Numerical modelling of the strain localization in soils ... · 3D digital image correlation applied...

Numerical modelling of the strain

localization in soils and rocks

Collin F., Levasseur S., B. Pardoen, F. Salehnia

R. Chambon, D. Caillerie, P. Bésuelle

P. Kotronis, G. Jouan, M. Soufflet

INTRODUCTION

Shear banding occurs frequently (at many scales) and is the source of

many soil and rock engineering problems:

natural or human-made slopes or excavations, unstable rock masses,

embankments or dams, tunnels and mine galleries, boreholes driven for

oil production, repositories for nuclear waste disposal

Failure in soils and rocks is almost always associated with fractures and/or shear bands developing in the geomaterial.

Introduction Experiment Theory Numerical Conclusions

In geomaterials, the understanding of failure processes is more complex by the fact that soils and rocks are multiphase porous materials where different multiphysical processes take place.

2

INTRODUCTION

In situ observation of shear banding

In situ observations of shear banding and/or faulting are made frequently

at many scales

Large scale: railway tracks after an earthquake in Turkey

Introduction Experiment Theory Numerical Conclusions 3


Bierset (Belgium) 1998 – Courtesy C. Schroeder

Human-made slope along E42 exit road



Fractures observed during the construction of the connecting gallery at the URL in Mol. Vertical cross section through the gallery showing the fracturation pattern around it, as deduced from the observations (from Alheid et al. 2005)

Nuclear waste disposal


Outline:

Introduction

Experimental observations

Theoretical tools

Numerical models

Conclusions

6



• Biaxial test

• Axisymetric triaxial test

• True triaxial test

To better understand the development of the shear band, experiments are necessary, which are no more element tests as far as the behavior becomes heterogeneous.

Different teams have performed experimental works devoted to the study of strain localization:

• Desrues and co-workers

• Finno and co-workers

• Vardoulakis and co-workers

• …

7

Experimental observations: triaxial test

Triaxial test:

In triaxial tests (and more generally in axi-symmetric tests), the localization zone may remain more or less hidden inside the sample (need for special techniques to see the process)



Localized rupture in sandstone samples under different confining pressures (Bésuelle et al., 2000)


Experimental characterisation of the localisation phenomenon inside a Vosges sandstone in a triaxial cell

P. BESUELLE, J. DESRUES, S. RAYNAUD, International Journal of Rock Mechanics & Mining Sciences 37 (2000) p. 1223-1237

Experimental observations: triaxial tests

Desrues, J. et al. (1996). Géotechnique 46, No. 3, 529–546

Tomodensitometry:

Localization pattern observed in sand sample during axisymetric triaxial test


Experimental observations: triaxial tests

Tomodensitometry:

Localization pattern observed in sand sample during axisymetric triaxial test



Increment 4-5

3D digital image correlation applied to X-ray micro tomography images from triaxial compression tests on argillaceous rock LENOIR N , Bornert M, DESRUES J, BESUELLE P, VIGGIANI G Strain vol:43 No 3 pp.193-205


Experimental observations: biaxial test

Biaxial test:

As in triaxial tests (and more generally in axi-symmetric tests), the localization zone may remain more or less hidden inside the sample, most of the experimental campaigns on localization have been performed in biaxial apparatus


s1 s1

e , s33



Experimental

set-up

&

a typical test

14


Localization and Peak

1-2 2-3 3-4 4-5 5-6


Outline:

Introduction


Theoretical tools

Numerical models

Conclusions

16

Theoretical concepts

Experimental evidence:


Initial state Homogeneous strain field Localized strain field

17


Theoretical background


Following the previous works by (Hadamard, 1903), (Hill, 1958) and (Mandel, 1966), Rice and co-workers (Rice, 1976, Rudnicki et al., 1975) have proposed the so-called Rice criterion.

18



1 00n s s

:C Ls

1 0 0 0: ( ) : 0n C L g n C L

det( ) 0nCn

19

Static condition:

Kinematic condition:

Constitutive law:

When it is assumed that C 1=C 0=C , no trivial solution if and only if:

LLL 01

ngLL 01

i

j

j

i

x

u

x

uL

2

1



11 11 12 11

22 21 22 22

12 12 12

0

0

0 0 2

C C L

C C L

G L

s

s

s

1 00ij ij jns s

1 0 1 0

11 11 1 12 12 2

1 0 1 0

21 21 1 22 22 2

0

0

n n

n n

s s s s

s s s s

1 0

ij ij i jL g nL

If C 1=C 0=C :

1 2

1 2

11 1 1 12 2 2 12 1 2 2 1

12 1 2 2 1 21 1 1 22 2 2

0

0

C C n n

n C C n

g n g n G g n g n

G g n g n g n g n

Static condition:

Constitutive law in principal axis:

Kinematic condition:

Combining the three previous relationship yields:



4 2

1 3 5

4

1 0a z a z an

4 3 2

1 2 3 4 5

4

1 0a z a z a z a z an

2

1 2

2 2

11 1 12 2 1 12 1 2 12 2 1

2 2

21 1 2 12 2 1 22 2 12 1

0

0

C C

C g C

n G n g n n G n n g

n n G n n n G n g

det( ) 0nCn When it is assumed that C 1=C 0=C , no trivial solution if and only if:

4 4 2 22

11 12 1 22 12 2 11 22 12 12 12 1 22 0C G n C G n C C C G C n n

For a constitutive law written in cartesian axis:

2

1

nz

n



ni

qf

2dB2dB

2dB

s1s

1

s2

s2

s1s1

s2

s2

22

Extension to multiphysical context, mainly in hydro mechanical coupling:

Loret and co-workers (Loret et al., 1991) showed that for hydromechanical problems the condition of localization depends only on the drained properties of the medium In coupled problems much more complex localization pattern can be obtained, at least theoretically (Vardoulakis, 1996)



Which information can provide these theoretical tools ?

For element test, the tools allow you to check if and when the constitutive model is able to predict the localization direction observed at the laboratory. For boundary value problems, they provide you the stress state when bifucation may arise and the direction of potential bifurcation (fracturation). Be aware that the Rice criterion is a local one ! This criterion could be activated for any constitutive model, if you make the connection in the ELEMB2 and POSPEC routines and some additional state variables have to be defined.

Example of EDZ around a cavity


Skeleton mechanical behaviour

Linear elasticity : E0 et n0

Associated softening plasticity (decrease of cohesion) :

Drucker Prager criterion : 0tan

3

2

3ˆ

ss

cImIIF

sin3

sin2

m c = c0 f(g p)

pR

p

pR

p

pR

pp

si

sif

gg

ggg

gg

2

2

0)1(1)(



Softening behaviour : localization effects are very important

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Vertical displacement [mm]

Axia

l lo

ad [M

N]

Sample with defect

Perfect sample

NL in the global curve

Bifurcation





Softening behaviour : localization effects are very important

Bifurcation analysis thanks to the Rice criterion (Acoustic tensor)

4 4 3 2

1 1 2 3 4 5det ( ) 0n n a z a z a z a z a



Plastic point Bifurcation dir. Bifurcation cones



Plastic point Bifurcation dir. Bifurcation cones



The Rice criterion provides us the information on when and how localization may appear. Do we have any problem to model such phenomenon with classical finite element method ? Let’s consider the modelling of a biaxial with a defect triggering the localization, first without any hydromechanical effect.

Bottom-left defect

Smooth and rigid boundary



50 elements 200 elements 300 elements

The post peak behaviour depends on the mesh size !



Cylindrical cavity without retaining

Anisotropic initial state of stress

Geometrical dimensions : Internal radius 3 m

Mesh length 60 m

Choice :

Symetry of the problem is assumed

894 elements – 2647 nodes – 7941 dof

Let’s consider now a coupled modelling:



MPap

MPa

MPa

w

yy

zzxx

7.4

64.11

74.7

'

''

s

ss

MPap

MPa

MPa

w

yy

xx

7.4

4.15

5.11

s

s

'

'

0

11.5 1

15.4 1

4.7 1

0

xx xx rw w

yy yy rw w

w

xx yy w

t T

tbS p MPa

T

tbS p MPa

T

tp MPa

T

t T

p

s s

s s

s s

T = 1.5 Ms (17 jours)

ttotal = 300 Ms (9.5 ans)

246 él.



Coupled modelling – Comparison Coarse mesh / Refined mesh

Deviatoric strains



• Localization study : Acoustic tensor determinent

• Mesh dependency of the results for classical FE

• Non-uniqueness of the results in both cases

The numerical modelling of strain localization with classical FE is not adequate. We need another numerical model to fix this mesh dependency problem !

Outline:

Introduction


Theoretical tools

Numerical models

Conclusions

36

Numerical models


• Classical FE formulation: mesh dependency

• Different regularization methods

Gradient plasticity Non-local approach Microstructure continuum Cosserat model Second gradient local model

Mainly for monophasic materials !

Enrichment of the law

Enrichment of the kinematics

Numerical models


In second gradient model, the continuum is enriched with microstructure effects. The kinematics include therefore the classical one but also microkinematics (See Germain 1973, Toupin 1962, Mindlin 1964).

Let us define first the classical kinematics:

Numerical models


Here is the enrichment:

Numerical models


• The internal virtual work (Germain, 1973)

• The external virtual work (simplified)

• The virtual work equations can be extended to large strain problems

Numerical models


• Balance equations

• Boundary conditions

written in the current configuration

Three constitutive equations needed !

• Local second gradient models: we add the kinematical constraint:

this implies:

the virtual work equation reads

Numerical models


Numerical models


• Local second gradient models

balance equations

boundary conditions

Numerical models


How do we introduce an internal length scale in second grade model ?

Let’s take a simple example of a 1D-bar in traction:

Numerical models


if u’<elim

if u’>elim

General differential equation of the problem

Where N1= N - M’ = Cst, K = Cst, A = A1 if u’ <elim and A = A2 if u’ >elim

General solution of the problem

Numerical models


Let’s take another example: thick-walled cylinder problem (elastic second gradient model)

General differential equation of the problem

Balance equation

General solution

Numerical models


• Local second gradient model : additional assumption * *

ij ijv F

* 2 *

*i i

ij ijk ext

j j k

u ud W

x x xs

Introduction of Lagrange multiplier field :

** *

* *iji i

ij ijk ij ij ext

j k j

vu ud v d W

x x xs

* 0i

ij ij

j

uv d

x

Local quantities

Finite element formulation of a second grade model

Numerical models


Local Second gradient Finite element

Numerical models


• Biaxial compression test

Strain rate : 0.18% / hour

No lateral confinement

Globally drained (upper and lower drainage)

Bottom-left defect


Numerical models


• First gradient law :

E = 5800 MPa n = 0,3

= 25° Y = 25°

Linear elasticity : E0 and n0

Associated softening plasticity (decrease of cohesion) :

Drucker Prager criterion : 0tan

3

2

3ˆ

ss

cImIIF

sin3

sin2

m c = c0 f(g p)

pR

p

pR

p

pR

pp

si

sif

gg

ggg

gg

2

2

0)1(1)(

c0 = 1 MPa = 0,01 gR = 0,015

Numerical models


• Second gradient law : Linear relationship deduced from Mindlin

D = 20 kN

Numerical models


First modelling: no HM coupling (no overpressure)

Before After

Bifurcation directions (Regularization : Second gradient)

Numerical models


Before After

Plastic loading point


(Regularization : Second gradient)

Numerical models



Before After

Velocitiy field (Regularization : Second gradient)

Numerical models


Initiation of localization (Directional research)

Numerical models


Non uniqueness of the solution



Numerical models





Numerical models





Sieffert et al., 2009

Numerical models


• Main assumptions

– Quasi static motion

– Fully saturated

– Incompressible solid grains

• Aims

– Equations written in the spatial configuration

– Full Newton Raphson method

Our goal is to extend the second gradient formulation for multiphysics conditions. In the following, we focus on the hydromechanical model but the same procedure can be applied for TM, THM or THMC problems.

Numerical models


• Classical poromechanics field equations

Saturated porous medium

Balance of linear momentum for the mixture

* * *

ij ij mix i i i id g u d t u ds e

ij j in ts

'

ij ij ijps s

Boundary condition

Terzaghi’s postulate

Numerical models



Fluid mass balance

*

* * *

i

i

pM p m d Q p d q p d

x

( )i w w i

i

pm g

x

w ww

pM

k

i iq m nBoundary condition

Darcy’s law

Storage law

Numerical models



Balance of momentum for the fluid phase

Mass balance equation for the solid

Viscous drag force :

Numerical models


• Coupled local second gradient model

Second gradient effects are assumed only for solid phase

For the mixture, there are stresses which obey the Terzaghi postulate and double stresses which are only the one of the solid phase

Boundary conditions for the mixture are enriched

Numerical models



* 2 *

* * *i iiij ijk mix i i i i i

j j k

u ud g u d t u T Du d

x x xs

*

* * *

i

i


x

Numerical models



** *

*

* * *

iji i

ij ijk ij ij

j k j

imix i i i i i

vu ud v d

x x x

g u d t u T Du d

s

* 0i

ij ij

j

uv d

x

*

* * *

i

i


x

Numerical models


Equations are assumed to be met at time t

We are looking for the values of the different fields at time: t+t=t1

using a full Newton Raphson method and an implicit scheme for the rate :

Finite element formulation of the coupled local second gradient model

Numerical models


• Field equations at time t+t

R, S and W : Residuals of the balance equations

Numerical models


•Linearization of field equations Auxiliary linear problem

*

( , ) ( , )

T

x y x yU E dU d R S W

R, S and W : Residuals of the balance equations

1 1 2 2

( , ) 1 2

1 2 1 2 1 2

11 11 12 22

11 22 11 22

1 2 1 2

x y

du du du du dp dpdU du du dp

x x x x x x

dv dv dv dvdv dv d d

x x x x

Numerical models


(4 4) (4 2) (4 8) (4 4) (4 4)(4 3)

(2 4) (2 2) (2 3) (2 8) (2 4) (2 4)

(3 2) (3 8) (3 4) (3 4)(3 4) (3 3)

(8 4) (8 2) (8 3) (8 8) (8 4) (8 4)

(4 4) (4 2) (4 3) (4 8) (4 4) (4 4)

(4 4

1 0 0 0

1 0 2 0 0 0

0 0 0 0

2 0 0 0 0

3 0 0 0 0

4

x x WM x x xx

x x x x x x

MW x WW x x xx x

x x x x x x

x x x x x x

x

E K I

G G

K KE

E D

E I

E

) (4 2) (4 3) (4 8) (4 4) (4 4)0 0 0 0

x x x x xI

E1, E2, E3, E4 and D : see monophasic local sec. Gradient model

G1 and G2 : related to gravity volume force

KWW : Classical flow matrix

KMW and KWM : Coupling terms including large strain effect

Numerical models


Isoparametric Finite Element :

8 nodes for macro-displacement and pressure field 4 nodes for microkinetic gradient field 1 node for Lagrange multipliers field

Numerical models


1 1

*

1 1

*

detT T T

node node

T

node node

U B T E T B J d d dU

U k dU

•FE element discretization of linear auxiliary problem

Local stiffness matrix

1 1

* *

1 1

*

detT T T

ext node

T

node HE

R S W P U B T J d d

U f

s

Elementary out of balance forces

Numerical models


• Biaxial compression test

Strain rate : 0.18% / hour

No lateral confinement

Globally drained (upper and lower drainage)

Bottom-left defect


Numerical models


• Second gradient law : Linear relationship deduced from Mindlin

• Flow model parameters

D = 20 kN

= 10-19 / 10-12 m2

w= 1000 kg/m³ = 0.15

kw = 510-10 Pa-1

w = 0.001 Pa.s

Numerical models


(20 x 10) (30 x 15) (40 x 20)

•Equivalent strain after 0.2 % of axial strain ( = 10-12 m²)

Second modelling: HM coupling

Numerical models


•Plastic loading point after 0.2 % of axial strain ( = 10-12 m²)

(20 x 10) (30 x 15) (40 x 20)

Numerical models


•Fluid flow after 0.2 % of axial strain ( = 10-12 m²)

Numerical models


•Load-displacement curve ( = 10-12 m²)

Numerical models


•Load-displacement curve ( = 10-19 m²)

‘Undrained’ behaviour

Second modelling: HM coupling

Numerical models


For = 10 -19 m², the behaviour is undrained, we recover the

experimental observation showing that for dilatant material, no localization is possible before cavitation.



MPap

MPa

MPa

w

yy

zzxx

7.4

64.11

74.7

'

''

s

ss

MPap

MPa

MPa

w

yy

xx

7.4

4.15

5.11

s

s

'

'

0

11.5 1

15.4 1

4.7 1

0

xx xx rw w

yy yy rw w

w

xx yy w

t T

tbS p MPa

T

tbS p MPa

T

tp MPa

T

t T

p

s s

s s

s s

T = 1.5 Ms (17 jours)

ttotal = 300 Ms (9.5 ans)

246 él.



Coupled modelling – Comparison Coarse mesh - Refined mesh

Deviatoric strains

Classical FE formulation



Coupled modelling – Comparison Coarse mesh - Refined mesh

Deviatoric strains

Coupled second gradient FE formulation



Coupled modelling


François et al., 2012



Coupled modelling





Coupled modelling





Coupled modelling





Coupled modelling



Outline:

Introduction


Theoretical tools

Numerical models

Conclusions

88

Conclusions


Strain localization in shear band mode can be observed in most laboratory tests leading to rupture in geomaterials.

Complex localization patterns may be the result of specific geometrical or loading conditions.

The numerical modelling of strain localization with classical FE is not adequate. Enhanced models are needed for a robust modelling of the post peak behaviour.

Many experimental works and numerical developments are necessary to improve the prediction of failure in boundary value problems

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Numerical modelling of the strain localization in soils ... · 3D digital image correlation applied...

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