numerical modelling of multiphysics couplings and the ...for element test, the tools allow you to...

Numerical modelling of multiphysics

couplings and the strain localization

Collin F., P. Kotronis, B. Pardoen

P. Bésuelle, D. Caillerie, R. Chambon

ALERT Doctoral School (6th-8th Octobre 2016)

Modelling of instabilities and bifurcation in Geomechanics

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 Theory Models Application Conclusions 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



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


Experimental observations: triaxial test

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: biaxial test

Experimental

set-up

&

a typical test


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 Theory Models Application 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. The application problem will be the nuclear waste disposal in deep geological layers.

8

Outline:

Introduction

Theoretical tools

Numerical models

Conclusions

9

Numerical application

Theoretical concepts

Experimental evidence:

Initial state Homogeneous strain field Localized strain field



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.



1 00n

:C L

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

det( ) 0nCn

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

1 00ij ij jn

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

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

1

1

2

2

11

2

2

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 bifurcation may arise and the direction of potential bifurcation (fracturation). Be aware that the Rice criterion is a local one !


Biaxial test modelling

Skeleton mechanical behaviour

Linear elasticity : E0 et 0

Associated softening plasticity (decrease of cohesion) :

Drucker Prager criterion : 0tan

3

2

3ˆ

cImIIF

sin3

sin2

m c = c0 f( p)

pR

p

pR

p

pR

pp

si

sif

2

2

0)1(1)(


Example n°1


Softening behaviour : localization effects are very important




First stress invariant [MPa]

Se

co

nd

de

via

tori

c s

tre

ss in

va

ria

nt [M

Pa]


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


Z

A Acoustic tensor determinent


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 test with a defect triggering the localization, first without any hydromechanical effect.

Bottom-left defect

Smooth and rigid boundary




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


50 elements 200 elements 300 elements

The post peak behaviour depends on the mesh size !


Example of EDZ around a cavity

Cylindrical cavity without retaining structure

Anisotropic initial state of stress

Geometrical dimensions : Internal radius 3 m

Mesh length 60 m

Choice :

Symmetry of the problem is assumed

894 elements – 2647 nodes – 7941 dof

Let’s consider now a coupled modelling:


Example n°2


MPap

MPa

MPa

w

yy

zzxx

7.4

64.11

74.7

'

''

MPap

MPa

MPa

w

yy

xx

7.4

4.15

5.11

'

'

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

T = 1.5 Ms (17 days)

ttotal = 300 Ms (9.5 years)

246 él.



Coupled modelling – Comparison Coarse mesh / Refined mesh

Deviatoric strains


Output

• 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:

30

Introduction

Theoretical tools

Numerical models

Conclusions


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

Enrichment of the kinematics :

The continuum is enriched with microstructure effects.

Macro-kinematics + micro-kinematics

Micro Ωm:

Macro Ω:


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 (Chambon et al., 1998;Matsushima et al., 2002)

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:


𝜎 𝜒

d = A1 du’ if u’ <elim d = A2 du’ if u’ >elim

d = 0 du’ if u’ >elim (A1 +A2)/ A2 d = B du’’

Numerical models

General differential equation of the 1D problem

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

After two integrations, we obtain:


Numerical models

if u’<elim

if u’>elim

General differential equation of the problem

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


𝛼 =𝐵

𝜆 + 2𝜇

B

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 x

Introduction of Lagrange multiplier field :

** *

* *iji i

ij ijk ij ij ext

j k j

vu ud v d W

x x x

* 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



Example n°1 (again)

Numerical models

• First gradient law :

E = 5800 MPa = 0,3

= 25° Y = 25°

Linear elasticity : E0 and 0

Associated softening plasticity (decrease of cohesion) :

Drucker Prager criterion : 0tan

3

2

3ˆ

cImIIF

sin3

sin2

m c = c0 f( p)

pR

p

pR

p

pR

pp

si

sif

2

2

0)1(1)(

c0 = 1 MPa = 0,01 R = 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

Velocity field (Regularization : Second gradient)


Numerical models

Initiation of localization (Directional research – Chambon et al., 2001)


Numerical models

Non uniqueness of the solution

Initiation of localization (Directional research)



Numerical models





Numerical models


Localization mode switching (Bésuelle et al.,2006)


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 d

ij j in t

'

ij ij ijp

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 x

*

* * *

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

* 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=1

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

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



Example n°1 (last time)

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.


Numerical models


Homogeneous solution

Localized solution

(random initialization)

0

5

10

15

20

25

30

35

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

q =

σ1-σ

3[M

Pa]

εa[-]

Random initialization (coupled problem)

Numerical models


0

5

10

15

20

25

30

35

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

q =

σ1-σ

3[M

Pa]

εa[-]

Deviatoric strain increment


Localisation/Biaxial/2d grad/Calibration 1/biaxial_2dgrad_tirage aléatoire - E2d=0.5 - ci=3mpa-cf=2mpa-Bc=0.01 - calibr1 - confin=12mpa - mail 1000elem/test1.pdf

../Localisation/Biaxial/2d grad/Calibration 1/biaxial_2dgrad_tirage aléatoire - E2d=0.5 - ci=3mpa-cf=2mpa-Bc=0.01 - calibr1 - confin=12mpa - mail 1000elem/test1.pdf



Numerical models



0

5

10

15

20

25

30

35

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

q =

σ1-σ

3[M

Pa]

εa[-]

Deviatoric strain increment

Decrease of q = disappearance of a shear band

8 9 10 15 16 17


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


Outline:

82

Introduction

Theoretical tools

Numerical models

Conclusions



Long-term management of radioactive wastes Underground structures = network of galleries

Deep geological

disposal

Repository in deep

geological media with

good confining properties

(Low permeability

K<10-12 m/s)

Disposal facility of Cigéo project in France

(Labalette et al., 2013)

Intermediate

(long-lived)

&

high activity

wastes



Repository phases

Construction

Excavation

Maintenance

Ventilation

Repository

Sealing

Long term

Corrosion,

heat generation

Type C wastes (Andra, 2005)

Tim

e



Excavation Damaged Zone (EDZ)

Fracturing & permeability increase

(several orders of magnitude)

Opalinus clay in Switzerland (Bossart et al., 2002)

Mechanical fracturing

Excavation

↓

Stress redistribution

↓

Damage / Fracturing

↓

Coupled processes

HM property modifications

↓

Safety function alteration

Water transfer

Galleries ventilation

↓

Air-rock interaction

↓

Drainage / desaturation

↓

Modification of the

water transfer

Construction Maintenance



Callovo-Oxfordian claystone (COx) Sedimentary clay rock (France). - Underground research laboratory Feasibility of a safe repository France (Meuse / Haute-Marne, Bure)

(Armand et al., 2014)

Borehole core samples (Andra, 2005)



Introduction Experiment Theory Numerical Conclusions 87

- Fracturing Anisotropies: - stress : σH > σh ~ σv

- material : HM cross-anisotropy. Galery // to σh

Galery // to σH Issues : Prediction of the fracturing.

Effect of anisotropies ?

Permeability evolution & relation to fractures ?

(Armand et al., 2014)


Constitutive models for COx - Mechanical law - 1st gradient model Isotropic elasto-plastic internal friction model Non-associated plasticity, Van Eeckelen yield surface :

φ hardening / c softening - Hydraulic law Fluid mass flow (advection, Darcy) : Water retention and permeability curves (Mualem - Van Genuchten’s model)

ˆ 'tan C

cF II m I

30

0

0

ˆ

ˆ

p

f eq

p

c eq

c cc c

B

Strain localisation

, ,

,

w ij r w w

w i w w j

w j

k k pf g

x



Gallery excavation modelling - Numerical model HM modelling in 2D plane strain state Gallery radius = 2.3 m - Gallery in COx // σh Effect of stress anisotropy

Anisotropic stress state

pw,0 = 4.5 [MPa]

σx,0 = σH = 1.3 σv = 15.6 [MPa]

σy,0 = σv = 12 [MPa]

σz,0 = σh = 12 [MPa]

- Excavation

σH

σv



- Localisation zone

Incompressible solid grains, b=1

For an isotropic mechanical behaviour, the appearance and shape of the strain localisation are

mainly due to mechanical effects linked to the anisotropic stress state.

3 days

σr/σr,0 = 0.4

4 days

σr/σr,0 = 0.2

5 days

σr/σr,0 = 0.0

100 days

σ/σ0 = 0.00

1000 days

σ/σ0 = 0.00

Total deviatoric strain Plasticity

0 0.06

4.6 m

0.5 m

2ˆ ˆ ˆ

3eq ij ij

End of

excavation



- Gallery air ventilation : Water phases equilibrium at gallery wall (Kelvin's law) Compressibility of the solid grains: b=0.6

Ventilation

pw = -30.7 [MPa]

No ventilation

pw = patm [MPa]

Total

deviatoric

strain

Plasticity

No ventilation

100 days 1000 days

0 0.184

Ventilation

100 days 1000 days

,0

v c v

v w

p p MRH exp

p RT

suction ↑

σ’ ↑

Elastic unloading

Inhibition of localisation

Restrain ε

,r w w ijij ij b S p



- Convergence: Important during the excavation

Anisotropic convergence

Influence of the ventilation

Experimental results (GED - Andra’s URL)

No strain localisation Calcul with large D or no loc SDZ? If large D ça rigidifie un peu Shear band position Shear bands are necessary for anisotropy



Large-scale experiment of gallery ventilation (SDZ) Characterise the effect of gallery ventilation

on the hydraulic transfer around it.

drainage / desaturation

exchange at gallery wall



Permeability variation in fractured zone HM coupling in the EDZ. Saturated permeability in boreholes

Fracture and rock matrix permeabilities Capture kw evolution

Relation to fractures



Evolution of intrinsic water permeability Various approaches: deformation, damage, cracks… - Relation to deformation

Volumetric effects = increase of porous space ()(

(Kozeny-Carman) - Fracture permeability Cubic law for parallel-plate approach (Witherspoon 1980; Snow 1969, Olivella and Alonso 2008) Poiseuille flow (laminar flow) equivalent Darcy’s media - Empirical law Related to strain localisation effect

Permeability variation threshold

w

bk

B

3

12

cr

w

bk

2

12

n nb b B 0 0

1 2

12

0,0

0

1

1w wk k

,

n n

w wk k A 3

0 01

Localised deformation

Fracture initiation

3

ii

v

, , ,ˆthr

w ij w ij per eqk k YI YI 3

0 1 ˆ

ˆ

p

IIYI

II



Modelling of excavation and SDZ experiment HM coupling in EDZ - Gallery excavation SDZ GED gallery // σh Anisotropic σij,0 and material Localisation zone dominated

by stress anisotropy - Intrinsic permeability evolution Cross-sections

Plastic strain and a part of the elastic one EDZ extension + kw increase

Plasticity Total deviatoric strain

kw,,ij / kw,ij,0 [-]

0.95thrYI

,

, ,

ˆw ij thr

eq

w ij

kYI YI

k 3

0

1



Modelling of the SDZ experiment - Water pressure in the cavity air

pw (suction) corresponding

to RH and T (Kelvin’s law).

Imposed at gallery wall through hydraulic boundary condition. Excavation (RH=100%) Initiation phase Ventilation

lnwair

w atm

v

RTp RH p

M

0air

v vRH



Experimental Numerical

- Drainage / pw reproduction

Oblique 45° Horizontal

αv = 10-3 m/s



- Desaturation EDZ / w reproduction

Horizontal boreholes Desaturation: overestimation in long term Vapour transfer (αv = 10-3 m/s)

Good reproduction at gallery wall

At gallery wall

w

s

Mw

M


Outline:

100

Introduction

Theoretical tools

Numerical models

Conclusions


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.


numerical modelling of multiphysics couplings and the ...for element test, the tools allow you to...

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