Meso-Scale Finite Element Simulation of Strain Localization ...geomechanics.civil.northwestern.edu/Presentations_files/...x 1 x 2 X x j f W 0 f Matrix deﬁned by solid grains and

MotivationBalance Laws: Continuum Formulation

Constitutive ModelsNumerical Implementation

Closure

Meso-Scale Finite Element Simulation of Strain

Localization in Fluid-Saturated Granular Media

José E. Andrade and Ronaldo [email protected]

Department of Civil and Environmental Engineering

Northwestern University

WCCM VIILos Angeles, July 2006

J.E. Andrade Simulating Localization in Saturated Granular Media



Closure

Outline

1 MotivationMeso-scale and collaborative research

2 Balance Laws: Continuum Formulation

3 Constitutive ModelsEffective stressDarcy’s Law

4 Numerical Implementation

5 Closure




Closure

Meso-scale and collaborative research

Outline





5 Closure




Closure


What is meso-scale?

Rechenmacher: 2005

Meso-scale

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

Still looking at continuumpicture




Closure


Collaborative research

��

Patient

Motivations

Quantify porosity atmeso-scale in the lab

X-Ray CTDIP

Develop meso-scale modelsfor sands

Analyze behavior as BVP




Closure


Collaborative research

1.61

1.62

1.63

1.64

1.65

1.66

1.67

SPECIFIC VOLUME

Motivations

Quantify porosity atmeso-scale in the lab

X-Ray CTDIP

Develop meso-scale modelsfor sands

Analyze behavior as BVP




Closure

Outline





5 Closure




Closure

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




Closure

Multi-phase system

6

Solid grain

AAU

Matrix

BB

BBBM

Fluid








Closure

Multi-phase system

js

W0

W

x1

x2

X

x

jf

W0f








Closure

Balance for multi-phase system

Localized mass balancefor mixture

ρ̇0 = −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




Closure

Effective stressDarcy’s Law

Outline





5 Closure




Closure


Phenomenological plasticityAn elastoplastic model for sands

Model main features

1 F = F e · F p

2 Tension/compression

3 Pressure dependence

4 Dr dependence via ψi

F F F=e p.

W0

WX

Wp

xp

Fp

Fe

x

Borja and Andrade: CMAME 2006

Andrade and Borja: IJNME 2006




Closure



Model main features

1 F = F e · F p




t’1 t’2

t’3

Loose sand Dense sand

Originalz(q)






Closure



Model main features

1 F = F e · F p




t =’1 t t’ = ’2 3






Closure



Model main features

1 F = F e · F p




v

v1

vc

v2

-pi -p’

ln-p’

l~

y

yi

CSL






Closure



Model main features

1 F = F e · F p




Hardening law

π̇i = h (π∗

i − πi) ε̇ps

Jefferies: Géotech 1993






Closure



Model main features

1 F = F e · F p




Hardening law

π̇i = h (π∗

i − πi) ε̇ps

Jefferies: Géotech 1993

-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






Closure


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

Kfφfnink


Necessary condition for localization

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




Closure


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




Closure

Outline





5 Closure




Closure

Stress integration and finite elements

Return mapping algorithm

Consistent tangent operator

Mixed u − p formulation

Trapezoidal time integration

Stable, optimal convergence

t1 t2

t3

tn

tn+1tn+1

tr

Fn+1

Fn






Closure







c =3

∑

a=1

3∑

b=1

cab

ma ⊗ mb

+3

∑

a=1

∑

b6=a

γabmab ⊗ mab

+

3∑

a=1

∑

b6=a

γabmab ⊗ mba






Closure







Pressure node

Displacement nodeAndrade and Borja: IJNME 2006





Closure







{

u

p

}

n+1

=

{

u

p

}

n

+ ∆t (1 − α)

{

u̇

ṗ

}

n

+ ∆tα

{

u̇

ṗ

}

n+1






Closure







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






Closure

Matrix form

Galerkin recipe

u ≈ Nd + N ξξ

p ≈ Np + N ζζ

Find d and p

{

GextHext

}

−

{

GintH int

}

≡

{

0

0

}

Gext (t) =∫

Γt0N tt dΓ0, Hext (t) = ∆t

∫

Γq

0

Nt

Qn+α dΓ0

Gint (d,p) =∫

Ω0

[

Bt (τ ′ − Jpδ) − ρ0Ntg

]

dΩ0

H int (d,p) =∫

Ω0

[

Nt

∆ρ0 − ∆t(

JΓtq)

n+α

]

dΩ0




Closure

Plane strain compression of sandsUndrained dense sample

From CT-scan





Closure


From CT-scan FE model

1.56

1.57

1.58

1.59

1.6

1.61

1.62

1.63

SPECIFIC VOLUME





Closure


Flow and shear bandDEVIATORIC STRAIN

0 .0 5

0 .1

0 .1 5

0 .2

0 .2 5

0 .3





Closure



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





Closure



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





Closure





Closure

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





Closure


Flow and shear band

0.05

0.1

0.15

0.2

DEVIATORIC STRAIN





Closure


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





Closure


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





Closure





Closure

Outline





5 Closure




Closure

Closure

Developed advanced elastoplastic model for sands

Density parameter ψi provides bridge to meso-scale

Global stability influenced by meso-scale inhomogeneities

Flow properties of sample influenced by localization

Collaborative research opens door for better understanding

Efficient and stable numerical implementation




Closure

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 discrete

granular 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. In press, 2006.


Modeling deformation banding in dense and loose fluid-saturated sands.

Finite Elements in Analysis and Design, 2006. In review.


Instabilities in geomechanics

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



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

G0t

G0q

G0p

G0d

P = τ · F−t

Q = JF−1 · q

Find u and p

∇X·P + ρ0g = 0 in Ω0

ρ̇0 + ∇X·Q = 0 in Ω0

u = u on Γd0

P · N = t on Γt0

p = p on Γp0Q · N = −Q on Γq0

Plus I.C.



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


MotivationMeso-scale and collaborative research

Balance Laws: Continuum FormulationConstitutive ModelsEffective stressDarcy's Law

Numerical ImplementationClosureInstabilities in geomechanics

Documents

Meso-Scale Finite Element Simulation of Strain Localization ...geomechanics.civil.northwestern.edu/Presentations_files/...x 1 x 2 X x j f W 0 f Matrix deﬁned by solid grains and