Accepted Manuscript Multiscale modeling of cohesive geomaterials with a polycrystalline approach T. Zeng, J.F. Shao, W.Y. Xu PII: S0167-6636(13)00211-1 DOI: http://dx.doi.org/10.1016/j.mechmat.2013.10.001 Reference: MECMAT 2201 To appear in: Mechanics of Materials Received Date: 21 November 2012 Revised Date: 28 September 2013 Please cite this article as: Zeng, T., Shao, J.F., Xu, W.Y., Multiscale modeling of cohesive geomaterials with a polycrystalline approach, Mechanics of Materials (2013), doi: http://dx.doi.org/10.1016/j.mechmat.2013.10.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Transcript
Accepted Manuscript
Multiscale modeling of cohesive geomaterials with a polycrystalline approach
Received Date: 21 November 2012Revised Date: 28 September 2013
Please cite this article as: Zeng, T., Shao, J.F., Xu, W.Y., Multiscale modeling of cohesive geomaterials with apolycrystalline approach, Mechanics of Materials (2013), doi: http://dx.doi.org/10.1016/j.mechmat.2013.10.001
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customerswe are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, andreview of the resulting proof before it is published in its final form. Please note that during the production processerrors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Multiscale modeling of cohesive geomaterials with a polycrystalline
approach
T. ZENGa,b, J.F. SHAOa,b,∗, W.Y. XUa
aHOHAI University, Nanjing 210098, ChinabLML, UMR CNRS 8107, University of Lille, 59655 Villeneuve d’Ascq, France
Abstract
The main objective of this paper is to investigate the macroscopic elastic-plastic behaviorsof a class of cohesive geomaterials with the aid of classical polycrystalline schemes. Specificlocal constitutive equations are proposed to describe the typical features of geomaterials.The local yield criterion for crystallographic sliding systems takes into account the pressuresensitivity and a non-associated plastic potential is introduced to properly describe theplastic dilatancy. Consequently, the concentration law is also modified in order to establishthe relationship between the macroscopic stress tensor and the local stress tensor in eachmineral grain. Computational aspects associated with the numerical implementation ofpolycrystalline model are revisited and discussed. The proposed model is applied to a typicalpolycrystalline rock, granite. After the identification of material parameters, its validity isverified through comparisons between model’s predictions and experimental data on bothconventional and true triaxial compression tests.
In many engineering applications, it is necessary to describe mechanical behaviors ofvarious geomaterials (soils, rocks and concretes) by appropriate constitutive models. Phe-nomenological models, generally formulated within the framework of thermodynamics ofirreversible processes, have been largely developed and applied, including plastic models,viscoplastic models, damage models and coupled models. A number of models are able tocorrectly reproduce main features of mechanical behaviors of geomaterials under differentloading conditions. However, such models generally do not properly take into account phys-ical mechanisms involved at pertinent material scales and their consequences on materialmacroscopic responses. Indeed, most geomaterials contain different kinds of heterogeneitiesat different material scales, for instance, mineral grains, voids, micro-cracks and interfaces.
Preprint submitted to Mechanics of Materials October 8, 2013
The macroscopic responses of these materials are inherently related to their heterogeneousmicrostructure. In particular, the inelastic deformation and failure process of geomaterialsare directly related to the evolution of their microstructure, e. g. debonding of interfaces,microcrack growth, pore expansion and collapse, frictional sliding along microcracks, plas-tic sliding along crystallographic planes. Therefore, the formulation of constitutive modelsshould reflect these various physical phenomena. For this purpose, significant advances havebeen made during the last decades on the development of micro-mechanical models. Gen-erally, two families of micro-mechanical models have been proposed, respectively for plasticdeformation in ductile porous materials (Gurson, 1977; Maghous et al., 2009; Monchietet al., 2008; Guo et al., 2008; Shen et al., 2012) and micro-crack induced damage in brittleones (Gambarotta and Lagomarsino, 1993; Pensee et al., 2002; Zhu et al., 2008a,b), justto mention a few. These models are formulated using either limit analysis technique, lin-ear fracture mechanics theory or homogenization procedures based on Eshelby’s inclusionsolution (Eshelby, 1957). The micro-mechanical models provide a completely new way todescribe the inelastic responses of geomaterials. Some of them are successfully applied innumerical analysis of engineering structures.
However, in most micro-mechanical models developed so far, the geomaterials are gener-ally represented by a matrix-inclusion system. This kind of description is indeed representa-tive for a number of materials such as concretes with the cement paste as the matrix phase,hard clayey rocks with a dominant clay matrix, etc. However, many other geomaterials, forinstance granite, sandstone and limestone, cannot be represented by such a matrix-inclusionsystem. Their micro-structures are generally constituted by cemented randomly distributedmineral grains. Therefore, there is a need to consider a new microstructural description formicro-mechanical modeling of this kind of materials.
For these materials, the macroscopic plastic deformation and damage are mostly gener-ated by the degradation of cementation interfaces and frictional sliding along crystallographicand weakness planes. The well-established polycrystalline theory for metal materials pro-vides a feasible framework to develop such micro-mechanical models. However, for the sakeof simplicity and as a first stage of development, we consider here only the plastic deforma-tion due to the sliding along crystallographic or weakness planes, while the effect of graininterfaces degradation will be taken into account in future works.
Concerning the modeling of inelastic deformation in relation with the sliding in weaknessplanes, some original models have been proposed, in particular the so-called multi-laminatemodels (Zienkiewicz and Pande, 1977) and micro-plane models (Bazant and Oh, 1983). Inthese models, the overall inelastic strains are related to the local ones due to the sliding anddegradation of weakness planes in some orientations. The relationship between the localand overall stresses and strains is established through static or kinematic constraints andenergy conditions.
As the fundamental difference with these models, the formulation of micro-mechanicalmodels based on a homogenization procedure is performed in three steps: the definition ofa representative elementary volume(REV), the determination of concentration law estab-lishing the relationship between the microscopic and macroscopic stresses or strains, thehomogenization averaging to find the macroscopic properties. The determination of the
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concentration law should take into account the morphology of material microstructure suchas granular geometry and orientation, texture, voids, etc. Further, through an appropriateconcentration law, it is possible to account for interactions between grains and effects ofspatial distribution of grains. It is also possible to consider effects of mineral compositionsin multi-phase materials. Therefore, the homogenization-based micro-mechanical approachprovides a more general framework than the multi-laminate and micro-plane models.
As mentioned above, one essential requirement for the homogenization-based micro-mechanical models lies in establishing the relationship between local stress and strain in theindividual crystals and the overall stress and strain of the polycrystal, namely the so-calledconcentration law which is usually called the interaction law in polycrystalline materials.Various laws have been proposed so far, for instance (Taylor, 1938a; Budiansky and Wu,1962; Hill, 1965), just to mention some basic ones here. Among these models, the modelsuggested by Kroner, Budiansky and Wu (KBW model), which takes into account the in-teraction between slipping crystals, is extensively used due to its simplicity. Still based onEshelby’s solution, the strains or stresses in a single crystal are approximately obtained bysolving a spherical single crystal embedded in a uniform infinite plastically deformed matrix.The macroscopic properties of the matrix, also named Homogeneous Equivalent medium(HEM), are unknown and taken as identical to those of the polycrystalline materail. Inthis paper, without losing the assumptions and simplifications made in KBW model and atthe same time, taking into account the specific features of geomaterials, a generalized formof KBW model is proposed. This model can physically relate the macroscopic volumetricdilatancy to the microscopic normal aperture caused by the sliding of slip systems.
Another requirement is the complete description of the behavior of each single crystalinside the REV. Since our study is restricted to infinitesimal deformation, elastic latticedistortion is therefore neglected. The plastic deformation of the single crystal is the sum ofcontributions from all active slip systems. To reflect the effect of confining pressure on eachslip system, the classical Schmid’s law for each crystallographic plane is replaced by a Mohr-Coulomb type yield criterion. A general hardening law is adopted to depict the self-hardeningand cross-hardening behavior of each slip system. Furthermore, a non-associate plasticflow rule is proposed in order to properly describe the volumetric deformation observed inexperimental investigations.
The present paper is organized as follows: The scale decomposition is firstly introducedand the representative elementary volume of studied materials is defined. Before introduc-ing the specific yield criterion and plastic potential for a single crystal, a general form ofinteraction law is proposed. The stress update algorithm for rate form of constitutive re-lations, both at local and macro levels, are described in detail. After the determination ofmodel’s parameters from laboratory tests, performance of the proposed model is evaluatedby conducting the comparisons between numerical results and experiment data for a typicalpolycrystalline rock.
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2. Notation
Unless otherwise specified, the formulae and numerical calculations are expressed andcarried out with respect to a fixed coordinate system. The local fields (in each single crystal)are denoted by lower case letters, and the overall ones (in polycrystalline REV) by capital let-ters. The following tensor notations and operations are adopted: first-order tensor (vector) a;second-order tensor a; fourth-order tensor A; and simple contraction a · b = aibi, (i = 1, 2, 3);double contraction a : b = aijbij, A : b = Aijklbkl; dyadic products a⊗ b = aibj. With secondrank identity tensor δ, two classical isotropic fourth-order projectors J and K are expressed
as Kijkl = 13δijδkl, J = I − K with Iijkl = 1
2(δikδjl + δilδjk).
3. Polycrystalline model
3.1. Scale decomposition and representative elementary volume
At macroscopic scale, a sample of polycrystalline material is considered as a continuummade of many material points, as shown in Figure 1. In order to study its mechanicalbehavior by micro-mechanical approaches, we first assume that the sample is statisticallyhomogeneous and its response can be studied through the examination of any materialpoint. At mesoscopic scale, each material point is a heterogeneous medium composed ofdifferent constituents or phases, for instance mineral grains and voids. In order to determinethe macroscopic response of the sample, it is needed to define a representative elementaryvolume (REV) which should be large enough to represent the heterogeneities of each materialpoint at the mesoscopic scale. The REV is usually composed of randomly or non-randomlydistributed constituents. In this paper, for the sake of simplicity, we adopt a random spatialdistribution of constituents. Further, in the case of hard rocks such as granite studied here,voids are neglected and only mineral grains are considered as the constituents of the REV. Asa specificity of rocks, each mineral grain contains a number of weakness planes at microscopicscale. In the present work, the weakness planes are assimilated to the crystallographicplanes of metal materials. For the sake of simplicity, the crystal structure with the highestsymmetry,FCC, is here adopted.
3.2. Approximation of local fields by KBW model
Considering a REV subjected to a macroscopic stress or strain, the local stress or stainvaries not only from crystal to crystal, but also from point to point within each single crystal.Since it is nearly impossible to obtain the exact information of each point, for numericalsimplicity, it is generally admitted that the local fields are uniform throughout each singlecrystal. Although such a strong simplification is far form the real nonuniform distributionof local fields, it makes the subsequent average process possible.
In many micro-mechanical models, the Eshelby’s elastic solution for the spherical inclu-sion problem is used to determine the local fields (strains and stresses) as functions of thoseapplied at infinity. Therefore, the strains or stresses in a single crystal are approximated byconsidering a spherical single crystal embedded in an infinite elastically deformed matrix.
4
Figure 1: Scale decomposition of polycrystalline model. The mesoscopic photograph ofgranite is from (Soulie et al., 2007). The black arrow in 2D numerical model indicates theorientation of each single crystal. A typical single crystal is highlighted in the REV withred line.
KBW model extended Eshelby’s solution to plastic range and accounted for grain interac-tion. Each single crystal is subsequently embedded in a Homogeneous Equivalent Medium(HEM) with unknown properties, as shown in Figure 2. Note that in the multi-laminateand micro-plane models, the local stresses or strains are obtained by the projection of theiroverall counterparts respectively using the static or kinematic constraints. Compared tothese models, the interaction law based on the Eshelby solution provides a more generalframework to determining the relationship between the local and macroscopic stresses andstrains.
The extended interaction law for the plastic range, different from the original KBWmodel, can be written in the following rate form:
σ − Σ = −L? :(ε− E
)(1)
where L? = Lhom : (S−1 − I) is the interaction tensor. Lhom is the homogenized elastoplastictangent operator of HEM. S is the well-known Eshelby tensor, which is a function of grainshape and mechanical properties of matrix (HEM). Generally, S cannot be expressed in aclosed form and should be evaluated by an appropriate numerical integration method.
Considering the additive decomposition of local strain and making use of the Hooke’slaw,
ε = εe + εp and σ = C : εe (2)
similarly, for the macroscopic strain and stress:
E = Ee+ E
pand Σ = Chom : E
e(3)
Chom is the homogenized elastic stiffness tensor of HEM. In this paper, we assume that theREV of polycrystal is elastically homogeneous and isotropic, therefore Chom = C. Substi-tuting (2) and (3) into (1) and after some simple algebra operations, one gets:
σ = Σ−(I + L? : C−1
)−1: L? :
(εp − E
p)
(4)
5
Further, for the sake of simplicity, Kroner, Budiansky and Wu proposed to replace thetangent elastoplastic operator of HEM by its elastic stiffness tensor for the calculation ofEshelby tenor, i. e. Lhom = C. With this simplification, the Eshelby tensor S can be expressedin the following closed form:
S = cK + dJ (5)
with
c =3k
3k + 4µand d =
6
5
k + 2µ
3k + 4µ
where k and u, respectively, being the bulk and shear modulus of single crystal. Then, with(5) and the assumption of elastic interaction Lhom = C, (4) can be simplified as:
σ = Σ− (3k (1− c) K + 2u (1− d) J) :(εp − E
p)
(6)
In order to account for plastic volumetric strain in geomaterials, the interaction law (6)is modified by decomposing the plastic strain into a volumetric part and a deviatoric part:
σ = Σ− k (1− c)(εp
v− E
p
v
)− 2u (1− d)
(εp
d− E
p
d
)(7)
where the subscripts v and d denote the volumetric and deviatoric components of the localand macroscopic plastic strain tensors, respectively. This extended interaction law willbe used in the present work. However, in some situations, in particular when the plasticdeformation is large, the use of such as interaction law may provide too stiff responses.Different forms of correction, for instance, evolution of some parameters with plastic strain,can be introduced. This issue is not discussed in the present work. The use of otherinteraction laws for geomaterials will be investigated in our future works.
Figure 2: Schematic representation of KBW model
3.3. Averaging process
The macroscopic stresses or strains are obtained by the volumetric averaging of corre-sponding local quantities. However, in polycrystalline materials, it is convenient to define
6
the local quantities in a local coordinate system, called the crystal system. In order to per-form the averaging process, the local quantities should be transformed to the fixed globalcoordinate system. This can be realized by a series of sequent axis rotations. Although sev-eral methods exist for doing those operations, the most popular one, called Bunge’s method,is adopted here. Three Euler angles ϕ1, Φ and ϕ2, as shown in Figure 3, are introduced tospecify the rotation of each single crystal. The explicit form of three rotation steps is givenas follows:
Rot (Z,ϕ1) =
cos ϕ1 sin ϕ1 0− sin ϕ1 cos ϕ1 0
0 0 1
Rot (X ′, Φ) =
1 0 00 cos Φ sin Φ0 − sin Φ cos Φ
Rot (Z ′′, ϕ2) =
cos ϕ2 sin ϕ2 0− sin ϕ2 cos ϕ2 0
0 0 1
(8)
where Rot(a, b) means a standard rotation around the axis a by b degree. The basis vectorsof crystal coordinate system {ei} are related to the basis vectors of the fixed coordinatesystem {Ei} by a rotation matrix R
ei = RijEj with R = Rot (Z ′′, ϕ2) Rot (X ′, Φ) Rot (Z,ϕ1) (9)
Using these rotations, any tensor quantities in the local coordinate system can be trans-formed to those in the global one. Then, according to the classical upscaling principle (Hill,1965), the macroscopic quantities are calculated by
Σ =⟨σ⟩
=
Ng∑h=1
fhσh (10)
E =⟨ε⟩
=
Ng∑h=1
fhεh (11)
Ep =⟨εp
⟩=
Ng∑h=1
fhεp,h (12)
where fh is the volume fraction of each single crystal. Ng is the total number of singlecrystals. In the case of a random distribution, each single crystal possesses the same relativeweight, i. e. fh = 1/Ng.
4. Constitutive equations of single crystal
In this section, we will present the constitutive equations to describe the plastic behaviorof single crystals or mineral grains in geomaterials within the framework of small deformationtheory. As mentioned above, the plastic deformation of mineral grains is due to slippingalong weakness planes which are here assimilated to crystallographic directions and planes.For the FCC single crystal adopted here, there are 12 octahedral slip systems (four {1 1 1}planes and on each of these planes in three 〈1 0 0〉 directions) to be considered. A typicalslip system in a FCC unit cell is shown in Figure 4.
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Figure 3: Schematic description of the rotations through three Euler angles in the order ofϕ1, Φ and ϕ2
Figure 4: Schematic diagram of a representative crystallographic plane in a FCC unit cell.Its normal n and slip direction s are given under the crystal coordinate system oxyz
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4.1. General form of constitutive law
Within the framework of thermodynamics, many approaches have been employed toformulate constitutive equations for polycrystal materials. Here, we adopt the approachproposed by Meric (Meric et al., 1991). The formulation of constitutive law is separatedinto two operating levels: meso-level (single crystal) and micro-level (Crystallographic SlipSystem, CSS). For the meso-level, the key state variables are composed of the total straintensor ε and the plastic strain tensor εp. While for the micro-level, a couple of internalvariables, qα and Rα, α ∈ g, are introduced to define the inter-granular isotropic hardening.The set g is defined as g = {1, . . . , N}, N is the total number of slip systems in each singlecrystal. For FCC single crystal, N = 12.
The total strain tensor of single crystal is divided into an elastic part and plastic part:
ε = εe + εp (13)
The free energy Ψ is also decomposed into an elastic part Ψe corresponding to the elasticstrain energy and a plastic part part Ψp which is related to the plastic hardening work ineach slip system
ρΨ(ε, εp, q1, . . . , qN
)= ρΨe
(ε, εp
)+ ρΨp
(q1, . . . , qN
)(14)
With
ρΨe(ε, εp
)=
1
2
(ε− εp
): C :
(ε− εp
)ρΨp
(q1, . . . , qN
)=
1
2bQ
∑α∈g
∑β∈g
hαβqαqβ
where ρ is the volumetric mass. hαβ is an interaction matrix allowing the introduction ofcross influence of βth slip system on the hardening of αth slip system. If there is no crosshardening, hαβ is an identity matrix. Two model’s parameters b and Q are taken as constantsfor all slip systems. The fourth order elastic stiffness tensor C is expressed in the crystalcoordinate system. In the case of isotropic elasticity, one gets:
C = 3kK + 2µJ (15)
The state equations are deduced from the differentiation of free energy (14) with respectto the internal state variables:
σ = ρ∂Ψ
∂εe= C :
(ε− εp
)= C : εe (16)
Rα = ρ∂Ψ
∂qα= bQ
∑β∈g
hαβqβ (17)
Rα is the thermodynamic force associated with the internal variable qα, and represents thesize of yield surface along a slip system. To complete the formulation of constitutive model,it is needed to define a yield function fα and a plastic potential Fα.
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4.2. Yield criterion
According to the classical Schmid’s law widely used in metal materials, the plastic slip-ping occurs along a slip system when the magnitude of shear stress on that system reachesits yield stress. As an important difference with metal materials, the plastic slipping alongweakness planes of geomaterials is not only controlled by the shear stress but also stronglyinfluenced by the normal stress which is related to the hydrostatic part of stress tensor.This phenomenon, usually called pressure sensitivity of geomaterials, should be taken intoaccount in the local constitutive model. Therefore, the standard Schmid’s law for a typicalslip system α is modified as
fα(σ, Rα
)= |σ : mα|+ µfH (−σα
n) σ : Nα − (ταc + Rα) α ∈ g (18)
with
mα =1
2(sα ⊗ nα + nα ⊗ sα) Nα = nα ⊗ nα and σα
n = σ : Nα
nα and sα are two orthonormal vectors defining the normal of slip plane and the slip direction.ταc is the commonly referred Critical Resolved Shear Stress (CRSS). The second term in (18)
reflects the effect of normal stress on the plastic slipping kinetics. µf can be interpreted asthe frictional coefficient of all slip systems. H (•) is the Heaviside step function, which hasthe properties that H (x) = 1 if x > 0, otherwise H (x) = 0. Therefore, H (−σα
n) indicatesthat only a compressive normal stress has effect on the yield function fα.
4.3. Plastic flow rule and hardening law
Volumetric dilatancy is a common feature of plastic deformation in many geomaterials.In the case of polycrystal rocks, such a volumetric dilatancy can be related to plastic slippingalong weakness planes. Indeed, there are a number of asperities along weakness planes ofgeomaterials. Therefore, the slip systems are not the smooth planes. As a consequence, thetangential slip along the surfaces of rough weakness planes can induce a normal aperture,which is the microscopic origin of macroscopic volumetric dilatancy. In order to properlydescribe such a phenomenon and inspired by some previous works (Abdul-Latif et al., 1998),a non-associate plastic potential is proposed for each slip system and written in the followingform:
Fα(σ, Rα
)= |σ : mα|+ vF H (−σα
n) σ : Nα − (Rα − bqαRα) α ∈ g (19)
b characterizes the non-linearity of the local hardening. The parameter vF controls the rateof volumetric dilatancy and is related to the roughness degree of slip planes. Note that inthe case of associated plasticity, vF = uf .
The plastic flow rule and hardening law are derived from the plastic potential as follows:
εp =∑α∈g
λα ∂Fα
∂σ=
∑α∈g
λαmαsign(σ : mα) +∑α∈g
λαvF H (−σαn) Nα (20)
qα = −∑β∈g
λβ ∂F β
∂Rα= λα (1− bqα) (21)
10
where sign (•) is a signum function, which has the properties that sign (x) = 1 if x > 0,otherwise sign (x) = −1. λα is the plastic multiplier for the slip system α and can bedetermined by the Kuhn-Tucker condition (22) and consistency condition (23):
λα ≥ 0 fα ≤ 0 λαfα = 0 α ∈ g (22)
λαfα = 0 if fα = 0 α ∈ g (23)
Solving the ordinary differential equations (21) with the initial condition qα(0) = 0, onegets:
qα =1
b(1− exp (−bλα)) (24)
Substituting now (24) into (17), we finally obtain an integrated form of the hardening forcefor each slip system:
Rα = Q∑β∈g
hαβ
(1− exp
(−bλβ
))(25)
5. Computational aspects
In this section, computational aspects are discussed for numerical implementation of theconstitutive model both at meso-level and macro-level. The non-smooth multi-surface plas-ticity problem encountered at the meso-level is the key point for the stability and efficiencyof the implementation. Some previous works have been devoted to this problem, just tolist a few, (Simo and Hughes, 1998). However, most methods cannot be directly applied toFCC single crystal due to the strong inter-dependency between the slip systems. Such aninter-dependency results in a singularity of Jacobian matrix appeared in Newton-Raphsonmethod and the non-unique choice of active slip systems. Some mathematical methodsbased on the objective function minimization, e. g. Singular Value Decomposition (SVD)(Anand and Kothari, 1996) and perturbation techniques (Miehe and Schroder, 2001), havebeen proved to be efficient in solving this kind of difficulty in polycrystal models of metalmaterials. However, after making several attempts, we find that they are not fully efficientfor the constitutive models of geomaterials, mainly due to the pressure dependency and non-associated plastic potential. Therefore, in this paper, an iterative procedure is proposed todetermine a group of linear independent slip systems from several potential ones.
5.1. General issue
To simplify the notation and also to avoid the case that a negative resolved shear stressresults in a negative plastic multiplier, each slip system is split into two mirrored systems(Hutchinson, 1970). Based on that convention, the number of slip systems is doubled andthe dual slip systems α and α + 12 verify the relation:
α = {nα, sα} and α + 12 = {nα,−sα} (26)
The complete Miller indexes of 24 slip systems are given in Appendix A. The set of all slipsystems within FCC is now redefined as g?, g? = {1, . . . , 2N}.
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Both the yield function (6) and plastic potential (7) for each slip system can be rewrittenas:
fα(σ, Rα
)= σ : M f,α − (τα
c + Rα) α ∈ g? (27)
Fα(σ, Rα
)= σ : MF,α − (Rα − bqαRα) α ∈ g? (28)
withM f,α = ∂σf
α = mα + ufH (−σαn) Nα
MF,α = ∂σFα = mα + vF H (−σα
n) Nα
The slip systems satisfying the condition fα(σtrial, Rα
)> 0 are named as potentially
active systems. σtrial is the trial stress state calculated by freezing the plastic strain incurrent load increment. These slip systems will form the set JPact
JPact ={α ∈ g? | fα(σtrial, Rα) > 0
}(29)
the cardinality of JPact is NPact. Furthermore, if some potentially active systems also satisfythe consistency conditions (23), they will form the active systems JAct
JAct ={
α ∈ JPact | fα(σ, Rα) = 0 and fα(σ, Rα) = 0}
(30)
The cardinality of JAct is NAct. Accordingly, the non-active systems can be easily defined asJNonact = {α | g? − JAct}
Unlike the classical multi-surface plasticity problem, JAct generally differs from JPact here.The reason is that certain slip systems which overshot by the elastic trial stress may in factremain inactive due to the cross hardening effect from other active slip systems (Knockaertet al., 2000). A simple geometrical illustration about the intersection of two yield surfacesis given in Figure 5. The first case corresponding to Figure 5(a) is common in multi-surfaceplasticity problem, while the second case corresponding to Figure 5(b) is occasional, whicheasily occurs at the knee of a load-displacement curve (Anand and Kothari, 1996). Thedetailed mathematical derivation about these issues have been given by Simo (Simo andHughes, 1998).
5.2. Local stress update
To solve a non-linear problem, the increment method is preferred. The loading history[0, T ] is divided into a limited number of intervals or increments, i. e. [0, T ] =
∑∞k=1 [tk, tk+1].
Then starting from the initial conditions at t = tk, the solution at t = tk+1 is obtained byintegrating the rate form of constitutive equations within the time interval ∆t = tk+1 − tk.
With the newly estimated σk+1
from the interaction law (7), and the solution dependentstate variables εp
k, λα
k , JAct,k, α ∈ JAct,k, we aim to update all the solution dependent statevariables at the end of the current increment. A semi-implicit or cutting plane method(Simo and Hughes, 1998) is employed to discretize all the constitutive equations. Then thecorresponding set of non-linear algebra equations is solved with Newton-Raphson technique.After a group of ∆λα
k+1 is solved, the Kuhn-Tucker complementarity conditions (22) and
12
(a) (b)
Figure 5: Geometrical illustration of active slip system at a corner. a) f 1,trialn+1 > 0 and
f 2,trialn+1 > 0 (black solid line). After correction (red dot line), f 1
n+1 = 0 and f 2n+1 < 0. b)
f 1,trialn+1 < 0 and f 2,trial
n+1 > 0. After correction, f 1n+1 = 0 and f 2
n+1 = 0
the consistency condition (23) should be checked in two aspects: 1) ∆λαk+1 > 0, for ∀α ∈
JAct,k+1. Any slip system violates those conditions should be removed from JAct,k+1; 2)fα(σ
k+1, Rα
k+1) ≤ 0, ∀α ∈ JNonact,k+1. Any slip system can not satisfy those conditionsshould be added to JAct,k+1. This procedure should be repeated a number of times until thecomponents of JAct,k+1 are fixed.
To make the outline of the numerical algorithm more clear, a flowchart is given as follows:
1. Given the macroscopic trial stress Σ?
k+11 and the solution dependent state variables
at tk: εpk
at the meso-level and λαk , JAct,k, α ∈ JAct,k at the micro-level, determine the
trial stress and the set of potential active slip systems JPact,k+1
σtrial
k+1= Σ?
k+1− M?,e : εp
kwith M?,e = 3k(1− c)K + 2u(1− d)J.
JPact,k+1 ={
α ∈ g? | fα(σtrial
k+1, Rα
k+1
)> 0
}2. Check whether plastic deformation takes place or not
IF NPact = 0 THENSet ( •)k+1 = (•)k and GOTO 10
1Its definition is given in section 5.3
13
ELSEεp,(0)
k+1= εp
k
λα,(0)k+1 = λα
k
∆λα,(0)k+1 = 0
J(0)Act,k+1 = JAct,k
GOTO 3
END IF
3. Outer loop (active slip system set update): DO WHILE j ≤ jmax 2
IF NAct = 0 AND j = 0 THEN J(0)Act,k+1 = JPact,k+1, NAct = NPact
IF j = 1 AND NAct 6= 0 THEN J(1)Act,k+1 = ∅, NAct = 0
Reinitialize: εp,(0)k+1
= εpk,λ
α,(0)k+1 = λα
k , ∆λα,(0)k+1 = 0, α ∈ J
(j)Act,k+1, i = 0
Usually, for the initial estimation of the set of active slip systems, J(0)Act,k+1 = JAct,kis
preferred. However, if the current increment is just the first loading step where theplastic deformation occurs, i. e. JAct,k = ∅, then J
(0)Act,k+1 = JPact,k+1 is assumed. If the
fixed JAct,k in the last increment can not hold in the current increment, the Newton-Raphson iteration will start with the system which is the mostly violated. Other slipsystems will then be added or removed successively.
4. Inner loop (stress update with Newton-Raphson technique): DO WHILE i ≤ imaxEvaluate residuals of yield functions
Rα,(i)k+1 = fα
(σ(i)
k+1, R
α,(i)k+1
)α ∈ J
(j)Act,k+1
5. Check the convergence
IF∣∣∣Rα,(i)
k+1
∣∣∣ < Tol, ∀α ∈ J(j)Act,k+1. Tol is a specified tolerance here. GOTO 8
6. Obtain the increments of internal variables
[g]αβ,(i)k+1
[∆∆λ
β,(i)k+1
]= −R
α,(i)k+1
gβα,(i)k+1 = M f,β,(i)
k+1: M?,e : MF,α,(i)
k+1+ hβαbQ exp
(−bλ
α,(i)k+1
)When [g] is a singular matrix, additional loop is necessary to search a possible combi-
nation of linear independent slip systems from J(j)Act,k+1. The SVD method is preferred
to determine whether the matrix is singular or not (Press et al., 1996). And the searchprocedure is an iterative one. It is worth noting that both the cardinality and elementsof J
(j)Act,k+1may change here (J
(j)Act,k+1 will certainly not change if [g] is invertible). To
avoid the ambiguity and simplify the notation, J(j)Act,k+1 will be retained.
2Parameters controlling the maximum number of iteration. Usually imax = jmax = 100
14
7. Update increments of internal variable
∆λα,(i+1)k+1 = ∆λ
α,(i)k+1 + ∆∆λ
α,(i)k+1 α ∈ J
(j)Act,k+1
εp,(i+1)
k+1= εp
k+
∑β∈J
(j)Act,k+1
∆λβ,(i+1)k+1 MF,β,(i)
k+1
σ(i+1)
k+1= Σ?
k+1− M?,e : εp,(i+1)
k+1
Then i = i + 1 and END DO (Inner Loop)
8. Active slip system set update: Aspect–IIF ∆λβ
k+1 < 0, ∀β ∈ J(j)Act,k+1 THEN
J(j+1)Act,k+1 =
{J
(j)Act,k+1/
(α = min
(fβ,trial
k+1
), β ∈ J
(j)Act,k+1
)}END IFj = j + 1 and GOTO 3
9. Active slip system set update:Aspect–IIIF fβ
k+1 > 0, ∀β ∈ J(j)Nonact,k+1 THEN
J(j+1)Act,k+1 =
{J
(j)Act,k+1 ∪
(α = max
(fβ,
k+1
), β ∈ J
(j)Nonact,k+1
)}END IF
10. Exit the subroutine and go to next single crystal
5.3. Macroscopic stress update
In view of numerical implementation of the proposed model in a standard finite elementcode, consider a strain (displacement) controlled loading process. At the increment k + 1,a uniform strain increment ∆E
k+1is given and applied to the REV. The objective is now
to update the macroscopic stress tensor Σ at the end of the current increment with theconverged macroscopic quantities Σ
k, E
kand Ep
k. An iterative procedure is needed since
the local plastic strain εp is a function of Σ and Ep, while on the other hand, Ep is also afunction of εp. The fixed point method is adopted and explained as follows: A trial overallplastic strain Ep
k+1= Ep
k(i. e. ∆Ep
k+1= 0) is firstly assumed. With the corresponding
trial stress Σk+1
, they are substituted into the right-hand side of the interaction law (7).Making use of the subroutine introduced in section 5.2, local plastic strain of each singlecrystal εp
k+1is determined. Then according to the average process described in section 3.3,
we got a new macroscopic plastic strain Ep
k+1. The most recently estimated Ep
k+1should
be compared to the trial one. This process will be continued until the difference betweentwo consecutive plastic strain tensors is within the specified tolerance. The outline of thenumerical algorithm is described as follows:
15
1. At the loading step t = tk, the variables listed below are known. These variables needto be updated at the step t = tk+1 for the prescribed overall strain increment ∆E
k+1Macroscopic variables: Σ
kMesoscopic variables: εp
kMicroscopic variables: λα
k , JAct,k, α ∈ JAct,k
2. Initialize i = 1, Ep,(i−1)
k+1= Ep
k=
⟨εp
k
⟩, Σ(i−1)
k+1= C :
(E
k+ ∆E
k+1− Ep,(i−1)
k+1
),
Σ?,(i−1)
k+1= Σ(i−1)
k+1+ M?,e : Ep,(i−1)
k+1.
3. Macroscopic plastic strain Ep loop: DO WHILE i ≤ imax
4. Loop for each single crystal: DOInput variables: εp
k, λα
k , JAct,k, α ∈ JAct,k and Σ?,(i−1)
k+1Call the stress update subroutine in section 5.2Output variables: εp,(i)
k+1, λ
α,(i)k+1 , J
(i)Act,k+1, α ∈ J
(i)Act,k+1
END DO (loop for each single crystal)
5. Calculate macroscopic plastic strain Ep based on the most recently estimated εp,(i)k+1
Ep,(i)
k+1=
⟨εp,(i)
k+1
⟩Verify the consistency of initialization
Res = Ep,(i)
k+1− Ep,(i−1)
k+1
IF ‖Res‖ < Tol, THEN
GOTO 7
ELSEΣ(i)
k+1= C :
(E
k+ ∆E
k+1− Ep,(i)
k+1
)Σ?,(i)
k+1= Σ(i)
k+1+ M?,e : Ep,(i)
k+1
Ep,(i−1)
k+1= Ep,(i)
k+1, i = i + 1
ENDIF
6. ENDDO (macroscopic plastic strain Ep loop)
7. After the convergence, the macroscopic stress is updated
Σk+1
= C :(E
k+ ∆E
k+1− Ep
k+1
)The solution dependent state variables: 1) Mesoscopic variables: εp
k+1; 2) Microscopic
variables: λαk+1, JAct,k+1, α ∈ JAct,k+1 are updated and passed to the next increment.
16
6. Numerical simulations
The numerical algorithm described in the previous section can be implemented in astandard finite element software, for instance ABAQUS, without any particular difficulty.Each integration point is regarded as a polycrystal material. Based on previous works(Pilvin, 1990) and preliminary simulations, a polycrystal REV with 40 selected orientationsis considered in the following simulations. The pole diagrams of the 40 grains are given inFigure 6.
For each single crystal, the following solution dependent state variables are considered:1) Mesoscopic variables: εp; 2) Microscopic variables: λα and JAct. In the case of a FCCsingle crystal with the 24 slip systems listed in Table A.1, there are 43 (6+24+13) variablesfor each integration point. Since two mirrored systems can not slip simultaneously, thenthe reserved space for JAct to store the index of slip system is reduced by half. And thethirteenth place in JAct stores the total number of slip systems (NAct) for each single crystal.For the polycrystal composed of 40 single crystals, the total number of solution dependentstate variables is 1720 (43× 40). These variables should be passed at the beginning of eachincrement and must be updated at the end.
Figure 6: 〈0 0 1〉, 〈1 1 0〉 and 〈1 1 1〉 pole figures corresponding to the 40 discrete grains withcubic symmetry
6.1. Simulation of conventional triaxial compression tests
The predictive capacity of the proposed model is now checked through the reproductionof laboratory tests on a quasi-brittle rock, Lac du Bonnet granite, which has been widelystudied in the context of the Underground Research Laboratory (URL) for nuclear wastestorage in Canada (Shao et al., 1999; Martino and Chandler, 2004; Shao et al., 2006).The experimental data include different kinds of loading paths, e. g. conventional triaxialcompression, proportional triaxial compression and lateral extension (Martin et al., 1997).
Before performing the simulation, the model’s parameters are firstly determined. Intotal, 7 material parameters are necessary to characterize the mechanical response of singlecrystal: 1) Elastic constants E and ν; 2) Parameters reflecting the effect of normal stress uf
and the roughness degree of slip planes vF ; 3) The CRSS parameter τc and two hardening
17
parameters b and Q. Due to the assumptions adopted on the symmetry and morphologyof polycrystal, some parameters of single crystal are related to those of polycrystal. Themacroscopic elastic constants, which can be obtained from uniaxial compression or triaxialcompression tests, are therefore equal to the microscopic ones. The macroscopic initial yieldstress can also be identified from laboratory data (Wang and Tonon, 2009). Hutchinsonprovided a simple expression (31), giving the relationship between the microscopic andmacroscopic initial yield stresses of a polycrystal with a cubic crystal structure (Hutchinson,1970):
1
2|σ1 − σ3| =
(3
2 (ρ22 + ρ2
3)
) 12
τc = τ c (31)
where
ρ2 =C11 − C12
2µ (1− d) + d (C11 − C12)with ρ3 =
C44
µ (1− d) + dC44
The parameter d is defined in (5). σ1 and σ3 are the major and minor macroscopic principalstresses. τc and τ c are the microscopic and macroscopic initial yield stresses, respectively.C11, C22, and C33 are the components of elastic stiffness tensor. k and µ are defined by (32)and (33)
For an isotropic single crystal, 2C44 = C11−C12, it is easy to obtain ρ2 = ρ3 = 1 and τc = τ c.The other 4 parameters uf , vf , b and Q are determined from an optimization process
based on the comparison between model’s predictions and experimental data, as shown inFigure 7. The loading axis is in the 3-direction. E33 and E11 represent the axial and lateralstrains, respectively. Since the tests are performed on cylinder samples, one has E22 = E11.The volumetric strain is calculated by Ev = E33 + 2E11. Finally, the representative valuesof parameters used in numerical simulations are given in Table 1.
Table 1: Simulation parameters for Lac du Bonnet granite
E (MPa) ν µf vf τc (MPa) hαβ b Q (MPa)6.8× 104 0.21 0.4 0.6 35 1.0 100 40
Note that in all numerical results and experimental data presented in this paper, weadopt the following sign convention: tensile stresses and strains are given in positive values.The results presented in Figure 7 clearly show that the proposed model correctly describesthe main features of mechanical behaviors of granite and it is indispensable to considerthe volumetric dilatency for this class of materials. A comparison between the associated(uf = vf ) and non-associated plastic flow rules is shown in Figure 8. It is obvious that thenon-associated model better describes the mechanical response of granite than the associatedone. Therefore, all the following simulations are performed by using the non-associated flowrule.
18
Figure 7: Comparison between experiment data and model’s predictions for a triaxial com-pression test with a confining pressure of 10 MPa
Figure 8: Comparisons between experiment data and model’s predictions with associatedand non-associated flow rule
19
The comparisons between experimental data and numerical results for both uniaxial andconventional triaxial compression tests are given in Figure 9 to Figure 12. Further, twoother typical loading paths are also simulated: proportional triaxial compression test andlateral extension test. These simulations are realized to provide a validation of the modelin different loading conditions.
For the proportional compression test, the ratio between axial stress and lateral stressis kept constant throughout the loading process. For the results in Figure 13, one hasΣ33
Σ22= Σ33
Σ11= 10. For the lateral extension test shown in Figure 14, the whole loading
process is divided into three steps. Firstly, the sample is subjected to a hydrostatic stress,Σ11 = Σ22 = Σ33 = −60MPa. Secondly, the axial stress Σ33 is increased to a prescribedvalue 160MPa while the lateral stresses Σ11 = Σ22 remain unchanged. Lastly, the lateralstresses Σ11 = Σ22 are decreased from −60MPa to 0MPa while the axial stress Σ33 remainsconstant at −160MPa. Usually, only the strain-stress relations in the last step are presented.
It is worth noting that the proposed micro-mechanical model provides a better predictionfor triaxial compression than for uniaxial compression. Moreover, it seems that the higherthe confining pressure, the better prediction the model provides. Therefore, the proposedmode is more suitable to describe ductile behaviors than brittle ones. Indeed, under lowconfining pressure, damage due to debonding of grain interfaces becomes an important issuein mechanical behaviors of brittle rocks such as granite. This issue is not considered in thepresent work and will be investigated in our future works. Finally, the elastic constantsE and ν remain unchanged during all the simulations. However, in practice, the elasticmodulus of many rocks generally increases with confining pressure. Taking into accountsuch a variation of elastic modulus with confining pressure would improve numerical results(Xie et al., 2011).
20
Figure 9: Comparison between experiment data and polycrystal model predictions–Uniaxialcompression
Figure 10: Comparison between experiment data and polycrystal model predictions–Triaxialcompression with confining pressure 2MPa
21
Figure 11: Comparison between experiment data and polycrystal model predictions–Triaxialcompression with confining pressure 20MPa
Figure 12: Comparison between experiment data and polycrystal model predictions–Triaxialcompression with confining pressure 40MPa
22
Figure 13: Comparison between experiment data and polycrystal model predictions for pro-portional triaxial compression test
Figure 14: Comparison between experiment data and polycrystal model predictions for alateral extension test
23
6.2. Simulation of true triaxial compression tests
The simulation of conventional compression tests in section 6.1 provide a first validationof the model. To check the ability of the proposed model in predicting mechanical responsesof granite for more complex loading paths, further validations are needed. Now, the proposedmodel will be applied to reproduce true triaxial compression tests on Westerly granite (USA).Such tests are performed on cubic specimens by independently controlling the three principalstresses (Haimson and Chang, 2000).
The elastic constants E and ν, and the CRSS τc are obtained according to the parameteridentification procedure introduced in section 6.1. Their macroscopic values can be found in(Lockner, 1995, 1998; Homand and Shao, 2000). The other 4 parameters are also determinedby numerical optimization of test data. The loading axis is still in the 3-direction. E33
represents the axial strain and E11, E22 the lateral strains. The volumetric strain is calculatedby Ev = E11 + E22 + E33. Finally, the values of parameters for numerical simulations aregiven in Table. 2
Table 2: Simulation parameters for Westerly granite
E (MPa) ν µf vf τc (MPa) hαβ b Q (MPa)7.5× 104 0.21 0.5 0.75 45 1.0 900 50
The loading process of a true triaxial compression test is divided into 3 steps: 1) Thesample is subjected to the hydrostatic stress, i. e. Σ11 = Σ22 = Σ33 = −60MPa; 2) The Σ11
being kept constant (−60MPa) while Σ22 = Σ33 is respectively increased to the prescribedvalues −60MPa, −113MPa, −180MPa, −249MPa; 3) Σ11 and Σ22 remain unchanged andΣ33 is increased to the prescribed values −747MPa, −822MPa, −860MPa, −861MPa. Com-parisons between numerical results and experimental data are presented in Figure 15 toFigure 18. One can see that the proposed model well describes the main features of me-chanical responses of granite in true triaxial compression.
It is useful to note that the numerical simulations presented here are stopped at thelevel of experimental peak stresses in different tests. The macroscopic failure related to thelocalization of plastic strain and damage is not considered at this stage of work. In manyrocks, the macroscopic failure is generated by the coalescence of micro-cracks. In the case ofpolycrystal rocks, the growth of micro-cracks is mainly related to the debonding of cementedinterfaces between grains, leading to the degradation of material stiffness and strength. Thisimportant feature is being considered in our ongoing works.
24
Figure 15: Comparison between experiment data and model prediction–True triaxial com-pression with Σ33 = −747MPa, Σ22 = −60MPa, Σ11 = −60MPa
Figure 16: Comparison between experiment data and model predictions–True triaxial com-pression with Σ33 = −822MPa, Σ22 = −113MPa, Σ11 = −60MPa
25
Figure 17: Comparison between experiment data and model predictions–True triaxial com-pression with Σ33 = −860MPa, Σ22 = −180MPa, Σ11 = −60MPa
Figure 18: Comparison between experiment data and model predictions–True triaxial com-pression with Σ33 = −861MPa, Σ22 = −249MPa, Σ11 = −60MPa
26
7. Conclusion
In this paper, a micro-mechanical model is proposed for the description of elastic-plasticbehaviors of a class of rocks with a polycrystalline microstructure. The model is formulatedusing a nonlinear homogenization procedure for heterogeneous materials. The classical yieldcriterion and plastic potential for single crystal of metal materials are modified in orderto account for the pressure sensitivity and volumetric dilatency. Accordingly, the classicalinteraction law used in KBW model is also extended by considering these specific features.An improved numerical algorithm is proposed, using the elastic predictor and plastic cor-rector scheme, for numerical implementation of constitutive equations with multiple yieldsurfaces. The efficiency of this algorithm, particularly suitable for polycrystalline materials,is demonstrated. The validity of the proposed model is verified by comparing numericalresults with experimental data on different loading paths. The future work will include theextension of the model to time-dependent behavior of geomaterials, the use of more rigorousinteraction laws and the consideration of interface damage between mineral grains.
Acknowledgements
The present work is jointly supported by the China Scholarship Council for the firstauthor and the Chinese state 973 program with the grant 2011CB013504.
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Appendix A. Complete information for 24 slip systems in FCC unit cell
