CDM of Geomaterials Subj to Large Transformations

8/2/2019 CDM of Geomaterials Subj to Large Transformations

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Introduction Elasto-visco-plasticity at finite strains Damage mechanism Applications Summary

.

Continuum damage mechanics of

geomaterialsSubjected to Large Transformations

Ali KARRECH1

K. Regenauer-Lieb2

and T. Poulet

1CSIRO: Earth Science and Resource Engineering, 26 Dick Perry Ave,Kensington, WA 6151 Australia. 2 CSIRO and WACOE

18-03-2010


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Outline

1 IntroductionMotivations

Framework

2 Elasto-visco-plasticity at finite strains

Multiplicative decomposition

Constitutive relations

3 Damage mechanism

Void growth under several control mechanisms

The limit theory approximation

4 Applications

Validation of the large transformations model

Damage of a notched plate and effects of pressure

Diffusion through a damaging rock

5 Summary


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Motivations

Plate tectonics

The predicted forces for

splitting continents apart

are much higher thenavailable from plate

tectonics.

Time and length scales

cant be achieved in thelaboratory.

Regenauer-Lieb et al 06


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Motivations

Plate tectonics

Several dissipation

feedbacks can help

predicting the plate

tectonics in a more

accurate manner;

How to introduce these

weakening mechanisms?

How to adapt damage,

as one of thesemechanisms, in a

geoscientific context?

What about the validity of

CDM?


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Motivations

Large deformations

Large transformations to describe earth systems

instabilities

I d i El i l i i fi i i D h i A li i S


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Motivations

Classic rates

*

How to formulate

thermo-mechanicalcoupled viscoplastic

models for frictional

materials in finite strain

How to overcome thesespurious oscillations?

I t d ti El t i l ti it t fi it t i D h i A li ti S


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Framework

Framework

RVE: statistical representation of typical material

properties;

Mass * , energy, and entropy variations ** through the

volumetric contributions and the surface fluxes;

As a state variable, our damage parameter contributes to

the energy dissipation;

Introduction Elasto visco plasticity at finite strains Damage mechanism Applications Summary


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Framework

Application of the principle of maximum dissipation



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Small perturbations versus large transformations

Small perturbations: du


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Gradient of deformation

The deformation gradient

at X is expressed by

FT /X. *

The multiplicativedecomposition Lee and

Lui (67,69):

Fto = FthFeFvp

The thermal gradient is: Fth = JthI

The athermal gradient is: F = FeFvp

As a measure of deformation we consider the Hencky

tensors: h= 12 ln (b) =12 ln FFt and he = 12 lnFeFet



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p y g pp y


Objective rates

The deformation rate tensor and the vorticity tensor can be

defined as follows (l= FF1):

d=1

2(l+ lt) and w=

1

2(l lt)

We use the logarithmic corotational rate for objectivity *

and stability ** considerations (to derive Eulerian

quantities):A =

A + A

A

A Corotational rate of an Eulerian strain measure defined

by is objective if and only if = w+ Y and Y(b,d) isisotropic Xiao et al 98-06.



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p y g pp y


Objective rates

The logarithmic spin tensor was derived by Xiao et al.

(98-06):

= w+

n

A,B=1,A=B

A + BA B

+2

ln A ln BpAdpB

where pA = nA nA

Among all possible strain tensor measures, only henjoys

the property d=

h, where

is used as a corotationalspin (Xiao et al. 98-06)

Logarithmic strain rates are the only objective conjugates

of Kirchhoff stress which produce self consistent

elastoplasticity models *



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Dissipation inequality

In light of (Simo 91), we consider that the free energy describes the stored energy related to the elastic lattice

deformation:

(he, , D, )

Applying the principle of virtual work, the first and second

principles of thermodynamics, Clausius-Duhem inequality

can be obtained:

D = : (h he) +

he

: he (s+

)

.

D.D

q

.grad() 0



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Helmholtz free energy and a dissipation function of the form

DI = : 0; DT =q

.grad() 0

=

he; s=

; Y =

D; =

Hence, the following constitutive relations can beobtained:We postulate the following free energy:

(he,T) = GA=1,3

ln2(eA) +

1

2K ln J+

2

3G

ln J

3K( 0) ln J (cx

20)( 0)

2

where eA = J1/3

eA, is the coefficient of thermal

expansion, c is the specific heat capacity,

x= 1 9K20/c. K = (1D)K0 G= (1 D)G0



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Dissipation

we also postulate a dissipation function

D =2a

1

2

3ij

ij + ((ii) a+ rii)

ii3r

+1

1 2(

2

3ij

ij) + (

c

1

2

3ij

ij) + F(D)D

Euler theorem of homogeneous functions on the plastic

part => Thermodynamic forces in the dissipative regime

ij, D, etc.

Expressions of these forces result in equalities of the form:

fp(ij) = fD(D) = 0.0



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Application of the principle of maximum dissipation

Legendre-Fenchel: optimisation problem under equality

constraint results in

ij = p fpij= p3

2

ij

q+ prij and D= p fDD

The maximum dissipation results in forces-velocity

orthogonality ==> relationships between ij = ij, Y = D

In hyperelasticity, the thermodynamic force, Y, wasdeduced from Helmholtz free energy.

What is the suitable potential fD ?



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Void growth controlled by independent mechanisms (Cocks and

Ashby)

where b0, s0 and are material properties and f = (rh/l)2

Validity of CDM

f can be interpreted as a damage parameter D



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Ashby)


Validity of CDM

To the first order, each one of the mechanisms derives to the

evolution laws of Lemaitre, Chaboche and Kachanov.



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Ashby)


Validity of CDM

The original Lemaitre interpretation is justified only if (i) a single

creep mechanism is used (ii) f is small



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Ashby)


Validity of CDM

A potential fD still need to be identified



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Void growth controlled by independent mechanisms

We consider the effects of diffusion and dislocation

mechanisms:

voids are of arbitrary axisymmetric generatrice and within

an RVE

spacing of min(2d, 2L), where d and L are distances in the

longitudinal and radial directions respectively

voids are assumed to be of small size as compared to the

RVE

Applying the upper bound theory (Cocks and Ashby, Chuang etal. 80s):

V

W() + W()

dV

Tiui d



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Upper bound

The decomposition of the strain rates results in:

W() = Ad2

2+ Ap

nn+1

n+ 1and W() = Ad

2

2+ Ap

n+1

n+ 1

The decomposition of the total volume into porous and solid

parts results in:V

V=

VsVs

+

1

It can be deduced that:

Dg =

1

(1D)n (1 D)

ineq



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Upper bound

Hence by identification, it can be seen that:

Dg =

1

(1 D)n+1

cY

Unlike the original model of Lemaitre and Chaboche, no

growth is possible if D =0

Nucleation can be taken into account through an additional

term which depends only on the thermodynamic force Y:

D=

1

(1 D)n+1

cY+ (

Y

H)



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Simple shear: elastoplasticity



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Elastoplastic response of frictional materials



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Viscoplasticity, damage and shear heating


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Experimental (Courtesy of Prof Arcady Dyskin)


Diff i h h d i k


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Model and Geometry (Poulet et al. 2010)

Thermal gradient, Rate dependency, damage*

Single diffusion flow (neglecting the advective effects)

Dirichlet boundary conditions in concentration


Diff i th h d i k


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Inelastic deformation




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Substance concentration



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Conclusion

The models were formulated within the framework of

thermodynamics of frictional materials (Houlsby and Puzrin

(06));New numerical techniques are used to integrate the model;

Robust algorithm developed where pressure, temperature,

damage and rate dependencies can be included;

Thermo-coupling is included in the models;



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Perspective

Advection terms (need for a transport code);

Coupling of the current formulations with the reactivetransport code (by Thomas Poulet)

Multi-scaling techniques are needed to estimate ranges of

validity.

References


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Consider the orthogonal time-dependent second order tensor

Q(t) that describes the relationship the positions with respectto two different observers:

x x0 = Q(t)(x x0)

A physical quantity should be invariant relative to a change ofobserver: v = Q(t)v and T = Q(t)TQ(t)t

objective rate tensors can be written as:

T = QTQt + QTQt + QTQt

notice that I= QQt then QQt + QQt = 0 and denote = QQt

QtTQ= T0 = T+ TTback

References


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Truesdell rate of the Cauchy stress: = l lT + tr(l) Green-Naghdi rate (F = R Uand

=R R

T

): = + Jaumann rate of the Cauchy stress = + ww

back

References


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Slef-consistent: exactly integrable to deliver an

hyperelastic relation whenever a process of purely elasticdeformation is involved (D= De).

The following relationship is usually used to characterize

common materials: De = C :

how to choose the objective rate so that the above

equation is self-consistent?.

Simo and Pister (1984) showed that the above equation is

inconsistent with elasticity when the classic corotational

rates (Jauman, Green-Naghdi, Truesdell etc).

It was recently shown that there only one (and strictly one)

rate that fulfils the self-consistent criteria. It is =

log

back

References


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Principles of conservation

Internal energy and entropy conservation (g= e, sspecificquantities):

dG

dt=

[ g+ div(u)g+ g]d +

kgu

k.nda (1)

Conservation of mass

dM

dt=

[ + div(u)]d +

kuk.nda= 0 (2)

where k and uk represent the density and the entering velocityof the kth substance in through the surface da.

back

References


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Energy balance and Clausius-Duhem inequality

Application of the first law of thermodynamics and the principle

of virtual work results in:

[ + div(u)]e+e+div(keuk) = : +kk+rdiv(q) (3)

where k and k generically denote the chemical potential and

number of moles or equivalent quantities in terms of concent.

and a dual thermodynamic force. r is a heat source and q is a

heat flux.

Application of the second law of thermodynamics results in:

[ + div(u)] +

+ T s

+ div(kuk) ksuk.gradTq.gradT

T : kk(4)

where is Helmholtz free energy.back

References


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Kinematics of a deformable body

Consider an open bounded domain 0 R3 representing

the reference configuration.

This body is embedded in three dimensional Euclidian

space.In Lagrangian coordinates, the material points are denoted

by X R3.

Motion is described by deformation map : (0,R+) R3

such that x= (X, t).The deformation gradient at X is expressed by F /X

back

References


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Material properties

Parameter Quartz

Density, (Kgm3) 2730Moduli, K/G(GPa) 52/31.2Initial Yield, Y0(MPa) 100

Diffusion Coefficient, k= k0 + aD(m2/s) 4.6 107, 4.6 105

Activation energies, Q(kJmol1) 135e3Exponent, n 4.0

Pre-Coefficient Disl A(MPans1) 6.32 1012

Friction/Dilatation Angle, 204.45

Table: Simulation Parameters.

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CDM of Geomaterials Subj to Large Transformations