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Lecture Notes Advanced Finite Element Methods Dr.-Ing. habil. D. Kuhl Univ. Prof. Dr. techn. G. Meschke May 2005 Ruhr University Bochum Institute for Structural Mechanics

Kuhl & Meschke - Advanced Finite Element Methods

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  • Lecture Notes

    Advanced Finite Element Methods

    Dr.-Ing. habil. D. KuhlUniv. Prof. Dr. techn. G. Meschke

    May 2005

    Ruhr University BochumInstitute for Structural Mechanics

  • Lecture Notes

    Advanced Finite Element Methods

    Dr.-Ing. habil. D. KuhlUniv. Prof. Dr. techn. G. Meschke

    May 2005

    Ruhr University BochumInstitute for Structural MechanicsUniversitatsstrae 150 IA6D-44780 BochumTelefon: +49 (0) 234 / 32 29055Telefax: +49 (0) 234 / 32 14149E-Mail: [email protected]: http://www.sd.ruhr-uni-bochum.de

  • Contents

    1 Fundamentals of Linear Structural Mechanics 1

    1.1 Continuum Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

    1.1.1 Displacement Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

    1.1.2 Definition of a Non-Linear Strain Measure . . . . . . . . . . . . . . . . . . . . . . . 2

    1.1.3 Definition of a Linear Strain Measure . . . . . . . . . . . . . . . . . . . . . . . . . 5

    1.2 Continuum Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

    1.2.1 Cauchys Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

    1.2.2 Balance of Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

    1.2.3 Initial Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

    1.3 Initial and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

    1.3.1 Classification of Initial and Boundary Conditions . . . . . . . . . . . . . . . . . . . 12

    1.3.2 Dirichlet Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

    1.3.3 Neumann Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

    1.3.4 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

    1.4 Hyperelastic Constitutive Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

    1.4.1 Fundamental Assumptions and Classification . . . . . . . . . . . . . . . . . . . . . 16

    1.4.2 Elastic Material Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

    1.4.3 Isotropic, Elastic Material Relation of Continuum . . . . . . . . . . . . . . . . . . 17

    1.4.4 Plane Stress State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

    1.4.5 Plane Strain State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

    1.4.6 The Classical Hookes Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

    1.5 Initial Boundary Value Problem of Elastomechanics . . . . . . . . . . . . . . . . . . . . . 24

    1.5.1 Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

    1.5.2 Geometrically and Materially Linear Elastodynamics . . . . . . . . . . . . . . . . . 25

    1.5.3 Geometrically and Materially Linear Elastostatics . . . . . . . . . . . . . . . . . . 26

    1.6 Weak Form of The Initial Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . . 26

    1.6.1 Principle of Virtual Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

    1.6.2 Properties of The Principle of Virtual Work . . . . . . . . . . . . . . . . . . . . . . 29

    2 Spatial Isoparametric Truss Elements 31

    2.1 Fundamental Equations of One-dimensional Continua . . . . . . . . . . . . . . . . . . . . 32

    2.1.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

    i

  • ii Kuhl & Meschke, Finite Element Methods in Linear Structural Mechanics

    2.1.2 Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

    2.1.3 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

    2.1.4 Constitutive Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

    2.1.5 Principle of Virtual Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

    2.1.6 Euler Differential Equation and Neumann Boundary Conditions . . . . . . . . . . 38

    2.2 Finite Element Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

    2.2.1 Partitioning of The Structure into Elements . . . . . . . . . . . . . . . . . . . . . . 39

    2.2.2 Approximation of Variables of One-dimensional Continua . . . . . . . . . . . . . . 40

    2.2.3 Truss Element with Linear Shape Functions . . . . . . . . . . . . . . . . . . . . . . 45

    2.2.4 Truss Element with Quadratic Shape Functions . . . . . . . . . . . . . . . . . . . . 52

    2.2.5 Truss Element with Cubic Shape Functions . . . . . . . . . . . . . . . . . . . . . . 55

    2.2.6 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

    2.3 Assembly of the Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

    2.3.1 Transformation of the Element Matrices and Vectors . . . . . . . . . . . . . . . . . 62

    2.3.2 Assembly of the Elements to the System . . . . . . . . . . . . . . . . . . . . . . . . 66

    2.4 Solution of the System Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

    2.4.1 Linear Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

    2.4.2 Linear Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

    2.4.3 Solution of the Linear System of Equations . . . . . . . . . . . . . . . . . . . . . . 78

    2.5 Postprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

    2.5.1 Separation and Transformation of the Element Degrees of Freedom . . . . . . . . . 79

    2.5.2 Computation of Strains, Stresses and Section Loads . . . . . . . . . . . . . . . . . 79

    2.5.3 Aspects of Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

    3 Plane Finite Elements 81

    3.1 Basic Equations of Planar Continua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

    3.1.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

    3.1.2 Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

    3.1.3 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

    3.1.4 Constitutive Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

    3.1.5 Principle of Virtual Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

    3.1.6 Euler Differential Equation and Neumann Boundary Conditions . . . . . . . . . . 86

    3.2 Finite Elemente Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

    3.2.1 Partitioning into Elements and Discretization . . . . . . . . . . . . . . . . . . . . . 88

    3.2.2 Classification of Plane Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

    3.2.3 Shape Functions of Plane Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

    3.3 Bilinear Lagrange element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

    3.3.1 Ansatz functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

    3.3.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

    3.3.3 Jacobi transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

    3.3.4 Approximation of element quantities . . . . . . . . . . . . . . . . . . . . . . . . . . 99

  • Institute for Structural Mechanics, Ruhr University Bochum, May 2005 iii

    3.3.5 Strain vector approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

    3.3.6 Appproximation of internal virtual work . . . . . . . . . . . . . . . . . . . . . . . . 101

    3.3.7 Approximation of dynamic virtual work . . . . . . . . . . . . . . . . . . . . . . . . 102

    3.3.8 Approximation of virtual work of external loads . . . . . . . . . . . . . . . . . . . . 103

    3.3.9 Rectangular Bilinear Lagrange Element . . . . . . . . . . . . . . . . . . . . . . . . 106

    3.4 Rectangular biquadratic Lagrange element . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

    3.4.1 Ansatz functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

    3.4.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

    3.4.3 Jacoby transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

    3.4.4 Approximation of element quantities . . . . . . . . . . . . . . . . . . . . . . . . . . 124

    3.4.5 Approximation of the strain vector . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

    3.4.6 Element matrices and vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

    3.5 Biquadratic serendipity element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

    3.5.1 Ansatz functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

    3.5.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

    3.5.3 Approximation of element quantities . . . . . . . . . . . . . . . . . . . . . . . . . . 128

    3.6 Triangular plane finite elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

    3.6.1 Natural coordinates of a triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

    3.6.2 Ansatz functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

    3.6.3 Isoparametric approximation of continuous quantities . . . . . . . . . . . . . . . . 133

    3.6.4 Element matrices and vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

    3.6.5 Constant Strain Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

    3.7 Numerical integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

    3.7.1 Quadrangular elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

    3.7.2 Triangular elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

    4 Finite volume elements 143

    4.1 Fundamental equations of three-dimensional continua . . . . . . . . . . . . . . . . . . . . 144

    4.2 Finite element discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

    4.2.1 Natural coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

    4.2.2 Ansatz Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

    4.2.3 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

    4.2.4 Jacobi transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

    4.2.5 Differential Operator B() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

    4.2.6 Element Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

    4.2.7 Element Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

    5 Basics of non-linear structural mechanics 151

    5.1 Non-linearities of structural mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

    5.2 Material non-linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

    5.2.1 Mathematical formulation of material non-linearity . . . . . . . . . . . . . . . . . . 153

  • iv Kuhl & Meschke, Finite Element Methods in Linear Structural Mechanics

    5.3 Geometrical non-linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

    5.3.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

    5.3.2 Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

    5.3.3 Constitutive Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

    5.3.4 Principle of virtual displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

    5.3.5 Internal virtual work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

    5.3.6 Elastic internal potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

    5.3.7 Remarks regarding combined material and geometric non-linearity . . . . . . . . . 163

    5.4 Consistent linearization of internal virtual work . . . . . . . . . . . . . . . . . . . . . . . . 163

    5.4.1 Linearization background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

    5.4.2 Gateaux derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

    5.4.3 Gateaux derivative of internal virtual work . . . . . . . . . . . . . . . . . . . . . . 164

    5.4.4 Linearization of Green Lagrange strains . . . . . . . . . . . . . . . . . . . . . . . . 166

    5.4.5 Linearization of variation of Green Lagrange strains . . . . . . . . . . . . . . . . . 167

    6 Finite element discretization of geometrically non-linear continua 171

    6.1 Finite volume elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

    6.1.1 Discretization of internal virtual work . . . . . . . . . . . . . . . . . . . . . . . . . 172

    6.1.2 Non-linear semi-discrete initial value problem . . . . . . . . . . . . . . . . . . . . . 177

    6.1.3 Non-linear discrete static equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 178

    6.1.4 Discretization of linearized internal virtual work . . . . . . . . . . . . . . . . . . . 178

    6.1.5 Linearization of internal forces vector . . . . . . . . . . . . . . . . . . . . . . . . . 181

    6.2 Finite truss elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

    6.2.1 Non-linear continuum-mechanical formulation . . . . . . . . . . . . . . . . . . . . . 182

    6.2.2 Truss elements of arbitrary polynomial degree . . . . . . . . . . . . . . . . . . . . . 183

    6.2.3 Linear truss element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

    7 Solution of non-linear static structural equations 189

    7.1 Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

    7.2 Iteration methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

    7.2.1 Single step method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

    7.2.2 Pure Newton-Raphson method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

    7.2.3 Modified Newton-Raphson method . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

    7.3 Control of iteration procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

    7.3.1 Load-incrementing and control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

    7.3.2 Arc-length controlling method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

    References 202

  • Preface

    These lecture notes, which in actual fact are an English translation of the German lecturenotes Finite Elemente Methoden I of the diploma study course, were created in the contextof the lecture Finite Element Methods I which was first held in this form during the winterterm 1998/1999. Finite Element Methods in Linear Structural Mechanics thus represents theteachings of finite element methods in the area of linear structural mechanics with the focuson showing of possibilities and limits of the numerical method as well as the development ofisoparametric finite elements. These notes are to support the students in following up the lectureand to prepare them for the exam. They cannot possibly substitute the lecture or the exerciseentities. In addition to the lecture and the notes, mathematical programmes for deepeningthe lecture contents are available at the homepage of the Institute for Structural Mechanicshttp://www.sd.ruhr-uni-bochum.de/.

    Here, the authors would like to thank Mr. Jorn Mosler and Mr. Stefan Jox for the excellentconduction of the theoretical and practical exercise entities accompanying the lecture FiniteElement Methods I. Moreover, the authors give their thanks to Ms. Barbara Kalkhoff, graphicaldesigner, for the high quality drawings as well as to Ms. Monika Rotthaus, Ms. Wiebke Breil,Ms. Sandra Krimpmann, Ms. Julia Mergenheim, Mr. Christian Becker, Mr. Alexander Beer,Mr. Sonke Carstens and Mr. Janosch Stascheit for their indispensable efforts in creating theselecture notes.

    Last but not least the authors would like to thank Mr. Ivaylo Vladimirov, Mr. Hrvoje Vucemilovicand Ms. Amelie Gray who helped to translate the notes into the English language. At the sametime we would like to excuse the fact that the description of the drawings are in German.Nevertheless, we believe that the meaning becomes clear. The authors are continually workingon improving the lecture notes. Therefore, please feel free to communicate your comments, ideasand corrections.

    For all students who intend to continue with the lecture Finite Element Methods II withthe emphasis on non-linear structural mechanics, the lecture notes are complemented by thecorresponding chapters 5 to 7 as well as by the indication of further literature. The chaptersconcerning the non-linear finite element methods are also available in the form of lecture notes(Finite Elemente Methoden II, 3. edition, October 2002, in German language) at the Institutefor Structural Mechanics, IA 6/127.

    Bochum, May 2005 Gunther Meschke and Detlef Kuhl

    v

  • Chapter 5

    Basics of non-linear structuralmechanics

    In the scope of Finite Element discretization in linear structural mechanics presented in theprevious chapters, two major very important simplifying assumptions were made which did notalways result in adequate modelling of real structural behaviour. In these assumptions, it waspostulated that the material behaviour is linear elastic (materially linear) and that deforma-tions are small (geometrically linear). The former assumption was prescribing the validity ofthe Hooke law and a priori excluding the modelling of irreversible material behaviour suchas plastification or damaging. From the latter assumption, it follows that equilibrium can beestablished on an undeformed structure and that non-linear terms of Green Lagrange straintensor E can be neglected. The so far discussed linear structural mechanics theory makes upthe classic fundamentals of static and dynamic analysis in civil engineering. Inspite of funda-mental restrictive assumptions just mentioned, this theory will find application in calculationsof deformations and stresses in engineering structures in the future as well, as long as the nec-essary conditions are met (small deformations and stresses which justify the assumptions oflinear material behaviour), since complexity of FE formulation as well as numerical effort forsolving non-linear problems are notably rising compared to linear problems. What makes a dif-ference in engineering praxis is that superposition principle is not valid in non-linear cases andconsequently every analysed load case requires a complete computation.

    Apart from the classic linear analysis established in engineering, the demands on models ofstructural engineering will notably increase due to growing replacement of development andverification experiments by cost-reducing, transparent and faster computer simulations. For ex-ample, structures which are slender and light for technical or aesthetic reasons can only beadequately simulated and examined regarding stability with the help of a geometrically non-linear calculus (see e.g. Kramer et al. [110]). On the other hand, concrete or reinforcedconcrete is a material characterized by distinct non-linear behaviour due to inevitable cracks(see e.g. Kratzig, Mancevski &Polling [113]), which has to be taken into account in thestructural analysis calculus. Cupping and shaping processes in the field of industrial produc-tion (see e.g. Glaser [104]) or car crashes (see e.g. Moller [67]) are application examplesfor simulations dependent on modelling of metal plastification with large deformations. On theone hand it is the lasting deformations and on the other hand the dissipated plastification en-ergy that is of crucial significance for the product quality, that is, the safety of the passenger.That shaping processes are impossible without large deformations is self-evident; also the after-math of a crash seldomly justifies the assumption of small deformations. Other structures areso intensively loaded that they are impossible with structural exclusion of non-linear material

    151

  • 152 Kuhl & Meschke, Adavanced Finite Element Methods, DRAFTversion

    behaviour. Plastic deformation during exploitation has to be accepted and simulated accord-ingly in the design and development process (see e.g. simulation of incineration chambers inKuhl [114], Kuhl, Woschnak &Haidn [117]). The previous list of necessities of non-linearsimulation techniques can be extended almost at will, but we nevertheless want to concentrate onelaboration of geometrically or/and materially non-linear problems, the modelling thereof, andon numeric-algorithmic conversion. If the body deformations are large, the carrying pattern canchange significantly with the deformation what can reflect on the change of structural stiffness,or in view of dynamics, on the change of eigenfrequencies. In the extreme case, this can lead toloss of stability where significant deformations occur without the load increase and the structureeventually collapses. Especially the slender, thin-walled structures or structures optimized withrespect to linear properties tend to behave in a pragmatic, non-linear way with endangered sta-bility. Besides the geometrical non-linearity, modelling and discretization of non-linear problemsought to be investigated, too. Metal materials for example display a linear behaviour until theyreach a certain stress level, the so called flow limit, above which their plastic deformations occurin connection with a notably reduced material stiffness. The consequence is that the aictedstructures undergo a load redistribution which still leaves them serviceable even though plas-tic deformations were already localized. A similar phenomenon can be observed at concrete orother quasi-brittle materials, particularly ceramics. In these materials microcracks develop dueto loading and degrade the material strength and stiffness.

    5.1 Non-linearities of structural mechanics

    Structural mechanical simulations can be classified according to the modelled non-linearities andtheir combinations in the order of increased complexity, and according to the numerical analysiseffort, as follows (see figure 1.13):

    materially non-linear and geometrically linear

    materially linear and geometrically non-linear

    materially and geometrically non-linear with assumed moderate strains

    materially and geometrically non-linear with finite strains

    Compared to figure 1.13, the differentiation of combined material and geometrical non-linearitywith finite strains is of crucial importance for the formulation and numerical conversion of struc-tural mechanical problematics, but should however not be studied in more detail within the scopeof these lectures. The lectures will limit themselves for most part to pure geometric non-linearlocal impulse equilibrium and its formulation, discretization, linearization, numerical solutionwith diverse algorithms, as well as to stability observations. The material non-linearity will bepresented only schematically and the corresponding FE discretization will be elaborated briefly.For material formulations on the material point level, which is equivalent to a Gauschenintegration point in the numerical realization of FE methods, and their algorithmic conver-sion, refer to lectures of Computational Plasticity and to technical literature (e.g. Gro [54],Hill [108], Krajcinovic [111], Krajcinovic &Lemaitre [112], Lemaitre [118], Lemaitre&Chaboche [61], Lubliner [119] and Simo &Hughes [133]).

  • Institute for Structural Mechanics, Ruhr University Bochum, May 2005, DRAFTversion 153

    5.2 Material non-linearity

    5.2.1 Mathematical formulation of material non-linearity

    The rate-independent material non-linearity is, contrary to materially linear formulations (seechapter 1.4 and equation (1.50)), characterized by the fact that the stress state

    = C : (5.1)

    cannot be obtained by linear mapping of the strain state with the help of material tensor C.The stress tensor or vector is rather an arbitrary function of the strain tensor or vector ,and other values , described as internal variables or as time history variables that characterizenon-elastic deformations or damage

    = (,) (5.2)

    where the partial derivative of the stress tensor/vector with respect to the strain tensor definesthe tangential material tensor or the tangential material matrix.

    Ctan(,) =(,)

    (5.3)

    In order to make the solution of mechanical boundary problems or initial value problems possible,the stress function ((5.2) has to be supplemented with the so called evolution equations of internalvariables in the form

    = (,) (5.4)

    In the special case of non-linear elastic material laws, the stress state is only a function of thestrain state

    = () Ctan() =()

    (5.5)

    and the evolution equations are dropped. For the simulation of the non-linear material lawmostly

    non-linear elastic,

    elasto-plastic

    and elasto-damaged

    material models and presented ground type combinations are in use. As shown in figure 5.1, thecurves of three materially non-linear phenomena can basically be one and the same for loadingsequence, whereas the differences in material formulations are decisive in the unloading sequence.In the non-linear elastic case, the stress-strain diagram for unloading runs along the loading pathand after full unloading a strain-free state is reached and the new cycle is identical to the first one.In case of the elasto-plastic material model, the unloading sequence runs parallel to the initialrate E and after full unloading the structure is not strain-free because plastic strains remain.The next cycle is therefore different from the first one. As opposed to that, upon unloading in thecase of elasto-damage model, no permanent strains remain. Unlike the linear-elastic model, the

  • 154 Kuhl & Meschke, Adavanced Finite Element Methods, DRAFTversion

    (non-linear elastic elasto-plastic elasto-damaged

    no change plastic strain elasticity modulus

    -

    6

    E

    ::

    99

    -

    6

    E

    ::

    - -p p-

    6

    E (1d)E

    ::

    Figure 5.1: Cyclic loading and unloading of elastic, elasto-plastic, and elasto-damaged bilinearmaterial models

    unloading sequence does not follow the loading path but it runs linearly to the diagram origin.A new load introduction is influenced by degradation of stiffness with the damage parameter d,due to which the repeated load cycles are not identical in their effects on the material and thestructure.

    5.3 Geometrical non-linearity

    Alterations regarding geometrically linear observation:

    Consideration of non-linear terms of the strain tensor E (kinematics)

    Establishing the forces equilibrium or application of the impulse theorem in the deformedconfiguration (kinetics)

    We use:

    (total) Lagrange point of view which is also described as the material point of view

    Stress and strain quantities in the undeformed configuration (second Piola Kirchhoffstress tensor and Green Lagrange strain tensor)

    Literature:

    Altenbach&Altenbach [38], Antman [84], Basar [88], Betten [43], de Boer [44], Bonet&Wood [46], Malvern [63], Marsden &Hughes [65], Smith [73], Stein &Barthold [74]and Truesdell &Noll [78]

    5.3.1 Kinematics

    The fundamental of the geometrically non-linear formulation of structural mechanics is based onthe material deformation gradient F , which was already used within the scope of deriving the

  • Institute for Structural Mechanics, Ruhr University Bochum, May 2005, DRAFTversion 155

    e 1

    e 2

    e 3

    Y

    XP

    Qd X q

    d xpyx

    R e f e r e n z k o n f i g u r a t i o nu n d e f o r m i e r t e L a g e

    d e f o r m i e r t e L a g eM o m e n t a n k o n f i g u r a t i o n

    A b b i l d u n g ( X , t )

    Figure 5.2: Current and reference configuration of a deformable material body

    linear strain tensor with the help of a non-linear deformation analysis of a material body, andwithin the scope of subsequent linearization in chapter 1.1.2, equation (1.5), but not elaboratedbecause in linear observations it possesses no further significance. Non-linear observations area different story where the material deformation gradient defines the transformation fromreference to current configuration or from undeformed to deformed state and vice versa. Thesetransformations are referred to in technical literature as push forward and pull back. Thematerial deformation gradient is defined by a transformation of a line element dX of thereference configuration to the current configuration dx (see figure 5.2).

    dx = F dX F =x

    X= x (5.6)

    The Green Lagrange strain tensor E was also already derived and is given in equation(1.12) as function of the displacement gradient u and its transpose Tu. If we describe themotion of the material point from the reference to the current configuration with help of thedisplacement vector x = X + u, the Green Lagrange strain tensor

    E =1

    2

    [F T F 1

    ]= symu +

    1

    2Tu u =

    1

    2

    [Tu +u +Tu u

    ](5.7)

    can be represented as a function of the material deformation gradient.

    F =

    X(X + u) = 1 +u (5.8)

  • 156 Kuhl & Meschke, Adavanced Finite Element Methods, DRAFTversion

    According to this relation, the components of the term

    symu =

    u1,112(u1,2 + u2,1)

    12(u1,3 + u3,1)

    12(u1,2 + u2,1) u2,2

    12(u2,3 + u3,2)

    12(u1,3 + u3,1)

    12(u2,3 + u3,2) u3,3

    (5.9)

    of the linear part of the strain tensor E, given explicitly in equations (1.13) and (1.14), aresupplemented by the term 1/2Tu u, in the scope of geometrically non-linear theory, whichcan again be calculated with the displacement vector gradient according to eq. (1.7)

    u =

    u1,1 u1,2 u1,3

    u2,1 u2,2 u2,3

    u3,1 u3,2 u3,3

    (5.10)

    by matrix multiplication.

    1

    2Tu u =

    1

    2

    uk,1 uk,1 uk,1 uk,2 uk,1 uk,3

    uk,2 uk,1 uk,2 uk,2 uk,2 uk,3

    uk,3 uk,1 uk,3 uk,2 uk,3 uk,3

    (5.11)

    It should be noted that the summation is performed over k = 1, 2, 3, respectively. The compo-nent presentation of the Green Lagrange strain tensor finally yields the following:

    Eij =1

    2(ui,j + uj,i + uk,i uk,j) E = Eij Ei Ej (5.12)

    In order to formulate the non-linear Finite Element methods, it remains to convert the calcula-tion rule of the strain tensor, given in eq. (5.7) or (5.12), into the calculation rule of the strainvector in a suitable way . The linear part of the strain tensor can be expressed as a vector byapplication of the differential operator D to the displacement vector u, as described in eq.(1.16). However, for the non-linear part of the strain tensor, no suitable operator presentationcan be found.

    E(u) =

    E11E22E332E122E232E13

    =

    u1,1u2,2u3,3

    u1,2 + u2,1u2,3 + u3,2u1,3 + u3,1

    +

    1/2 (u1,1 u1,1 + u2,1 u2,1 + u3,1 u3,1)

    1/2 (u1,2 u1,2 + u2,2 u2,2 + u3,2 u3,2)

    1/2 (u1,3 u1,3 + u2,3 u2,3 + u3,3 u3,3)

    u1,1 u1,2 + u2,1 u2,2 + u3,1 u3,2u1,2 u1,3 + u2,2 u2,3 + u3,2 u3,3u1,1 u1,3 + u2,1 u2,3 + u3,1 u3,3

    (5.13)

    According to eq (5.13), the Green Lagrange strain tensor of geometrically non-lineardeformations is obtained by addition of the well-known linear part Du and the non-linear partEnl(u),

    E(u) = D u + Enl(u) (5.14)

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    where Du and Enl(u) are defined as follows (summation over k = 1, 2, 3).

    D u =

    u1,1u2,2u3,3

    u1,2 + u2,1u2,3 + u3,2u1,3 + u3,1

    Enl(u) =

    1/2 uk,1 uk,11/2 uk,2 uk,21/2 uk,3 uk,3

    uk,1 uk,2uk,2 uk,3uk,1 uk,3

    (5.15)

    Formulation of Dirichlet boundary conditions (eq. (1.33))

    u(X) = u?(X) X u (5.16)

    and of initial conditions remains unaltered compared to linear structural mechanics (eq. (1.45)).

    u(X , t = 0) = u?(X)

    oder

    u(X , t = 0) = u?(X)

    X (5.17)

    5.3.2 Kinetics

    Unlike the linear structural analysis, its non-linear counterpart requires that the dynamic orthe static forces equilibrium be observed in the deformed configuration. This firstly calls forthe evaluation of mass distribution which gives the relation between density in the currentconfiguration c and the one in the reference configuration , with the help of determinant |F |of the material deformation gradient (see e.g. Marsden &Hughes [65]).

    = |F | c (5.18)

    The forces equilibrium of a geometrically linear approximation, presented in chapter 1.2, eq.(1.25), was obtained by pure kinetic analysis of a differential volume element. Analoguous anal-ysis of a volume element in a deformed configuration gives us the Cauchy motion equation inthe so called spatial or Euler formulation.

    c u = div + c b x (5.19)

    div symbolises the tensor divergence of the real stresses or the Cauchy stress tensor, relatedto the current configuration. As the first description of this stress tensor might lead us to antic-ipate, this stress quantity is defined by a differential load in the current configuration, effectinga deformed surface element da, arbitrarily oriented with a normal vector n inside the body,leading to the consequence that the actual stresses occurring in the material can be described.

    It is of advantage to numerical conversion in structural mechanics to utilize the motion equationin the materialor Lagrange formulation. To perform this it is necessary to relate the Cauchymotion equation to the undeformed configuration. Multiplication of motion equation (5.19)by determinant of the material deformation gradient already transforms the density in thereference configuration, according to (5.18).

  • 158 Kuhl & Meschke, Adavanced Finite Element Methods, DRAFTversion

    |F | c u = |F | div + |F | c b u = |F | div + b (5.20)

    It remains to transform the middle term of the equation. Since this transformation requires theprofound familiarity with non-linear continuum mechanics, we shall leave out the derivationprocedure and resort to the related literature for the result (e.g. Malvern [63], Marsden& Hughes [65], Smith [73]). To clarify the outcome,we shall give the resulting identity insymbolic presentation and it component notation.

    |F | div = DIV(F S) |F |ijxj

    =(Fik Skj)

    Xj(5.21)

    DIV symbolises the divergence operator with respect to the reference configuration. Tensor 1

    S, which is here used for the first time, is the second Piola Kirchhoff stress tensor defined withrespect to the reference configuration. It should be noted that the Piola Kirchhoff stresstensor S, unlike the Cauchy stress tensor , does not refer to actual stresses but to pseudostresses. They are defined with respect to the reference configuration by a differential loadeffecting a surface element of the reference configuration dA, which is oriented with a normalvector N . Introduction of equation (5.21) into equation (5.20) eventually yields the material orthe Lagrange formulation of impulse rule, that is, the Cauchy motion equation.

    u = DIV(F S) + b X (5.22)

    In order to formulate a well-defined problem, it is necessary to supplement the impulse rulewith the static or Neumann boundary condition. The latter is given in spatial formulationanalogous to equation (1.40). It is possible to apply the procedure for derivation of equilibriumat the boundary, demonstrated in the geometrically linear case, to the deformed configuration

    n = t? x (5.23)

    and thereupon to transform it. Material formulation of the Neumann boundary conditions isin the geometrically non-linear case defined by

    F S N = T ? X (5.24)

    (see e. g. Marsden & Hughes [65]), where N presents the normal vector of referenceconfiguration. Stress vector T ? (first Piola Kirchhoff stress tensor) is defined by a differentialload vector of current configuration acting on a surface element of the reference configurationdA parallel to t?.

    T ? dA = t? da (5.25)

    5.3.3 Constitutive Law

    For moderate strains, to which we shall limit ourselves here, we can introduce the SaintVenant Kirchhoff material model to project the Green Lagrange strains on to second

    1The Product P = F S presents the first Piola Kirchhoff stress tensor, which is not used or discussed in the

    lectures

  • Institute for Structural Mechanics, Ruhr University Bochum, May 2005, DRAFTversion 159

    Piola Kirchhoff stresses in analogy with equation (1.56) of linear structural mechanics.

    S = C : E C = 2 I + 1 1 (5.26)

    5.3.4 Principle of virtual displacements

    5.3.4.1 Weak formulation of initial value problem

    Weak formulation of the initial value problem of geometrically non-linear structural mechanicscan be obtained in a way analogous with linear formulation in chapter 1.6.1, by choice of dis-placement vector transformation u as special test function, and by applying the calculationrule (1.87) to u DIV(F S),= DIV(u (F S))u : (F S) , where the volume integralof the first right-hand side term DIV(u (F S)) can be transformed to upper surface integralvia u F S N, by applying the Gau integral law (see chapter 1.6.1). With the help ofmentioned transformations we get the virtual displacements principle.

    u u dV +

    u : (F S) dV =

    u b dV +

    u T ? d (5.27)

    dV and d describe the volume element, that is, the line element in the reference configuration.First term in equation (5.30) is the virtual work of inertial forces Wdyn, the second term isthe virtual work Wint. The sum of the third and the fourth term makes up the virtual workof external loads Wext. Internal virtual work in equation (5.27) should be transformed in sucha way that one obtains the form of virtual displacements equivalent to equation (1.94). To getthis form, we first vary the term E : S according to definition of E in equation (5.7), where inorder to achieve further transformation we use the symmetry of the stress tensor.

    E : S =1

    2(F T F 1

    ): S =

    1

    2

    (F T F + F T F

    ): S = F T F : S = F : F S (5.28)

    With the definition of the material deformation gradient in equation (5.6) and the unchangingcoordinates of the reference configuration (X = 0),

    F =

    X(X + u) =

    u

    X= u (5.29)

    we can write the virtual displacement principle with equations (5.27), (5.28) and (5.29) inthe form preferred for further derivations (in analogy with equation (1.94) of geometricallynon-linear structural mechanics).

    u u dV +

    E : S dV =

    u b dV +

    u T ? d (5.30)

    It may be noticed once again that the Lagrange formulation of the impulse rule is used, andas a consequence, the integration over the volume, that is, over the boundary of a material bodyhas to be performed in the reference configuration. From equation (5.30),it can be concluded

  • 160 Kuhl & Meschke, Adavanced Finite Element Methods, DRAFTversion

    that virtual work of inertial forces Wdyn, and of external loads Wext, did not undergoany formal change in comparison with linear observations in chapter 1.6.1. This practicallymeans that for isoparametric one-, two- and three-dimensional finite elements elaborated inchapters 2 to 4, calculation of the mass matrix me, and of the kinematically equivalent loadsor the consistent element load vectors rep and r

    en, can be inferred from the linear formulation.

    This claim is valid for structural elements only in special cases due to rotational degrees offreedom (see e.g. Argyris [85] or Betsch, Menzel &Stein [91]), used to describe kinematicsand deformation (displacement-based description of rotation parameters, isoparametric shear-susceptible elements - Timoshenko, Kirchhoff-Love and Nagdi), and needs to be discussednext. Within the scope of a generalised approach to geometrically non-linear formulation offinite element methods we shall limit ourselves to finite elements which allow the assumptionsmade. The necessary changes are occasionally considered, discussed, and in some cases replacedby approximations.

    As opposed to the terms of virtual work of inertial forces and of external loads, the internalvirtual work Wint for geometrically non-linear observations is crucially different. Instead ofvariation of Green strain tensor that is strain vector , comes the Green Lagrange straintensor E that is strain vector E, given in equation (5.7) or equation (5.14). The Cauchy stresstensor/vector has to be replaced with the second Piola Kirchhoff stress tensor/vector Swhich is linked with the Green Lagrange strain tensor/vector through equation (5.26) in thematerially linear case.

    Wint =

    E : S dV =

    E : C : E dV =

    E C E dV (5.31)

    5.3.4.2 Variation of Green Lagrange strains

    Equation (5.31) contains the variation of Green Lagrange strain tensor E. Before we de-rive this variation, we first review the geometrically non-linear theory. In case of geometricnon-linearity, the Green strain vector could be computed with the help of the deformation-independent differential operator D and the displacement vector u , = Du (see equation(1.16)). Accordingly, the variation of Green strain vector was obtained by variation of the linearmapping just mentioned = Du. The corresponding relation should be derived as prepara-tion for the variation of Green Lagrange strain vector, needed to discretize and formulatefinite elements in chapter 6.

    E =

    [E11 E22 E33 2E12 2E23 2E13

    ]T(5.32)

    Variation of strain components Eij with i, j = 1, 2, 3 can be calculated as follows with equation(5.12)

    Eij =1

    2

    (uiXj

    +ujXi

    +ukXi

    ukXj

    )(5.33)

    and with the exchangeability of sequence of variation and partial derivatives when applying theproduct rule.

    Eij =1

    2

    (uiXj

    +ujXi

    +ukXj

    ukXi

    +ukXi

    ukXj

    )(5.34)

  • Institute for Structural Mechanics, Ruhr University Bochum, May 2005, DRAFTversion 161

    Here it is summated over k = 1, 2, 3. Setting i, j = 1, 2, 3 yields the components of the strainvector variation.

    E11 =u1X1

    +ukX1

    ukX1

    E22 =u2X2

    +ukX2

    ukX2

    E33 =u3X3

    +ukX3

    ukX3

    2E12 =u1X2

    +u2X1

    +ukX2

    ukX1

    +ukX1

    ukX2

    2E23 =u2X3

    +u3X2

    +ukX3

    ukX2

    +ukX2

    ukX3

    2E13 =u1X3

    +u3X1

    +ukX3

    ukX1

    +ukX1

    ukX3

    (5.35)

    Having precised the components of the variation of strain vector Eij in matrix form takinginto account the summation convention of k and l, yields the following presentation which isexaminable by multiplication:

    E11

    E22

    E33

    2E12

    2E23

    2E13

    =

    X1

    +u1,1

    X1u2,1

    X1u3,1

    X1

    u1,2

    X2

    X2+u2,2

    X2u3,2

    X2

    u1,3

    X3u2,3

    X3

    X3+u3,3

    X3

    X2+u1,2

    X1+ u1,1

    X2

    X1+u2,2

    X1+ u2,1

    X2u3,2

    X1+ u3,1

    X2

    u1,3

    X2+ u1,2

    X3

    X3+u2,3

    X2+ u2,2

    X3

    X2+u3,3

    X2+ u3,2

    X3

    X3+u1,3

    X1

    + u1,1

    X3u2,3

    X1+ u2,1

    X3

    X1+u3,3

    X1+ u3,1

    X3

    u1

    u2

    u3

    (5.36)

    If we have a closer look at the equation, it becomes evident that the differential operator whichmaps the displacement vector variation to the Green Lagrange strain vector is composed byaddition of the constant part already defined in equation (1.16)

    D =

    X10 0

    0

    X20

    0 0

    X3

    X2

    X10

    0

    X3

    X2

    X30

    X1

    (5.37)

  • 162 Kuhl & Meschke, Adavanced Finite Element Methods, DRAFTversion

    and of the deformation-dependent part.

    Dnl (u) =

    u1,1

    X1u2,1

    X1u3,1

    X1

    u1,2

    X2u2,2

    X2u3,2

    X2

    u1,3

    X3u2,3

    X3u3,3

    X3

    u1,2

    X1+ u1,1

    X2u2,2

    X1+ u2,1

    X2u3,2

    X1+ u3,1

    X2

    u1,3

    X2+ u1,2

    X3u2,3

    X2+ u2,2

    X3u3,3

    X2+ u3,2

    X3

    u1,3

    X1+ u1,1

    X3u2,3

    X1+ u2,1

    X3u3,3

    X1+ u3,1

    X3

    (5.38)

    The connection between variation of continuous displacements and variation of the GreenLagrange strain tensor can be demonstrated with assistance of the differential operator Dand Dnl (u), where the product D

    nl u stands for the variation of the non-linear strain vector

    term Enl(u).

    E(u) = D u + Enl(u) =

    (D + D

    nl (u)

    )u (5.39)

    5.3.5 Internal virtual work

    With developments of strains and their variation in linear and non-linear parts, we can nowtransform the internal virtual work according to equation (5.31). Parts of Green Lagrangestrain vector E that arise and their variation E are described in equations (5.13) and (5.39).We can also infer the material matrix of a three-dimensional continuum from equation (1.62)or equation (1.63).

    Wint =

    E C E dV =

    [(D + D

    nl (u)

    )u

    ]C

    [D u + E

    nl(u)]dV (5.40)

    5.3.6 Elastic internal potential

    Because of its importance for derivation of solution algorithms in geometrically non-linear struc-tural mechanics and for development of direct methods which determine singular points (sta-bility), we should write down the internal potential int shall be written as function of elasticpotential function W (E) or of Green Lagrange strain tensor E in connection with the ma-

  • Institute for Structural Mechanics, Ruhr University Bochum, May 2005, DRAFTversion 163

    terial tensor C.

    int =

    W (E) dV =1

    2

    E : S dV =1

    2

    E : C : E dV (5.41)

    The variation of internal potential is identical with the internal virtual work, with assumptionof a material law having a potential character (see equation (5.31)).

    int =1

    2

    (E : C : E + E : C : E) dV =

    E : C : E dV =

    E : S dV = Wint (5.42)

    5.3.7 Remarks regarding combined material and geometric non-linearity

    If one should combine the non-linearities dealt with in this and the previous chapter 5.2, namelymaterial and geometric non-linearity, one should note that non-linear material models are for-mulated in true, that is in Cauchy stresses but also in strains (Euler strain tensor) relatedto current configuration. For this reason, the actual Green Lagrange strain tensor generallyhas to be related to the momentary configuration with a push forward. There, the Cauchystress tensor is computed with the help of the used material model and thereafter related to thereference configuration with a pull back. The second Piola Kirchhoff stress tensor, which isthe result of this procedure, is thereby a function of the Green Lagrange strain tensor, of thematerial deformation gradient (push forward and pull back) and of internal variables dependenton the material model.

    S = S(E,F ,) (5.43)

    Compared to a pure material non-linearity, the reference configuration related stress and strainquantities have to be utilized here instead of and as well as the material deformationgradient.

    5.4 Consistent linearization of internal virtual work

    The basis for the solution of geometrically non-linear finite element systems is the linearizationof internal virtual work. Before we perform the linearization in the next chapter 6, we mustdefine the directional or the Gateaux derivative of a scalar or a vector.

    5.4.1 Linearization background

    5.4.2 Gateaux derivative

    Gateaux derivative of a scalar, vector, matrix or a tensor will henceforth be designated witha symbol. The definition is given with the help of an arbitrary scalar function f(u).

    f(u) =d

    d[f(u + u)]|=0 =

    f(u)

    uu (5.44)

    In this definition, u stands for the actual displacement vector and u stands for an incremental

  • 164 Kuhl & Meschke, Adavanced Finite Element Methods, DRAFTversion

    change of u. dd

    [f(u + u)] is the increase or the derivative of function f along a direction

    or a straight line set by u. If this derivative is evaluated at = 0, we get the function fderivative of the displacement state u in direction u. On the other hand, the scalar product ofthe normal vector defined by f(u)/u gives, along with the vector of incremental displacementu , the change f of function f for the change u in a tangential plane at f(u). The secondidentity can be shown if f(u + u) is replaced by f(u()) with u() = u + u, and if weapply the chain rule.

    d

    df(u + u) =

    d

    df(u()) =

    f(u())

    uu()

    =f(u + u)

    uu (5.45)

    Equation (5.45) evaluated at = 0 finally gives the sought second identity of equation (5.44).

    5.4.3 Gateaux derivative of internal virtual work

    Application of Gateaux derivative definition (5.44) to the internal virtual work Wint, accordingto equation (5.40) with variation of the strain tensor according to equation (5.39), determinesthe linearization of internal virtual work.

    Wint =Wintu

    u =d

    d

    [

    [D (u + u) + E

    nl(u + u)]C

    [D (u + u) + E

    nl(u + u)]dV

    ]=0

    (5.46)

    Gateaux derivative is assembled by differentiation with respect to scalar and by applyingthe chain rule and evaluating at = 0.

    Wint =

    [D u +d

    d(Enl(u + u))] C

    [D (u + u) + Enl(u + u)] dV

    =0

    +

    [D (u + u) + Enl(u + u)] C

    [D u +d

    d(Enl(u + u))] dV

    =0

    =

    [D u +d

    d(Enl(u + u))|=0] C [D u + E

    nl(u)] dV

    +

    [D u + Enl(u)] C [D u +

    d

    dEnl(u + u)|=0] dV

    (5.47)

    If we observe the Gateaux derivative definition in equation (5.44) applied to the derivativesyet to be performed in equation (5.47), and if we take into consideration that, according to itsproperties formulated in chapter 1.6.1, the virtual displacement u is arbitrary that is indepen-dent of the displacement state u, making u = u = 0, we get the Gateaux derivative of

  • Institute for Structural Mechanics, Ruhr University Bochum, May 2005, DRAFTversion 165

    internal virtual work,

    Wint =

    Enl(u) C [D u + Enl(u)] dV

    +

    [D u + Enl(u)] C [D u + E

    nl(u)] dV

    (5.48)

    which can further be transformed after introduction of equation (5.39) applied to the termEnl(u).

    Wint =

    Enl(u) C [D u + Enl(u)] dV

    +

    [(D + Dnl (u)) u] C [D u + E

    nl(u)] dV

    (5.49)

    This equation represents the Gateaux derivative or the directional derivative of internal virtualwork related to the incremental change of displacement state by u. The Gateaux derivativeis also called the consistent linearization of internal virtual work due to its strict mathematicalderivation. This consistent linearized virtual work presents a milestone of non-linear structuralmechanics since it is the basis of all iterative, incremental solution strategies (see chapter 7).

    We should notice that Du + Enl(u) presents the linearization of the Green Lagrange

    strain vector E(u) and that the Gateaux derivative of the second Piola Kirchhoff stressvector for materially non-linear models is given by

    S(u) = C [D u + Enl(u)] = C E(u) (5.50)

    We should further notice the identity of linearization of the Green Lagrange strain vectorvariation and its non-linear part

    E(u) = D u + Enl(u) = Enl(u) (5.51)

    due to the vanishing linearization of the linear part. Consequently we can write the linearizedinternal virtual work (5.49) in a compact form with equations (5.14), (5.50) and (5.51).

    Wint =

    [E(u) S(u) + E(u) S(u)] dV (5.52)

    In order to discretisize the directional derivative or linearize the internal virtual work, accordingto equation (5.49) or (5.52), the linearization of the non-linear part of the Green Lagrangestrain vector Enl(u) and its variation Enl(u) needs to be determined. The non-linear partof the Green Lagrange strain vector Enl(u) and differential operators D and D

    nl (u) are

    already given in equations (5.14), (5.38) and (5.39).

  • 166 Kuhl & Meschke, Adavanced Finite Element Methods, DRAFTversion

    5.4.4 Linearization of Green Lagrange strains

    When using a linear material law, the linearization of the Green Lagrange strain vector,according to equation (5.50), multiplied by the material matrix is equivalent to the linearizationof the second Piola Kirchhoff stress vector. In order to determine the linearization of theGreen Lagrange strain vector, it suffices to investigate the non-linear part since the linearpart is already known to be D.u.

    E(u) = D u + Enl(u) E

    nl(u) =d

    d(Enl(u + u))

    =0

    (5.53)

    In order to generate Enl, the components of the non-linear part of the strain vector should beobserved in index notation and linearized according to equation (5.12) that is equation (5.33).

    Enlij =1

    2

    Xiuk

    Xjuk (5.54)

    The application of formalism of the Gateaux derivative yields:

    Enlij =1

    2

    d

    d

    [

    Xi(uk + uk)

    Xj(uk + uk)

    ]=0

    =1

    2

    [

    Xiuk

    Xj(uk + uk) +

    Xi(uk + uk)

    Xjuk

    ]=0

    =1

    2

    [

    Xiuk

    Xjuk +

    Xiuk

    Xjuk

    ]=

    1

    2

    [uk,j

    Xiuk + uk,i

    Xjuk

    ](5.55)

    By comparing the last row of the above equation with the variation of the corresponding straincomponent Enlij in equation (5.34), we showed the equivalence of variation and linearizationof strains. As a consequence, we conclude that linearization of the Green Lagrange strainvector is given directly with equation (5.39), where the variation symbol needs to be replacedby the linearization symbol .

    E(u) = D u + Enl(u) =

    (D + D

    nl (u)

    )u (5.56)

  • Institute for Structural Mechanics, Ruhr University Bochum, May 2005, DRAFTversion 167

    5.4.5 Linearization of variation of Green Lagrange strains

    The linearization of variation of Green Lagrange strains follows from considering equation (5.51)based on equation (5.34) with the help of the Gateaux derivative in components.

    Eij = Enlij

    =1

    2

    d

    d

    [

    Xi(uk + uk)

    Xj(uk + uk)

    +

    Xi(uk + uk)

    Xj(uk + uk)

    ]=0

    =1

    2

    [

    Xiuk

    Xj(uk + uk) +

    Xi(uk + uk)

    Xjuk

    +

    Xiuk

    Xj(uk + uk) +

    Xi(uk + uk)

    Xjuk

    ]=0

    =1

    2

    [

    Xiuk

    Xjuk +

    Xiuk

    Xjuk

    +

    Xiuk

    Xjuk +

    Xiuk

    Xjuk

    ]

    (5.57)

    As already elaborated in 5.4.3, the linearization of the virtual displacement uk = 0 vanishesand therefore also all terms in equation (5.57) which contain this term.

    Eij = Enlij =

    1

    2

    [

    Xiuk

    Xjuk +

    Xjuk

    Xiuk

    ](5.58)

    Executing equation (5.58) for all permutations of i, j = 1, 2, 3 and summation over k = 1, 2, 3,we get components of linearized variation of the Green Lagrange strain vector.

    E11 =

    X1uk

    X1uk

    E22 =

    X2uk

    X2uk

    E33 =

    X3uk

    X3uk

    E12 =1

    2

    [

    X1uk

    X2uk +

    X2uk

    X1uk

    ]E23 =

    1

    2

    [

    X2uk

    X3uk +

    X3uk

    X2uk

    ]E13 =

    1

    2

    [

    X1uk

    X3uk +

    X3uk

    X1uk

    ]

    (5.59)

    In order to obtain greater clarity on linearization of variation of the strain vector we should, asalternative to the above derivations, vary Green Lagrange strains in tensor notation accordingto equation (5.7), with exchangeability of variation and linearization sequence,

    E =1

    2

    [Tu +u +Tu u +Tu u

    ](5.60)

  • 168 Kuhl & Meschke, Adavanced Finite Element Methods, DRAFTversion

    and thereupon linearize them with the help of Gateaux derivative definition according to (5.44),where the identity u = 0 must be heeded.

    E =1

    2

    d

    d

    [T(u + u) +(u + u)

    +T(u + u) (u + u) +T(u + u) (u + u)

    ]=0

    =1

    2

    [Tu u +Tu u

    ] (5.61)

    With that E is determined in a written form. It is evident that this expression does not dependon the given state of displacement u. For the following discretization of internal virtual workin a pure and linearized form in chapter 6, some additional considerations and transformationsshould by all means be done beforehand. The linearized variation of Green Lagrange strainsappears in the linearized internal virtual work (equation (5.52)) as a scalar valued product withthe second Piola Kirchhoff stresses. In tensor notation, this can be written as follows andtransformed and summarized due to the symmetry of the stress tensor.

    E : S =1

    2

    [Tu u

    ]: S +

    1

    2

    [Tu u

    ]: S

    =[Tu u

    ]: S =

    [Tu u

    ]: S = E S

    (5.62)

    The proof of equation (5.62) comes in components, where symmetry of the stress tensor Sij = Sjiand exchangeability of dummy (summation) indices enable the particular steps of the proof.

    Eij =1

    2[uk,juk,i + uk,iuk,j]

    EijSij =1

    2[uk,juk,iSij + uk,iuk,jSij] =

    1

    2[uk,juk,iSij + uk,iuk,jSji]

    =1

    2[uk,juk,iSij + uk,juk,iSij] = uk,juk,iSij

    (5.63)

    After this, it remains to specify the expression (analogy with calculation of Tu u, given inequation (5.11))

    [Tu u] : S =

    uk,1 uk,1 uk,1 uk,2 uk,1 uk,3

    uk,2 uk,1 uk,2 uk,2 uk,2 uk,3

    uk,3 uk,1 uk,3 uk,2 uk,3 uk,3

    :

    S11 S12 S13

    S12 S12 S23

    S13 S23 S33

    (5.64)

    and to transfer it into a suitable form in matrix notation.

    [Tu u] : S = u S u (5.65)

    Here, u defines a vector component of the displacement vector gradient u.

    u =

    [u1,1 u1,2 u1,3 u2,1 u2,2 u2,3 u3,1 u3,2 u3,3

    ]Tu =

    [u1,1 u1,2 u1,3 u2,1 u2,2 u2,3 u3,1 u3,2 u3,3

    ]T (5.66)

  • Institute for Structural Mechanics, Ruhr University Bochum, May 2005, DRAFTversion 169

    or

    u =

    u1Xu2Xu3X

    =

    u1

    u2

    u3

    u =

    u1Xu2Xu3X

    =

    u1

    u2

    u3

    (5.67)

    and S defines the hyper-diagonal matrix of the second Piola Kirchhoff stress components.

    S =

    S11 S12 S13S12 S22 S23S13 S23 S33

    S11 S12 S13S12 S22 S23S13 S23 S33

    S11 S12 S13S12 S22 S23S13 S23 S33

    , S =

    S S

    S

    (5.68)

    The validity of identity (5.65) can be examined by calculation of the corresponding scalaraccording to both left and right side of this equation, and also by applying definitions (5.66)and (5.68). Intotal the identity relevant to discretization of linearized internal work is obtainedwith equations (5.62) and (5.65).

    E S = u S u (5.69)

  • 170 Kuhl & Meschke, Adavanced Finite Element Methods, DRAFTversion

  • Chapter 6

    Finite element discretization ofgeometrically non-linear continua

    The finite element discretization of the weak form of impulse balance, or of the principle ofvirtual work (5.30), gives the static or dynamic equilibrium in the form of a non-linear vectorequation or vector differential equation. For the finite elements dealt with in this chapter, thelatter differs from the discrete formulation of the principle of virtual work for small deformations(geometrically linear observation) only in the term of discretisized internal virtual work. Theresult of discretization of this different term is the deformation-dependent vector of internalforces.

    Besides the discretization of principle of virtual work, when considering definite deformations(geometrically non-linear), the discretization of linearized internal virtual work (5.49) or (5.52)is of crucial significance for numerical solution of geometrically non-linear elasto-mechanics.Discretization of the linearized internal virtual work defines the so called tangential stiffnessmatrix. This tangential stiffness, identical to linearization of vector of internal forces, forms thebasis of iterative Newton procedures on the one hand, and is of importance for characterizationof stability properties of the structure on the other hand.

    Based on the fundamental understanding of finite element development in linear structuralmechanics, it is effective to first derive the discretization of the three-dimensional non-linearcontinuum with isoparametric finite volume elements, and thereafter develop the correspondingone- and two-dimensional finite elements analogously.

    6.1 Finite volume elements

    The discretization of the geometrically non-linear three-dimensional continuum can be doneanalogously to the linear case (chapter 4). In comparison to geometrically linear observation,the following developing steps remain unaltered:

    Approximation of continuous values with the ansatz function matrix N() and of elementvectors Xe, ue, ue and ue of a NN -noded hexagon element according to equations (4.10)and (4.11), where the generation of ansatz functions is performed according to table ??.

    The generation of Jacobi matrix J() and Jacobi determinant |J()| according to equa-tions (4.12) and (4.14) remains the same. Also, the inversion for calculation of Jacobimatrix inverse J1(), which is defined in equation (4.13).

    171

  • 172 Kuhl & Meschke, Adavanced Finite Element Methods, DRAFTversion

    The spatial Gau-Legendre integration of element matrices and vectors.

    The mass matrix me is identical to linear observation in equation (4.19).

    The calculation of consistent equivalent loads of volume loads rep (4.20) remains unmodifiedas well, and one obtains the consistent equivalent load of boundary loads ren by replacingboundary stresses t? by boundary stresses T ?, see equations (5.23) and (5.24).

    In other words, the internal virtual work and its linearization remain to be discretisized, wherein order to discretisize Wint only the virtual displacement by ansatz functions and variation ofelement free values are approximated. Contrarily, to discretisize Wint, the approximation tou and u is used. All further displacement-dependent terms remain undiscretisized.

    6.1.1 Discretization of internal virtual work

    After the preceding derivations, it remains to discuss the discretization if internal virtual workof a finite element e according to equation (5.31). In order to formulate the isoparametric finitevolume element, the equation (5.40) should be applied to the element volume, and the differentialvolume d should be transformed into natural coordinates d = |J()| d1d2d3 according toequation (4.14), with the help of the Jacobi determinant |J()|. In the differential operatorsD, D

    nl (u) as well as in the non-linear part of the Green Lagrange strain vector E

    nl(u),the derivatives of physical coordinates must be substituted by derivatives of natural coordinateswith the help of the Jacobi transformation. Formally this is designated by index Jacobi. Thedifferential operator D was already given in equation (4.15). The explicit generation of furtherterms follows in the next two sections. For the further development of internal virtual work,equation (5.40) is resolved.

    W eint =

    11

    11

    11

    E(u, ) C E(u, ) |J()| d1 d2 d3

    =

    11

    11

    11

    (D() u()) C D() u() |J()| d1 d2 d3

    +

    11

    11

    11

    (D() u()) C Enl (u, ) |J()| d1 d2 d3

    +

    11

    11

    11

    (Dnl(u, ) u()

    )C D() u() |J()| d1 d2 d3

    +

    11

    11

    11

    (Dnl(u, ) u()

    )C Enl (u, ) |J()| d1 d2 d3

    (6.1)

    The first summand corresponds to the internal virtual work of geometrically linear elastome-

    chanics W e linint = ue keeu

    e according to (1.96), which can again be discretisized with thestiffness matrix (equation (4.18)), which is marked as elastic stiffness matrix kee for distinctionpurposes, and with the element displacement vector ue as well as its variation ue (equation(4.10)). Independent of this realization, the simplification of the scientific notation for purposesof a uniform presentation of all terms, and most of all, in order to effectively compute thetangential stiffness matrix (should at first not be considered). Since the non-linear part of the

  • Institute for Structural Mechanics, Ruhr University Bochum, May 2005, DRAFTversion 173

    strain vector Enl(u) cannot be obtained by a linear mapping of the displacement vector (seeequation (5.13)), a strategy alternative to linear theory must be applied for discretization ofinternal virtual work. For that reason, the strain variation E (u, ) is discretisized with thehelp of the approximation of the displacement variation based on the product of matrix ofansatz functions N() (equation (4.11)) and variation of the element displacement vector ue

    according to equation (4.10).

    u() u() = N() ue ue =[ue11 u

    e12 u

    e13 u

    eNN3

    ]T(6.2)

    D() u D() N() ue = B() ue

    Enl (u, ) Dnl(u, ) N() u

    e = Bnl(u, )ue

    E(u, ) (D() + D

    nl(u, )

    )N() ue = B(u, ) ue

    (6.3)

    The remaining linear and non-linear terms of the strain vector D()u and Enl (u, ) are not

    subject to discretization. The reason for that is evident in chapter 7 in the scope of iterativesolution procedures of non-linear structural mechanics. Here, the result of investigations thatare yet to come should only be sketched: In order to realize the sought algorithms, the strainsare calculated just for one estimated or approximated displacement state uek or continuousdisplacement state uk(), where the estimated continuous displacements are calculable fromthe displacement vector with the help of approximation (4.10). 1

    uk() uk() = N() uek uek =[uek11 u

    ek12 u

    ek13 u

    ekNN3

    ]T(6.4)

    Accordingly, also the Green Lagrange strains Enl (uk, ) can be determined with equation

    (5.13). These statements are analogously valid for deformation-dependent terms Dnl(uk, ) that

    is Bnl(uk, ) which are contained in equations (6.3). They are determined with the displacementstate uk and equations (5.38) or (6.3). After this elaboration, from now on the so called iterationindex k will be omitted; all undiscretisized displacements should be understood as iterativelyapproximated displacements. If we apply the approximation (6.3) to the element-specific

    1Already in advance, in the scope of non-linear algorithms, uek presents the element displacement vector of

    the last iteration step

  • 174 Kuhl & Meschke, Adavanced Finite Element Methods, DRAFTversion

    formulation of internal virtual work (6.1), we get its discrete approximation.

    W eint ue

    11

    11

    11

    BT () C D() u() |J()| d1 d2 d3

    + ue

    11

    11

    11

    BT () C Enl (u, ) |J()| d1 d2 d3

    + ue

    11

    11

    11

    Bnl T (u, ) C D() u() |J()| d1 d2 d3

    + ue

    11

    11

    11

    Bnl T (u, ) C Enl (u, ) |J()| d1 d2 d3

    (6.5)

    The internal virtual work can also be calculated with definition of the deformation-dependentelement vector of internal forces rei ,

    rei (ue) =

    11

    11

    11

    BT () C D() u() |J()| d1 d2 d3

    +

    11

    11

    11

    BT () C Enl (u, ) |J()| d1 d2 d3

    +

    11

    11

    11

    Bnl T (u, ) C D() u() |J()| d1 d2 d3

    +

    11

    11

    11

    Bnl T (u, ) C Enl (u, ) |J()| d1 d2 d3

    (6.6)

    and with variation of the displacement vector ue.

    W eint ue rei (u

    e) (6.7)

    Alternatively, the vector of internal forces can be presented in a more compact form if thepartitioning of the strain vector and the differential operator is taken back and the equation(5.14) is utilized.

    rei (ue) =

    11

    11

    11

    [BT () + Bnl T (u, )

    ]C E(u, )|J()|d1d2d3 (6.8)

    Generally, the internal forces are calculable by integration of the product of the non-lineardifferential operator B(u, ) = B() + Bnl(u, ) and the stress vector S(u, ) = CE(u, ),where of course the Jacobi transformation needs to be included.

    rei (ue) =

    11

    11

    11

    BT (u, ) S(u, ) |J()| d1 d2 d3 (6.9)

  • Institute for Structural Mechanics, Ruhr University Bochum, May 2005, DRAFTversion 175

    The deformational dependence of internal forces was characterised by the functional dependenceof the displacement vector ue. This implies that for calculation of internal forces in general, andfor calculation of Bnl, Du as well as E

    nl , especially the element displacement vector has to

    be either known or assessable.

    6.1.1.1 Jacoby transformation of E

    When previously discretisizing the internal virtual work, the variation of the strain vectorE(u, ) formulated in natural coordinates was already used, but not determined. By means ofthe Jacobi transformation of equation (5.39), E (u, ) can be expressed with the assistanceof D() and D

    nl(u, ).

    E(u) = D u + Enl (u) =

    (D + D

    nl(u)

    )u (6.10)

    The differential operator D and consequently also the B-operator B() are already knownfrom the development of a finite volume element of linear structural mechanics (equations (4.15)to (4.17)). It now only remains to proceed with the development of the operator Dnl(u, ) and

    of the result of its application to the matrix of ansatz functions Dnl(u, )N()=Bnl(u, ). The

    differential operator Dnl (u, ), defined in equation (5.38), contains derivatives of displacementcomponents with respect to physical coordinates as well as differentiation rules with respect tophysical coordinates of a generalised form:

    uj,k

    Xl=

    ujXk

    Xl(6.11)

    With the help of components of the inverse Jacobi matrix according to equation (4.13), thetransformation can be performed in natural coordinates

    Xl=mXl

    muj,k =

    ujXk

    =nXk

    ujn

    (6.12)

    where the summation is over m = 1, 2, 3, that is over n = 1, 2, 3. Derivatives of the displacementvector with respect to natural coordinates are set by the derivative of ansatz functions.

    u()

    n= N;n() u

    e uj()

    n=

    NNi=1

    N i;n() ueij (6.13)

    For generation of the differential operator Dnl, we write only the derivative /Xl. The deriva-tives of displacement components with respect to natural coordinates are implicitly included

  • 176 Kuhl & Meschke, Adavanced Finite Element Methods, DRAFTversion

    according to equations (6.12)2 and (6.13).

    Dnl(u, ) =

    u1,1m

    X1

    mu2,1

    m

    X1

    mu3,1

    m

    X1

    m

    u1,2m

    X2

    mu2,2

    m

    X2

    mu3,2

    m

    X2

    m

    u1,3m

    X3

    mu2,3

    m

    X3

    mu3,3

    m

    X3

    m

    u1,2m

    X1

    m+ u1,1

    m

    X2

    mu2,2

    m

    X1

    m+ u2,1

    m

    X2

    mu3,2

    m

    X1

    m+ u3,1

    m

    X2

    m

    u1,3m

    X2

    m+ u1,2

    m

    X3

    mu2,3

    m

    X2

    m+ u2,2

    m

    X3

    mu3,3

    m

    X2

    m+ u3,2

    m

    X3

    m

    u1,3m

    X1

    m+ u1,1

    m

    X3

    mu2,3

    m

    X1

    m+ u2,1

    m

    X3

    mu3,3

    m

    X1

    m+ u3,1

    m

    X3

    m

    (6.14)

    Application of the differential operator Dnl to the matrix of ansatz functions Ni() belonging

    to node i, yields the non-linear part of B-operator Bnli (u, ) = Dnl(u, )N

    i(),

    Bnlij (u, ) =

    uj,1mX1

    N i;m

    uj,2mX2

    N i;m

    uj,3mX3

    N i;m

    uj,2mX1

    N i;m + uj,1mX2

    N i;m

    uj,3mX2

    N i;m + uj,2mX3

    N i;m

    uj,3mX1

    N i;m + uj,1mX3

    N i;m

    Bnli (u, ) =

    [Bnli1 B

    nli2 B

    nli3

    ](6.15)

    where the non-linear operator of the finite volume element is assembled analogously to equation(4.17).

    Bnl(u, ) =

    [Bnl1 (u, ) B

    nl2 (u, ) B

    nlNN(u, )

    ](6.16)

    6.1.1.2 Jacoby transformation of E

    The derivatives of physical coordinates of the non-linear part of the strains Enl can be replacedby derivatives of natural coordinates, too. If the observed strains are composed from the linearand the non-linear part of equation (5.14), one gets the strains in natural coordinates after theJacobi transformation,

    E(u) = D u + Enl (u) (6.17)

    in case of which the linear part Du is already determined (equation (4.15)). The vectorEnl(u) contains, according to equation (5.12) that is (5.13), terms of the form uj,k that can be

  • Institute for Structural Mechanics, Ruhr University Bochum, May 2005, DRAFTversion 177

    transformed with the help of components of the Jacobi matrix according to equation (6.12).

    Enlij =1

    2

    Xiuk

    Xjuk E

    nlij =

    1

    2

    mXi

    muk

    nXj

    nuk (6.18)

    As an example of generation of Enl , the normal component Enl11 is derived.

    Enl11(u, ) =1

    2(u1,1 u1,1 + u2,1 u2,1 + u3,1 u3,1)

    Enl11(u, ) =1

    2

    (mX1

    u1m

    nX1

    u1n

    +mX1

    u2m

    nX1

    u2n

    +mX1

    u3m

    nX1

    u3n

    ) (6.19)Further components of the strain tensor can be derived in the same manner.

    6.1.2 Non-linear semi-discrete initial value problem

    Due to the spatial discretization presented in preceding sections, the principle of virtual workcan be approximated in the element plane.

    ue me ue + ue rei (ue) = ue (rep + r

    en) (6.20)

    By summing equation (6.20) or explicitly, the vector of internal loads,

    ri(u) =NE

    e = 1

    rei (ue) (6.21)

    with the vector of external loads rep + ren and the mass matrix m

    e in analogy with equation(2.146), we obtain the system-related spatially discrete formulation of the principle of virtualwork,

    u M u + u ri(u) = u r (6.22)

    which can be transferred to the semi-discrete initial value problem of non-linear elasto-dynamicsby application of the fundamental lemma of variation calculus (see chapter 2.2 and 2.3). Theproblem is defined by the semi-discrete differential equation of motion of the second order

    M u + ri(u) = r (6.23)

    and by the discretisized initial conditions.

    u(0) = u?

    ri(u(0)) = [r(0)M u(0)]

    u(0) = u?

    u(0) = M1 [r(0) ri(u(0))](6.24)

    The substantial difference to linear elasto-dynamics is that due to geometric non-linearity, thereis a non-linear equation of motion with the deformation-dependent vector of internal loadsri(u), instead of the product of constant stiffness matrix, and with the displacement vectorKu. As a further characteristic, we should bring up the non-linearity of the initial conditionwhen prescribing accelerations u(0) = u?.

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    6.1.3 Non-linear discrete static equilibrium

    For static or quasi-static problems of structural mechanics we can formulate the discreteequation of non-linear static equilibrium by neglecting the inertial forces Mu = 0.

    ri(u) = r (6.25)

    This equation presents a non-linear algebraic vector equation. The solution of the non-linearcoupled vector equation (6.25) requires the discretization of linearized internal virtual work(chapter 5.4) and the solution of incremental equilibrium relations with special numericalprocedures, elaborated in chapter 7.

    6.1.4 Discretization of linearized internal virtual work

    For finite element discretization of the linearized internal virtual work, formed in equation (5.49)and (5.56) that is equation (5.52), the integrands first have to be applied to the element volumeand thereafter transformed to natural coordinates 1 2 and 3.

    W eint =

    11

    11

    11

    [E(u, ) S(u, ) + E(u, ) S(u, )] |J()| d1 d2 d3

    =

    11

    11

    11

    Enl (u, ) C E(u, ) |J()| d1 d2 d3

    +

    11

    11

    11

    [(D() + Dnl(u, )) u] C

    [D() + Dnl(u, )] u |J()| d1 d2 d3

    (6.26)

    After the derivations of the previous chapter 6 and 5.4.3, only the discretization of thisexpression requires the discretization of the non-linear part of strain vector linearizationE(u, ), that is of stress vector S(u, ) and of term E(u, ) S(u, ). Due to analogywith the variation and linearization, the linearization of the Green Lagrange strain vectorE(u, ) can be written directly in natural coordinates, in analogy with equation (6.10),

    E(u) = D u + Enl (u) =

    (D + D

    nl(u)

    )u (6.27)

    and be discretisized.

    D() u B() ue Enl (u, ) B

    nl(u, ) ue (6.28)

    Thereby, the second summand of W eint can be written according to the operators, theirvariation, linearization, linearized variation and discretization, in its spatially discretisized formalready derived in equation (6.26). This should however be postponed until the first summandof W eint is discretisized, too. Therefore, the transformation is used which was derived in the

    preceding section (equation (5.69)) in which the stress matrix S and the vectors u that is

  • Institute for Structural Mechanics, Ruhr University Bochum, May 2005, DRAFTversion 179

    u, must be given in natural coordinates (S, u and u). The components of the stress

    matrix S can be computed based on strains E, according to equation (6.17) that is (6.18)with the help of the constitutive law (5.26). The Jacobi transformation from u is analoguousto those with u, which is why this vector u is shown as an example. The derivatives ofphysical coordinates ui,j are replaced by derivatives of natural coordinates with the Jacobitransformation according to equation (6.12) (summation over k = 1, 2, 3).

    u =

    kX1

    ku1

    kX2

    ku1

    kX3

    ku1

    kX1

    ku2

    kX2

    ku2

    kX3

    ku2

    kX1

    ku3

    kX2

    ku3

    kX3

    ku3

    =

    kX1

    kkX2

    kkX3

    kkX1

    kkX2

    kkX3

    kkX1

    kkX2

    kkX3

    k

    u1

    u2

    u3

    (6.29)

    With the discretization of vector u, which was defined in equation (4.10) with the help of thematrix of ansatz functions N() (equation (4.11)), and variation of the element displacementvector ue (u = N()ue), we obtain the approximation of u. The same can be done foru analogous to u.

    u Bg() ue u B

    g() ue (6.30)

    The differential operator Bg(), which is dependent solely on natural coordinates, can becomposed with the differential operator

    Bgi () =

    N i()

    XN i()

    XN i()

    X

    N i()

    X=

    kX1

    N i;k()

    kX2

    N i;k()

    kX3

    N i;k()

    (6.31)

    associated with the element node i.

    Bg() =

    [B

    g1() B

    g2() B

    gNN ()

    ](6.32)

    Therefore, it is possible to approximate the scalar product of the linearized variation of GreenLagrange strains and the second Piola Kirchhoff stresses with the help of differential

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    operator Bg(), and with the hyper-matrix of stresses S(u, ) (equations (5.69) and (6.30)):

    E(u, ) S(u, ) = u S(u, ) u ue Bg T () S(u, ) B

    g() ue (6.33)

    The linear internal virtual work can finally be entirely discretisized. Furthermore, we takeadvantage of the independence of element displacement vector ue from natural coordinates in order to extract ue and ue from the integral expressions.

    W eint =

    11

    11

    11

    E(u, ) S(u, ) |J()| d1 d2 d3

    +

    11

    11

    11

    [(D() + Dnl(u, )) u] C

    [D() u + Enl (u, )] |J()| d1 d2 d3

    (6.34)

    W eint ue

    11

    11

    11

    Bg T () S(u, ) Bg() |J()| d1 d2 d3 u

    e

    + ue

    11

    11

    11

    (BT () + Bnl T (u, )) C

    (B() + Bnl(u, )) |J()| d1 d2 d3 ue

    (6.34)

    The compact form of linearized internal virtual work is obtained with definitions of thedeformation-dependent geometric element stiffness matrix

    keg(ue) =

    11

    11

    11

    Bg T () S(u, ) Bg() |J()| d1 d2 d3 (6.35)

    and the deformation-dependent material element stiffness matrix

    kem(ue) =

    11

    11

    11

    [BT () + Bnl T (u, )

    ]C

    [B() + Bnl(u, )

    ]|J()| d1 d2 d3 (6.36)

    with the sum of the two yielding the tangential element stiffness matrix:

    W eint ue

    (keg(u

    e) + kem(ue)

    )ue = ue ket (u

    e) ue (6.37)

    If we use the additive composition of the B-operator of the material stiffness matrix, we

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    can break up the elastic element stiffness matrix kee, identical to the linear theory, and thedeformation-dependent matrix of initial displacements keu(u

    e).

    kem(ue) = kee + k

    eu(u

    e)

    kee =

    11

    11

    11

    BT () C B() |J()| d1 d2 d3

    keu(ue) =

    11

    11

    11

    [BT () C Bnl(u, ) + Bnl T (u, ) C B()

    ]|J()| d1 d2 d3

    +

    11

    11

    11

    Bnl T (u, ) C Bnl(u, ) |J()| d1 d2 d3

    (6.38)

    It is obvious that all parts of the tangential stiffness matrix are symmetric. Symmetry of elementstiffness matrices keg(u

    e), kem(ue) and kee can be inferred directly from respective definitions. The

    symmetry of keu(ue) follows from the symmetry of kem(u

    e) and kee and equation (6.38).

    6.1.5 Linearization of internal forces vector

    In the scope of the iterative solution of non-linear dynamic differential vector equations (6.23)or the non-linear static vector equation (6.25), the linearization of internal forces will be ofgreat importance. In the element plane, the linearization of internal forces r i(u) is defined bylinearization of the internal virtual work in equation (6.7), what, by comparison to equation(6.37), can be put down to the tangential element stiffness and the increment of the elementdisplacement vector.

    W eint ue rei (u

    e) = ue rei (u

    e)

    ueue = ue

    (keg(u

    e) + kem(ue)

    )ue (6.39)

    By summing the linearized internal forces and parts of the tangential stiffness matrix

    ri(u) =NE

    e = 1

    rei (ue)

    Kt(u) =NE

    e = 1

    ket (ue)

    Kg(u) =NE

    e = 1

    keg(ue)

    Km(u) =NE

    e = 1

    kem(ue)

    (6.40)

    we eventually get the linearization of the system vector of internal forces r i(u) as function ofthe material system stiffness matrix Km(u), the geometric system stiffness matrix Kg(u), thetangential system stiffness matrix Kt(u) and the increment of the system displacement vectoru.

    ri(u) = (Kg(u) + Km(u)) u = Kt(u) u (6.41)

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    6.2 Finite truss elements

    6.2.1 Non-linear continuum-mechanical formulation

    6.2.1.1 Kinematics

    The Green Lagrange strain in the longitudinal direction of the truss E11 is determined basedon equation (5.12) that is (5.13) with u2,1 = u3,1 = 0.

    E11 =1

    2(u1,1 + u1,1 + u1,1 u1,1) = u1,1 +

    1

    2u1,1 u1,1 (6.42)

    The presentation of the Green Lagrange truss strain, analoguous with the presentation of thedifferential operator (5.14) can be acquired by the following transformation.

    E11 =

    X1u1 +

    1

    2u1,1 u1,1 = D u1 +E

    nl11(u1) (6.43)

    6.2.1.2 Kinetics

    The Lagrange formulation of the forces equilibrium of a truss element is obtained with equation(5.22) for S22 = S33 = S12 = S23 = S13 = 0, and the vanishing derivatives in e2 and e3 directions(divergence DIV)

    u1 =

    X1

    (x1X1

    S11

    )+ b1 (6.44)

    6.2.1.3 Constitutive law

    The one-dimensional special case of the constitutive equation (5.26) can be described only withthe assistance of the elasticity modulus.

    S11 = E E11 (6.45)

    6.2.1.4 Principle of virtual work

    The principle of virtual displacements of a one-dimensional continuum is obtained by reductionof the three-dimensional formulation (5.30), or by adjusting the linear formulation ((2.24). Incase of the latter procedure, the Green strain 11 and the Cauchy stress 11 have to be replacedby the Green Lagrange strain E11 and the second Piola Kirchhoff stress S11, respectively,and the non-specified length l by the reference length L.

    L2

    L2

    u1 u1 A dX1 +

    L2

    L2

    E11 S11 A dX1 = ue11 N

    e11 + u

    e21 N

    e21 +

    L2

    L2

    u1 p1 dX1 (6.46)

    The inertial term as well as terms of internal and external loads are the same as in the linearobservation. For the internal virtual work, the integrand is already known, with the exception of

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    the variation of the Green Lagrange strain. E11 is obtained by reduction of equation (5.34)for the one-dimensional continuum or by variation of equation (6.42).

    E11 =

    X1u1 +

    1

    2

    X1u1

    X1u1 +

    1

    2

    X1u1

    X1u1 =

    X1u1 + u1,1

    X1u1 (6.47)

    This can be presented by means of analogy to equation (5.39) with the help of the differentialoperator D and D

    nl (u1).

    E11 = D u1 + Enl11(u1) =

    (D +D

    nl (u1)

    )u1 (6.48)

    6.2.1.5 Linearization of internal virtual work

    The linearized internal virtual work can be extracted from the internal virtual work of the three-dimensional continuum (5.49), or much simpler, by linearization of internal virtual work givenin equation (6.46).

    Wint =

    L2

    L2

    A (E11 S11 + E11 S11) dX1 (6.49)

    In this expression, we need to generate the linearization of variation of the Green Lagrangestrain and of the second Piola Kirchhoff stress. The first linearization yields:

    E11 = D u1 + Enl11(u1) = E

    nl11(u1) = u1,1 u1,1 + u1,1 u1,1 = u1,1 u1,1 (6.50)

    The linearization of the second Piola Kirchhoff stress can be put down to the linearization ofthe Green Lagrange strain with the help of the constitutive equation (6.45), which is definedby the equivalence of variation and linearization in analogy to equation (6.48).

    S11 = E E11 = E(D +D

    nl (u1)

    )u1 (6.51)

    For the discretization of linearized internal virtual work of the three-dimensional continuum,it has turned out to be favourable to properly transform the expression E S equivalentto E11S11. For a deeper understanding of this transformation, we should do it for the one-dimensional continuum. Since E11 and S11 are scalar expressions, this transformation is ofno further significance.

    E11 S11 = u1,1 u1,1 S11 = u1,1 S11 u1,1 =

    X1u1 S11

    X1u1 = u1 S11 u1 (6.52)

    6.2.2 Truss elements of arbitrary polynomial degree

    In order to develop a family of truss elements for the modelling of geometrically non-linearstructural behaviour, we should assume equidistant element nodes. The development is separatedinto determination of the element vector of internal forces by discretization of the internal virtualwork, and generation of the tangential element stiffness matrix based