Meso-Scale Finite Element Simulation of Deformation ...geomechanics.civil.northwestern.edu/Presentations_files/seminar.en… · Constitutive Models Numerical Implementation Numerical

MotivationBalance Laws: Continuum Formulation

Constitutive ModelsNumerical Implementation

Numerical ExamplesClosure and Future Work

Meso-Scale Finite Element Simulation of

Deformation Banding in Fluid-Saturated Sands

Jose E. [email protected]

Department of Civil and Environmental EngineeringNorthwestern University

University of Michigan, November 2006

J.E. Andrade Deformation Banding in Saturated Sands




Outline

1 MotivationInstabilities in geomechanicsMeso-scale and collaborative research

2 Balance Laws: Continuum Formulation

3 Constitutive ModelsEffective stressDarcy’s Law

4 Numerical Implementation

5 Numerical ExamplesDrained responseUndrained response

6 Closure and Future Work





Outline











Outline











Outline











Outline











Outline











Instabilities in geomechanicsMeso-scale and collaborative research

Outline












Catastrophic instabilitiesLiquefaction






Catastrophic instabilitiesLiquefaction






Catastrophic instabilitiesStrain localization

Lab response

n

Alshibli et al: JGGE 2003






Catastrophic instabilitiesStrain localization

Lab response

n

In situ response

Alshibli et al: JGGE 2003






What is meso-scale?

Rechenmacher: 2005

Meso-scale

Smaller than specimen(macro) but larger thangrain (particle)

Still looking at continuum

picture






What is meso-scale?

Hafner et al: C&S 2006

Meso-scale

Smaller than specimen(macro) but larger thangrain (particle)

Still looking at continuum

picture






Collaborative research

Patient

Motivations

Quantify porosity atmeso-scale in the lab

X-Ray CTDIP

Develop meso-scale modelsfor sands

Analyze behavior as BVP







1.61

1.62

1.63

1.64

1.65

1.66

1.67

SPECIFIC VOLUME

Motivations


X-Ray CTDIP









1.61

1.62

1.63

1.64

1.65

1.66

1.67

SPECIFIC VOLUME

Motivations


X-Ray CTDIP








Triaxial compression of dense sandInitial specific volume

0

0.5

1

0

0.5

1

0

0.5

1

1.5

2

0

0.5

1

0

0.5

1

0

0.5

1

1.5

2

1.59

1.595

1.6

1.605

1.61

1.615

1.62

1.625

1.63

1.635

1.64

INHOMOGENEOUS 1 INHOMOGENEOUS 2 SPECIFIC VOLUME

Andrade and Borja: IJNME 2006






Triaxial compression of dense sandAcoustic tensor at localization

0

0.5

1

0

0.5

1

0

0.5

1

1.5

0

0.5

1

0

0.5

1

0

0.5

1

1.5

0.5

1

1.5

2

2.5

x 1011INHOMOGENEOUS 1 INHOMOGENEOUS 2 DETERMINANT







Triaxial compression of dense sandDeviatoric strains at localization

0

0.5

1

0

0.5

1

0

0.5

1

1.5

0

0.5

1

0

0.5

1

0

0.5

1

1.5

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

INHOMOGENEOUS 1 INHOMOGENEOUS 2 DEVIATORIC STRAIN







Triaxial compression of dense sandMacroscopic behavior

Reactive stresses

0 5 10 15 20100

150

200

250

300

350

400

450

500

NOMINAL AXIAL STRAIN, %

NO

MIN

AL

AX

IAL

ST

RE

SS

, kP

a

HOMOGENEOUSINHOMOGENEOUS 1INHOMOGENEOUS 2

LOCALIZATION







Triaxial compression of dense sandMacroscopic behavior

Reactive stresses

0 5 10 15 20100

150

200

250

300

350

400

450

500


NO

MIN

AL

AX

IAL

ST

RE

SS

, kP

a


LOCALIZATION

Volume change

0 5 10 15 20−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14


∆ V

OLU

ME

, m3


LOCALIZATION






Outline











Multi-phase system

6

Solid grain

AAU

Matrix

BB

BBBM

Voids

Matrix defined by solidgrains and voids

Mixture theory: saturationφs + φf = 1

Follow matrix deformationu = x − X

Apply continuum mechanicslaws on each phase





Multi-phase system

6

Solid grain

AAU

Matrix

BB

BBBM

Fluid









Multi-phase system

js

W0

W

x1

x2

X

x

jf

W0

f









Multi-phase system

js

W0

W

x1

x2

X

x

jf

W0

f









Balance for multi-phase system

Localized mass balancefor mixture

ρ0 = −J ∇x· q

where

ρ0 = Jρ

J = det F

F = ∂ϕs/∂X

q = ρf (vf − v)

Localized momentum formixture (quasi-static)

∇x·σ + ρg = 0

Coupled system

Need constitutive models forσ and q

Andrade and Borja: FEAD 2006





Effective stressDarcy’s Law

Outline












Phenomenological plasticityFeatures of sand behavior

log - ’p

v

Dvp

A

B

C

Schofield and Wroth: 1968

Material models for sandshould capture

1 Nonlinearity andirrecoverable deformations

2 Pressure dependence

3 Different tensile/compressivestrength

4 Relative density dependence

5 Nonassociative plastic flow







0 100 200 300

100

200

-p’ (kPa)

q(k

Pa)

s’vc

191 kPa

241

310380

Graham et al: CGJ 1983













t’1

Von Mises

Mohr Coulomb

t’2

t’3

Loose sand Dense sand

Lade and Duncan: JGGE 1975













0 2 4 6 8 10

200

0

400

600

800

1000

1

2

3

4

0

-1

sr

sa

|s -

sa

r(k

Pa)

|

ea (%)

e v(%

)

Dense SandLoose Sand

Cornforth: Geotech 1964













-p’

q

Yield FunctionPlastic PotentialFlow vector

Poorooshasb et al: CGJ 1967












Phenomenological plasticityAn elastoplastic model for sands

Model main features

1 F = F e · F p

2 Tension/compression


4 Dr dependence via ψi

5 Nonassociative flow

F F F=e p.

W0

WX

Wp

xp

Fp

Fe

x

Lee: JAM 1969Borja and Andrade: CMAME 2006








Model main features

1 F = F e · F p





Original yield surface

t =’1 t t’ = ’2 3

Jefferies: Geotech 1993

Borja and Andrade: CMAME 2006








Model main features

1 F = F e · F p





Enhanced yield surface

F (τ , πi) = ζ (θ) q + pη (p, πi)

where

p =1

3tr τ q =

√

3

2‖ξ‖

1√6

cos 3θ =tr ξ3

χ3ξ = τ − p1









Model main features

1 F = F e · F p





Enhanced yield surface

F (τ , πi) = ζ (θ) q + pη (p, πi)

E.g., Gudehus-Argyris

ζ (θ, ρ) =(1 + ρ) + (1 − ρ) cos 3θ

2ρ

η =

M [1 + ln (πi/p)] if N = 0

M/Nh

1 − (1 − N) (p/πi)N/(1−N )

i

if N > 0









Model main features

1 F = F e · F p




5 Nonassociative flowt’1 t’2

t’3

Loose sand Dense sand

Originalz(q)









Model main features

1 F = F e · F p





t =’1 t t’ = ’2 3









Model main features

1 F = F e · F p





v

v1

vc

v2

-pi -p’

ln-p’

l~

y

yi

CSL

Been and Jefferies: Geotech 1985









Model main features

1 F = F e · F p





Hardening law

πi = h (π∗i − πi) εps










Model main features

1 F = F e · F p





Hardening law

πi = h (π∗i − πi) εps


-0.2 -0.15 -0.1 -0.05 0 0.050.8

1

1.2

1.4

1.6

1.8

2

N=0

N

N=0.5

yi

p*

i/p









Model main features

1 F = F e · F p





Plastic potential

Q (τ , πi) = ζ (θ) q + pη (p, πi)

whereζ = ζ (θ, ρ) and η = η

(

p, πi,N)

So, when

ρ = ρ and N = N , associative flow









Model main features

1 F = F e · F p




5 Nonassociative flow-200 pi=-100 0

0

100

N=0

N=0.5q (kPa)

p’ (kPa)








Localization criteria

Tangent operators

Lv (τ ′) = cep : d and aep = cep + 1 ⊕ τ ′

Drained case

Aik = Adik = nja

epijklnl

Rudnicki and Rice: JMPS 1975

Undrained case

Aik = Adik + J Kf

φf nink


Necessary condition for localization

F (A) = inf |n det (A) = 0







Tangent operators


Drained case

Aik = Adik = nja

epijklnl


Undrained case

Aik = Adik + J Kf

φf nink



F (A) = inf |n det (A) = 0







Tangent operators


Drained case

Aik = Adik = nja

epijklnl


Undrained case

Aik = Adik + J Kf

φf nink



F (A) = inf |n det (A) = 0







Tangent operators


Drained case

Aik = Adik = nja

epijklnl


Undrained case

Aik = Adik + J Kf

φf nink



F (A) = inf |n det (A) = 0






Permeability and porosity

Eulerian Darcy’s law

q = −1

gk · [∇x p− γf ]

k = kγf/µ1 (isotropic) [L/T]

k = intrinsic permeability [L2]

γf = fluid specific weight [F/L3]

µ = fluid dynamic viscosity [FT/L2]

Kozeny-Carman

k(

φf)

=1

180

φf 3

(1 − φf)2d2

d = grain diameter

Kozeny: 1927 and Carman: TICEL 1937





Outline











Stress integration and finite elements

Return mapping algorithm

Consistent tangent operator

Localization searchalgorithm

Mixed u − p formulation

Trapezoidal time integration

Stable, optimal convergence

t1 t2

t3

tn

tn+1

tn+1

tr

Fn+1

Fn














c =

3∑

a=1

3∑

b=1

cabma ⊗ mb

+3

∑

a=1

∑

b6=a

γabmab ⊗ mab

+

3∑

a=1

∑

b6=a

γabmab ⊗ mba














q

f

n1

n2

n3

nR

3

0 50 100 1500

20

40

60

80

100

120

140

160

180

THETA, degP

HI,

deg

0

0.5

1

1.5

2

2.5

3

3.5

x 1011

ALGO SOLN














Pressure node

Displacement nodeAndrade and Borja: IJNME 2006













u

p

n+1

=

u

p

n

+ ∆t (1 − α)

u

p

n

+ ∆tα

u

p

n+1














1 2 3 4 5 6 7 810

−15

10−10

10−5

100

ITERATION

NO

RM

ALI

ZE

D R

ES

IDU

AL

STEP NO. 45STEP NO. 90STEP NO. 135STEP NO. 180TOLERANCE







Drained responseUndrained response

Outline












Triaxial compression: calibrationDense Brasted sand, e0 = 0.57, ψi 0 = −0.2

q = |σa − σr|

0 1 2 3 4 5 6 7400

500

600

700

800

900

1000

AXIAL STRAIN, %

DE

VIA

TO

RIC

ST

RE

SS

, kP

a

FE RESULTEXPERIMENTAL







Triaxial compression: calibrationDense Brasted sand, e0 = 0.57, ψi 0 = −0.2

q = |σa − σr|

0 1 2 3 4 5 6 7400

500

600

700

800

900

1000

AXIAL STRAIN, %

DE

VIA

TO

RIC

ST

RE

SS

, kP

a


Volume change

0 1 2 3 4 5 6 7−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

AXIAL STRAIN, %

VO

LUM

ET

RIC

ST

RA

IN, %








Plane strain compression: predictionDense Brasted sand, e0 = 0.57, ψi 0 = −0.2

Principal stresses

0 0.5 1 1.5 2200

400

600

800

1000

1200

1400

1600

AXIAL STRAIN, %

PR

INC

IPA

L S

TR

ES

S, k

Pa

FE PRINCIPALEXPERIMENTAL PRINCIPALFE INTERMEDIATEEXPERIMENTAL INTERMEDIATE







Plane strain compression: predictionDense Brasted sand, e0 = 0.57, ψi 0 = −0.2

Principal stresses

0 0.5 1 1.5 2200

400

600

800

1000

1200

1400

1600

AXIAL STRAIN, %

PR

INC

IPA

L S

TR

ES

S, k

Pa

FE PRINCIPALEXPERIMENTAL PRINCIPALFE INTERMEDIATEEXPERIMENTAL INTERMEDIATE

Volume change

0 0.5 1 1.5 2-1

-0.5

0

0.5

1

1.5

2

2.5x 10

-3

VO

LU

ME

TR

IC S

TR

AIN

, %

AXIAL STRAIN, %

FE RESULT

EXPERIMENTAL







Plane strain compression of sandsUndrained dense sample

From CT-scan








From CT-scan FE model

1.56

1.57

1.58

1.59

1.6

1.61

1.62

1.63

SPECIFIC VOLUME








Flow and shear bandDEVIATORIC STRAIN

0 .0 5

0 .1

0 .1 5

0 .2

0 .2 5

0 .3









0 .0 5

0 .1

0 .1 5

0 .2

0 .2 5

0 .3

Pressure and deformation

46

48

50

52

54

56

58

60

62

64

FLUID PRESSURE









0 .0 5

0 .1

0 .1 5

0 .2

0 .2 5

0 .3

Reactive stresses

0 0.5 1 1.5 2 2.5 3 3.5 4100

120

140

160

180

200

220

240

260

NOMINAL AXIAL STRAIN, %N

OM

INA

LA

XIA

LS

TR

ES

S,

kP

a

INHOMOGENEOUS

HOMOGENEOUS

LOCALIZATION













Plane strain compression of sandsUndrained loose sample

From CT-scan FE modelSPECIFIC VOLUME

1.635

1.64

1.645

1.65

1.655

1.66

1.665








Flow and shear band

0.05

0.1

0.15

0.2

DEVIATORIC STRAIN








Flow and shear band

0.05

0.1

0.15

0.2

DEVIATORIC STRAIN

Pressure and deformation

45

50

55

60

65

70

FLUID PRESSURE








Flow and shear band

0.05

0.1

0.15

0.2

DEVIATORIC STRAIN

Reactive stresses

0 0.5 1 1.5 2 2.5 3 3.5 4100

120

140

160

180

200

220

NOMINAL AXIAL STRAIN, %N

OM

INA

LA

XIA

LS

TR

ES

S,

kP

a

INHOMOGENEOUS

HOMOGENEOUS

LOCALIZATION












Outline











Closure

Enhancements to elastoplastic model for sands

Large deformationsThree stress-invariantNonassociative flow

Density parameter ψi provides bridge to meso-scale

Model shown to capture signature behavior of sand

Global stability influenced by meso-scale inhomogeneities

Flow properties of sample influenced by localization

Collaborative research opens door for better understanding

Robust and stable numerical implementation





Closure













Closure













Closure













Closure













Closure













Closure













Future work

Liquefaction

Reliability of geostructures

Enhanced & stabilized FEM

Energy-related applications

Multi-scale modeling





Future work

Liquefaction




Multi-scale modeling0.00

0.25

0.50

0.75

1.00

0.10.001E

CD

F

ARIAS INTENSITY, g-seconds

0.01

RECORDED

SIGMASIGMA/2

SPECTRA

MEDIAN

SIGMASIGMA/2

SHAKE

MEDIAN

Andrade and Borja: AG 2006





Future work

Liquefaction









Future work

Liquefaction









Future work

Liquefaction









References

J. E. Andrade and R. I. Borja.

Fully implicit numerical integration of a hyperelastoplastic model for sands based on critical state plasticity.In K. J. Bathe, editor, Computational Fluid and Solid Mechanics 2005, pages 52–54. Elsevier Science Ltd.,2005.

R. I. Borja and J. E. Andrade.

Critical state plasticity, Part VI: Meso-scale finite element simulation of strain localization in discretegranular materials.Computer Methods in Applied Mechanics and Engineering, 195:5115–5140, 2006.


Capturing strain localization in dense sands with random density.International Journal for Numerical Methods in Engineering, 67:1531–1564, 2006.


Modeling deformation banding in dense and loose fluid-saturated sands.Finite Elements in Analysis and Design, 2006.In review for the 18th Annual Melosh Competition Special Issue.


Quantifying sensitivity of local site response models to statistical variations in soil properties.Acta Geotechnica. In press, 2006.


The effective stress concept

Definition

Effective Cauchy stress

σ′ = σ + p1

But the Kirchhoff stress τ = Jσ hence

τ ′ = τ + Jp1

Large deformation plasticity in terms ofτ ′


Strong form

W0

G0

t

G0

qG0

p

G0

d

P = τ · F−t

Q = JF−1 · q

Find u and p

∇X·P + ρ0g = 0 in Ω0

ρ0 + ∇X·Q = 0 in Ω0

u = u on Γd0

P · N = t on Γt0

p = p on Γp0

Q · N = −Q on Γq0

Plus I.C.


Matrix form

Mixed formulation: u − p

Isoparametric discretization

Different order interpolation(e.g., Q9P4)



Find d and p

Gext

Hext

−

Gint

H int

≡

0

0

Gext (t) =∫

Γt0N tt dΓ0, Hext (t) = ∆t

∫

Γq0N

t

Qn+α dΓ0

Gint (d,p) =∫

Ω0

[

Bt (τ ′ − Jpδ) − ρ0Ntg

]

dΩ0

H int (d,p) =∫

Ω0

[

Nt

∆ρ0 − ∆t(

JΓtq)

n+α

]

dΩ0


1D ConsolidationTerzaghi’s theory revisited

H0

?H

w

POROUS SOLID MATRIX

IMPERVIOUS BOUNDARY

(a) (b)

w

DRAINAGE

BOUNDARY


Terzaghi Solution RevisitedIsochrones for incompressible fluid

For incompressible fluid i.e., Kf → ∞ and w = 90 kPa

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3

3.5

4

PRESSURE, kPa

VE

RT

ICA

L C

OO

RD

INA

TE

, m

FE SOLN

ANAL SOLN


Terzaghi Solution RevisitedSettlement

For incompressible fluid i.e., Kf → ∞ and w = 90 kPa

0 5 10 15 20−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

TIME, sec

VE

RT

ICA

L D

ISP

LAC

EM

EN

T, m

FE SOLNANAL SS SOLN


Terzaghi Solution RevisitedIsochrones for compressible fluid

For compressible fluid i.e., Kf → 2 × 104 kPa and w = 90 kPa

0 10 20 30 40 50 60 70 80 900

0.5

1

1.5

2

2.5

3

3.5

4

PRESSURE, kPa

VE

RT

ICA

L C

OO

RD

INA

TE

, m

FE SOLN

ANAL SOLN


Documents

Meso-Scale Finite Element Simulation of Deformation ...geomechanics.civil.northwestern.edu/Presentations_files/seminar.en… · Constitutive Models Numerical Implementation Numerical