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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Introduction Elasto-visco-plasticity at finite strains Damage mechanism Applications Summary
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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Introduction Elasto-visco-plasticity at finite strains Damage mechanism Applications Summary
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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Introduction Elasto-visco-plasticity at finite strains Damage mechanism Applications Summary
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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Introduction Elasto-visco-plasticity at finite strains Damage mechanism Applications Summary
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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Introduction Elasto-visco-plasticity at finite strains Damage mechanism Applications Summary
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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Introduction Elasto-visco-plasticity at finite strains Damage mechanism Applications Summary
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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Introduction Elasto-visco-plasticity at finite strains Damage mechanism Applications Summary
Framework
Application of the principle of maximum dissipation
Introduction Elasto visco plasticity at finite strains Damage mechanism Applications Summary
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Introduction Elasto-visco-plasticity at finite strains Damage mechanism Applications Summary
Multiplicative decomposition
Small perturbations versus large transformations
Small perturbations: du
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Introduction Elasto-visco-plasticity at finite strains Damage mechanism Applications Summary
Multiplicative decomposition
Small perturbations versus large transformations
Small perturbations: du
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Introduction Elasto visco plasticity at finite strains Damage mechanism Applications Summary
Multiplicative decomposition
Small perturbations versus large transformations
Small perturbations: du
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Introduction Elasto visco plasticity at finite strains Damage mechanism Applications Summary
Multiplicative decomposition
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
Introduction Elasto-visco-plasticity at finite strains Damage mechanism Applications Summary
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p y g pp y
Multiplicative decomposition
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.
Introduction Elasto-visco-plasticity at finite strains Damage mechanism Applications Summary
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p y g pp y
Multiplicative decomposition
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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Constitutive relations
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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Constitutive relations
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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Constitutive relations
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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Constitutive relations
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 under several control mechanisms
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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Void growth under several control mechanisms
Void growth controlled by independent mechanisms (Cocks and
Ashby)
where b0, s0 and are material properties and f = (rh/l)2
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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Void growth under several control mechanisms
Void growth controlled by independent mechanisms (Cocks and
Ashby)
where b0, s0 and are material properties and f = (rh/l)2
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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Void growth under several control mechanisms
Void growth controlled by independent mechanisms (Cocks and
Ashby)
where b0, s0 and are material properties and f = (rh/l)2
Validity of CDM
A potential fD still need to be identified
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The limit theory approximation
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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The limit theory approximation
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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The limit theory approximation
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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Validation of the large transformations model
Introduction Elasto-visco-plasticity at finite strains Damage mechanism Applications Summary
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Validation of the large transformations model
Simple shear: elastoplasticity
Introduction Elasto-visco-plasticity at finite strains Damage mechanism Applications Summary
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Validation of the large transformations model
Elastoplastic response of frictional materials
Introduction Elasto-visco-plasticity at finite strains Damage mechanism Applications Summary
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Damage of a notched plate and effects of pressure
Introduction Elasto-visco-plasticity at finite strains Damage mechanism Applications Summary
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Damage of a notched plate and effects of pressure
Viscoplasticity, damage and shear heating
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Damage of a notched plate and effects of pressure
Experimental (Courtesy of Prof Arcady Dyskin)
Introduction Elasto-visco-plasticity at finite strains Damage mechanism Applications Summary
Diff i h h d i k
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Diffusion through a damaging rock
Model and Geometry (Poulet et al. 2010)
Thermal gradient, Rate dependency, damage*
Single diffusion flow (neglecting the advective effects)
Dirichlet boundary conditions in concentration
Introduction Elasto-visco-plasticity at finite strains Damage mechanism Applications Summary
Diff i th h d i k
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Diffusion through a damaging rock
Inelastic deformation
Introduction Elasto-visco-plasticity at finite strains Damage mechanism Applications Summary
Diffusion through a damaging rock
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Diffusion through a damaging rock
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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