

x

17

12

1613

6

18

210

7

8

20

19

1

3

2

15

Figure 2.3: Isoparametric 8–20 node brick element in global and local coordinate systems

H20 =isp (20 ) (1 + r1) (1 − r2)

(1 − r23

)

4H19 =

isp (19 ) (1 − r1) (1 − r2)(1 − r23

)

4

H18 =isp (18 ) (1 − r1) (1 + r2)

(1 − r23

)

4H17 =

isp (17 ) (1 + r1) (1 + r2)(1 − r23

)

4

H16 =isp (16 ) (1 + r1)

(1 − r22

)(1 − r3)

4H15 =

isp (15 )(1 − r21

)(1 − r2) (1 − r3)

4

H14 =isp (14 ) (1 − r1)

(1 − r22

)(1 − r3)

4H13 =

isp (13 )(1 − r21

)(1 + r2) (1 − r3)

4

H12 =isp (12 ) (1 + r1)

(1 − r22

)(1 + r3)

4H11 =

isp (11 )(1 − r21

)(1 − r2) (1 + r3)

4

H10 =isp (10 ) (1 − r1)

(1 − r22

)(1 + r3)

4H9 =

isp (9 )(1 − r21

)(1 + r2) (1 + r3)

4

20Einstein’s summation rule is implied unless stated differently, all lower case indices (i, j, p, q, m, n, o, r, s, t, . . . ) can have values

of 1, 2, 3, and values for capital letter indices will be specified where need be.21for more details see Bathe (1982).



H8 =(1 + r1) (1 − r2) (1 − r3)

8+

−h15 − h16 − h20

2

H7 =(1 − r1) (1 − r2) (1 − r3)

8+

−h14 − h15 − h19

2

H6 =(1 − r1) (1 + r2) (1 − r3)

8+

−h13 − h14 − h18

2

H5 =(1 + r1) (1 + r2) (1 − r3)

8+

−h13 − h16 − h17

2

H4 =(1 + r1) (1 − r2) (1 + r3)

8+

−h11 − h12 − h20

2

H3 =(1 − r1) (1 − r2) (1 + r3)

8+

−h10 − h11 − h19

2

H2 =(1 − r1) (1 + r2) (1 + r3)

8+

−h10 − h18 − h9

2

H1 =(1 + r1) (1 + r2) (1 + r3)

8+

−h12 − h17 − h9

2

where r1, r2 and r3 are the axes of natural, local, curvilinear coordinate system and isp (nod num) is boolean

function that returns +1 if node number (nod num) is present and 0 if node number (nod num) is not present.

To be able to evaluate various important element tensors22, we need to calculate the strain–displacement

transformation tensor23. The element strains are obtained in terms of derivatives of element displacements with

respect to the local coordinate system. Because the element displacements are defined in the local coordinate

system, we need to relate global x1, x2 and x3 derivatives to the r1, r2 and r3 derivatives. In order to obtain

derivatives with respect to global coordinate system, i.e. ∂∂xa

we need to use chain rule for differentiation in the

following form:

∂

∂xk=∂ra∂xk

∂

∂ra= J−1

ak

∂

∂ra(2.45)

while the inverse relation is:

∂

∂rk=∂xa∂rk

∂

∂xa= Jak

∂

∂xa(2.46)

where Jak is the Jacobian operator relating local coordinate derivatives to the global coordinate derivatives:

Jak =∂xa∂rk

=

∂x1∂r1

∂x2∂r1

∂x3∂r1

∂x1∂r2

∂x2∂r2

∂x3∂r2

∂x1∂r3

∂x2∂r3

∂x3∂r3

(2.47)

22i.e. (m)KIacJ , (m)F BIa, (m)F S

Ia, (m)Fǫ0mnIa

, (m)Fτ0

mnIa

, that are defined in chapter (2.2).23from the equation ǫab = 1

2

``

HI,b uIa

´

+`

HI,a uIb

´´



The existence of equation (2.45) requires that the inverse of Jak exists and that inverse exists provided that

there is a one–to–one24 correspondence between the local and the global coordinates of element.

It should be pointed out that except for the very simple cases, volume and surface element tensor25 integrals

are evaluated by means of numerical integration26 Numerical integration rules is quite a broad subject and will

not be covered here27.

24unique.25as defined in chapter (2.2) by equations (2.30), (2.31), (2.32), (2.33) and (2.34).26Gauss–Legendre, Newton–Coates, Lobatto are among most used integration rules.27nice explanation with examples is given in Bathe (1982).




Chapter 3

Small Deformation Elasto–Plasticity

(1991–1994–2002–2006–2010–)

(In collaboration with Prof. Zhaohui Yang, Dr. Zhao Cheng, Prof. Mahdi Taiebat and Doctoral Student Nima Tafazzoli)

3.1 Elasticity

In linear elasticity the relationship between the stress tensor σij and the strain tensor ǫkl can be represented in

the following form:

σij = σ (ǫij) (3.1)

If we assume the existence of a strain energy function1 W (ǫij) then the stress strain relation is:

σij =∂W (ǫij)

∂ǫij(3.2)

The introduction of the strain energy density function into elasticity is due to Green, and elastic solids for which

such a function is assumed to exist are called Green elastic or hyperelastic solids.

Linearization of an elastic continuum is carried out with respect to a reference configuration which is stress free

at temperature T0, so that 0σij = 0. If we denote as Eijkl an isothermal modulus tensor, then under isothermal

conditions, we obtain the generalized Hooke’s law:

σij = Eijklǫkl (3.3)

1 per unit volume.

39


The Eijkl is called the elastic constants tensor of fourth order2. It has 81 independent components in total. It

has symmetries with respect to pairs of indices ij and kl and these symmetries reduce the number of independent

components to 36. There is an additional symmetry conditions:

Eijkl =

∣∣∣∣

∂2W

∂ǫij ∂ǫkl

∣∣∣∣ǫ=0

=

∣∣∣∣

∂2W

∂ǫkl ∂ǫij

∣∣∣∣ǫ=0

(3.4)

thus we have Eijkl = Eklij and the number of independent components is reduced to 21 (Spencer, 1980).

We will restrain our considerations to the isotropic case. The most general form of the isotropic tensor of

rank 4 has the following representation:

I4 = λδijδkl + µδikδjl + νδilδjk (3.5)

If Eijkl has this form then in order to satisfy the symmetry condition3 Eijkl = Ejikl we must have ν = µ. The

symmetry condition4 Eijkl = Eklji is then automatically satisfied. The elastic constant tensor has the following

form:

Eijkl = λδijδkl + µ (δikδjl + δilδjk) (3.6)

where λ and µ are the Lame coefficients:

λ =νE

(1 + ν) (1 − 2ν); µ =

E

2 (1 + ν)(3.7)

and E and ν are Young’s Modulus and Poisson’s ratio respectively. The symmetric part of the fourth order unit

tensor is :

Isymijkl =1

2(δikδjl + δilδjl) (3.8)

and can be found as multiplier of µ in equation (3.186). Equation (3.186) can be written in terms of E and ν

as:

Eijkl =E

2 (1 + ν)

(2ν

1 − 2νδijδkl + δikδjl + δilδjk

)

(3.9)

The same relation in terms of bulk modulus K and shear modulus G is:

Eijkl = Kδijδkl +G

(

−2

3δijδkl + δikδjl + δilδjk

)

(3.10)

where K and G are given as:

K = λ+2

3µ ; G = µ (3.11)

2also stiffness tensor.3 symmetry in stress tensor.4 existence of strain energy function.



The relation between the strain tensor, ǫkl and the stress tensor, σij is:

ǫkl = Dklpqσpq (3.12)

where Dklpq is the elastic compliance fourth order tensor, defined as:

Dklpq =−λ

2µ (3λ+ 2µ)δklδpq +

1

4µ(δkpδlq + δkqδlp) (3.13)

or in terms of E and ν:

Dklpq =1 + ν

2E

( −2ν

1 + νδklδpq + δkpδlq + δkqδlp

)

(3.14)

of in terms of K and G:

Dklpq =1

9K(δklδpq) +

1

2G

(

−1

3δklδpq +

1

2(δkpδlq + δkqδlp)

)

(3.15)

It is worthwhile noting that the part adjacent to the inverse of the bulk modulus K:

(δklδpq)

controls the volumetric response and that the part adjacent to the inverse of the shear modulus G:(

−1

3δklδpq +

1

2(δkpδlq + δkqδlp)

)

controls the shear response! This note will prove useful later on. Linear transformation of the stress tensor σpq

into itself, i.e. σij is defined as:

σij = Eijklǫkl = EijklDklpqσpq (3.16)

where

EijklDklpq =1

2(δipδjq + δiqδjp) = Isymijpq (3.17)

Linear Elastic Model. Linear elastic law is the simplest one and assumes constant Young’s modulus E and

constant Poisson’s Ration ν.

Non–linear Elastic Model #1. This nonlinear model (Janbu, 1963), (Duncan and Chang, 1970) assumes

dependence of the Young’s modulus on the minor principal stress σ3 = σmin in the form

E = Kpa

(σ3

pa

)n

(3.18)

Here, pa is the atmospheric pressure in the same units as E and stress. The two material constants K and n are

constant for a given void ratio.



Non–linear Elastic Model #2. If Young’s modulus and Poisson’s ratio are replaced by the shear modulus G

and bulk modulus K the non–linear elastic relationship can be expressed in terms of the normal effective mean

stress p as

G and/or K = AF (e,OCR)pn (3.19)

where e is the void ratio, OCR is the overconsolidation ratio and p = σii/3 is the mean effective stress (Hardin,

1978).

Lade’s Non–linear Elastic Model. Lade and Nelson (1987) and Lade (1988a) proposed a nonlinear elastic

model based on Hooke’s law in which Poisson ratio ν is kept constant. According to this model, Young’s modulus

can be expressed in terms of a power law as:

E = M pa

((I1pa

)2

+

(

61 + ν

1 − 2ν

)J2D

p2a

)λ

(3.20)

where I1 = σii is the first invariant of the stress tensor and J2D = (sijsij)/2 is the second invariant of the

deviatoric stress tensor sij = σij − σkkδij/3. The parameter pa is atmospheric pressure expressed in the same

unit as E, I1 and√J2D and the modulus number M and the exponent λ are constant, dimensionless numbers.

3.2 Elasto–plasticity

3.2.1 Constitutive Relations for Infinitesimal Plasticity

A wide range of elasto–plastic materials can be characterized by means of a set of constitutive relations of the

general form:

ǫij = ǫeij + ǫpij (3.21)

σij = Eijklǫekl (3.22)

dǫpij = dλ∂Q

∂σij= dλ mij(σij , q∗) (3.23)

dq∗ = dλ h∗(τij , q∗) (3.24)

where, following standard notation ǫij , ǫeij and ǫpij denotes the total, elastic and plastic strain tensor, σij is

the Cauchy stress tensor, and q∗ signifies some suitable set of internal variables5. The asterisk in the place of

indices in q∗ replaces n indices6. Equation (3.21) expresses the commonly assumed additive decomposition of

the infinitesimal strain tensor into elastic and plastic parts. Equation (3.22) represents the generalized Hooke’s

5In the simplest models of plasticity the internal variables are taken as either plastic strain components ǫpij or the hardening

variables κ defined, for example as a function of inelastic (plastic) work, i.e. κ = f (W p). See Lubliner (1990) page 115.6 for example ij if the variable is ǫp

ij , or nothing if the variable is a scalar value, i.e. κ .



law7 which linearly relates stresses and elastic strains through a stiffness modulus tensor Eijkl. Equation (3.23)

expresses a generally associated or non-associated flow rule for the plastic strain and (3.24) describes a suit-

able set of hardening laws, which govern the evolution of the plastic variables. In these equations, mij is the

plastic flow direction, h∗ the plastic moduli and dλ is a plastic parameter to be determined with the aid of the

loading—unloading criterion, which can be expressed in terms of the Karush–Kuhn–Tucker condition (Karush,

1939; Kuhn and Tucker, 1951) as:

F (σij , q∗) ≤ 0 (3.25)

dλ ≥ 0 (3.26)

F dλ = 0 (3.27)

In the previous equations F (σij , q∗) denotes the yield function of the material and (3.25) characterizes the

corresponding elastic domain, which is presumably convex. Along any process of loading, conditions (3.25),

(3.26) and (3.27) must hold simultaneously. For F < 0, equation (3.27) yields dλ = 0, i.e. elastic behavior,

while plastic flow is characterized by dλ > 0, which with (3.27) is possible only if the yield criterion is satisfied,

i.e. F = 0. From the latter constraint, in the process of plastic loading the plastic consistency conditions8 is

obtained in the form:

dF =∂F

∂σijdσij +

∂F

∂q∗dq∗ = nijdσij + ξ∗dq∗ = 0 (3.28)

where :

nij =∂F

∂σij(3.29)

ξ∗ =∂F

∂q∗(3.30)

Equation (3.28) has the effect of confining the stress trajectory to the yield surface9. It is worthwhile noting that

nij and ξ∗ are normals to the yield surface in stress space and the plastic variable space respectively.

An interesting alternative way of representing non–associated flow rules can be found in Runesson (1987). A

fictitious plastic strain derived from associated flow rule, epij is introduced. This fictitious plastic strain is assumed

to be related to the real plastic strain ǫpij , which is derived from a non–associated flow rule10 through the linear

transformation:

epij = Aijklǫpkl (3.31)

Linear transformation tensor Aijkl may be state dependent in general case, and it reduces to the symmetric part

of the fourth order identity tensor11 for the case of associated plasticity.

7also Eq. 3.1858first order accuracy condition.9Since it is only linear expansion stress trajectory is confined to the tangential plane only.

10as in equation 3.23.11Aijkl ≡ Isym

ijkl≡ 1

2

`

δikδjl + δilδjk

´

.



It is often of interest to model deviatoric strains by an associated flow rule while the volumetric part is

non–associated. For this case, Aijkl can be formulated as:

Aijkl =

(

β1

3(δijδkl) +

1

2(δikδjl + δilδjk)

)

(3.32)

A−1ijkl =

(

− β

1 + β

1

3(δijδkl) +

1

2(δikδjl + δilδjk)

)

(3.33)

and it is obvious that the non–associated flow rule is obtained with β 6= 0 and the associated flow rule with β = 0.

It is useful to choose β ≥ 0 and retain nice, positive definite properties of adjusted constitutive tensors later.

Let the ‖ · ‖ norm, be the complementary energy norm12:

‖σij‖2 = σijDijklσkl (3.34)

where Dijkl is the elastic compliance tensor ( Dijkl = E−1ijkl ), and let us introduce the adjusted complementary

energy norm as:

A‖σij‖2 = σij (AijklDklmn)σmn = σij(ADijmn

)σmn (3.35)

where ADijmn is the elastic compliance tensor transformed with respect to the non–associativity involved. It is

clear that when Aijkl ≡ Isymijkl =⇒ A‖σij‖2 ≡ ‖σij‖2

3.2.2 On Integration Algorithms

In the section Constitutive Relations for Infinitesimal Plasticity we have summarized constitutive equations that

are capable of representing a wide variety of elasto–plastic materials. The problem in Computational Elasto–

plasticity is to devise accurate and efficient algorithms for the integration of such constitutive relations. In the

context of finite element analysis using isoparametric elements, the integration of constitutive equations is carried

out at Gauss points. In each step the deformation increments are given or known, and the unknowns to be found

are updated stresses and plastic variables. According to Ortiz and Popov (1985) an acceptable algorithm should

satisfy:

• consistency with the constitutive relations to be integrated or first order accuracy,

• Numerical stability,

• incremental plastic consistency

A non—required but desirable feature to be added to the above list is:

• higher13 order accuracy

First two conditions are needed for attaining convergence for the numerical solution as the step or increment

becomes vanishingly small. The third condition is the algorithmic counterpart of the plastic consistency condition

and requires that the state of stress computed from the algorithm be contained within the elastic domain.

12This norm will be reintroduced later on!13at least second order accuracy.



3.2.3 Midpoint Rule Algorithm

A class of algorithms for integrating constitutive equations with potential to satisfy the above mentioned conditions

are the Generalized Midpoint rule algorithms. They are given in the following form:

n+1σij = Eijkl(n+1ǫkl − n+1ǫpkl

)(3.36)

n+1ǫpij = nǫpij + λ n+αmij (3.37)

n+1q∗ = nq∗ + λ n+αh∗ (3.38)

Fn+1 = 0 (3.39)

where:

n+αmij = mij

((1 − α) nσij + α

(n+1σij

), (1 − α) nq∗ + α

(n+1q∗

))(3.40)

n+αh∗ = h∗((1 − α) nσij + α

(n+1σij

), (1 − α) nq∗ + α

(n+1q∗

))(3.41)

It is quite clear that the case α = 0 corresponds to the Forward Euler approach14, the case α = 1 corresponds

to the Backward Euler approach15, and the case α = 1/2 to the Crank – Nicholson scheme. Equations (3.36),

(3.37), (3.38), (3.39), (3.40) and (3.41) are the nonlinear algebraic equations to be solved for the unknowns

n+1σij ,n+1ǫpij ,

n+1q∗ and λ. From the Figure (3.1)16 it can be seen that the Generalized Midpoint rule may be

regarded as a returning mapping algorithm in which the elastic predictor predσij is projected on the updated yield

surface along the flow direction evaluated at the midpoint (n+ασij ,n+αq∗).

Accuracy Analysis

Bearing in mind the context of the displacement based finite element analysis the integration of constitutive

equations is performed for the given strain increment. The updated strains n+1ǫij = ǫij (tn + ∆t) may be viewed

as the known function of the step size ∆t. The remaining updated variables n+1σij ,n+1ǫpij ,

n+1q∗, as well as

the incremental plastic parameter λ become functions of ∆t implicitly defined through equations (3.36), (3.37),

(3.38) and (3.39). It should be clear from (3.36), (3.37), (3.38) and (3.39) that as ∆t→ 0 than n+1ǫij → nǫij ,

and thus the limiting values of n+1σij ,n+1ǫpij ,

n+1q∗ and λ are obtained:

14explicit scheme.15implicit scheme.16it should be pointed out that the vectors, as drawn on this figure, are pointing in the right direction only if we assume that

Eijkl ≡ Iijkl. For any general elasticity tensor Eijkl all vectors are defined in the Eijkl metric, so the term ”normal”, as we are

used to it, does not apply here.



σn+1

ij

n+1σij

n

F=0 F=0αn+

Elastic

region

ij

n+1

n+α

ijσ n+1σij

σ1

σ2

σ3

σn ij

σij

σpredictor

ij

cross

mcross

ij

mαn+

mij

n+1

F=0

n+1

Q=0

Figure 3.1: integration algorithms in elasto–plasticity

lim∆t→0

(n+1σij

)= nσij

lim∆t→0

(n+1ǫpij

)= nǫpij

lim∆t→0

(n+1q∗

)= nq∗

lim∆t→0

λ = 0 (3.42)

It can also be argued that, by virtue of the implicit function theorem ((Abraham et al., 1988) Chapter 2.5), n+1σij ,

n+1ǫpij ,n+1q∗ and λ are differentiable functions of ∆t, if the functions n+αmij ,

n+αh∗ and F are sufficiently smooth.

Sufficient smoothness will be assumed as needed.

First Order Accuracy. First order accuracy17 of the algorithm, defined by the equations (3.36), (3.37), (3.38)

and (3.39) with the constitutive equations given by (3.21), (3.22), (3.23) and (3.24) necessitates that the nu-

merically integrated variables n+1σij ,n+1ǫpij and n+1q∗ agree with their exact values σij(t+ ∆t), ǫpij (t+ ∆t)

and q∗ (t+ ∆t) to within second order terms in the Taylor’s expansion around the initial state nσij = σij(t),

nǫpij = ǫpij (t) and nq∗ = q∗ (t) in ∆t. First order accuracy can be written in the following form:

lim∆t→0

d(n+1σij

)

d (∆t)=d (nσij)

d (∆t)= Eijkl

(

d (nǫij)

d (∆t)−d(nǫpij)

d (∆t)

)

(3.43)

17first order consistency.



lim∆t→0

d(n+1ǫpij

)

d (∆t)=d(nǫpij)

d (∆t)=d (nλ)

d (∆t)nmij (3.44)

lim∆t→0

d(n+1q∗

)

d (∆t)=d (nq∗)

d (∆t)=d (nλ)

d (∆t)nh∗ (3.45)

lim∆t→0

d (λ)

d (∆t)=d (nλ)

d (∆t)(3.46)

and the plastic parameter d (nλ) /d (∆t) is determined with the aid of the plastic consistency condition at t:

d (nF )

d (∆t)=∂ (nF )

∂σij

dσijd (∆t)

+∂ (nF )

∂q∗

dq∗d (∆t)

= nnijdσijd (∆t)

+ nξ∗dq∗d (∆t)

= 0 (3.47)

It is now rather straightforward to check whether the Generalized Midpoint rule satisfies the consistency conditions

as given by (3.43), (3.44), (3.45) and (3.46). We can proceed further on by differentiating (3.36), (3.37), (3.38)

and (3.39) with respect to ∆t18:

d(n+1σij

)

d (∆t)= Eijkl

(

d(n+1ǫkl

)

d (∆t)− d

(n+1ǫpkl

)

d (∆t)

)

(3.48)

d(n+1ǫpij

)

d (∆t)=

dλ

d (∆t)

(n+αmij

)+ λ

d (n+αmij)

d (∆t)=

dλ

d (∆t)

(n+αmij

)+ λα

(

∂mij

∂σij

∣∣∣∣n+1

d(n+1σij

)

d (∆t)+∂mij

∂q∗

∣∣∣∣n+1

d(n+1q∗

)

d (∆t)

)

(3.49)

d(n+1qp∗

)

d (∆t)=

dλ

d (∆t)

(n+αh∗

)+ λ

d (n+αh∗)

d (∆t)=

dλ

d (∆t)

(n+αh∗

)+ λα

(

∂h∗∂σij

∣∣∣∣n+1

d(n+1σij

)

d (∆t)+∂h∗∂q∗

∣∣∣∣n+1

d(n+1q∗

)

d (∆t)

)

(3.50)

d(n+1F

)

d (∆t)=

∂(n+1F

)

∂ (n+1σij)

d(n+1σij

)

d (∆t)+∂(n+1F

)

∂ (n+1q∗)

d(n+1n+1q∗

)

d (∆t)= 0 (3.51)

18 bearing in mind that values at t are constants and that only variables at t + ∆t are changing with respect to ∆t.



where n+αmij and n+αh∗ are defined by the equations (3.40) and (3.41).

By taking ∆t to the limit value, ∆t→ 0, in the (3.48), (3.49), (3.50) and (3.51) and using the relations from

(3.42) one finds:

lim∆t→0

d(n+1σij

)

d (∆t)= Eijkl

(

d (nǫkl)

d (∆t)−d(n+1ǫpkl

)

∆t=0

d (∆t)

)

(3.52)

lim∆t→0

d(n+1ǫpij

)

d (∆t)=

dλ

d (∆t)(nmij)

(3.53)

lim∆t→0

d(n+1q∗

)

d (∆t)=

dλ

d (∆t)(nh∗)

(3.54)

lim∆t→0

d(n+1F

)

d (∆t)=∂ (nF )

∂σij

(

lim∆t→0

d(n+1σij

)

d (∆t)

)

+∂ (nF )

∂q∗

(

lim∆t→0

d(n+1q∗

)

d (∆t)

)

= 0 (3.55)

In the previous equations it is quite clear that since ∆t = 0, then equations (3.42) hold and since the variables

nσij ,nǫpij and nq∗ are constant with respect to the change in ∆t, the result follows readily, i.e. the Midpoint rule

satisfies first order accuracy.

Second Order Accuracy To investigate second order accuracy of the algorithm given by (3.36), (3.37), (3.38)

and (3.39) together with the constitutive equations given by (3.21), (3.22), (3.23) and (3.24) we shall proceed

in the following manner. Second order accuracy actually means that the numerically integrated variables n+1σij ,

n+1ǫpij and n+1q∗ agree with their ”exact” values σij(t+ ∆t), ǫpij (t+ ∆t) and q∗ (t+ ∆t) to within third order

terms in the Taylor’s expansion around the initial state nσij = σij(t),nǫpij = ǫpij (t) and nq∗ = q∗ (t) in ∆t. This

verbal statement can be written in the following mathematical form:

lim∆t→0

d2(n+1σij

)

d (∆t)2 =

Eijkl

(

lim∆t→0

d2(n+1ǫkl

)

d (∆t)2 − lim

∆t→0

d2 (nǫpkl)

d (∆t)2

)

= Eijkl

(

d2 (nǫkl)

d (∆t)2 − d2 (nǫpkl)

d (∆t)2

)

(3.56)



lim∆t→0

d2(n+1ǫpij

)

d (∆t)2 =

d2λ

d (∆t)2 lim

∆t→0

(n+1mij

)+ lim

∆t→0

d(n+1λ

)

d (∆t)

d (n+αmij)

d (∆t)=

d2λ

d (∆t)2 (nmij) +

d (nλ)

d (∆t)

d (n+αmij)

d (∆t)=

d2λ

d (∆t)2 (nmij) +

d (nλ)

d (∆t)

(∂mij

∂σij |nd (nσij)

d (∆t)+

∂mij

∂q∗ |nd (nq∗)

d (∆t)

)

(3.57)

lim∆t→0

d2(n+1qp∗

)

d (∆t)2 =

d2λ

d (∆t)2 lim

∆t→0

(n+1h∗

)+ lim

∆t→0

d(n+1λ

)

d∆t

d(n+1h∗

)

d (∆t)=

d2λ

d (∆t)2 (nh∗) +

d(n+1λ

)

d∆t

d (nh∗)

d (∆t)=

d2λ

d (∆t)2 (nh∗) +

d(n+1λ

)

d∆t

(∂h∗

∂σij |nd (nσij)

d (∆t)+

∂h∗∂q∗ |n

d (nq∗)

d (∆t)

)

(3.58)

lim∆t→0

d2 (λ)

d (∆t)2 =

d2 (nλ)

d (∆t)2 (3.59)

and the plastic parameter d2 (nλ) /d (∆t)2

is determined with the aid of the second order oscillatory satisfaction

of the plastic consistency condition:

d2 (nF )

d (∆t)2 =

dnijd∆t

∣∣∣∣n

dσijd (∆t)

+ nnijd2 (σij)

d (∆t)2

∣∣∣∣∣n

+d (ξ∗)

d∆t

∣∣∣∣n

dnq∗d (∆t)

+ nξ∗d2 (nq∗)

d (∆t)2 = 0 (3.60)

Now we can proceed by taking the second derivative of the equations (3.36), (3.37), (3.38) and (3.39) or use

the already derived first derivatives from equations (3.48), (3.49), (3.50) and (3.51), and then differentiate them

again so that we get:

d2(n+1σij

)

d (∆t)2 = Eijkl

(

d2(n+1ǫkl

)

d (∆t)2 − d2

(n+1ǫpkl

)

d (∆t)2

)

(3.61)



d2(n+1ǫpij

)

d (∆t)2 =

d2λ

d (∆t)2

(n+αmij

)+

2dλ

d (∆t)α

(

∂mij

∂σij

∣∣∣∣n+1

d(n+1σij

)

d (∆t)+∂mij

∂q∗

∣∣∣∣n+1

d(n+1q∗

)

d (∆t)

)

+

λαd

d (∆t)

(

∂mij

∂σij

∣∣∣∣n+1

d(n+1σij

)

d (∆t)+∂mij

∂q∗

∣∣∣∣n+1

d(n+1q∗

)

d (∆t)

)

(3.62)

d2(n+1qp∗

)

d (∆t)2 =

d2λ

d (∆t)2

(n+αh∗

)+

2dλ

d (∆t)α

(

∂h∗∂σij

∣∣∣∣n+1

d(n+1σij

)

d (∆t)+∂h∗∂q∗

∣∣∣∣n+1

d(n+1q∗

)

d (∆t)

)

+

λαd

d (∆t)

(

∂h∗∂σij

∣∣∣∣n+1

d(n+1σij

)

d (∆t)+∂h∗∂q∗

∣∣∣∣n+1

d(n+1q∗

)

d (∆t)

)

(3.63)

d2(n+1F

)

d (∆t)2 =

d(n+1nij

)

dσij

d(n+1σij

)

d (∆t)+ n+1nij

d2(n+1σij

)

d (∆t)2 +

+d(n+1ξ∗

)

dσij

d(n+1q∗

)

d (∆t)+ n+1ξ

d2(n+1q∗

)

d (∆t)2 = 0 (3.64)

If we drive ∆t to the limit, namely by taking lim∆t→0 and keeping in mind equations (3.42) and the assumed

consistency of the algorithm19 as given by the equations (3.43), (3.44), (3.45) and (3.46) one finds:

lim∆t→0

d2(n+1σij

)

d (∆t)2 = Eijkl

(

d2 (nǫkl)

d (∆t)2 − lim

∆t→0

d2(n+1ǫpkl

)

d (∆t)2

)

(3.65)

lim∆t→0

d2(n+1ǫpij

)

d (∆t)2 =

lim∆t→0

d2(n+1λ

)

d (∆t)2

(n+αmij

)+ 2

d (nλ)

d (∆t)α

(∂mij

∂σij

∣∣∣∣n

d (nσij)

d (∆t)+∂mij

∂q∗

∣∣∣∣n

d (nq∗)

d (∆t)

)

(3.66)

19actually the first order accuracy that is already proven.



lim∆t→0

d2(n+1qp∗

)

d (∆t)2 =

lim∆t→0

d2(n+1λ

)

d (∆t)2

(n+αh∗

)+ 2

d (nλ)

d (∆t)α

(∂h∗∂σij

∣∣∣∣n

d (nσij)

d (∆t)+∂h∗∂q∗

∣∣∣∣n

d (nq∗)

d (∆t)

)

(3.67)

lim∆t→0

d2(n+1F

)

d (∆t)2 =

d (nnij)

dσij

d (nσij)

d (∆t)+ nnij lim

∆t→0

d2(n+1σij

)

d (∆t)2 +

+d (nξ∗)

dσij

d (nq∗)

d (∆t)+ nξ lim

∆t→0

d2(n+1q∗

)

d (∆t)2 = 0 (3.68)

By comparing equations (3.65), (3.66), (3.67) and (3.68) with the second order accuracy condition stated in

equations (3.56), (3.57), (3.58) and (3.59) it is quite clear that the second order accuracy is obtained iff20

α = 1/2 !

The conclusion is that the Midpoint–rule algorithm is consistent21 for all α ∈ [0, 1] and it is second order

accurate for α = 1/2. However, one should not forget that these results are obtained for the limiting case ∆t→ 0,

i.e. the strain increments are small and tend to zero.

Numerical Stability Analysis

Numerical stability of an algorithm plays a central role in approximation theory for initial value problems. In

fact, it can be stated that consistency and stability are necessary and sufficient conditions for convergence of an

algorithm as the time step tends to zero. In the approach presented by Ortiz and Popov (1985) a new methodology

is proposed by which the stability properties of an integration algorithm for elasto–plastic constitutive relations

can be established. Our attention is confined to perfect plasticity and a smooth yield surface.

The purpose of the following stability analysis is to determine under what conditions a finite perturbation in

the initial stresses is diluted by the algorithm. In other words:

d(n+1σ

(2)ij ,

n+1σ(1)ij

)

≤ d(nσ

(2)ij ,

nσ(1)ij

)

(3.69)

where d (·, ·) is some suitable distance on the yield surface and n+1σ(1)ij and n+1σ

(2)ij are two sets of updated

stresses corresponding to arbitrary initial stress values nσ(1)ij and nσ

(2)ij , respectively, and all of the previous stress

values are assumed to lie on the yield surface. Stability in the sense of equation (3.69) is referred to as large scale

20 if and only if ( ⇐⇒ ).21first order accurate.



stability. It is shown in Helgason (1978)22 that for nonlinear initial value problems defined on Banach manifolds,

consistency and large scale stability with respect to a complete metric are sufficient for convergence.

The task of directly establishing estimates of the type expressed in (3.69) is rather difficult, and so despite the

conceptual appeal of large scale stability, simplified solutions are sought. It should be recognized that attention

can be restricted to infinitesimal perturbation in the initial conditions of the type nσij → nσij + d (nσij). This

simplification is founded on the fact that the dilution or attenuation, by the algorithm of infinitesimal perturbations:

‖d n+1σij‖ ≤ ‖d nσij‖ (3.70)

with respect to some suitable norm ‖·‖, of small scale stability, implies large scale stability in the sense of equation

(3.69).

Let the ‖ · ‖ norm, be the energy norm:

‖σij‖2 = σijDijklσkl (3.71)

where Dijkl is the elastic compliance tensor (Dijkl = E−1ijkl ), and let the distance d (·, ·) on the yield surface be

defined as

d(

σ(1)ij , σ

(2)ij

)

= infγ

∫

γ

‖σ′

ij (s) ‖ds (3.72)

where the infinum is taken over all possible stress paths γ on the yield surface that are joining two stress states,

namely σ(1)ij and σ

(2)ij . It can be found in Helgason (1978) that for a smooth yield surface, equation (3.72) defines

the geodesic distance which endows the yield surface with a complete metric structure.

Suppose that we have any two initial states of stress nσ(1)ij and nσ

(2)ij and let n+1σ

(1)ij and n+1σ

(2)ij be the

corresponding updated values, respectively, and all the previous stress states are assumed to lie on the yield

surface. Then, according to Helgason (1978), there exists a unique geodesic curve that joins nσ(1)ij and nσ

(2)ij for

which the infinum in equation (3.72) is attained. If γn is such a curve, then by definition:

d(nσ

(1)ij ,

nσ(2)ij

)

=

∫

γn

‖σ′

ij (s) ‖ds (3.73)

Let the new curve γn+1 be the transform of curve γn by the algorithm. By definition γn+1 lies on the yield

surface and joins two stress states n+1σ(1)ij and n+1σ

(2)ij . By the definition given in (3.72), it follows that:

d(n+1σ

(1)ij ,

n+1σ(2)ij

)

=

∫

γn+1

‖σ′

ij (s) ‖ds (3.74)

Under the assumption of small scale stability of the algorithm one can write:

‖σ′

ij (sn+1) ‖ds = ‖dσij (sn+1) ‖ ≤ ‖dσij (sn) ‖ = ‖σ′

ij (sn) ‖ds (3.75)

22the first Chapter of Helgason’s book.



for every pair of corresponding points sn and sn+1 on γn and γn+1 respectively, so it follows:

∫

γn+1

‖σ′

ij (sn+1) ‖ds ≤∫

γn

‖σ′

ij (sn) ‖ds (3.76)

By combining equations (3.74), (3.75) and (3.76) it is concluded that:

d(n+1σ

(1)ij ,

n+1σ(2)ij

)

≤ d(nσ

(1)ij ,

nσ(2)ij

)

(3.77)

which proves large scale stability. The main conclusion of the above argument may be stated as follows: small

scale stability in the energy norm is equivalent to large scale stability in the associated geodesic distance.

The previous result is of practical importance, since it shows that the stability analysis for the integration

algorithm in elasto–plasticity can be carried out by the assessment of small scale stability. The small scale stability

analysis of the Generalized Midpoint rule is necessary to determine how the algorithm propagates infinitesimal

perturbations in the initial conditions. By differentiating equations (3.36), (3.37), (3.38) and (3.39)and considering

that we are dealing with perfectly plastic case here so that(n+1q∗

)= (nq∗) = constants, it follows:

d(n+1σij

)= −Eijkl d

(n+1ǫpkl

)(3.78)

d (nσij) = −Eijkl d (nǫpkl) (3.79)

d(n+1ǫpij

)− d

(nǫpij)

= d λ(n+αmij

)+ λ d

(n+αmij

)(3.80)

d(n+1F

)=

∂F

∂σij

∣∣∣∣n+1

d(n+1σij

)= n+1nij d

(n+1σij

)= 0 (3.81)

Let us now examine the shape of d (n+αmij) having in mind the original definition23 given in equation (3.40):

n+αmij = mij

((1 − α) nσij + α

(n+1σij

), (1 − α) nq∗ + α

(n+1q∗

))

and the differential of the previous equation is:

d(n+αmij

)= (1 − α)

∂mij

∂σkl

∣∣∣∣n+α

d (nσkl) + α∂mij

∂σkl

∣∣∣∣n+α

d(n+1σkl

)

23 the remark about restraining analysis to perfectly plastic case still holds, so that`

n+1q∗´

and (nq∗) are constant.



To ease writing let us introduce the following fourth order tensor:

Mijkl =∂mij

∂σkl

The equation (3.80) now reads:

d(n+1ǫpij

)− d

(nǫpij)

=

dλ(n+αmij

)+ λ

((1 − α)

(n+αMijkl

)d (nσkl) + α

(n+αMijkl

)d(n+1σkl

))

(3.82)

By using equations (3.78) and (3.79) and knowing that E−1ijkl = Dijkl one can write:

d(n+1ǫpij

)= −Dijkl d

(n+1σkl

)

d(nǫpij

)= −Dijkl d(

nσkl)

so that the equation (3.82) now reads:

−Dijkl d(n+1σkl

)+Dijkl d (nσkl) =

dλ(n+αmij

)+ λ

((1 − α)

(n+αMijkl

)d (nσkl) + α

(n+αMijkl

)d(n+1σkl

))

Now we are proceeding by solving the previous equation for d(n+1σkl

):

(Dijkl + λ α

(n+αMijkl

))d(n+1σkl

)=

(Dijkl − λ (1 − α)

(n+αMijkl

))d (nσkl) − dλ

(n+αmij

)

and by denoting :

Ψijkl = Dijkl − λ (1 − α)(n+αMijkl

)



Γijkl = Dijkl + λ α(n+αMijkl

)

it follows:

d(n+1σkl

)= Γ−1

ijkl

(Ψijkl d (nσkl) − dλ

(n+αmij

))(3.83)

Then by inserting the solution for d(n+1σkl

)in the consistency condition (3.81):

d(n+1F

)= n+1nkl d

(n+1σkl

)= 0

one gets:

d(n+1F

)=

n+1nkl Γ−1ijkl

(Ψijkl d (nσkl) − dλ

(n+αmij

))= 0 (3.84)

then if we solve for dλ:

dλ n+αmijn+1nkl Γ−1

ijkl = n+1nkl Γ−1ijklΨijkl d (nσkl) (3.85)

or24:

dλ =n+1nrs Γ−1

pqrsΨpqrs d (nσrs)

(n+αmpq) (n+1nrs) Γ−1pqrs

(3.86)

then by using the solution for d(n+1σkl

)from (3.83) and the solution for dλ from (3.86) one can find:

d(n+1σkl

)= Γ−1

ijklΨijkl d (nσkl) − Γ−1pqrs Ψpqrs

n+1nrs Γ−1ijkl (n+αmij)

n+αmpqn+1nrs Γ−1

pqrs

d (nσrs)

(3.87)

24 where the change in dummy indices is possible because dλ is scalar.



d(n+1σkl

)= Γ−1

ijklΨijkl

(

δksδrl −n+1nrs Γ−1

ijkl (n+αmij)


pqrs

)

d (nσrs) (3.88)

to ease the writing we can define the following notation:

Φklrs = δksδrl −n+1nrs Γ−1

ijkl (n+αmij)


pqrs

(3.89)

so that the equation (3.88) now reads:

d(n+1σkl

)= Γ−1

ijklΨijkl Φklrs d (nσrs) (3.90)

In order to derive the estimate of the type (3.70) from (3.90) we shall proceed in the following way. The norm of

a tensor is defined as:

‖Aijkl‖ = supσ

‖Aijklσkl‖‖σkl‖

(3.91)

If we take the norm of (3.90), while recalling the inequalities:

‖Aijklσkl‖ ≤ ‖Aijkl‖ ‖σkl‖ ; ‖AijklBijkl‖ ≤ ‖Aijkl‖ ‖Bijkl‖ (3.92)

it follows:

‖d(n+1σkl

)‖ = ‖Γ−1

ijklΨijkl Φklrs d (nσrs) ‖ (3.93)

then by using equations (3.92), we are able to write:

‖d(n+1σkl

)‖ ≤ ‖Γ−1

ijkl Ψijkl‖ ‖Φklrs‖ ‖d (nσrs) ‖ (3.94)

Considering the norm of ‖Φklrs‖ it should be noted that Φklrs defines a projection along the direction of

Γ−1ijkl

n+αmij onto the hyperplane that is orthogonal to n+1nrs, so that the following properties hold:



(Φklrs)(

Γ−1ijkl

n+αmij

)

= ∅ (3.95)

(Φklrs) (σrs) = σrs (3.96)

for every σrs that is orthogonal to n+1nrs. From these properties and the definition in equation (3.91) it follows

that:

‖Φklrs‖ ≡ 1 (3.97)

In what follows it is assumed that the fourth order tensor field

Mijkl = ∂mij/∂σkl

is symmetric and positive definite everywhere on the yield surface. The assumption is valid, if the flow direction

mij is derived from the convex potential function, which is a rather common feature among yield criteria. It is

now clear that :

‖Γ−1ijkl Ψijkl‖ =

∣∣∣∣

maxγij Ψijklmaxγkl

maxγij Γijkl maxγkl

∣∣∣∣

(3.98)

where maxγij is the eigentensor corresponding to the maximum eigenvalue of the eigenproblem:

(Ψijkl − µ Γijkl) γkl = 0 (3.99)

which is normalized to satisfy:

‖maxγij‖‖maxγij‖ = maxγij Dijklmaxγkl = 1 (3.100)

If we denote:



n+αβ = maxγijn+αMijkl

maxγkl (3.101)

as the maximum eigenvalue of the fourth order tensor n+αMijkl and that value is a positive real number25, then

from equations (3.98), (3.100), (3.101) and from the definition26 of Ψijkl and Γijkl, it follows:

‖Γ−1ijkl Ψijkl‖ =

∣∣∣∣

1 − (1 − α) λ (n+αβ)

1 + α λ (n+αβ)

∣∣∣∣

(3.102)

which, when inserted in the equation (3.94) yields:

‖d(n+1σkl

)‖ ≤

∣∣∣∣

1 − (1 − α) λ (n+αβ)

1 + α λ (n+αβ)

∣∣∣∣‖d (nσrs) ‖ (3.103)

Since it is said that n+αβ is a positive real number it follows that:

∣∣∣∣

1 − (1 − α) λ (n+αβ)

1 + α λ (n+αβ)

∣∣∣∣≤∣∣∣∣

1 − α

α

∣∣∣∣

n+αβn+αβ

=

∣∣∣∣

1 − α

α

∣∣∣∣

(3.104)

and α ∈ [0, 1]. The new form of equation (3.103) is now:

‖d(n+1σkl

)‖ ≤

∣∣∣∣

1 − α

α

∣∣∣∣‖d (nσrs) ‖ (3.105)

which in conjunction with the requirement for unconditional stability27 yields:

∣∣∣∣

1 − α

α

∣∣∣∣≤ 1 (3.106)

and so it is necessary that:

α ≥ minα =1

2(3.107)

25because n+αMijkl is derived from a convex potential function.26 Ψijkl = Dijkl − λ (1 − α)

`

n+αMijkl

´

and Γijkl = Dijkl + λ α`

n+αMijkl

´

27 that is ‖d n+1σij‖ ≤ ‖d nσij‖



The conclusion is that the Generalized Midpoint rule is unconditionally stable for α ≥ 1/2. In the case when

α < 1/2 the Generalized Midpoint rule is only conditionally stable. To obtain a stability condition for α ≤ 1/2

one has to go back to equation (3.103), and we conclude that:

∣∣∣∣

1 − (1 − α) λ (n+αβ)

1 + α λ (n+αβ)

∣∣∣∣≤ 1 ⇒ λ ≤ 2

maxβ (1 − 2α)for α ≤ 1

2(3.108)

and when α = 1/2, then criticalλ→ ∞, and thus the unconditional stability is recovered.

3.2.4 Crossing the Yield Surface

Midpoint rule algorithms in computational elasto–plasticity require28 the evaluation of the intersection29 stress.

Despite the appeal of the closed form solution, as found in Bicanic (1989), and numerical iterative procedures

as found in Marques (1984) and Nayak and Zienkiewicz (1972), for some yield criteria30 the solution is not that

simple to find. Special problems arises, even with the numerical iterative methods in the area of a apex. The

apex area problems are connected to the derivatives of yield a function.

Having in mind the before mentioned problems, a different numerical scheme, that does not need derivatives,

was sought for solving this problem. One possible solution was found in Press et al. (1988b) in the form of

an excellent algorithm that combines root bracketing, bisection, and inverse quadratic interpolation to converge

from a neighborhood of a zero crossing. The algorithm was developed in the 1960s by van Wijngaarden, Dekker

and others at the Mathematical Center in Amsterdam. The algorithm was later improved by Brent, and so it

is better known as Brent’s method. The method is guaranteed to converge, so long as the function31 can be

evaluated within the initial interval known to contain a root. While the other iterative methods that do not require

derivatives32 assume approximately linear behavior between two prior estimates, inverse quadratic interpolation

uses three prior points to fit an inverse quadratic function33, whose value at y = 0 is taken as the next estimate

of the root x. Lagrange’s classical formula for interpolating the polynomial of degree N − 1 through N points

y1 = f(x1), y2 = f(x2), . . . y3 = f(x3) is given by:

P (x) =(x− x2) (x− x3) · · · (x− xN )

(x1 − x2) (x1 − x3) · · · (x1 − xN )y1 +

+(x− x1) (x− x3) · · · (x− xN )

(x2 − x1) (x2 − x3) · · · (x2 − xN )y2 + · · ·

· · · +(x− x1) (x− x2) · · · (x− xN )

(xN − x1) (xN − x3) · · · (xN − xN−1)yN (3.109)

28 except for the fully implicit Backward Euler algorithm.29contact, penetration point, i.e the point along the stress path where F = 0 or the point where stress state crosses from the

elastic to the plastic region.30 namely for the MRS-Lade elasto–plastic model.31 in our case yield function F (σij).32 false position and secant method.33x as a quadratic function of y.



F( σn

ij)

elastic region

ijσcontact

F( )=0F( σ

predictor

ij )

predictor

σij

ijσ

n

F=0

σ

σσ

2

3

1

Figure 3.2: The pictorial representation of the intersection point problem in computational elasto–plasticity: which

must be resolved for the Forward and Midpoint schemes

If the three point pairs are [a, f(a)], [b, f(b)], [c, f(c)], then the interpolating formula (3.109) yields:

x =(y − f (a)) (y − f (b))

(f (c) − f (a)) (f (c) − f (b))c+

+(y − f (b)) (y − f (c))

(f (a) − f (b)) (f (a) − f (c))a+

+(y − f (c)) (y − f (a))

(f (b) − f (a)) (f (b) − f (a))b (3.110)

By setting y = 0, we obtain a result for the next root estimate, which can be written as:

x = b+

f(b)f(a)

(f(a)f(c)

(f(b)f(c) −

f(a)f(c)

)

(c− b) −(

1 − f(b)f(c)

)

(b− a))

(f(a)f(c) − 1

)(f(b)f(c) − 1

)(f(b)f(a) − 1

) (3.111)

In practice b is the current best estimate of the root and the term:



f(b)f(a)

(f(a)f(c)

(f(b)f(c) −

f(a)f(c)

)

(c− b) −(

1 − f(b)f(c)

)

(b− a))

(f(a)f(c) − 1

)(f(b)f(c) − 1

)(f(b)f(a) − 1

)

is a correction. Quadratic methods34 work well only when the function behaves smoothly. However, they run

serious risk of giving bad estimates of the next root or causing floating point overflows, if divided by a small

number

(f (a)

f (c)− 1

)(f (b)

f (c)− 1

)(f (b)

f (a)− 1

)

≈ 0

Brent’s method prevents against this problem by maintaining brackets on the root and checking where the

interpolation would land before carrying out the division. When the correction of type (3.112) would not land

within bounds, or when the bounds are not collapsing rapidly enough, the algorithm takes a bisection step. Thus,

Brent’s method combines the sureness of bisection with the speed of a higher order method when appropriate.

3.2.5 Singularities in the Yield Surface

Corner Problem

Some yield criteria are defined with more that one yield surface35. We will restrict our attention to a two–surface

yield criterion36. Koiter has shown in Koiter (1960) and Koiter (1953) that in the case when two yield surfaces

are active, the plastic strain rate from equation (3.23) can be derived as follows:

dǫpij = dλconeconemij (σij , q∗) + dλcap

capmij (σij , q∗) (3.112)

where conemij (σij , q∗) and capmij (σij , q∗) are normals to the potential functions at a corner, which belongs to

the yield functions that are active, i.e. Fcone and Fcap. We now observe that we have two non–negative plastic

multipliers dλcone and dλcap instead of one. We must require that at the end of the loading step37, neither of

the two yield functions is violated. These multipliers dλcone and dλcap can be determined from the conditions:

Fcone(n+1σij ,

n+1q∗)

= 0 (3.113)

Fcap(n+1σij ,

n+1q∗)

= 0 (3.114)

34 Newton’s method for example.35for example MRS-Lade yield criterion has two surfaces.36having in mind MRS-Lade cone-cap yield criterion.37after stress correction, i.e. return to the yield surface(s).



ncapm

cap=ij ij

p0

pcapp

capα

cornergrayregion

corner

q

p

n

m

cone

cone

ij

ij

F Q

Fcone

Qcone cap

cap

=0 =0 =0=0

σ

σpredictor

ij

ij

start

two vectorreturn path

σij

n+1

Figure 3.3: Pictorial representation of the corner point problem in computational elasto–plasticity: Yield surfaces

with singular points

noting that by virtue of equation (3.112) we have at the corner singular point:

n+1σij = predσij − dλcone Eijklconemkl − dλcap Eijkl

capmkl (3.115)

The full algorithm for the Backward Euler scheme is derived in Section (??).

Apex Problem

The apex problem, as depicted in Figure (3.4) is solved in an empirical fashion. Rather than facing the complexity

of solving a complex differential geometry problem38 the stress point that is situated in the gray apex region is

immediately returned to the apex point.

In the case when the hardening rule for the cone portion has developed to the stage that it affects the size

of that cone portion of the yield criterion and not the position of intersection with the hydrostatic axis, then all

stress returns from any part of apex gray region will be to the apex point itself. This strategy was used by Crisfield

38using Koiter’s work described in Koiter (1960) and Koiter (1953) and the fact that the sum dǫpij =

P

k dλk (∂Fk/∂σij) can be

transformed into the integral equation dǫpij =

R

dλ (∂F/∂σij)|aroundapexwhere the integration should be carried out infinitesimally

close to, but in the vicinity of the apex point.



regiongray

p

q

σij

nσpredictor

ij

p0 σ

ij

n+1

Fcone=0

straightreturn

apex

apex

to apex

Figure 3.4: The pictorial representation of the apex point problem in computational elasto–plasticity: Yield

surfaces with singular points

(1987). Nevertheless, the problem of integrating the rate equations in the apex gray region is readily solvable for

the piecewise flat yield criteria39 by using Koiter’s conditions as found in Koiter (1960) and Koiter (1953). The

apex problem for yield criteria that are smooth and differentiable everywhere except at the apex point, is solvable

by means of differential geometry. Further work is needed for solving the problem, when the yield surface is not

piecewise flat in the apex vicinity.

Influence Regions in Meridian Plane

In order to define which surface is active and which is not for the current state of stress, a simple two dimensional

analysis will be conducted. The fortunate fact for the MRS-Lade material model is that such an analysis can be

conducted in the p− q meridian plane, only, i.e. the value for θ can be ”frozen”. The concept is to calculate the

stress invariants p, q and θ for the current state of stress40, calculate the position of the apex and corner points in

p− q space for given the θ, calculate the two dimensional gradients at these points, perform linear transformation

of the current stress state41 to the new coordinate systems, and then check for the values of p′i, i = 1, 2, 3, 4,

where p′i is the transformed pi axis.

The angle ψ is defined as the angle between the p axis and the tangent to the potential function. In Appendix

39Mohr - Coulomb for example.40by using equations (3.167) and (3.168) as defined in section (3.4.5).41now in p, q and θ space.



([p,q] v [p’,q’])

p0

q

pcap

4

p

q’

q’

q’

p’

p’

p’

p’

2

1

1

4p

capα

φ

σφ+90o

q

p

q

p

q’2

q’3

p’3

tran

tran

Figure 3.5: Influence regions in the meridian plane for the cone/cap surface of the MRS-Lade material model.

(??) the gradients to the cone portion of the potential surface are defined as:

∂Qcone∂p

= −n ηcone

∂Qcone∂q

= g(θ)

(

1 +q

qa

)m

+g(θ)mq

(

1 + qqa

)−1+m

qa

while, in Appendix (??) the gradients of the cap portion of the yield/potential surface is defined as:

∂Qcap∂p

=2 (p− pm)

p2r

∂Qcap∂q

=2 g(θ)2 q

(

1 + qqa

)2m

fr2 +

2 g(θ)2mq2(

1 + qqa

)−1+2m

fr2 qa



The vector of gradients in p− q space is defined as:

∂Q∂p

∂Q∂q

(3.116)

and the angle φ is calculated as:

φ = arctan

(∂Q∂p

)

(∂Q∂q

)

− 90 (3.117)

Care must be exercised with regard to which potential function is to be used in angle calculations. It should be

mentioned that for the cap portion, the angle at the corner is φ = 0, while at the tip of the cap, the angle is

φ = −90. If a new definition, as found in Ferrer (1992), is used for the cone potential function, where n is

variable and n→ 0 as p→ αpcap, then the corner gray region is empty.

The linear transformation42 between coordinate systems p′ − q′ and p− q is defined as:

p′

q′

=

cosφ sinφ

− sinφ cosφ

p− tranp

q − tranq

(3.118)

and by using that linear transformation, one can check the region where our current stress state, in p, q and θ

space, belongs. Figure (3.5) depicts the transformation scheme and the new coordinate systems at three important

points43.

3.3 A Forward Euler (Explicit) Algorithm

The explicit algorithm (Forward Euler) is based on using the starting point (the state stress σnij and internal

variable space qn∗ on the yield surface) for finding all the relevant derivatives and variables.

The Explicit algorithm can be derived by starting from a first order Taylor expansion about starting point

(σnij , qn∗ ):

Fnew = F old +∂F

∂σmn

∣∣∣∣n

d (nσmn) +∂F

∂q∗

∣∣∣∣n

dq∗ =

= nnmn dσmn + ξ∗h∗dλ = 0 (3.119)

From the differential form of equation (3.36) it follows:

d(fEσmn

)= Emnpq

(d (ǫpq) − d

(ǫppq))

=

= Emnpqd (ǫpq) − Emnpq d(ǫppq)

= Emnpqd (ǫpq) − Emnpq dλ (crossmpq)

42translation and rotation.43at the apex point, corner point and the cap tip point.



so that equation (3.119) becomes:

nnmn Emnpq dǫpq − nnmn Emnpq dλnmpq + ξ∗h∗dλ = 0

and it follows, after solving for dλ

dλ =nnmn Emnpq dǫpq

crosnmn Emnpq crosmpq − ξ∗h∗

With this solution for dλ one can obtain the increments in stress tensor and internal variables as

dσmn = Emnpq dǫpq − Emnpqnnrs Erstu dǫtu

nnab Eabcd nmcd − ξAhAnmpq (3.120)

dqA =

(nnmn Emnpq dǫpq

crosnmn Emnpq crosmpq − ξAhA

)

hA (3.121)

where n() denotes the starting elastic–plastic point for that increment where the combined stress–internal variable

state nes the yield surface. It should be noted that the explicit algorithm performs only one step of the computation

and does not check on the convergence of the provided solutions. This usually results in the slow drift of the

stress–internal variable point from the yield surface for monotonic loading. It also results in spurious plastic

deformations during elastic unloading during cycles of loading–unloading.

Continuum Tangent Stiffness Tensor. The continuum tangent stiffness tensor (contEeppqmn) is obtained from

the explicit (forward Euler) integration procedure (Jeremic and Sture, 1997):

contEeppqmn = Epqmn − Epqklnmkl

nnijEijmnnnotEotrs nmrs − nξA hA

(3.122)

It is important to note that continuum tangent stiffness (contEeppqmn) posses minor symmetries (contEeppqmn =

contEepqpmn = contEeppqnm), while major symmetry (contEeppqmn = contEepmnpq), is only retained for associated elastic–

plastic materials, when nij ≡ mij .

3.4 A Backward Euler (Implicit) Algorithm

In previous sections, the general theory of elasto–plasticity was presented. The accuracy and stability for the

general Midpoint rule algorithm has been shown. In this chapter, the focus is on the Backward Euler algorithm,

which is derived from the general Midpoint algorithm by setting α = 1. The advantage of the Backward Euler

scheme over other midpoint schemes is that the solution is sought by using the normal44 at the final stress state.

By implicitly assuming that such a stress state exists, the Backward Euler scheme is guaranteed to provide a

solution, despite the size of the strain step45. However, it was shown in section (3.2.3) that the Backward Euler

algorithm is only accurate to the first order.

44mij = ∂Q∂σij

45large strain step increments were tested, the scheme converged to the solution even for deviatoric strain steps of 20% in magnitude.



The full implicit Backward Euler algorithm is based on the equation:

n+1σij = predσij − ∆λ Eijkln+1mkl (3.123)

where predσij = Eijkl ǫkl is the elastic trial stress state, Q is the plastic potential function and n+1mkl = ∂Q∂σkl

∣∣∣n+1

is the gradient to the plastic potential function in the stress space at the final stress position, and

predσij = nσij + Eijklpred∆ǫkl (3.124)

is the elastic predicted (trial) stress state.

An initial estimate for the stress n+1σij can be obtained using various other methods. This estimate generally

does not satisfy the yield condition, so some kind of iterative scheme is necessary to return the stress to the yield

surface.

3.4.1 Single Vector Return Algorithm.

If the predictor stress predσij is not in a corner or apex gray regions, a single vector return to the yield surface is

possible. In order to derive such a scheme for a single vector return algorithm, a tensor of residuals rij will be

defined as 46 :

rij = σij −(predσij − ∆λ Eijkl mkl

)(3.125)

This tensor represents the difference between the current stress state σij and the Backward Euler stress state

predσij − ∆λ Eijkl mkl.

The trial stress state predσij is kept fixed during the iteration process. The first order Taylor series expansion

can be applied to Equation 3.125 to obtain the new residual newrij from the old one oldrij

newrij = oldrij + dσij + d(∆λ) Eijkl mkl + ∆λ Eijkl

(∂mkl

∂σmndσmn +

∂mkl

∂qAdqA

)

(3.126)

where dσij is the change in σij , d(∆λ) is the change in ∆λ, and ∂mkl

∂σmndσmn + ∂mkl

∂qAdqA is the change in mkl.

The goal is let newrij = ∅, so one can write

∅ = oldrij + dσij + d(∆λ) Eijkl mkl + ∆λ Eijkl

(∂mkl

∂σmndσmn +

∂mkl

∂qAdqA

)

(3.127)

Similarly,

qA = nqA + ∆λ hA (3.128)

rA will be defined as:

rA = qA − (nqA + ∆λ hA) (3.129)

and nqA is kept fixed during iteration, that

∅ = oldrA + dqA − d(∆λ) hA − ∆λ

(∂hA∂σij

dσij +∂hA∂qB

dqB

)

(3.130)

46By default at increment n + 1, and n+1() is omitted for simplicity.



From equation 3.127 and 3.130, one obtains

Isijmn + ∆λEijkl

∂mkl

∂σmn∆λEijkl

∂mkl

∂qA

−∆λ ∂hA

∂σijδAB − ∆λ∂hA

∂qB

dσmn

dqB

+d(∆λ)

Eijklmkl

−hA

+

oldrijoldrA

= ∅ (3.131)

Since f(σij , qA) = 0, one obtains

∅ = oldf + nmndσmn + ξBdqB (3.132)

From equations 3.131 and 3.132,

d(∆λ) =

oldf −

nmn ξB

Isijmn + ∆λEijkl

∂mkl

∂σmn∆λEijkl

∂mkl

∂qA

−∆λ ∂hA


∂qB

−1

oldrijoldrA

nmn ξB

Isijmn + ∆λEijkl

∂mkl

∂σmn∆λEijkl

∂mkl

∂qA

−∆λ ∂hA


∂qB

−1

Eijklmkl

−hA

(3.133)

The iteration of ∆λ is then

∆λk+1 = ∆λk + d(∆λ)k (3.134)

The iterative procedure is continued until the yield criterion f = 0, ‖rij‖ = ∅, and ‖rA‖ = ∅ are satisfied within

some tolerances at the final stress state 47.

In Equation 3.133, the generalized matrix C, which is defined by

C =

Isijmn + ∆λEijkl

∂mkl

∂σmn∆λEijkl

∂mkl

∂qA

−∆λ ∂hA


∂qB

−1

(3.135)

plays an important role in the implicit algorithm. It should be mentioned here that the above definition is a

simplified expression for very general model with various isotropic and kinematic hardening. Specifically, if there

is no hardening,

C =[

Isijmn + ∆λEijkl∂mkl

∂σmn

]−1

(3.136)

If there is only one isotropic internal variable q,

C =

Isijmn + ∆λEijkl

∂mkl

∂σmn∆λEijkl

∂mkl

∂q

−∆λ ∂h∂σij

1 − ∆λ∂h∂q

−1

(3.137)

For only one kinematic internal variable αij ,

C =

Isijmn + ∆λEijkl

∂mkl

∂σmn∆λEijkl

∂mkl

∂αmn

−∆λ∂hmn

∂σijIsijmn − ∆λ∂hmn

∂αij

−1

(3.138)

47‖‖ is some normal of the tensor



For one isotropic variable q and one kinematic variable αij ,

C =


∂σmn∆λEijkl

∂mkl

∂q ∆λEijkl∂mkl

∂αmn

−∆λ ∂h∂σij

1 − ∆λ∂h∂q −∆λ ∂h∂αij

−∆λ∂hmn

∂σij−∆λ∂hmn

∂q Isijmn − ∆λ∂hmn

∂αij

−1

(3.139)

or for two kinematic variables zij and αij ,

C =


∂σmn∆λEijkl

∂mkl

∂zmn∆λEijkl

∂mkl

∂αmn

−∆λ ∂hz

∂σijIsijmn − ∆λ ∂hz

∂zmn−∆λ ∂hz

∂αij

−∆λ∂hα

mn

∂σij−∆λ

∂hαmn

∂zmnIsijmn − ∆λ

∂hαmn

∂αij

−1

(3.140)

If we define

n =

nmn

ξB

(3.141)

m =

Eijklmkl

−hA

(3.142)

oldr =

oldσijoldrA

(3.143)

Equation 3.134 can be simplified as

d(∆λ) =oldf − nT C

oldr

nT C M(3.144)

and

dσmn

dqB

= −C

(oldr + d(∆λ)m

)(3.145)

3.4.2 Backward Euler Algorithms: Starting Points

Some remarks are necessary in order to clarify the Backward Euler Algorithm. It is a well known fact that the

rate of convergence of the Newton - Raphson Method , or even obtaining convergence at all, is closely tied to the

starting point for the iterative procedure. Bad initial or starting points might lead our algorithm to an oscillating

solution, i.e. the algorithm does not converge. In the following, starting points for the Newton - Raphson iterative

procedure will be established for one– and two–vector return algorithms.

Single Vector Return Algorithm Starting Point.

One of the proposed starting points (Crisfield, 1991) uses the normal at the elastic trial point48 predσij . A first

order Taylor expansion about point predσij yields:

predFnew = predF old +∂F

∂σmn

∣∣∣∣pred

d(predσmn

)+

∂F

∂qA

∣∣∣∣pred

dqA =

= predF old + prednmn dσmn + ξAhAdλ = 0 (3.146)

48I have named this scheme as semi Backward Euler scheme.



It is assumed that the total incremental strain ǫkl is applied in order to reach the point predσij , i.e. predσij =

Eijkl ǫkl so that any further stress ”relaxation” toward the yield surface takes place under zero total strain

condition ǫkl = ∅ . From the differential form of equation (3.36) it follows:

d(predσmn

)= Emnpq

(d(predǫpq

)− d

(predǫppq

))=

= −Emnpq d(predǫppq

)= −Emnpq dλ

(predmpq

)

and equation (3.146) becomes:

predF old − prednmn Emnpq dλpredmpq + ξAhAdλ = 0

and it follows:

dλ =predF old

prednmn Emnpq predmpq − ξAhA

With this solution for dλ we can obtain the starting point for the Newton-Raphson iterative procedure

startσmn = Emnpqpredǫpq − Emnpq

predF old

prednmn Emnpq predmpq − ξAhApredmpq (3.147)

This starting point in six dimensional stress space will in general not satisfy the yield condition F = 0, but it will

provide a good initial guess for the upcoming Newton-Raphson iterative procedure.

It should be mentioned, however, that this scheme for returning to the yield surface is the well known Radial

Return Algorithm , if the yield criterion under consideration is of the von Mises type. In the special case the

normal at the elastic trial point predσij coincides with the normal at the final stress state n+1σij , the return is

exact, i.e. the yield condition is satisfied in one step.

Another possible and readily available starting point can be obtained by applying one Forward Euler step49.

To be able to use the Forward Euler integration scheme, an intersection point has to be found. The procedure

for calculating intersection points is given in section (3.2.4).

A first order Taylor expansion about intersection point crossσij yields:

Fnew = F old +∂F

∂σmn

∣∣∣∣cross

d (crossσmn) +∂F

∂qA

∣∣∣∣cross

dqA =

= crossnmn dσmn + ξAhAdλ = 0 (3.148)

From the differential form of equation (3.36) it follows:

d(fEσmn

)= Emnpq

(d (ǫpq) − d

(ǫppq))

=

= Emnpqd (ǫpq) − Emnpq d(ǫppq)

= Emnpqd (ǫpq) − Emnpq dλ (crossmpq)

49or more steps for really large strain increments, for example over 10% in deviatoric direction. What has actually been done is to

divide the θ region into several parts and depending on the curvature of the yield surface in deviatoric plane, use different schemes

and different number of subincrements ( the more curved, the more subincrements) to get the first, good initial guess. In the region

around θ = 0, one step of the semi Backward Euler scheme is appropriate, but close to θ = π/3 the Forward Euler subincrementation

works better.



and equation (3.148) becomes:

−crossnmn Emnpq dǫpq − crossnmn Emnpq dλcrossmpq + ξAhAdλ = 0

and it follows

dλ =crossnmn Emnpq dǫpq

crosnmn Emnpq crosmpq − ξAhA

With this solution for dλ we can obtain the starting point for the Newton-Raphson iterative procedure

startσmn = Emnpq dǫpq − Emnpqcrossnrs Erstu dǫtu

crossnab Eabcd crossmcd − ξAhAcrossmpq (3.149)

This starting point in six–dimensional stress space will again not satisfy the yield condition50 F = 0, but will

provide a good initial estimate for the upcoming Newton-Raphson iterative procedure.

3.4.3 Numerical Analysis

In this section, the accuracy analysis of the implicit algorithm is assessed. Examples of simple models (Von-

Mises and Drucker-Prager) for accuracy analysis are demonstrated to verify the validation of the general implicit

algorithm. Convergence performance analysis is conducted. More details on accuracy analysis and consistent tan-

gent stiffness are explained. Numerical simulation examples are demonstrated using the implemented framework.

Special concerns are on the comparison of experimental data and numerical results of Dafalias-Manzari model.

Error Assessment

There are various error measures for the integration algorithms. Simo and Hughes (1998), Manzari and Prachathananukit

(2001) used the relative stress norm by Equation 3.150,

δr =

√

(σij − σ∗ij)(σij − σ∗

ij)√σ∗pqσ

∗pq

(3.150)

where σ∗ij is the ‘exact’ stress solution, and σij the calculated stress solution. Alternatively, Jeremic and Sture

(1997) used the normalized energy norm by Equation 3.151,

δn =

∥∥σij − σ∗

ij

∥∥

‖punit‖ (3.151)

where ‖σij‖2= σijDijklσkl, and Dijkl is the elastic compliance fourth-order tensor, punit is the ‘unit’ energy

norm for normalization.

The relative stress norm by Equation 3.150 is more reasonable since two points having the same∥∥σij − σ∗

ij

∥∥

but different σ∗pqσ

∗pq should have different error measures. However, this norm becomes singular and possible

meaningless when σ∗pqσ

∗pq close to zero. The normalized energy norm by Equation 3.151 have no such singularity

50except for the yield criteria that have flat yield surfaces ( in the stress invariant space) so that the first order Taylor linear

expansion, is exact.



problem but it may give the same error index for two points having the same∥∥σij − σ∗

ij

∥∥ but different σ∗

pqσ∗pq.

In this work, we use these two error measure methods, but for simplicity, Equation 3.151 is modified into

δr =

√

(σij − σ∗ij)(σij − σ∗

ij)√

σ0pqσ

0pq

(3.152)

where σ0pqσ

0pq is evaluated at some non-zero initial isotropic stress state. That is, the normalized error is evaluated

by Equation 3.152, and the relative error is evaluated by Equation 3.150.

In our examples, the initial stress state point is set p0 = 100 kPa, q0 = 0 kPa, θ0 = 0, which is the σ0pq

in Equation 3.152. The one-step predicted stress state point for the implicit algorithm is within the range of

0.1 ≤ p ≤ 100 kPa, 0 ≤ q ≤ 100 kPa, 0 ≤ θ ≤ π/3. The ‘exact’ solution is actually unknown for most

elastoplastic problems. Here the ‘exact’ solution is simply replaced by 50 substep solution of the explicit algorithm

in the same one-step prediction incremental. All these error evaluations are within the material constitutive level.

The first test examples are Von-Mises models with the uniaxial yield strength k = 50 kPa, with linear elasticity

parameters are Young’s modulus E = 1×105 kPa, and Poisson’s ratio ν = 0.25.

Figures 3.6 and 3.7 show the iso-error maps for the Von-Mises model with linear isotropic hardening. The

linear hardening modulus H = 2×104 kPa. The blue lines represents the yield surface boundary. It can be seen

that the error magnitudes are as small as 10−10 to 10−9, which implies that the solutions by implicit algorithm for

this linear isotropic hardening Von-Mises model are numerically accurate if one realized that the machine floating

errors cannot be avoided.

0 50 100 150 2000

20

40

60

80

100

p (kPa)

q (k

Pa)

5e−010

5e−0101e−009

1e−009

1.5e−009

(a) At θ = 0

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

θ (rad)

q (k

Pa)

1e−010

2e−0103e−0104e−010

5e−0106e−010

(b) At p = 100 kPa

Figure 3.6: Normalized iso-error maps of Von-Mises model with linear isotropic hardening.

Figures 3.8 and 3.9 show the iso-error maps for the Von-Mises model with Armstrong-Frederick translational

kinematic hardening. The hardening parameters are ha = 5×104 kPa and Cr = 2.5×103. It can be seen that

errors are very small which proves the good performance of the implicit algorithm. The iso-error map gives a good

trend, i.e., the further away from the yield surface, the errors become more pronounced; the normalized errors

are pressure-independent, which fits well the feature of Von-Mises model; the iso-error lines in the q − θ figure

are parallel to the yield surface and are independent of the Lode’s angle θ, which again fits well with Von-Mises



0 50 100 150 2000

20

40

60

80

100

p (kPa)

q (k

Pa)

5e−0

10

5e−010 5e−010

1e−00

9

1e−0

09

1.5e

−009

2e−0

09

(a) At θ = 0

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

θ (rad)

q (k

Pa)

1e−010

2e−0103e−0104e−010

5e−0106e−010

(b) At p = 100 kPa

Figure 3.7: Relative iso-error maps of Von-Mises model with linear isotropic hardening.

model which is only q-related.

0 50 100 150 2000

20

40

60

80

100

p (kPa)

q (k

Pa)

0.002

0.0040.006

0.0080.01 0.010.0120.0140.016

(a) At θ = 0

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

θ (rad)

q (k

Pa)

0.002

0.0040.0060.008

0.01 0.010.0120.0140.016

(b) At p = 100 kPa

Figure 3.8: Normalized iso-error maps of Von-Mises model with Armstrong-Frederick kinematic hardening.

The second test examples are Drucker-Prager model with yield surface constant q/p = 0.8. Linear elasticity

parameters are Young’s modulus E = 1×105 kPa, and Poisson’s ratio ν = 0.25.

The iso-error maps for perfectly plastic Drucker-Prager model are shown in Figures 3.10 and 3.11. The blue

lines represents the yield surface boundary. It can be seen that the error magnitudes are as small as 10−11 to

10−9. Again, these errors are so small that we can consider that the implicit algorithm give accurate solutions

numerically.

Another Drucker-Prager model is with Armstrong-Frederick rotational kinematic hardening, and the parameters

are ha = 20, Cr = 2. The iso-error maps are shown in Figures 3.12 and 3.13. Unlike Von-Mises model, the

normalized errors are pressure-dependent, which fits well the feature of Drucker-Prager model; the iso-error lines



0 50 100 150 2000

20

40

60

80

100

p (kPa)

q (k

Pa)

0.01

0.01

0.020.

030.04

(a) At θ = 0

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

θ (rad)

q (k

Pa)

0.002

0.0040.006

0.008 0.0080.01 0.010.012 0.0120.014 0.014

(b) At p = 100 kPa

Figure 3.9: Relative iso-error maps of Von-Mises model with Armstrong-Frederick kinematic hardening.

0 50 100 150 2000

20

40

60

80

100

p (kPa)

q (k

Pa)

2e−

010

2e−0

10

4e−

010

4e−0

106e

−01

0

6e−

010

8e−0

108e

−01

01e

−00

9

1e−

0091.2e−

009

1.4e−009

(a) At θ = 0

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

θ (rad)

q (k

Pa)

2e−0114e−0116e−0118e−0111e−0101.2e−0101.4e−0101.6e−010

(b) At p = 100 kPa

Figure 3.10: Normalized iso-error maps of Drucker-Prager perfectly plastic model.

in the q − θ figure are parallel to the yield surface and are independent of the Lode’s angle θ, which still fits well

with Drucker-Prager model which does not consider the third stress invariant, Lode’s angle θ. From the relative

iso-error maps in Figure 3.13, very dense iso-error lines are investigated in the region of small pressure, which is

evidently due to the cone apex singularity of Drucker-Prager yield surface.

From the error analysis by the above Von-Mises and Drucker-Prager models, One finds that the implemented

implicit algorithm can offer accurate solutions for simple models with simple hardening laws, e.g. Von-Mises model

with linear hardening and Drucker-Prager model with perfectly plastic hardening (no hardening). Complicated

hardening laws increases the error even for simple plastic models, although the errors are still small. These

observations match the well known conclusion that the error of the implicit algorithm is pretty dependent on the

smoothness of the solution. The implemented implicit algorithm proves very robust for Von-Mises and Drucker-

Prager model with simple or complicated hardening laws.



0 50 100 150 2000

20

40

60

80

100

p (kPa)

q (k

Pa)

1e−

009

1e−

009

1e−0

09

2e−009

2e−

009

2e−

0093e

−00

93e

−00

94e

−00

94e

−00

9

5e−0096e−

009 (a) At θ = 0

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

θ (rad)

q (k

Pa)

2e−0114e−0116e−0118e−0111e−010

1.2e−0101.4e−010

(b) At p = 100 kPa

Figure 3.11: Relative iso-error maps of Drucker-Prager perfectly plastic model.

0 50 100 150 2000

20

40

60

80

100

p (kPa)

q (k

Pa)

0.05

0.05

0.1

0.1

0.15

0.15

0.2

0.25

0.3

(a) At θ = 0

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

θ (rad)

q (k

Pa)

0.0010.0020.0030.0040.0050.0060.0070.0080.009

(b) At p = 100 kPa

Figure 3.12: Normalized iso-error maps of Drucker-Prager model with Armstrong-Frederick kinematic hardening.

Figures 3.14 and 3.15 present the iso-error maps of Dafalias-Manzari model. The initial void ratio is 0.8, and

the other parameters are from Dafalias and Manzari (2004). The blue lines represents the yield surface boundary

(slope ratio m = 0.01). Unlike Von-Mises and Drucker-Prager models, the iso-error lines in the q − θ figure of

Dafalias-Manzari model are not parallel to the yield surface and are dependent of the Lode’s angle θ, which was

one of the highlighting improvements upon the previous version (Manzari and Dafalias, 1997). From Figure 3.15,

when the predicted stress q close to 100 kPa, or or about 100 times the yield strain increment, the relative errors

can reach up to 100%, which implies that even for implicit algorithm, Dafalias-Manzari model still requires small

strain increments. However, when q < 30 kPa, or about 30 times of the yield strain increment, the relative errors

are less than 5%, excepts at the region close to the yield surface apex.

It should be pointed out that errors for the complex Dafalias-Manzari model are much bigger than those of

simple models (e.g. Von-Mises and Drucker-Prager), due to its high non-linearity. However, if the predicted stress



0 50 100 150 2000

20

40

60

80

100

p (kPa)

q (k

Pa)

0.1

0.1

0.1

0.2

0.2

0.2

0.3

0.3

0.3

0.4

0.4

0.4

0.5

0.5

0.5

0.6

0.6

0.6

0.7

0.7

0.7

0.8

0.8

0.8

0.9

0.9

(a) At θ = 0

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

θ (rad)

q (k

Pa)

0.0010.0020.0030.0040.0050.0060.0070.008

(b) At p = 100 kPa

Figure 3.13: Relative iso-error maps of Drucker-Prager model with Armstrong-Frederick kinematic hardening.

(or in other words, the strain increment) is small enough, the algorithm errors are within a small tolerant range.

0 50 100 150 2000

20

40

60

80

100

p (kPa)

q (k

Pa)

0.05

0.05

0.1

0.1

0.1

0.15

0.15

0.15

0.2

0.2

0.2

0.20.25

0.25

(a) At θ = 0

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

θ (rad)

q (k

Pa)

0.05 0.05

0.10.1

0.150.15

0.20.2

0.25

(b) At p = 100 kPa

Figure 3.14: Normalized iso-error maps of Dafalias-Manzari model with average elastic moduli.

Figures 3.14 and 3.15 are based on an approach of averaged elastic moduli. Instead, Figures 3.16 and 3.17

present iso-error maps based on constant elastic moduli approach. The averaged elastic moduli approach improves

the accuracy against the averaged elastic moduli approach. More explanation on these two approaches and other

performance analysis of Dafalias-Manzari model will be in section ??.

Constitutive Level Convergence

In the implemented implicit algorithm, the iteration continues until the absolute value of yield function and the

residue norm of considering variables are less than some small tolerances, or if by equations,

|f | ≤ Tol1; rnorm = ‖r‖ ≤ Tol2 (3.153)



0 50 100 150 2000

20

40

60

80

100

p (kPa)

q (k

Pa)

0.2

0.2

0.2

0.4

0.4

0.4

0.6

0.6

0.6

0.8

0.8

0.81

(a) At θ = 0

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

θ (rad)

q (k

Pa)

0.05 0.05

0.10.1

0.150.15

0.20.2

0.250.25

(b) At p = 100 kPa

Figure 3.15: Relative iso-error maps of Dafalias-Manzari model with average elastic moduli.

0 50 100 150 2000

20

40

60

80

100

p (kPa)

q (k

Pa)

0.05

0.05

0.1

0.1

0.1

0.1

0.15

0.15

0.15

0.15

0.15

0.2

0.2

0.2

0.2

0.2

0.2

0.25

0.25

0.25

0.3

0.3

0.35

0.4

(a) At θ = 0

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

θ (rad)

q (k

Pa)

0.05 0.05

0.10.1

0.150.15

0.20.2

0.25

(b) At p = 100 kPa

Figure 3.16: Normalized iso-error maps of Dafalias-Manzari model with constant elastic moduli.

Three examples including simple Von-Mises model with linear isotropic hardening, relative complicated Drucker-

Prager model with Armstrong-Frederick kinematic hardening, and even more complicated Dafalias-Manzari model

considering fabric dilation effect are presented here to show the constitutive level convergence performances for

the implemented implicit algorithm. In all these examples, both |f | and rnorm v.s. iteration numbers are plotted.

Iteration number 0 represents the ‘virtual’ iteration number before return mapping implicit iteration cycle. |f | at

iteration number 0 thus means |f | at the first predicted stress for each load increment; there is no value of rnorm

at iteration number 0. A tolerance of Tol1 = Tol2 = 1×10−7 is for both |f | and rnorm. The iteration stops

when |f | ≤ Tol1 and rnorm =≤ Tol2 are satisfied, even if there is only one iteration number. The initial stress

is an isotropic stress state of p0 = 100 kPa. The undrained-like load increment is adopted by strain control as

ǫ11 = −2ǫ22 = −2ǫ33 = n×∆ǫ, where n is the load increment number and ∆ǫ is the strain increment interval,

ǫij are strain components.



0 50 100 150 2000

20

40

60

80

100

p (kPa)

q (k

Pa)

0.2

0.2

0.2

0.4

0.4

0.4

0.6

0.6

0.6

0.8

0.8

0.8

1

11

1.2

1.2

1.41.6

(a) At θ = 0

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

θ (rad)

q (k

Pa)

0.05 0.05

0.10.1

0.150.15

0.20.2

0.250.25

(b) At p = 100 kPa

Figure 3.17: Relative iso-error maps of Dafalias-Manzari model with constant elastic moduli.

Figure 3.18 shows the typical constitutive level convergence performance for Von-Mises model with linear

isotropic hardening. The input parameters are Young’s Modulus E = 1×105 kPa, Poisson’s ratio ν = 0.25, the

material strength k = 50 kPa, and the linear isotropic hardening modulus H = 2×104 kPa. The strain increment

interval ∆ǫ is set 2×10−4. It can be seen that for this simple example, only two iteration steps are needed and

|f | and rnorm are far smaller than the tolerances and in fact close to the machine floating error value, or in other

words, the stresses are exactly at the yield surface and the residue norm is zero.

0 1 210

−15

10−10

10−5

100

101

Iteration Number

Tolerance Line

| f |rnorm

Figure 3.18: Typical convergence for Von-Mises model with linear isotropic hardening (tolerance value 1×10−7).

Figure 3.19 shows the typical constitutive level convergence performance for Drucker-Prager model with



Armstrong-Frederick kinematic hardening. The input parameters are Young’s Modulus E = 1×104 kPa, Poisson’s

ratio ν = 0.25, the material q/p ratio is 0.8, and the Armstrong-Frederick parameters are ha = 20, Cr = 2. The

strain increment interval ∆ǫ is set −2×10−4. For this example, both |f | and rnorm are stably decreasing with

the increasing iteration number; However, |f | and rnorm show different rates; |f | needs 5 iteration steps while

rnorm needs 7 iteration steps; The convergence rate of rnorm lags behind that of |f |.

0 1 2 3 4 5 6 7

10−15

10−10

10−5

100

101

Iteration Number

Tolerance Line

| f |rnorm

Figure 3.19: Typical convergence for Drucker-Prager model with Armstrong-Frederick kinematic hardening (tol-

erance value 1×10−7).

Figure 3.20 shows the typical constitutive level convergence performance for the complicated Dafalias-Manzari

model considering fabric dilation effect. The input parameters are as in Table ??, and the initial void ration is

set as 0.8. Different from the above examples, The strain increment interval ∆ǫ is set a much smaller value of

−1×10−5. In this example, again, both |f | and rnorm are stably decreasing with the increasing iteration number;

However, |f | and rnorm show different rates; |f | needs less iteration steps than rnorm; The convergence rate of

rnorm lags behind that of |f |. It should be mentioned here for this complicated Dafalias-Manzari model considering

fabric dilation effect, the typical constitutive level convergence performance is similar to that of Drucker-Prager

model with Armstrong-Frederick kinematic hardening, but with much smaller strain increment interval.

From the above examples, it is clear that the simpler the model is , the better constitutive level convergence

performances are observed. This is consistent to the error assessment in section 3.4.3. Generally, the implemented

implicit algorithm shows stable constitutive level convergence performances provided an appropriate small strain

increment interval for the material model.



0 1 2 3 4 5 6 710

−15

10−10

10−5

100

101

Iteration Number

Tolerance Line

| f |rnorm

Figure 3.20: Typical convergence for Dafalias-Manzari model (tolerance value 1×10−7).

3.4.4 Consistent Tangent Stiffness Tensor

The final goal in deriving the Backward Euler scheme for integration of elasto–plastic constitutive equations is to

use that scheme in finite element computations. If the Newton – Raphson iterative scheme is used at the global

equilibrium level then the use of the so called traditional tangent stiffness tensor51 Eepijkl destroys the quadratic

rate of asymptotic convergence of the iterative scheme. In order to preserve such a quadratic rate, a consistent,

also called algorithmic, tangent stiffness tensor is derived. The consistent tangent stiffness tensor make use of

derivatives of direction52 normal to the potential function, and they are derived at the final, final at each iteration,

that converges to the final stress point on the yield surface, stress point. The traditional forward scheme has a

constant derivative, mij that is evaluated at the intersection point.

It appears that Simo and Taylor (1985) and Runesson and Samuelsson (1985) have first derived the consistent

tangent stiffness tensor. Other interesting articles on the subject can be found in Simo and Taylor (1986),

Simo and Govindjee (1988), Jetteur (1986), Braudel et al. (1986), Crisfield (1987), Ramm and Matzenmiller

(1988) and Mitchell and Owen (1988). As a consequence of consistency, the use of the consistent tangent

stiffness tensor significantly improves the convergence characteristics of the overall equilibrium iterations, if a

Newton - Raphson scheme is used for the latter. Use of the consistent tangent stiffness tensor yields a quadratic

convergence rate of Newton - Raphson equilibrium iterations. In what follows, two derivations are given, namely

the consistent tangent stiffness tensor for single– and two–vector return algorithms.

The concept of consistent linearization was introduced by Hughes and Pister (1978), while detailed explanation

is given by Simo and Hughes (1998). The consistent tangent stiffness leads to quadratic convergence rates at

51the one obtained with the Forward Euler method, i.e. where parameter α = 0.52mij = ∂Q/∂σij , i.e. ∂mij/∂σkl = ∂2Q/∂σij∂σkl .



global level.

It should be mentioned that there are various ‘equivalent’ forms of consistent tangent stiffness depending on

the specific implicit algorithm equations. For instance, Simo and Hughes (1998), and Belytschko et al. (2001)

derived the consistent tangent stiffness by taking current plastic strain as unknown and seeking its derivatives

in the stress space; Perez-Foguet and Huerta and Perez-Foguet et al. (2000) used the numerical differentiation

to calculate the consistent tangent stiffness in a compact matrix-vector form; Choi (2004) adopted the compact

matrix-vector form by Perez-Foguet and Huerta and Perez-Foguet et al. (2000) but taking current plastic strain

as unknown and seeking its derivatives in the elastic strain space. Slightly different from the above strategies,

in this work the implicit algorithm is adopting the traditional form but taking current stress as unknown and

seeking its derivatives in the stress space. Provided these differences, the consistent tangent stiffness in this work

is slightly different from those in the above work.

Single Vector Return Algorithm.

In implicit algorithm, a very important advantage is that the implicit algorithm may lead to consistent (algorithmic)

tangent stiffness (Equation 3.161), a concept against continuum tangent stiffness (Equation ??). The concept of

consistent linearization was introduced in Hughes and Pister (1978), more details on consistent tangent stiffness

were explained in Simo and Hughes (1998). The consistent tangent stiffness leads to quadratic convergence rates

at global level.

It should be mentioned that there are various ‘equivalent’ forms of consistent tangent stiffness depending on

the specific implicit algorithm equations. For instance, Simo and Hughes (1998), and Belytschko et al. (2001)

derived the consistent tangent stiffness by taking current plastic strain as unknown and seeking its derivatives

in the stress space; Perez-Foguet and Huerta and Perez-Foguet et al. (2000) used the numerical differentiation

to calculate the consistent tangent stiffness in a compact matrix-vector form; Choi (2004) adopted the compact

matrix-vector form by Perez-Foguet and Huerta and Perez-Foguet et al. (2000) but taking current plastic strain

as unknown and seeking its derivatives in the elastic strain space. Slightly different from the above strategies, in

this work (section ??) the implicit algorithm is adopting the traditional form but taking current stress as unknown

and seeking its derivatives in the stress space. Provided these differences, the consistent tangent stiffness in this

work is slightly different from those in the above work. The detail derivation will be followed.

When seeking the algorithmic tangent stiffness, we look into the explicit expression of dσij/dǫpredmn . At the

same time, the internal variables are initialized the values at the previous time step, in other words, they are fixed

within the time step when seeking the algorithmic tangent stiffness.

Linearize Equation 3.123, one obtains

dσij = Eijkl dǫpredkl − d(∆λ) Eijkl mkl − ∆λ Eijkl

(∂mkl

∂σmndσmn +

∂mkl

∂qAdqA

)

(3.154)

Similarly, linearize Equation 3.128, one obtains

dqA = d(∆λ) hA + ∆λ

(∂hA∂σij

dσij +∂hA∂qB

dqB

)

(3.155)



From equation 3.154 and 3.155, one obtains

Isijmn + ∆λEijkl

∂mkl

∂σmn∆λEijkl

∂mkl

∂qA

−∆λ ∂hA


∂qB

dσmn

dqB

+d(∆λ)

Eijklmkl

−hA

=

Eijkl dǫpredkl

0

(3.156)

If one use the definitions of 3.135, 3.142 and 3.141, Equation 3.156 can be simplified to

C−1

dσmn

dqB

+ d(∆λ)m =

Eijkl dǫpredkl

0

(3.157)

Linearize the yield function f(σij , qA) = 0, one obtains

nmndσmn + ξBdqB = 0 (3.158)

or in a simplified form

nT

dσmn

dqB

= 0 (3.159)

From Equations 3.157 and 3.159, one obtain

d(∆λ) =nT

C

nTCm

Eijmn dǫpredmn

0

(3.160)

Substitute expression 3.160 into 3.157, one obtains

dσij

dqA

=

C − CmnTC

nT C m

Eijmn dǫpredmn

0

(3.161)

This equation gives the explicit expression of the consistent tangent stiffness dσij/dǫpredmn for the implicit algorithm.

From section ??, if there are interactions between internal variables, the implicit algorithm will become very

complicated. Simple models (e.g. Von-Mises model, or sometimes termed as J2 model) have been proved efficient

and good performance by the implicit algorithm (Simo and Hughes, 1998). Evidently, the implicit algorithm

is mathematically based on the Newton-Raphson nonlinear equation solving method as well as the Eulerian

backward integration method. Theoretically, the Newton-Raphson method may have quadratic convergence

rate. However, Newton-Raphson method is not unconditional stable, and sometimes the iteration will diverge

(Press et al., 1988a). Any bad starting point, non-continuous derivatives around solution, high nonlinearity, and

interactions between internal variables, will deteriorate the implicit algorithm performance. A complicated model

cannot guarantee good performance or quadratic convergence by the implicit algorithm Crisfield (1997). The task

to obtain the analytical expressions (Equations 3.136 to 3.140) may prove exceeding laborious for complicated

plasticity models Simo and Hughes (1998).

The explicit and implicit algorithm performances for the simple model such as Von-Mises and Drucker-Prager

models should not be considered novel, however, these examples can verify the validation of the implemented

algorithms in the general framework of NewTemplate3Dep. To demonstrate the implicit algorithm performance,



an one-element (1×1×1) triaxial compression example of Drucker-Prager model with linear kinematic hardening,

the elastic Young’s modulus is 20000 kPa, Poisson’s ratio is 0.25, yield surface constant q/p = 0.8, linear kinematic

hardening modulus is 2000 kPa. the number of the total incremental steps is 100 with 0.0005 axial displacement

increment in each step. Table 3.1 and Figure 3.21 shows the number of iteration per step and the typical current

residual norms by output interface of OpenSees are shown in Table 3.2. It can be seen that for this problem, only

1-3 global Newton iteration steps are needed and most iteration steps are 2. The iteration convergence rates are

quadratic and almost quadratic.

Table 3.1: Number of Newton iteration/step.

Step 1-4 5-71 72-100

Iterations 3 2 1

Table 3.2: Residual norms for typical iteration steps (tolerance value 1×10−6).

Step 4 20 40 80

Residual Norm 6.80E-05 4.50E-05 1.26E-06 9.48E-09

1.32E-06 4.73E-09 2.36E-10

3.47E-07

1 2 310

−10

10−9

10−8

10−7

10−6

10−5

10−4

Iteration Number

Res

idua

l Nor

m

Setp 4Step 20Step 40Step 80

Tolerance Line

Figure 3.21: Residual norms for typical iteration steps (tolerance value 1×10−6).

The performances analysis of the implemented algorithms for the much more complicated Dafalias-Manzari

model need additional efforts. The reasons are (1) the elasticity of Dafalias-Manzari model is nonlinear; (2)

the yield surface of Dafalias-Manzari model has very small slope; (3) the third stress invariant, or Lode’s angle

θ is considered; (4) the void ratio (volumetric strain) is involved in the constitutive relations; (5) complicated

formulations.



The elastic nonlinearity creates an elastic stiffness that is not a constant during prediction and return mapping

iterations. Borja and Lee (1990); Borja (1991) used a secant elastic stiffness for the fully implicit algorithm of

Cam-Clay elastic-plastic model. Similar strategy was presented by Manzari and Prachathananukit (2001) in the

fully implicit algorithm of Manzari-Dafalias model (1997), which was the simplified version of Dafalias-Manzari

model (2004). However, this method requires a direct integration for predicted elastic bulk or shear modulus

before calculating the secant stiffness. Different models have different integration equations, which makes this

method lose generality. The simplest way is to assume the elastic stiffness is constant. However, constant elastic

stiffness assumption may lead to too much errors. In our implementation, we use average elastic stiffness instead

of secant elastic stiffness for general non-linear elasticity. The elastic stiffness can be expressed by:

E1ijkl = E1(σ

(k)ij ) (3.162)

σ(k+1)ij = σ

(k)ij + E1

ijkl∆ǫkl (3.163)

E2ijkl = E2(σ

(k+1)ij ) (3.164)

Eijkl = 0.5(E1ijkl + E2

ijkl) (3.165)

From Figures 3.14 and 3.15 to Figures 3.16 and 3.17, The averaged elastic moduli approach proves improved

accuracy against the averaged elastic moduli approach.

The very small slope of yield surface of Dafalias-Manzari model can introduce another problem. Suppose

the initial stress point is at p = p0, q = 0, θ = 0, the volumetric strain increment is zero, and the deviatoric

strain increment is ∆ǫq. The small slope of yield surface of Dafalias-Manzari model means a small value of

∆ǫqyield (Manzari and Prachathananukit, 2001) named it yield strain increment) will make the material yield. For

Dafalias-Manzari model,

∆ǫqyield =mp0

G(3.166)

where G is defined by Equation 3.297. For Toyoura sand, ∆ǫqyield can be easily calculated by the material

parameters given by Dafalias and Manzari (2004), e.g. ∆ǫqyield = kyieldp0.50 = 3×10−6p0.5

0 for e = 0.8 if p is in

terms of kPa. Although other soils may have different kyield from Toyoura sand, it should be no far away from

that of Toyoura sand. The small value of kyield can hint us that Dafalias-Manzari model requires very small

strain increment for good performance of elastoplastic calculation, which should be considered as one featured

shortcoming of Dafalias-Manzari model.

Although the simple models, e.g. Von-Mises and Drucker-Prager model, the explicit formulation of the

algorithmic (consistent) tangent stiffness can be easily obtained by Equation 3.161, the explicit expression of the

algorithmic tangent stiffness for Dafalias-Manzari model is very laborious to be obtained, for the reason that the

void ratio is involved in various constitutive relations, the elasticity is not linear, and the αinij heavily depends on

the reverse loading paths. Its algorithmic tangent stiffness is only approximate by using the general calculation

framework for all material models. The global performance will be effected although the local performance proves

well, which again limit the permissible strain increments.



3.4.5 Gradients to the Potential Function

In the derivation of the Backward Euler algorithm and the Consistent Tangent Matrix it is necessary to derive

the first and the second derivatives of the potential function. The function Q is the function of the stress tensor

σij and the plastic variable tensor qA. Derivatives with respect to the stress tensor σij and plastic variable tensor

qA are given here. It is assumed that any stress state can be represented with the three stress invariants p, q and

θ given in the following form:

p = −1

3I1 q =

√

3J2D cos 3θ =3√

3

2

J3D√

(J2D)3(3.167)

I1 = σkk J2D =1

2sijsij J3D =

1

3sijsjkski sij = σij −

1

3σkkδij (3.168)

Stresses are here chosen as positive in tension. The definition of Lode’s angle θ in equation (3.167) implies that

θ = 0 defines the meridian of conventional triaxial extension (CTE), while θ = π/3 denotes the meridian of

conventional triaxial compression (CTC).

The Potential Function is given in the following form:

Q = Q(p, q, θ) (3.169)

The complete derivation of the closed form gradients is given in Appendix C.

Analytical Gradients

The first derivative of the function Q in stress space is:

∂Q

∂σij=∂Q

∂p

∂p

∂σij+∂Q

∂q

∂q

∂σij+∂Q

∂θ

∂θ

∂σij(3.170)

and subsequently the first derivatives of the chosen stress invariants are

∂p

∂σij= −1

3δij (3.171)

∂q

∂σij=

3

2

1

qsij (3.172)



∂θ

∂σij=

3

2

cos (3θ)

q2 sin (3θ)sij −

9

2

1

q3 sin (3θ)tij (3.173)

where:

tij =∂J3D

∂σij

The second derivative of the function Q in stress space is

∂2Q

∂σpq∂σmn=

(∂2Q

∂p2

∂p

∂σmn+∂2Q

∂p∂q

∂q

∂σmn+∂2Q

∂p∂θ

∂θ

∂σmn

)∂p

∂σpq+∂Q

∂p

∂2p

∂σpq∂σmn+

+

(∂2Q

∂q∂p

∂p

∂σmn+∂2Q

∂q2∂q

∂σmn+∂2Q

∂q∂θ

∂θ

∂σmn

)∂q

∂σpq+∂Q

∂q

∂2q

∂σpq∂σmn+

+

(∂2Q

∂θ∂p

∂p

∂σmn+∂2Q

∂θ∂q

∂q

∂σmn+∂2Q

∂θ2∂θ

∂σmn

)∂θ

∂σpq+∂Q

∂θ

∂2θ

∂σpq∂σmn(3.174)

and the second derivatives of the stress invariants are

∂2p

∂σpq∂σmn= ∅ (3.175)

∂2q

∂σpq∂σmn=

3

2

1

q

(

δpmδnq −1

3δpqδnm

)

− 9

4

1

q3smnspq (3.176)

∂2θ

∂σpq∂σmn=

−(

9

2

cos 3θ

q4 sin (3θ)+

27

4

cos 3θ

q4 sin3 3θ

)

spq smn +81

4

1

q5 sin3 3θspq tmn +

+

(81

4

1

q5 sin 3θ+

81

4

cos2 3θ

q5 sin3 3θ

)

tpq smn − 243

4

cos 3θ

q6 sin3 3θtpq tmn +

+3

2

cos (3θ)

q2 sin (3θ)ppqmn − 9

2

1

q3 sin (3θ)wpqmn (3.177)

where:



wpqmn =∂tpq∂σmn

= snpδqm + sqmδnp −2

3sqpδnm − 2

3δpqsmn

and:

ppqmn =∂spq∂σmn

=

(

δmpδnq −1

3δpqδmn

)

Another important gradient is:

∂2Q

∂σij∂qA=∂mij

∂qA=

=∂ ∂Q∂p∂qA

∂p

∂σij+∂ ∂Q∂q∂qA

∂q

∂σij+∂ ∂Q∂θ∂qA

∂θ

∂σij=

=∂2Q

∂p∂qA

∂p

∂σij+

∂2Q

∂q∂qA

∂q

∂σij+

∂2Q

∂θ∂qA

∂θ

∂σij(3.178)

Finite Difference Gradients

After having developed the closed form, analytical derivatives53 the author of this thesis asked himself: ”is there

a simpler way of finding these derivatives?” One of the proposed ways to check the analytical solution is found

in Dennis and Schnabel (1983). Dennis and Schnabel proposes the finite difference method for approximating

derivatives if these derivatives are not analytically available and as a tool to check your analytical derivatives if

they are derived.

Another good reason for developing alternative gradients is that for θ = 0, π/3 gradients are not defined, i.e.

indefinite terms as 0/0 are appearing. One possible solution is the use of l’Hospital’s rule. This has been done in

Peric (1991). The solution to the problem in this work went in a different direction, i.e. instead of aiming for the

analytical form, numerical derivatives are derived.

We should recall that for a function f of a single variable, the finite difference approximation to f ′(x), by

using forward finite difference approach, is given by:

a =f(x+ h) − f(x)

h(3.179)

where h is a vanishingly small quantity. The same definition was used in deriving the finite difference approximation

for the first derivative of the yield function F and potential function Q. The first derivative of F ( or Q ) with

respect to the stress tensor σij for diagonal elements is54 :

53see Appendix (C).54no sum convention implied, just the position of the element.



approx.F,ii =F (σii + hii) − F (σii)

hii(3.180)

and for non-diagonal elements55:

approx.F,ij =F (σij + hij + hji) − F (σij)

2hij(3.181)

where hij is the step size which, because of finite precision arithmetic, is a variable56.

The accuracy of the finite difference approximation to the analytical derivatives is closely bound to the step

size hij . It is suggested in Dennis and Schnabel (1983)[section 5.4.] that for functions given by the simple

formula, the number h should be h =√macheps, while for more complicated functions that number should

be larger. Here macheps is the so called machine epsilon. It is defined as the smallest distinguishable positive

number57, such that 1.0 + macheps > 1.0 on the given platform. For example58, on the Intel x86 platform59

macheps = 1.08E − 19 while on the SUNSparc and DEC platforms macheps = 2.22E − 16. It has been found

that in the case of yield or potential functions the best approximation of analytical gradients is obtained by using

h =√macheps 103. The three order of magnitude increase in the finite difference step is due to a rather

complicated60 formula for yield and potential functions. The error in the approximation, approx.F,ij is found to

be after the N th decimal place, where N is the order of macheps, i.e. macheps = O(N).

Second derivative approximations for one variable function are given in the form:

a =(f(x+ hiei + hjej) − f(x+ hiei)) − (f(x+ hjej) − f(x))

hihj(3.182)

If the first derivatives are available in closed form, one could use equations (3.180) and (3.181) just by replacing

the function values with tensor values for analytical derivatives61.

However, if the analytic derivatives are not available, one has to devise a formula that will create a fourth

order tensor from the changes in two dimensional stress tensors, σij and σkl. Using the scheme employed in

equation (3.182) the following scheme has been devised:

approx.Q,ijkl =

(Q(σmn + hij + hkl) −Q(σmn + hij)) − (Q(σmn + hkl) −Q(σmn))

hijhkl

(3.183)

55since the stress tensor σij is symmetric, change in one non-diagonal element triggers the other to be changed as well.56it is actually one small number, h, that is multiplied with the current stress value so that the relative order of magnitude is

retained.57in a given precision, i.e. float ( real*4 ), double ( real*8 ) or long double ( real*10 ).58the precision sought was double ( real*8 ).59PC computers.60One should not forget that we work with six dimensional tensor formulae directly.61see Dennis and Schnabel (1983), section 5.6.



Special considerations are necessary in order to retain symmetry of the fourth order tensor. At the moment

it has not been possible to figure out how to build the finite difference approximation to the second derivatives

of yield/potential functions for a general stress state. The only finite difference approximation of the second

derivatives that appears to have worked was the one devised in principal stress space. Namely, diagonal elements

of the analytical and the approximate gradients matched exactly, but development of non-diagonal elements, and

the whole scheme of symmetrizing the fourth order approximation, still remain a mystery. However, some pattern

was observed in non–diagonal elements, and the work on symmetrizing it is in progress.

For many different potential functions (or yield functions) the only task left would be the derivation of the first

derivatives of F and Q and the second derivatives of Q with respect to p, q and θ, namely the first derivatives ∂Q∂p ,

∂Q∂q and ∂Q

∂θ and ∂Q∂p , ∂Q∂q and ∂Q

∂θ and the second derivatives ∂2Q∂p2 , ∂2Q

∂p∂q ,∂2Q∂p∂θ ,

∂2Q∂q∂p ,

∂2Q∂q2 , ∂2Q

∂q∂θ ,∂2Q∂θ∂p ,

∂2Q∂θ∂q and

∂2Q∂θ2 . If the potential function is twice differentiable with respect to the stress tensor σij , and if it is continuous

then the Hessian matrix is symmetric.

3.5 Elastic–Plastic Material Models

In this section we present elements of general elastic–plastic material models for geomaterials. We describe various

forms of the yield functions, plastic flow directions and hardening and softening laws.

3.5.1 Elasticity

In elasticity the relationship between the stress tensor σij and the strain tensor ǫkl can be represented in the

following form:

σij = σ (ǫij) (3.184)

In it’s simplest (incremental) form it reads

∆σij = Eijkl∆ǫkl (3.185)

where Eijkl is the fourth order elastic stiffness tensor with 81 independent components in total. The elas-

tic stiffness tensor features both minor symmetry Eijkl = Ejikl = Eijlk and major symmetry Eijkl = Eklij

(Jeremic and Sture, 1997). The number of independent components for such elastic stiffness tensor is 21 (Spencer,

1980).

Most of the models used to describe elastic behavior of soils assume isotropic behavior. The most general

form of the isotropic elastic stiffness tensor of rank 4 has the following representation:

Eijkl = λδijδkl + µ (δikδjl + δilδjk) (3.186)

where λ and µ are the Lame coefficients:

λ =νE

(1 + ν) (1 − 2ν); µ =

E

2 (1 + ν)(3.187)

and E and ν are Young’s Modulus and Poisson’s ratio respectively.



The elastic isotropic behavior of soils obeys Hooke’s law with a constant Poisson’s ratio. The variation of the

Young’s modulus E is usually assumed to be a function of the stress state. To this end we use four different

elastic laws.

Linear Elastic Model. Linear elastic law is the simplest one and assumes constant Young’s modulus E and

constant Poisson’s Ration ν.

Non–linear Elastic Model #1. This non–linear model (Janbu, 1963), (Duncan and Chang, 1970) assumes

dependence of the Young’s modulus on the minor principal stress σ3 = σmin in the form

E = Kpa

(σ3

pa

)n

(3.188)

Here, pa is the atmospheric pressure in the same units as E and stress. The two material constants K and n are

constant for a given void ratio. It should be noted that this material model was developed in order to capture the

reduction of soil stiffness with increasing strain without any reference to plasticity. However, the model can be

used (with some care) to represent non–linear elastic behavior of soils.

Non–linear Elastic Model #2. If Young’s modulus and Poisson’s ratio are replaced by the shear modulus G

and bulk modulus K the non–linear elastic relationship can be expressed in terms of the normal effective mean

stress p as

G and/or K = AF (e,OCR)pn (3.189)

where e is the void ratio, OCR is the overconsolidation ratio (related to the overburden stress) and p = σii/3 is

the mean effective stress (Hardin, 1978).

Lade’s Non–linear Elastic Model. Lade and Nelson (1987) and Lade (1988a) proposed a non–linear elastic

model based on Hooke’s law in which Poisson ratio ν is kept constant. According to this model, Young’s modulus

can be expressed in terms of a power law as:

E = M pa

((I1pa

)2

+

(

61 + ν

1 − 2ν

)J2D

p2a

)λ

(3.190)

where I1 = σii is the first invariant of the stress tensor and J2D = (sijsij)/2 is the second invariant of the

deviatoric stress tensor sij = σij − σkkδij/3. The parameter pa is atmospheric pressure expressed in the same

unit as E, I1 and√J2D and the modulus number M and the exponent λ are constant, dimensionless numbers.

3.5.2 Yield Functions

The typical plastic behavior of frictional materials is influenced by both normal and shear stresses. It is usually

assumed that there exists a yield surface F in the stress space that encompasses the elastic region. States of

stress inside the yield surface are assumed to be elastic (linear or non–linear). Stress states on the surface are

assumed to produce plastic deformations. Yield surfaces for geomaterials are usually shaped as asymmetric tar

drops with smoothly rounded triangular cross sections. In addition to that, simpler yield surfaces, based on



the Drucker–Prager cone or Mohr–Coulomb hexagon can also be successfully used if matched with appropriate

hardening laws. Yield surface shown in Figure 3.22 Lade (1988b) represent typical meridian plane trace for an

isotropic granular material. Line BC represents stress path for conventional triaxial compression test. Figure 3.23

Figure 3.22: Yield surface patterns in the meridian plane for isotropic granular materials (from Lade (1988b))

represents the view of the yield surface traces in the deviatoric plane.

σ2 σ3

σ1

γ 1

γ 0

σ3σ2

σ1

θ

low confinment trace

high confinment trace

Figure 3.23: Deviatoric trace of typical yield surface for pressure sensitive materials.

3.5.3 Plastic Flow Directions

Plastic flow directions are traditionally derived from a potential surface which to some extent reassembles the yield

surface. Potential surfaces for metals are the same as their yield surfaces but experimental evidence suggests that

it is not the case for geomaterials. The non–associated flow rules, used in geomechanics, rely on the potential

surface, which is different from the yield surface, to provide the plastic flow directions. It should be noted that



the potential surface is used for convenience and there is no physical reason to assume that the plastic strain

rates are related to a potential surface Q (Vardoulakis and Sulem, 1995). Instead of defining a plastic potential,

one may assume that the plastic flow direction is derived from an tensor function which does not have to possess

a potential function.

3.5.4 Hardening–Softening Evolution Laws

The change in size and/or shape of the yield and potential surfaces is controlled by the hardening–softening

evolution laws. Physically, these laws control the hardening and/or softening process during loading. Depending

on the evolution type they control, these laws can be in general separated into isotropic and kinematic (also called

anisotropic). The isotropic evolution laws control the size of the yield surface through a single scalar variable. This

is usually related to the Coulomb friction or to the mean stress values at isotropic yielding. The non–isotropic

evolution laws can be further specialized to rotational, translational kinematic and distortional. It should be

noted that all of the kinematic evolution laws can be treated as special case of the general, distortional laws

(Baltov and Sawczuk, 1965). Figure 3.24 depicts various types of evolution laws (for the control of hardening–

softening) in the meridian plane62.

q

p

q

ppc

q

p

q

p

a) b)

d)c)

Figure 3.24: Various types of evolution laws that control hardening and/or softening of elastic–plastic material

models: (a) Isotropic (scalar) controlling equivalent friction angle and isotropic yield stress. (b) Rotational

kinematic hardening (second order tensor) controlling pivoting around fixed point (usually stress origin) of the

yield surface. (c) Translational kinematic hardening (second order tensor) controlling translation of the yield

surface. (d) Distortional (fourth order tensor) controlling the shape of the yield surface.

3.5.5 Tresca Model

The first yield criteria in the metal plasticity is Tresca yield criteria. Tresca yield criteria states that when the

maximum shear stress or, the half difference of the maximum and minimum principal stresses, reaches the shear

62The meridian plane is chosen just for illustration purposes, similar sketch can be produced in deviatoric plane as well.



strength, τs, the material will begin yielding. It is can be expressed by the yield function

f = |τmax| − τs =1

2|σ1 − σ3| − τs = 0 (3.191)

Tresca yield surface in the principal stress space is a regular hexagonal cylinder. It is implied that the intermediate

principal stress plays no role in the yielding for Tresca yield criteria.

3.5.6 von Mises Model

Experimental data showed that for most metals, von Mises yield criteria is more accurate than Tresca criteria.

von Mises yield function can be expressed by

f = 3J2 − k2 = 0 (3.192)

or if extended to include the kinematic hardening,

f =3

2(sij − αij)(sij − αij) − k2 = 0 (3.193)

where k is the scalar internal variable; its initial value is the uniaxial tension strength. αij is the tensor internal

variable termed back stress. Similar to sij , αij is also a deviatoric symmetric tensor.

Although von Mises model is mainly for the metal plasticity analysis, for undrained analysis in geomechanics,

von Mises model can be approximately used to simulate the undrained behaviors, (Yang and Jeremic, 2002),

(Yang and Jeremic, 2003).

The stress derivative of the yield function is

∂f

∂σij= 3(sij − αij) (3.194)

From Equation 3.194, it is easily to derive that

∂f

∂αij= −3(sij − αij) (3.195)

and

∂f

∂k= −2k (3.196)

They will be used in Equation ?? as specific forms of ∂f/∂qA.

If the associated plastic flow rule g = f is assumed, then

mij =∂g

∂σij= 3(sij − αij) (3.197)

∂mij

∂σmn= 3Isijmn − δijδmn (3.198)

∂mij

∂αmn= −3Isijmn (3.199)

where Isijmn is the symmetric unit rank-4 tensor.



It is interesting that from the Equation 3.197, von Mises model gives

ǫpv = ǫpii = λmii = 3λ(sii − αii) = 0 (3.200)

which accords with the phenomena that no plastic volumetric strain occurs for metals. It is implied that the

isotropic stress (hydrostatic pressure) can never make the metal yield for this yield criteria. von Mises model is

therefore pressure-independent.

If k is assumed a linear relation to the equivalent plastic strain ǫpq , or by the equation

k = Hsǫpeq = λHs

(2

3mdevij mdev

ij

)0.5

(3.201)

where Hs is the linear hardening/softening modulus to the equivalent plastic strain, the corresponding hA in

Equation ?? for the evolution law is then

h = Hs

(2

3mdevij mdev

ij

)0.5

(3.202)

where mdevij is the ‘deviatoric’ plastic flow, and if it is associated plasticity,

h = 2Hsk (3.203)

If αij is assumed a linear relation to the plastic strain tensor ǫpij , or by the equation

αij = Htǫpij = λHtmij (3.204)

if it is associated plasticity,

αij = 3λHt(sij − αij) (3.205)

where Ht is the linear hardening/softening modulus to plastic strain tensor, the corresponding hA in Equation ??

for the evolution law is then

hij = Htmij (3.206)


hij = 3Ht(sij − αij) (3.207)

A saturation-type kinematic hardening rule is the Armstrong-Frederick hardening (Armstrong and Frederick,

1966),

αij =2

3haǫ

pij − cr ǫ

peqαij (3.208)


αij = λ[ha(sij − αij) − 2crkαij ] (3.209)

where ha and cr are material constants. The corresponding hA in Equation ?? for the Armstrong-Frederick

evolution law is then

hij =2

3hamij − crmeqαij (3.210)

where meq is the ‘equivalent’ plastic flow, and if it is associated plasticity,

hij = 2hasij − 2(ha + crk)αij (3.211)



Yield and Plastic Potential Functions: von Mises Model (form I)

Yield function and related derivatives

f =3

2[(sij − αij) (sij − αij)] − k2 = 0 (3.212)

∂f

∂σij= 3

∂skl∂σij

(skl − αkl)

= 3

(

δkiδlj −1

3δklδij

)

(skl − αkl)

= 3 (sij − αij) (3.213)

∂f

∂αij= −3

∂αkl∂αij

(skl − αkl)

= −3δkiδlj (skl − αkl)

= −3 (sij − αij) (3.214)

∂f

∂k= −2k (3.215)

Plastic flow (associated plasticity) and related derivatives

mij =∂f

∂σij= 3 (sij − αij) (3.216)

∂mij

∂σmn= 3δimδjn − δijδmn (3.217)

∂mij

∂k= 0 (3.218)

∂mij

∂αmn= −3δimδjn (3.219)

Yield and Plastic Potential Functions: von Mises Model (form II)


f = [(sij − αij) (sij − αij)]0.5 −

√

3

2k = 0 (3.220)



∂f

∂σij=∂skl∂σij

(skl − αkl) [(smn − αmn) (smn − αmn)]−0.5

=

(

δkiδlj −1

3δklδij

)

(skl − αkl) [(smn − αmn) (smn − αmn)]−0.5

= (sij − αij) [(smn − αmn) (smn − αmn)]−0.5

(3.221)

∂f

∂αij= − (sij − αij) [(smn − αmn) (smn − αmn)]

−0.5(3.222)

∂f

∂k= −

√

3

2(3.223)


mij =∂f

∂σij= (sij − αij) [(smn − αmn) (smn − αmn)]

−0.5(3.224)

∂mij

∂σmn=

(

δimδjn − 1

3δijδmn

)

[(srs − αrs) (srs − αrs)]−0.5−(sij − αij) (smn − αmn) [(srs − αrs) (srs − αrs)]

−1.5

(3.225)

∂mij

∂k= 0 (3.226)

∂mij

∂αmn= −δimδjn [(srs − αrs) (srs − αrs)]

−0.5+(sij − αij) (smn − αmn) [(srs − αrs) (srs − αrs)]

−1.5(3.227)

3.5.7 Drucker-Prager Model

Drucker and Prager (1950) proposed a right circle cone to match with the Mohr-Coulomb irregular hexagonal

pyramid, which can be expressed by

f = αI1 +√

J2 − β = 0 (3.228)

or if considering the kinematic hardening,

f = αI1 + [1

2(sij − pαij)(sij − pαij)]

12 − β = 0 (3.229)

where α and β are material constants.

By coinciding Drucker-Prager cone with the outer apexes of the Mohr-Coulomb hexagon locus, we get the

compressive cone of Drucker-Prager model, with the constants as

α =2 sinφ√

3(3 − sinφ), β =

6 cosφ√3(3 − sinφ)

c (3.230)



By coinciding Drucker-Prager cone with the inner apexes of the Mohr-Coulomb hexagon locus, we get the

tensile cone of Drucker-Prager model, with the constants as

α =2 sinφ√

3(3 + sinφ), β =

6 cosφ√3(3 + sinφ)

c (3.231)

We can also get the mean cone of the compressive and tensile cone, with the constants as

α =

√3 sinφ

9 − sin2 φ, β =

2√

3 cosφ

9 − sin2 φc (3.232)

Another inner-tangent cone to the the Mohr-Coulomb pyramid, with the constants as

α =tanφ

√

9 + 12 tan2 φ, β =

3c√

9 + 12 tan2 φ(3.233)

Obviously, in practice α and β are not directly obtained from experiments. They are functions of Mohr-

Coulomb parameters, the cohesion c and the friction angle φ, which can be determined by experiments. The

shape of Drucker-Prager yield surface has different types. They only partially satisfy the above requirements for

locus in the π plane: they do not coincide with both compressive and tensile experimental points.

A useful formulation on Equation 3.228 is

∂f

∂σij= αδij +

sij

2√J2

(3.234)

For cohesionless sands, k = 0, Drucker-Prager yield function can thus be simplified as

f = αI1 +√

J2 = 0 (3.235)

or in terms of p and q,

f = q −Mp = 0 (3.236)

If Equation 3.230 is adopted, then M can be easily derived as

M =6 sinφ

3 − sinφ(3.237)

If the kinematic hardening is taken account, Equation 3.236 can be extended into

f =3

2[(sij − pαij)(sij − pαij)] −M2p2 = 0 (3.238)

Useful formulations for this yield function are

∂f

∂σij= 3sij +

(

smnαmn +2

3M2p

)

δij (3.239)

∂f

∂αij= −3psij (3.240)

where sij = sij − pαij .

If the plastic flow is assumed associated, g = f , then

mij =∂g

∂σij= 3sij +

(

smnαmn +2

3M2p

)

δij (3.241)

the ‘deviatoric’ plastic flow is therefore

meq = 2Mp (3.242)



Yield and Plastic Potential Functions: Drucker-Prager Model (form I)


f =3

2[(sij − pαij) (sij − pαij)] − k2p2 = 0 (3.243)

∂f

∂σij=

3

2

[

2∂smn∂σij

(smn − pαmn)

]

+3

2

[

−2αmn∂p

∂σij(smn − pαmn)

]

− 2k2p∂p

∂σij

= 3

(

δmiδnj −1

3δmnδij

)

(smn − pαmn) + 3

[

αmn1

3δij (smn − pαmn)

]

+2

3k2pδij

= 3 (sij − pαij) + αmn (smn − pαmn) δij +2

3k2pδij (3.244)

∂f

∂αij= −3p (sij − pαij) (3.245)

∂f

∂k= −2kp2 (3.246)


mij =∂f

∂σij= 3 (sij − pαij) + αrs (srs − pαrs) δij +

2

3k2pδij (3.247)

∂mij

∂σmn= 3

((

δimδjn − 1

3δijδmn

)

− 1

3δmnαij

)

+ αrs∂ (srs − pαrs)

∂σmnδij +

2

3k2 ∂p

∂σmnδij

= 3δimδjn − δijδmn − δmnαij + αrs

(

δrmδsn − 1

3δrsδmn +

1

3δmnαrs

)

δij +2

3k2 ∂p

∂σmnδij

= 3δimδjn − δijδmn − δmnαij + αmnδij +1

3δmnαrsαrsδij −

2

9k2δmnδij

= 3δimδjn +

(

−1 +1

3αrsαrs −

2

9k2

)

δijδmn − δmnαij + αmnδij (3.248)

∂mij

∂k=

4

3kpδij (3.249)

∂mij

∂αmn= −3pδimδjn + δrmδsn (srs − pαrs) δij − αrspδrmδsnδij

= −3pδimδjn + (smn − pαmn) δij − αmnpδij

= −3pδimδjn + smnδij − 2pαmnδij (3.250)



Yield and Plastic Potential Functions: Drucker-Prager Model (form II)


f = [(sij − pαij) (sij − pαij)]0.5 −

√

2

3kp = 0 (3.251)

∂f

∂σij=

(∂smn∂σij

− αmn∂p

∂σij

)

(smn − pαmn) [(srs − pαrs) (srs − pαrs)]−0.5 −

√

2

3k∂p

∂σij

=

(

δmiδnj −1

3δmnδij +

1

3αmnδij

)

(smn − pαmn) [(srs − pαrs) (srs − pαrs)]−0.5

+

√

2

27kδij

=

[

(sij − pαij) +1

3αmnδij (smn − pαmn)

]

[(srs − pαrs) (srs − pαrs)]−0.5

+

√

2

27kδij (3.252)

∂f

∂αij= −pδmiδnj (smn − pαmn) [(srs − pαrs) (srs − pαrs)]

−0.5

= −p (sij − pαij) [(srs − pαrs) (srs − pαrs)]−0.5

(3.253)

∂f

∂k= −

√

2

3p (3.254)


mij =∂f

∂σij=

[

(sij − pαij) +1

3αpqδij (spq − pαpq)

]


+

√

2

27kδij (3.255)

∂mij

∂σmn=

[(

δmiδnj −1

3δmnδij +

1

3δmnαij

)

+1

3αpqδij

(

δmpδnq −1

3δmnδpq +

1

3δmnαpq

)]


−[

(sij − pαij) +1


](

δmrδns −1

3δmnδrs +

1

3δmnαrs

)

(srs − pαrs) [(stu − pαtu) (stu − pαtu)]−1.5

(3.256)

∂mij

∂k=

√

2

27δij (3.257)

∂mij

∂αmn=

[

−pδmiδnj +1

3δmpδnqδij (spq − pαpq) −

1

3pαpqδijδmpδnq

]


−[

(sij − pαij) +1


]

[−pδrmδsn (srs − pαrs)] [(stu − pαtu) (stu − pαtu)]−1.5

(3.258)



Hardening and Softening Functions

Isotropic Hardening and related derivatives

Linear Isotropic Hardening (Linear Eeq)

k = Hmequivalent = H

(2

3mijmij

)0.5

(3.259)

∂k

∂σij=

2

3Hmpq

∂mpq

∂σij

(2

3mmnmmn

)−0.5

(3.260)

∂k

∂k=

2

3Hmpq

∂mpq

∂k

(2

3mmnmmn

)−0.5

(3.261)

∂k

∂αij=

2

3Hmpq

∂mpq

∂αij

(2

3mmnmmn

)−0.5

(3.262)

Kinematic Hardening and related derivatives

Linear Kinematic Hardening (Linear Eij)

αij = Hmdevij = H

(

mij −1

3mklδklδij

)

(3.263)

∂αij∂σmn

= H

(∂mij

∂σmn− 1

3

∂mkl

∂σmnδklδij

)

(3.264)

∂αij∂k

= H

(∂mij

∂k− 1

3

∂mkl

∂kδklδij

)

(3.265)

∂αij∂αmn

= H

(∂mij

∂αmn− 1

3

∂mkl

∂αmnδklδij

)

(3.266)

Armstrong-Frederick Kinematic Hardening (AF Eij)

αij =2

3hamij − cr

(2

3mrsmrs

)0.5

αij (3.267)

∂αij∂σmn

=2

3ha∂mij

∂σmn− 2

3crmrs

∂mrs

∂σmn

(2

3mklmkl

)−0.5

αij (3.268)

∂αij∂αmn

= Ht∂mij

∂αmn(3.269)



3.5.8 Modified Cam-Clay Model

The pioneering research work on the critical state soil mechanics by the researchers in Cambridge University

(Roscoe et al., 1958), (Roscoe and Burland, 1968), (Wood, 1990)) has made great contribution on the modern

soil elastoplastic models. The original Cam Clay model (Roscoe et al., 1963), and later the modified Cam Clay

model (Schofield and Wroth, 1968) were within the critical state soil mechanics framework. We focus on only

the modified Cam Clay model and herein the word ‘modified’ is omitted to shorten writing.

Critical State The critical state line (CSL) takes the form

ec = ec,r − λc ln pc (3.270)

where ec is the critical void ratio at the critical mean effective stress pc63, ec,r is the reference critical void ratio,

λc is the normal consolidation slope.

The critical state soil mechanics assumes that the normal consolidation line (NRL) is parallel to the CSL,

which is expressed by

e = eλ − λ ln p (3.271)

where eλ is the intercept on the NRL at p = 1. λ is the normal consolidation slope or the elastoplastic slope of

e− ln p relation, and λc = λ.

The unloading-reloading line (URL) take the similar form but with different slope by

e = eκ − κ ln p (3.272)

where eκ is the intercept on the URL at p = 1. λc is the normal consolidation slope or the elastoplastic slope of

e− ln p relation.

Elasticity The elastic bulk modulus K can be directly derived from the Equation 3.272 and takes the form

K =(1 + e)p

κ(3.273)

If a constant Poisson’s ratio ν is assumed, since the isotropic elasticity needs only two material constants, the

shear elastic modulus can be obtained in terms of K and ν by

G =3(1 − 2ν)

2(1 + ν)K =

3(1 − 2ν)(1 + e)

2(1 + ν)κp (3.274)

Alternatively, a constant shear elastic modulus G can be assumed and then the Poisson’s ratio ν is expressed

in terms of K and G as

ν =3K − 2G

2(G+ 3K)(3.275)

63In this chapter, only single-phase (dry phase) is studied, the total and effective stresses are thus identical, e.g. p′c = pc.



Yield Function The yield function of the Cam Clay model is defined by

f = q2 −M2c [p(p0 − p)] = 0 (3.276)

where Mc is the critical state stress ration in the q − p plane, and the p0 is the initial internal scalar variable,

which is controlled by the change of the plastic volumetric strain.

The gradient of the yield surface to the stress can be obtained as

∂f

∂σij= 2q

∂q

∂σij−M2

c (2p− p0)∂p

∂σij= 3sij +

1

3M2c (p0 − 2p)δij (3.277)

where ∂q/∂σij and ∂p/∂σij are independent of the yield function.

The gradient of the yield surface to p0 will be used in the integration algorithm, and can be expressed by

∂f

∂p0= M2

c p (3.278)

Plastic Flow The plastic flow of the Cam Clay model is associated with its yield function, in other words, the

plastic flow is defined by the potential function, g, which is assumed the same as the yield function, f .

g = f = q2 −M2c [p(p0 − p)] = 0 (3.279)

The stress gradient to the yield surface can be obtained as

mij =∂g

∂σij= 2q

∂q

∂σij+M2

c (2p− p0)∂p

∂σij= 3sij +

1

3M2c (p0 − 2p)δij (3.280)

It can define the plastic dilation angle β, which is related to the ratio of plastic volumetric and deviatoric

strain (Wood, 1990), by

tanβ = −∆ǫpv∆ǫpq

=M2c (p0 − 2p)

2q(3.281)

It is interesting to find that from Equation 3.281, when p < p0/2, the plastic dilation angle is positive; when p >

p0/2, the plastic dilation angle is negative. If p = p0/2, the plastic dilation angle is zero, which is corresponding

to the critical state. This is evidently more realistic than Drucker-Prager model, whose associated plastic flow

always gives positive plastic dilation angle.

Evolution Law The evolution law of the Cam Clay model is a scalar one, which can be expressed by

p0 =(1 + e)p0

λ− κǫpv (3.282)

With this scalar evolution law, the change of p0 is decided by the change of plastic volumetric strain. When

it reaches the critical state, or when there is no plastic volumetric strain, the evolution of p0 will cease. From

Equation 3.282, one gets

p0 = λ(1 + e)p0

λ− κmii (3.283)

so if using Equation 3.280 further, one obtains

h =(1 + e)p0

λ− κM2c (2p− p0) (3.284)

or by dilation angle,

h =2(1 + e)p0q

λ− κtanβ (3.285)



Yield and Plastic Potential Functions: Cam-Clay Model


f = q2 −M2c [p(p− p0)] = 0 (3.286)

∂f

∂σij= 2q

∂q

∂σij−M2

c (2p− p0)∂p

∂σij

= 3sij +1

3M2c (p0 − 2p)δij (3.287)

∂f

∂p0= M2

c p (3.288)


mij =∂f

∂σij= 3sij +

1

3M2c (p0 − 2p) δij (3.289)

∂mij

∂σmn= 3

∂sij∂σmn

− 2

3M2c δij

∂p

∂σmn

= 3δimδjn − δijδmn +2

9M2c δijδmn

= 3δimδjn +

(2

9M2c − 1

)

δijδmn (3.290)

∂mij

∂p0=

1

3M2c δij (3.291)

Isotropic Hardening and related derivatives (CC Ev) Note the due to the current definition of p (i.e.

p = − 13σii), a minus sign appears in from of the evolution of p0 as follows:

p0 = − (1 + e) p0

λ− κmii (3.292)

=(1 + e) p0

λ− κM2c (2p− p0) (3.293)

∂p0

∂σij=

(1 + e) p0

λ− κM2c

(−2

3δij

)

(3.294)

∂p0

∂p0=

2 (1 + e)

λ− κM2c (p− p0) (3.295)



3.5.9 Dafalias-Manzari Model

Within the critical state soil mechanics framework, Manzari and Dafalias (1997) proposed a two-surface sand

model. This model considered the effects of the state parameter on the behaviors of the dense or loose sands.

The features of this model include successfully predicting the softening at the dense state in drained loading,

and also softening at the loose state but in the undrained loading. Dafalias and Manzari (2004) later presented

an improved version. This version introduced the fabric dilatancy tensor which has a significant effect on the

contraction unloading response. It is also considered the Lode’s angle effect on the bounding surface, which

produces more realistic responses in non-triaxial conditions. Here only the new version is summarized. The

compression stress is assumed negative here, which is different from the original reference by Dafalias and Manzari

(2004).

Critical State Instead of using the most common linear line of critical void ration vs. logarithmic critical mean

effective stress, the power relation recently suggested by Li and Wang (1998) was used:

ec = ec,r − λc

(pcPat

)ξ

(3.296)

where ec is the critical void ratio at the critical man effective stress p′c, ec,r is the reference critical void ratio, λc

and ξ (for most sands, ξ = 0.7) are material constants, and Pat is the atmospheric pressure for normalization.

Elasticity The elastic incremental moduli of shear and bulk, are following Richart et al. (1970):

G = G0(2.97 − e)2

(1 + e)

(p

Pat

)0.5

Pat, K =2(1 + ν)

3(1 − 2ν)G (3.297)

where G0 is a material constant, e is the void ratio, and ν is the Poisson’s ratio.

The isotropic hypoelasticity is then defined by

eeij =sij2G

, ǫev =p

K(3.298)

Yield Function The yield function is defined by

f = |Λ| −√

2

3mp = 0 (3.299)

where sij is the deviatoric stress tensor, αij is the deviatoric back stress-ratio tensor, m is a material constant,

and

|Λ| = ‖sij − pαij‖ = [(sij − pαij)(sij − pαij)]0.5 (3.300)

The gradient of the yield surface to the stress can be obtained as

∂f

∂σij= nij +

1

3(αpqnpq +

√

2

3m)δij (3.301)

where rij = sij/p is the normalized deviatoric stress tensor, and nij is the unit gradient tensor to the yield surface

defined by

nij =sij − pαij

|Λ| (3.302)



It is evident that nii ≡ 0 and nijnij ≡ 1.

The gradient of the yield surface to αij can be easily obtained as

∂f

∂αij= −pnij (3.303)

The tensor of nij is to defined θn, the Lode’s angle of the yield gradient, by the equation

cos 3θn = −√

6nijnjknki (3.304)

where 0 ≤ θn ≤ π/6 and θn = 0 at triaxial compression and θn = π/6 at triaxial extension.

The critical stress ratio M at any stress state can be interpolated between Mc, the triaxial compression critical

stress ratio, and Me, the triaxial extension critical stress ratio.

M = Mcg(θn, c), g(θn, c) =2c

(1 + c) − (1 − c) cos 3θn, c =

Me

Mc(3.305)

The line from the origin of the π plane parallel to nij will intersect the bounding, critical and dilation surfaces

at three ‘image’ back-stress ratio tensor αbij , αcij , and αdij respectively (Figure 3.25), which are expressed as

αbij =

√

2

3[M exp (−nbψ) −m]nij =

(√

2

3αbθ

)

nij (3.306)

αcij =

√

2

3[M −m]nij =

(√

2

3αcθ

)

nij (3.307)

αdij =

√

2

3[M exp (ndψ) −m]nij =

(√

2

3αdθ

)

nij (3.308)

where ψ = e− ec is the state parameter; nb and nd are material constants.

Plastic Flow The plastic strain is given by

ǫpij = λRij = λ(R′ij +

1

3Dδij) (3.309)

The deviatoric plastic flow tensor is

R′ij = Bnij + C(niknkj −

1

3δij) (3.310)

where

B = 1 +3

2

1 − c

cg cos 3θn, C = 3

√

3

2

1 − c

cg (3.311)

The volumetric plastic flow part is

D = −Ad(αdij − αij)nij = −Ad(√

2

3αdθ − αijnij

)

(3.312)

where

Ad = A0(1 + 〈zijnij〉) (3.313)

A0 is a material constant, and zij is the fabric dilation tensor. The Macauley brackets 〈〉 is defined that 〈x〉 = x,

if x > 0 and 〈x〉 = 0, if x ≤ 0.



Figure 3.25: Schematic illustration of the yield, critical, dilatancy, and bounding surfaces in the π-plane of

deviatoric stress ratio space (after Dafalias and Manzari 2004).

Evolution Laws This model has two tensorial evolution internal variable, namely, the back stress-ratio tensor

αij and the fabric dilation tensor zij .

The evolution law for the back stress-ratio tensor αij is

αij = λ[2

3h(αbij − αij)] (3.314)

with

h =b0

(αij − αij)nij(3.315)

where αij is the initial value of αij at initiation of a new loading process and is updated to the new value when

the denominator of Equation 3.315 becomes negative. b0 is expressed by

b0 = G0h0(1 − che)

(p

Pat

)−0.5

(3.316)

where h0 and ch are material constants.

The evolution law for the fabric dilation tensor zij is

zij = −cz⟨

D⟩

(zmaxnij + zij) (3.317)

where cz and zmax are material constants.

Analytical Derivatives for the Implicit Algorithm When implemented into an implicit algorithm for the

Dafalias-Manzari model, some complicated additional analytical derivatives are needed. This section gives the

analytical derivatives expressions based on the tensor calculus.



Analytical expression of∂mij

∂σkl:

∂mij

∂σmn= B

∂nij∂σmn

+ nij∂B

∂σmn+ C

∂nik∂σmn

nkj + (niknkj −1

3δij)

∂C

∂σmn

+1

3δij

∂D

∂σmn(3.318)

where

∂nij∂σmn

=1

|Λ|

[

Isijmn − 1

3δijδmn +

1

3αijδmn − nijnmn − 1

3(αabnab)nijδmn

]

(3.319)

∂D

∂σmn= − ∂Ad

∂σmn

(√

2

3αdθ − αabnab

)

−Ad

(√

2

3

∂αdθ∂σmn

− αab∂nab∂σmn

)

(3.320)

and

∂B

∂σmn=

3

2

(1 − c

c

)(∂g

∂σmncos 3θ + g

∂ cos 3θ

∂σmn

)

(3.321)

∂C

∂σmn= 3

√

3

2

(1 − c

c

)∂αdθ∂σmn

(3.322)

∂αdθ∂σmn

= Mc exp (ndψ)

(

gnd∂ψ

∂σmn+

∂g

∂σmn

)

(3.323)

∂ψ

∂σmn= − ξλc

3Pat

(p

Pat

)(ξ−1)

δmn (3.324)

∂g

∂σmn= g2

(1 − c

2c

)∂ cos 3θ

∂σmn(3.325)

∂ cos 3θ

∂σmn= −3

√6∂nij∂σmn

(njknki) (3.326)

∂Ad∂σmn

= A0zab∂nab∂σmn

zabnab (3.327)

and defineX

= 1 if X > 0, andX

= 0 if X ≤ 0.


∂αkl:

∂mij

∂αmn= B

∂nij∂αmn

+ nij∂B

∂αmn+ C

∂nik∂αmn

nkj + (niknkj −1

3δij)

∂C

∂αmn

+1

3δij

∂D

∂αmn(3.328)

where

∂nij∂αmn

=p

|Λ|(nijnmn − Isijmn

)(3.329)



∂D

∂αmn= − ∂Ad

∂αmn

(√

2

3αdθ − αabnab

)

−Ad

(√

2

3

∂αdθ∂αmn

− nmn − αab∂nab∂αmn

)

(3.330)

and

∂B

∂αmn=

3

2

(1 − c

c

)(∂g

∂αmncos 3θ + g

∂ cos 3θ

∂αmn

)

(3.331)

∂C

∂αmn= 3

√

3

2

(1 − c

c

)∂αdθ∂αmn

(3.332)

∂αdθ∂αmn

= Mc exp (ndψ)∂g

∂αmn(3.333)

∂g

∂αmn= g2

(1 − c

2c

)∂ cos 3θ

∂αmn(3.334)

∂ cos 3θ

∂αmn= −3

√6∂nij∂αmn

(njknki) (3.335)

∂Ad∂αmn

= A0zab∂nab∂αmn

zabnab (3.336)


∂zmn:

∂mij

∂zmn=

1

3δij

∂D

∂zmn(3.337)

where

∂D

∂zmn= − ∂Ad

∂zmn

(√

2

3αdθ − αabnab

)

(3.338)

and

∂Ad∂zmn

= A0nmn zabnab (3.339)

Analytical expression of∂Aij∂σmn

:

∂Aij∂σmn

=2

3

[

∂h

∂σmn

(√

2

3αbθnij − αij

)

+

√

2

3h

(

nij∂αbθ∂σmn

+ αbθ∂nij∂σmn

)]

(3.340)

where

∂αbθ∂σmn

= Mc exp (−nbψ)

(∂g

∂σmn− nbg

∂ψ

∂σmn

)

(3.341)

∂h

∂σmn=

1

(αab − αinab)nab

[∂b0∂σmn

− h(αpq − αinpq)∂npq∂σmn

]

(3.342)



and

∂b0∂σmn

=b06pδmn (3.343)

Analytical expression of∂Aij∂αmn

:

∂Aij∂αmn

=2

3

[(√

2

3αbθnij − αij

)

∂h

∂αmn

+

√

2

3h

(

nij∂αbθ∂αmn

+ αbθ∂nij∂αmn

− Isijmn

)]

(3.344)

where

∂αbθ∂αmn

= Mc exp (−nbψ)∂g

∂αmn(3.345)

∂h

∂αmn= − h

(αab − αinab)nab

[

nmn + (αpq − αinpq)∂npq∂αmn

]

(3.346)

Analytical expression of∂Aij∂zmn

:

∂Aij∂zmn

= ∅ (3.347)

Analytical expression of∂Zij∂σmn

:

∂Zij∂σmn

= −cz[

(zmaxnij + zij)∂D

∂σmn+ zmaxD

∂nij∂σmn

]D

(3.348)

Analytical expression of∂Zij∂αmn

:

∂Zij∂αmn

= −cz[

(zmaxnij + zij)∂D

∂αmn+ zmaxD

∂nij∂αmn

]D

(3.349)

Analytical expression of∂Zij∂zmn

:

∂Zij∂zmn

= −cz(

DIsijmn + zmaxnij∂D

∂zmn

)D

(3.350)




Chapter 4

Probabilistic Elasto–Plasticity (2004–)

(In collaboration with Dr. Kallol Sett)

111



Chapter 5

Stochastic Elastic–Plastic Finite Element

Method (2006–)

(In collaboration with Dr. Kallol Sett)

113



Chapter 6

Large Deformation Elasto–Plasticity

(1996–2004–)

(In collaboration with Dr. Zhao Cheng)

6.1 Continuum Mechanics Preliminaries: Kinematics

6.1.1 Deformation

In modeling the material nonlinear behavior of solids, plasticity theory is applicable primarily to those bodies that

can experience inelastic deformations considerably greater than the elastic deformation. If the resulting total

deformation, including both translations and rotations, are small enough, we can apply small deformation theory

in solving these problems. If, however strains and rotations are finite, one must resort to the theory of large

deformations. In doing so, we will be using two sets of representations1, namely:

• Material coordinates in the undeformed configuration, also called Lagrangian coordinates,

• Spatial coordinates in the deformed configuration, also called Eulerian coordinates.

Figure 6.1 shows the displacement of a particle from its initial position XI to the current position xi, defined by

the deformation equation:

xi = xi (X1,X2,X3, t) (6.1)

The initial position XI of the particle now occupying the position xi is given by the Eulerian equation:

XI = XI (x1, x2, x3, t) (6.2)

The two positions are connected by the displacement uI :

1See Malvern (1969).

115


d

d

X

X u

x

x i

i

iI

I

Figure 6.1: Displacement, stretch and rotation of material vector dXI to new position dxi.

xi = XI + ui ; XI = xi − ui (6.3)

6.1.2 Deformation Gradient

The deformation gradients are the gradients of the functions on the right–hand side of equations (6.1) and (6.2).

To emphasize the difference between the material, Lagrangian setting and the spatial, Eulerian setting, we will

use capital letters for the material coordinate indices and lower case letters for the spatial coordinate indices.

We limit our work to the rectangular Cartesian coordinates, thus simplifying the tensor notation to the covariant

indices only.

The deformation gradient is defined as the two–point tensor whose rectangular Cartesian components are the

partial derivatives:

FkK =∂xk∂XK

= xk,K (6.4)

The deformation gradient FkK transforms (convects) on an arbitrary infinitesimal material vector dXI at XI to

associate it with a vector dxi at xi:

dxk = FkKdXK =∂xk∂XK

dXK = xk,KdXK (6.5)

The the spatial deformation gradients are tensors referred to the deformed, Eulerian configuration:

(FKk)−1

=∂XK

∂xk= XK,k (6.6)

Similarly to the deformation gradient FkK , spatial deformation gradient (FKk)−1

operates on an arbitrary in-

finitesimal material vector dxi at xi to associate it with a vector dXI at XI :

dXK = (FKk)−1dxk =

∂XK

∂xkdxk = XK,kdxk (6.7)



The spatial deformation gradient (FKk)−1

at xi is the inverse to the two–point tensor FkK at XI :

FiJ (FJk)−1

= δik and (FIj)−1FjK = δIK (6.8)

The Jacobian of the mapping (6.4) can be represented as:

J = det (FkK) =1

6eijkePQRFiPFjQFkR (6.9)

The relative deformation gradient fkm is the gradient for the relative motion function:

ξ = χt (xi, τ) (6.10)

and is defined as:

fkm = ξk,m ≡ ∂ξk∂xm

(6.11)

If the fixed reference position XI , the current position xi and the variable position ξi are all referred to the

rectangular Cartesian coordinate system, the chain rule of differentiation yields:

∂ξk∂XI

=∂ξk∂xm

∂xm∂XI

or FkI = fkm FmI (6.12)

The polar decomposition theorem permits the unique representation2:

Fij = RikUkj = vikRkj (6.13)

where Ukj , vik are positive definite symmetric tensors, called right stretch tensors and left stretch tensors,

respectively, and Rkj is an orthogonal tensor such that:

RikRjk = δij and also RkiRkj = δij (6.14)

Equation (6.13), as well as Figure 6.1.2 demonstrate that the motion and deformation of an infinitesimal

volume element at Xi consist of consecutive applications of:

• a stretch by Ukj ,

• a rigid body rotation by Rik,

• a rigid body translation to xi

or alternatively:

• a rigid body translation to xi

• a rigid body rotation by Rkj ,

• a stretch by vik,

2referring xi and Xi to the same reference axes and using lower case indices for both. This reference to the same coordinate

system will be applied only for the polar decomposition example presented here.



d

d

X

X u

x

x i

i

i

I

I

Figure 6.2: Illustration of the equation Fij = RikUkj = vikRkj .

6.1.3 Strain Tensors, Deformation Tensors and Stretch

The strain tensors EIJ and eij are defined so that they give the change in the square length of the material vector

dXI . For the Lagrangian formulation we write:

(ds)2 − (dS)2 = 2dXIEIJdXJ (6.15)

and for the Eulerian formulation:

(ds)2 − (dS)2 = 2dxieijdxj (6.16)

The deformation tensors CIJ and cij are connecting the squared lengths in Lagrangian and Eulerian config-

urations. The Green deformation tensor3 CIJ , referred to the undeformed configuration, gives the new squared

length (ds)2 of the element into which the given element dXI is deformed:

(ds)2 = dXICIJdXJ (6.17)

The Cauchy deformation tensor cij , sometimes also denoted as4 (bij)−1

, gives the initial squared length (dS)2 of

an element dxi identified in the deformed configuration:

(dS)2 = dxicijdxj (6.18)

Substituting equation (6.17) into (6.15) yield:

2EIJ = CIJ − δIJ (6.19)

3Also called right Cauchy–Green tensor.4Another name for bij is Finger deformation tensor or left Cauchy–Green tensor.



and similarly, substituting equation (6.18) into (6.16) we obtain:

2eij = δij − cij (6.20)

By using equation (6.5) we can express (ds)2 as:

(ds)2 = dxkdxk = (FkIdXI)(FkJdXJ) =

(xk,IdXI)(xk,JdXJ) = dXI(FkIFkJ)dXK = dXICIJdXK (6.21)

so we have obtained the connection between the deformation tensor CIJ and the deformation gradient FkI in the

form:

CIJ = (FkIFkJ) = xk,IdXIxk,JdXJ (6.22)

Similarly, by using equation (6.7) and the expression for (dS)2 we can establish the connection between the

deformation tensor cij and the deformation gradient FKi as:

(dS)2 = dSKdXK = (FKidxi)(FKjdxj) =

(XK,idxi)(XK,jdxJ) = dxi(FKiFKj)dxk = dxicijdxk ⇒

cij = (FKi)−1F−1Kj (6.23)

The expressions for the strain tensors in Lagrangian and Eulerian description5 is obtained from equations

(6.19) and (6.20):

L: EIJ =1

2((FkIFkJ) − δIJ ) ; E: eij =

1

2

(

δij − (FKi)−1

(FKj)−1)

(6.24)

If one starts from the displacement equation (6.3), referenced to the same axes for both XI and xi

xI = XI + uI ; XI = xI − uI

the general expression for the Lagrangian strain tensor EIJ in terms of displacements is:

EIJ =1

2((FKIFKJ) − δIJ ) =

1

2((δKI + uK,I) (δKJ + uK,J) − δIJ ) =

1

2(δKIδKJ + δKIuK,J + uK,IδKJ + uK,IuK,J − δIJ ) =

1

2(δIJ + uI,J + uJ,I + uK,IuK,J − δIJ ) =

1

2(uI,J + uJ,I + uK,IuK,J) (6.25)

Similarly, the general expression for the Eulerian strain tensor eij in terms of displacements is:

eij =1

2

(

δij − (Fki)−1

(Fkj)−1)

=

1

2(δij − (δki − uk,i) (δkj − uk,j)) =

1

2(δij − δkiδkj + δkiuk,j + uk,iδkj − uk,iuk,j) =

1

2(δij − δij + ui,j + uj,i − uk,iuk,j) =

1

2(ui,j + uj,i − uk,iuk,j) (6.26)

5Lagrangian format will be denoted by L: while Eulerian format by E:.



It is worthwhile noting that equations (6.25) and (6.26) represent the complete finite strain tensor. They involve

only linear and quadratic terms in the components of displacement gradients.

The stretch is a measure of extension of an infinitesimal element and is a function of direction of an element,

in either deformed or undeformed configuration. By denoting NI a unit vector in the undeformed configuration

and ni a unit vector in the deformed configuration, we denote material stretch as Λ(N) of those elements whit

initial direction NI and spatial stretch λ(n) of those elements with initial direction ni. By dividing equations

(6.15) and (6.16) by (ds)2 and (dS)2 respectively and by using:

NI =dXI

dSand ni =

dxids

(6.27)

we obtain the Cartesian form of stretch in the Lagrangian and Eulerian descriptions:

L: Λ2(N) =

dXI

dSCIJ

dXJ

dSand E: λ2

(n) =dxids

cijdxjds

(6.28)

General strain tensors can be defined by considering a scale function (Hill, 1978) for the stretch. Scale function

is any smooth, monotonic function of stretch f(λ) such that:

f(λ) ; λ ∈ [0,∞) subject to f(1) = 0, f ′(1) = 1 (6.29)

Scale function is often taken in the form (λ2m − 1)/2m, where m may have any value. If we choose m to be an

integer, the corresponding strain tensor is:

EIJ =

(U2mIJ − δIJ

)

2mwhere FIJ = RIKUKJ = vIKRKJ (6.30)

Table 6.1 shows different Lagrangian strain measures obtained for a particular choice of parameter m.

Table 6.1: Different Lagrangian strain measures.

Strain measure name parameter m expression for EmIJ

Green–Lagrange 1 EGLIJ =(U2IJ − δIJ

)/2

Almansi -1 EAIJ =(δIJ − U−2

IJ

)/2

Biot 1/2 EBIJ = (UIJ − δIJ)

Hencky 0 EHIJ = ln (UIJ )

In the Eulerian setting, generalized strain tensor is defined as

eij =

(δij − v2m

ij

)

2m; FIJ = RIKUKJ = vIKRKJ (6.31)



6.1.4 Rate of Deformation Tensor

The rate of deformation tensor6 describes the tangent motion in terms of velocity components vi = dxi/dt. The

spatial coordinates are:

vi = vi (x1, x2, x3, t) (6.32)

P

Q

p

q

d

d

Xi

vi

i

i i+dv v

vi

dx

u

i

Figure 6.3: Relative velocity dvi of particle Q at point q relative to particle P at point p.

In Figure 6.3 the dashed lines represents the trajectories of particles P and Q. The velocity vectors vi at p and

vi + dvi at q are tangent to the two trajectories. The relative velocity components dvi of particle at q relative to

the particle at p are given by:

dvk =∂vk∂xm

dxm = vk,mdxm = Lkmdxm (6.33)

The spatial gradient of the velocity Lkm can be decomposed as the sum of the symmetric, rate of deformation

tensor Dkm, and a skew symmetric spin tensor Wkm as follows:

Lkm =1

2(Lkm + Lmk) +

1

2(Lkm − Lmk) = Dkm +Wkm (6.34)

where:

Dkm =1

2(Lkm + Lmk) = Dmk and Wkm =

1

2(Lkm − Lmk) = −Wmk (6.35)

An alternate way of deriving the rate of deformation tensor goes as follows. The rate of change of squared length

(ds)2

is given as:

d (ds)2

dt= 2

d (ds)

dtds (6.36)

since (ds)2

= dxkdxk it follows:

d (ds)2

dt= 2

d (dxk)

dtdxk (6.37)

6Also called stretch tensor or velocity strain.



and with dxk = (∂xk/∂Xm)dXm it follows:

d (dxk)

dt=d(∂xk

∂XmdXm

)

dt=d(∂xk

∂Xm

)

dtdXm +

d (dXm)

dt

∂xk∂Xm

=d(∂xk

∂Xm

)

dtdXm (6.38)

since d (dXm) /dt ≡ 0, because the initial relative position vector dXm does not change with time. By inter-

changing the order of differentiation we get:

d (dxk)

dt= dvk =

d(∂xk

∂XM

)

dtdXm =

∂vk∂Xm

dXm where vk =∂xkdt

(6.39)

From equation (6.33) dvk = Lkmdxm and equation (6.39) it follows that:

∂vk∂Xm

dXm = Lkmdxm ⇒ d (dxk)

dt= dvk = Lkmdxm = vk,mdxm (6.40)

and then the equation (6.37) becomes:

d (ds)2

dt= 2

d (dxk)

dtdxk = 2dxkvk,mdxmdxk = 2dxkLkmdxmdxk =

= 2dxkDkmdxmdxk + 2dxkWkmdxmdxk = 2dxkDkmdxmdxk (6.41)

since dxkdxm ≡ dxmdxk and Wkm is skew symmetric such that Wkm = −Wmk. Finally we obtain:

d (ds)2

dt= 2dxkDkmdxm (6.42)

and thus it follows that the rate of change of the squared length (ds)2

of the material instantaneously occupying

any infinitesimal relative position dxk at point p is determined by the tensor Dkm at point p.

In order to compare the strain rate to the rate of deformation, we differentiate equation (6.15) with respect

to time:

d((ds)2 − (dS)2

)

dt= 2

d (dXIEIJdXJ )

dt=

=d((ds)2

)

dt= 2dXI

d (EIJ)

dtdXJ (6.43)

since (dS)2 and dXI are constant through time. From the equations (6.42) and (6.43) it follows that:

d (ds)2

dt= 2dxkDkmdxmd = 2 (dXIFIk)Dkm (FmJdXJ ) = 2dXI (FIkDkmFmJ) dXJ

(6.44)

and from equations (6.43) and (6.44) it follows that:

dEIJdt

= FIkDkmFmJ (6.45)

or inversely:

Dkm = (FIk)−1 dEIJ

dt(FmJ)

−1(6.46)

To obtain the rate of change of the deformation gradient we start from equations (6.4) and differentiate it

with respect to time:

dFkKdt

=d(∂xk

∂XK

)

dt=∂(dxk

dt

)

∂XK=

∂vk∂XK

=∂vk∂xm

∂xm∂XK

= vk,mxm,K =dxk,Kdt

=

= LkmFmK = FkK (6.47)



or inversely:

vk,m =dxk,Kdt

XK,m =dFkKdt

(FKm)−1

=

= FkK (FKm)−1

= Lkm (6.48)

6.2 Constitutive Relations: Hyperelasticity

6.2.1 Introduction

A material is called hyperelastic or Green elastic, if there exists an elastic potential function W , also called the

strain energy function per unit volume of the undeformed configuration, which represents a scalar function of

strain of deformation tensors, whose derivatives with respect to a strain component determines the corresponding

stress component. The most general form of the elastic potential function, is described in equation 6.49, with

restriction to pure mechanical theory, by using the axiom of locality and the axiom of entropy production7:

W = W (XK , FkK) (6.49)

By using the axiom of material frame indifference8, we conclude that W depends only on XK and CIJ , that is:

W = W (XK , CIJ ) or: W = W (XK , cij) (6.50)

By assuming hyperelastic response, the following are the constitutive equations for the material stress tensors:

• 2. Piola–Kirchhoff stress tensor:

SIJ = 2∂W

∂CIJ(6.51)

• Mandel stress tensor:

TIJ = CIKSKJ = 2CIK∂W

∂CKJ(6.52)

• 1. Piola–Kirchhoff stress tensor

PiJ = SIJ (FiI)t = 2

∂W

∂CIJ(FiI)

t (6.53)

and the spatial, Kirchhoff stress tensor is defined as:

7See Marsden and Hughes (1983) pp. 190.8See Marsden and Hughes (1983) pp. 194.



• Kirchhoff stress tensor

τij = 2∂W

∂bij= 2 FiA(FjB)t

∂W

∂CAB= FiA(FjB)tSAB (6.54)

Material tangent stiffness relation is defined from:

dSIJ = 2∂2W

∂CIJ ∂CKLdCKL =

1

2LIJKL dCKL (6.55)

where

LIJKL = 4∂2W

∂CIJ ∂CKL(6.56)

The spatial tangent stiffness tensor Eijkl is obtained by the following push–forward operation with the defor-

mation gradient:

Eijkl = FiIFjJ(FkK)t(FlL)tLIJKL (6.57)

6.2.2 Isotropic Hyperelasticity

In the case of material isotropy, the strain energy function W (XK , CIJ ) belongs to the class of isotropic, invariant

scalar functions. It satisfies the relation:

W (XK , CKL) = W(

XK , QKICIJ (QJL)t)

(6.58)

where QKI is the proper orthogonal transformation. If we choose QKI = RKI , where RKI is the orthogonal

rotation transformation, defined by the polar decomposition theorem in equation (6.13), then:

W (XK , CKL) = W (XK , UKL) = W (XK , vkl) (6.59)

Right and left stretch tensors, UKL, vkl have the same principal values9 λi ; i = 1, 3 so the strain energy

function W can be represented in terms of principal stretches, or similarly in terms of principal invariants of

deformation tensor:

W = W (XK , λ1, λ2, λ3, ) = W (XK , I1, I2, I3) (6.60)

where:

I1def= CII

I2def=

1

2

(I21 − CIJCJI

)

I3def= det (CIJ) =

1

6eIJKePQRCIPCJQCKR = J2 (6.61)

9Principal stretches.



Left and right Cauchy–Green tensors were defined by equations (6.22) and (6.23), respectively as:

CIJ = (FkI)tFkJ ; (c−1)ij = bij = FiK(FjK)t (6.62)

The spectral10 decomposition theorem for symmetric positive definite tensors11 states that:

CIJ = λ2A

(

N(A)I N

(A)J

)

Awhere A = 1, 3 (6.63)

and NI are the eigenvectors12 of CIJ . Values λ2A are the roots of the characteristic polynomial

P (λ2A)

def= −λ6

A + I1 λ6A − I2 λ

4A + I3 = 0 (6.64)

It should be noted that no summation is implied over indices in parenthesis13.

The mapping of the eigenvectors can be deduced from equation (6.5) and is given by

λ(A) n(A)i = FiJ N

(A)J (6.65)

where ‖n(A)i ‖ ≡ 1. The spectral decomposition of FiJ , RiJ and bij is then given by

FiJ = λA

(

n(A)i N

(A)J

)

A(6.66)

RiJ =3∑

A=1

n(A)i N

(A)J (6.67)

bij = λ2A

(

n(A)i n

(A)j

)

A(6.68)

Spectral decomposition from equation (6.63) is valid for the case of non–equal principal stretches, i.e. λ1 6=λ2 6= λ3. If two or all three principal stretches are equal, we shall introduce a small perturbation to the numerical

values for principal stretches in order to make them distinct. The case of two or all three values of principal

stretches being equal is theoretically possible and results for example from standard triaxial tests or isotropic

compression tests. However, we are never certain about equivalence of two numerical numbers, because of the

finite precision arithmetics involved in calculation of these numbers. From the numerical point of view, two

number are equal if the difference between them is smaller than the machine precision (macheps) specific to the

computer platform on which computations are performed. Our perturbation will be a function of the macheps.

The characteristic polynomial P (λ2A) from equation (6.64) can be solved14 for λA:

λA =1√3

√

I1 + 2√

I21 − 3I2 cos

(Θ + 2πA

3

)

(6.69)

where

Θ = arccos2I3

1 − 9I1I2 + 27I3

2

√

(I21 − 3I2)

3(6.70)

10See Simo and Taylor (1991).11Cauchy–Green tensor CIJ for example.12So that ‖NI‖ = 1.13For example, in the present case N

(A)I

is the Ath eigenvector with members N(A)1 , N

(A)2 and N

(A)3 , so that the actual equation

CIJ = λ2A

“

N(A)I

N(A)J

”

Acan also be written as CIJ =

PA=3A=1 λ2

(A)N

(A)I

N(A)J

. In order to follow the consistency of indicial notation

in this work, we shall make an effort to represent all the tensorial equations in indicial form.14See also Morman (1986) and Schellekens and Schellekens and Parisch (1994).



Recently, Ting (1985) and Morman (1986) have used Serrin’s representation theorem in order to devise a

useful representation for generalized strain tensors15 EIJ and eij through CmIJ and bmij . Morman (1986) has

shown that bmij can be stated as

bmij = λ2mA

(b2)ij −

(

(I1 − λ2(A)

)

bij + I3λ−2(A)δij

2λ4(A) − I1λ2

(A) + I3λ−2(A)

A

(6.71)

By comparing equations (6.71) and (6.68) it follows that the Eulerian eigendiad n(A)i n

(A)j can be written as

n(A)i n

(A)j =

(b2)ij −(

I1 − λ2(A)

)

bij + I3λ−2(A)δij

2λ4(A) − I1λ2

(A) + I3λ−2(A)

(6.72)

The Lagrangian eigendiad N(A)I N

(A)J , from equation (6.63), can be derived, if one substitutes mapping of the

eigenvectors, (6.65), into equation (6.72) to get:

N(A)I N

(A)J = λ2

(A)

CIJ −(

I1 − λ2(A)

)

δIJ + I3λ−2(A)(C

−1)IJ

2λ4(A) − I1λ2

(A) + I3λ−2(A)

(6.73)

where it was used that:

CIJ = (FiI)−1 (b2)ij (FjJ)−t (6.74)

δIJ = (FiI)−1 bij (FjJ)−t (6.75)

(C−1)IJ = (FiI)−1 δij (FjJ)−t (6.76)

It should be noted that the denominator in equations (6.72) and (6.73) can be written as:

2λ4(A) − I1λ

2(A) + I3λ

−2(A) =

(

λ2(A) − λ2

(B)

)(

λ2(A) − λ2

(C)

)def= D(A) (6.77)

where indices A,B,C are cyclic permutations of 1, 2, 3. It follows directly from the definition of D(A) in equation

(6.77) that λ1 6= λ2 6= λ3 ⇒ D(A) 6= 0 for equations (6.72) and (6.73) to be valid. Similarly to equations (6.63)

and (6.68) we can obtain:

(C−1)IJ = λ−2A

(

N(A)I N

(A)J

)

A(6.78)

(b−1)ij = λ−2A

(

n(A)i n

(A)j

)

A(6.79)

6.2.3 Volumetric–Isochoric Decomposition of Deformation

It proves useful to separate deformation in volumetric and isochoric parts by a multiplicative split of a deformation

gradient as

FiI = FiβvolFβI where Fiβ = FiIJ

− 13 ; volFβI = J

13 δβI (6.80)

where xβ represents an intermediate configuration such that deformation XI → xβ is purely volumetric and

xβ → xi is purely isochoric. It also follows from equation (6.80) that FβI and FiI have the same eigenvectors.

15Defined by equations (6.30) and (6.31).



FiI

Fiβ

isoFβI

vol

xβ

xi

XI

Figure 6.4: Volumetric isochoric decomposition of deformation.

The isochoric part of the Green deformation tensor CIJ , defined in equation (6.63) can be defined as

CIJ = J− 23CIJ = λ2

A

(

N(A)I N

(A)J

)

A(6.81)

while the isochoric part of the Finger deformation tensor bij can be defined similarly as

bij = J− 23 bij = λ2

A

(

n(A)i n

(A)j

)

A(6.82)

where the isochoric principal stretches are defined as

λA = J− 13λA = (λ1λ2λ3)

− 13λA (6.83)

The free energy W is then decomposed additively as:

W(XK , λ(A)

)= isoW

(

XK , λ(A)

)

+ volW (XK , J) (6.84)

6.2.4 Simo–Serrin’s Formulation

In Section (6.2.2) we have presented the most general form of the isotropic strain energy function W in terms of

of principal stretches:

W = W (XK , λ1, λ2, λ3, ) (6.85)

It was also shown in Section (6.2.1) that it is necessary to calculate the gradient ∂W/∂CIJ in order to obtain

2. Piola–Kirchhoff stress tensor SIJ and accordingly other stress measures. Likewise, it was shown that the

material tangent stiffness tensor LIJKL (as well as the spatial tangent stiffness tensor Eijkl) requires second order

derivatives of strain energy function ∂2W/(∂CIJ ∂CKL). In order to obtain these quantities we introduce16 a

second order tensor M(A)IJ

16See Runesson (1996).



M(A)IJ

def= λ−2

(A) N(A)I N

(A)J (6.86)

= (FiI)−1(

n(A)i n

(A)j

)

(FjJ)−t

=1

D(A)

(

CIJ −(

I1 − λ2(A)

)

δIJ + I3λ−2(A)(C

−1)IJ

)

from (6.73)

where D(A) was defined by equation (6.77). With M(A)IJ defined by equation (6.86), we get from equation (6.63)

that:

CIJ = λ4A

(

M(A)IJ

)

A(6.87)

and also from equation (6.78) it follows that:

(C−1)IJ = M(1)IJ +M

(2)IJ +M

(3)IJ (6.88)

It can also be concluded that:

δIJ = λ2(1)M

(1)IJ + λ2

(2)M(2)IJ + λ2

(3)M(3)IJ = λ2

A

(

M(A)IJ

)

A(6.89)

since, from the orthogonal properties of eigenvectors

δIJ =3∑

A=1

N(A)I N

(A)J =

(

N(A)I

)

A

(

N(A)J

)

A(6.90)

We are now in a position to define the Simo–Serrin fourth order tensor MIJKL as:

M(A)IJKL

def=

∂M(A)IJ

∂CKL=

1

D(A)

(

IIJKL − δKLδIJ + λ2(A)

(

δIJ M(A)KL +M

(A)IJ δKL

)

+

+ I3λ−2(A)

(

(C−1)IJ (C−1)KL +1

2

((C−1)IK(C−1)JL + (C−1)IL(C−1)JK

))

−

− λ−2(A) I3

(

(C−1)IJ M(A)KL +M

(A)IJ (C−1)KL

)

−D′(A) M

(A)IJ M

(A)KL

)

(6.91)

Complete derivation of MIJKL is given in Appendix (D.1).

6.2.5 Stress Measures

In Section (6.2.1) we have defined various stress measures in terms of derivatives of the free energy function W .

With the free energy function decomposition, as defined in equation (6.84) we can appropriately decompose all

the previously defined stress measures:

• 2. Piola–Kirchhoff stress tensor:

SIJ = 2∂W

∂CIJ= 2

∂isoW

∂CIJ+ 2

∂volW

∂CIJ

= isoSIJ + volSIJ (6.92)



• Mandel stress tensor:

TIJ = CIKSKJ = 2CIK∂W

∂CKJ= 2CIK

∂isoW

∂CKJ+ 2CIK

∂volW

∂CKJ

= isoTIJ + volTIJ (6.93)

• 1. Piola–Kirchhoff stress tensor

PiJ = SIJ(FiI)t = 2

∂W

∂CIJ(FiI)

t = 2∂isoW

∂CIJ(FiI)

t + 2∂volW

∂CIJ(FiI)

t

= isoPiJ + volPiJ (6.94)

• Kirchhoff stress tensor

τab = 2∂W

∂eij= FaI(FbJ )tSIJ = 2FaI(FbJ)t

∂isoW

∂CIJ+ 2FaI(FbJ )t

∂volW

∂CIJ

= FaI(FbJ )tisoSIJ + FaI(FbJ)tvolSIJ

= isoτab + volτab (6.95)

The derivative of the volumetric part of the free energy function is

∂volW (J)

∂CIJ=∂volW (J)

∂J

∂J

∂CIJ=

1

2

∂volW (J)

∂JJ (C−1)IJ (6.96)

where equation (D.9) was used, while the derivative of the isochoric part of the free energy function yields

∂isoW (λ(A))

∂CIJ=∂isoW (λ(A))

∂λ(A)

∂λ(A)

∂CIJ=

1

2

∂isoW (λ(A))

∂λ(A)λ(A)(M

(A)IJ )A =

1

2wA(M

(A)IJ )A

(6.97)

where equation (D.7) was used and wA is derived in Appendix D.4 as:

wA =∂isoW (λ(A))

∂λB

∂λB∂λ(A)

λ(A) = −1

3

∂isoW (λ(A))

∂λBλB +

∂isoW (λ(A))

∂λ(A)

λ(A) (6.98)

The decomposed 2. Piola–Kirchhoff stress tensor is

SIJ = volSIJ + isoSIJ

=∂volW (J)

∂JJ (C−1)IJ + wA (M

(A)IJ )A (6.99)

The derivative of the free energy is then:

∂W (λ(A))

∂CIJ=

∂volW (λ(A))

∂CIJ+∂isoW (λ(A))

∂CIJ

=1

2

∂volW (J)

∂JJ (C−1)IJ +

1

2wA (M

(A)IJ )A (6.100)

It is obvious that the only material dependent parts are derivatives in the form ∂volW/∂J and wA, while the

rest is independent of which hyperelastic material model we choose.



6.2.6 Tangent Stiffness Operator

The free energy function decomposition (6.84) is used together with the appropriate definitions made in section

(6.2.1) toward the tangent stiffness operator decomposition

LIJKL = volLIJKL + isoLIJKL = 4∂2(volW

)

∂CIJ ∂CKL+ 4

∂2(isoW

)

∂CIJ ∂CKL(6.101)

The volumetric part ∂2(volW

)/(∂CIJ ∂CKL) can be written as:

∂2volW

∂CIJ ∂CKL=

1

4

(

J2 ∂2(volW

)

∂J∂J+ J

∂(volW

)

∂J

)

(C−1)KL(C−1)IJ +1

2J∂(volW

)

∂JI(C−1)IJKL

(6.102)

and the complete derivation is again given in appendix D.2.

The isochoric part ∂2(isoW

)/(∂CIJ ∂CKL) can be written in the following form:

∂2isoW (λ(A))

∂CIJ∂CKL=

1

4YAB (M

(B)KL )B (M

(A)IJ )A +

1

2wA (M(A)

IJKL)A (6.103)

and the complete derivation is given in the appendix (D.3).

Finally, one can write the volumetric and isochoric parts of the tangent stiffness tensors as:

volLIJKL =

J2 ∂2volW (J)

∂J∂J(C−1)KL(C−1)IJ + J

∂volW (J)

∂J(C−1)KL(C−1)IJ + 2J

∂volW (J)

∂JI(C−1)IJKL

(6.104)

LisoIJKL = YAB (M(B)KL )B (M

(A)IJ )A + 2 wA (M(A)

IJKL)A (6.105)

In a similar manner to the stress definitions it is clear that the only material model dependent parts are YAB

and wA. The remaining second and fourth order tensors M(A)IJ and M(A)

IJKL are independent of the choice of the

material model. This observation has a practical consequence in that it is possible to create a template derivations

for various hyperelastic isotropic material models. Only first and second derivatives of strain energy function with

respect to isochoric principal stretches (λA) and Jacobian (J) are needed in addition to the independent tensors,

for the determination of various stress and tangent stiffness tensors.

6.2.7 Isotropic Hyperelastic Models

The strain energy function for isotropic solid in terms of principal stretches is represented as:

W = W (λ1, λ2, λ3) (6.106)



The only restriction is that W is a symmetric function of λ1, λ2, λ3, although an appropriate natural configuration

condition requires that:

W (1, 1, 1) = 0 and∂W (1, 1, 1)

∂λi= 0 (6.107)

The strain energy function W can either be regarded as a function of principal stretches or the principal invariants

of stretches17:

I1 = λ21 + λ2

2 + λ23

I2 = λ22λ

23 + λ2

3λ21 + λ2

1λ22

I3 = λ21λ

22λ

23 (6.108)

A slightly more general formulation is obtained by using principal stretches in the strain energy function

definition. A widely exploited family of compressible hyperelastic models18 are defined (Ogden, 1984) as an

infinite series in powers of (I1 − 3), (I2 − 3) and (I3 − 1) as:

W =

N→∞∑

p,q,r=0

cpqr (I1 − 3)p(I2 − 3)

q(I3 − 1)

r(6.109)

The regularity condition that W is continuously differentiable an infinitely number of times is satisfied. The

requirement that energy vanishes in the reference configuration is met provided c000 = 0. Reference configuration

is stress free iff c100 + 2c010 + c001 = 0. Isochoric deviatoric decoupling is possible by setting cpqr = 0 (r =

1, 2, 3, ...) and cpqr = 0 (p, q = 1, 2, 3, ..) to obtain:

W = isoW + volW (6.110)

where:

isoW =

N→∞∑

p,q=0

cpq0 (I1 − 3)p(I2 − 3)

q

volW =

N→∞∑

r=0

c00r (I3 − 1)r

(6.111)

In what follows, we will present a number of widely used strain energy functions for isotropic elastic solids.

Ogden Model

A very general set of hyperelastic models was defined by Ogden (1984). The strain energy is expressed as a

function of principal stretches as:

W =

N→∞∑

r=1

crµr

(λµr

1 + λµr

2 + λµr

3 − 3) (6.112)

The isochoric strain energy function can be written as:

isoW =

N→∞∑

r=1

crµr

(

λµr

1 + λµr

2 + λµr

3 − 1)

(6.113)

17See also equation (6.61).18Used mainly for rubber–like materials.



where the following was used λi = J− 13λi.

Derivatives needed for building tensors wA and YAB are given by the following formulae:

∂isoW

∂λA=

N→∞∑

r=1

cr

(

λA

)µr−1

(6.114)

∂2(isoW

)

∂λ2A

=

N→∞∑

r=1

cr (µr − 1)(

λA

)µr−2

(6.115)

∂2(isoW

)

∂λA∂λB= 0 (6.116)

Neo–Hookean Model

The general isotropic hyperelastic model defined in terms of invariants of principal stretches contains the Neo–

Hookean model as special cases. The isochoric part of Neo–Hookean isotropic elastic model can be obtained by

selecting N = 1, q = 0, cp00 = G/2, to get:

isoW =G

2

(

λ21 + λ2

2 + λ23 − 3

)

(6.117)

while the volumetric part can be defined by choosing N = 2, c001 = 0, c002 = Kb/2, as:

volW =Kb

2

(λ2

1λ22λ

23 − 1

)2=Kb

2

(J2 − 1

)2(6.118)

where G and Kb are the shear and bulk moduli respectively.


∂isoW

∂λA= G λA (6.119)

∂2(isoW

)

∂λ2A

= G (6.120)

∂2(isoW

)

∂λA∂λB= 0 (6.121)

Mooney–Rivlin Model

Mooney proposed a strain energy function for isochoric behavior of the form:

isoW =N→∞∑

n=0

(

an

(

λ2n1 + λ2n

2 + λ2n3 − 3

)

+ an

(

λ−2n1 + λ−2n

2 + λ−2n3 − 3

))

=N→∞∑

n=0

(

an

(

λ2n1 + λ2n

2 + λ2n3 − 3

)

+ an

((

λ2λ3

)−2n

+(

λ3λ1

)−2n

+(

λ1λ2

)−2n

− 3

))

(6.122)

with an and bn being the material parameters and a volume preserving constrain λa = 1/(λbλc, and a, b, c are

cyclic permutations of (1, 2, 3). A more general form was proposed by Rivlin:

isoW =

N→∞∑

p,q=0

cpq (I1 − 3)p(I2 − 3)

q(6.123)



which is actually quite similar to the isochoric part of the general isotropic representation from equation (6.111).

Both Mooney and Rivlin strain energy functions become similar, if one chooses to set N = 1 and c10 = C1 and

c01 = C2 to obtain:

isoW =(

C1

(

λ21 + λ2

2 + λ23 − 3

)

+ C2

(

λ−21 + λ−2

2 + λ−23 − 3

))

=(

C1

(

I1 − 3)

+ C2

(

I2 − 3))

(6.124)


∂isoW

∂λA= 2 C1 λA − 2 C2 λ

−3A (6.125)

∂2(isoW

)

∂λ2A

= 2 C1 + 6 C2 λ−4A (6.126)

∂2(isoW

)

∂λA∂λB= 0 (6.127)

Logarithmic Model

By choosing an alternative set of isochoric principal stretch invariants in the form:

I ln1 = 2(

ln λ1

)2

+ 2(

ln λ2

)2

+ 2(

ln λ3

)2

=(

λln1

)2

+(

λln2

)2

+(

λln3

)2

I ln2 = 4(

ln λ2

)2 (

ln λ3

)2

+ 4(

ln λ3

)2 (

ln λ1

)2

+ 4(

ln λ1

)2 (

ln λ2

)2

=(

λln2

)2 (

λln3

)2

+(

λln3

)2 (

λln1

)2

+(

λln1

)2 (

λln2

)2

(6.128)

where the isochoric logarithmic stretch λlni was used:

λlni =

√2 ln λi =

1√2

ln λ2i (6.129)

The general representation of the isochoric part of the strain energy function in terms of I ln1 and I ln2 was

proposed by Simo and Miehe (1992). A somewhat simpler isochoric strain energy function can be presented in

the form:

isoW = G

((

ln λ1

)2

+(

ln λ2

)2

+(

ln λ3

)2)

(6.130)

while the volumetric part is suggested in the form:

volW =Kb

2(lnJ)

2(6.131)


∂isoW

∂λA= 2 G

(

λA

)−1

(6.132)

∂2(isoW

)

∂λ2A

= −2 G(

λA

)−2

(6.133)



∂2(isoW

)

∂λA∂λB= 0 (6.134)

d(volW

)

dJ= Kb J

−1 lnJ (6.135)

d2(volW

)

dJ2= Kb J

−2 −Kb J−2 lnJ (6.136)

Simo–Pister Model

Another form or a volumetric part of strain energy function was proposed by Simo and Pister (1984) in the form:

Wvol(J) =1

4Kb

(J2 − 1 − 2 ln J

)(6.137)

The first and second derivatives with respect to J are then given as:

dvolW (J)

dJ=

(−2J + 2J

)Kb

4(6.138)

d2volW (J)

dJ2=

(2 + 2

J2

)Kb

4(6.139)

6.3 Finite Deformation Hyperelasto–Plasticity

6.3.1 Introduction

The mathematical structure and numerical analysis of classical small deformation elasto–plasticity is generally well

established. However, development of large deformation elastic–plastic algorithms for isotropic and anisotropic

material models is still a research area. Here, we present a new integration algorithm, based on the multiplicative

decomposition of the deformation gradient into elastic and plastic parts. The algorithm is novel in that it is

designed to be used with isotropic as well as anisotropic material models. Consistent derivation is based on the

idea from the infinitesimal strain algorithm developed earlier by Jeremic and Sture (1997). The algorithm is not

an extension of earlier developments, but rather a novel development which consistently utilizes Newton’s method

for numerical solution scheme for integrating pertinent constitutive equations. It is also shown that in the limit,

the proposed algorithm reduces to the small strain counterpart.

In what follows, we briefly introduce the multiplicative decomposition of the deformation gradient and pertinent

constitutive relations. We then proceed to present the numerical algorithm and the algorithmic tangent stiffness

tensor consistent with the presented algorithm.

6.3.2 Kinematics

An appropriate generalization of the additive strain decomposition is the multiplicative decomposition of displace-

ment gradient. The motivation for the multiplicative decomposition can be traced back to the early works of

Bilby et al. (1957), and Kroner (1960) on micromechanics of crystal dislocations and application to continuum

modeling. In the context of large deformation elastoplastic computations, the work by Lee and Liu (1967), Fox

(1968) and Lee (1969) stirred an early interest in multiplicative decomposition.



The appropriateness of multiplicative decomposition technique for soils may be justified from the particulate

nature of the material. From the micromechanical point of view, plastic deformation in soils arises from slipping,

crushing, yielding and plastic bending19 of granules comprising the assembly20. It can certainly be argued that

deformations in soils are predominantly plastic, however, reversible deformations could develop from the elasticity

of soil grains, and could be relatively large when particles are locked in high density specimens.

Fp

Fe

Reference

Configuration

X

d X

dx

Current

Configuration

x

Fe

-1

Ω0

Intermediate

Configuration

σ x

F

X u

Ω

dx

Ω

x

Figure 6.5: Multiplicative decomposition of deformation gradient: schematics.

The reasoning behind multiplicative decomposition is a rather simple one. If an infinitesimal neighborhood

of a body xi, xi + dxi in an inelastically deformed body is cut–out and unloaded to an unstressed configuration,

it would be mapped into xi, xi + dxi. The transformation would be comprised of a rigid body displacement21

and purely elastic unloading. The elastic unloading is a fictitious one, since materials with a strong Baushinger’s

effect, unloading will lead to loading in an other stress direction, and, if there are residual stresses, the body

19For plate like clay particles.20See also Lambe and Whitman (1979) and Sture (1993).21Translation and rotation.



must be cut–out in small pieces and then every piece relieved of stresses. The unstressed configuration is thus

incompatible and discontinuous. The position xi is arbitrary, and we may assume a linear relationship between

dxi and dxi, in the form22:

dxk = (F eik)−1dxi (6.140)

where (F eik)−1

is not to be understood as a deformation gradient, since it may represent the incompatible, dis-

continuous deformation of a body. By considering the reference configuration of a body dXi, then the connection

to the current configuration is23:

dxk = FkidXi ⇒ dxk = (F eik)−1FijdXj (6.141)

so that one can define:

F pkjdef= (F eik)

−1Fij ⇒ Fij

def= F ekiF

pkj (6.142)

The plastic part of the deformation gradient, F pkj represents micro–mechanically, the irreversible process of

slipping, crushing dislocation and macroscopically the irreversible plastic deformation of a body. The elastic part,

F eki represents micro–mechanically a pure elastic reversal of deformation for the particulate assembly, macroscop-

ically a linear elastic unloading toward a stress free state of the body, not necessarily a compatible, continuous

deformation but rather a fictitious elastic unloading of small cut outs of a deformed particulate assembly or

continuum body.

6.3.3 Constitutive Relations

We propose the free energy density W , which is defined in Ω, as follows

ρ0W (Ceij , κα) = ρ0We(Ceij) + ρ0W

p(κα) (6.143)

where W e(Ceij) represents a suitable hyperelastic model in terms of the elastic right deformation tensor Ceij ,

whereas W p(κα) represents the hardening. It has been shown elsewhere (Runesson, 1996), that the pertinent

dissipation inequality becomes

D = Tij Lpij +

∑

α

Kα κα ≥ 0 (6.144)

where Tij is the Mandel stress24 and Lpij is the plastic velocity gradient defined on Ω.

We now define elastic domain B as

B = Tij , Kα | Φ(Tij , Kα) ≤ 0 (6.145)

When Φ is isotropic in Tij (which is the case here) in conjunction with elastic isotropy, we can conclude that Tij

is symmetrical and we may replace Tij by τij in Φ.

22referred to same Cartesian coordinate system.23See section 6.1.2.24See section 6.2.1.



As to the choice of elastic law, it is emphasized that this is largely a matter of convenience since we shall be

dealing with small elastic deformations. Here, the Neo–Hookean elastic law is adopted. The generic situation

is Tij = Tij(˜Uekl, J

e), where we have used the isochoric/volumetric split of the elastic right stretch tensor as

Uekl = ˜Uekl (Je)1/3.

The constitutive relations can now be written as

Tij = Tij(˜Uekl, J

e) (6.146)

Lpij := F pik

(

F pjk

)−1

= µ∂Φ∗

∂Tij= µMij (6.147)

Kα = Kα(κβ) (6.148)

˙κβ = µ∂Φ∗

∂Kβ, κβ(0) = 0 (6.149)

where F pik = (F eli)−1Flk is the plastic part of the deformation gradient.

6.3.4 Implicit Integration Algorithm

The incremental deformation and plastic flow are governed by the system of evolution equations (6.147) and

(6.149):

F pik

(

F pjk

)−1

= µMij (6.150)

˙κβ = µ∂Φ∗

∂Kβ, κβ(0) = 0 (6.151)

The flow rule from equation (6.150) can be integrated to give

n+1F pij = exp(∆µn+1Mik

)nF pkj (6.152)

By using the multiplicative decomposition

Fij = F eik Fpkj ⇒ F eik = Fij

(

F pkj

)−1

(6.153)

and equation (6.152) we obtain

n+1F eij = n+1Fim (nF pmk)−1

exp(−∆µn+1Mkj

)

= n+1F e,trik exp(−∆µn+1Mkj

)(6.154)

where we used that

n+1F e,trik = n+1Fim (nF pkm)−1

(6.155)

The elastic deformation is then

n+1Ceijdef=

(n+1F eim

)T n+1F emj

= exp(−∆µn+1MT

ir

) (n+1F e,trrk

)T n+1F e,trkl exp(−∆µn+1Mlj

)

= exp(−∆µn+1MT

ir

)n+1Ce,trrl exp

(−∆µn+1Mlj

)(6.156)



By recognizing that the exponent of a tensor can be expanded in Taylor series25

exp(−∆µn+1Mlj

)= δlj − ∆µn+1Mlj +

1

2

(∆µn+1Mls

) (∆µn+1Msj

)+ · · · (6.157)

and by using the first order expansion in the equation (6.156), we obtain

n+1Ceij =(δir − ∆µn+1Mir

)n+1Ce,trrl

(δlj − ∆µn+1Mlj

)

=(n+1Ce,tril − ∆µn+1Mir

n+1Ce,trrl

) (δlj − ∆µn+1Mlj

)

= n+1Ce,trij − ∆µn+1Mirn+1Ce,trrj − ∆µ n+1Ce,tril

n+1Mlj

+∆µ2 n+1Mirn+1Ce,trrl

n+1Mlj (6.158)

Remark 6.3.1 The Taylor’s series expansion from equation (6.157) is a proper approximation for the general

nonsymmetric tensor Mlj . That is, the approximate solution given by equation (6.158) is valid for a general

anisotropic solid. This contrasts with the spectral decomposition family of solutions26 which are restricted to

isotropic solids.

Remark 6.3.2 Taylor’s series expansion27 is proper for “small” values of plastic flow tensor ∆µn+1Mlj . This is

indeed the case for small increments, when ∆µ → 0 which are required for following the equilibrium path for

path–dependent solids.

Remark 6.3.3 In the limit, when the displacements are sufficiently small, the solution (6.158) collapses to

limFij→δij

δij + 2n+1ǫij = + δij + 2n+1ǫe,trij

− ∆µn+1Mir

(δrj + 2n+1ǫe,trrj

)

− ∆µ(δil + 2n+1ǫe,tril

)n+1Mlj

+ ∆µ2 n+1Mir

(δrl + 2n+1ǫe,trrl

)n+1Mlj

= + δij + 2n+1ǫe,trij

− ∆µn+1Mij − 2∆µn+1Mirn+1ǫtrrj

− ∆µ n+1Mij − 2∆µ n+1ǫtriln+1Mlj

+ ∆µ2 n+1Miln+1Mlj + 2∆µ2 n+1Mir

n+1ǫtrrln+1Mlj

= δij + 2n+1ǫe,trij − 2∆µ n+1Mij

⇒ n+1ǫij = n+1ǫtrij − ∆µ n+1Mij (6.159)

which is a small deformation elastic predictor–plastic corrector equation in strain space. In working out the small

deformation counterpart (6.159) it was used that

limFij→δij

n+1Ceij = δij + 2n+1ǫij

2∆µ n+1ǫtriln+1Mlj ≪ n+1Mij

∆µ ≪ 1 (6.160)

25See for example Pearson (1974).26See Simo (1992).27It should be called MacLaurin’s series expansion, since expansion is about zero plastic flow state (no incremental plastic defor-

mation).



By neglecting the higher order term with ∆µ2 in equation (6.158), the solution for the right elastic deformation

tensor n+1Ceij can be written as

n+1Ceij = n+1Ce,trij − ∆µ(n+1Mir

n+1Ce,trrj + n+1Ce,triln+1Mlj

)(6.161)

The hardening rule (6.151) can be integrated to give

n+1κα = nκα + ∆µ∂Φ∗

∂Kα

∣∣∣∣n+1

(6.162)

Remark 6.3.4 It is interesting to note that equation (6.161) resembles the elastic predictor–plastic corrector

equation for small deformation elastic–plastic incremental analysis. That resemblance will be used to build an

iterative solution algorithm in the next section.

The incremental problem is defined by equations (6.161), (6.162), and the constitutive relations

n+1SIJ = 2∂W

∂CIJ

∣∣∣∣n+1

(6.163)

n+1Kα = − ∂W

∂κα

∣∣∣∣n+1

(6.164)

and the Karush–Kuhn–Tucker (KKT) conditions

∆µ < 0 ; n+1Φ ≤ 0 ; ∆µ n+1Φ = 0 (6.165)

where

Φ = Φ(Tij ,Kα) (6.166)

Remark 6.3.5 The Mandel stress tensor Tij can be obtained from the second Piola–Kirchhoff stress tensor Skj

and the right elastic deformation tensor Ceik as

Tij = Ceik Skj (6.167)

This set of nonlinear equations will be solved with a Newton type procedure, described in the next section.

For a given n+1Fij , or n+1Ce,trij , the upgraded quantities n+1SIJ and n+1Kα can be found, then the appropriate

pull–back to B0 or push–forward28 to B will give n+1SIJ and n+1τij

n+1SIJ =(n+1F piI

)−1 n+1SIJ

(n+1F pjJ

)−T(6.168)

n+1τij = n+1F eiIn+1SIJ

(n+1F ejJ

)−1(6.169)

The elastic predictor, plastic corrector equation

n+1Ceij = n+1Ce,trij − ∆µ(n+1Mir

n+1Ce,trrj + n+1Ce,triln+1Mlj

)

= n+1Ce,trij − ∆µ n+1Zij (6.170)

28See Appendix ??.



is used as a starting point for a Newton iterative algorithm. In previous equation, we have introduced tensor Zij

to shorten writing. The trial right elastic deformation tensor is defined as

n+1Ce,trij =(n+1F e,trri

)T (n+1F e,trrj

)

(n+1FrM (nF piM )

−1)T

(

n+1FrS

(nF pjS

)−1)

(6.171)

We introduce a tensor of deformation residuals

Rij = Ceij︸︷︷︸

current

−(n+1Ce,trij − ∆µ n+1Zij

)

︸︷︷︸

BackwardEuler

(6.172)

Tensor Rij represents the difference between the current right elastic deformation tensor and the Backward Euler

right elastic deformation tensor. The trial right elastic deformation tensor n+1Ce,trij is maintained fixed during the

iteration process. The first order Taylor series expansion can be applied to the equation (6.172) in order to obtain

the iterative change, the new residual Rnewij from the old Roldij

Rnewij = Roldij + dCeij + d(∆µ) n+1Zij + ∆µ∂n+1Zij∂Tmn

dTmn + ∆µ∂n+1Zij∂Kα

dKα (6.173)

By using that

Tmn = Cemk Skn ⇒(Cesk

)−1Tsn = Skn (6.174)

we can write

dTmn = dCemk Skn + Cemk dSkn

= dCemk Skn +1

2Cemk Leknpq dCepq from (6.55)

= dCemk(Cesk

)−1Tsn +

1

2Cemk Leknpq dCepq from (6.174) (6.175)



and the equation (6.173) can be rewritten as

Rnewij = Roldij + dCeij + d(∆µ) n+1Zij +

+ ∆µ∂n+1Zij∂Tmn

(

dCemk(Cesk

)−1Tsn +

1

2Cemk Leknpq dCepq

)

+

+ ∆µ∂n+1Zij∂Kα

dKα

= Roldij + dCeij + d(∆µ) n+1Zij +

+ ∆µ∂n+1Zij∂Tmn

dCemk(Cesk

)−1Tsn +

+1

2∆µ

∂n+1Zij∂Tmn

Cemk Leknpq dCepq +


dKα

= Roldij + dCeij + d(∆µ) n+1Zij +

+ ∆µ∂n+1Zmn∂Tik

(Cesj)−1

Tsk dCeij + dummy indices rearrangement

+1

2∆µ

∂n+1Zpq∂Tmn

Cemk Leknij dCeij + dummy indices rearrangement


dKα (6.176)

The goal is to have Rnewij = 0 so one can write

0 = Roldij + dCeij + d(∆µ) n+1Zij +


(Cesj)−1

Tsk dCeij +

+1

2∆µ

∂n+1Zpq∂Tmn

Cemk Leknij dCeij +


dKα

= Roldij + d(∆µ) n+1Zij


dKα +

+ dCeij +


(Cesj)−1

Tsk dCeij +

+1

2∆µ

∂n+1Zpq∂Tmn

Cemk Leknij dCeij

= Roldij + d(∆µ) n+1Zij + ∆µ∂n+1Zij∂Kα

dKα +

+ (δimδnj + ∆µ∂n+1Zmn∂Tik

(Cesj)−1

Tsk +1

2∆µ

∂n+1Zmn∂Tpq

Cepk Lekqij ) dCeij

(6.177)



Upon introducing notation

Tmnij = δimδnj + ∆µ∂n+1Zmn∂Tik

(Cesj)−1

Tsk +1

2∆µ

∂n+1Zmn∂Tpq

Cepk Lekqij (6.178)

we can solve (6.177) for dCeij

dCeij = (Tmnij)−1

(

−Roldmn − d(∆µ) n+1Zmn − ∆µ∂n+1Zmn∂Kα

dKα

)

(6.179)

or, by rearranging indices

dCepq = (Tmnpq)−1

(

−Roldmn − d(∆µ) n+1Zmn − ∆µ∂n+1Zmn∂Kα

dKα

)

(6.180)

By using that

dKα =∂Kα

∂κβdκβ = −d(∆µ)

∂Kα

∂κβ

∂Q

∂Kβ= −d(∆µ) Hαβ

∂Q

∂Kβ(6.181)

it follows from (6.180)

dCepq = (Tmnpq)−1

(

−Roldmn − d(∆µ) n+1Zmn + ∆µ∂n+1Zmn∂Kα

d(∆µ) Hαβ∂Q

∂Kβ

)

(6.182)

A first order Taylor series expansion of a yield function yields

newΦ(Tij ,Kα) = oldΦ(Tij ,Kα) +

+∂Φ(Tij ,Kα)

∂TmndTmn

+∂Φ(Tij ,Kα)

∂KαdKα

= oldΦ(Tij ,Kα) +

+∂Φ(Tij ,Kα)

∂Tmn

(

dCemk(Cesk

)−1Tsn +

1

2Cemk Leknpq dCepq

)

+∂Φ(Tij ,Kα)

∂KαdKα

= oldΦ(Tij ,Kα) +

+∂Φ(Tij ,Kα)

∂Tpn

(Cesq)−1

Tsn dCepq dummy indices rearrangement

+1

2

∂Φ(Tij ,Kα)

∂TmnCemk Leknpq dCepq

+∂Φ(Tij ,Kα)

∂KαdKα

= oldΦ(Tij ,Kα) +

+

(∂Φ(Tij ,Kα)

∂Tpn

(Cesq)−1

Tsn +1

2

∂Φ(Tij ,Kα)

∂TmnCemk Leknpq

)

dCepq

+∂Φ(Tij ,Kα)

∂KαdKα (6.183)

By using (6.181), equation (6.183) becomes


+

(∂Φ(Tij ,Kα)

∂Tpn

(Cesq)−1

Tsn +1

2

∂Φ(Tij ,Kα)

∂TmnCemk Leknpq

)

dCepq

− d(∆µ)∂Φ(Tij ,Kα)

∂KαHαβ

∂Φ∗

∂Kβ(6.184)



Upon introducing the following notation

Fpq =∂Φ(Tij ,Kα)

∂Tpn

(Cesq)−1

Tsn +1

2

∂Φ(Tij ,Kα)

∂TmnCemk Leknpq (6.185)

and with the solution for dCepq from (6.182), (6.184) becomes


+ Fpq(

(Tmnpq)−1

(

−Roldmn − d(∆µ) n+1Zmn + d(∆µ) ∆µ∂n+1Zmn∂Kα

Hαβ∂Φ∗

∂Kβ

))

− d(∆µ)∂Φ(Tij ,Kα)

∂KαHαβ

∂Φ∗

∂Kβ(6.186)

After setting newΦ(Tij ,Kα) = 0 we can solve for the incremental inconsistency parameter d(∆µ)

d(∆µ) =oldΦ −Fpq (Tmnpq)−1

Roldmn

Fpq (Tmnpq)−1 n+1Zmn − ∆µ Fpq (Tmnpq)−1 ∂n+1Zmn∂Kα

Hαβ∂Φ∗

∂Kβ+

∂Φ

∂KαHαβ

∂Φ∗

∂Kβ

(6.187)

Remark 6.3.6 In the perfectly plastic case, the increment inconsistency parameter d(∆µ) is

d(∆µ) =oldΦ −Fpq (Tmnpq)−1

Roldmn

Fpq (Tmnpq)−1 n+1Zmn

(6.188)

Remark 6.3.7 In the limit, for small deformations, isotropic response, the increment inconsistency parameter

d(∆µ) becomes

d(∆µ) =

oldΦ − (nmn Emnpq)

(

δpmδnq + ∆µ∂mmn

∂σijEijpq

)−1

Roldmn

nmn Emnpq

(

δmpδqn + ∆µ∂mpq

∂σijEijmn

)−1n+1mmn +

∂Φ

∂KαHαβ

∂Φ∗

∂Kβ

(6.189)

since in the limit, as deformations are getting small

Tmnpq → δpmδnq + ∆µ∂mmn

∂σijEijpq

Fpq → 1

2

∂Φ

∂σmnEmnpq

Zpq → 2 mpq

Rpq → 2 ǫpq (6.190)

Upon noting that residual Rpq is defined in strain space, the increment inconsistency parameter d(∆µ) compares

exactly with it’s small strain counterpart (Jeremic and Sture, 1997).

The procedure described below summarizes the implementation of the return algorithm.



Trial State Given the right elastic deformations tensor nCepq and a set of hardening variables nKα at a specific

quadrature point in a finite element, compute the relative deformation gradient n+1fij for a given displacement

increment ∆n+1ui, and the right deformation tensor

n+1fij = δij + ui,j (6.191)

n+1Ce,trij =(n+1fir

nF erk)T (n+1fkl

nF elj)

= (nF erk)T (n+1fir

)T (n+1fklnF elj

)(6.192)

Compute the trial elastic second Piola–Kirchhoff stress and the trial elastic Mandel stress tensor

n+1Se,trij = 2∂W

∂n+1Ce,trij

(6.193)

n+1T e,trij = n+1Ce,triln+1Se,trlj (6.194)

Evaluate the yield function n+1Φtr(T e,trij ,Kα). If n+1Φtr ≤ 0 there is no plastic flow in current increment

n+1Ceij = n+1Ce,trij

n+1Kα = nKα

n+1Tij = nT e,trij

and exit constitutive integration procedure.

Return Algorithm If yield criteria has been violated (n+1Φtr > 0) proceed to step 1.

step 1. kth iteration. Known variables

n+1Ce(k)ij ; n+1κ(k)

α ; n+1K(k)α ; n+1T

(k)ij ; n+1∆µ(k)

evaluate the yield function and the residual

Φ(k) = Φ(n+1Te(k)ij , n+1K(k)

α )

R(k)ij = n+1C

e,(k)ij −

(n+1Ce,trij − n+1∆µ(k)n+1Z

(k)ij

)

step 2. Check for convergence, Φ(k) ≤ NTOL and ‖R(k)ij ‖ ≤ NTOL. If convergence criteria is satisfied set

n+1Ceij = n+1Ce(k)ij

n+1κα = n+1κ(k)α

n+1Kα = n+1K(k)α

n+1Tij = n+1T(k)ij

n+1∆µ = n+1∆µ(k)

Exit constitutive integration procedure.

step 3.29 If convergence is not achieved, i.e. Φ(k) > NTOL or ‖R(k)ij ‖ > NTOL then compute the elastic

stiffness tensor Lijkl

L(k)ijkl = 4

∂2W

∂Ce(k)ij ∂C

e(k)kl

(6.195)

29From step 3. to step 9. all of the variables are in intermediate n + 1 configuration. For the sake of brevity we are omitting

superscript n + 1.



step 4. Compute the incremental inconsistency parameter d(∆µ(k+1))

d(∆µ(k+1)) =Φ(k) − F (k)

mn Rmn(k)

F (k)mn Zmn(k) − ∆µ(k) Fmn(k)

∂Zmn(k)

∂KαHα

(k) +∂Φ(k)

∂KαHα

(k)

(6.196)

where

Hα(k) = Hαβ

(k) ∂Φ∗,(k)

∂Kβ; Fmn(k) = Fpq(k)

(

Tmnpq(k))−1

Fpq =∂Φ(T

(k)ij ,K

(k)α )

∂Tpn

(

Ce,(k)sq

)−1

T (k)sn +

1

2

∂Φ(T(k)ij ,K

(k)α )

∂TmnCe,(k)mk Le,(k)knpq

Tmnij = δimδnj + ∆µ(k) ∂Z(k)mn

∂T(k)ik

(

Ce,(k)sj

)−1

T(k)sk +

1

2∆µ(k) ∂Z

(k)mn

∂TpqCe,(k)pk Le,(k)kqij

step 5. Updated the inconsistency parameter ∆µ(k+1)

∆µ(k+1) = ∆µ(k) + d(∆µ(k+1)) (6.197)

step 6. Updated the right deformation tensor, the hardening variable and the Mandel stress

dCe,(k+1)pq =

(

T (k)mnpq

)−1(

−R(k)mn − d(∆µ(k+1)) n+1Z(k)

mn + ∆µ(k) ∂Z(k)mn

∂Kαd(∆µ(k+1)) H(k)

α

)

(6.198)

dκ(k+1)α = d(∆µ(k+1))

∂Φ∗,(k)

∂Kβ(6.199)

dK(k+1)α = −d(∆µ(k+1)) H

(k)αβ

∂Φ∗,(k)

∂Kβ(6.200)

dT (k+1)mn = dC

e,(k+1)mk

(

Ce,(k)sk

)−1

T (k)sn +

1

2Ce,(k)mk Le,(k)knpq dCe,(k+1)

pq (6.201)

step 7. Update right deformation tensor Ce,(k+1)pq , hardening variable K

(k+1)α and Mandel stress T

(k+1)mn

Ce,(k+1)pq = Ce,(k)pq + d(Ce,(k+1)

pq )

κ(k+1)α = κ(k)

α + d(κ(k+1)α )

K(k+1)α = K(k)

α + d(K(k+1)α )

T (k+1)mn = T (k)

mn + d(T (k+1)mn ) (6.202)

step 8. evaluate the yield function and the residual

Φ(k+1) = Φ(Te(k+1)ij ,K(k+1)

α ) ; R(k+1)ij = C

e,(k+1)ij −

(

Ce,trij − ∆µ(k+1)Z(k+1)ij

)

(6.203)



step 9. Set k = k + 1

∆µ(k) = ∆µ(k+1)

Ce,(k)pq = Ce,(k+1)pq

κ(k)α = κ(k+1)

α

K(k)α = K(k+1)

α

T (k)mn = T (k+1)

mn (6.204)

and return to step 2.

6.3.5 Algorithmic Tangent Stiffness Tensor

Starting from the elastic predictor–plastic corrector equation

n+1Ceij = n+1Ce,trij − ∆µ n+1Zij (6.205)

and taking the first order Taylor series expansion we obtain

dCeij = dCe,trij − d(∆µ) Zij − ∆µ∂Zij∂Tmn

dTmn − ∆µ∂Zij∂Kα

dKα

= dCe,trij − d(∆µ) Zij

−∆µ∂Zij∂Tmn

(

dCemk(Cesk

)−1Tsn +

1

2Cemk Leknpq dCepq

)

from (6.175)

−∆µ∂Zij∂Kα

dKα (6.206)

Previous equation can be written as

dCeij + ∆µ∂Zij∂Tmn

(Cesk

)−1Tsn dC

emk + ∆µ d(∆µ)

∂Zij∂Tmn

1

2Cemk Leknpq dCepq

= dCe,trij − d(∆µ) Zij + ∆µ d(∆µ)∂Zij∂Kα

Hαβ∂Φ∗

∂Kβ(6.207)

or as

dCeij (Tmnij) = dCe,trij − d(∆µ) Zij + ∆µ d(∆µ)∂Zij∂Kα

Hαβ∂Φ∗

∂Kβ(6.208)

where

Tmnij = δimδnj + ∆µ(k) ∂Z(k)mn

∂T(k)ik

(

Ce,(k)sj

)−1

T(k)sk +

1

2∆µ(k) ∂Z

(k)mn

∂TpqCe,(k)pk Le,(k)kqij

The solution for the increment in right elastic deformation tensor is then

dCeij = (Tmnij)−1

(

dCe,trij − d(∆µ) Zij + ∆µ d(∆µ)∂Zij∂Kα

Hαβ∂Φ∗

∂Kβ

)

(6.209)



We next use the first order Taylor series expansion of yield function dΦ(Tij ,Kα) = 0

∂Φ

∂TmndTmn +

∂Φ

∂KαdKα =

∂Φ

∂Tmn

(

dCemk(Cesk

)−1Tsn +

1

2Cemk Leknpq dCepq

)

+∂Φ

∂KαdKα =

∂Φ

∂Tpn

(Cesq)−1

Tsn dCepq +

1

2

∂Φ

∂TmnCemk Leknpq dCepq +

∂Φ

∂KαdKα =

(∂Φ

∂Tpn

(Cesq)−1

Tsn +1

2

∂Φ

∂TmnCemk Leknpq

)

dCepq −∂Φ

∂Kαd(∆µ) Hαβ

∂Φ∗

∂Kβ=

Fpq dCepq −∂Φ


∂Φ∗

∂Kβ= 0

(6.210)

where

Fpq =∂Φ

∂Tpn

(Cesq)−1

Tsn +1

2

∂Φ

∂TmnCemk Leknpq (6.211)

By using solution for dCeij from 6.209 we can write

Fpq (Tmnpq)−1

(

dCe,trmn − d(∆µ) Zmn + ∆µ d(∆µ)∂Zmn∂Kα

Hαβ∂Φ∗

∂Kβ

)

− ∂Φ


∂Φ∗

∂Kβ= 0

(6.212)

We are now in the position to solve for the incremental inconsistency parameter d(∆µ)

d(∆µ) =Fpq (Tmnpq)−1

dCe,trmn

Γ(6.213)

where we have used Γ to shorten writing

Γ = Fpq (Tmnpq)−1 n+1Zmn − ∆µFpq (Tmnpq)−1 ∂n+1Zmn∂Kα

Hαβ∂Φ∗

∂Kβ+

∂Φ

∂KαHαβ

∂Φ∗

∂Kβ

(6.214)

Since

dSkn =1

2Leknpq dCepq (6.215)

and by using 6.209 we can write

dCepq =

(Tmnpq)−1

(

δmv δnt −Fop (Trsop)−1

δrv δstΓ

Zmn+

∆µFop (Trsop)−1

δrv δstΓ

∂Zij∂Kα

Hαβ∂Φ∗

∂Kβ

)

dCe,trvt (6.216)

Then

dCepq = Ppqvt dCe,trvt (6.217)



where

Ppqvt = (Tmnpq)−1δmv δnt −

Fop (Trsop)−1δrv δst

ΓZmn

+ ∆µFop (Trsop)−1

δrv δstΓ

∂Zij∂Kα

Hαβ∂Φ∗

∂Kβ

= (Tmnpq)−1

(

δmvδnt −Fab(Tvtab)−1

Γ

(

Zmn − ∆µ∂n+1Zmn∂Kα

Hαβ∂Φ∗

∂Kβ

))

(6.218)

Algorithmic tangent stiffness tensor Lijkl (in intermediate configuration Ω) is then defined as

LATSknvt = Leknpq Ppqvt (6.219)

Pull–back to the reference configuration Ω0 yields the algorithmic tangent stiffness tensor Lijkl in reference

configuration Ω0

n+1LATSijkl = n+1F pimn+1F pjn

n+1F pkrn+1F pls

n+1LATSmnrs (6.220)

Remark 6.3.8 In the limit, for small deformations, isotropic response, the Algorithmic Tangent Stiffness tensor

LATSijkl becomes

lim LATSvtpq = EATSvtpq = Eknpq

(

Υ−1mnpq

(

δmvδnt −ncdEcdabΥ

−1vtabHmn

Γ

))

= Eknpq

(

Υ−1vtpq −

Υ−1mrpqncdEcdabΥ

−1vtabHmr

Γ

)

= EknpqΥ−1vtpq −

EknpqΥ−1mrpqncdEcdabΥ

−1vtabHmr

Γ

= Rknvt −ncdRcdvtRknmrHmr

Γ(6.221)

since

lim Tmnpq = Υmnpq = δpmδnq + ∆µ∂Zmn∂Tpk

(Cesq)−1

Tsk +1

2∆µ

∂Zmn∂Trs

CerkLekspq

= δpmδnq + ∆µ∂mmn

∂σrsEekspq (6.222)

limFab = lim

(∂Φ

∂Tad

(Cesb)−1

Tsd +1

2

∂Φ

∂TcdCeckLekdab

)

=1

2ncdE

ecdab (6.223)

Hmn = mmn − ∆µ∂mmn

∂KαHαβ

∂Φ∗

∂Kβ(6.224)

lim Γ = lim

(

Fpq (Tmnpq)−1 n+1Zmn − ∆µFpq (Tmnpq)−1 ∂n+1Zmn∂Kα

Hαβ∂Φ∗

∂Kβ+

∂Φ

∂KαHαβ

∂Φ∗

∂Kβ

)

= nabEabpqΥ−1mnpqmmn − ∆µnabEabpqΥ

−1mnpq

∂mmn

∂KαHαβ

∂Φ∗

∂Kβ+

∂Φ

∂KαHαβ

∂Φ∗

∂Kβ

= nabEabpqΥ−1mnpq

(

mmn − ∆µ∂mmn

∂KαHαβ

∂Φ∗

∂Kβ

)

+∂Φ

∂KαHαβ

∂Φ∗

∂Kβ

= nabRabmnHmn +∂Φ

∂KαHαβ

∂Φ∗

∂Kβ(6.225)



It is noted that the Algorithmic Tangent Stiffness tensor given by 6.221 compares exactly with it’s small strain

counterpart (Jeremic and Sture, 1997).

6.4 Material and Geometric Non–Linear Finite Element Formulation

6.4.1 Introduction

We here present a detailed formulation of a material and geometric non–linear static finite element system of

equations. The configuration of choice is material or Lagrangian. Eulerian and mixed Eulerian–Lagrangian

configuration will be mentioned as need be.

6.4.2 Equilibrium Equations

The local form of equilibrium equations in material format (Lagrangian) for static case can be written as:

PiJ,J − ρ0bi = 0 (6.226)

where PiJ = SIJ(FiI)t and SIJ are first and second Piola–Kirchhoff stress tensors, respectively and bI are body

forces.

Weak form of equilibrium equations is obtained by premultiplying 6.226 with virtual displacements δui and

integrating by parts on the initial configuration B0 (initial volume V0):

∫

V0

δui,jPijdV =

∫

V0

ρ0δuibidV −∫

S0

δuitidS (6.227)

It proves beneficial to rewrite Lagrangian format of weak form of equilibrium equilibrium by using symmetric

second Piola–Kirchhoff stress tensor Sij :

∫

V0

δui,jFjlSildV =

∫

V0

1

2(δui,jFjl + Fljδuj,i+)SildV =

∫

V0

1

2(δui,j (δjl + uj,l) + (δlj + ul,j) δuj,i)SildV =

∫

V0

1

2(δui,l + δui, juj,l) + (δul,i + ul,jδuj,i)SildV =

∫

V0

1

2((δui,l + δul,i) + (δui,juj,l + ul,jδuj,i))SildV =

(6.228)

where we have used the symmetry of Sil, definition for deformation gradient Fki = δki + uk,i. We have also

conveniently defined differential operator Eil(δui, ui)

Eil(δui, ui) =1

2(δul,i + δui,l) +

1

2(ul,jδuj,i + δui,juj,l) (6.229)



6.4.3 Formulation of Non–Linear Finite Element Equations

Consider the motion of a general body in a stationary Cartesian coordinate system, as shown in Figure (6.6),

and assume that the body can experience large displacements, large strains, and nonlinear constitutive response.

The aim is to evaluate the equilibrium positions of the complete body at discrete time points 0,∆t, 2∆t, . . . ,

where ∆t is an increment in time. To develop the solution strategy, assume that the solutions for the static and

kinematic variables for all time steps from 0 to time t inclusive, have been obtained. Then the solution process

for the next required equilibrium position corresponding to time t+∆t is typical and would be applied repetitively

until a complete solution path has been found. Hence, in the analysis one follows all particles of the body in their

motion, from the original to the final configuration of the body. In so doing, we have adopted a Lagrangian ( or

material ) formulation of the problem.

t+ t∆

t+ t∆ ui

tui

t+ t∆u ix i0

ix0

ixt

Ω 0

Ωn

Ωn+1

1

22 20 t

2 3)P(

0

0V

t

t

AV

V

0

3330 tx x x

x x x

x x x,

, ,

t+ t

t+ t t+ t t+ t

t+ t

t+ t

t+ t

1tP( x , 2

tx , )t3

x

01P( x )

03x0

2x

0

Configuration

at time

t

Configuration

at time

Configuration

at time t+ t

∆

∆ ∆ ∆

∆

∆

∆

∆

, ,

,,

,

t+ t∆

i

i

ixx

x

1x,1xt,1x

t

A

A

i=1,2,3

++

+=

==

Figure 6.6: Motion of body in stationary Cartesian coordinate system

Weak format of the equilibrium equations can be obtained by premultiplying 6.226 with virtual displacements

δui and integrating by parts. We obtain the virtual work equations in the Lagrangian format:∫

V0

δui,jPijdV =

∫

V0

ρ0δuibidV −∫

S0

δuitidV (6.230)

Virtual work equations can also be written in terms of second Piola–Kirchhoff stress tensor SIJ as:∫

V0

δui,jFjlSildV =

∫

V0

ρ0δuibidV +

∫

S0

δuitidV (6.231)

which after some algebraic manipulations, and after observing that SIJ = SJI yields (SEE ABOVE!) By intro-

ducing a differential operator E(u1, u2) as:

Eil(1ui,

2ui) =1

2

(1ui,l +

1ul,i)

+1

2

(1ul,j

2uj,i + 2ui,j1uj.l

)(6.232)



virtual work equation 6.229 can be written as:∫

V0

Eil(δui, ui)SildV =

∫

V0

ρ0δuibidV +

∫

S0

δuitidV (6.233)

or as:

W (δui, u(k)i )int +W ext(δui) = 0 (6.234)

with:

W int(δui,n+1

0 u(k)i ) =

∫

Ωc

Eij(δui,n+1

0 u(k)i ) n+1

0 S(k)ij dV (6.235)

=

∫

Ωc

((δuj,i + δui,j) + (uj,rδur,i + δui,rur,j)) S(k)ij dV

W ext(δui) = −∫

Ωc

ρ0 δuin+1

0 bi dV −∫

∂Ωc

δuin+1

0 ti dS (6.236)

6.4.4 Computational Domain in Incremental Analysis

In this Chapter we elaborate on the choice of Total Lagrangian (TL) formulations as computational domain. We

also choose Newton type procedure for satisfying equilibrium, i.e. virtual work for a given computational domain.

Given the displacement field u(k)i (Xj), in iteration k, the iterative change δui

u(k+1)i = u

(k)i + ∆ui (6.237)

is obtained from the linearized virtual work expression

W (δui, u(k+1)i ) ≃W (δui, u

(k)i ) + ∆W (δui,∆ui;u

(k)i ) (6.238)

Here, W (δui, u(k)i ) is the virtual work expression

W (δui, u(k)i ) = W (δui, u

(k)i )int +W ext(δui) (6.239)

with

W int(δui,n+1

0 u(k)i ) =

∫

Ωc

Eij(δui,n+1

0 u(k)i ) n+1

0 S(k)ij dV (6.240)


Ωc

ρ0 δuin+1

0 bi dV −∫

∂Ωc

δuin+1

0 ti dS (6.241)

and the ∆W (δui,∆ui;u(k)i ) is the linearization of virtual work

∆W (δui,∆ui;u(k)i ) = lim

ǫ→0

∂W (δui, ui + ǫ∆ui)

∂ǫ

=

∫

Ωc

Eij(δui, ui) dSij dV +

∫

Ωc

∆Eij(δui, ui) SijdV

=

∫

Ωc

Eij(δui, ui) Lijkl Ekl(∆ui, ui) dV +

∫

Ωc

∆Eij(δui, ui) Sij dV

(6.242)

Here we have used dSij = 1/2 LijkldCkl = LijklEkl(∆ui, ui).



In order to obtain expressions for stiffness matrix we shall work on 6.242 in some more details. To this end,

(6.242) can be rewritten by expanding definitions for E as

∆W (δui,∆ui;u(k)i ) =

1

4

∫

Ωc

((δuj,i + δui,j) + (uj,rδur,i + δui,rur,j)) Lijkl ((∆uk,l + ∆ul,k) + (uk,s∆us,l + ∆ul,sus,k)) dV +

+

∫

Ωc

1

2(∆uj,lδul,i + δui,l∆ul,j)Sij dV

(6.243)

Or, by conveniently splitting the above equation we can write

∆1W (δui,∆ui;u(k)i ) =

1

4

∫

Ωc

((δuj,i + δui,j) + (uj,rδur,i + δui,rur,j)) Lijkl ((∆uk,l + ∆ul,k) + (uk,s∆us,l + ∆ul,sus,k)) dV

(6.244)


∫

Ωc

1

2(∆uj,lδul,i + δui,l∆ul,j)Sij dV (6.245)

By further working on 6.244 we can write:


∫

Ωc

(1

2(δuj,i + δui,j)

)

Lijkl(

1

2(∆uk,l + ∆ul,k)

)

dV

+

∫

Ωc

(1

2(δuj,i + δui,j)

)

Lijkl(

1

2(uk,s∆us,l + ∆ul,sus,k)

)

dV

+

∫

Ωc

1

2(uj,rδur,i + δui,rur,j)Lijkl

1

2(uk,s∆us,l + ∆ul,sus,k) dV

+

∫

Ωc

1

2(uj,rδur,i + δui,rur,j)Lijkl

(1

2(∆uk,l + ∆ul,k)

)

dV (6.246)

It should be noted that the Algorithmic Tangent Stiffness (ATS) tensor Lijkl poses both minor symmetries

(Lijkl = Ljikl = Lijlk). However, Major symmetry cannot be guaranteed. Non–associated flow rules in elasto-

plasticity lead to the loss of major symmetry (Lijkl 6 =Lklij). Moreover, it can be shown (i.e. Jeremic and Sture

(1997)) that there is algorithmic induced symmetry loss even for associated flow rules.

With the minor symmetry of Lijkl one can write (6.246) as:


∫

Ωc

δui,j Lijkl ∆ul,kdV

+

∫

Ωc

δui,j Lijkl uk,s∆ul,sdV

+

∫

Ωc

δui,rur,j Lijkl uk,s∆ul,sdV

+

∫

Ωc

δui,rur,j Lijkl ∆ul,kdV (6.247)

Similarly, by observing symmetry of second Piola–Kirchhoff stress tensor Sij we can write


∫

Ωc

δui,l∆ul,j SijdV (6.248)



Weak form of equilibrium expressions (i.e. (6.236) and (6.236) ) for internal (W int) and external (W ext) virtual

work, with the above mentioned symmetry of Sij can be written as

W int(δui,n+1

0 u(k)i ) =

∫

Ωc

δui,j SijdV +

∫

Ωc

δui,rur,j SijdV (6.249)


Ωc

ρ0 δuibi dV −∫

∂Ωc

δui ti dS (6.250)

Standard finite element discretization of displacement field yields:

ui ≈ ui = HI uIi (6.251)

where ui is the approximation to exact, analytic (if it exists) displacement field ui, HI are standard FEM shape

functions and uIi are nodal displacements. With this approximation, we have:


∫

Ωc

(HI,jδuIi) Lijkl (HQ,k∆uQl) dV

+

∫

Ωc

(HI,jδuIi) Lijkl (HJ,kuJs) (HQ,s∆uQl) dV

+

∫

Ωc

(HI,rδuIi) (HJ,j uJr)Lijkl (HJ,kuJs) (HQ,s∆uQl) dV

+

∫

Ωc

(HI,rδuIi) (HJ,j uJr)Lijkl (HQ,k∆uQl) dV (6.252)


∫

Ωc

(HI,lδuIi) (HQ,j∆uQl)SijdV (6.253)

W int(δui,n+1

0 u(k)i ) =

∫

Ωc

(HI,jδuIi) SijdV +

∫

Ωc

(HI,rδuIi) (HJ,j uJr) SijdV (6.254)


Ωc

ρ0 (HIδuIi) bi dV −∫

∂Ωc

(HIδuIi) ti dS (6.255)

Upon noting that virtual nodal displacements δuIi are any non–zero, continuous displacements, and since they

occur in all expressions for linearized virtual work (from Equations (6.238), (6.239), (6.240), (6.241) and (6.242))

they can be factored out so that we can write (while remembering that ∆W 1 + ∆W 2 +W ext +W int = 0:∫

Ωc

(HI,j) Lijkl (HQ,k∆uQl) dV

+

∫

Ωc

(HI,j) Lijkl (HJ,kuJs) (HQ,s∆uQl) dV

+

∫

Ωc

(HI,r) (HJ,j uJr)Lijkl (HJ,kuJs) (HQ,s∆uQl) dV

+

∫

Ωc

(HI,r) (HJ,j uJr)Lijkl (HQ,k∆uQl) dV

+

∫

Ωc

(HI,l) (HQ,j∆uQl)SijdV

+

∫

Ωc

(HI,j) SijdV +

∫

Ωc

(HI,r) (HJ,j uJr) SijdV

=

∫

Ωc

ρ0 (HI) bi dV +

∫

∂Ωc

(HI) ti dS (6.256)



By rearranging previous equations one can write:(∫

Ωc

HI,jLijklHQ,kdV +

∫

Ωc

HI,jLijklHJ,kuJsHQ,sdV +

∫

Ωc

HI,rHJ,j uJrLijklHJ,kuJsHQ,sdV

+

∫

Ωc

HI,rHJ,j uJrLijklHQ,kdV +

∫

Ωc

HI,lHQ,jSijdV

)

∆uQl

+

∫

Ωc

(HI,j) SijdV +

∫

Ωc

(HI,r) (HJ,j uJr) SijdV

=

∫

Ωc

ρ0 (HI) bi dV +

∫

∂Ωc

(HI) ti dS (6.257)

The vectors of external and internal forces are

fint =∂(W int(δui,

n+10 u

(k)i ))

∂(δui)(6.258)

fext =∂(W ext(δui))

∂(δui)(6.259)

The Algorithmic Tangent Stiffness (ATS) tensor LATSijkl is defined as a linearization of second Piola–Kirchhoff

stress tensor Sij with respect to the right deformation tensor Ckl

dSij =1

2Lijkl dCkl with dCkl = 2 Ekl(dui, ui) (6.260)

Then, the global algorithmic tangent stiffness matrix (tensor) is given as

Kt =∂(∆W (δui,∆ui;u

(k)i ))

∂(δui)(6.261)

The iterative change in displacement vector ∆ui is obtained by setting a linearized virtual work to zero

W (δui, u(k+1)i ) = 0 ⇒ W (δui, u

(k)i ) = −∆W (δui,∆ui;u

(k)i ) (6.262)

Total Lagrangian Format

The undeformed configuration Ω0 is chosen as the computational domain (Ωc = Ω0). The iterative displacement

∆ui is obtained from the equation

W (δui,n+1u

(k)i ) = −∆W (δui,∆ui;

n+1u(k)i ) (6.263)



where

W (δui,n+1u

(k)i ) =

∫

Ωc

Eij(δui,n+1u

(k)i ) n+1S

(k)ij dV

−∫

Ωc

ρ0 δuin+1bi dV −

∫

∂Ωc

δuin+1ti dS (6.264)

and

∆W (δui,∆ui;n+1u

(k)i ) =

∫

Ωc

Eij(δui,n+1u

(k)i ) n+1L(k)

ijkl Ekl(∆ui,n+1u

(k)i ) dV

+

∫

Ωc

dEij(δui,∆ui)n+1S

(k)ij dV (6.265)

In the case of hyperelastic–plastic response, second Piola–Kirchhoff stress n+1S(k)ij is obtained by integrating the

constitutive law, described in Chapter 6.3. It should be noted that by integrating in the intermediate configuration,

we obtain Mandel stress n+1Tij and subsequently30 the second Piola–Kirchhoff stress Skj . The ATS tensor Lijklis then obtained based on Skj . In order to obtain second Piola–Kirchhoff stress Skj and ATS tensor in initial

configuration we need to perform a pull-back from the intermediate configuration to the initial one

n+1Sij = n+1F pipn+1F pjq

n+1Spq (6.266)

n+1Lijkl = n+1F pimn+1F pjn

n+1F pkrn+1F pls

n+1Lmnrs (6.267)

30Skj =`

Cik

´−1Tij




Chapter 7

Solution of Static Equilibrium Equations

(1994–)

7.1 The Residual Force Equations

In previous Chapters we have derived the basic equations of materially nonlinear analysis of solids. Discretization

of such problems by finite element methods results in a set of nonlinear algebraic equations called residual force

equations:

r(u, λ) = fint(u) − λfext = 0 (7.1)

where fint(u) are the internal forces which are functions of the displacements, u, the vector fext is a fixed external

loading vector and the scalar λ is a load–level parameter that multiplies fext. Equation (7.1) describes the case

of proportional loading in which the loading pattern is kept fixed.

All solution procedures of practical importance are strongly rooted in the idea of ”advancing the solution” by

continuation. Except in very simple problems, the continuation process is multilevel and involves hierarchical

breakdown into stages, incremental steps and iterative steps. Processing a complex nonlinear problem generally

involves performing a series of analysis stages. Multiple control parameters are not varied independently in each

stage and may therefore be characterized by a single stage control parameter λ. Stages are only weakly coupled

in the sense that end solution of one may provide the starting point for another.

7.2 Constraining the Residual Force Equations

Various forms of path following methods1 have stemmed from the original work of Riks (1972), Riks (1979) and

Wempner (1971). They aimed at finding the intersection of equation (7.1) with s = constant where s is the

1also called arc-length methods with various methods of approximating the exact length of an arc.

157


arc-length , defined as2:

s =

∫

ds (7.2)

where:

ds =

√

ψ2u

u2ref

duTSdu + dλ2ψ2f (7.3)

Differential form (7.3) can be replaced with an incremental form:

a = (∆s)2 − (∆l)2 =

(

ψ2u

u2ref

∆uTS∆u + ∆λ2ψ2f

)

− (∆l)2

(7.4)

where ∆l is the radius of the desired intersection3 and represents an approximation to the incremental arc length.

Scaling matrix S is usually diagonal non-negative matrix that scales the state vector ∆u and uref is a reference

value with the dimension of√

∆uTS∆u. It is important to note that the vector ∆u and scalar ∆λ are incremental

and not iterative values, and are starting from the last converged equilibrium state.

The main essence of the arc-length methods is that the load parameter λ becomes a variable. With load

parameter λ variable we are dealing with n+ 1 unknowns4. In order to solve this problem we have n equilibrium

equations (7.1) and the one constraint equation (7.4). We can solve the augmented system of n + 1 equations

by applying the Newton-Raphson5 method to equations (7.1) and (7.4)

rnew(u, λ) = rold(u, λ) +∂r(u, λ)

∂uδu +

∂r(u, λ)

∂λδλ =

= rold(u, λ) + Kt δu − fext δλ =

= 0 (7.5)

anew = aold + 2ψ2u

u2ref

∆uTSδu + 2∆λδλ ψ2f = 0 (7.6)

where Kt = ∂r(u,λ)∂u is the tangent stiffness matrix. The aim is to have rnew(u, λ) = 0 and anew = 0 so the

previous system can be written as:

Kt −fext

2ψ2

u

u2ref

∆uTS 2∆λ ψ2f

δu

δλ

= −

rold

aold

(7.7)

2A bit different form in that it is scaled with scaling matrix S, introduced by Felippa (1984).3See Figure (7.1).4n unknown displacement variables and on extra unknown in the form of load parameter.5By using a truncated Taylor series expansion.



∆ l

u∆ 1

Load

fδλ

1fδλ

fδλ

0

2

λf

ext

ext

ext

(u λ f, ext0 0

∆λ1fext

∆λ f2 ext

∆λ f3 ext

δu0δu1

δu2

(u λ f,2 2 ext(u λ f,3 3 ext

Displacement

u

ConstraintHypersurface

)

)

fλ0 ext

EquilibriumPath

)

or(u λ f, ext

(u1 λ1f, ext)

)p p

u0 2

u∆3

u∆

Figure 7.1: Spherical arc-length method and notation for one DOF system.

One can solve previous system of two equations for δu and δλ:

δu

δλ

= −

Kt −fext

2ψ2

u

u2ref

∆uTS 2∆λ ψ2f

−1

rold

aold

(7.8)

or by defining the augmented stiffness matrix6 K as:

K =

Kt −fext

2ψ2

u

u2ref

∆uTS 2∆λ ψ2f

(7.9)

the equation (7.8) can be written as:

δu

δλ

= −K−1

rold

aold

(7.10)

It should be mentioned that the augmented stiffness matrix remains non-singular even if Kt is singular.

6Or augmented Jacobian.



7.3 Load Control

7.4 Displacement Control

7.5 Generalized, Hyper–Spherical Arc-Length Control

In section (7.2) we have introduced an constraining equation that is intended to reduce the so called drift error

in the incremental nonlinear finite element procedure. The constraining equation was given in a rather general

form. Some further comments and observations are in order. By assigning various numbers to parameters ψu,

ψf , S and uref one can obtain different constraining schemes from (7.4).

ConstraintHypersurfaces

EquilibriumPath

ψuψf =

ψuψf < ψuψf <<

∆ l

Load λf

Displacement

u

EquilibriumPath

ψuψf = ψuψf >

ψuψf >>

ConstraintHypersurfaces

∆ l

Load λf

Displacement

u

Figure 7.2: Influence of ψu and ψf on the constraint surface shape.

Coefficients ψu and ψf may not be simultaneously zero. Useful choices for S are I, Kt and diag (Kt). If

S = I and uref = 1 the method is called the arclength method7. If we choose S = diag (Kt) nice scaling is

obtained8 but otherwise no physical meaning can be attributed to this scaling type. With S = Kt and ψf ≡ 0

one ends up with something very similar to the external work constraint of Bathe and Dvorkin (1983). A rather

general equation (7.4) can be further specialized to load (λ) control with ψu ≡ 0;ψf ≡ 1 and state control9 with

ψu ≡ 1;ψf ≡ 0 and S = I. In the finite element literature, the term displacement control has been traditionally

associated with the case in which only one of the components of the displacement vector u10 is specified. This

may be regarded either as a variant of state control, in which a norm that singles out the ith component is used,

or as a variant of the λ control if the control parameter is taken as λui. It is, of course, possible to make the

previous parameters variable, functions of different unknowns. For example if one defines uref = ∆uTS∆u then

close to the limit point ∆u → 0 ⇒ ψ2u

u2ref

≫ ψ2f that makes our constraint from equation (7.4) behave like state

7It actually reduces to the original work of fRiks (1972), Riks (1979) and Wempner (1971).8For example if FEM model includes both translational and rotational DOFs.9That is the cylindrical constraint, or general displacement control.

10Say ui.



control. One important aspect of scaling constraint equations by using S = diag (Kt) or S = Kt is the possibility

of non–positive definite stiffness matrix Kt. It usually happens that after the limit point is passed, at least one

of the eigenvalues of Kt is non–positive, thus rendering the constraint hypersurface non–convex.

In order to get better control of the solution to the system of equations (7.10) one may directly introduce the

constraint from equation (7.6) by following the approach proposed by Batoz and Dhatt (1979), as described by

Crisfield (1991) and Felippa (1993).

The iterative displacement δu is split into two parts, and with the Newton change at the new unknown load

level:

λnew = λold + δλ (7.11)

becomes:

δu = −K−1t r

(uold, λ

)= −K−1

t

(fint(u

old) − λnewfext)

= −K−1t

(fint(u

old) −(λold + δλ

)fext)

= −K−1t

((fint(u

old) − λoldfext)− δλfext

)

= −K−1t

(r(uold, λold

)− δλfext

)= −K−1

t rold + δλK−1t fext = δu + δλδut (7.12)

where δut = K−1t fext is the displacement vector corresponding to the fixed load vector fext, and δu is an iterative

change that would stem from the standard load-controlled Newton-Raphson, at a fixed load level λold. With the

solution11 for the δu from (7.12), the new incremental displacements are:

∆unew = ∆uold + δu = ∆uold + δu + δλδut (7.13)

where δλ is the only unknown. The constraint from equation (7.4) can be used here, and by rewriting it as:

(

ψ2u

u2ref

(∆unew)T

S (∆unew) + (∆λnew)2ψ2f

)

= (∆l)2

(7.14)

then by substituting ∆unew from equation (7.13) into equation (7.14) and by recalling that λnew = λold + δλ

one ends up with the following quadratic scalar equation:

(

ψ2u

u2ref

(∆uold + δu + δλδut

)TS(∆uold + δu + δλδut

)+(∆λold + δλ

)2ψ2f

)

= (∆l)2

(7.15)

or, by collecting terms:

11But having in mind that δλ is still unknown!



(

ψ2u

u2ref

δuTt Sδut + ψ2f

)

δλ2 +

+2

(

ψ2u

u2ref

δuTt S(∆uold + δu

)+ ∆λoldψ2

f

)

δλ +

+

(

ψ2u

u2ref

(∆uold + δu

)TS(∆uold + δu

)− ∆l2 +

(∆λold

)2ψ2f

)

= 0 (7.16)

or:

a1δλ2 + 2a2δλ+ a3 = 0 (7.17)

where:

a1 =ψ2u

u2ref

δuTt Sδut + ψ2f

2a2 = 2

(

ψ2u

u2ref

δuTt S(∆uold + δu

)+ ∆λoldψ2

f

)

a3 =ψ2u

u2ref

(∆uold + δu

)TS(∆uold + δu

)− ∆l2 +

(∆λold

)2ψ2f

The quadratic scalar equation (7.17) can be solved for δλ:

δλ = δλ1 =−a2 +

√

a22 − a1a3

a1; δλ = δλ2 =

−a2 −√

a22 − a1a3

a1(7.18)

or, if a1 = 0, then:

δλ = − a3

2a2(7.19)

and then the complete change is defined from equation (7.13):

∆unew = ∆uold + δu + δλδut (7.20)

An ambiguity is introduced in the solution for δλ from (7.18). The tangent at the regular point on the

equilibrium path has two possible directions, which generally intersect the constraint hypersurface at two points.

However, some exceptions from that rule are possible, so the solutions from (7.18) can be categorized as:

• Real roots of opposite signs. This occurs when the iteration process converges normally and there is no

limit or turning point enclosed by the constraint hypersurface. The root is chosen by applying one of the

schemes proposed below.

• Real roots of equal sign opposite to that of ∆λold. This usually happens when going over a ”flat” limit

point.



• Real roots of equal sign same as that of ∆λold. This is an unusual case. It may signal a turning point or

be triggered by erratic iteration behavior.

• Complex roots. This is an unusual case too. It may signal a sharp turning point, a bifurcation point, erratic

or divergent iterates.

For the first two cases, the correct ∆λ can be chosen by applying one of the following schemes.

7.5.1 Traversing Equilibrium Path in Positive Sense

Positive External Work

The simplest rule requires that the external work expenditure over the predictor step be positive:

∆W = fText∆u = fTextK−1t fextδλ > 0 (7.21)

The simple conclusion is that δλ should have the sign of fTextK−1t fext. This condition is particularly effective at

limit points. However, it fails when fext and K−1t fext are orthogonal:

fTextK−1t fext = 0 (7.22)

This can happen at:

• Bifurcation points,

• Turning points,

The treatment of bifurcation points is of a rather special nature and is left for the near future. Turning

points12 can be traversed by a modification of a previous rule, as described in the next section.

Angle Criterion

Near a turning point application of the positive work rule (7.21) causes the path to double back upon itself. When

it crosses the turning point it reverses so the turning point becomes impassable. Physically, a positive work rule

is incorrect because in passing a turning point the structure releases external work until another turning point is

encountered.

To pass a turning point imposing a condition on the angle of the prediction vector proves more effective. The

idea is to compute both solutions δλ1 and δλ2 and then both ∆pnew1 and ∆unew1 :

∆unew1 = ∆uold + δu + δλ1δut (7.23)

∆unew2 = ∆uold + δu + δλ2δut (7.24)

12One might ask ”why treating turning points in a material nonlinear analysis?”. The answer is rather simple: ”try to prevent all

unnecessary surprises”. For a good account of some of surprises in material nonlinear analysis one might take a look at some examples

in Crisfield (1991) pp. 270.



PointLimit

PointTurning

Loadfλ

Displacement

u

PrimaryEquilibriumPath

BifurcationPoint

Secondary

PathEquilibrium

Figure 7.3: Simple illustration of Bifurcation and Turning point.

The one that lies closest to the old incremental step direction ∆uold is the one sought. This should prevent the

solution from double backing. The procedure can be implemented by finding the solution with the minimum angle

between ∆uold and ∆pnew, and hence the maximum cosine of the angle:

cosφ =

(∆uold

)T(∆unew)

‖∆uold‖ ‖∆unew‖ =

(∆uold

)T (∆uold + δu + δλδut

)

‖∆uold‖ ‖∆uold + δu + δλδut‖(7.25)

where δu = −K−1t rold and δut = K−1

t fext. Once the turning point has been crossed, the work criterion should

be reversed so the external work is negative.

By directly introducing the constraint from equation (7.6) and following the method through equations (7.12)

to (7.25) a limitation is introduced. Precisely at the limit point, Kt is singular and the equations cannot be solved.

However, Batoz and Dhatt (1979) and Crisfield (1991) report that no such problem has occurred, because one

appears never to arrive precisely at limit point.

7.5.2 Predictor step

The predictor solution is achieved by applying one forward Euler, explicit step from the last obtained equilibrium

point:

∆up = K−1t ∆qe = ∆λpK

−1t fext = ∆λpδut (7.26)



where Kt is the tangent stiffness matrix at the beginning of increment. Substituting equation (7.26) into the

constraint equation (7.14) one obtains:

(

ψ2u

u2ref

(∆unew)T

S (∆unew) + (∆λnew)2ψ2f

)

=

(

ψ2u

u2ref

∆λ2pδu

Tt Sδut

)

+ (∆λp)2ψ2f =

∆λ2p

(

ψ2u

u2ref

δuTt Sδut + ψ2f

)

= (∆l)2

(7.27)

The solution for ∆λp is readily found:

∆λp = ± ∆l√

ψ2u

u2ref

|δuTt Sδut| + ψ2f

(7.28)

where ∆l > 0 is the given increment length. The absolute value of |δuTt Sδut| is needed if the stiffness matrix

is chosen as a scaling matrix, i.e. S = Kt, since, after passing limit point, the stiffness matrix is non–positive

definite so δuTt Sδut ≤ 0. The question of choosing the right sign + or − in (7.28) is still a vigorous research

topic. In the simplified procedure13 the negative sign − is chosen with respect to the occurrence of one negative

pivot during factorization of the tangent stiffness matrix Kt. If more than one pivot happens to be negative, one

is advised14 to stop the analysis and try to restart from previously converged solution with smaller step size.

7.5.3 Automatic Increments

A number of workers have advocated different strategies for controlling the step length size. In this work we will

follow the strategy advocated by Crisfield (1991). The idea is to find the new incremental length by applying:

∆lnew = ∆lold(IdesiredIold

)n

(7.29)

where ∆lold is the old incremental factor for which Iold iterations were required, Idesired is the input, desired

number of iterations15 and the parameter n is set to 12 as suggested by Ramm (1982) Ramm (1981).

7.5.4 Convergence Criteria

Introduction of an iterative scheme calls for the introduction of an iteration termination test. There are several

convergence criteria that can be applied.

13Which is not guaranteed to work if one takes into account bifurcation phenomena.14For more details see Crisfield (1991).15Say Idesired ≈ 3.



• Displacement Convergence Criteria. The change in the last correction δu of the state vector u, measured

in an appropriate norm, should not exceed a given tolerance ǫu. For example, if we use Euclidean norm16

the termination criteria can be written as:

‖δu‖scaled =

√

(δu)T

S (δu) ≤ ǫu (7.30)

Scaling matrix S is used in order to ensure that for a problem involving mixed variables17, all parameters

have the same dimension. Here, an obvious choice for the scaling matrix is S = diag(K−1t ). If, on the

other hand we don’t have mixed variables in state vector u then the simplest choice for scaling matrix is

S = I.

• Residual Convergence Criteria. Since the residual r(u, λ) measures the departure from the equilibrium path,

an appropriate convergence test would be to compare Euclidean norm of residual with some predefined

tolerance:

‖r(u, λ)‖scaled =

√

(r)T

S (r) ≤ ǫr (7.31)

Here, an obvious choice for scaling matrix is S = diag(Kt)

• Energy Based Convergence Criteria. The two previous convergence criteria can be combined in a single

work change criterion:

‖ (δu)T

(r) ‖ =

√

(δu)T

(r) ≤ ǫuǫr (7.32)

A word of caution is appropriate at this point. As pointed out by Crisfield (1991), it follows that:

‖ (δu)T

(r) ‖ = ‖ (δu)T (

K−1t Kt

)(r) ‖ = ‖ − (δu)

TKt (δu) ‖ ≤ ǫuǫr (7.33)

where the iterative change was (δu) = −K−1t r. It should be noted that equations (7.33) give a measure of

the ”stiffness” of Kt. This merely implies that a stationary energy position has been reached in the current

iterative direction, δu. This can occur when the solution is still far away from equilibrium.

Since u and r usually have physical dimensions, so do necessarily ǫu and ǫr. For a general purpose implemen-

tation of Newton–Raphson iteration this dependency on physical units is undesirable and it is more convenient

to work with ratios that render the ǫu and ǫr dimensionless. Displacement Convergence Criteria can be rendered

dimensionless by using ratio of scaled Euclidean norm of iterative displacement ‖δu‖scaled and scaled Euclidean

norm of total displacement ‖u‖scaled:

‖δu‖scaled‖u‖scaled

≤ ǫu (7.34)

The similar approach can be used for Residual Convergence Criteria:

‖r‖scaled‖rpredictor‖scaled

≤ ǫr (7.35)

16The so called 2–norm.17For example, if rotations and displacements are involved.



Another important thing to be considered is Divergence. The Newton–Raphson scheme is not guaranteed

to converge. Some sort of divergence detection scheme is therefor necessary in order to interrupt an erroneous

iteration cycle. Divergence can be diagnosed if either of following inequalities occur:

‖δu‖scaled‖u‖scaled

≥ gu (7.36)

‖r‖scaled‖rpredictor‖scaled

≥ gr (7.37)

where gu and gr are dangerous growth factors that can be set to, for example gu = gr = 100.

In some cases, the Newton–Raphson iteration scheme will neither diverge nor converge, but rather exhibit

oscillatory behavior. To avoid excessive bouncing around, a good practice is to put upper limit to the number of

iterations performed in one iteration cycle. Typical limits to the iteration number are 20 to 50.

7.5.5 The Algorithm Progress

The progress of the scheme will be briefly described, in relation with the Figure (7.1). The procedure starts from

a previously converged solution (u0, λ0fext). An incremental, tangential predictor step ∆u1,∆λ1 is obtained18

and the next point obtained is (u1, λ1fext). The first iteration would then use quadratic equation 7.17 where

constants a1, a2 and a3 should be computed with ∆uold = ∆u1 and ∆λold = ∆λ1, to calculate δλ1 and19

δu1 = −K−1t r (u1, λ1)+ δλ1K

−1t fext. After these values are calculated, the updating procedure20 would lead to:

∆λ2 = ∆λ1 + δλ1 and ∆u2 = ∆u1 + δu1 (7.38)

When added to the displacements u0 and load level λ0, at the end of the previous increment this process would

lead to the next point (u2, λ2fext).

The next iteration would then again use quadratic equation 7.17 where constants a1, a2 and a3 should be

computed with ∆uold = ∆u2 and ∆λold = ∆λ2, to calculate δλ3 and δu3 = δu + δλ2δut. After these values

are calculated, the updating procedure would lead to:

∆λ3 = ∆λ2 + δλ2 and ∆u3 = ∆u2 + δu2 (7.39)

When added to the displacements u0 and load level λ0, at the end of the previous increment this process would

lead to the next point (u3, λ3fext).

18As explained in Section (7.5.2).19From equation (7.12).20See (7.11) and (7.13)




Chapter 8

Solution of Dynamic Equations of

Motion (1989–2006–)

8.1 The Principle of Virtual Displacements in Dynamics

(see section 2.1 on page 23).

Argyris and Mlejnek (1991b)

8.2 Direct Integration Methods for the Equations of Dynamic Equilib-

rium

8.2.1 Newmark Integrator

The Newmark time integration method (Newmark, 1959) uses two parameters, β and γ, and is defined by the

following two equations:

n+1x = nx+ ∆t nx+ ∆2t [(1

2− β) nx+ β n+1x] (8.1)

n+1x = nx+ ∆t [(1 − γ) nx+ γ n+1x] (8.2)

These equatons give relationship between knowns variables at time step n to the unknown variables at next time

step n+ 1. Method is in general an implicit one, except when both β and γ are zero.

There are several possible implementation methods for Newmark Integrator. One of them, the predictor-

corrector method is defined throuh a predictors:

n+1x = nx+ ∆t nx+ ∆t2 (1

2− β) nx (8.3)

n+1x = nx+ ∆t (1 − γ) nx (8.4)

and then a correctors:

n+1x = n+1x + ∆t2 β n+1x (8.5)

n+1x = n+1x + ∆t γ n+1x (8.6)

169


The target is to find n+1x, n+1x, n+1x based on nx, nx, nx which satisfies

n+1R = M n+1x+ C n+1x+K ′ n+1x+ F (n+1x) − n+1f (8.7)

It can be solved using Newton integration method

[M + γ∆tC + β∆t2K

]∆x = −n+1R (8.8)

Equation 8.3 to 8.8 constitute an iterative solving procedure.

An alternative Newmark integration approach is to use displacement as the basic unknowns, and the following

difference relations are used to relate n+1x and n+1x to n+1x and the response quantities are

n+1x =γ

β∆t

(n+1x− nx

)+

(

1 − γ

β

)

nx+

(

1 − γ

2β

)

nx (8.9)

n+1x =1

β∆t2(n+1x− nx

)− 1

β∆tnx+

(

1 − 1

2β

)

nx (8.10)

The predictors are then:

n+1x⋄ = − γ

β∆tnx+

(

1 − γ

β

)

nx+

(

1 − γ

2β

)

nx (8.11)

n+1x⋄ = − 1

β∆t2nx− 1

β∆tnx+

(

1 − 1

2β

)

nx (8.12)

and the correctors:

n+1x = n+1x⋄ +γ

β∆tn+1x (8.13)

n+1x = n+1x⋄ +1

β∆t2n+1x (8.14)

The Newton integration method becomes[M

β∆t2+

C

γ∆t+K

]

∆x = −n+1R (8.15)

Equation 8.11 to 8.15 constitute an iterative solving procedure Argyris and Mlejnek (1991b).

If the parameters β and γ satisfy

γ ≥ 1

2, β =

1

4(γ +

1

2)2 (8.16)

it is unconditionally stable and second-order accurate. Any γ value greater than 0.5 will introduce numerical

damping. Well-known members of the Newmark time integration method family include: trapezoidal rule or

average acceleration method for β = 1/4 and γ = 1/2, linear acceleration method for β = 1/6 and γ = 1/2, and

(explicit) central difference method for β = 0 and γ = 1/2. If and only if γ = 1/2, the accuracy is second-order

Hughes (1987).

8.2.2 HHT Integrator

Numerical damping introduced in the Newmark time integration method will degrade the order of accuracy.

The Hilber-Hughes-Tailor (HHT) time integration α-method (Hilber et al., 1977), (Hughes and Liu, 1978a) and



(Hughes and Liu, 1978b) using an alternative residual form by introducing an addition parameter α to improve

the performance:

n+1R = M n+1x+ (1 + α)F (n+1x, n+1x) − αF (nx, nx) − n+1f (8.17)

but retaining the Newmark finite-difference formulas 8.1 and 8.2 or 8.9 and 8.10. If α = 0, Equation 8.17 is

reduced to Equation 8.7, and this special case of the HHT time integration method is exactly the Newmark time

integration method. Decreasing α value increase numerical dissipation (Hughes, 1987).

The iteration method for HHT time integration is similar to that of Newmark time integration. Due to the

change of Equation 8.17, Equations 8.8 and 8.15 respectively become

[M + (1 + α)γ∆tC + (1 + α)β∆t2K

]∆x = −n+1R (8.18)

for acceleration iteration and[M

β∆t2+

(1 + α)C

γ∆t+ (1 + α)

]

∆x = −n+1R (8.19)

for displacement iteration.

If the parameters α, β and γ satisfy

−1/3 ≤ α ≤ 0, γ =1

2(1 − 2α), β =

1

4(1 − α)2 (8.20)

it is unconditionally stable and second-order accurate (Hughes, 1987).

There are different denotation meaning of the parameter α, e.g. the parameter α in the HHT integrator codes

of OpenSees equals to the conventional α plus one.




Chapter 9

Finite Element Formulation for Porous

Solid–Pore Fluid Systems (1999–2005–)

(In collaboration with Dr. Zhao Cheng and Prof. Mahdi Taiebat)

9.1 General form of u–p–U Governing Equations

9.1.1 Background

For single-phase material encountered in structural mechanics, the response under ultimate load (failure) can be

predicted using very simple calculations, at least for static problems. But for soil mechanics, simple, limit-load

calculations can not be fully justified under static situation. However, for problems of soil dynamics, the use of

simplified methods is almost never admissible.

As the strength of the soil can be determined once the pore water pressures are known, it is possible to reduce

the soil mechanics problem to that of the behavior a single phase. Then we can use again the simple, single-phase

analysis approaches. Now we will introduce the concept of effective stress.

The relationship between effective stress, total stress and pore pressure is (assume tensile components of stress

as positive and compressive pressure, p is positive) (Zienkiewicz et al., 1999a)

σ′′

ij = σij + αδijp (9.1)

where σ′′

ij is effective stress tensor, σij is total stress tensor, δij is Kronecker delta. δij = 1, when i=j, and

δij = 0, when i 6= j. For isotropic materials,α = 1 −KT /KS . KT is the total bulk modulus of the solid matrix,

KS is the bulk modulus of the solid particle. For most of the soil mechanics problems, as the bulk modulus KS of

the solid particles is much larger that that of the whole material, α ≈ 1 can be assumed. Equation (9.1) becomes

σ′′

ij = σij + δijp (9.2)

9.1.2 Governing Equations of Porous Media

Before proceeding with the general equations, we introduce the associated notation. In this we have

173


• σij the total Cauchy stress in the mixture at any instant

• ui the displacement of the solid skeleton

• wi the pseudo-displacement of the fluid phase relative to the skeleton of solid

• p the pore water pressure

• εij = 12 (dui,j + duj,i) the strain increment of the solid phase

• ωij = 12 (dui,j − duj,i) the rotation increment of the solid phase

• ρ, ρs, ρf the densities of the mixture, and the solid phase and water respectively

• n the porosity

• θ = −wi,i the rate at which volume of water changes per unit total volume of mixture

With these definitions, and others introduced as necessary, we can write the governing equation of the coupled

system.

The equilibrium equation of the mixture

The overall equilibrium or momentum balance equation for the soil-fluid ’mixture’ can be written as (Zienkiewicz et al.,

1999a)

σij,j − ρui − ρf [wi + wjwi,j ] + ρbi = 0 (9.3)

Where ui is the acceleration of the solid part, bi is the body force per unit mass, wi + wjwi,j is the fluid

acceleration relative to the solid part, wi is local acceleration, wjwi,j is convective acceleration.

The underlined terms in the above equation represent the fluid acceleration relative to the solid and convective

terms of this acceleration. Generally this acceleration is so small that we shall frequently omit it. And for static

problems, equation(9.3) only consists of the first and last terms.

For fully saturated porous media (no air inside), from definition

ρ =Mt

Vt

=Ms +Mf

Vt

=Vsρs + Vfρf

Vt

=VfVtρf +

Vt − VfVt

ρs

= nρf + (1 − n)ρs

ρ = nρf + (1 − n)ρs (9.4)

where Mt, Ms and Mf are the mass of total, solid part and fluid part respectively. Vt, Vs and Vf are the volume

of total, solid part and fluid part respectively.



The equilibrium equation of the fluid

For the pore fluid, the equation of momentum balance can be written as (Zienkiewicz et al., 1999a)

− p,i −Ri − ρf ui − ρf [wi + wjwi,j ]/n+ ρfbi = 0 (9.5)

where R is the viscous drag forces. It should be noted that the underlined terms in (9.5) represent again the

convective fluid acceleration and are generally small. Also note that, throughout report, the permeability k is

used with dimensions of [length]3[time]/[mass], which is different from the usual soil mechanics convention, the

permeability has the same dimension of velocity, i.e. [length]/[time]. Their values are related by k = K/ρfg,

where g is the gravitational acceleration at which the is measured. Assuming the Darcy seepage law: nw = Ki,

here i is the head gradient. Seepage force is then R = ρfgi. R can be written as

Ri = k−1ij wj or Ri = k−1wi (9.6)

kij or k are Darcy permeability coefficients for anisotropic and isotropic conditions respectively.

Flow conservation equation

The final equation is supplied by the mass conservation of the fluid flow (Zienkiewicz et al., 1999a)

wi,i + αεii +p

Q+ n

ρfρf

+ s0 = 0 (9.7)

The first term of equation(9.7) is the flow divergence of a unit volume of mixture. The second term is the volume

change of the mixture. In the third term, Q is relative to the compressibility of the solid and fluid. The underlined

terms represent change of density and rate of volume expansion of the solid in case of thermal changes. They are

generally negligible.

1

Q≡ n

Kf+α− n

Ks

∼= n

Kf+

1 − n

Ks(9.8)

where Ks and Kf are the bulk moduli of the solid and fluid phases respectively.

Thus, we got the total mixture equilibrium equation (9.3), fluid equilibrium equation(9.5) and the flow con-

servation equation (9.7) for saturated soil. By omitting the convective acceleration (the underline terms in (9.3)

and (9.5)), density variation and the volume expansion due to the thermal change (the underline terms in (9.7)),

the equations of the total coupled system can be further simplified, they are summarized as below

σij,j − ρui − ρf wi + ρbi = 0 (9.9)

− p,i −Ri − ρf ui −ρf win

+ ρfbi = 0 (9.10)

wi,i + αεii +p

Q= 0 (9.11)



Figure 9.1: Fluid mechanics of Darcy’s flow (wi) versus real flow (Ui = wi/n).

9.1.3 Modified Governing Equations

Solid part equilibrium equation

First of all, a new variable Ui is introduced in place of the relative pseudo-displacement wi, it is defined as

Ui = ui + URi = ui +win

(9.12)

Insertion of definition (9.12) into equation (9.9)(9.10) and subtraction of [n × (9.10)] from equation (9.9)

leads to the equation of skeleton equilibrium

σij,j − ρui + ρbi + np,i + nRi + nρf ui − nρfbi = 0 (9.13)

By substituting ρ = (1 − n)ρs + nρf

σij,j − (1 − n)ρsui − nρf ui + (1 − n)ρsbi + nρfbi + np,i + nRi + nρf ui − nρfbi = 0

σij,j + np,i + (1 − n)ρsbi − (1 − n)ρsui + nRi = 0 (9.14)

By using the definition of effective stress (9.1),(9.13) becomes

σ′′

ij,j − (α− n)p,i + (1 − n)ρsbi − (1 − n)ρsui + nRi = 0 (9.15)

Fluid part equilibrium equation

The fluid part equilibrium equation can be obtained simply by [n× (9.10],i.e.

− np,i − nRi − nρf ui − ρf wi + nρf bi = 0

−np,i − nRi − nρf (ui +win

) + nρf bi = 0 (9.16)

From equation (9.12), we have

Ui = ui +win

(9.17)

so that equation (9.16) becomes:

− np,i + nρfbi − nρf Ui − nRi = 0 (9.18)



Mixture balance of mass

By differentiating equation (9.12) in time space, we have

wi,i = nUi,i − nui,i (9.19)

Notice that εii = ui,i, so that equation (9.19) becomes

wi,i = nUi,i − nεii (9.20)

By substituting (9.20) to (9.11), we obtain

nUi,i − nεii + αεii +p

Q= 0 (9.21)

or:

− nUi,i = (α− n)εii +1

Qp (9.22)

Thus we got the whole set of modified governing equations (9.15),(9.18) and (9.22). They are summarized as

below

σ′′

ij,j − (α− n)p,i + (1 − n)ρsbi − (1 − n)ρsui + nRi = 0 (9.23)

− np,i + nρf bi − nρf Ui − nRi = 0 (9.24)


Qp (9.25)

From the modified equation set (9.23),(9.24) and (9.25), we can see that only ui occurs in the first equation,and

only Ui in the second, thus leading to a convenient diagonal form in discretization.

Now we have a complete equation system given by (9.23),(9.24) and (9.25). With the basic definitions

introduced earlier, there are three essential

1. solid displacement u

2. pore pressure p

3. absolute fluid displacement U

The boundary conditions imposed on these variables will complete the problem. These boundary conditions

are: For the momentum balance part, on boundary Γt, traction ti(t)(or σijnj), where ni is the i-th component

of the normal to the boundary is specified. On boundary Γu, the displacement ui is given. For the fluid part,

again the boundary is divided into two parts. On Γp, the pressure p is specified, on Γw, the normal outflow wn

is specified. For impermeable boundary a zero value for the outflow should be specified.

The boundary conditions can be summarized below

Γ = Γt ∪ Γu

ti = σijnj = ti on Γ = Γw

ui = ui on Γ = Γu (9.26)



and

Γ = Γp ∪ Γw

p = p on Γ = Γp

nTw = wn on Γ = Γw (9.27)

9.2 Numerical Solution of the u–p–U Governing Equations

The solutions to the problems governed by the modified governing equation set (9.23),(9.24) and (9.25) can be

found by solving partial differential equations, which can be written as

AΦ +BΦ + L(Φ) = 0 (9.28)

where A, B are matrices of constant, and L is an operator involving spatial differentials. The dot notation

represents the time differentiation. Φ is a vector of dependent variables, in this problem, we say it represents the

displacement u or the pressure p.

The finite element solution of a problem will always proceed as the following procedures:

1. Discretize or approximate the unknown functions Φ by a finite set of parameters Φk and shape function

Nk. They are specified in space dimensions. Thus, we can write

Φ ∼= Φh =

n∑

k=1

NkΦk (9.29)

2. Insert the value of the approximating function Φ into the differential equations to obtain a residual, then

we can write a set of weighted residual equations in the form

∫ Ω

WTj (AΦh +BΦh + L(Φh))dΩ = 0 (9.30)

In finite element method, the weighting functions Wj are usually identical to the shape functions.

The solid displacement ui, the pore pressure p, and the absolute fluid displacement Ui can be approximated

using shape functions and nodal values.

ui = NuKuKi

p = NpKpK

Ui = NUKUKi (9.31)

where NuK , Np

K and NUK are shape functions for solid displacement, pore pressure and fluid displacement respec-

tively, uKi,pK ,UKi are nodal values of solid displacement, pore pressure and fluid displacement respectively.



9.2.1 Numerical Solution of solid part equilibrium equation

To obtain the numerical solution of the first equation, we premultiply (9.23) by NuK and integrate over the domain.

First term of (9.23) becomes∫

Ω

NuKσ

′′

ij,jdΩ =

∫

Γt

NuKnjσ

′′

ijdΓ −∫

Ω

NuK,jσ

′′

ijdΩ

=

∫

Γt

NuK(ti + niαp)dΓ −

∫

Ω

NuK,jσ

′′

ijdΩ

= (fu1 )Ki −∫

Ω

NuK,jDijmlεmldΩ

= (fu1 )Ki − [

∫

Ω

NuK,jDijmlN

uP,ldΩ]uPm

= (fu1 )Ki −KEPKimPuPm

= (fu1 )Ki −KEPKijLuLj

= (fu1) − KEP u (9.32)

where KEP is the stiffness matrix of the solid part,ni is the direction of the normal on the boundary.

Second term of (9.23) becomes

−∫

Ω

NuK(α− n)p,idΩ = −

∫

Γp

NuK(α− n)nipdΓ +

∫

Ω

NuK,i(α− n)pdΩ

= −∫

Γp

NuK(α− n)nipdΓ + [

∫

Ω

NuK,i(α− n)Np

MdΩ]pM

= −(fu4 )Ki + (G1)KiMpM

= −fu4

+ (G1)p (9.33)

Third term of (9.23) (Solid body force) is then∫

Ω

NuK(1 − n)ρsbidΩ = (fu5 )Ki (9.34)

Forth term of (9.23) can be written as

−∫

Ω

NuK(1 − n)ρsδij ujdΩ = −[

∫

Ω

NuK(1 − n)ρsδijN

uLdΩ]uLj

= −(Ms)KijLuLj

= Msu (9.35)

whereMs is the mass matrix of solid part. By substituting equations (9.6) and (9.12),last term of (9.23) (Damping

Matrix) becomes∫

Ω

NuKRidΩ =

∫

Ω

NuKnk

−1ij wjdΩ

=

∫

Ω

NuKn

2k−1ij UjdΩ −

∫

Ω

NuKn

2k−1ij ujdΩ

= [

∫

Ω

NuKn

2k−1ij N

UL dΩ]ULj − [

∫

Ω

NuKn

2k−1ij N

UL dΩ]uLj

= (C2)KijLULj − (C1)KijLuLj

= C2U − C1u (9.36)



Equation (9.23) becomes

−KEP u + fu1

− fu4

+ G1p + fu5

− Msu + C2U − C1u = 0 (9.37)

or

KEP u − G1p − C2U + C1u + Msu = fs (9.38)

where

fs = fu1 − fu4 + fu5 (9.39)

in index form

KEPKijL − (G1)KiLpL + (C2)KijLULj − (C1)KijLuLj + (Ms)KijLuKi = (fs)Ki (9.40)

Where

KEP = (KEP )KimP =

∫

Ω

NuK,jDijmlN

uP,ldΩ

G1 = (G1)KiM =

∫

Ω

NuK,i(α− n)Np

MdΩ

C2 = (C2)KijL =

∫

Ω

NuKn

2k−1ij N

UL dΩ

C1 = (C1)KijL =

∫

Ω

NuKn

2k−1ij N

uLdΩ

Ms = (Ms)KijL =

∫

Ω


uLdΩ

f = (fs)Ki = (fu1 )Ki − (fu4 )Ki + (fu5 )Ki (9.41)

9.2.2 Numerical Solution of fluid part equilibrium equation

From equations (9.6) and (9.12), we obtain

Ri = n2k−1ij (Uj − uj) (9.42)

By substituting (9.42) into equation (9.24), we obtain

− np,i + nρfbi − nρf Ui − n2k−1ij (Uj − uj) = 0 (9.43)

By premultiplying (9.43) by NUK and integrating over the domain

First term of (9.43) becomes

−∫

Ω

nNUKp,idΩ = −

∫

Γp

nNUKnipdΓ +

∫

Ω

nNUK,ipdΩ

= −(f1)Ki + [

∫

Ω

nNUK,iN

pMdΩ]pM

= −(f1)Ki + (G2)KiMpM

= −(f1)Ki + (G2)p (9.44)

Second term of (9.43) is then∫

Ω

NUKρfbidΩ = (f2)Ki (9.45)



Third term of (9.43) (Lumped mass matrix by multiplying δij) becomes

−∫

Ω

NUKnρfδijUjdΩ = −[

∫

Ω

NUKnρfδijN

UL dΩ]ULj

= −(Mf )KijLULj

= −Mf U (9.46)

Forth term of (9.43) becomes

−∫

Ω

NUKn

2k−1ij UjdΩ +

∫

Ω

NUKn

2k−1ij ujdΩ = −[

∫

Ω

NUKn

2k−1ij N

UL dΩ] ˙ULj (9.47)

+[

∫

Ω

NUKn

2k−1ij N

uLdΩ] ˙uLj

= −(C3)KijL˙ULj + (C2)

TLjiK

˙uLj

= C3U + CT2

u (9.48)

Equation (9.43) becomes

−f1 + G2p + f2 − Mf U − C3U + CT2

u = 0 (9.49)

or

−G2p − CT2

u + C3U + Mf U = ff (9.50)

where

ff = f2 − f1 (9.51)

in index form

− (G2)KiMpM − (C2)TLjiK uLj + (C3)KijLULj + (Mf )KijLULj = (ff )Ki (9.52)

where

(ff)Ki = (f1)Ki − (f2)Ki

G2 = (G2)KiN =

∫

Ω

nNUK,iN

pMdΩ

CT2

= (CT2 )KijL =

∫

Ω

NUKn

2k−1ij N

uLdΩ

C3 = (C3)KijL =

∫

Ω

NUKn

2k−1ij N

UL dΩ

Mf = (Mf )KijL =

∫

Ω

NUKnρfδijN

UL dΩ (9.53)

9.2.3 Numerical Solution of flow conservation equation

By integrating (9.25) in time and noticing εii = ui,i, we can obtain


Qp (9.54)

By multiplying (9.54) by NpM and integrating over the domain, first term of (9.54) becomes

− [

∫

Ω

NpMnN

UL,jdΩ]ULj = −(G2)MLjULi = −GT

2U (9.55)



Second term of (9.54) is

∫

Ω

NpM (α− n)ui,idΩ = [

∫

Ω

NpM (α− n)Nu

L,jdΩ]uLj

= (G1)LjMuLj

= GT1

u (9.56)

Third term of (9.54) becomes

[

∫

Ω

NpN

1

QNpMdΩ]pN = PNMpM = Pp (9.57)

The equation (9.54) becomes

GT2

U + GT1

u + Pp = 0 (9.58)

in index form

(G2)LiKULi + (G1)LiKuLi + PKLpL = 0 (9.59)

9.2.4 Matrix form of the governing equations

The numerical forms of governing equations (9.38),(9.50) and (9.58) can be written together in the matrix form

as

Ms 0 0

0 0 0

0 0 Mf

u

p

U

+

C1 0 −C2

0 0 0

−CT2 0 C3

u

p

U

+

KEP −G1 0

−GT1 −P −GT20 −G2 0

u

p

U

=

fs

0

ff

(9.60)

or in index form

(Ms)KijL 0 0

0 0 0

0 0 (Mf )KijL

uLj

pN

ULj

+

(C1)KijL 0 −(C2)KijL

0 0 0

−(C2)LjiK 0 (C3)KijL

uLj

pN

ULj

+

(KEP )KijL −(G1)KiM 0

−(G1)LjM −PMN −(G2)LjM

0 −(G2)KiL 0

uLj

pM

ULj

=

fsolid

Ki

0

ffluid

Ki

(9.61)



where

Ms = (Ms)KijL =

∫

Ω


uLdΩ

Mf = (Mf )KijL =

∫

Ω

NUKnρfδijN

UL dΩ

C1 = (C1)KijL =

∫

Ω

NuKn

2k−1ij N

uLdΩ

C2 = (C2)KijL =

∫

Ω

NuKn

2k−1ij N

UL dΩ

C3 = (C3)KijL =

∫

Ω

NUKn

2k−1ij N

UL dΩ

KEP = (KEP )KijL =

∫

Ω

NuK,mDimjnN

uL,ndΩ

G1 = (G1)KiM =

∫

Ω

NuK,i(α− n)Np

MdΩ

G2 = (G2)KiM =

∫

Ω

nNUK,iN

pMdΩ

P = PNM =

∫

Ω

NpN

1

QNpMdΩ

(9.62)

fsolid

Ki = (fu1 )Ki − (fu4 )Ki + (fu5 )Ki

ffluid

Ki = −(fU1 )Ki + (fU2 )Ki

(fu1 )Ki =

∫

Γt

NuKnjσ

′′

ijdΓ

(fu4 )Ki =

∫

Γp

NuK(α− n)nipdΓ

(fu5 )Ki =

∫

Ω

NuK(1 − n)ρsbidΩ

(fU1 )Ki =

∫

Γp

nNUKnipdΓ

(fU2 )Ki =

∫

Ω

nNUKρf bidΩ (9.63)

Here we have Nu,Np,NU as the shape functions of skeleton, pore pressure and fluid, ρ,ρs,ρf are the density of the

total, the solid and the fluid phases, respectively, n is the porosity, and by its definition ρ = (1− n)ρs + nρf , the

symbol ni is the direction of the normal on the boundary, u is the displacement of the solid part, p is pore pressure

and U is the absolute displacement of the fluid part. Equation (9.60) represents the general form (u− p−U) for

coupled system which can be written in a familiar form as

Mx + Cx + Kx = f (9.64)

where x represents the generalized unknown variable. Solution of this equation for each time step will render

unknown field for given initial and boundary conditions.



9.2.5 Choice of shape functions

Isoparametric elements are used in previous sections, where the coordinates are interpolated using the same shape

functions as for the unknown. This mapping allows the use of elements of more arbitrary shape than simple forms

such as rectangles and triangles. But in static or dynamic undrained analysis the permeability (and compressibility)

matrices are zero, i.e.(Q→ ∞,and P → 0),resulting in a zero-matrix diagonal term in the equation(9.61).

The matrix to be solved is the same as that in the solutions of problems of incompressible elasticity or fluid

mechanics. Actually a wide choice of shape functions is available if the limiting(undrained) condition is never

imposed. Due to the presence of first derivatives in space in all the equations it is necessary to use ”Co-continuous”

interpolation functions and the suitable element forms are shown in Fig.9.2.

Figure 9.2: Shape functions used for coupled analysis, displacement u and pore pressure p formulation

9.3 Examples

In the derivation of the governing equations of the governing equations of the coupled system, there are no

limitations on the compressibility of the fluid involved, so a general complete u-p-U form is presented. While

in the general soil mechanics, the notion of fluid incompressibility are often assumed. As a consequence of the

incompressible fluid, the compressive waves in the fluid can be neglected and there is only one dilational wave

existing in the mixture. And the pore pressure generation in the compressible fluid usually is slightly higher than

that of the incompressible fluid because of the existence of the oscillatory waves (Zienkiewicz and Shiomi, 1984).

Special attention should also be paid to the concept of incompressible fluid. The compressibility of the porous

media is governed by the compressibility of the solid skeleton, permeability and volume factions, instead of the

fluid compressibility.

Before proceeding to the analysis, we have to make the following assumptions: For high-frequency components,

the permeability remains constant, the dependency of the permeability on the frequency should be neglected.

Unless specified, all the models in this report are elastic isotropic.



In order to illustrate the performance of the formulation and versatility of the numerical implementation,

several elastic one-dimensional problems are presented in the following sections, including:

1. Drilling of a borehole

2. The case of a spherical cavity

3. Consolidation of a soil layer

4. Line injection of a fluid in a reservoir

5. Shock wave propagation

9.3.1 Verification Example: Drilling of a well

The Problem Let us consider an infinite half space domain composed of an isotropic, homogeneous and sat-

urated thermoporoelastic material. At its reference state, it is assumed that the temperature, fluid pressure and

stress fields are uniform and equal respectively, to T0, p0 and σ0 = σ01(with σ0 < 0). At time = 0, an infinite

cylinder of radius r0 is instantaneously drilled parallel to the vertical axis Oz. It is filled with a fluid of the same

nature as that saturating the porous medium but at a different pressure and temperature at the values of p1 and

T1 respectively. The interface r = r0 between the well and the porous medium is assumed to be in thermodynamic

equilibrium.

In cylindrical coordinates (r, θ, z), the boundary conditions can be summarized as follows (see Fig.9.3):

t ≤ 0 → σ0 = σ01 p(r) = p0 T (r) = T0 (9.65)

t > 0 → σrr(r0) = −p1 σrθ(r0) = σrz(r0) = 0

σrr(r → ∞) → σ0 σrθ(r → ∞) = σrz(r → ∞) → 0

p(r0, t) = p1 p(r → ∞) → p0

T (r0) = T1 T (r → ∞) → T0 (9.66)

Analytical Solution Since the well is assumed to be infinite long in its vertical axis Oz, the analysis is performed

under plane strain hypothesis(ǫzz = 0). Therefore,

ξ = ξr(r)er p = p(r) T = T (r) (9.67)

in which ξr is the radial displacement. In cylindrical coordinates, Eqn. 9.103 yields

ǫrr =∂ξr

∂rǫθθ =

ξr

rother ǫij = 0 (9.68)

Based on the constitutive equations from Coussy (1995), it follows that

σrr = σ0 + λ0(∂ξr

∂r+ξr

r) + 2µ

∂ξr

∂r− b(p− p0) − 3αK0(T − T0) (9.69)

σθθ = σ0 + λ0(∂ξr

∂r+ξr

r) + 2µ

ξr

∂r− b(p− p0) − 3αK0(T − T0) (9.70)



Figure 9.3: Boundary Conditions for Drilling of a Borehole

σzz = σ0 + λ0(∂ξr

∂r+ξr

r) − b(p− p0) − 3αK0(T − T0) (9.71)

other σij = 0 (9.72)

Finally combined with the Eqns. 9.69-9.72, it yields the near field or long-term solution (Coussy, 1995)

ξr =σ0 + p1

2µ

r02

r+r0[b(p1 − p0) + 3α0K0(T1 − T0)]

2(λ0 + 2µ)(r

r0− r0

r) (9.73)

σrr = −p1r0

2

r2− σ0 −

µ[b(p1 − p0) + 3αK0(T1 − T0)]

λ0 + 2µ(1 − r0

2

r2) (9.74)

σθθ = (2σ0 + p1)r0

2

r2− µ[b(p1 − p0) + 3αK0(T1 − T0)]

λ0 + 2µ(1 +

r02

r2) (9.75)

σzz = σ0 − 2µ

λ0 + 2µ[b(p1 − p0) + 3αK0(T1 − T0)] (9.76)

And the diffusion process can be achieved if that the time are large enough with respect to the characteristics

diffusion time relative to point r. When the boundary conditions for r = r0 in fluid pressure and temperature

which are p = p1 and T = T1 apply for the whole model, the following equations correspond to the undrained

solution of the instantaneous drilling of a borehole in an infinite elastic medium.

ξr =σ0 + p1

2µ

r02

rσrr = σ0 − (σ0 + p1)

r02

r2

σθθ = σ0 + (σ0 + p1)r0

2

r2σzz = σ0 (9.77)



Parameter Symbol Value Units

Poisson Ratio ν 0.2 -

Young’s Modulus E 1.2E+6 kN/m2

Solid Bulk Modulus Ks 3.6E+7 kN/m2

Fluid Bulk Modulus Kf 1.0E+17 kN/m2

Solid Density ρs 2.7 ton/m3

Fluid Density ρf 1.0 ton/m3

Porosity n 0.4 -

Table 9.1: Material Properties used to study borehole problem

Discussion of the Results As the problem is Axisymmetric, we construct the model as a quarter of a donut.

The inside diameter of the donut is 10 cm and the outside diameter is 1 m. To accommodate both the plain

strain hypothesis and the geometry of the element for finite element, the thickness of the model is chosen to

be 5 cm. The final mesh is generated as Fig.9.4. And the boundary conditions is as follows: As a consequence

of plain strain problem, all the movements for solid and fluid in vertical direction Oz are suppressed; the solid

and fluid displacement for the nodes along the X axis and Y axis are fixed in Y and X direction respectively for

the reason of axisymmetry; the nodes along the outside perimeter are fixed in the solid and fluid displacement

with the assumption of infinite medium. the pressure is translated into nodal forces and applied on the nodes

along the inside perimeter. For simplicity, the hydrostatic stress σ0 is equal to zero and with the assumption of

thermodynamic equilibrium through the process, the temperature factor can be neglected. Also the initial fluid

pressure p0 is set to be 0 kPa. The analytical solution is studied below using the following set of parameters

shown in Table 9.1.

Figure 9.4: The mesh generation for the study of borehole problem

In the analysis, ten loading cases for final fluid pressure from 10 kPa to 100 kPa are studied. And by



manipulating the permeability, it is possible to investigate both the drained behavior and undrained behavior. For

the drained behavior, we choose the permeability as k = 3.6 × 10−4m/s, which is a typical value for sand, the

comparison between the close solution and experimental result is shown in Fig.9.5. From the results, we can see

that along the inside perimeter, the close solution and experimental result provide very good agreement to each

other. But as the increase of the radius, we can see the analytical solution is getting more and more distant from

the experimental results. In another word, the analytical solution can be interpreted as that with the increase from

the loading surface, the radial displacement is larger. This is unreasonable in the point of view in soil mechanics.

While the experimental result show the effect that with the increase of the radial distance, the radial displacement

is decreasing.

(The input files for the above examples are:

• ValidationExamples/WellDrilling/Drained

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

2

2.5

3

3.5x 10

−3

Pressure (kPa)

Rad

ial D

ispl

acem

ent (

cm)

r=0.1m, close solutionr=0.1m, experimentalr=0.5m, close solutionr=0.5m, experimentalr=0.9m, close solutionr=0.9m, experimental

Figure 9.5: The comparison of radial solid displacement between analytical solution and experimental result for

drained behavior

For the undrained behavior, the permeability of k = 3.6 × 10−8m/s is selected as a representative value for

typical clayey soil. The comparison between the close solution and experimental result is provided as well. From

the Fig.9.6 we can see that, the analytical solution is linearly away from the experimental result by a ratio of

approximately 1.6. It should also be noticed that the close solution of the drained and undrained behavior for

the nodes along the inside perimeter are exactly the same, which is contradictory to the definition of drained and

undrained behavior. For the drained behavior, as the water easily dissipate from the soil body, the problem can

be treated with the knowledge of continuum mechanics using the parameters of the solid skeleton. While for


http://sokocalo.engr.ucdavis.edu/~jeremic/Example_Input_Files/u-p-U/ValidationExamples/WellDrilling/Drained


the undrained behavior, with the involvement of the pore water, the elastic parameters for the mixture should be

different, so the response will not be the same as well. As a result of this, the experimental results give a more

reasonable conclusion.


• ValidationExamples/WellDrilling/Undrained

0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1x 10

−3

Pressure (kPa)

Rad

ial D

ispl

acem

ent (

cm)



undrained behavior

As for the drained behavior, the fluid totally flows out of the soil body and all excessive pore pressure

dissipates, there is small coupling between the solid and fluid phase. We can use the continuum mechanics to

treat this problem. Here introduces a problem of an infinite cylindrical tube, with the inner radius R1 and outer

radius R0, subjected to an internal pressure P1 and an external pressure P2. The displacement field as follows

(S.Timoshenko and D.H.Young, 1940):

ξr =R1

2P1

2(R02 −R1

2)(

r

λ+ µ+R0

2

µr) (9.78)

With P0 = 0 and take the limit of R0 → ∞, we can obtain the following equation:

ξr =P1

2µ

R12

r(9.79)

which is identical to Eq.9.74. Also to minimize the effect of infinite boundary, we introduce the result from

another model which is exactly the same as the previous one besides the expansion of the outer radius to 30m. At

the final fluid pressure of 50 kPa, the results are shown in Fig.9.7. From the plot we can make a conclusion that


http://sokocalo.engr.ucdavis.edu/~jeremic/Example_Input_Files/u-p-U/ValidationExamples/WellDrilling/Drained


the undrained analytical solution from Coussy (1995) is actually the drained solution and the undrained solution

still needs to be investigated.


• ValidationExamples/WellDrilling/Comparison

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6x 10

−4

Radius (m)

Rad

ial D

ispl

acem

ent (

cm)

Coussy Undrained SolutionTimoshenko SolutionExperimental(1m Boundary)Experimental(30m Boundary)

Figure 9.7: The comparison of radial solid displacement between two analytical solutions and expanded boundary

9.3.2 Verification Example: The Case of a Spherical Cavity

The Problem Considering a medium composed of an isotropic, homogeneous, saturated thermoporoelastic

material. In its initial state, it is assumed that the temperature, fluid pressure and stress fields are uniform and

equal respectively, to T0, p0 and σ0 = σ01(with σ0 < 0). At time t= 0, a spherical cavity of radius r0 is

immediately drilled and filled with the same saturating fluid in the medium. For t > 0, the temperature and the

pressure of the fluid are kept constant with the value of T1 and p1 respectively. The interface r = r0 between the

well and the porous medium is assumed to be in the thermodynamic equilibrium.

In spherical coordinates (r, θ, ϕ), the boundary conditions can be summarized as follows:


http://sokocalo.engr.ucdavis.edu/~jeremic/Example_Input_Files/u-p-U/ValidationExamples/WellDrilling/Comparison


t ≤ 0 → σ0 = σ01 p(r) = p0 T (r) = T0 (9.80)

t > 0 → σrr(r0) = −p1 σrθ(r0) = σrϕ(r0) = 0

σrr(r → ∞) → σ0 σrθ(r → ∞) = σrϕ(r → ∞) → 0

p(r0, t) = p1 p(r → ∞) → p0

T (r0) = T1 T (r → ∞) → T0 (9.81)

Strictly speaking, the expressions for r → ∞ are not boundary conditions. They are complementary conditions

to be satisfied by the solution. It is used to model that at the point far from the disturbed area, the state of the

medium are held as its initial state.

Analytical Solution This is a problem of spherical symmetry. The radial displacement is the only non-zero

displacement and all the fields are r and t dependent. Therefore,

ξ = ξr(r)er p = p(r) T = T (r) (9.82)

in which ξr is the radial displacement. In spherical coordinates, Eqn.9.82 yields

ǫrr =∂ξr

∂rǫθθ =

ξr

rother ǫij = 0 (9.83)


σrr = σ0 + λ0(∂ξr

∂r+ξr

r) + 2µ

∂ξr

∂r− b(p− p0) − 3αK0(T − T0) (9.84)

σθθ == σϕϕ = σ0 + λ0(∂ξr

∂r+ξr

r) + 2µ

ξr

∂r− b(p− p0) − 3αK0(T − T0) (9.85)

other σij = 0 (9.86)

Finally combined with the Eqns. 9.83-9.86, it yields the near field or long-term solution (Coussy, 1995)

ξr =σ0 + p1

4µ

r03

r2+r0[b(p1 − p0) + 3αK0(T1 − T0)]

2(λ0 + 2µ)(1 − r0

2

r2) (9.87)

σrr = −p1r0

3

r3+ σ0(1 − r0

3

r3) − 2µ[b(p1 − p0) + 3αK0(T1 − T0)]

λ0 + 2µ(r0r

− r03

r3) (9.88)

σθθ = σϕϕ = p1r0

3

2r3− σ0(1 +

r03

2r3) − µ[b(p1 − p0) + 3αK0(T1 − T0)]

λ0 + 2µ[r0r

(1 +r0

2

r2)]

(9.89)

And the diffusion process can be achieved if that the time are large enough with respect to the characteristics

diffusion time relative to point r. When the boundary conditions for r = r0 in fluid pressure and temperature

which are p = p1 and T = T1 apply for the whole model, the following equations correspond to the undrained

solution of the instantaneous drilling of a borehole in an infinite elastic medium.

ξr =σ0 + p1

4µ

r03

r2σrr = −p1

r03

r3+ σ0(1 +

r03

r3)

σθθ = σϕϕ = p1r0

3

2r3+ σ0(1 +

r03

2r3) (9.90)



Discussion of the Results The model is constructed as a quarter of a half ball. The cavity radius is 10cm. As

the outside boundary is fixed, to minimize the possibility of the sudden increase of the fluid bulk modulus, the

outside radius of the sphere is set to be 2 m. The final mesh is generated as Fig.9.8. And the following boundary

conditions apply: The nodes on XZ and Y Z plane are fixed for solid and fluid displacement in Y and X direction

respectively; the vertical solid and fluid displacement for the nodes on the XY plane are suppressed; for the nodes

along the outside surface, to satisfy the complementary conditions, all the solid and fluid displacements are set to

be zero as well. The pressure is translated in to nodal forces and applied in the radial direction. For simplicity, the

hydrostatic stress σ0 is equal to zero and with the assumption of thermodynamic equilibrium through the process,

the temperature factor can be neglected. Also the initial fluid pressure p0 is set to be 0 kPa. The analytical

solution is studied below using the following set of parameters shown in Table 9.2.


• ValidationExamples/SphericalCavity/Drained

• ValidationExamples/SphericalCavity/Undrained

Figure 9.8: The mesh generation for the study of spherical cavity

As the same procedure in the previous drilling of borehole problem, we compared both the drained and

undrained behavior. The drained and undrained behavior are tested by the permeability of k = 3.6 × 10−4m/s

and k = 3.6 × 10−8m/s respectively. In drained behavior, we can see along the cavity surface, the experimental

result of the radial displacement match the analytical solution very well. While with the increase of the radius,

the decrease of the radial displacement for close solution is much smaller that of the experimental results. For

the undrained behavior, we can see the radial displacement of the experimental results are always smaller than

the close solution. Again it should be noted that the close solutions for the drained and undrained behavior

along the cavity surface are exactly the same. This can be explained in the same way as the previous drilling

of the borehole problem. When the experimental results from drained behavior are compared with the analytical

undrained solution, it is observed they provide good agreement to each other as well.


http://sokocalo.engr.ucdavis.edu/~jeremic/Example_Input_Files/u-p-U/ValidationExamples/SphericalCavity/Drained

http://sokocalo.engr.ucdavis.edu/~jeremic/Example_Input_Files/u-p-U/ValidationExamples/SphericalCavity/Undrained









Porosity n 0.4 -

Table 9.2: Material Properties used to study spherical cavity problem

0 10 20 30 40 50 60 70 80 90 1000

1

2

3

4

5

6x 10

−4

Pressure (kPa)

Rad

ial D

ispl

acem

ent (

cm)



drained behavior

9.3.3 Verification Example: Consolidation of a Soil Layer

The Problem The consolidate process can be defined as follows: When a soil layer is subjected to an external

loading, immediately the water will alone sustain this load and cause the build-up the excessive pore water pressure.

In the progress of the flow of the water to the surface, the load is gradually transferred to the soil skeleton and

the excessive pore water pressure will dissipate. At the same time, the settlement of the soil layer occurs. As

settlement is usually a major concern in geotechnical engineering, this is a key problem in soil mechanics.

Consider a soil layer composed of an isotropic, homogeneous and saturated thermoporoelastic material. The

layer has a thickness of h in the Oy direction and of infinite extent in the two other directions Ox and Oy. The



0 1 2 3 4 5 6 7 8 9 10

x 104

0

1

2

3

4

5

6x 10

−4

Pressure (kPa)

Rad

ial D

ispl

acem

ent (

cm)



undrained behavior

layer is underlain by a rigid and impervious base at y = 0. And the top surface at y = h is so perfectly drained

that the pore pressure is held constant as zero.

At the initial state of the soil layer, the thermal effects are neglected so that the boundary conditions follow

that:

t ≤ 0 → y = h p = 0

y = 0∂p

∂z= 0 (9.91)

At time t = 0, tan instantaneous vertical load −ey is suddenly applied on the top surface y = h, the induced

boundary conditions require that

t > 0 → y = h σey = −ey (9.92)

The undeformability of the substratum reads

y = 0 ξ = 0 (9.93)



0 1 2 3 4 5 6 7 8 9 10

x 104

0

1

2

3

4

5

6x 10

−4

Pressure (kPa)

Rad

ial D

ispl

acem

ent (

cm)

r=0.1m, undrained(c)r=0.1m, drained(e)r=0.15m, undrained(c)r=0.15m, drained(e)r=0.25m, undrained(c)r=0.25m, drained(e)

Figure 9.11: The comparison of radial solid displacement between analytical solution for undrained behavior and

experimental result for drained behavior

Figure 9.12: Consolidation of a Soil Layer



The impermeability implies

y = 0 − w · ey = −wy = 0 (9.94)

The problem is then to determine the new fields of fluid pressure, stress and displacement induced by the external

loading.

Analytical Solution Since this is a one-dimensional problem, the only non-zero displacement is the vertical

displacement ξy. But in particular the fluid pressure depends only on y and t.

ξ = ξy(y, t)ey p = p(y, t) (9.95)


σyy = (λ0 + 2µ)∂ξy

∂y+ bp

σxx = σzz =λ0

(λ+ 2µ)σyy −

2µb

λ0 + 2µp (9.96)

And because the fluid pressure p must be an ordinary function of time t, although the derivative of p is infinite

at time t = 0 according to the consolidation equation (Coussy, 1995), the discontinuity of the fluid pressure p at

time t = 0 must satisfy

p(y, t = 0+) = η =ν − ν0

(1 − ν)(1 − 2ν0)b(9.97)

where ν and ν0 are the drained and undrained Poisson ratio, respectively. For time t > 0, the vertical stress

σyy = − is constant in time and space, therefore the diffusion equation reads

t > 0 cm∂2

∂y2p =

∂

∂tp (9.98)

Collecting the above results, finally the fluid pressure reads

p(y, t) = η∞∑

n=0

4(−1)n

π(2n+ 1)cos[

(2n+ 1)π

2

y

h]exp[− (2n+ 1)2π2

4

t

τ] (9.99)

Each term of the series decreases exponentially with respect to the ratio tτ , in which τ is a characteristics

consolidation time



τ =h2

cmcm = kM

λ0 + 2µ

λ+ 2µ(9.100)

where λ and λ0 are the drained and undrained Lame coefficient, respectively. Given by the Eqn.9.96, the only

non-zero displacement ξy satisfies

∂ξy

∂y=

1

λ0 + 2µ(σyy + bp) (9.101)

By substituting the value of − of the vertical stress and expression of (9.101), the series converges and it can

integrated term by term yielding

ξy(y, t) =

λ0 + 2µ(y

h+

8ηb

π2)

∞∑

n=0

(−1)n

(2n+ 1)2sin[

(2n+ 1)π

2

y

h]exp(− (2n+ 1)2π2

4

t

τ]

(9.102)

Using Eqn.9.102 and substitute y = h, the settlement can be expressed as

s(t) = s∞ + (s0+ − s∞)∞∑

n=0

8

π2(2n+ 1)2exp− (2n+ 1)2π2

4

t

τ

s0+ =h

λ+ 2µs∞ =

h

λ0 + 2µ(9.103)

Discussion of the Results Twenty 1D finite elements are used to model the horizontal layer. The height of

the soil column is 20 m and the height of each element is 1 m. The material properties, shown in Table 9.3, are

chosen as representative values for natural soil deposit. A uniform vertical pressure of 200 kPa is applied on the

top surface of the soil column. The following boundary conditions apply: As the bottom of the soil column is

modelled as an undeformable and impermeable layer, both the solid and fluid displacements are fixed. The pore

pressure is kept constant as zero at the top surface of the soil column because of the perfectly drained condition.

For the reason of 1D consolidation problem, all the lateral movement of the solid and fluid phase are suppressed

so that the vertical displacement is the only non-zero displacement for the intermediate nodes. To capture both

the long term(t >0.1 sec) and short term (t <0.1 sec) response of the soil column, two different time steps are

adopted: 0.1 sec and 0.005 sec. In order to observe the dissipation of the excessive pore water pressure in a

reasonable and convenient period, we select k = 3.6 × 10−4m/s as the value for the permeability. To cure the

artificial oscillation, some numerical damping is introduced into the analysis by using γ = 0.6 and β = 0.3025 in

the Newmark algorithm within the OpenSees.

Based on the above parameters, the other relative parameters can be calculated as follows:

The bulk modulus of the mixture:

K =E

3(1 − 2ν)= 6.67 × 105kPa










Porosity n 0.4 -

Table 9.3: Material Properties used to study consolidation of a soil layer

λ =Eµ

(1 − ν)(1 − 2ν)= 5 × 105kPa µ =

E

2(1 + ν)= 5 × 105kPa

The Biot coefficient:

b = 1 − K

Ks= 0.98

The undrained bulk modulus of the mixture:

N =Ks

b− n= 6.19 × 107kPa M =

KfN

Kf +Nn= 6.19 × 107kPa

Ku = K + b2M = 6.03 × 107kPa

The diffusion coefficient and characteristic time of consolidation:

cf =kM

γw

K + 4µ3

Ku + 4µ3

= 49.7m2/s τ =h2

cf= 8.04s

In Fig.9.13, the normalized fluid pressure p is plotted against the location z for various normalized times t.

For normalized time t = 0.001 (natural time t=0.008 sec), only the nodes close to the top free flow surface

display the dissipation of the pore pressure, for normalized depth z < 0.15, the pore pressure equals to the applied

external pressure. The experimental result provide good agreement with the analytical solution except at the end

of the curve, the reason could be the small time step we adopt in this analysis to show the undrained behavior.

With the increase of the normalized time, we can clearly see the tendency of the dissipation of the water. At

normalized time t = 1.0 (natural time t= 8 sec), the maximum normalized pore pressure is only about 0.11. We

can say that our model can effectively show the process of the dissipation of the pore pressure. Now let us turn

to the settlement(see Fig.9.14), the experimental results match the analytical solution very well, and at natural

time t= 20 sec, the soil layer almost finish the primary consolidation and reach the primary settlement.


• ValidationExamples/Consolidation


http://sokocalo.engr.ucdavis.edu/~jeremic/Example_Input_Files/u-p-U/ValidationExamples/Consolidation


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalized Pressure

Nor

mal

ized

Dep

thNormalized Time = 0.001(c)Normalized Time = 0.001(e)Normalized Time = 0.1(c)Normalized Time = 0.1(e)Normalized Time = 0.5(c)Normalized Time = 0.5(e)Normalized Time = 1.0(c)Normalized Time = 1.0(e)

Figure 9.13: The comparison between analytical solution and experimental results for the normalized p during the

consolidation process against normalized depth z = z/h for various normalized t = cf t/h2

0 2 4 6 8 10 12 14 16 18 20

0

0.5

1

1.5

2

2.5

3

x 10−3

Time (sec)

Set

tlem

ent (

m)

Analytical SolutionExperimental Results

Figure 9.14: The comparison between analytical solution and experimental results for the settlement



9.3.4 Verification Example: Line Injection of a fluid in a Reservoir

The Problem Liquid water is usually injected into a reservoir from a primary well in order to recover the oil

from a secondary well in petroleum engineering. This induces a problem of injecting a fluid into a cylindrical well

of negligible dimensions.

Consider a reservoir of infinite extent composed of an isotropic, homogeneous and saturated poroelastic

material. Through a cylindrical well of negligible dimensions, the injection of the same fluid is performed in all

directions orthogonal to the well axis forming the Oz axis of coordinates. As a result of the axisymmetry and

cylindrically infinite, all quantities spatially depends on r only. The injection starts at time t = Γ and stops at

time t = Γ. The flow rate of fluid mass injection is constant and equal to q.As a finite amount of Ω of fluid mass

is injected instantaneously(i.e. Γq → ΩasΓ → 0).

Analytical Solution This is a problem of cylindrically symmetry. Consequently the cylindrical coordinates(r,θ,z)is

adopted. The vector of relative flow of fluid mass w reads

w = w(r, t)er (9.104)

where er is the unit vector along the radius. Using the fluid mass balance relationship, it yields

∫ r

0

∂m

∂t(r, t)2πrdr = q − 2πrw(r, t) ∀r, t (9.105)

In addition, we require the fluid flow to reduce to zero infinitely far from the well

rw → 0 r → ∞ t <∞∫ ∞

0

∂m

∂t(r, t)rdr =

q

2r∀0 < t <∞ (9.106)

Based on above Eqs.9.105-9.106, the radial displacement is derived in the form

p =Ω

4πρfl0 ktexp(− r2

4cmt)

ξr =bMΩ

2πρfl0 (λ+ 2µ)r[1 − exp(− r2

4cmt)] (9.107)

Using the constitutive equation, the stress field can be derived as follows:

σrr = −2µξrr

σθθ = 2µξr

r− 2µb

λ0 + 2µp

σzz = − 2µb

λ0 + 2mup (9.108)



Discussion of the Results As a result of axisymmetry, the model can be constructed as a quarter of a pie.

The radius of the pie is 1 m and the thickness of the pie is 5 cm. A cylindrical well is drilled at the center of the

pie, and its radius is 1 cm, which can be neglected in dimension when compared with the whole pie. The final

mesh is shown as Fig.9.15. And the boundary conditions is as follows: As a consequence of plain strain problem,

all the movements for solid and fluid in vertical direction Oz are suppressed; the solid and fluid displacement for

the nodes along the X axis and Y axis are fixed in Y and X direction respectively for the reason of axisymmetry;

the nodes along the outside perimeter are fixed in the solid and fluid displacement with the assumption of infinite

medium. To the difference with the previous problems, the traction boundary conditions are applied on the fluid

displacement. It should be noted that the Ω mention in the above equations is the volume of the fluid injected

per unit of vertical well length and has a unit of m3/m. In order to generate the volume of 1 cm3/m, the

corresponding fluid displacement of the nodes along the well has been calculated and applied as a step function

at the time of 0 sec. For simplicity, the initial fluid pressure p0 is set to be 0 kPa. The analytical solution is

studied below using the following set of parameters shown in Table 9.4.


• ValidationExamples/LineInjection

Figure 9.15: The mesh generation for the study of line injection problem

In the analysis, the pore pressure and the radial displacement are studied. The results are recorded from

three points at the radius of 10 cm, 50 cm and 85 cm. The close solution and experimental results are shown

in Fig.9.22 and Fig.9.23. As the time step is set to be 1 sec, the first data point starts at the time of 1 sec.

From the pore pressure plot we can see that the build-up of the pore pressure reach the peak value of 34 kPa at

the radius of 85 cm. With the decrease of the radius, the pore pressure decreases as well. This can be explained

by the fact that the closer the point to the injection location, the earlier and the larger load is applied, so the

pore pressure dissipates faster. And as time passes by, we can see the pore pressure progressively dissipates and

finally almost reaches the same value within the model. The same phenomena can be been observed from the

radial solid displacement. The maximum solid displacement occurs at the radius of 85 cm, which means more

coupling between the solid and fluid phase, as consequence, the pore pressure should have the largest value. This


http://sokocalo.engr.ucdavis.edu/~jeremic/Example_Input_Files/u-p-U/ValidationExamples/LineInjection







Undrained Bulk Modulus Ku 6.0E+7 kN/m2

Bulk Modulus K 6.7E+5 kN/m2



Fluid Diffusivity coefficient cf 0.4973 m2/s

Porosity n 0.4 -

Permeability k 3.6E-6 m/s

Table 9.4: Material Properties used to study the line injection problem

corresponds to the previous result. With the increase of the time, the radial solid displacement get closer to zero,

which means the fluid moves out the solid skeleton.

0 5 10 15 20 25 300

5

10

15

20

25

30

35

Time (sec)

Por

e P

ress

ure

(kP

a)

Radius = 10cm(c)Radius = 10cm(e)Radius = 50cm(c)Radius = 50cm(e)Radius = 85cm(c)Radius = 85cm(e)

Figure 9.16: The comparison between analytical solution and experimental result for pore pressure



0 5 10 15 20 25 300

1

2

3

4

5

6

7

8x 10

−4

Time (sec)

Rad

ial D

ispl

acem

ent (

cm)


Figure 9.17: The comparison between analytical solution and experimental result for radial displacement

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8x 10

−4

Time (sec)

Rad

ial D

ispl

acem

ent (

cm)


Figure 9.18: The comparison between analytical solution and experimental result for radial displacement



Table 9.5: Simulation parameters used for the shock wave propagation verification problem.

Parameter Symbol Value

Poisson ratio ν 0.3

Young’s modulus E 1.2 × 106 kN/m2

Solid particle bulk modulus Ks 3.6 × 107 kN/m2

Fluid bulk modulus Kf 2.17 × 106 kN/m2

Solid density ρs 2700 kg/m3

Fluid density ρf 1000 kg/m3

Porosity n 0.4

Newmark parameter γ 0.6

9.3.5 Verification: Shock Wave Propagation in Saturated Porous Medium

In order to verify the dynamic behavior of the system, an analytic solution developed by Gajo (1995) and

Gajo and Mongiovi (1995) for 1D shock wave propagation in elastic porous medium was used. A model was

developed consisting of 1000 eight node brick elements, with boundary conditions that mimic 1D behavior. In

particular, no displacement of solid (ux = 0, uy = 0) and fluid (Ux = 0, Uy = 0) in x and y directions is allowed

along the height of the model. Bottom nodes have full fixity for solid (ui = 0) and fluid (Ui = 0) displacements

while all the nodes above base are free to move in z direction for both solid and fluid. Pore fluid pressures

are free to develop along the model. Loads to the model consist of a unit step function (Heaviside) applied as

(compressive) displacements to both solid and fluid phases of the model, with an amplitude of 0.001 cm. The

u–p–U model dynamic system of equations was integrated using Newmark algorithm (see section ??). Table 9.5

gives relevant parameters for this verification.

Two set of permeability of material were used in our verification. The first model had permeability set

k = 10−6 cm/s which creates very high coupling between porous solid and pore fluid. The second model had

permeability set to k = 10−2 cm/s which, on the other hand creates a low coupling between porous solid and

pore fluid. Comparison of simulations and the analytical solution are presented in Figure 9.19.

9.4 u-p Formulation

9.4.1 Governing Equations of Porous Media

The formulation given here is based on Zienkiewicz et al. (1999b).

The first governing equation of porous media is total momentum balance equation:

σij,j − ρui + ρbi = 0 (9.109)

where σij = σ′′ij − αpδij and ρ = (1 − n)ρs + nρf .

The second governing equation is the fluid mass balance equation:

(kij(−p,j + ρfbj)),i + αui,i +p

Qsf= 0 (9.110)



4 6 8 10 12 140

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

time (µsec)

Sol

id D

ispl

. (x1

0−3 cm

)

4 6 8 10 12 140

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

time (µsec)

Flu

id D

ispl

. (x1

0−3 cm

)

K=10−6cm/s, FEM

K=10−6cm/s, Closed Form

K=10−2cm/s, FEM

K=10−2cm/s, Closed Form

Figure 9.19: Compressional wave in both solid and fluid, comparison with closed form solution.

where

kij =k′ijgρf

=k′ijγf

(9.111)

and k′ij is the permeability in Darcy’s law with the same unit as velocity.

Qsf =KsKf

Ks +Kf(9.112)

is the total compression modulus, Ks and Kf are the solid and fluid compression modulus respectively.

The boundary conditions are

σijnj = ti on Γ = Γt (9.113)

ui = ui on Γ = Γu (9.114)

niwi = nikij(−p,j + ρfbj) = w = −q on Γ = Γw (9.115)

p = p on Γ = Γp (9.116)

where w is the outflow and q is the influx.

9.4.2 Numerical Solutions of the Governing Equations

The solid displacement ui and the pore pressure p can be approximated using shape functions and nodal values:

ui = NuK uKi (9.117)

p = NpLpL (9.118)

Similar approximations are applied to ui, ui, p and p.



Numerical solution of the total momentum balance

The numerical solution of the total momentum balance is

∫

Ω

NuK (σij,j − ρui + ρbi) dΩ = 0 (9.119)


Ω

NuKσij,jdΩ =

∫

Γt

NuKσij,jnjdΩ −

∫

Ω

NuK,jσijdΩ

=

∫

Γt

NuK tidΩ −

∫

Ω

NuK,j(σ

′′ij − αpδij)dΩ

= (fu1 )Ki −∫

Ω

NuK,jσ

′′ijdΩ +

∫

Ω

NuK,iαpdΩ

= (fu1 )Ki −∫

Ω

NuK,jDijmlεmldΩ + [

∫

Ω

αNuK,iN

pNdΩ]pN

= (fu1 )Ki − [

∫

Ω

NuK,jDijmlN

uP,mdΩ]uPm + [

∫

Ω

αNuK,iN

pNdΩ]pN

= (fu1 )Ki − (KepKimP )uPm + (QKiN )pN

= fu1

− (Kep)u + Qp (9.120)

Second term of (9.119) becomes

−∫

Ω

NuKρuidΩ = −

∫

Ω

NuKρN

uLdΩuLi

= −[

∫

Ω

NuKρN

uLdΩ]üLi

= −[δij

∫

Ω

NuKρN

uLdΩ]üLj

= −(MKijL)üLj

= −M ü (9.121)

Third term of (9.119) becomes∫

Ω

NuKρbidΩ = (fu2 )Ki

= fu2

(9.122)

The equation (9.119) thus becomes

(MKijL)üLj − (QKiN )pN + (KepKijL)uLj = (fu1 )Ki + (fu2 )Ki = (fu)Ki (9.123)

or

M ü − Qp + (Kep)u = fu1

+ fu2

= fu (9.124)

Numerical solution of the fluid mass balance

The numerical solution of the fluid mass balance is

∫

Ω

NpM

(

kij(−p,j + ρfbj),j + αui,i +p

Qsf

)

dΩ = 0 (9.125)




Ω

NpM (kij(−p,j + ρfbj)),i dΩ (9.126)

=

∫

γw

NpMwinidΩ −

∫

Ω

NpM,ikij(−p,j + ρfbj)dΩ

=

∫

γw

NpM wdΩ +

∫

Ω

NpM,ikijp,jdΩ −

∫

Ω

NpM,ikijρf bjdΩ

= (fp1 )M +

∫

Ω

NpM,ikijp,jdΩ −

∫

Ω

NpM,ikijρf bjdΩ

= (fp1 )M + [

∫

Ω

NpM,ikijN

pN,jdΩ]pN − (fp2 )M

= (fp1 )M + (HMN )pN − (fp2 )M

= fp1

+ Hp − fp2

(9.127)

Second term of (9.125) becomes∫

Ω

NpMαui,idΩ = [

∫

Ω

NpMαN

uL,jdΩ] ˙uLj

= (QLjM ) ˙uLj

= QT ˙u (9.128)

Third term of (9.125) becomes∫

Ω

NpM

p

QsfdΩ = [

∫

Ω

NpM

1

QsfNpNdΩ] ˙pN

= (SMN ) ˙pN

= S ˙p (9.129)

The equation (9.125) thus becomes

(HMN )pN + (QLjM ) ˙uLj + (SMN ) ˙pN = −(fp1 )M + (fp2 )M = (fp)M (9.130)

or

Hp + QT ˙u + S ˙p = −fp1

+ fp2

= fp (9.131)

Matrix form of the governing equations

Combine equation (9.123) and (9.130), we obtain

MKiLj 0

0 0

uLj

pN

+

0 0

QLjM SMN

uLj

pN

+

(Kep)KiLj −QKiN

0 HMN

uLj

pN

=

fuKi

fpM

(9.132)

or, by combining equations (9.124) and (9.131), we obtain

M 0

0 0

u

p

+

0 0

QT S

u

p

+

Kep Q

0 H

u

p

=

fu

fp

(9.133)



where

fu ↔ fuKi = (fu1 )Ki + (fu2 )Ki (9.134)

fp ↔ fpM = −(fp1 )M + (fp2 )M (9.135)

and

fu1 ↔ (fu1 )Ki =

∫

Γt

NuK tidΓ (9.136)

fu2 ↔ (fu2 )Ki =

∫

Ω

NuKρbidΩ (9.137)

fp1 ↔ (fp1 )M =

∫

Γw

NpM wdΓ (9.138)

fp2 ↔ (fp2 )M =

∫

Ω

NpM,ikijN

pN,jdΩ (9.139)

M ↔ MKiLj = δij

∫

Ω

NuKρN

uLdΩ (9.140)

Q ↔ QKiN =

∫

Ω

αNuK,iN

pNdΩ (9.141)

S ↔ SMN =

∫

Ω

NpM

1

QsfNpNdΩ (9.142)

H ↔ HMN =

∫

Ω

NpM,ikijN

pN,jdΩ (9.143)


Chapter 10

Earthquake–Soil–Structure Interaction

(2002–)

(In collaboration with Dr. Matthias Preisig and Dr. Guanzhou Jie)

10.1 Dynamic Soil-Foundation-Structure Interaction

Current design practice for structures subject to earthquake loading regards dynamic SFSI to be mainly beneficial

to the behavior of structures (Jeremic and Preisig, 2005). Including the flexibility of the foundation reduces the

overall stiffness of a system and therefore reduces peak loads caused by a given ground motion. Even if this is

true in most cases there is the possibility of resonance occurring as a result of a shift of the natural frequencies

of the SFS-system. This can lead to large inertial forces acting on a structure.

As a result of these large inertial forces caused by the structure oscillating in it’s natural frequency the structure

as well as the soil surrounding the foundation can undergo plastic deformations. This in turn further modifies the

overall stiffness of the SFS-system and makes any prediction on the behavior very difficult.

Dynamic SFSI also becomes important in the design of large infrastructure projects. As authorities and

insurance companies try to introduce the concept of performance based design to the engineering community

more sophisticated models are needed in order to obtain the engineering demand parameters (EDP’s). A good

numerical model of a soil-foundation-structure system can therefore not only prevent the collapse or damage of a

structure but also help to save money by optimizing the design to withstand an earthquake with a certain return

period.

A variety of methods of different levels of complexity are currently being used by engineers. In the following

an overview over the most important ones is presented. A more thorough discussion on methods and specific

aspects of dynamic SFSI is available in Wolf (1985) and more recently in Wolf and Song (2002).

• No SFSI

The ground motion is applied directly to the base of the building. Alternatively, instead of applying the

ground motion directly to the base of the structure, effective earthquake forces proportional to the base

209


acceleration can be applied to the nodes.

This procedure is reasonable only for flexible structures on very stiff soil or rock. In this case the displacement

of the ground doesn’t get modified by the presence of the structure. For stiffer structures on soil the ground

motion has to be applied to the soil. The model has to incorporate propagation of the motion through the

soil, its interaction with the structure and the radiation away from the structure.

• Direct methods

Direct methods treat the SFS-system as a whole. The numerical model incorporates the spatial discretization

of the structure and the soil. The analysis of the entire system is carried out in one step. This method

provides most generality as it is capable of incorporating all nonlinear behavior of the structure, the soil and

also the interface between those two (sliding, uplift).

• Substructure methods

Substructure methods refer to the principle of superposition. The SFS-system is generally subdivided into

a structure part and a soil part. Both substructures can be analyzed separately and the total displacement

can be obtained by adding the contributions at the nodes on the interface.

This method reduces the size of the problem considerably. As the time needed for an analysis doesn’t

increase linearly with an increasing number of equations the substructure method is much faster than the

direct method. The biggest drawback of the method however is the fact that linearity is assumed. For

nonlinear systems the substructure method cannot be used.

For the direct method different levels of sophistication are possible:

• Foundation stiffness approach

The behavior of the soil is accounted for by simple mechanical elements such as springs, masses and dash

pots. Different configurations of the subsoil can be taken into account by connecting several springs,

masses and dash pots whose parameters have been determined by a curve fitting procedure Wolf (1994).

This approach is very popular among structural engineers as it is relatively easy to be integrated in a

commonly used finite element code.

Other methods use frequency dependent springs and dash pots and therefore require an analysis in frequency

domain. Relatively complex configurations of layered subsoil and embedded foundations can be modeled

with good accuracy by replacing the (elastic) soil with a sequence of conical rods Wolf and Song (2002)

and Wolf and Preisig (2003).

• p-y methods

Attempts have been made to apply the static p-y approach for evaluating lateral loading on pile foundations

to dynamic problems. E. and H. (2002) lists several references and provides a parametric study of single

piles and pile groups in different soil types under simplified loading cases.

Even if p-y curves are widely used for estimating lateral loading on piles they are rarely used in full dynamic

soil-structure interaction analysis. Current work trying to implement these methods into finite element

codes is likely to make them more popular with the engineering community.



• Full 3d

Full nonlinear three-dimensional modeling of dynamic soil-foundation-structure interaction can be regarded

as the ’brute force’ approach. Displacements and forces can be obtained not only for the structure as in

the above mentioned methods but also for the soil. In spite of the computational resources and modeling

effort required for an analysis it is the only method that remains valid for all kinds of problems involving

material nonlinearities, contact problems, different loading cases and complex geometries.

10.2 Introduction

The Domain Reduction Method (DRM) was developed recently by Bielak et al. (2003); Yoshimura et al. (2003)).

It is a modular, two-step dynamic procedure aimed at reducing the large computational domain to a more

manageable size. The method was developed with earthquake ground motions in mind, with the main idea to

replace the force couples at the fault with their counterpart acting on a continuous surface surrounding local

feature of interest. The local feature can be any geologic or man made object that constitutes a difference from

the simplified large domain for which displacements and accelerations are easier to obtain. The DRM is applicable

to a much wider range of problems. It is essentially a variant of global–local set of methods and as formulated

can be used for any problems where the local feature can be bounded by a continuous surface (that can be closed

or not). The local feature in general can represent a soil–foundation–structure system (bridge, building, dam,

tunnel...), or it can be a crack in large domain, or some other type of inhomogeneity that is fairly small compared

to the size of domain where it is found.

In what follow, the DRM is developed in a somewhat different way than it was done in original papers by

Bielak et al. (2003); Yoshimura et al. (2003)). The main features of the DRM are then analyzed and appropriate

practical modeling issues addressed.

10.3 The Domain Reduction Method

A large physical domain is to be analyzed for dynamic behavior. The source of disturbance is a known time history

of a force field Pe(t). That source of loading is far away from a local feature which is dynamically excited by

Pe(t) (see Figure 10.1).

The system can be quite large, for example earthquake hypocenter can be many kilometers away from the

local feature of interest. Similarly, the small local feature in a machine part can be many centimeters away from

the source of dynamic loading which influences this local feature. In this sense the term large domain is relative

to the size of the local feature and the distance to the dynamic forcing source.

It would be beneficial not to analyze the complete system, as we are only interested in the behavior of the

local feature and its immediate surrounding, and can almost neglect the domain outside of some relatively close

boundaries. In order to do this, we need to somehow transfer the loading from the source to the immediate

vicinity of the local feature. For example we can try to reduce the size of the domain to a much smaller model

bounded by surface Γ as shown in Figure 10.1. In doing so we must ensure that the dynamic forces Pe(t) are

appropriately propagated to the much smaller model boundaries Γ.



Pe(t)

ui

ub

ue

Γ

Large scale domain

Local feature

Ω+

Ω

Figure 10.1: Large physical domain with the source of load Pe(t) and the local feature (in this case a soil–

foundation–building system.

10.3.1 Method Formulation

In order to appropriately propagate dynamic forces Pe(t) one actually has to solve the large scale problem which

will include the effects of the local feature. Most of the time this is impossible as it involves all the complexities

of large scale computations and relatively small local feature. Besides, the main goal of presented developments

is to somehow reduce the large scale domain as to be able to analyze in details behavior of the local feature.

In order to propagate consistently the dynamic forces Pe(t) we will make a simplification in that we will replace

a local feature with a simpler domain that is much easer to be analyzed. That is, we replace the local feature

(bridge, building. tunnel, crack) with a much simpler geometry and material. For example, Figure 10.2 shows a

simplified model, without a foundation–building system. The idea is to simplify the model so that it is much easier

0ub

ui0

Pe(t)0ue

Ω 0

Γ

Simplified large scale domain

Ω+

Figure 10.2: Simplified large physical domain with the source of load Pe(t) and without the local feature (in this

case a soil–foundation–building system. Instead of the local feature, the model is simplified so that it is possible

to analyze it and simulate the dynamic response as to consistently propagate the dynamic forces Pe(t).

to consistently propagate the dynamic forces to the boundary Γ. The notion that it is much easier to propagate

those dynamic forces is of course relative. This is still a very complex problem, but at least the influence of local

feature is temporarily taken out.

It is convenient to name different parts of domain. For example, the domain inside the boundary Γ is named

Ω0. The rest of the large scale domain, outside boundary Γ, is then named Ω+. The outside domain Ω+ is still



the same as in the original model, while the change, simplification, is done on the domain inside boundary Γ.

The displacement fields for exterior, boundary and interior of the boundary Γ are ue, ub and ui, on the original

domain.

The equations of motions for the complete system can be written as

[

M]

u

+[

K]

u

=

Pe

(10.1)

or if written for each domain (interior, boundary and exterior of Γ) separately, the equations obtain the following

form:

MΩii MΩ

ib 0

MΩbi MΩ

bb +MΩ+bb MΩ+

be

0 MΩ+eb MΩ+

ee

ui

ub

ue

+

KΩii KΩ

ib 0

KΩbi KΩ

bb +KΩ+bb KΩ+

be

0 KΩ+eb KΩ+

ee

ui

ub

ue

=

0

0

Pe

(10.2)

In these equations, the matrices M and K denote mass and stiffness matrices respectively; the subscripts i, e,

and b refer to nodes in either the interior or exterior domain or on their common boundary; and the superscripts

Ω and Ω+ refer to the domains over which the various matrices are defined.

The previous equation can be separated provided that we maintain the compatibility of displacements and

equilibrium. The resulting two equations of motion are

MΩii MΩ

ib

MΩbi MΩ

bb

ui

ub

+

KΩii KΩ

ib

KΩbi KΩ

bb

ui

ub

=

0

Pb

, inΩ (10.3)

and

MΩ+bb MΩ+

be

MΩ+eb MΩ+

ee

ub

ue

+

KΩ+bb KΩ+

be

KΩ+eb KΩ+

ee

ub

ue

=

−PbPe

, inΩ+ (10.4)

Compatibility of displacements is maintained automatically since both equations contain boundary displacements

ub (on boundary Γ), while the equilibrium is maintained through action–reaction forces Pb.

In order to simplify the problem, the local feature is removed from the interior domain. Thus, the interior

domain is significantly simplified. In other words, the exterior region and the material therein are identical to

those of the original problem as the dynamic force source. On the other hand, the interior domain (denoted as

Ω0), is simplified, the localized features is removed (as seen in figure 10.2).

For this simplified model, the displacement field (interior, boundary and exterior, respectively) and action–

reaction forces are denoted by u0i , u

0b , u

0e and P 0

b . The entire simplified domain Ω0 and Ω+ is now easier to

analyze.

The equations of motion in Ω+ for the auxiliary problem can now be written as:

MΩ+bb MΩ+

be

MΩ+eb MΩ+

ee

u0b

u0e

+

KΩ+bb KΩ+

be

KΩ+eb KΩ+

ee

u0b

u0e

=

−P 0b

Pe

(10.5)

Since there was no change to the exterior domain Ω+ (material, geometry and the dynamic source are still the

same) the mass and stiffness matrices and the nodal force Pe are the same as in Equations (10.3) and (10.4).



Second part of previous equation (10.5) can be used to obtain the dynamic force Pe as

Pe = MΩ+eb u0

b +MΩ+ee u0

e +KΩ+eb u

0b +KΩ+

ee u0e (10.6)

The total displacement, ue, can be expressed as the sum of the free field u0e (from the background, simplified

model) and the residual field we (comming from the local feature) as following:

ue = u0e + we (10.7)

It is important to note that this is just a change of variables and not an application of the principle of superposition.

The residual displacement field, we is measured relative to the reference free field ue0.

By substituting Equation (10.7) in Equation (10.2) one obtains:

MΩii MΩ

ib 0

MΩbi MΩ

bb +MΩ+bb MΩ+

be

0 MΩ+eb MΩ+

ee

ui

ub

u0e + we

+

KΩii KΩ

ib 0

KΩbi KΩ

bb +KΩ+bb KΩ+

be

0 KΩ+eb KΩ+

ee

ui

ub

u0e + we

=

0

0

Pe

(10.8)

which, after moving the free field motions u0e to the right hand side, becomes

MΩii MΩ

ib 0

MΩbi MΩ

bb +MΩ+bb MΩ+

be

0 MΩ+eb MΩ+

ee

ui

ub

we

+

KΩii KΩ

ib 0

KΩbi KΩ

bb +KΩ+bb KΩ+

be

0 KΩ+eb KΩ+

ee

ui

ub

we

=

0

−MΩ+be u0

e −KΩ+be u0

e

−MΩ+ee u0

e −KΩ+ee u

0e + Pe

(10.9)

By substituting Equation (10.6) in previous Equation (10.9), the right hand side can now be written as

MΩii MΩ

ib 0

MΩbi MΩ

bb +MΩ+bb MΩ+

be

0 MΩ+eb MΩ+

ee

ui

ub

we

+

KΩii KΩ

ib 0

KΩbi KΩ

bb +KΩ+bb KΩ+

be

0 KΩ+eb KΩ+

ee

ui

ub

we

=

0

−MΩ+be u0

e −KΩ+be u0

e

MΩ+eb u0

b +KΩ+eb u

0b

(10.10)

The right hand side of equation (10.10) is the dynamically consistent replacement force (so called effective

force, P eff for the dynamic source forces Pe. In other words, the dynamic force Pe was consistently replaced by

the effective force P eff :

P eff =

P effi

P effb

P effe

=

0

−MΩ+be u0

e −KΩ+be u0

e

MΩ+eb u0

b +KΩ+eb u

0b

(10.11)



10.3.2 Method Discussion

Single Layer of Elements used for P eff . The Equation (10.11) shows that the effective nodal forces P eff

involve only the sub-matrices Mbe, Kbe, Meb, Keb. These matrices vanish everywhere except the single layer of

finite elements in domain Ω+ adjacent to Γ. The significance of this is that the only wavefield (displacements and

accelerations) needed to determine effective forces P eff is that obtained from the simplified (auxiliary) problem

at the nodes that lie on and between boundaries Γ and Γe, as shown in Figure 10.3.

0ub

ui0

Pe(t)

Ω+

0ue

0ue Γe

Γ +

Local feature

Γ

Ω

Figure 10.3: DRM: Single layer of elements between Γ and Γe is used to create P eff .

Only residual waves outgoing. Another interesting observation is that the solution to problem described in

Equation (10.10) comprises full unknowns (displacements and accelerations) inside and on the boundary Γ (ui and

ub respectively). On the other hand, the solution for the domain outside single layer of finite elements (outside

Γe) is obtained for the residual unknown (displacement and accelerations) field, we only. This residual unknown

field is measured relative to the reference free field of unknowns (see comments on page 214). That effectively

means that the solution to the equation Equation (10.10) outside the boundary Γe will only contain additional

waves field resulting from the presence of a local feature. This in turn means that if the interest is in behavior of

local feature and the surrounding media (all within boundary Γ) one can neglect the behavior of the full model

(outside Γe in Ω+) and provide appropriate supports (including fixity and damping) at some distance from the

boundary Γe into region Ω+. This is significant for a number of reasons:

• large models can be reduced in size to encompass just a few layers of elements outside boundary Γe

(significant reduction for, say earthquake problems where the size of a local feature is orders of magnitudes

smaller then the distance to the dynamic source force Pe (earthquake hypocenter).

• the residual unknown field can be monitored and analyzed for information about the dynamic characteristics

of the local feature. Since the residual wave field is we is measured relative to the reference free field ue0,

the solution for we has all the characteristics of the additional wave field stemming from the local feature.

Inside domain Ω can be inelastic. In all the derivations in section 10.3 no restriction was made on the type

of material inside the plastic bowl (inside Γe). That is, the assumption that the material inside is linear elastic



is not necessary as the DRM is not relying on principle of superposition. The Equation 10.7 was only describing

the change of variables, and clearly there was no use of the principle of superposition, which is only valid for

linear elastic solids and structures. It is therefor possible to assume that the derivations will still be valid with any

type of material (linear or nonlinear, elastic or inelastic) inside Γe. With this in mind, the DRM becomes a very

powerful method for analysis of soil–foundation–structure systems.

10.4 Numerical Accuracy and Stability

The accuracy of a numerical simulation of dynamic SFSI is controlled by two main parameters: a) the spacing of

the nodes of the finite element model (∆h) and b) the length of the time step ∆t. Assuming that the numerical

method converges toward the exact solution as ∆t and ∆h go toward zero the desired accuracy of the solution

can be obtained as long as sufficient computational resources are available.

10.4.1 Grid Spacing ∆h

In order to represent a traveling wave of a given frequency accurately about 10 nodes per wavelength λ are

required (Bathe and Wilson, 1976; Hughes, 1987; Argyris and Mlejnek, 1991b). Fewer than 10 nodes can lead to

numerical damping as the discretization misses certain peaks of the wave. In order to determine the appropriate

maximum grid spacing the highest relevant frequency fmax that is present in the model needs to be found by

performing a Fourier analysis of the input motion. Typically, for seismic analysis fmax is about 10 Hz. By

choosing the wavelength λmin = v/fmax, where v is the wave velocity, to be represented by 10 nodes the

smallest wavelength that can still be captured partially is λ = 2∆h, corresponding to a frequency of 5 fmax.

The maximum grid spacing should not exceed

∆h ≤ λ

10=

v

10 fmax(10.12)

where v is the lowest wave velocity that is of interest in the simulation. Generally this is the shear wave velocity.

10.4.2 Time Step Length ∆t

The time step ∆t used for numerically solving nonlinear vibration or wave propagation problems has to be limited

for two reasons. The stability requirement depends on the numerical procedure in use and is usually formulated

in the form ∆t/Tn < value. Tn denotes the smallest fundamental period of the system. Similar to the spatial

discretization Tn needs to be represented by about 10 time steps. While the accuracy requirement provides a

measure on which higher modes of vibration are represented with sufficient accuracy, the stability criterion needs

to be satisfied for all modes. If the stability criterion is not satisfied for all modes of vibration, then the solution

may diverge. In many cases it is necessary to provide an upper bound to the frequencies that are present in a

system by including frequency dependent damping to the model.

The second stability criterion results from the nature of the finite element method. As a wave front progresses

in space it reaches one point after the other. If the time step in the finite element analysis is too large the wave

front can reach two consecutive elements at the same moment. This would violate a fundamental property of



wave propagation and can lead to instability. The time step therefore needs to be limited to

∆t <∆h

v(10.13)

where v is the highest wave velocity.

10.4.3 Nonlinear Material Models

If nonlinear material models are used the considerations for stability and accuracy as stated above don’t necessarily

remain valid. Especially modal considerations need to be examined further for these cases. It is however save

to assume that the natural frequencies decrease as plastic deformations occur. The minimum time step required

to represent the natural frequencies of the dynamic system can therefore taken to be the same as in an elastic

analysis.

3.8 4−8

−6

−4

−2

0

2

4

6

8

10

Time [s]

Acc

eler

atio

n [m

/s2 ]

Linear algorithmNewton−Raphson algorithm

Figure 10.4: Resulting acceleration using Linear and Newton-Raphson algorithms

A high frequency component is introduced due to plastic slip and counter balancing of the resulting displace-

ment. This is especially true if a linear algorithm with no iterations within one time step is used. Figure 10.4

shows a part of an acceleration time history from an analysis involving elastic-plastic material. It can be seen

that the out-of-balance forces at the end of a time step can be quite large if a linear algorithm is used. While

the Newton-Raphson algorithm minimizes out-of-balance forces within one time step the linear algorithm requires

several time steps to return to a stable equilibrium path.

The frequencies corresponding to these peaks are typically of the order of 1 /(a few ∆t). Normally the time

step is small enough so that these frequencies don’t interfere with the input motion. They can be prevented from

propagating through the model by an appropriate choice of algorithmic or material damping.

For stability the time step used in a nonlinear analysis needs to be smaller than in a linear elastic analysis.

By how much it has to be reduced is difficult to predict as this depends on many factors such as the material

model, the applied loading or the numerical method itself. Argyris and Mlejnek (1991a) suggest the time step to

be reduced by 60% or more compared to the time step used in an elastic analysis. The best way to determine

whether the time step is appropriate for a given analysis consists in running a second analysis with a reduced time

step.



10.5 Domain Boundaries

One of the biggest problems in dynamic SFSI in infinite media is related to the modeling of domain boundaries.

Because of limited computational resources the computational domain needs to be kept small enough so that

it can be analyzed in a reasonable amount of time. By limiting the domain however an artificial boundary is

introduced. As an accurate representation of the soil-structure system this boundary has to absorb all outgoing

waves and reflect no waves back into the computational domain. The most commonly used types of domain

boundaries are presented in the following:

• Fixed or free

By fixing all degrees of freedom on the domain boundaries any radiation of energy away from the structure is

made impossible. Waves are fully reflected and resonance frequencies can appear that don’t exist in reality.

The same happens if the degrees of freedom on a boundary are left ’free’, as at the surface of the soil.

A combination of free and fully fixed boundaries should be chosen only if the entire model is large enough

and if material damping of the soil prevents reflected waves to propagate back to the structure.

• Absorbing Lysmer Boundaries

A way to eliminate waves propagating outward from the structure is to use Lysmer boundaries. This method

is relatively easy to implement in a finite element code as it consists of simply connecting dash pots to all

degrees of freedom of the boundary nodes and fixing them on the other end (Figure 10.5).

Cs

CsCp

Cs

CsCp

Cs

CsCp

Cs

CsCp

Figure 10.5: Absorbing boundary consisting of dash pots connected to each degree of freedom of a boundary

node

Lysmer boundaries are derived for an elastic wave propagation problem in a one-dimensional semi-infinite

bar. It can be shown that in this case a dash pot specified appropriately has the same dynamic properties

as the bar extending to infinity (Wolf, 1988). The damping coefficient C of the dash pot equals

C = Aρ c (10.14)



where A is the section of the bar, ρ is the mass density and c the wave velocity that has to be selected

according to the type of wave that has to be absorbed (shear wave velocity cs or compressional wave velocity

cp).

In a 3d or 2d model the angle of incidence of a wave reaching a boundary can vary from almost 0 up to

nearly 180. The Lysmer boundary is able to absorb completely only those under an angle of incidence of

90. Even with this type of absorbing boundary a large number of reflected waves are still present in the

domain. By increasing the size of the computational domain the angles of incidence on the boundary can

be brought closer to 90 and the amount of energy reflected can be reduced.

In Section 10.7.1 the Lysmer boundary is tested on a two-dimensional model.

• More sophisticated boundaries modeling wave propagation toward infinity

For a spherical cavity involving only waves propagating in radial direction a closed form solution for radiation

toward infinity, analogous to the Lysmer boundary for wave propagation in a prismatic rod, exists (Sections

3.1.2 and 3.1.3 in Wolf (1988)). Since this solution, in contrast to the Lysmer boundary, includes radiation

damping it can be thought of as an efficient way of eliminating reflections on a semi-spherical boundary

surrounding the computational domain.

More generality in terms of absorption properties and geometry of the boundary are provided by the various

boundary element methods (BEM) available in the literature.

10.6 Verification using one-dimensional Wave Propagation

10.6.1 Problem Statement

The model used for verification of the DRM consists of 13 8-node brick elements (Figure 10.6). Each element

ElementBoundary

Mod

el 1

(F

ree−

Fie

ld)

Mod

el 2

(S

oil−

Str

uctu

re In

tera

ctio

n)

ElementBoundary

ExteriorDomain

InteriorDomain

Figure 10.6: The analyzed models


http://cml00.engr.ucdavis.edu/~matthias/OpenSees/index.html#1dDRM


is a cube of 1 meter side length. The four nodes in a horizontal plane are constrained to move together in all

directions.

Model 1 represents the free field whereas Model 2 has a column with a lumped mass attached to its top. The

translational degrees of freedom of the base of the beam-column are connected to the nodes at the top of the

soil column, the rotational degrees of freedom are fixed. The natural period of the pile, assuming it is fixed at

the soil surface, is 0.5 seconds. As input motion, a sinusoidal horizontal ground motion with a period of 1 second

has been applied to the base of the soil column, resulting in an upward propagating shear wave.

The following material properties have been used in the analysis: Young’s modulus E = 1944 kPa, Poisson’s

ratio ν = 0.35 and specific density ρ = 1.8 kg/m3. The resulting shear wave velocity is 20m/s. The lowest

natural frequencies are 3 Hz for model 1 and 0.31 Hz for model 2.

10.6.2 Results

As a first test Model 2 is analyzed by prescribing the sinusoidal motion to the base of the model. The displacement

and acceleration time histories of the nodes of the boundary layer are recorded. In a second analysis these time

histories are used to calculate P eff which then is applied to the nodes of the boundary layer. As can be seen

in Figure 10.7 the results are identical in the interior domain. In the exterior domain the result from the DRM-

analysis shows no motion at all. This is the expected result since the ’large domain’ used to obtain ue and ue

and the ’local domain’ are identical. The residual field in the exterior domain we therefore vanishes.

As a second test the same analysis as above is performed, but this time ue and ue from a free field analysis

have been used to calculate P eff (Figure 10.8). In the interior domain the results from the DRM- and from the

base-shaking analysis are identical at the beginning and start to diverge after about 2 seconds. Since the free-field

motions aren’t compatible with the soil-structure system, the motions in the exterior domain don’t vanish. Those

downward propagating waves, which result from interaction with the structure, will not be canceled out by P eff

in the boundary layer and will therefore leave the interior domain. At the base they will eventually be reflected

and sent upward into the interior domain, where undesired interference occurs.

In order to reduce this undesired interference absorbing boundaries of the Lysmer type can be used (Section

10.5). The result of the previous analysis, this time with absorbing boundaries, is given in Figure 10.9. The

Lysmer boundary reduces the waves in the exterior domain considerably. The displacement in the exterior domain

is now entirely due to the outgoing wave caused by interaction of the free-field motion with the beam-column

at the surface. It can be seen that the residual displacement field contains the frequency of the sinusoidal input

motion and the natural frequency of the SFSI-model (0.31 Hz).



0 2 4 6−1

−0.5

0

0.5

1

Time [s]

x−D

isp.

[m]

Base, z = −13 m

0 2 4 6−4

−3

−2

−1

0

1

2

3

Time [s]

x−D

isp.

[m]

Exterior domain, z = −9 m

0 2 4 6−3

−2

−1

0

1

2

Time [s]

x−D

isp.

[m]

Exterior node of boundary layer

0 2 4 6−1.5

−1

−0.5

0

0.5

1

1.5

Time [s]

x−D

isp.

[m]

Interior node of boundary layer

0 2 4 6−3

−2

−1

0

1

2

3

Time [s]x−

Dis

p. [m

]

Interior domain, z = −3 m

0 2 4 6−4

−2

0

2

4

Time [s]

x−D

isp.

[m]

Surface

Base shakingDRM

1D soil column with structure subject to sinusoidal motion

Figure 10.7: Using total motions to calculate P eff



0 2 4 6−1

−0.5

0

0.5

1

Time [s]

x−D

isp.

[m]

Base, z = −13 m

0 2 4 6−4

−3

−2

−1

0

1

2

3

Time [s]

x−D

isp.

[m]


0 2 4 6−3

−2

−1

0

1

2

3

Time [s]

x−D

isp.

[m]


0 2 4 6−1.5

−1

−0.5

0

0.5

1

1.5

Time [s]

x−D

isp.

[m]


0 2 4 6−3

−2

−1

0

1

2

3

Time [s]

x−D

isp.

[m]


0 2 4 6−4

−2

0

2

4

Time [s]

x−D

isp.

[m]

Surface

Base shakingDRM

1D soil column with structure subject to sinusoidal motion

Figure 10.8: ue and ue obtained from free-field model



0 2 4 6−1

−0.5

0

0.5

1

Time [s]

x−D

isp.

[m]

Base, z = −13 m

0 2 4 6−4

−3

−2

−1

0

1

2

3

Time [s]

x−D

isp.

[m]


0 2 4 6−3

−2

−1

0

1

2

Time [s]

x−D

isp.

[m]


0 2 4 6−1.5

−1

−0.5

0

0.5

1

1.5

Time [s]

x−D

isp.

[m]


0 2 4 6−3

−2

−1

0

1

2

3

Time [s]x−

Dis

p. [m

]


0 2 4 6−4

−2

0

2

4

Time [s]

x−D

isp.

[m]

Surface

Base shakingDRM with Absorbing Boundary

1D soil column subject to sinusoidal motion

Figure 10.9: As in Figure 10.8 but with absorbing boundary at the base



10.7 Case History: Simple Structure on Nonlinear Soil

10.7.1 Simplified Models for Verification

Due to the complexity of full scale finite element models it is helpful to perform preliminary tests on simplified

models in order to verify the adequacy of the time and mesh discretization with respect to the input motion. It

also provides good insight in the performance of the nonlinear material model. To achieve this a series of tests

on a one-dimensional soil column have been proposed:

• Static pushover test on nonlinear soil column

Through the static pushover test the behavior of the nonlinear material model can be verified.

• Dynamic test of elastic soil column

By applying an earthquake motion to the elastic soil column it can be tested whether the selected grid

spacing is capable of representing the motion correctly without filtering out any relevant frequencies. This

test also allows to choose appropriate damping parameters. It should be noted that this is additional (small)

damping that is used for stability of the numerical scheme and should not be relied upon to provide major

energy dissipation. Major energy dissipation should be coming from inelastic deformations of the SFS

system.

• Dynamic test of nonlinear soil column

Finally the stability and the accuracy of the numerical method can be examined by applying the earthquake

motion to the nonlinear column of soil. A second analysis with a time step reduced by 50% should not give

a significantly different result.

Furthermore it will be examined how propagation through an elastic-plastic material will change the fre-

quency content of the motion.

Model Description

The one-dimensional soil column used for verification has the same depth and element sizes as the 2d and 3d

models that will be addressed later. Its total depth is 10.5 meters and it consists of a single stack of 8-node brick

elements of 1.5 meters side length. In order to achieve one-dimensional wave propagation in vertical direction the

movement of four nodes at each level of depth is constrained to be equal. The input motion is applied to the

four nodes at the base of the model. As input motions four time histories from the Northridge Earthquake are

selected (Figure 10.10).

The material properties of the soil are given in Table 10.7.1.

Friction angle φ′ 37

Undrained shear strength cu 10 kPa

Mass density ρ 1800 kg/m3

Shear wave velocity vs 200 m/s



0 10 20 30 40

−1

0

1

LA University Hospital

Acc

eler

atio

n [g

]

0 5 10 15 20 250

0.01

0.02

0.03

Fou

rier

Am

plitu

de [g

]

0 10 20 30 40

−1

0

1

Lake Hughes

Acc

eler

atio

n [g

]

0 5 10 15 20 250

0.01

0.02

0.03

Fou

rier

Am

plitu

de [g

]

0 10 20 30 40

−1

0

1

2

Century City 090

Acc

eler

atio

n [g

]

0 5 10 15 20 250

0.01

0.02

0.03

0.04

Fou

rier

Am

plitu

de [g

]

0 10 20 30 40

−1

0

1

Century City 360

Acc

eler

atio

n [g

]

Time [s]0 5 10 15 20 25

0

0.02

0.04

0.06

Fou

rier

Am

plitu

de [g

]

Frequency [Hz]

Figure 10.10: Acceleration time histories and Fourier amplitude spectra’s of the selected ground motions

The discretization parameters, the time step ∆t and the maximum grid spacing ∆h, are determined following

the guidelines outlined in Section 10.4. This yields a maximum grid spacing of

∆h ≤ vs10 fmax

=200

10 10= 2m (10.15)

For the following analysis ∆h = 1.5 m is selected. The maximum time step is

∆t ≤ ∆h

vs=

1.5

200= 0.0075 s (10.16)

Taking into account a further reduction of the time step by about 60% due to the use of nonlinear material models

∆t = 0.002 s is chosen.

Static Pushover Test on Elastic-Plastic Soil Column

For the static pushover test, an elastic perfectly plastic Drucker-Prager material model as specified in Table 10.7.1

is used.

After applying self weight a horizontal load of 100 kN is applied to a surface node in increments of 0.1 kN.

The system of equations is solved using a full Newton-Raphson algorithm. The predicted shear strength of the



first element that is expected to fail, the one at the surface, is:

τf = cu + z ρ g tanφ′

= 10 + 0.75x 1.8x 9.81 tan 37

= 19.98 kPa (10.17)

where z is the depth of the center of the first element.

Self weight produces the following stresses in the element at the surface:

σx = σy = 8.83 kPa

σz = 13.24 kPa

The maximum shear stress is

τmax =

√(σz − σx

2

)2

+ τ2xz (10.18)

The theoretical failure load can be obtained as follows:

Pf = τxzA

=

√

τ2f −

(σz − σx

2

)2

A

= 44.7 kN (10.19)

The static failure load is underestimated by about 6%. This accuracy is acceptable for the given model because

the boundary conditions cannot assure constant stresses at a given depth (no shear stress is applied to the lateral

surfaces).

Dynamic Test on Elastic Soil Column

In order to test the spatial discretization of the model an earthquake motion is propagated through an elastic

soil column. The grid spacing of the finite element mesh can be considered sufficiently fine if frequencies up to

fmax = 10 Hz are represented accurately in the numerical analysis. A good way to verify this is to calculate

transfer functions between the base and the surface of the soil column. Because transfer functions don’t depend

on the input motion they can easily be compared with closed form solutions.

The transfer function of a soil deposit describes the amplification between the frequencies of the motion at

the base and at the soil surface:

TF (ω) =u(z = 0, ω)

u(z = H,ω)(10.20)

where z is the depth measured from the surface and H is the thickness of the soil deposit above the bedrock.

ω = 2π f is the circular frequency.

For elastic soil with viscous damping the wave equation can be written as (Kramer, 1996)

ρ∂2u

∂t2= G

∂2u

∂z2+ η

∂3u

∂z2∂t(10.21)



η is the damping coefficient, defined as

η =2G

ωξ (10.22)

where ξ is the frequency independent hysteretic material damping.

After solving the wave equation the transfer function can be written as

TF (ω) =1

cosωH/v∗s(10.23)

where v∗s is the complex shear wave velocity

v∗s =

√

G∗

ρ=

√

G(1 + i 2 ξ)

ρ(10.24)

In a finite element model with mass- and stiffness proportional Rayleigh damping the damping coefficient η is

constant. Therefore the hysteretic material damping ratio ξ needs to be frequency dependent in order to satisfy

Equation 10.22. Solving Equation 10.22 for ξ and substituting it into Equation 10.24 and then into Equation

10.23 yields a new transfer function:

TF (ω) =1

cos

(

ωH

√ρ

G + iωη

) (10.25)

Figure 10.11 shows a comparison between the closed form solution and the numerical transfer functions

obtained from the finite element analysis. Rayleigh damping is used to obtain the damping matrix C:

C = αM + βK (10.26)

The analysis are performed using stiffness proportional Rayleigh damping of β = 0.001 and β = 0.01. No mass

proportional damping is applied (α = 0). The damping coefficients of the closed form solution are chosen to be

η = β G .

It can be seen that the numerical transfer functions are very close to the closed form solutions for η = β G.

The peak corresponding to the second natural frequency of the soil layer is slightly shifted to the right in the

result of the FE analysis. For the FE analysis the Rayleigh damping cannot be reduced any further as the solution

would become unstable. This result proves that a FE analysis involving Rayleigh damping with α = 0 and

β = η /G is equivalent to the closed form solution of the wave equation with frequency-dependent hysteretic

material damping.

Based on the above observations a stiffness proportional Rayleigh damping of β = 0.01 is selected for the

finite element analysis. This choice damps frequencies above 10 Hz appropriately.

Dynamic Test on Elastic-Plastic Soil Column

As the next step an elastic-plastic material model of Drucker-Prager type with kinematic strain hardening has been

selected. Previous analysis involving material with isotropic hardening have proved to be unsuitable because energy

can only be dissipated as the yield surface expands. For dynamic problems this can lead to an unreasonably large

extension of the yield surface, especially if resonance frequencies are present. Therefore only kinematic hardening

has been selected in this analysis.



0 5 10 150

5

10

15Solution obtained from FE analysis

Frequency [Hz]

Stiffness proportional Rayleigh Damping: β = 0.001Stiffness proportional Rayleigh Damping: β = 0.01

0 5 10 150

5

10

15Closed Form Solution

Frequency [Hz]

Frequency dependent Hysteretic Material Damping η = 0.001*GFrequency dependent Hysteretic Material Damping η = 0.01*GNo Damping

Figure 10.11: Transfer Function between Surface and Base of Soil Layer

The analysis were performed with four different ground motions using time steps of ∆t = 0.002s and ∆t =

0.001s. A linear integrator without iterations within a time step was used. All ground motions were scaled to a

maximum acceleration of 1g. For comparison the analysis were also performed on elastic material. Figure 10.12

shows the displacement time histories at the surface for all four ground motions. While the overall shapes of the

displacements are the same as for the elastic case there is some residual plastic displacement resulting in the time

histories of the Century City motions.

The Fourier amplitude spectra’s of the acceleration recorded at the surface (Figure 10.13) have the same

general shape for the case of elastic and elastic-plastic material. The amplification at the first resonance frequency

(f = 4.75Hz) is bigger in the elastic analysis. Higher frequencies resulting from plastic slip are damped out

effectively in the nonlinear analysis.

Figure 10.14 shows the acceleration time history at the upper node of the lowest element, that is the first

free node above the base. The record shows large peaks of the order of about 6 g. These peaks are caused by

plastic slip and counter balancing of the resulting plastic deformation. The periods of the peaks are of the order

of a few time steps, they add a very high frequency component to the acceleration. Because these frequencies

are due to a purely numerical phenomenon, they should not be allowed to propagate through the model. This

can be achieved easily by specifying an appropriate numerical procedure (Newmark with appropriate combination

of γ and β) or with Rayleigh Damping.

As for the elastic model transfer functions were also computed for the nonlinear model. In Figure 10.15 the

transfer functions between the acceleration at the soil surface and the base are compared. The functions for the



0 5 10 15 20 25 30

−0.1

0

0.1

Time [s]

Dis

plac

emen

t [m

] ElasticElastic−Plastic


0 5 10 15 20

−0.1

0

0.1

0.2

Time [s]

Dis

plac

emen

t [m

]

Lake Hughes

0 5 10 15 20 25

−0.2

0

0.2

Time [s]

Dis

plac

emen

t [m

]

Century City 090

0 5 10 15 20 25

−0.2

0

0.2

Time [s]

Dis

plac

emen

t [m

]

Century City 360

Figure 10.12: Displacement Time-Histories at surface of 1d Soil Column, elastic and elastic-plastic material

nonlinear model are not smooth anymore but the general shape is the same as for the linear elastic model, i.e.

the first natural frequency of the layer is clearly visible. The peaks that are present in the range of 25 Hz are

purely numerical as they appear due to the division by a very small value.

A second set of analysis performed with half the time step of the previous analysis gives an idea of the accuracy

of the numerical method. In Figure 10.16 the difference between the displacement (or acceleration) of the analysis

with ∆t = 0.002s and ∆t = 0.001s, divided by the corresponding maximum value is given for the entire time

history:

∆d =d0.002(t) − d0.001(t)

max d0.001and ∆a =

a0.002(t) − a0.001(t)

max a0.001(10.27)

In Figure 10.17 an integral measure for the difference in displacements and accelerations between the two analysis

is given for all depths. The integral measures are defined as

diffd =1

max |d|1

T

T∑

0

|d0.002(t) − d0.001(t)| dt (10.28)

diffa =1

max |a|1

T

T∑

0

|a0.002(t) − a0.001(t)| dt (10.29)

(10.30)

The integral differences in accelerations are quite large in the elements that are close to the base, that is where

the motion is applied. Toward the surface the difference becomes smaller than 1%. This is a result of the fact



0 5 10 150

0.2

0.4

0.6

Frequency [Hz]

Fou

rier

Am

plitu

de [m

/s2 ]


0 5 10 150

0.2

0.4

0.6

Frequency [Hz]F

ourie

r A

mpl

itude

[m/s

2 ]

Lake Hughes

0 5 10 150

0.5

1

Frequency [Hz]

Fou

rier

Am

plitu

de [m

/s2 ]

Century City 090

0 5 10 150

0.2

0.4

0.6

Frequency [Hz]

Fou

rier

Am

plitu

de [m

/s2 ]

ElasticElastic−Plastic

Century City 360

Figure 10.13: Fourier Amplitudes at surface of 1d Soil Column

that most of the plastic deformation occurs near the base which represents an undesired boundary effect. Again

this result underlines the importance of an appropriate choice of the size of the computational domain.

With a point wise difference not exceeding 5% for accelerations and 2% for displacements the time step

∆t = 0.002s is sufficiently small to ensure stable and accurate results.

2d Model

A 2d-model is proposed as a simplification of the full 3d-model. Representing a cross section of the full model it

is expected to provide insight into its dynamic behavior while requiring considerably less computational resources.

The 2d-model consists of one slice of eight-node brick elements as shown in Figure 10.18. The nodes of the two

lateral faces are constrained to move together in x- and z-direction, the out-of-plane displacement in y-direction

is fixed. The model approximates a plane strain situation.

The earthquake motion is applied to the model by the DRM method.

Input Motions

As input for the 2d model the motion from the Northridge earthquake recorded at LA University Hotel (Figure

10.10) is used. The acceleration time history is scaled to a peak ground acceleration of 1 g. Motion is applied in

x-direction only, that is, this is a 1–D wave propagation.



12.6 12.65 12.7 12.75−40

−20

0

20

40

60

Time [s]

Acc

eler

atio

n [m

/s2 ]

0 5 10 15 20 25−60

−40

−20

0

20

40

60

Time [s]

Acc

eler

atio

n [m

/s2 ]

∆ t = 0.002 s∆ t = 0.001 s

Figure 10.14: Acceleration time history at lowest free node

Acceleration time histories at all the nodes of the boundary layer are obtained by vertically propagating a

plane wave using the program SHAKE91 (Idriss and Sun, 1992). Because the free-field model has to match

the properties of the free field as represented by the finite element model for the reduced domain, only linear

elastic material without strain dependent reduction of shear modulus and a constant amount of hysteretic material

damping is used in the SHAKE91-analysis. The earthquake motion obtained in this way corresponds to a shear

wave propagating upward through a homogeneous linearly elastic half space.

The acceleration time histories from the SHAKE91-analysis are then integrated twice to obtain displacements.

Before integration the acceleration and velocity time histories are transformed into Fourier space, multiplied with a

high pass filter and transformed back into time domain. Then a simple parabolic baseline correction is performed

in order to obtain zero initial, final and mean values.

Boundary Conditions

Different boundary conditions are tested on the free-field model. First all outside boundary nodes are fully fixed

as shown in Figure 10.19 a). Then they are released and attached to dash pots that are both perpendicular

and tangential to the boundary (schematically shown in Figure 10.19 b)). The dash pots perpendicular to the

boundary are specified to absorb p-waves, those tangential to the boundary to absorb s-waves. Because in this

configuration no displacement constraint is imposed to the model on the faces at x = ± 10.5 meter the horizontal

at-rest soil pressure has to be applied to the corresponding nodes manually. This is done by recording the reaction

forces in the model with fixed boundaries and applying them with opposite sign to the model with absorbing

boundaries. The horizontal displacements after applying self weight should be very small.

This configuration of boundary conditions has no fixed point in x-direction. Because the dash pots only provide



0 5 10 15 20 250

1

2

3

4

5

Frequency [Hz]

TF

[−]

ElasticElastic−Plastic


0 5 10 15 20 250

1

2

3

4

5

Frequency [Hz]T

F [−

]

Lake Hughes

0 5 10 15 20 250

1

2

3

4

5

Frequency [Hz]

TF

[−]

Century City 090

0 5 10 15 20 250

1

2

3

4

5

Frequency [Hz]

TF

[−]

Century City 360

Figure 10.15: Transfer functions between acceleration at the soil surface and the base

resistance to high velocity motions the model is very sensitive to low frequency components of the motion. The

slightest imbalance in acceleration causes the entire model to move as a rigid body in x-direction. To avoid this

to happen the node at the center of the base (x = 0.0 m, z = -10.5 m) is fully fixed in the following analysis.

Figure 10.20 shows results from a free-field analysis on a homogeneous elastic model. Displacements on an

exterior boundary node as well as transfer functions between a point at the surface and a point on the exterior

boundary of the plastic bowl are presented for the two configurations of boundary conditions shown in Figure

10.19. It can be seen that the displacements outside the plastic bowl in the model using absorbing boundary

conditions are much larger compared to the model with fixed boundaries. This result is as expected considering

the immediate proximity of the boundary. It also gives an idea about the constraints the fixed boundary imposes

on the motions. The transfer function in Figure 10.20 b) is defined as the ratio between the Fourier amplitude

spectra of a point at the surface and a point on the exterior boundary:

TF (ω) =A1(ω)

A2(ω)(10.31)

where A1(ω) is the Fourier amplitude spectrum of the acceleration time history at the point (x,z) = (0,0)m

and A2(ω) the corresponding spectrum at the point (x,z) = (9.0,-7.5)m. The figure shows that the large peak

representing the first natural frequency of the system, corresponding to a standing shear wave in a soil layer of

10.5 meter depth, gets reduced considerably by the absorbing boundary. An energy build-up in the model due



0 5 10 15 20 25 30−5

0

5

Time [s]

Diff

eren

ce [%

] DisplacementAcceleration


0 5 10 15 20−5

0

5

Time [s]

Diff

eren

ce [%

]

Lake Hughes

0 5 10 15 20 25−5

0

5

Time [s]

Diff

eren

ce [%

]

Century City 090

0 5 10 15 20 25−5

0

5

Time [s]

Diff

eren

ce [%

]

Century City 360

Figure 10.16: Difference between results of analysis with different time steps, in percent of the maximum value

to reflection of waves on the model boundaries can be reduced effectively with the configuration of boundary

conditions shown in Figure 10.19 b). By releasing the fixed node at (x,z) = (0,-10.5)m the resonance peak could

be reduced by another 10% approximately, however at the cost of remaining permanent displacements at the end

of the analysis.

Alternatively to imposing a rigid constraint to a single node at the base the model can be prevented to move

horizontally as a rigid body through uniaxial springs. This gives the possibility to adjust the frequency of the

eigenmode that corresponds to a vertically propagating plane shear wave. By appropriately choosing the spring

constants the model can therefore be adjusted in such a way that it represents the natural frequency of a soil

deposit on bedrock.

Structure

Four very simple structures are chosen to illustrate the effects of dynamic SFSI. A beam-column element of length

L and moment of inertia Iy is fixed to a footing. A lumped mass of M = 100, 000 kg is added to the translational



LA University HospitalCentury City 090

Lake HughesCentury City 360

−9−7.5

−6−4.5

−3−1.5

0

0

0.5

1

Displacements

Depth [m]

Diff

eren

ce [%

]

LA University HospitalLake Hughes

Century City 360Century City 090

−9−7.5

−6−4.5

−3−1.5

0

0

2

4

6

8

Accelerations

Depth [m]

Diff

eren

ce [%

]

Figure 10.17: Averaged differences between results of analysis with different time steps

14 elements

y x

10.5 m

21 m

0

1.5 m

z

outside layer

boundary layer

plastic bowl

7 elements

Figure 10.18: Two-dimensional quasi-plane-strain model

degrees of freedom of the top of the structure. The footing is 0.5 m deep, spans over four soil elements and

is rigidly connected to the adjacent soil nodes. Its Young’s modulus is chosen large enough so that the footing



b)a)

Figure 10.19: The boundary conditions of the 2d model

0 5 10 15 20 25 30−5

0

5

10x 10

−3

Time [s]

Dis

plac

emen

t [m

]

Fixed boundaryAbsorbing boundary

0 2 4 6 8 10 12 14 16 18 200

5

10

15

Frequency [Hz]

[−]

maximum value = 5

maximum value = 60

Fixed boundaryAbsorbing boundary

a)

b)

Figure 10.20: Elastic homogeneous free-field model: a) Displacements of an exterior boundary node (x,z) =

(9.0,0.0) m, b) Acceleration transfer function between surface and depth A(ω)1A(ω)2

can be considered rigid. The mass density of the footing is ρ = 2400 kg/m3, the column is considered massless.

The moment connection between the nodes of the footing, having 3 (translational) degrees of freedom, and the

6 degrees of freedom of the nodes of the column is assured by a very stiff beam element that is connected to a

node at the bottom and a node at the top of the footing. The column is then simply connected to the upper

node of this auxiliary beam element.

The parameters of the four columns are chosen such that the second natural frequency, that is the natural

frequency attributed to bending of the column (Figure 10.22 b)), is evenly distributed over the frequency range

of the input motion (Figure 10.23). Structure 4 is designed such that it’s second natural frequency matches the

largest spike in the input motion. Table 10.1 lists the properties of the structures used in the analysis. For the

nonlinear columns a strain hardening material is chosen that consists of an initial elastic branch with tangent

modulus E and a post-yield branch with tangent modulus 0.2E. The Young’s modulus for all four structures is

E = 210GPa. The yield stress fy for structures 1, 2 and 4 is 20MPa and for structure 3 it is 2MPa.



mass

Figure 10.21: The 2d SFSI-model

b)a)

Figure 10.22: a) First eigenmode, b) second eigenmode of SFSI-system

0 1 2 3 4 5 60

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Fou

rier

Am

plitu

de [g

]

Frequency [Hz]

fn 1 f

n 2 f

n 3f

n 4

Figure 10.23: Fourier amplitude spectrum of input motion with second natural frequencies of the 4 SFSI-systems



Structure Length Stiffness E Iy Mass Yield Moment

[m] [MN m2] [kg] [kNm]

1 5.5 1680 100,000 800

2 3.5 5670 100,000 1,800

3 2.5 13440 100,000 320

4 5.0 5670 100,000 1,800

Table 10.1: Properties of the analyzed structures

01

23

45

6 2.5

3

3.5

4

4.50

2

4

6

f0 [Hz]

Linear structure

Input motions

Frequency [Hz]

Fou

rier

Acc

eler

atio

n A

mpl

itude

Figure 10.24: Parametric study of 15 linear structures with varying natural frequency.

Structure with Fixed Base

To begin with a parametric study of a series of structures with varying stiffnesses is analyzed. The stiffness is

varied by changing the width of the column section. The different structures are expected to respond specifically

to the frequency range of the input motion that is in the neighborhood of the natural frequency of the column.

The input motion that is applied at the base of the structure has been recorded in a previous free-field analysis

of the 2d-model.

The results of this parametric study are shown in Figures 10.24 and 10.25 for linear and nonlinear structures,

respectively. The Fourier amplitudes spectra’s of the acceleration at the top of the structure are plotted for 15

structures with variable natural frequency fn. A line of equal frequency is also provided. The input motion is

plotted in the background of the figure. It can be seen that the maxima of the frequency spectra’s are almost

perfectly aligned along the line of equal frequency. This is even more obvious in the case of a linear structure. In



01

23

45

6 2.5

3

3.5

4

4.50

0.5

1

1.5

f0 [Hz]

Nonlinear structure

Input motions

Frequency [Hz]

Fou

rier

Acc

eler

atio

n A

mpl

itude

Figure 10.25: Parametric study of 15 nonlinear structures with varying natural frequency

that case the responses of the structures are very narrow banded. As the structure remains elastic the top of the

structure oscillates mainly in its initial natural frequency. Lower and higher frequencies are eliminated to a great

extent.

In the case of the inelastic structure there is clearly more damping and reduction of the responses at some

(most) frequencies. The nonlinearity in structure is producing a longer effective period for the structure, and that

effective period changes during shaking. This in turn widens the frequency range of structural response. That is,

the response is lower, but the frequency characteristic is (much) wider.

A series of fixed-base analysis is also performed on the four structures mentioned in Section 10.7.1. The first

natural frequencies of the four structures with its base fixed, corresponding to the second mode of vibration of

the SFSI-model, are given in Table 10.2. It can be seen that the influence of the soil on the natural frequency of

the SFSI-system increases as the overall stiffness of the structure increases.

Structure 1st natural frequency 2nd natural frequency

of fixed-base system [Hz] of SFSI-system [Hz]

1 2.71 2.07

2 9.82 3.89

3 25.1 5.53

4 5.75 3.52

Table 10.2: Eigenfrequencies of the analyzed models



Results

The results of the SFSI- as well as the fixed-base analysis are presented in the following. The displacements at

the top of the nonlinear structures are recorded and plotted in Figure 10.26. It can be seen that the results from

0 5 10 15 20 25 30

−0.1

0

0.1

Time [s]

Dis

plac

emen

t [m

] Structure 1 Fixed BaseSFSI

0 5 10 15 20 25 30

−0.1

0

0.1

Time [s]

Dis

plac

emen

t [m


0 5 10 15 20 25 30

−0.1

0

0.1

Time [s]

Dis

plac

emen

t [m


0 5 10 15 20 25 30

−0.1

0

0.1

Time [s]

Dis

plac

emen

t [m


Figure 10.26: Displacements in x-direction at the top of the nonlinear structures

the SFSI- and the fixed-base-model differ considerably in terms of maximum as well as permanent displacement.

In contrast to this the displacements at the base of the column are almost identical for the two models (results

not plotted). Figure 10.27 displays the displacements at the top of structures 1 and 2 for all the combinations of

linear and nonlinear soil and structures that have been analyzed. Due to the low yield moment the permanent

displacement for structure 1 is relatively large in the analysis involving nonlinear columns. The results involving

nonlinear columns on linear and on nonlinear soil are very similar in their overall shape, however permanent

deformations are very different. It seems that the forces that trigger plastic deformations in the column strongly

depend on the behavior of the soil beneath the foundation.

In order to investigate the forces causing plastic deformations in the structures we look at the base moments

between foundation and column. In Figure 10.28 the moments at the base of the linear structures are plotted.

For structures 1 and 4 the moments for the fixed-base model are higher than for the SFSI-model. This means

that in this case neglecting the effects of SFSI leads to a conservative design. Structures 2 and 3 however have

to resist higher moments when SFSI is taken into account. Because the SFSI-system is more flexible than the

fixed-base structure its modes of vibration are excited by a different range of frequencies contained in the input



05

1015

2025

30

−0.

15

−0.

1

−0.

050

0.050.

1

0.15

Tim

e [s

]

Displacement [m]

Str

uctu

re 1

Fix

ed B

ase

− n

onlin

ear

colu

mn

SF

SI −

non

linea

r so

il, li

near

col

umn

SF

SI −

non

linea

r so

il, n

onlin

ear

colu

mn

SF

SI −

line

ar s

oil,

nonl

inea

r co

lum

n

05

1015

2025

30

−0.

15

−0.

1

−0.

050

0.050.

1

0.15

Tim

e [s

]

Displacement [m]

Str

uctu

re 2

Fix

ed B

ase

SF

SI −

non

linea

r so

il, li

near

col

umn

SF

SI −

non

linea

r so

il, n

onlin

ear

colu

mn

SF

SI −

line

ar s

oil,

nonl

inea

r co

lum

n

Figure 10.27: Displacements in x-direction at the top of structures 1 and 2

motion. For a particular motion this can lead to resonance of the SFSI-system. This result is in contradiction

with current engineering practice suggesting that neglecting SFSI in general leads to a more conservative design.

Figure 10.29 shows the moments at the base of structures 1 to 4, this time for the analysis involving nonlinear

column elements. The evolution of the second natural frequency of the SFSI-system is also provided as a qualitative

indication for when plastic deformations occur. The base moments for structures 1 and 3 in the fixed-base- and

the SFSI-analysis are very similar. Due to the low yield moment of the structure no resonance with the input

motion occurs as a lot of energy is dissipated through plastic deformation.

Figure 10.30 shows an interesting aspect of nonlinear SFSI. In the analysis involving elastic-plastic soil the

Fourier amplitudes of the moment at the base of the structure are reduced in the neighborhood of the natural

frequency of the system. This is most likely due to dissipation of energy caused by elastic-plastic deformations

in the soil that, in their turn, are a result of large loads provoked by resonance between the SFSI-system and the

input motion.

As a measure of the plastic strain occurring beneath the footing the equivalent plastic strains averaged over



5 10 15 20−10

−5

0

5

10

Time [s]

Mom

ent [

MN

m]

Structure 1Fixed BaseSFSImax/min(Fixed Base)max/min(SFSI)

5 10 15 20−4

−2

0

2

4

Time [s]

Mom

ent [

MN

m]

Structure 2 Fixed BaseSFSImax/min(Fixed Base)max/min(SFSI)

5 10 15 20−3

−2

−1

0

1

2

3

Time [s]

Mom

ent [

MN

m]


5 10 15 20−10

−5

0

5

10

Time [s]

Mom

ent [

MN

m]


Figure 10.28: Moments at the base of the linear column

all the Gauss points are calculated. The results are given at t = 12 s and at t = 14 s, that is shortly before and

after the largest plastic deformation occurs (Figures 10.31 and 10.32).

Plastic strains are larger in the analysis involving an elastic structure. This reflects the fact that elastic

structures don’t dissipate any energy by themselves. For structure 2 no significant difference can be observed

because of its high yield moment. Structure 4 is characterized by the same yield moment, its slightly smaller

natural frequency however causes resonance with the input motion which leads to larger plastic strains beneath

the footing. The largest plastic strains develop in the layer of elements adjacent to the boundary layer. This can

be due to an input motion that isn’t fully compatible with the elastic properties of the DRM-model. It should be

possible to reduce these undesired plastic strains by either increasing the size of the soil model or by selecting a

method to obtain the free-field motions that represents the soil properties of the DRM-model more closely.



5 10 15 20−10

−5

0

5

10

Time [s]

Mom

ent [

MN

m]


0

2

4

6

8

Nat

ural

Fre

quen

cy [H

z]

5 10 15 20−4

−2

0

2

4

Time [s]

Mom

ent [

MN

m]


0

4

8

12

16

Nat

ural

Fre

quen

cy [H

z]

5 10 15 20−3

−2

−1

0

1

2

3

Time [s]

Mom

ent [

MN

m]


0

2.5

5

7.5

10

12.5

15

Nat

ural

Fre

quen

cy [H

z]

5 10 15 20−10

−5

0

5

10

Time [s]

Mom

ent [

MN

m]


0

4

8

12

16

Nat

ural

Fre

quen

cy [H

z]

Figure 10.29: Moments at the base of the nonlinear column

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3x 10

5

Frequency [Hz]

Fou

rier

Am

plitu

de [N

m]

Elastic soilElastic−plastic soil

Figure 10.30: Fourier amplitude spectra of moments at the base of nonlinear column, SFSI-analysis



Structure 1

Ela

stic

Col

umn

Inel

astic

Col

umn

Structure 2 Structure 3 Structure 4

Figure 10.31: Average equivalent plastic strain at time t = 12 s

Structure 1

Ela

stic

Col

umn

Inel

astic

Col

umn

Structure 2 Structure 3 Structure 4

Figure 10.32: Average equivalent plastic strain at time t = 14 s

10.7.2 Full nonlinear 3d Model

The 2d SFSI-model presented in the previous section is extended to a 3d model in the following. The goal is

to show that the considerations for accuracy and stability of the numerical method obtained from the 1d-model

remain valid for the 3d-model. Even if the simplicity of the analyzed problem doesn’t necessarily justify the

additional computational effort it is important to show that it is possible to obtain reliable results for a problem

that involves the following elements:

• 3d model with about 700 elements, 960 nodes and 2700 equations

• Elastic-plastic soil (Drucker-Prager with kinematic hardening)

• Nonlinear structure (bilinear material model)

• Ground motion applied through the Domain Reduction Method (DRM)

• Absorbing boundary of Lysmer type

Description of Model

The 3d model is based on the 2d model shown in Figure 10.21. In y-direction 6 more slices of 7 x 14 elements are

added (Figure 10.33). The x-z plane at y = 0 represents a plane of symmetry. Lysmer boundaries are attached



14 elements

7 elements

boundary layer

outside layer

plastic bowl

10.5 m

10.5 m

21 m

7 elements

xy

z

0

mass

Figure 10.33: The full 3d-model

to all outside boundaries with the exception of the plane of symmetry and the soil surface. The main difference

to the 2d model is that 3d wave propagation is possible which leads to higher radiation damping.

The structure was chosen to have the same geometric and material properties as Structure 4 in the previous

section.

Results

Some results of the 3d-analysis together with the corresponding data of the 2d analysis are presented in Figure

10.34. Due to limited memory only the first 20 seconds of the time history were processed. A more efficient

implementation of the application of effective forces for the DRM-method inside the finite element code should

solve this problem. The analysis took 66 hours to finish.

The displacements obtained at the top of the structure as well as the moments at its base are very close to

the results of the 2d-analysis. This shows that the analysis provides reliable results for a full 3d nonlinear SFSI

problem. The amplitude of the base moment is at several instances larger for the 3d-model than for the 2d-model.

This can be explained with the fact that more energy is present in the 3d-model whereas the energy the structure

can absorb is the same as in the 2d-model. Also it is obvious that the natural frequencies of the 3d-model are

not exactly the same as for the 2d-model and therefore changes the dynamic behavior in a way that is almost

impossible to predict beforehand.

Because of the simple geometry of the problem the 2d-model is absolutely sufficient for analyzing the forces

acting on the structure. If one is interested in the stress history in the soil surrounding the footing then the

3d-model can provide valuable additional information.



0 5 10 15 20 25

−0.1

0

0.1

Time [s]

Dis

plac

emen

t [m

]

2d SFSI3d SFSI

0 5 10 15 20 25−4

−3

−2

−1

0

1

2

3

4

Time [s]

Mom

ent [

MN

m]

2d SFSI3d SFSI

Figure 10.34: Top: Displacements at the top of Structure 4, Bottom: Moments at the base of Structure 4




Chapter 11

Parallel Computing in Computational

Geomechanics (1998–2000-2005–)

(In collaboration with Dr. Guanzhou Jie)

247



Chapter 12

Practical Applications (1994–)

12.1 Consolidation of Clays

(In collaboration with Mr. Zhao Cheng)

the 1 dimensional model used for consolidation verification is shown in Figure 12.2 below.

P0

A

B

C

D

F

G

H

I

J

K

E

Figure 12.1: Finite element for 1D consolidation analysis.

The model was verified against closed form solutions for elastic material available in

249


P0

A

B

C

D

F

G

H

I

J

K

E

Figure 12.2: Finite element for 1D consolidation analysis.



12.2 Staged Construction Analysis

(In collaboration with Ms. Guanzhou Jie)



12.3 Seismic Wave Propagation in Soils (Ground Motions)

(In collaboration with Mr. Matthias Preisig, Mr. Guanzhou Jie and Mr. Kallol Sett)



12.4 Static and Dynamic Behavior of Pile Foundations in Dry and Sat-

urated Soils

(In collaboration with Mr. Guanzhou Jie and Mr. Zhao Cheng)



12.5 Static and Dynamics Behavior of Shallow Foundations

(In collaboration with Mr. George Hue and Mr. Guanzhou Jie)


Bibliography

R. Abraham, J. Marsden, and T. Ratiu. Manifolds, Tensor Analysis, and Applications, volume 75 of Applied

Mathematical Sciences. Springer Verlag, second edition, 1988.

Working Paper for Draft Proposed International Standard for Information Systems–Programming Language C++.

ANSI/ISO, Washington DC, April 1995. Doc. No. ANSI X3J16/95-0087 ISO WG21/N0687.

J. Argyris and H.-P. Mlejnek. Dynamics of Structures. North Holland in USA Elsevier, 1991a.

J. Argyris and H.-P. Mlejnek. Dynamics of Structures. North Holland in USA Elsevier, 1991b.

P. Armstrong and C. Frederick. A mathematical representation of the multiaxial bauschinger effect. Technical

Report RD/B/N/ 731,, C.E.G.B., 1966.

A. Baltov and A. Sawczuk. A rule of anisotropic hardening. Acta Mechanica, I(2):81–92, 1965.

K.-J. Bathe. Finite Element Procedures in Engineering Analysis. Prentice Hall Inc., 1996. ISBN 0-13-301458-4.

K.-J. Bathe. Finite Element Procedures in Engineering Analysis. Prentice Hall Inc., 1982.

K.-J. Bathe and E. Dvorkin. On the automatic solution of nonlinear finite element equations. Computers &

Structures, 17(5-6):871–879, 1983.

K.-J. Bathe and L. Wilson, Edward. Numerical Methods in Finite Element Analysis. Prentice Hall Inc., 1976.

J. Batoz and G. Dhatt. Incremental displacement algorithms for nonlinear problems. International Journal for

Numerical Methods in Engineering, 14:1262–1267, 1979. Short Communications.

T. Belytschko, W. K. Liu, and B. Moran, editors. Nonlinear Finite Element for Continua and Structures. John

Wiley & Sons Ltd., New York, 2001.

P. N. Bicanic. Exact evaluation of contact stress state in computational elasto plasticity. Engineering Computa-

tions, 6:67–73, 1989.

J. Bielak, K. Loukakis, Y. Hisada, and C. Yoshimura. Domain reduction method for three–dimensional earthquake

modeling in localized regions. part I: Theory. Bulletin of the Seismological Society of America, 93(2):817–824,

2003.

255


B. A. Bilby, L. R. T. Gardner, and A. N. Stroh. Continuous distributions of dislocations and the theory of plasticity.

In IXe Congres International de Mechanique Appliquee, volume VIII, pages 35–44, Universite de Bruxelles, 50.

Avenue Franklin Roosevelt, 1957.

G. Booch. Object Oriented Analysis and Design with Applications. Series in Object–Oriented Software Engineering.

Benjamin Cummings, second edition, 1994.

R. I. Borja. Cam – clay plasticity, part II: Implicit integration of constitutive equation based on a nonlinear elastic

stress predictor. Computer Methods in Applied Mechanics and Engineering, 88:225–240, 1991.

R. I. Borja and S. R. Lee. Cam–clay plasticity, part I: Implicit integration of elasto–plastic constitutive relations.

Computer Methods In Applied Mechanics and Engineering, 78:49–72, 1990.

H. J. Braudel, M. Abouaf, and J. L. Chenot. An implicit and incremental formulation for the solution of elasto-

plastic problems , by the finite element method. Computers Structures, 22(5):801–814, 1986.

W. F. Chen and D. J. Han. Plasticity for Structural Engineers. Springer Verlag, 1988.

C. H. Choi. Physical and Mathematical Modeling of Coarse-Grained Soils. PhD thesis, Department of Civil and

Environmental Engineering, University of Washington, Seattle, Washington, 2004.

J. O. Coplien. Advanced C++, Programming Styles and Idioms. Addison – Wesley Publishing Company, 1992.

O. Coussy. Mechanics of Porous Continua. John Wiley and Sons, 1995. ISBN 471 95267 2.

M. A. Crisfield. Non–Linear Finite Element Analysis of Solids and Structures Volume 1: Advanced Topics. John

Wiley and Sons, Inc. New York, 605 Third Avenue, New York, NY 10158–0012, USA, 1997.

M. A. Crisfield. Consistent schemes for plasticity computation with the Newton Raphson method. Computational

Plasticity Models, Software, and Applications, 1:136–160, 1987.

M. A. Crisfield. Non–Linear Finite Element Analysis of Solids and Structures Volume 1: Essentials. John Wiley

and Sons, Inc. New York, 605 Third Avenue, New York, NY 10158–0012, USA, 1991.

Y. F. Dafalias and M. T. Manzari. Simple plasticity sand model accounting for fabric change effects. Journal of

Engineering Mechanics, 130(6):622–634, 2004.

J. E. Dennis, Jr. and R. B. Schnabel. Numerical Methods for Unconstrained Optimization and Nonlinear Equations.

Prentice Hall , Engelwood Cliffs, New Jersey 07632., 1983.

P. Donescu and T. A. Laursen. A generalized object–oriented approach to solving ordinary and partial differential

equations using finite elements. Finite Elements in Analysis and Design, 22:93–107, 1996.

J. M. Duncan and C.-Y. Chang. Nonlinear analysis of stress and strain in soils. Journal of Soil Mechanics and

Foundations Division, 96:1629–1653, 1970.

S. Dunica and B. Kolundzija. Nelinarna Analiza Konstrukcija. Gradjevinski fakultet Univerziteta u Beogradu,

Bulevar revolucije 73 i IRO ”Naucna knjiga”, Beograd, Uzun–Mirkova 5, 1986. Nonlinear Analysis of Structures,

In Serbian.



M. Y. E. and E. N. M. H. Dynamic analysis of laterally loaded pile groups in sand and clay. Canadian Geotechnical

Journal, 39(6):1358–1383, 2002.

B. Eckel. Using C++. Osborne McGraw – Hill, 1989.

D. Eyheramendy and T. Zimmermann. Object–oriented finite elements II. a symbolic environment for automatic

programming. Computer Methods in Applied Mechanics and Engineering, 132:277–304, 1996.

C. A. Felippa. Dynamic relaxation under general incremental control. In W. K. Liu, T. Belytschko, and K. C. Park,

editors, Innovative Methods for Nonlinear Problems, pages 103–133. Pineridge Press, Swansea U.K., 1984.

C. A. Felippa. Nonlinear finite element methods. Lecture Notes at CU Boulder, January-May 1993.

P. A. Ferrer. Elastoplatic characterization of granular materials. Master’s thesis, University of Puerto Rico, 1992.

B. W. R. Forde, R. O. Foschi, and S. F. Steimer. Object – oriented finite element analysis. Computers and

Structures, 34(3):355–374, 1990.

N. Fox. On the continuum theories of dislocations and plasticity. Quarterly Journal of Mechanics and Applied

Mathematics, XXI(1):67–75, 1968.

A. Gajo. Influence of viscous coupling in propagation of elastic waves in saturated soil. ASCE Journal of

Geotechnical Engineering, 121(9):636–644, September 1995.

A. Gajo and L. Mongiovi. An analytical solution for the transient response of saturated linear elastic porous media.

International Journal for Numerical and Analytical Methods in Geomechanics, 19(6):399–413, 1995.

B. O. Hardin. The nature of stress–strain behavior of soils. In Proceedings of the Specialty Conference on

Earthquake Engineering and Soil Dynamics, volume 1, pages 3–90, Pasadena, 1978.

S. Helgason. Differential Geometry, Lie Groups, and Symmetric Spaces. Pure and Applied Mathematics. Academic

Press, 1978.

H. M. Hilber, T. J. R. Hughes, and R. L. Taylor. Improved numerical dissipation for time integration algorithms

in structural dynamics. Earthquake Engineering and Structure Dynamics, pages 283–292, 1977.

R. Hill. Aspects of invariance in solid mechanics. In C.-S. Yih, editor, Advances in Applied Mechanics, volume 18,

pages 1–72. Academic Press, 1978.

K. Hjelmstad. Fundamentals of Structural Mechanics. Prentice–Hall, 1997. ISBN 0-13-485236-2.

T. Hughes. The Finite Element Method ; Linear Static and Dynamic Finite Element Analysis. Prentice Hall Inc.,

1987.

T. Hughes and K. Pister. Consistent linearization in mechanics of solids and structures. Computers and Structures,

8:391–397, 1978.

T. J. R. Hughes and W. K. Liu. Implicit-explicit finite elements in transient analaysis: implementation and

numerical examples. Journal of Applied Mechanics, pages 375–378, 1978a.



T. J. R. Hughes and W. K. Liu. Implicit-explicit finite elements in transient analaysis: stability theory. Journal of

Applied Mechanics, pages 371–374, 1978b.

I. M. Idriss and J. I. Sun. Excerpts from USER’S Manual for SHAKE91: A Computer Program for Conducting

Equivalent Linear Seismic Response Analyses of Horizontally Layered Soil Deposits. Center for Geotechnical

Modeling Department of Civil & Environmental Engineering University of California Davis, California, 1992.

N. Janbu. Soil compressibility as determined by odometer and triaxial tests. In Proceedings of European Conference

on Soil Mechanics and Foundation Engineering, pages 19–25, 1963.

B. Jeremic. nDarray programming tool. Object Oriented Approach to Numerical Computations in Elastoplactity,

Reference Manual University of Colorado at Boulder, December 1993.

B. Jeremic and M. Preisig. Seismic soil foundation structure interaction: Numerical modeling issues. In SEI

Structures Congress, 2005.

B. Jeremic and S. Sture. Implicit integrations in elasto–plastic geotechnics. International Journal of Mechanics

of Cohesive–Frictional Materials, 2:165–183, 1997.

B. Jeremic and S. Sture. Tensor data objects in finite element programming. International Journal for Numerical

Methods in Engineering, 41:113–126, 1998.

P. Jetteur. Implicit integration algorithm for elastoplasticity in plane stress analysis. Engineering Computations,

3:251–253, 1986.

W. Karush. Minima of functions of several variables with inequalities as side constraints. Master’s thesis, University

of Chicago, Chicago, IL., 1939.

A. Koenig. C++ columns. Journal of Object Oriented Programming, 1989 - 1993.

W. T. Koiter. General theorems for elastic-plastic solids. In I. N. Sneddon and R. Hill, editors, Progress in Solid

Mechanics, pages 165–221. North Holland, 1960.

W. T. Koiter. Stress - strain relations, uniqueness and variational theorems for elastic - plastic materials with

singular yield surface. Quarterly of Applied Mathematics, 2:350–354, 1953.

M. Kojic. Computational Procedures in Inelastic Analysis of Solids and Structures. Center for Scientific Research

of Srbian Academy of Sciences and Arts and University of Kragujevac and Faculty of Mechanical Engineering

University of Kragujevac, 1997. ISBN 86-82607-02-6.

S. L. Kramer. Geotechnical Earthquake Engineering. Prentice Hall, Inc, Upper Saddle River, New Jersey, 1996.

E. Kroner. Algemeine kontinuumstheorie der versetzungen und eigenspanungen. Archive for Rational Mechanics

and Analysis, 4(4):273–334, 1960.

H. W. Kuhn and A. W. Tucker. Nonlinear programming. In J. Neyman, editor, Proceedings of the Second Berkeley

Symposium on Mathematical Statistics and Probability, pages 481 – 492. University of California Press, July

31 – August 12 1950 1951.



P. V. Lade. Model and parameters for the elastic behavior of soils. In Swoboda, editor, Numerical Methods in

Geomechanics, pages 359–364, Innsbruck, 1988a. Balkema, Rotterdam.

P. V. Lade. Effects of voids and volume changes on the behavior of frictional materials. International Journal for

Numerical and Analytical Methods in Geomechanics, 12:351–370, 1988b.

P. V. Lade and R. B. Nelson. Modeling the elastic behavior of granular materials. International Journal for

Numerical and Analytical Methods in Geomechanics, 4, 1987.

W. T. Lambe and R. V. Whitman. Soil Mechanics, SI Version. John Wiley and Sons, 1979.

E. H. Lee. Elastic–plastic deformation at finite strains. Journal of Applied Mechanics, 36(1):1–6, 1969.

E. H. Lee and D. T. Liu. Finite–strain elastic–plastic theory with application to plane–wave analysis. Journal of

Applied Physics, 38(1):19–27, January 1967.

X. S. Li and Y. Wang. Linear representation of steady-state line for sand. Journal of Geotechnical and Geoenvi-

ronmental Engineering, 124(12):1215–1217, 1998.

J. Lubliner. Plasticity Theory. Macmillan Publishing Company, New York., 1990.

L. E. Malvern. Introduction to the Mechanics of a Continuous Medium. In Engineering of the Physical Sciences.

Prentice Hall Inc., 1969.

M. T. Manzari and Y. F. Dafalias. A critical state two–surface plasticity model for sands. Geotechnique, 47(2):

255–272, 1997.

M. T. Manzari and R. Prachathananukit. On integartion of a cyclic soil plasticity model. International Journal

for Numerical and Analytical Methods in Geomechanics, 25:525–549, 2001.

J. M. M. C. Marques. Stress computations in elastoplasticity. Engineering Computations, 1:42–51, 1984.

J. E. Marsden and T. J. R. Hughes. Mathematical Foundations of Elasticity. Prentice Hall Inc,, 1983. local

CM65; 4QA 931.M42 ; ISBN 0-13-561076-1.

G. R. Miller. An object – oriented approach to structural analysis and design. Computers and Structures, 40(1):

75–82, 1991.

G. P. Mitchell and D. R. J. Owen. Numerical solution for elasto - plastic problems. Engineering Computations,

5:274–284, 1988.

K. N. Morman. The generalized strain meassures with application to nonhomogeneous deformation in rubber–like

solids. Journal of Applied Mechanics, 53:726–728, 1986.

G. C. Nayak and O. C. Zienkiewicz. Elasto - plastic stress analysis a generalization for various constitutive relations

including strain softening. International Journal for Numerical Methods in Engineering, 5:113–135, 1972.

N. M. Newmark. A method of computation for structual dynamics. Journal of the Engineering Mechanics Division,

ASCE, pages 67–94, 1959.



W. L. Oberkampf, T. G. Trucano, and C. Hirsch. Verification, validation and predictive capability in computational

engineering and physics. In Proceedings of the Foundations for Verification and Validation on the 21st Century

Workshop, pages 1–74, Laurel, Maryland, October 22-23 2002. Johns Hopkins University / Applied Physics

Laboratory.

R. W. Ogden. Non–Linear Elastic Deformations. Series in mathematics and its applications. Ellis Horwood

Limited, Market Cross House, Cooper Street, Chichester, West Sussex, PO19 1EB, England, 1984.

M. Ortiz and E. P. Popov. Accuracy and stability of integration algorithms for elastoplastic constitutive relations.

International Journal for Numerical Methods in Engineering, 21:1561–1576, 1985.

C. E. Pearson, editor. Handbook of Applied Mathematics. Van Nostrand Reinhold Company, 1974.

A. Perez-Foguet and A. Huerta. Plastic flow potetnial for the cone region of the MRS–Lade model.

A. Perez-Foguet, A. Rodrıguez-Ferran, and A. Huerta. Numerical differentiation for non-trivial consistent tangent

matrices: an application to the mrs-lade model. International Journal for Numerical Methods in Engineering,

48:159–184, 2000.

D. Peric. Localized Deformation and Failure Analysis of Pressure Sensitive Granular Material. PhD thesis,

University of Colorado at Boulder, 1991.

R. M. V. Pidaparti and A. V. Hudli. Dynamic analysis of structures using object–oriented techniques. Computers

and Structures, 49(1):149–156, 1993.

W. H. Press, B. P. Flannery, S. A. teukolsky, and W. T. Vetterling. Numerical Recipes in C. Cambridge University

Press, Cambridge, 1988a.

W. H. Press, B. P. Flannery, S. A. Teukolvsky, and W. T. Vetterling. Numerical Recipes in C, The Art of Scientific

Computing. Cambridge University Press, 1988b.

E. Ramm. Strategies for tracing the nonlinear response near limit points. In W. Wunderlich, E. Stein, and

K.-J. Bathe, editors, Nonlinear Finite Element Analysis in Structural Mechanics, pages 63–89. Springer–Werlag

Berlini Heidelberg New York, 1981.

E. Ramm. The Riks/Wempner approach – an extension of the displacement control method in nonlinear analysis.

In E. Hinton, D. Owen, and C. Taylor, editors, Recent Advances in Non–Linear Computational Mechanics,

chapter 3, pages 63–86. Pineridge Press, Swansea U.K., 1982.

E. Ramm and A. Matzenmiller. Consistent linearization in elasto - plastic shell analysis. Engineering Computations,

5:289–299, 1988.

F. E. Richart, J. J. R. Hall, and R. D. Woods. Vibration of Soils and Foundations, International Series in

Theoretical and Applied Mechanics. Prentice-Hall, Englewood Cliffs, N.J., 1970.

E. Riks. The application of Newton’s method to the problems of elastic stability. Journal of Applied Mechanics,

39:1060–1066, December 1972.



E. Riks. An incremental approach to the solution of snapping and buckling problems. International Journal for

Solids and Structures, 15:529–551, 1979.

A. D. Robison. C++ gets faster for scientific computing. Computers in Physics, 10(5):458–462, Sept/Oct 1996.

K. H. Roscoe and J. B. Burland. On the generalized stress–strain behavior of wet clays. In J. Heyman and F. A.

Leckie, editors, Engineering Plasticity, pages 535–609, 1968.

K. H. Roscoe, A. N. Schofield, and C. P. Wroth. On the yielding of soils. Geotechnique, 8(1):22–53, 1958.

K. H. Roscoe, A. N. Schofield, and A. Thurairajah. Yielding of clays in states wetter than critical. Geotechnique,

13(3):211–240, 1963.

K. Runesson. Constitutive theory and computational technique for dissipative materials with emphasis on plas-

ticity, viscoplasticity and damage: Part III. Lecture Notes, Chalmers Technical University, Goteborg, Sweden,

September 1996.

K. Runesson. Implicit integration of elastoplastic relations with reference to soils. International Journal for

Numerical and Analytical Methods in Geomechanics, 11:315–321, 1987.

K. Runesson and A. Samuelsson. Aspects on numerical techniques in small deformation plasticity. In G. N. P.

J. Middleton, editor, NUMETA 85 Numerical Methods in Engineering, Theory and Applications, pages 337–347.

AA.Balkema., 1985.

J. C. J. Schellekens and H. Parisch. On finite deformation elasticity. Technical Report ISD Nr. 94/3, Institute for

Statics and Dynamics of Aerospace Structures, Pfaffenwaldring 27, 70550 Stutgart, 1994.

A. N. Schofield and C. P. Wroth. Critical State Soil Mechanics. McGraww– Hill, 1968.

S.-P. Scholz. Elements of an object – oriented FEM++ program in C++. Computers and Structures, 43(3):

517–529, 1992.

J. C. Simo. A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplica-

tive decomposition: Part i. continuum formulation. Computer Methods in Applied Mechanics and Engineering,

66:199–219, 1988. TA345. C6425.

J. C. Simo. Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping

schemes of the infinitesimal theory. Computer Methods in Applied Mechanics and Engineering, 99:61–112,

1992.

J. C. Simo and S. Govindjee. Exact closed - form solution of the return mapping algorith in plane stress elasto -

viscoplasticity. Engineering Computations, 5:254–258, 1988.

J. C. Simo and T. J. R. Hughes, editors. Computational Inelasticity. Springer-verlag, New York, 1998.

J. C. Simo and C. Miehe. Associative coupled thermoplasticity at finite strains: Formulation, numerical analysis

and implementation. Computer Methods in Applied Mechanics and Engineering, 98:41–104, 1992.



J. C. Simo and K. S. Pister. Remarks on rate constitutive equations for finite deformations problems: Computa-

tional implications. Computer Methods in Appliied Mechanics and Engineering, 46:201–215, 1984.

J. C. Simo and R. L. Taylor. Quasi–incompressible finite elasticity in principal stretches. continuum basis and

numerical algorithms. Computer Methods in Applied Mechanics and Engineering, 85:273–310, 1991.

J. C. Simo and R. L. Taylor. Consistent tangent operators for rate-independent elastoplasticity. Computer Methods

in Applied Mechanics and Engineering, 48:101–118, 1985.

J. C. Simo and R. L. Taylor. A returning mapping algorithm for plane stress elastoplasticity. International Journal

for Numerical Methods in Engineering, 22:649–670, 1986.

A. Spencer. Continuum Mechanics. Longman Mathematical Texts. Longman Group Limited, 1980.

S.Timoshenko and D.H.Young. Engineering Mechanics. McGraw-Hill Book Company, London, 1940.

B. Stroustrup. The Design and Evolution of C++. Addison–Wesley Publishing Company, 1994.

S. Sture. Engineering properties of soils. Lecture Notes at CU Boulder, January - May 1993.

T. C. T. Ting. Determination of C1/2, C−1/2 and more general isotropic tensor functions of C. Journal of

Elasticity, 15:319–323, 1985.

I. Vardoulakis and J. Sulem. Bifurcation Analysis in Geomechanics. Blackie Academic & Professional, 1995. ISBN

0-7514-0214-1.

Various Authors. The C++ report: Columns on C++, 1991-.

T. Veldhuizen. Expression templates. C++ Report, 7(5):26–31, June 1995.

T. Veldhuizen. Rapid linear algebra in C++. Dr. Dobb’s Journal, August 1996.

G. A. Wempner. Discrete approximations related to nonlinear theories of solids. International Journal for Solids

and Structures, 7:1581–1599, 1971.

J. P. Wolf. Dynamic Soil-Structure-Interaction. Prentice-Hall, Englewood Cliffs (NJ), 1985.

J. P. Wolf. Soil-Structure-Interaction Analysis in Time Domain. Prentice-Hall, Englewood Cliffs (NJ), 1988.

J. P. Wolf. Foundation Vibration Analysis Using Simple Physical Models. Prentice-Hall, Englewood Cliffs (NJ),

1994.

J. P. Wolf and M. Preisig. Dynamic stiffness of foundation embedded in layered halfspace based on wave

propagation in cones. Earthquake Engineering and Structural Dynamics, 32(7):1075–1098, JUN 2003.

J. P. Wolf and C. M. Song. Some cornerstones of dynamic soil-structure interaction. Engineering Structures, 24

(1):13–28, 2002.

D. M. Wood. Soil Behaviour nad Critical State Soil Mechanics. Cambridge University Press, 1990.



Z. Yang and B. Jeremic. Numerical analysis of pile behaviour under lateral loads in layered elastic-plastic soils.

International Journal for Numerical and Analytical Methods in Geomechanics, 26(14):1385–1406, 2002.

Z. Yang and B. Jeremic. Numerical study of the effective stiffness for pile groups. International Journal for

Numerical and Analytical Methods in Geomechanics, 27(15):1255–1276, Dec 2003.

C. Yoshimura, J. Bielak, and Y. Hisada. Domain reduction method for three–dimensional earthquake modeling in

localized regions. part II: Verification and examples. Bulletin of the Seismological Society of America, 93(2):

825–840, 2003.

G. W. Zeglinski, R. S. Han, and P. Aitchison. Object oriented matrix classes for use in a finite element code using

C++. International Journal for Numerical Methods in Engineering, 37:3921–3937, 1994.

O. Zienkiewicz, A. Chan, M. Pastor, B. Schrefler, and T. Shiomi. Computational Geomechanics–with Special

Reference to Earthquake Engineering. John Wiley and Sons Ltd., Baffins Lane, Chichester, West Sussex PO19

IUD, England, 1999a.

O. C. Zienkiewicz and T. Shiomi. Dynamic behaviour of saturated porous media; the generalized Biot formulation

and its numerical solution. International Journal for Numerical Methods in Engineering, 8:71–96, 1984.

O. C. Zienkiewicz and R. L. Taylor. The Finite Element Method, volume 1. McGraw - Hill Book Company, fourth

edition, 1991a.

O. C. Zienkiewicz and R. L. Taylor. The Finite Element Method, volume 2. McGraw - Hill Book Company, Fourth

edition, 1991b.

O. C. Zienkiewicz, A. H. C. Chan, M. Pastor, B. A. Schrefler, and T. Shiomi. Computational Geomechanics with

Special Reference to Earthquake Engineering. John Wiley and Sons., 1999b. ISBN 0-471-98285-7.




Appendix A

nDarray

Material in this chapter is based on the following publications Jeremic (1993); Jeremic and Sture (1998).

This section describes a programming tool, nDarray, which is designed using an Object Oriented Paradigm

(OOP) and implemented in the C++ programming language. Finite element equations, represented in terms of

multidimensional tensors are easily manipulated and programmed. The usual matrix form of the finite element

equations are traditionally coded in FORTRAN, which makes it difficult to build and maintain complex program

systems. Multidimensional data systems and their implementation details are seldom transparent and thus not

easily dealt with and usually avoided. On the other hand, OOP together with efficient programming in C++

allows building new concrete data types, namely tensors of any order, thus hiding the lower level implementation

details. These concrete data types prove to be quite useful in implementing complicated tensorial formulae

associated with the numerical solution of various elastic and elastoplastic problems in solid mechanics. They

permit implementing complex nonlinear continuum mechanics theories in an orderly manner. Ease of use and the

immediacy of the nDarray programming tool in constitutive driver programming and in building finite element

classes will be shown.

A.1 Introduction

In implementing complex programming systems for finite element computations, the analyst is usually faced with

the challenge of transforming complicated tensorial formulae to a matrix form. Considerable amount of time

in solving problems by the finite element method is often devoted to the actual implementation process. If one

decides to use FORTRAN, a number of finite element and numerical libraries are readily available. Although quick

results can be produced in solving simpler problems, when implementing complex small deformation elastoplastic

or large deformation elastic and elastoplastic algorithms, C++ provides clear benefits.

Some of the improvements C++ provides over C and FORTRAN are classes for encapsulating abstractions, the

possibility of building user–defined concrete data types and operator overloading for expressing complex formulae

in a natural way. In the following we shall show that the nDarray tool will allow analysts to be a step closer to

the problem space and a step further away from the underlying machine.

As most analysts know, the intention (Stroustrup, 1994) behind C++ was not to replace C. Instead, C was

265


extended with far more freedom given to the program designer and implementor. In C and FORTRAN, large

applications become collections of programs and functions, order and the structure are left to the programmer.

The C++ programming language embodies the OOP, which can be used to simplify and organize complex

programs. One can build a hierarchy of derived classes and nest classes inside other classes. A concern in C

and FORTRAN programming languages is handling data type conflicts and data which are being operated on or

passed. The C++ programming language extends the definition of type to include abstract data types. With

abstract data types, data can be encapsulated with the methods that operate on it. The C++ programming

language offers structure and mechanisms to handle larger, more complex programming systems. Object Oriented

technology, with function and operator overloading, inheritance and other features, provides means of attacking a

problem in a natural way. Once basic classes are implemented, one can concentrate on the physics of a problem.

By building further abstract data types one can describe the physics of a problem rather that spend time on

the lower level programming issues. One should keep in mind the adage, credited to the original designer and

implementor of C++ programming language, Bjarne Stroustrup: “C makes it easy to shoot yourself in the foot,

C++ makes it harder, but when you do, it blows away your whole leg”.

Rather than attempting here to give a summary of Object Oriented technology we will suggest useful references

for readers who wish to explore the subject in greater depth (Booch, 1994). The current language definition is

given in the Working Paper for Draft Proposed International Standard for Information Systems–Programming

Language C++ (ANS, 1995). Detailed description of language evolution and main design decisions are given

by Stroustrup (1994). Useful sets of techniques, explanations and directions for designing and implementing

robust C++ code are given in books (Coplien, 1992) (Eckel, 1989) and journal articles (Koenig, 1989 - 1993)

(Various Authors, 1991-).

Increased interest in using Object Oriented techniques for finite element programming has resulted in a

number (Donescu and Laursen, 1996) (Eyheramendy and Zimmermann, 1996) (Forde et al., 1990) (Miller, 1991)

(Pidaparti and Hudli, 1993) (Scholz, 1992) (Zeglinski et al., 1994) of experimental developments and implemen-

tations. Programming techniques used in some of the papers are influenced by the FORTRAN programming style.

Examples provided in some of the above mentioned papers are readable by C++ experts only. It appears that

none of the authors have used Object Oriented techniques for complex elastoplasticity computations.

A.2 nDarray Programming Tool

A.2.1 Introduction to the nDarray Programming Tool

The nDarray programming tool is a set of classes written in the C++ programming language. The main purpose

of the package is to facilitate algebraic manipulations with matrices, vectors and tensors that are often found in

computer codes for solving engineering problems. The package is designed and implemented using the Object

Oriented philosophy. Great care has been given to the problem of cross–platform and cross–compiler portability.

Currently, the nDarray set of classes has been tested and running under the following C++ compilers:

• Sun CC on SunOS and Solaris platforms,

• IBM xlC on AIX RISC/6000 platforms,



• Borland C++ and Microsoft C++ on DOS/Windows platforms,

• CodeWarrior C++ on Power Macintosh platform,

• GNU g++ on SunOS, SOLARIS, LINUX, AIX, HPUX and AMIGA platforms.

A.2.2 Abstraction Levels

nDarray tool has the following simple class hierarchy:

nDarray_rep, nDarray

matrix

vector

tensor

Indentation of class names implies the inheritance level. For example, class vector is derived from class matrix,

which, in turn is derived from classes nDarray and nDarray rep. The idea is to subdivide classes into levels of

abstraction, and hide the implementation from end users. This means that the end user can use the nDarray

tool on various levels.

• At the highest level of abstraction, one can use tensor, matrix and vector objects without knowing anything

about the implementation and the inner workings. They are all designed and implemented as concrete data

types. In spite of the very powerful code that can be built using Object Oriented technology, it would be

unwise to expect proficiency in Object Oriented techniques and the C++ programming language from end

users. It was our aim to provide power programming with multidimensional data types to users with basic

knowledge of C.

• At a lower abstraction level, users can address the task of the actual implementation of operators and

functions for vector, matrix and tensor classes. A number of improvements can be made, especially in

optimizing some of the operators.

• The lowest level of abstraction is associated with nDarray and nDarray rep classes. Arithmetic operators1

are implemented at this level.

Next, classes are described from the base and down the inheritance tree. Later we focus our attention on

nDarray usage examples. Our goal is to provide a useful programming tool, rather than to teach OOP or to

show C++ implementation. For readers interested in actual implementation details, source code, examples and

makefiles are available at http://sokocalo.engr.ucdavis.edu/~jeremic

nDarray rep class

The nDarray rep class is a data holder and represents an n dimensional array object. A simple memory manager,

implemented with the reference counting idiom (Coplien, 1992) is used. The memory manager uses rather

inefficient built–in C memory allocation functions. Performance can be improved if one designs and implements

1Like addition and subtraction.


http://sokocalo.engr.ucdavis.edu/~jeremic


specially tailored allocation functions for fast heap manipulations. Another possible improvement is in using

memory resources other than heap memory. Sophisticated memory management introduced by the reference

counting is best explained by Coplien (1992). The nDarray rep class is not intended for stand–alone use. It is

closely associated with the nDarray class.

The data structure of nDarray rep introduces a minimal amount of information about a multidimensional

array object. The actual data are stored as a one–dimensional array of double numbers. Rank, total number

of elements, and array of dimensions are all that is needed to represent an multidimensional object. The data

structure is allocated dynamically from the heap, and memory is reclaimed by the system after the object has

gone out of scope.

nDarray class

The nDarray class together with the nDarray rep class represents the abstract base for derived multidimensional

data types: matrices, vectors and tensors. Objects derived from the nDarray class are generated dynamically by

constructor functions at the first appearance of an object and are destroyed at the end of the block in which the

object is referenced. The reference counting idiom provides for the object’s life continuation after the end of the

block where it was defined. To extend an object’s life, a standard C++ compiler would by default call constructor

functions, thus making the entire process of returning large objects from functions quite inefficient. By using

reference counting idiom, destructor and constructor functions manipulate reference counter which results in a

simple copying of a pointer to nDarray rep object. By using this technique, copying of large objects is made very

efficient.

constructor function description

nDarray(int rank_of_nD=1, double initval=0.0) default

nDarray(int rank_of_nD, const int *pdim, double *val) from array

nDarray(int rank_of_nD, const int *pdim, double initval) from scalar value

nDarray(const char *flag, int rank_of_nD, const int *pdim) unit nDarrays

nDarray(const nDarray & x) copy-initializer

nDarray(int rank_of_nD, int rows, int cols, double *val) special for matrix

nDarray(int rank_of_nD, int rows, int cols, double initval) special for matrix

Table A.1: nDarray constructor functions.

Objects can be created from an array of values, or from a single scalar value, as shown in Table A.1. Some

of the frequently used multidimensional arrays are predefined and can be constructed by sending the proper

flag to the constructor function. For example by sending the “I” flag one creates Kronecker delta δij and by

sending “e” flag, one creates a rank 3 Levi-Civita permutation tensor eijk. Functions and operators common

to multidimensional data types are defined in the nDarray class, as described in Table A.2. These common

operators and functions are inherited by derived classes. Occasionally, some of the functions will be redefined,

overloaded in derived classes. In tensor multiplications we need additional information about indices. For example

Cil = (Aijk +Bijk) ∗Djklcoded−→ C=(A("ijk")+B("ijk"))*D("jkl"), the temporary in brackets will receive



operator or function left value right value description

= nDarray nDarray nDarray assignment

+ nDarray nDarray nDarray addition

+= nDarray nDarray nDarray addition

- nDarray unary minus

- nDarray nDarray nDarray subtraction

-= nDarray nDarray nDarray subtraction

* double nDarray scalar multiplication (from left)

* nDarray double scalar multiplication (from right)

== nDarray nDarray nDarray comparison

val(...) nDarray reference to members of nDarray

cval(...) nDarray members of nDarray

trace() nDarray trace of square nDarray

eigenvalues() nDarray eigenvalues of rank 2 square nDarray

eigenvectors() nDarray eigenvectors of rank 2 square nDarray

General_norm() nDarray general p-th norm of nDarray

nDsqrt() nDarray square root of nDarray

print(...) nDarray generic print function

Table A.2: Public functions and operators for nDarray class.

ijk indices, to be used for multiplication with Djkl. It is interesting to note (Koenig, 1989 - 1993) that operator

+= is defined as a member and + is defined as an inline function in terms of += operator.

Matrix and Vector Classes

The matrix class is derived from the nDarray class through the public construct. It inherits common operators

and functions from the base nDarray class, but it also adds its own set of functions and operators. Table (A.3)

summarizes some of the more important additional functions and operators for the matrix class. The vector class


= matrix matrix matrix assignment

* matrix matrix matrix multiplication

transpose() matrix matrix transposition

determinant() matrix determinant of a matrix

inverse() matrix matrix inversion

Table A.3: Matrix class functions and operators (added on nDarray class definitions).

defines vector objects and is derived and inherits most operators and data members from the matrix class. Some



functions, like copy constructor, are overloaded in order to handle specifics of a vector object.

Tensor Class

The main goal of the tensor class development was to provide the implementing analyst with the ability to write

the following equation directly into a computer program:

dσmn = −oldrijT−1ijmn − dλ Eijkl

n+1mklT−1ijmn

as:

dsigma = -(r("ij")*Tinv("ijmn")) - dlambda*((E("ijkl")*dQods("kl"))*Tinv("ijmn"));

Instead of developing theory in terms of indicial notation, then converting everything to matrix notation and then

implementing it, we were able to copy formulae directly from their indicial form to the C++ source code.

In addition to the definitions in the base nDarray class, the tensor class adds some specific functions and

operators. Table A.4 summarizes some of the main new functions and operators. The most significant addition is


+ tensor tensor tensor addition

- tensor tensor tensor subtraction

* tensor tensor tensor multiplication

transpose0110() tensor Aijkl → Aikjl

transpose0101() tensor Aijkl → Ailkj

transpose0111() tensor Aijkl → Ailjk

transpose1100() tensor Aijkl → Ajikl

transpose0011() tensor Aijkl → Aijlk

transpose1001() tensor Aijkl → Aljki

transpose11() tensor aij → aji

symmetrize11() tensor symmetrize second order tensor

determinant() tensor determinant of 2nd order tensor

inverse() tensor tensor inversion (2nd, 4th order)

Table A.4: Additional and overloaded functions and operators for tensor class.

the tensor multiplication operator. With the help of a simple indicial parser, the multiplication operator contracts

or expands indices and yields a resulting tensor of the correct rank. The resulting tensor receives proper indices,

and can be used in further calculations on the same code statement.



A.3 Finite Element Classes

A.3.1 Stress, Strain and Elastoplastic State Classes

The next step in our development was to use the nDarray tool classes for constitutive level computations. The

simple extension was design and implementation of infinitesimal stress and strain tensor classes, namely stresstensor

and straintensor. Both classes are quite similar, they inherit all the functions from the tensor class and we add

some tools that are specific to them. Both stress and strain tensors are implemented as full second order 3 × 3

tensors. Symmetry of stress and strain tensor was not used to save storage space. Table A.5 summarizes some

of the main functions added on for the stresstensor class.

operator or function description

Iinvariant1() first stress invariant I1

Iinvariant2() second stress invariant I2

Iinvariant3() third stress invariant I3

Jinvariant2() second deviatoric stress invariant J2

Jinvariant3() third deviatoric stress invariant J3

deviator() stress deviator

principal() principal stresses on diagonal

sigma_octahedral() octahedral mean stress

tau_octahedral() octahedral shear stress

xi() Haigh–Westergard coordinate ξ

rho() Haigh–Westergard coordinate ρ

p_hydrostatic() hydrostatic stress invariant

q_deviatoric() deviatoric stress invariant

theta() θ stress invariant (Lode’s angle)

Table A.5: Additional methods for stress tensor class.

Further on, we defined an elastoplastic state, which according to incremental theory of elastoplasticity with

internal variables, is completely defined with the stress tensor and a set of internal variables. This definition led

us to define an elastoplastic state termed class ep state. Objects of type ep state contain a stress tensor and a

set of scalar or tensorial internal variables2.

A.3.2 Material Model Classes

With all the previous developments, the design and implementation of various elastoplastic material models was

not a difficult task. A generic class Material Model defines techniques that form a framework for small deformation

elastoplastic computations. Table A.6 summarizes some of the main methods defined for the Material Model class

2Internal variables can be characterized as tensors of even order, where, for example, zero tensor is a scalar internal variable

associated with isotropic hardening and second order tensors can be associated with kinematic hardening.



in terms of yield (F ) and potential (Q) functions.

operator or function description

F F Yield function value

dFods ∂F/∂σij

dQods ∂Q/∂σij

d2Qods2 ∂2Q/∂σij∂σkl

dpoverds ∂p/∂σij

dqoverds ∂q/∂σij

dthetaoverds ∂θ/∂σij

d2poverds2 ∂2p/∂σij∂σkl

d2qoverds2 ∂2q/∂σij∂σkl

d2thetaoverds2 ∂2θ/∂σij∂σkl

ForwardPredictorEPState Explicit predictor elastoplastic state

BackwardEulerEPState Implicit return elastoplastic state

ForwardEulerEPState Explicit return elastoplastic state

BackwardEulerCTensor Algorithmic tangent stiffness tensor

ForwardEulerCTensor Continuum tangent stiffness tensor

Table A.6: Some of the methods in material model class.

It is important to note that all the material model dependent functions are defined as virtual functions.

Integration algorithms are designed and implemented using template algorithms, and each implemented material

model appends its own yield and potential functions and appropriate derivatives. Implementation of additional

material models requires coding of yield and potential functions and respective derivative functions.

A.3.3 Stiffness Matrix Class

Starting from the incremental equilibrium of the stationary body, the principle of virtual displacements and with

the finite element approximation of the displacement field u ≈ ua = HI uIa, the weak form of equilibrium can be

expressed as (Zienkiewicz and Taylor, 1991a)

⋃

m

∫

Vm

HI,b Eabcd HJ,d dVm uJc =

⋃

m

∫

Vm

fa HI dVm or (fIa (uJc))int = λ (fIa)ext

where Eabcd is the constitutive tangent stiffness tensor3. The element stiffness tensor is recognized as

keaIcJ =

∫

Vm

HI,btanEabcd HJ,d dV

m

This generic form for the finite element stiffness tensor is easily programmed with the help of the nDarray tool.

A simple implementation example is provided later. It should be noted that the element stiffness tensor in this

case is a four–dimensional tensor. It is the task of the assembly function to collect proper terms for addition in a

global stiffness matrix.

3Which may be continuum or algorithmic (Jeremic and Sture, 1997) tangent stiffness tensor



A.4 Examples

A.4.1 Tensor Examples

Some of the basic tensorial calculations with tensors are presented. Tensors have a default constructor that creates

a first order tensor with one element initialized to 0.0:

tensor t1;

Tensors can be constructed and initialized from a given set of numbers:

static double t2values[] = 1,2,3,

4,5,6,

7,8,9 ;

tensor t2( 2, DefDim2, t2values); // order 2; 3x3 tensor (like matrix)

Here, DefDim2, DefDim3 and DefDim4 are arrays of dimensions for the second, third and fourth order tensor4.

A fourth order tensor with 0.0 value assignment and dimension 3 in each order (3 × 3 × 3 × 3) is constructed in

the following way:

tensor ZERO(4,DefDim4,0.0);

Tensors can be multiplied using indicial notation. The following example will do a tensorial multiplication of

previously defined tensors t2 and t4 so that tst1 = t2ijt4ijklt4klpqt2pq. Note that the memory is dynamically

allocated to accept the proper tensor dimensions that will result from the multiplication5

tensor tst1 = t2("ij")*t4("ijkl")*t4("klpq")*t2("pq");

Inversion of tensors is possible. It is defined for 2 and 4 order tensors only. The fourth order tensor inversion is

done by converting it to matrix, inverting that matrix and finally converting matrix back to tensor.

tensor t4inv_2 = t4.inverse();

There are two built–in tensor types, Levi-Civita permutation tensor eijk and Kronecker delta tensor δij

tensor e("e",3,DefDim3); // Levi-Civita permutation tensor

tensor I2("I", 2, DefDim2); // Kronecker delta tensor

Trace and determinant functions for tensors are used as follows

double deltatrace = I2.trace();

double deltadet = I2.determinant();

Tensors can be compared to within a square root of machine epsilon6 tolerance

tensor I2again = I2;

if ( I2again == I2 )

printf("I2again == I2 TRUE (OK)");

else

printf("I2again == I2 NOTTRUE");

4In this case dimensions are 3 in every order.5In this case it will be zero dimensional tensor with one element.6Machine epsilon (macheps) is defined as the smallest distinguishable positive number (in a given precision, i.e. float (32 bits),

double (64 bits) or long double (80 bits), such that 1.0 + macheps > 1.0 yields true on the given computer platform. For example,

double precision arithmetics (64 bits), on the Intel 80x86 platform yields macheps=1.08E-19 while on the SUN SPARCstation and

DEC platforms macheps=2.22E-16.



A.4.2 Fourth Order Isotropic Tensors

Some of the fourth order tensors used in continuum mechanics are built quite readily. The most general repre-

sentation of the fourth order isotropic tensor includes the following fourth order unit isotropic tensors7

tensor I_ijkl = I2("ij")*I2("kl");

The resulting tensor I_ijkl will have the correct indices, I ijklijkl = I2ijI2kl. Note that I_ijkl is just a name

for the tensor, and the _ijkl part reminds the implementor what that tensor is representing.

The real indices, ∗ijkl in this case, are stored in the tensor object, and can be used further or changed

appropriately. The next tensor that is needed is a fourth order unit tensor obtained by transposing the previous

one in the minor indices,

tensor I_ikjl = I_ijkl.transpose0110();

while the third tensor needed for representation of general isotropic tensor is constructed by using similar transpose

function

tensor I_iljk = I_ijkl.transpose0111();

The inversion function can be checked for fourth order tensors:

tensor I_ikjl_inv_2 = I_ikjl.inverse();

if ( I_ikjl == I_ikjl_inv_2 )

printf(" I_ikjl == I_ikjl_inv_2 (OK) !");

else

printf(" I_ikjl != I_ikjl_inv_2 !");

Creating a symmetric and skew symmetric unit fourth order tensors gets to be quite simple by using tensor addition

and scalar multiplication

tensor I4s = (1./2.)*(I_ikjl+I_iljk);

tensor I4sk = (1./2.)*(I_ikjl-I_iljk);

Another interesting example is a numerical check of the e−δ identity (Lubliner, 1990) (eijmeklm = δikδjl−δilδjk)

tensor id = e("ijm")*e("klm") - (I_ikjl - I_iljk);

if ( id == ZERO )

printf(" e-delta identity HOLDS !! ");

A.4.3 Elastic Isotropic Stiffness and Compliance Tensors

The linear isotropic elasticity tensor Eijkl can be built from Young’s modulus E and Poisson’s ratio ν

double Ey = 20000; // Young’s modulus of elasticity

double nu = 0.2; // Poisson’s Ratio

tensor E = ((2.*Ey*nu)/(2.*(1.+nu)*(1-2.*nu)))*I_ijkl + (Ey/(1.+nu))*I4s;

Similarly, the compliance tensor is

tensor D = (-nu/Ey)*I_ijkl + ((1.0+nu)/Ey)*I4s;

One can multiply the two and check if the result is equal to the symmetric fourth order unit tensor

7Remember that I2 was constructed as the Kronecker delta tensor δij .



tensor test = E("ijkl")*D("klpq");

if ( test == I4s )

printf(" test == I4s TRUE (OK up to sqrt(macheps)) ");

else

printf(" test == I4s NOTTRUE ");

The linear isotropic elasticity and compliance tensors can be obtained in a different way, by using Lame constants

λ and µ

double lambda = nu * Ey / (1. + nu) / (1. - 2. * nu);

double mu = Ey / (2. * (1. + nu));

tensor E = lambda*I_ijkl + (2.*mu)*I4s; // stiffness tensor

tensor D = (-nu/Ey)*I_ijkl + (1./(2.*mu))*I4s; // compliance tensor

A.4.4 Second Derivative of θ Stress Invariant

As an extended example of nDarray tool usage, the implementation for the second derivative of the stress invariant

θ (Lode angle) is presented. The derivative is used for implicit constitutive integration schemes applied to three

invariant material models. The original equation reads:

∂2θ

∂σpq∂σmn=

−(

9

2

cos 3θ

q4 sin (3θ)+

27

4

cos 3θ

q4 sin3 3θ

)

spq smn +81

4

1


+

(81

4

1

q5 sin 3θ+

81

4

cos2 3θ

q5 sin3 3θ

)

tpq smn − 243

4

cos 3θ


+3

2

cos (3θ)


2

1

q3 sin (3θ)wpqmn

where:

q =

√

3

2sijsij ; cos 3θ =

3√

3

2

13sijsjkski√

( 12sijsij)

3; sij = σij −

1

3σkkδij

wpqmn = snpδqm + sqmδnp −2

3sqpδnm − 2

3δpqsmn ; ppqmn = δmpδnq −

1

3δpqδmn

and the implementation follows:

tensor Yield_Criteria::d2thetaoverds2(stresstensor & stress)

tensor ret( 4, DefDim4, 0.0);

tensor I2("I", 2, DefDim2);

tensor I_pqmn = I2("pq")*I2("mn");

tensor I_pmqn = I_pqmn.transpose0110();

double J2D = stress.Jinvariant2();

tensor s = stress.deviator();

tensor t = s("qk")*s("kp") - I2*(J2D*(2.0/3.0));



double theta = stress.theta();

double q_dev = stress.q_deviatoric();

//setting up some constants

double c3t = cos(3*theta);

double s3t = sin(3*theta);

double s3t3 = s3t*s3t*s3t;

double q3 = q_dev * q_dev * q_dev;

double q4 = q3 * q_dev;



double tempss = -(9.0/2.0)*(c3t)/(q4*s3t)-(27.0/4.0)*(c3t/(s3t3*q4));

double tempst = +(81.0/4.0)*(1.0)/(s3t3*q5);

double tempts = +(81.0/4.0)*(1.0/(s3t*q5))+(81.0/4.0)*(c3t*c3t)/(s3t3*q5);

double temptt = -(243.0/4.0)*(c3t/(s3t3*q6));

double tempp = +(3.0/2.0)*(c3t/(s3t*q_dev*q_dev));

double tempw = -(9.0/2.0)*(1.0/(s3t*q3));

tensor s_pq_d_mn = s("pq")*I2("mn");

tensor s_pn_d_mq = s_pq_d_mn.transpose0101();

tensor d_pq_s_mn = I2("pq")*s("mn");

tensor d_pn_s_mq = d_pq_s_mn.transpose0101();

tensor p = I_pmqn - I_pqmn*(1.0/3.0);

tensor w = s_pn_d_mq+d_pn_s_mq - s_pq_d_mn*(2.0/3.0)-d_pq_s_mn*(2.0/3.0);

// finally

ret = (s("pq")*s("mn")*tempss + s("pq")*t("mn")*tempst +

t("pq")*s("mn")*tempts + t("pq")*t("mn")*temptt +

p*tempp + w*tempw );

return ret;

A.4.5 Application to Computations in Elastoplasticity

A useful application of the previously described classes is for elastoplastic computations. If the Newton iterative

scheme is used at the global equilibrium level, then in order to preserve a quadratic rate, a consistent, algorithmic

tangent stiffness (ATS) tensor should be used. For a general class of three–invariant, non–associated, hardening

or softening material models, ATS is defined (Jeremic and Sture, 1997) as:

consEeppqmn = Rpqmn − Rpqkln+1Hkl

n+1nijRijmnn+1notRotpq n+1Hpq + n+1ξ∗ h∗

where

mkl =∂Q

∂σkl; nkl =

∂F

∂σkl; ξ∗ =

∂F

∂q∗; Tijmn = δimδnj + ∆λ Eijkl

∂mkl

∂σmn

Hkl = n+1mkl + ∆λ∂mkl

∂q∗h∗ ; Rmnkl =

(n+1Tijmn

)−1Eijkl

A straightforward implementation of the above tensorial formula follows:

double Ey = Criterion.E();

double nu = Criterion.nu();



tensor Eel = StiffnessTensorE(Ey,nu);

tensor I2("I", 2, DefDim2);

tensor I_ijkl = I2("ij")*I2("kl");

tensor I_ikjl = I_ijkl.transpose0110();

tensor m = Criterion.dQods(final_stress);

tensor n = Criterion.dFods(final_stress);

double lambda = current_lambda_get();

tensor d2Qoverds2 = Criterion.d2Qods2(final_stress);

tensor T = I_ikjl + Eel("ijkl")*d2Qoverds2("klmn")*lambda;

tensor Tinv = T.inverse();

tensor R = Tinv("ijmn")*Eel("ijkl");

double h_ = h(final_stress);

double xi_ = xi(final_stress);

double hardMod_ = h_ * xi_;

tensor d2Qodqast2 = d2Qoverdqast2(final_stress);

tensor H = m + d2Qodqast2 * lambda * h_;

//

tensor upper = R("pqkl")*H("kl")*n("ij")*R("ijmn");

double lower = (n("ot")*R("otpq"))*H("pq")).trace();

lower = lower + hardMod_;

tensor Ep = upper*(1./lower);

tensor Eep = R - Ep; // elastoplastic ATS constitutive tensor

This ATS tensor can be used further in building finite element stiffness tensors, as will be shown in our next

example.

A.4.6 Stiffness Matrix Example

By applying a numerical integration technique to the stiffness matrix equation

keaIcJ =

∫

Vm

HI,b Eabcd HJ,d dVm

individual contributions are summed into the element stiffness tensor. This process can be implemented on a

integration point level by using the nDarray tool as

K = K + H("Ib") * E("abcd") * H("Jd") * weight ;

It is interesting to note the lack of loops at this level of implementation. However, there exists a loop over

integration points which contributes stiffness to the element tensor.

A.5 Performance Issues

In the course of developing the nDarray tool, execution speed was not a priority or issue that we tried to perfect.

The benefit of being able to implement and test various numerical algorithms in a straightforward manner was

the main concern. The efficiency of the nDarray tool when compared with FORTRAN or C was never assessed.

In all honesty, some of the formulae implemented in C++ with the help of the nDarray tool would be difficult



to implement in FORTRAN or C. The entire question of efficiency of the nDarray as compared to FORTRAN or

C codes might thus remain unanswered for the time being.

The efficiency of C++ for numerical computations has been under consideration (Robison, 1996) for some time

now. Poor efficiency and possible remedies for improving efficiency of C++ computations has been reported in

literature (Robison, 1996) (Veldhuizen, 1995) (Veldhuizen, 1996). Novel techniques, such as Template Expressions

(Veldhuizen, 1995) can be used to achieve and sometimes surpass the performance of hand–tuned FORTRAN or

C codes.

A.6 Summary and Future Directions

A novel programming tool, named nDarray, has been presented which facilitates implementation of tensorial

formulae. It was shown how OOP and efficient programming in C++ allows building of new concrete data

types, in this case tensors of any order. In a number of examples these new data types were shown to be useful in

implementing tensorial formulae associated with the numerical solution of various elastic and elastoplastic problems

with the finite element method. The nDarray tool is been used in developing of the FEMtools tools library. The

FEMtools tools library includes a set of finite elements, various solvers, solution procedures for non–linear finite

element system of equations and other useful functions.


Appendix B

Useful Formulae

B.1 Stress and Strain

This section reviews small deformation stress and strain measures used in this report.

B.1.1 Stress

In this work, the tensile stress is assumed positive, and in general we follow classical strength of materials

(mechanics of materials) conventions for stress and strain. The stress tensor σij is given as

σ =

σxx σxy σxz

σyx σyy σyz

σzx σzy σzz

=

σx σxy σzx

σxy σy σyz

σzx σyz σz

(B.1)

In small deformation theory, this stress is symmetric, that is, σxy = σyx, σyz = σzy, and σzx = σxz. There are

only six independent components and sometimes the stress can be expressed in the vector form

σ = σxx, σyy, σzz, σxy, σyz, σzx (B.2)

The principle stresses σ1, σ2, and σ3 (σ1 ≥ σ2 ≥ σ3) are the eigenvalues of the symmetric tensor σij in

Equation B.1 and can be obtained by solving the equation

∣∣∣∣∣∣∣∣

σxx − σ σxy σxz

σyx σyy − σ σyz

σzx σzy σzz − σ

∣∣∣∣∣∣∣∣

= 0 (B.3)

or in alternative form

σ3 − I1σ2 − I2σ − I3 = 0 (B.4)

279


The three first-type stress invariants are then

I1 = σii

= σxx + σyy + σzz

= σ1 + σ2 + σ3 (B.5)

I2 =1

2σijσji

= −(σxxσyy + σyyσzz + σzzσxx) + (σ2xy + σ2

yz + σ2zx)

= −(σ1σ2 + σ2σ3 + σ3σ1) (B.6)

I3 =1

3σijσjkσki = det (σij)

= σxxσyyσzz + 2σxyσyzσzx − (σxxσ2yz + σyzσ

2zx + σzzσ

2xy)

= σ1σ2σ3 (B.7)

The stress σij can be decomposed into the hydrostatic stress σmδij and deviatoric stress sij as σij =

σmδij + sij , with the definitions

σm =1

3I1, sij = σij −

1

3σkkδij (B.8)

where δij is the Kronecker operator such that δij = 1 for i = j and δij = 0 for i 6= j.

Since both hydrostatic and deviatoric stresses are stress tensors, they have their own coordinate-independent

stress invariants respectively. The three invariants of the hydrostatic stress are

I1 = 3σm = I1, I2 = −3σ2m = −1

3I21 , I3 = σ3

m =1

27I31 (B.9)

Since I1, I2 and I3 are all simple functions of I1, the hydrostatic stress state can therefore be represented by only

one variable I1.

The three eigenvalues of the deviatoric stresses sij are called principal deviatoric stresses, with the order

s1 ≥ s2 ≥ s3. The three invariants of the deviatoric stress are

J1 = sii = 0 (B.10)

J2 =1

2sijsji

=1

3I21 + I2

=1

6[(σ1 − σ2)

2 + (σ2 − σ3)2 + (σ3 − σ1)

2]

= −(sxxsyy + syyszz + szzsxx) + (s2xy + s2yz + s2zx)

=1

2(s21 + s22 + s23) = −(s1s2 + s2s3 + s3s1) (B.11)



J3 =1

3sijsjkski = det(sij)

= I3 +1

3I1I2 +

2

27I31 = I3 −

1

3I1J2 −

1

27I31

=1

27(2σ1 − σ2 − σ3)(2σ2 − σ3 − σ1)(2σ3 − σ1 − σ2)

= sxxsyyszz + 2sxysyzszx − (sxxs2yz + syys

2zx + szzs

2xy)

= s1s2s3 (B.12)

The deviatoric stress state can therefore be represented by only two variables J2 and J3.

Combining hydrostatic and deviatoric stress, we can conclude that the stress state can be be represented

by three variables I1, J2 and J3. Using the three invariants (I1, J2, J3) or its equivalents instead of the nine

components of σij is widely used in geomechanics.

The stress state may also be described in three dimensional space (p, q, θσ), defined as

p = −1

3I1 (B.13)

q =√

3J2 (B.14)

θσ =1

3arccos

(

3√

3

2

J3√

J32

)

(B.15)

where θσijis the stress Lode’s angle (0 ≤ θσij

≤ π/3). A stress state with θσ = 0 corresponds to the meridian

of conventional triaxial compression (CTC), while θσ = π/3 to the meridian of conventional triaxial extension

(CTE). The relationship between (σ1, σ2, σ3) and (p, q, θσ) is

σ1

σ2

σ3

= −p+2

3q

cos θσ

cos (θσ − 2

3π)

cos (θσ +2

3π)

(B.16)

The line of the principal stress space diagonal is called hydrostatic axis. Any plane perpendicular to the

hydrostatic axis is an deviatoric plane, or π plane. The Haigh-Westergaard three dimensional stress coordinate

system (ξ, ρ, θσ) Chen and Han (1988), is defined as

ξ =I1√3

= −√

3p (B.17)

ρ =√

2J2 =

√

2

3q (B.18)

The Haigh-Westergaard invariants have physical meanings. ξ is the distance of the deviatoric plane to the origin

of the Haigh-Westergaard coordinates, and ρ is the distance of a stress point to the hydrostatic line and represents

the magnitude of the deviatoric stress. The projections of the axes σ1, σ2 and σ3 on the deviatoric plane are

assumed σ′1, σ

′2 and σ′

3 respectively. (ρ, θσ) is the polar coordinate system in the deviatoric plane with the σ′1 the

polar axis and θσ the polar angle. The relationship between (σ1, σ2, σ3) and (ξ, ρ, θσ) is

σ1

σ2

σ3

=1√3ξ +

√

2

3ρ

cos θσ

cos (θσ − 2

3π)

cos (θσ +2

3π)

(B.19)



B.1.2 Strain

Point P (xi) and nearby point Q(xi + dxi) displace due to applied loading to new positions P (xi + Ui) and

Q(ui + (∂ui/∂xj)dxj). We can define a displacement gradient tensor ui,j as

ui,j =∂ui∂xj

(B.20)

Matrix form of the displacement gradient can decomposed into the symmetric and antisymmetric parts

u1,1 u1,2 u1,3

u2,1 u2,2 u2,3

u3,1 u3,2 u3,3

=

u1,112 (u1,2 + u2,1)

12 (u1,3 + u3,1)

12 (u2,1 + u1,2) u2,2

12 (u2,3 + u3,2)

12 (u3,1 + u1,3)

12 (u3,2 + u2,3) u3,3

+

0 12 (u1,2 − u2,1)

12 (u1,3 − u3,1)

12 (u2,1 − u1,2) 0 1

2 (u2,3 − u3,2)

12 (u3,1 − u1,3)

12 (u3,2 − u2,3) 0

(B.21)

or

ui,j = ǫij + wij (B.22)

where

ǫij =1

2(ui,j + uj,i) (B.23)

wij =1

2(ui,j − uj,i) (B.24)

The symmetric part of the deformation gradient tensor, ǫij , is the small deformation strain tensor 1, while the

antisymmetric part of the deformation gradient tensor, wij , is the rotation motion tensor. The matrix form of

the strain ǫij is

ǫ =

ǫxx ǫxy ǫzx

ǫxy ǫyy ǫyz

ǫzx ǫyz ǫzz

=

ǫx12γxy

12γzx

12γxy ǫy

12γyz

12γzx

12γyz ǫz

(B.25)

The engineering strain is usually expressed in the vector form

ǫ = ǫx, ǫy, ǫz, γxy, γyz, γzxT (B.26)

Note that the engineering shear strain γij is the double of the corresponding strain component ǫij .

Similar to the stress tensor, the strain tensor also has three principle strains ǫi(ǫ1 ≥ ǫ2 ≥ ǫ3), and three strain

invariants I ′1, I′2, and I ′3, defined as

I ′1 = ǫii

= ǫxx + ǫyy + ǫzz

= ǫ1 + ǫ2 + ǫ3 (B.27)

1Here the second and higher order derivative terms are neglected due to the small deformation assumption.



I ′2 =1

2ǫijǫji

= −(ǫxxǫyy + ǫyyǫzz + ǫzzǫxx) + (ǫ2xy + ǫ2yz + ǫ2zx)

= −(ǫ1ǫ2 + ǫ2ǫ3 + ǫ3ǫ1) (B.28)

I ′3 =1

3ǫijǫjkǫki = det (ǫij)

= ǫxxǫyyǫzz + 2ǫxyǫyzǫzx − (ǫxxǫ2yz + ǫyzǫ

2zx + ǫzzǫ

2xy)

= ǫ1ǫ2ǫ3 (B.29)

The first strain invariant is also called the volumetric strain ǫv.

The strain ǫij can be decomposed into the hydrostatic strain ǫmδij and deviatoric strain eij through ǫij =

ǫmδij + eij where:

ǫm =1

3I ′1, eij = ǫij −

1

3ǫkkδij (B.30)

Since both hydrostatic and deviatoric strains are strain tensors, they have their own strain invariants respec-

tively. The three invariants of the hydrostatic strain are

I ′1 = 3ǫm = I ′1, I ′2 = −3ǫ2m = −1

3(I ′1)

2, I ′3 = ǫ3m =1

27(I ′1)

3 (B.31)

The hydrostatic strain state can therefore be represented by only one variable I ′1.

The three eigenvalues of the deviatoric strains eij are called principal deviatoric strains, with the order e1 ≥e2 ≥ e3. The three invariants of the deviatoric strain are

J ′1 = eii = 0 (B.32)

J ′2 =

1

2eijeji

=1

3(I ′1)

2 + I ′2

=1

6[(ǫ1 − ǫ2)

2 + (ǫ2 − ǫ3)2 + (ǫ3 − ǫ1)

2]

= −(exxeyy + eyyezz + ezzexx) + (e2xy + e2yz + e2zx)

=1

2(e21 + e22 + e23) = −(e1e2 + e2e3 + e3e1) (B.33)

J ′3 =

1

3eijejkeki = det(eij)

= I ′3 +1

3I ′1I

′2 +

2

27(I ′1)

3 = I3 −1

3I ′1J

′2 −

1

27(I ′1)

3

=1

27(2ǫ1 − ǫ2 − ǫ3)(2ǫ2 − ǫ3 − ǫ1)(2ǫ3 − ǫ1 − ǫ2)

= exxeyyezz + 2exyeyzezx − (exxe2yz + eyys

2zx + ezze

2xy)

= e1e2e3 (B.34)

The deviatoric strain state can therefore be represented by only two variables J ′2 and J ′

3.



Combining the hydrostatic and deviatoric strain, we can conclude that the strain state can be be represented

by three variables I ′1, J′2 and J ′

3.

Strain state may also be represented with another three invariant (ǫp, ǫq, θǫ), defined as

ǫp = −I ′1 = −ǫv (B.35)

ǫq = 2

√

J ′2

3(B.36)

θǫ =1

3arccos

(

3√

3

2

J ′3

√

(J ′2)

3

)

(B.37)

where θǫ is the strain Lode’s angle and 0 ≤ θǫ ≤ π/3. The relationship between (ǫ1, ǫ2, ǫ3) and (ǫp, ǫq, θǫ) is

ǫ1

ǫ2

ǫ3

= −1

3ǫp +

√

3

2ǫq

cos θǫ

cos (θǫ −2

3π)

cos (θǫ +2

3π)

(B.38)

B.2 Derivatives of Stress Invariants

In this part of the Appendix, we shall derive some useful formulae, that are rarely found2 in texts treating elasto–

plastic problems in mechanics of solid continua.

First derivative of I1 with respect to stress tensor σij :

∂I1∂σij

=∂σkk∂σij

= δij

First derivative of J2D with respect to stress tensor σij :

∂J2D

∂σij=∂( 1

2smnsnm)

∂σij=

1

2

∂smn∂σij

snm +1

2

∂snm∂σij

smn =

=∂snm∂σij

smn =∂(σnm − 1

3σkkδnm)

∂σijsmn = (δniδjm − 1

3δnmδkiδjk)smn =

= (δniδjm − 1

3δnmδij)smn = δniδjmsnm − 1

3δnmδijsmn = sij

†

First derivative of J3D with respect to stress tensor σpq:

2if found at all.†because δnmδijsmn ≡ 0



∂J3D

∂σpq=∂( 1

3sijsjkski)

∂σpq=

1

3

∂sij∂σpq

sjkski +1

3

∂sjk∂σpq

sijski +1

3

∂ski∂σpq

sijsjk =

=∂sij∂σpq

sjkski =∂(σij − 1

3σkkδij)

∂σpqsjkski = (δipδqj −

1

3δijδkpδqk)sjkski =

= δipδqjsjkski −1

3δijδkpδqksjkski = sqkskp −

2

3δpqJ2D = tpq

‡

First derivative of spq with respect to stress tensor σmn, or second derivative of J2D with respect to stress tensors

σpq and σmn:

∂spq∂σmn

=∂(σpq − 1

3δpqσkk)

∂σmn=∂((δmpδnq − 1

3δpqδmn)σmn

)

∂σmn=

=

(

δmpδnq −1

3δpqδmn

)

= ppqmn

Second derivative of J3D with respect to stress tensors σpq and σmn:

∂tpq∂σmn

=∂(sqkskp − 2

3δpqJ2D

)

∂σmn=∂ (sqkskp)

∂σmn− ∂

(23δpqJ2D

)

∂σmn=

=∂ (sqkskp)

∂σmn− 2

3δpq

∂J2D

∂σmn=

∂sqk∂σmn

skp + sqk∂skp∂σmn

− 2

3δpqsmn =

=

(

δqmδnk −1

3δqkδnm

)

skp + sqk

(

δkmδnp −1

3δkpδnm

)

− 2

3δpqsmn =

=

(

δqmsnp −1

3sqpδnm

)

+

(

sqmδnp −1

3sqpδnm

)

− 2

3δpqsmn =


3sqpδnm − 2

3δpqsmn = wpqmn

Multiplying stiffness tensor Eijkl with compliance tensor Dklpq:

‡since 13δijδkpδqksjkski = 1

3δkpδqksikski = 1

3δqpsikski = 2

3δpqJ2D see also Chen and Han (1988) page 222



EijklDklpq =

E

2 (1 + ν)

1 + ν

2E

(2ν

1 − 2νδijδkl + δikδjl + δilδjk

)(−2ν

1 + νδklδpq + δkpδlq + δkqδlp

)

=

1

4(δikδjlδkpδlq + δilδjkδkpδlq + δikδjlδkqδlp + δilδjkδkqδlp) +

+ν

2 (1 − 2ν)(δijδklδkpδlq + δijδklδkqδlp) −

ν

2 (1 + ν)(δikδjlδklδpq + δilδjkδklδpq) −

− ν2

(1 − 2ν) (1 + ν)δijδklδklδpq =

1

2(δipδjq + δiqδjp) +

+ν

2 (1 − 2ν)(δijδkqδkp + δijδkpδkq) −

ν

2 (1 + ν)(δilδjlδpq + δilδjlδpq) −

− 3ν2

(1 − 2ν) (1 + ν)δijδpq =

1


+ν

2 (1 − 2ν)(δijδpq + δijδpq) −

ν

2 (1 + ν)(δijδpq + δijδpq) −

− 3ν2

(1 − 2ν) (1 + ν)δijδpq =

1


+ν

(1 − 2ν)δijδpq −

ν

2 (1 + ν)δijδpq −

− 3ν2

(1 − 2ν) (1 + ν)δijδpq =

1


+ν (1 + ν) − ν (1 − 2ν) − 3ν2

(1 − 2ν) (1 + ν)δijδpq =

1

2(δipδjq + δiqδjp) = Isymijpq


Appendix C

Closed Form Gradients to the Potential

Function

A complete derivation of the gradients to the Potential and Yield function follows. The potential function Q is a

function of the stress tensor σij and plastic variable tensor q∗. Only derivatives with respect to the stress tensor

σij are given here. It is assumed that any stress state can be represented with three stress invariants p, q and θ

given in the following form:

p = −1

3I1 q =

√

3J2D cos 3θ =3√

3

2

J3D√

(J2D)3(C.1)

I1 = σkk J2D =1

2sijsij J3D =

1

3sijsjkski sij = σij −

1

3σkkδij (C.2)

and stresses are chosen as positive in tension. One can write the Potential Function in the following form:

Q = Q(p, q, θ) (C.3)

and the derivation follows. Hopefully the pace of derivation is rather slow, thus little explanation will be given

until the end of the derivation. Chain rule of differentiation yields:

∂Q

∂σij=∂Q

∂p

∂p

∂σij+∂Q

∂q

∂q

∂σij+∂Q

∂θ

∂θ

∂σij(C.4)

and the intermediate derivatives are:

287


∂p

∂σij=∂(− 1

3σkk)

∂σij= −1

3δij (C.5)

∂q

∂σij=∂√

3J2D

∂σij=

√3

2

1√J2D

∂J2D

∂σij=

√3

2

1√J2D

sij =3

2

1

qsij (C.6)

∂θ

∂σij= (C.7)

=1

3

−1√

1 − ( 3√

32

J3D

J3/22D

)2

3√

3

2

(

∂J3D

∂σij

1√

(J2D)3− 3

2J3D

∂J2D

∂σij

1√

(J2D)5

)

=

=1

3

√

1 −(

3√

32

J3D

J3/22D

)2

3√

3

2

(

−tij1

√

(J2D)3+

3

2J3Dsij

1√

(J2D)5

)

=

=1

sin 3θ

√3

2

(

3

2J3D

1√

(J2D)5sij −

1√

(J2D)3tij

)

=

=

√3

2

1

sin (3θ)

(√3 cos (3θ)

q2sij −

3√

3

q3tij

)

=

=3

2

cos (3θ)

q2 sin (3θ)sij −

9

2

1

q3 sin (3θ)tij (C.8)

Second derivatives of the potential function Q using again the chain rule of differentiation are as follows:



∂2Q

∂σpq∂σmn=∂(∂Q∂σpq

)

∂σmn=

∂(∂Q∂p

∂p∂σpq

+ ∂Q∂q

∂q∂σpq

+ ∂Q∂θ

∂θ∂σpq

)

∂σmn=

∂(∂Q∂p

)

∂σmn

∂p

∂σpq+∂Q

∂p

∂2p

∂σpq∂σmn+

+∂(∂Q∂q

)

∂σmn

∂q

∂σpq+∂Q

∂q

∂2q

∂σpq∂σmn+

+∂(∂Q∂θ

)

∂σmn

∂θ

∂σpq+∂Q

∂θ

∂2θ

∂σpq∂σmn=

(∂2Q

∂p2

∂p

∂σmn+∂2Q

∂p∂q

∂q

∂σmn+∂2Q

∂p∂θ

∂θ

∂σmn

)∂p

∂σpq+∂Q

∂p

∂2p

∂σpq∂σmn+

+

(∂2Q

∂q∂p

∂p

∂σmn+∂2Q

∂q2∂q

∂σmn+∂2Q

∂q∂θ

∂θ

∂σmn

)∂q

∂σpq+∂Q

∂q

∂2q

∂σpq∂σmn+

+

(∂2Q

∂θ∂p

∂p

∂σmn+∂2Q

∂θ∂q

∂q

∂σmn+∂2Q

∂θ2∂θ

∂σmn

)∂θ

∂σpq+∂Q

∂θ

∂2θ

∂σpq∂σmn

and the intermediate derivatives are as follows:

∂2p

∂σpq∂σmn=∂2(− 1

3σkk)

∂σpq∂σmn=∂(− 1

3δkpδqk)

∂σmn= ∅

∂2q

∂σpq∂σmn=∂(

∂q∂σpq

)

∂σmn=

∂(√

32

1√J2D

spq

)

∂σmn=

√3

2

1√J2D

∂spq∂σmn

+

√3

2

∂ 1√J2D

∂σmnspq =

√3

2

1√J2D

(

δpmδnq −1

3δpqδkmδnk

)

+

√3

2

(

−1

2

(

1(√J2D

)3

)

smn

)

spq =

√3

2

1√J2D

(

δpmδnq −1

3δpqδnm

)

−√

3

4

(1√J2D

)3

smnspq =

3

2

1

q

(

δpmδnq −1

3δpqδnm

)

− 9

4

1

q3smnspq

Let us introduce a slightly different form for the equation ∂2θ∂σpq∂σmn

in order to simplify writing:



∂θ

∂σpq=

3

2

cos (3θ)

q2 sin (3θ)spq −

9

2

1

q3 sin (3θ)tpq =

= AS spq +AT tpq

where:

AS =3

2

cos (3θ)

q2 sin (3θ)

AT = −9

2

1

q3 sin (3θ)

Now the problem will be separated in two smaller problems, namely:

∂2θ

∂σpq∂σmn=∂ ∂θ∂σpq

∂σmn=

∂(

32

cos (3θ)q2 sin (3θ)spq − 9

21

q3 sin (3θ) tpq

)

∂σmn=

∂ (AS spq +AT tpq)

∂σmn=

∂ (AS spq)

∂σmn+∂ (AT tpq)

∂σmn

Now let us take a look at∂(AS spq)∂σmn

. Since:



∂ (AS spq)

∂σmn=

∂AS

∂σmnspq +AS

∂spq∂σmn

=

(∂AS

∂q

∂q

∂σmn+∂AS

∂θ

∂θ

∂σmn

)

spq +AS∂spq∂σmn

=

(−3. cot(3 θ)

q33

2

1

qsmn +

+−4.5 csc(3 θ)

2

q2

(3

2

cos (3θ)

q2 sin (3θ)smn − 9

2

1

q3 sin (3θ)tmn

))

spq +

3

2

cos (3θ)

q2 sin (3θ)ppqmn =

−9

2

cos 3θ

q4 sin (3θ)spq smn − 27

4

cos 3θ

q4 sin3 3θspq smn +

81

4

1


3

2

cos (3θ)

q2 sin (3θ)ppqmn =

−(

9

2

cos 3θ

q4 sin (3θ)+

27

4

cos 3θ

q4 sin3 3θ

)

spq smn +81

4

1


3

2

cos (3θ)

q2 sin (3θ)ppqmn

where:

ppqmn =∂spq∂σmn

=

(

δmpδnq −1

3δpqδmn

)

is the projection tensor and:

∂AS

∂q=

−3. cot(3 θ)

q3

∂AS

∂θ=

−4.5 csc(3 θ)2

q2

The second member is∂(AT tpq)∂σmn

:



∂ (AT tpq)

∂σmn=

∂AT

∂σmntpq +AT

∂tpq∂σmn

=

(∂AT

∂q

∂q

∂σmn+∂AT

∂θ

∂θ

∂σmn

)

tpq +AT∂tpq∂σmn

=

(13.5 csc(3 θ)

q43

2

1

qsmn+

+13.5 cot(3 θ) csc(3 θ)

q3

(3

2

cos (3θ)

q2 sin (3θ)smn − 9

2

1

q3 sin (3θ)tmn

))

tpq +

+ − 9

2

1

q3 sin (3θ)wpqmn =

81

4

1

q5 sin 3θtpq smn +

81

4

cos2 3θ

q5 sin3 3θtpq smn − 243

4

cos 3θ

q6 sin3 3θtpq tmn −

−9

2

1

q3 sin (3θ)wpqmn =

(81

4

1

q5 sin 3θ+

81

4

cos2 3θ

q5 sin3 3θ

)

tpq smn − 243

4

cos 3θ

q6 sin3 3θtpq tmn −

−9

2

1

q3 sin (3θ)wpqmn

where:

wpqmn =∂tpq∂σmn


3sqpδnm − 2

3δpqsmn

∂AT

∂q=

13.5 csc(3 θ)

q4

∂AT

∂θ=

13.5 cot(3 θ) csc(3 θ)

q3



Then finally by collecting terms back again we have:

∂2θ

∂σpq∂σmn=

∂ (AS spq)

∂σmn+∂ (AT tpq)

∂σmn=

−(

9

2

cos 3θ

q4 sin (3θ)+

27

4

cos 3θ

q4 sin3 3θ

)

spq smn +81

4

1


+

(81

4

1

q5 sin 3θ+

81

4

cos2 3θ

q5 sin3 3θ

)

tpq smn − 243

4

cos 3θ


+3

2

cos (3θ)


2

1

q3 sin (3θ)wpqmn




Appendix D

Hyperelasticity: Detailed Derivations

D.1 Simo–Serrin’s Formula

In order to derive the analytical gradient of the fourth order tensor

MIJKL =∂MIJ

∂CKL

(D.1)

we shall proceed by using the third equation in (6.86).

∂MIJ

∂CKL

=1

D(A)

„

IIKJL −∂I1

∂ CKL

δIJ + 2λ(A)

∂λ(A)

∂CKL

δIJ +

+∂I3

∂CKL

λ−2(A) (C−1)IJ − 2λ−3

(A)

∂λ(A)

∂CKL

I3 (C−1)IJ +∂(C−1)IJ

∂CKL

λ−2(A) I3

«

−

−1

D2(A)

∂D(A)

∂CKL

“

CIJ −`

I1 − λ2(A)

´

δIJ + I3 λ−2(A) (C−1)IJ

”

(D.2)

where it was used that

∂CIJ

∂CKL

= IIKJL (D.3)

Derivatives ∂λ(A)/∂CKL can be found by starting from equation for CIJ (6.63) and differentiating it

dCIJ = 2λAdλ(A)

“

N(A)I N

(A)J

”

A+ λ2

A

“

dN(A)I N

(A)J

”

A+ λ2

A

“

N(A)I dN

(A)J

”

A(D.4)

By premultiplying previous equation with N(A)J and post-multiplying with N

(A)I , and by noting that

N(A)I dN

(A)I ≡ 0 ; ‖N

(A)I ‖ ≡ 1 (D.5)

we get

N(A)J dCIJ N

(A)I = 2λAdλ(A) (D.6)

or

dCIJN(A)I N

(A)J = dCIJλ(A) M

(A)IJ = 2λAdλ(A) ⇒

∂λA

∂CKL

=1

2λ(A) (M

(A)KL )A (D.7)

295


It can be proved1 that

∂I1

∂ CKL

= δIJ ;∂I2

∂ CKL

= I1 δKL − CKL ;∂I3

∂ CKL

= I3 (C−1)KL (D.8)

and since I3 = J2

∂J

∂ CKL

=1

2J (C−1)KL (D.9)

With this in mind, equation (D.2) can be rewritten as:

∂MIJ

∂CKL

=1

D(A)

„

IIKJL − δKL δIJ + 2λ2(A)

1

2M

(A)KL δIJ +

+ I3 λ−2(A) (C−1)IJ (C−1)KL − λ−2

(A) I3 (C−1)IJ M(A)KL +

+1

2

`

(C−1)IK(C−1)JL + (C−1)IL(C−1)JK

´

λ−2(A) I3

«

−

−1

D(A)

∂D(A)

∂CKL

MIJ (D.10)

where the definition of MIJ from equation (6.86) was used and also:

∂(C−1)IJ

∂CKL

= −1

2

`

(C−1)IK(C−1)JL + (C−1)IL(C−1)JK

´

= I(C−1)IJKL (D.11)

Relation (D.11) can be obtained if one starts from the identity:

CIJ (C−1)JK = δIK (D.12)

which after differentiation reads:

dCIJ (C−1)JK + CIJ d(C−1)JK = 0 ⇒

⇒ d(C−1)JK = −(C−1)JM dCMN (C−1)NK =

= −1

2

`

(C−1)JM (C−1)KN + (C−1)JN (C−1)KM

´

dCMN ⇒

⇒∂(C−1)JK

∂CMN

= −1

2

`

(C−1)JM (C−1)KN + (C−1)JN (C−1)KM

´

(D.13)

The derivative of D(A), that was defined in equation (6.77)as

D(A) = 2λ4(A) − I1λ

2(A) + I3λ

−2(A) (D.14)

1See Marsden and Hughes (1983)



is given by:

∂D(A)

∂CKL

= 8λ3(A)

∂λ(A)

∂CKL

−∂I1

∂CKL

λ2(A) − 2λ(A)I1

∂λ(A)

∂CKL

+∂I3

∂CKL

λ−2(A) − 2λ−3

(A)I3∂λ(A)

∂CKL

= 4λ4(A) M

(A)KL − δKL λ2

(A) − λ2(A)I1 M

(A)KL + I3(C

−1)KL λ−2(A) − λ−2

(A)I3 M(A)KL

=“

4λ4(A) − λ2

(A)I1 − λ−2(A)I3

”

M(A)KL − δKL λ2

(A) + I3(C−1)KL λ−2

(A)

= D′(A) M

(A)KL − δKL λ2

(A) + I3(C−1)KL λ−2

(A)

(D.15)

where D′(A) = 4λ4

(A) − λ2(A)I1 − λ−2

(A)I3. With the previous derivations, equation (D.10) can be written in expanded form

as:

∂MIJ

∂CKL

=1

D(A)

„

IIKJL − δKL δIJ + 2λ2(A)

1

2M

(A)KL δIJ +

+ I3 λ−2(A) (C−1)IJ (C−1)KL − λ−2

(A) I3 (C−1)IJ M(A)KL +

+1

2

`

(C−1)IK(C−1)JL + (C−1)IL(C−1)JK

´

λ−2(A) I3 −

−“

D′(A) M

(A)KL − δKL λ2

(A) + I3(C−1)KL λ−2

(A)

”

MIJ

”

(D.16)

If one collects similar terms, equation (D.16), also known as Simo–Serrin’s formula can be written in the final form

as:

∂MIJ

∂CKL

= MIJKL =

1

D(A)

“

IIKJL − δKL δIJ + λ2(A)

“

δIJ M(A)KL + M

(A)IJ δKL

”

+

+ I3 λ−2(A)

„

(C−1)IJ (C−1)KL +1

2

`

(C−1)IK(C−1)JL + (C−1)IL(C−1)JK

´

«

−

− λ−2(A) I3

“

(C−1)IJ M(A)KL + M

(A)IJ (C−1)KL

”

− D′(A) M

(A)IJ M

(A)KL

”

(D.17)

D.2 Derivation of ∂2volW/(∂CIJ ∂CKL)

The volumetric part ∂2volW/(∂CIJ ∂CKL) can be derived by starting from the equation (6.96):



∂2volW

∂CIJ ∂CKL

=

∂“

12

∂volW∂J

J (C−1)IJ

”

∂CKL

=

1

2

∂“

∂volW∂J

”

∂CKL

J (C−1)IJ +1

2

∂volW

∂J

∂ (J)

∂CKL

(C−1)IJ +1

2

∂volW

∂JJ

∂`

(C−1)IJ

´

∂CKL

=

1

2

∂2`

volW´

∂J ∂J

∂J

∂CKL

J (C−1)IJ +1

2

∂ volW

∂J

1

2J (C−1)KL (C−1)IJ +

1

2

∂volW

∂JJ I

(C−1)IJKL

=

1

4J2 ∂2 volW

∂J ∂J(C−1)KL(C−1)IJ +

1

4J

∂volW

∂J(C−1)KL (C−1)IJ +

1

2J

∂volW

∂JI(C−1)IJKL

=

1

4

„

J2 ∂2 volW

∂J ∂J+ J

∂volW

∂J

«

(C−1)KL (C−1)IJ +1

2J

∂volW

∂JI(C−1)IJKL

(D.18)

where equations (D.11) and (D.9) were used.

D.3 Derivation of ∂2isoW/(∂CIJ ∂CKL)

The isochoric part ∂2isoW/(∂CIJ ∂CKL) can be derived by starting from equation (6.97)

∂2isoW (λ(A))

∂CIJ∂CKL

=

1

2

∂“

wA (M(A)IJ )A

”

∂CKL

=

1

2

∂wA

∂CKL

(M(A)IJ )A +

1

2wA

∂(M(A)IJ )A

∂CKL

=

1

2

∂wA

∂λB

∂λB

∂CKL

(M(A)IJ )A +

1

2wA (M

(A)IJKL)A =

1

2

∂wA

∂λB

1

2λ(B) (M

(B)KL )B(M

(A)IJ )A +

1

2wA (M

(A)IJKL)A =

1

4YAB (M

(B)KL )B (M

(A)IJ )A +

1

2wA (M

(A)IJKL)A

(D.19)



where equation (D.7) was used and tensor YAB is defined as:

YAB =∂wA

∂λB

λ(B) (D.20)

D.4 Derivation of wA

wA =∂isoW

∂λ(A)

λA =∂isoW

∂λB

∂λB

∂λ(A)

λA (D.21)

where λB is the isochoric part of the stretch defined as

λB = J− 13 λB (D.22)

From the definition of λB in equation (D.22) it follows

∂λB

∂λ(A)

=∂J− 1

3

∂λ(A)

λB + J− 13

∂λB

∂λ(A)

= −1

3J− 1

3 λ−1(A) λB + J− 1

3 δB(A) (D.23)

since

∂J− 13

∂λ(A)

= −1

3J− 4

3∂λ1λ2λ3

∂λ(A)

= −1

3J− 4

3 J λ−1(A) = −

1

3J− 1

3 λ−1(A) (D.24)

then

wA = −1

3J− 1

3∂isoW

∂λB

λ−1(A) λB λ(A) + J− 1

3∂isoW

∂λB

δB(A) λ(A)

= −1

3

∂isoW

∂λB

λB +∂isoW

∂λ(A)

λ(A) (D.25)

D.5 Derivation of YAB

By starting from equation D.20

YAB =∂wA

∂λB

λ(B) (D.26)

and by using equation D.25

wA = −1

3

∂isoW

∂λC

λC +∂isoW

∂λ(A)

λ(A) (D.27)

we can write:

YAB =∂wA

∂λD

∂λD

∂λB

λ(B) (D.28)



By first considering ∂wA/∂λD we get:

∂wA

∂λD

=

∂

„

− 13

∂isoW

∂λCλC + ∂isoW

∂λ(A)λ(A)

«

∂λD

= −1

3

∂2isoW

∂λC∂λD

λC −1

3

∂isoW

∂λC

∂λC

∂λD

+∂2isoW

∂λ(A)∂λD

λ(A) +∂isoW

∂λ(A)

∂λ(A)

∂λD

= −1

3

∂2isoW

∂λC∂λD

λC −1

3

∂isoW

∂λC

δCD +∂2isoW

∂λ(A)∂λD

λ(A) +∂isoW

∂λ(A)

δ(A)D

= −1

3

∂2isoW

∂λC∂λD

λC −1

3

∂isoW

∂λD

+∂2isoW

∂λ(A)∂λD

λ(A) +∂isoW

∂λ(A)

δ(A)D (D.29)

Next, from equation D.23, we have that

∂λD

∂λ(B)= −

1

3J− 1

3 λ−1(B) λD + J− 1

3 δD(B) (D.30)

and by multiplying the result for ∂wA/∂λD from equation D.29 and the result for ∂λD/∂λ(B) from equation D.30 we

obtain:

∂wA

∂λD

∂λD

∂λ(B)= +

1

9

∂2isoW

∂λC∂λD

λC J− 13 λ−1

(B) λD −1

3

∂2isoW

∂λC∂λD

λCJ− 13 δD(B)

+1

9

∂isoW

∂λD

J− 13 λ−1

(B) λD −1

3

∂isoW

∂λD

J− 13 δD(B)

−1

3

∂2isoW

∂λ(A)∂λD

λ(A)J− 1

3 λ−1(B) λD +

∂2isoW

∂λ(A)∂λD

λ(A)J− 1

3 δD(B)

−1

3

∂isoW

∂λ(A)

δ(A)DJ− 13 λ−1

(B) λD +∂isoW

∂λ(A)

δ(A)DJ− 13 δD(B)

= +1

9

∂2isoW

∂λC∂λD

λC λ−1(B) λD −

1

3

∂2isoW

∂λC∂λ(B)

λC J− 13

+1

9

∂isoW

∂λD

λ−1(B) λD −

1

3

∂isoW

∂λ(B)

J− 13

−1

3

∂2isoW

∂λ(A)∂λD

λ(A)λ−1(B) λD +

∂2isoW

∂λ(A)∂λ(B)

λ(A) J− 13

−1

3

∂isoW

∂λ(A)

λ−1(B) λ(A) +

∂isoW

∂λ(A)

δ(A)(B) J− 13 (D.31)

where equation D.22 was used. The final form for YAB is obtained by multiplying equation D.31 with λ(B) to obtain:

YAB =∂wA

∂λD

∂λD

∂λB

λ(B) =

+1

9

∂2isoW

∂λC∂λD

λC λ−1(B) λD λ(B) −

1

3

∂2isoW

∂λC∂λ(B)

λC λ(B)

+1

9

∂isoW

∂λD

λ−1(B) λD λ(B) −

1

3

∂isoW

∂λ(B)

λ(B)

−1

3

∂2isoW

∂λ(A)∂λD

λ(A)λ−1(B) λD λ(B) +

∂2isoW

∂λ(A)∂λ(B)

λ(A) λ(B)

−1

3

∂isoW

∂λ(A)

λ−1(B) λ(A) λ(B) +

∂isoW

∂λ(A)

δ(A)(B) λ(B) (D.32)

By recognizing that λ−1(B)λ(B) ≡ 1 and after rearranging elements, we can finally write the equation for YAB as:



YAB =∂isoW

∂λ(A)

δ(A)(B) λ(B) +∂2isoW

∂λ(A)∂λ(B)

λ(A) λ(B)

−1

3

∂2isoW

∂λC∂λ(B)

λC λ(B) +∂isoW

∂λ(B)

λ(B) +∂2isoW

∂λ(A)∂λD

λ(A)λD +∂isoW

∂λ(A)

λ(A)

!

+1

9

„

∂2isoW

∂λC∂λD

λC λD +∂isoW

∂λD

λD

«

(D.33)


Documents

62DD18B7d01