Homogenization of Cohesive-Frictional StrengthProperties of Porous Composites: Linear

Comparison Composite ApproachJ. Alberto Ortega1; Benjamin Gathier2; and Franz-Josef Ulm, M.ASCE3

Abstract: This paper introduces a novel micromechanics method for strength homogenization of cohesive-frictional porous composites.Within a yield design formulation, the inherently nonlinear homogenization problem associated with strength upscaling is treated by the linearcomparison composite (LCC) theory, which resolves the strength properties of the heterogeneous medium by estimating the effective proper-ties of a suitable linear comparison composite with similar underlying microstructure. The LCC homogenization method rationalizes thedevelopment of strength criteria for cohesive-frictional materials affected by the presence of porosity and rigidlike inclusions. Modelingresults for benchmark microstructures improve existing micromechanics formulations by allowing the consideration of the complete rangeof frictional behaviors for the Drucker-Prager solid and by lifting the restriction on the incompressibility of the solid for the estimation ofmorphology factors that describe the mechanical interaction between material phases. The LCC strength homogenization is implemented in amultiscale thought model applicable to geomaterials, which serves as a generalized framework for quantitative assessment of effects ofmaterial composition, grain-scale properties, microstructure, and interface conditions on the overall strength of the porous composite.DOI: 10.1061/(ASCE)NM.2153-5477.0000025. © 2011 American Society of Civil Engineers.

CE Database subject headings: Strength; Composite materials; Material properties; Friction; Micromechanics.

Author keywords: Strength homogenization; Linear comparison composite; Cohesive-frictional material.

Introduction

Geomaterials such as rocks, soils, and concrete have recentlyemerged as crucial components in the development of energysolutions (e.g., gas shale, enhanced geothermal systems), environ-mental management alternatives (e.g., carbon sequestration), andsustainable infrastructure (e.g., green concrete). These engineeringchallenges require predictive models that consider the multiscaleand compositionally diverse nature of these materials to adequatelyreproduce their mechanical behavior. Whereas numerous effortsin multiscale modeling of poroelastic behavior, for example,have been accomplished, the strength modeling of compositematerials remains an active research field. The current knowledgeof strength behaviors has been exclusively derived from macro-scopic testing techniques, and most theoretical models cannotrigorously link the observed overall responses to grain-scaleproperties. Hence, conventional criteria designed to capture thecohesive-frictional strength of geomaterials (e.g., Coulomb,Drucker-Prager, Cam-Clay, Hoek-Brown) must be calibrated foreach investigated material system. Developments in continuum

micromechanics have enabled the prediction of macroscopicstrength criteria for composite materials from considerations ofthe strength behaviors of material constituents, effects of hetero-geneities, and microstructure. In the field of strength homogeniza-tion, most contributions have treated composite materials withpurely cohesive strength attributes (e.g., Lee and Mear 1992a, b;Suquet 1997). Recently, the micromechanics approach hasbeen extended to modeling cohesive-frictional composites.Dormieux and coworkers (Lemarchand et al. 2002; Barthélémyand Dormieux 2003; Barthélémy and Dormieux 2004; Dormieuxet al. 2006; Maghous et al. 2009) have derived predictions ofstrength domains for cohesive-frictional solids intermixed withporosity or reinforced with rigid grains. The methodology relieson the mathematical similitude between the determination ofthe strength-compatible macroscopic stress states based on limitanalysis and the solution for the effective behavior of a compositewith a fictitious nonlinear viscous solid phase. The effective strain-rate method has been implemented for assessing cohesive-frictional properties of porous materials from indentation hardnessmeasurements (Ganneau et al. 2006; Cariou et al. 2008) andthe modeling of gravel-reinforced sands (Barthélémy andDormieux 2004).

This work develops a strength homogenization method forcohesive-frictional composite materials based on the applicationof the linear comparison composite (LCC) theory (PonteCastañeda 1991, 1992, 1996, 2002). In the proposed framework,the nonlinear problem of upscaling strength properties for a hetero-geneous material is treated by modeling a suitable linearthermoelastic comparison composite with similar underlyingmicrostructure. The LCC strength homogenization method isimplemented in a multiscale thought model for geomaterials.The multiscale structure follows a two-step upscaling scheme, inwhich a porous solid exhibiting cohesive-frictional properties is

1Postdoctoral Associate, Dept. of Civil and Environmental Engineering,Massachusetts Institute of Technology, Cambridge, MA 02139. E-mail:ortega@mit.edu

2Research Assistant, Dept. of Civil and Environmental Engineering,Massachusetts Institute of Technology, Cambridge, MA 02139.

3George Macomber Professor, Dept. of Civil and EnvironmentalEngineering, Massachusetts Institute of Technology, Cambridge, MA02139 (corresponding author). E-mail: ulm@mit.edu

Note. This manuscript was submitted on April 10, 2010; approved onSeptember 14, 2010; published online on July 29, 2010. Discussion periodopen until August 1, 2011; separate discussions must be submitted for in-dividual papers. This paper is part of the Journal of Nanomechanics andMicromechanics, Vol. 1, No. 1, March 1, 2011. ©ASCE, ISSN 2153-5434/2011/1-11–23/$25.00.

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intermixed with rigid inclusions. The micromechanics solutions,which are shown to improve on existing formulations, enablethe investigation of the effects of composition, microstructure, solidstrength properties, and interface conditions on the overall macro-scopic strength. By way of application, the model is employed forthe back-analysis of strength properties for the load-bearing clayphase in shale materials.

Elements of Strength Homogenization

Yield Design Framework

The problem of strength homogenization is framed within the yielddesign approach, which relates the strength capacity of a materialsystem in response to an applied load to the state of plastic collapse(Suquet 1982; de Buhan 1986; Salençon 1990). As a consequence,the supplied external work is entirely dissipated in the material bulk.Consider the strength behavior of a material phase within a macro-scopic composite. At plastic collapse, the microscopic stress σðzÞ isassociated with the maximum dissipation capacity of the materialphase that is defined by the support function

πiðdÞ ¼ supσ∈Gi

fσ∶dg ð1Þ

where dðvÞ is the microscopic strain rate corresponding to the veloc-ity field v, and sup is the supremum, or least-upper bound of the setGi. The microscopic stress is on the boundary of the local strengthdomain Gi, which establishes the set of strength-compatible statescharacterized by a convex criterion F iðσÞ (Ulm and Coussy2003). At the scale of the representative elementary volume (rev)of composite material, the homogenization problem is expressedby the following set of equations describing the strength responseof the material system subjected to uniform strain-rate boundaryconditions and the static admissibility of the stress-field solutionformulated in strain rates:

divσ ¼ 0 ðΩÞ ð2a Þ

σ ¼ ∂πiðdÞ∂d ðΩiÞ ð2b Þ

d ¼ 12ðgradvþ tgradvÞ ðΩÞ ð2c Þ

vðzÞ ¼ D · z ð∂ΩÞ ð2d ÞThe condition σ ¼ 0 must be added if the heterogeneity is a porespace. The application of the Hill lemma provides a link betweenthe microscopic dissipation function and its counterpart at the scaleof the rev

ΠhomðDÞ ¼ supΣ∈Ghom

fΣ∶Dg ¼ infv0∈VðDÞπðz;dðv

0ÞÞ ð3Þ

where the macroscopic stress Σ, which satisfies the averaging ruleΣ ¼ σðzÞ, is situated at the boundary of the macroscopic strengthdomain Ghom, inf is the infimum, or greatest lower bound ofthe set VðDÞ, and VðDÞ is the set of kinematically admissiblemicroscopic velocity fields v0ðzÞ for a given macroscopic strain-ratetensor D

VðDÞ ¼ fv0; v0ðzÞ ¼ D · zð∀ z ∈ ∂ΩÞg ð4ÞThe focus of the yield design approach is the evaluation of themacroscopic support function ΠhomðDÞ; and by applying the dual

definition of the macroscopic strength domain Ghom, the determina-tionof themacroscopic stress at the boundary∂Ghom of the domain ofstrength-compatible stress states

Σ ¼ ∂ΠhomðDÞ∂D ð5Þ

Linear Comparison Composite Approach

The constitutive behavior in Eq. (2b), defined by the plastic dissi-pation capacity of the solid phase, introduces a nonlinearity in theformulation of the boundary value problem. Being the supportfunction πiðdÞ, a homogeneous function of degree one, the imple-mentation of yield design in strength homogenization prevents theuse of classical linear homogenization techniques. The nonlinearhomogenization of strength properties is pursued in this workthrough implementing the LCC theory developed by PonteCastañeda (1991, 1992, 1996, 2002). The LCC method estimatesthe behavior of the nonlinear composite in terms of a suitablychosen linear comparison composite with a similar underlyingmicrostructure. In particular, the second-order method (PonteCastañeda 2002), which makes use of more general types of linearcomparison composites, is employed. Reformulating the varia-tional problem Eq. (3) based on the LCC method begins withdefining a linear thermoelastic comparison composite with piece-wise-constant strain-rate energy

ψðz;dÞ ¼Xi

χðzÞψiðdÞ ð6Þ

where χðzÞ is the characteristic function (χ ¼ 1 if z ∈ Ωi; χ ¼ 0 ifz ∉Ωi), and ψiðdÞ is the strain-rate energy density from which themicroscopic stress derives

ψiðdÞ ¼ 12d∶Ci∶dþ τi∶d⟶σ ¼ ∂ψiðdÞ

∂d ¼ Ci∶dþ τi ð7Þ

where Ci is a positive-definite modulus tensor and τi is a prestress.Noting the lower-than-quadratic character of the dissipationfunction πðz;dÞ compared with the strain-rate energy ψðz;dÞ,the function ðψ� πÞ is bounded from below. Applying the classicalinequality infxff ðxÞ þ gðxÞg ≥ infxff ðxÞg þ infxfgðxÞg to thefunction ψ ¼ π þ ðψ� πÞ yields the following relation (PonteCastañeda 2002):

infv0∈VðDÞψðz;dðv

0ÞÞ ≥ infv0∈VðDÞπðz; dðv

0ÞÞ

þ infv0∈VðDÞψðz;dðv

0ÞÞ � πðz;dðv0ÞÞ ð8Þ

The left-hand side of Eq. (8) represents the effective macroscopicstrain-rate energy of the linear comparison composite

ΨðDÞ ¼ infv0∈VðDÞψðz; dðv

0ÞÞ ð9Þ

The first term on the right-hand side of Eq. (8) is readily recognizedas the macroscopic dissipation function, as in Eq. (3). The secondterm on the right-hand side of Eq. (8) can be reformulated byrelaxing the constraint on the kinematic boundary condition

infv0∈VðDÞψðz;dðv

0ÞÞ � πðz;dðv0ÞÞ

≥ infv0ψðz;dðv0ÞÞ � πðz;dðv0ÞÞ ≥ �

Xi

f iΥi ð10Þ

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where function Υi is constant in each phase of volumefraction f i

Υi ¼ supdfπiðdÞ � ψiðdÞg ð11Þ

The latter expression measures the nonlinearity of the compositematerial and contains the information about local strength domainsthrough the material dissipation function πiðdÞ. RearrangingEq. (8) and using Eqs. (3), (9), and (10) delivers an expressionfor an upper bound of the macroscopic dissipation capacity ofthe comparison composite material

ΠhomðDÞ ≤ infCi;τi

�ΨðDÞ þ

Xi

f iΥi

�ð12Þ

The objective of the LCC method is to determine the modulusand prestress parameters ðCi; τiÞ of the comparison compositethat lead to the lowest possible upper bound for Πhom.However, preserving an upper-bound status can prove to bedifficult for some applications (Ponte Castañeda 2002). Instead,it is of interest to consider a generalized version of Eq. (12)by replacing the extremal points by stationary points. Theresulting estimates are then stationary estimates and not boundsin general

~ΠhomðDÞ ¼ statCi;τi

�ΨðDÞ þ

Xi

f iΥi

�ð13Þ

where the nonlinearity function [Eq. (11)] is redefinedaccordingly

Υi ¼ statdfπiðdÞ � ψiðdÞg ð14Þ

The stationary operation (stat) involves solving theset of equations obtained by setting the partial derivatives of theargument with respect to the applicable variables equal to zero.The procedure to implement the LCC methodology to deliveran estimate of the macroscopic strength criterion follows thesesteps:1. Compute the expression for the macroscopic strain-rate energy

ΨðDÞ of the linear comparison composite. Results from linearmicromechanics are used to furnish the macroscopic strain-rate energy.

2. Compute the function Υi for each phase. This expression mea-sures the nonlinearity of the original material and contains theinformation about local strength domains through the materialdissipation function πiðdÞ.

3. Generate the stationary expressions Eqs. (13) and (14) andsolve the system in terms of Ci, τi.

4. Use the estimated dissipation capacity ~ΠhomðDÞ to derive themacroscopic strength domain through the yield designdefinition Eq. (5).

Multiscale Strength Upscaling

Implementing the LCC methodology requires revisiting twoclassical modeling problems in strength homogenization: a poroussolid (e.g., porous clay fabric in shale) and a solid intermixedwith rigidlike inclusions (e.g., clinker grains embedded in cementpaste). These two reference microstructures are incorporatedinto a generalized multiscale structure thought model shownschematically in Fig. 1. The macroscopic material system iscomposed of three phases: a cohesive-frictional solid, rigid inclu-sions, and porosity, which follow a volumetric partitioning of theform

Macroscale: f s þ f inc þ ϕ0 ¼ 1 ð15Þ

where f s is the solid volume fraction, f inc is the inclusionvolume fraction, and ϕ0 is the total porosity. The characteristicsize of the rigid inclusions is assumed to be much larger thanthe pores and agglomerates of particles forming the solidphase. Based on the scale separability in continuum micro-mechanics (Zaoui 2002), it is appropriate to first determine thehomogenized response of the porous solid composite. Conse-quently, the volumetric description at the mesoscale involvesthe solid packing density η and the (micro)porosity φ0 ¼ ϕ0=ð1� f incÞ such that

Mesoscale: η þ φ0 ¼ 1 ð16Þ

The multiscale material in Fig. 1 serves as a reference fordeveloping the strength homogenization model. The LCCmethodology will be applied in the derivation of thestrength criteria for the cohesive-frictional porous solid (Level I)and for the porous solid–rigid inclusion composite (Level II).The model development begins by establishing the form ofthe strength properties for the solid phase at the microscale(Level 0).

Level 0: Cohesive-Frictional Solid

The pressure-sensitive behavior of the solid phase is characterizedby the Drucker-Prager strength criterion, which depends ontwo invariants of the stress sustained by the material: the meanstress σm ¼ ð1=3ÞI1 ¼ ð1=3Þ tr ðσÞ, and the deviatoric stresss ¼ σ � σm1

F sðσÞ ¼ σd þ ασm � cs ≤ 0 ð17Þ

where σd ¼ffiffiffiffiffiJ2

p ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1=2Þs∶spand α, cs are the Drucker-Prager

friction coefficient and cohesion describing the intrinsic strengthof the solid. The Drucker-Prager friction coefficient is limited toα <

ffiffiffi3

p=2, corresponding to a Mohr-Coulomb friction angle of

90° (Desrues 2002). The support function associated with the dualdefinition of the Drucker-Prager strength domain [Eq. (17)] is givenby (Salençon 1990)

πsðdÞ ¼� ðcs=αÞdv if dv ≥ 2αdd∞ else

ð18Þ

where dv ¼ tr ðdÞ and dd ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1=2Þδ∶δp

are the invariantsof the strain-rate tensor d ¼ δþ ð1=3Þdv1. It is convenient toformulate Eqs. (17) and (18) as strictly convex functions inview of the forthcoming application of the LCC approach to

Cohesive-frictionalsolid

Porous solidPorous solid –

inclusion composite

Microscale(Level 0)

Mesoscale(Level I)

Macroscale(Level II)

snoisulcni digirserop

Fig. 1. Multiscale structure thought model applicable togeomaterials

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strength homogenization. A regularized strength criterion thatcircumvents the point of singularity at (σd ¼ 0, σm ¼ cs=α) isconsidered

F ðσÞ ¼ 1� ðσm � S0Þ2A

þ σ2d

B≤ 0 ð19Þ

The set of hyperbolas in Eq. (19) defined by the scalar quantities A,B, and S0 (which do not posses specific physical meanings),allow retrieval of the Drucker-Prager criterion [Eq. (17)] assumedfor the cohesive-frictional solid by specifying the followingconditions:

B ¼ α2A; S0 ¼cs

α; A → 0 ð20Þ

The regularized dual definition of the strength domain in terms ofthe support function πðdÞ is classically obtained through the con-ditions of associated plasticity (Ulm and Coussy 2003), leading tothe following expression:

πðdÞ ¼ πðdv; ddÞ ¼ S0dv �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAd2v � 4Bd2d

qsubject to ðAd2v � 4Bd2dÞ ≥ 0

ð21Þ

The Drucker-Prager support function [Eq. (18)] is retrieved whenEq. (21) is subjected to the set of conditions in Eq. (20). Theregularized support function [Eq. (21)] will be employed in thesubsequent derivations of homogenized strength criteria, whichare ultimately specified for the case of the Drucker-Prager solidphase by implementing the conditions in Eq. (20).

Level I: Porous Solid Composite

The first application of the LCC method is to establish the strengthbehavior of the porous medium composed of a Drucker-Prager-type solid and a pore space. The procedure outlined in the“Linear Comparison Composite Approach” section is followedhereafter.

Strain-Rate Energy FunctionThe first step in the LCC method consists of determining thestrain-rate energy function ΨðDÞ of the linear thermoelastic com-parison composite. For this, consider a continuous description ofthe microscopic stress field within the rev of the porous solid Ω

σðzÞ ¼ CðzÞ: dðzÞ þ τðzÞ ð∀ z ∈ ΩÞ ð22Þ

where the microscopic modulus tensor CðzÞ and the prestress τðzÞobserve the following spatial distributions within the solid-porecomposite:

CðzÞ ¼�Cs ¼ 3ksJþ 2gsK ðΩsÞ0 ðΩpÞ τðzÞ ¼

�τ1 ðΩsÞ0 ðΩpÞ

ð23Þand Ωs ¼ ηΩ, Ωp ¼ φ0Ω ¼ ð1� ηÞΩ are the domains occupiedby the solid and pore space, respectively. In addition, Jijkl ¼ð1=3ÞðδijδklÞ, K ¼ I� J are tensor projections, and I is thefourth-order identity tensor. The proposed distribution inEq. (23) assumes drained conditions (i.e., a nonpressurized porespace). Using classical results of linear micromechanics, the corre-sponding macroscopic stress equation of state is

Σ ¼ CIhom: Dþ TI ð24Þ

where CIhom and TI are the macroscopic modulus tensor and

prestress

CIhom ¼ CðzÞ∶AðzÞ ¼ ηCs∶�As ¼ 3kIhomJþ 2gIhomK ð25a Þ

TI ¼ τðzÞ∶AðzÞ ¼ ητ1∶�As ¼ τ1∶ðCsÞ�1∶CIhom ¼ τ

kIhomks

1 ð25b Þ

with

kIhom ¼ ηksJ∶�As ¼ gsKI

�ks

gs; η�;

gIhom ¼ ηgsK∶�As ¼ gsMI

�ks

gs; η� ð26Þ

The term �As is the fourth-order strain (rate) localization tensoraveraged over the volume of the solid phase. From dimen-sional analysis, the micromechanics results for kIhom, gIhom inEq. (26) are expressed in terms of the inclusion morphologyfactors KI , MI . These dimensionless functions, which willsubsequently be specified, depend on the bulk-to-shear modulusratio ks=gs, the solid packing density η, and the microstructuralfeatures of the solid-pore composite. The strain-rate energy func-tion ΨðDÞ for a two-phase composite is obtained from classicalmicromechanics of thermoelasticity (Levin 1967; Laws 1973),which adapted for the isotropic composite defined in Eq. (23)simplifies to

ΨIðDv;DdÞ ¼12kIhomD

2v þ 2gIhomD

2d þ

kIhomks

τDv

þ 12ks

�kIhomks

� η�τ2

¼ 12gsKID2

v þ 2gsMID2d þ

gs

ksτKIDv

þ 12ks

�gs

ksKI � η

�τ2 ð27Þ

where Dv ¼ trðDÞ and Dd ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1=2ÞΔΔ∶ΔΔp

are the macroscopicstrain rate invariants, with ΔΔ ¼ D� ð1=3ÞDv1.

Measure of NonlinearityThe second step consists in determining the Υs function for thesolid, which provides a measure of the nonlinearity for the solidphase compared with the linear comparison composite. The valueof the function Υp for the pore domain is zero. The evaluation of Υs

[Eq. (14)] requires the expressions for the regularized supportfunction [Eq. (21)] and the strain-rate energy of the solid[Eq. (7)]. The application of the stationarity condition to Υs withrespect to the microscopic strain-rate invariants delivers the follow-ing relations:

∂Υs

∂dv ¼ S0 �Advffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Ad2v � 4Bd2dp � ksdv � τ ¼ 0 ð28a Þ

∂Υs

∂dd ¼ 4BddffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAd2v � 4Bd2d

p � 4gsdd ¼ 0 ð28b Þ

The consideration of the prestress ensures that the microscopicmoduli ks, gs are positive by manipulating Eq. (28)

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τ ¼ S0 �2Advffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Ad2v � 4Bd2dp ; ks ¼ Affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Ad2v � 4Bd2dp > 0;

gs ¼ BffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAd2v � 4Bd2d

p > 0 ð29Þ

The positivity of the expressions for ks, gs is a direct consequenceof the definition of the regularized hyperbolic criterion (withpositive scalars A and B) and of the form of the regularized supportfunction [Eq. (21)]. As a result, only two of the three parametersrelated to the behavior of the linear comparison composite areindependent, since ks=gs ¼ A=B ¼ cst. This relation for theDrucker-Prager case [Eq. (20)] takes the form

ks

gs¼ A

B¼ 1

α2 ð30Þ

The previous results deliver the expression of the nonlinearityfunction Υs for the hyperbolic criterion [Eq. (21)] as a functionof the independent parameters gs, τ

Υs ¼ B2gs

�ðS0 � τÞ22A

� 1

�ð31Þ

Stationarity of the Dissipation FunctionThe third step consists in evaluating the stationarity condition forthe homogenized dissipation function of the solid-pore composite.The result in Eq. (30) reduces the degrees of freedom for the linearcomparison composite to two ðgs; τÞ, such that the evaluation ofEq. (13) takes the explicit form

∂ ~ΠIhom

∂gs ¼ ∂ΨI

∂ks∂ks∂gs þ

∂ΨI

∂gs þ η∂Υs

∂gs ¼ 0 ð32a Þ

∂ ~ΠIhom

∂τ ¼ ∂ΨI

∂τ þ η∂Υs

∂τ ¼ 0 ð32b Þ

Using Eqs. (27) and (31) in Eq. (32b), while using Eq. (30)yields

τ ¼ Að2gsKIDv � ηS0ÞηA� 2BKI ð33Þ

Substituting Eq. (33) into Eq. (32a) delivers

ðgsÞ2 ¼ ηB½ηAðS20 � AÞ þ BKIð2A� S20Þ�A½ηAKID2

v þ 4ðηAMI � 2BKIMIÞD2d�

ð34Þ

Finally, using the latter two expressions in Eq. (13) provides anestimate for the homogenized dissipation function

~ΠIhomðDv;DdÞ ¼ ΣI

homDv

� sgnð2BKI � ηAÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAIhomD

2v þ 4BI

homD2d

qsubject to ðAI

homD2v þ 4BI

homD2dÞ ≥ 0

ð35Þ

where

AIhom ¼ η2BKI ½ηAðS20 � AÞ þ BKIð2A� S20Þ�

ðηA� 2BKIÞ2 ð36a Þ

BIhom ¼ ηBMI ½ηAðS20 � AÞ þ BKIð2A� S20Þ�

AðηA� 2BKIÞ ð36b Þ

ΣIhom ¼ ηBKIS0

2BKI � ηAð36c Þ

The comparison of Eqs. (21) and (35) reveals that ~Πhom is thesupport function of a hyperbolic criterion provided that2BKI � ηA > 0, whereas the case 2BKI � ηA < 0 correspondsto an elliptical strength criterion. The expressions in Eq. (36) reduceto the following homogenization factors for the Drucker-Pragercase:

AIhom

ðcsÞ2 ¼η2KIðη � α2KIÞðη � 2α2KIÞ2 ;

BIhom

ðcsÞ2 ¼ηMIðη � α2KIÞ

η � 2α2KI ;

ΣIhom

cs¼ ηαKI

2α2KI � ηð37Þ

The form of the criterion is determined by the sign ofϱ ¼ 2α2KI � η: ϱ > 0 for a hyperbola, ϱ < 0 for an ellipse.

Strength Criterion for the Porous Solid CompositeThe strength criterion for the porous solid composite is derivedby using the homogenized support function and the yield designdefinition [Eq. (5)]

Σm ¼ 13tr ðΣÞ ¼ ∂ ~Πhom

∂Dv; Σd ¼

12∂ ~Πhom

∂Ddð38Þ

where Σd ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1=2ÞS∶Sp

, and S ¼ Σ� Σm1. The application toEq. (35) yields the sought homogenized strength criterion

F IhomðΣÞ ¼ sgnðBI

homÞ�ðΣm � ΣI

homÞ2AIhom

þ Σ2d

BIhom

� 1

�≤ 0 ð39Þ

Inclusion Morphology Factors for Level IThe only terms missing for evaluating Eq. (39) are theinclusion morphology factors KI , MI , which characterize theunderlying microstructure of the porous solid. Based on theirdefinition in Eq. (26), linear micromechanics theory is used fortheir determination. In particular, two characteristic microstructuresare considered. The first case relates to a solid matrix–pore inclu-sion morphology, which is well represented by the Mori-Tanakahomogenization scheme (Mori and Tanaka 1973). The matrix-inclusion composite can develop mechanical responses for theentire range of solid concentrations. The second case relates to ahighly disordered composite, in which none of the mechanicalphases plays the specific role of matrix or inclusion. This micro-structure, which resembles granular and polycrystal morphologies,is captured by the self-consistent scheme (Hershey 1954; Kröner1958). In applications to porous solids, the self-consistent schemepredicts a solid percolation threshold η0 below which the compositeexhibits no mechanical response to loading. The Mori-Tanaka andself-consistent estimates for the inclusion morphology factors arereported hereafter for the case of spherical morphologies ascribedto the heterogeneities

KIMT ¼ 4η

3ð1� ηÞ þ 4α2 ð40a Þ

MIMT ¼ ð9þ 8α2Þη

3ð5� 2ηÞ þ 4ð5� 3ηÞα2 ð40b Þ

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KISC ¼ 4ηMI

SC

4α2MISC þ 3ð1� ηÞ ð41a Þ

MISC ¼ 1

2� 54ð1� ηÞ � 3

16α2 ð2þ ηÞ þ 116α2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi144ðα4 � α2Þ � 480α4η þ 400α4η2 þ 408α2η � 120α2η2 þ 9ð2þ ηÞ2

qð41b Þ

Level II: Porous Solid–Rigid Inclusion Composite

The second homogenization step defines the strength properties ofthe porous solid intermixed with rigid inclusions. Developed sim-ilarly to the Level I homogenization, the strength modeling followsthe procedure outlined in the “Linear Comparison CompositeApproach” section.

Strain-Rate Energy FunctionConsider the composite formed by a (homogenized) porous solidand rigid inclusions, which occupy the volumes Ωps ¼ ð1� f incÞΩand Ωinc ¼ f incΩ, respectively. Following a similar continuousdescription of the microscopic stress field in the rev as inEq. (22), the spatial distributions of the modulus tensor CðzÞand prestress τðzÞ are defined as

CðzÞ ¼�Cps ¼ 3kpsJþ 2gpsK ðΩpsÞ

∞ ðΩincÞ

τðzÞ ¼� τps1 ðΩpsÞ0 ðΩincÞ ð42Þ

The stress state equation for the macroscopic composite follows theform of Eq. (24), where the corresponding homogenized moduliand prestress are given by

CIIhom ¼ 3kIIhomJþ 2gIIhomK; TII ¼ τps1 ð43Þ

Analogously to Eq. (26), the homogenized moduli are expressed indimensionless form

kIIhom ¼ gpsKII

�kps

gps; f inc

�; gIIhom ¼ gpsMII

�kps

gps; f inc

�ð44Þ

where the inclusion morphology factors KII, MII will be specifiedsubsequently. With this information, the strain-rate energy is speci-fied for the porous solid reinforced with rigid inclusions

ΨIIðDv;DdÞ ¼12gpsKIID2

v þ 2gpsMIID2d þ τ psDv ð45Þ

Measure of NonlinearityIn determining the function Υps for the (homogenized) porous solidphase (Υinc for the rigid inclusion phase is zero), recall that the sup-port function of the porous solid at Level I can be adapted fromEq. (35):

πpsðdÞ ¼ ~ΠIhomðD → dÞ

¼ ΣIhomdv � sgnðϱÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAIhomd

2v þ 4BI

homd2d

qð46Þ

where ϱ ¼ 2α2KI � η distinguishes between the hyperbolic(ϱ > 0) and elliptical (ϱ < 0) strength domains. The strain-rateenergy function for the porous solid phase ψpsðdÞ is adapted fromEq. (7). Given the equivalent forms between πpsðdÞ, ψpsðdÞ, andtheir Level I counterparts, previous developments can be used toderive the Υps function, provided that the form of the support func-

tion ~ΠIhom is hyperbolic. Noting that the derivation of Eq. (31)

began with a support function corresponding to a hyperboliccriterion [see Eq. (21)], one can adapt Eq. (31) to the form ofEq. (46) for the case of a hyperbolic criterion governing the poroussolid by replacing BI

hom by �BIhom

Υpsðϱ > 0Þ ¼ BIhom

2gps

�1� ðΣI

hom � τ psÞ22AI

hom

�ð47Þ

together with

β ¼ kps

gps¼ �sgnðϱÞA

Ihom

BIhom

¼ cst ð48Þ

where the term �sgnðϱÞ accounts for BIhom < 0 in the hyperbolic

case. For the case of an elliptical strength domain governing theporous solid, evaluating the stationarity conditions for Υps with re-spect to the microscopic strain-rate invariants delivers the followingexpression:

Υpsðϱ < 0Þ ¼ BIhom

2gpsð49Þ

In the elliptical case, the consideration of the prestress τps ensuresthe positivity of the moduli ðkps; gpsÞ, which are also relatedby Eq. (48).

Stationarity of the Dissipation Function and StrengthCriteriaThe dissipation capacity of the porous solid-rigid inclusioncomposite is obtained from the stationarity estimates of thestrain-rate energy and measure of nonlinearity functions for thelinear comparison composite

~ΠIIhom ¼ statgps;τpsfΨIIðDv;DdÞ þ ð1� f incÞΥpsg ð50Þ

where the two degrees of freedom for the application of the statio-narity conditions are gps, τ ps given the relation in Eq. (48). Notethat two different cases are recognized in the evaluation of Eq. (50)depending on the particular estimate of Υps [Eq. (47) or Eq. (49)related to the form of the strength domain for the porous solid atLevel I]. Following procedures similar to those used for the Level Imodeling, the resulting dissipation function from Eq. (50) isemployed to deliver the macroscopic strength criterion

F IIhomðΣÞ ¼ sgnðBII

homÞ�ðΣm � ΣII

homÞ2AIIhom

þ Σ2d

BIIhom

� 1

�≤ 0 ð51Þ

where the homogenization factors associated with a hyperbolicstrength criterion (ϱ > 0) governing Level I are

AIIhom ¼ 2AI

hom þ ð1� f incÞKIIBIhom;

BIIhom ¼ ð1� f incÞMIIBI

hom; ΣIIhom ¼ ΣI

hom ð52Þ

whereas those for an elliptical criterion (ϱ < 0) are

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AIIhom ¼ ð1� f incÞKIIBI

hom; BIIhom ¼ ð1� f incÞMIIBI

hom;

ΣIIhom ¼ ΣI

hom ð53Þ

The proportionality observed between the factors BIIhom and BI

homimplies that the form of the macroscopic criterion is defined by thatof the porous solid. Thus, a hyperbolic strength domain for Level I(with BI

hom < 0) specifies a macroscopic hyperbolic domain forLevel II. A similar observation holds for the elliptical case.

Inclusion Morphology Factors for Level IIEvaluating the homogenized strength criterion [Eq. (51)]requires the definition of the inclusion morphology factors KII ,MII . To this end, the microstructure of the porous solid–rigidinclusion composite is represented by spherical morphologies.The mechanical interactions between phases are described bythe Mori-Tanaka and self-consistent schemes, which approximatematrix-inclusion and granular materials, respectively. Results fromlinear micromechanics are used to determine the homogenizedmoduli for the two-phase composite, whose solid is defined byEq. (48) and the homogenization results for Level I [Eq. (37)]

β ¼ kps

gps¼ �sgnðϱÞA

Ihom

BIhom

¼ ηKI

MI jη � 2α2KI j > 0 ð54Þ

The inclusion morphology factors KII , MII in the context of theMori-Tanaka and the self-consistent schemes and for the case ofperfect adherence (A) between interfaces are given by

KIIMT;A ¼ 4f inc þ 3β

3ð1� f incÞ ð55a Þ

MIIMT;A ¼ 3ð3f inc þ 2Þβ þ 4ð2f inc þ 3Þ

6ð1� f incÞð2þ βÞ ð55b Þ

KIISC;A ¼ ½18� 42f inc þ 15ðf incÞ2�β þ 4f incð3� f incÞ þ f incκA

18ð1� f incÞð1� 2f incÞð56a Þ

MIISC;A ¼ ð15f inc � 6Þβ þ 4ð3� f incÞ þ κA

24ð1� 2f incÞ ð56b Þ

with κA ¼ ½9ð5f inc � 2Þ2β2 � 24ðf inc þ 2Þð5f inc � 3Þβ þ 16ðf inc�3Þ2�1=2. The inclusion volume fraction corresponding to the self-consistent estimates for rigid inclusions with perfect bonding isbounded by f inc < 1=2. In contrast to perfect adherence, the limitcase of slip-type (S) or nonfrictional interfaces is characterizedby a purely normal stress vector acting on the interface(Christensen and Lo 1979; Barthélémy and Dormieux 2004).The inclusion morphology factors related to rigid inclusions withslip interface conditions are (Barthélémy 2005)

KIIMT;S ¼

3β þ 4f inc

3ð1� f incÞ ð57a Þ

MIIMT;S ¼

3ð3f inc þ 5Þβ þ 8ðf inc þ 3Þ3ð5� 2f incÞβ þ 12ð2� f incÞ ð57b Þ

KIISC;S ¼

3½12� 23f inc þ 8ðf incÞ2�β þ 8f incð3� 2f incÞ þ f incκS

18ð1� f incÞð2� 3f incÞð58a Þ

MIISC;S ¼

8ð3� 2f incÞ � ð15� 24f incÞβ þ κS24ð2� 3f incÞ ð58b Þ

with κS ¼ ½9ð8f inc � 5Þ2β2 þ ½720� 1392f inc þ 528ðf incÞ2�βþ64ð2f inc � 3Þ2�1=2. The inclusion volume fraction correspondingto the self-consistent estimates for rigid inclusions with slip inter-faces is bounded by f inc < 2=3.

Comparison with Micromechanics Solutions

The strength criteria for the different levels of the structure thoughtmodel developed in the section “Multiscale Strength Upscaling”are compared with existing micromechanics solutions for twoclassical microstructures of interest in strength homogenizationproblems: a porous solid and a solid reinforced with rigid inclu-sions. The porous solid configuration is addressed in Level I ofthe multiscale model, whereas the solid–rigid inclusion compositeis retrieved from the evaluation of Level II for the limit case of apure solid phase (η → 1). The upscaling of strength properties forcohesive-frictional composites exhibiting the two aforementionedmicrostructures has been pioneered by Dormieux and coworkers.The strength homogenization is treated akin to a viscous flowproblem, in which the behavior of the solid phase is describedby a viscous state equation

σðzÞ ¼ ∂πsðdðzÞÞ∂d ¼ CsðdðzÞÞ: dðzÞ ð∀ z ∈ ΩsÞ ð59Þ

The heterogeneous character of the viscosity tensor CsðdðzÞÞ andits dependence on load level are captured by applying theso-called secant methods in nonlinear homogenization (e.g., Suquet1997). The secant method aims at approximating the heterogeneousresponse of the solid phase by a uniform value related to theappropriate choice of a reference strain rate CsðdðzÞÞ ∼ CsðdrÞ.Typically, higher-order averages have delivered relevantestimates for the strain-rate fields involved in solving strengthhomogenization problems (Dormieux et al. 2006).

Porous Solid

The strength homogenization of a Drucker-Prager solid intermixedwith porosity has been addressed by Barthélémy and Dormieux(2003) (see also Dormieux et al. 2006; Cariou et al. 2008). Onthe basis of the effective strain-rate approach, the derived strengthdomain follows an elliptical form. Using a notation similar to that inEq. (39), the homogenization factors for the elliptical criterion are(Cariou et al. 2008)

AIhom

ðcsÞ2 ¼η3bKI

ðη � α2bKIÞ2;

BIhom

ðcsÞ2 ¼η2cMI

η � α2bKI ;

ΣIhom

cs¼ αηbKI

α2bKI � ηð60Þ

where bKI ¼ limα→0KI , cMI ¼ limα→0M

I are the inclusionmorphology factors evaluated for an incompressible solid(ks=gs → ∞⟺α → 0). The incompressibility of the viscous solidis a trait inherent to the effective strain-rate approach. The LCCmethodology removes this restriction in evaluating the nonlinearhomogenization problem, because the linear comparison compositemoduli are explicitly related by ks=gs ¼ 1=α2 [see Eq. (30)]. Byway of illustration, a set of comparisons between the effective strainrate and LCC models is presented in Fig. 2, which displays thedifference in predictions of unconfined compressive strength asfunctions of the packing density η and the friction coefficient α.

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The unconfined compressive strength (UCS) represents a particularstress state in the macroscopic strength domain (Σm × Σd), and it isobtained by substituting Σm ¼ �ð1=3ÞΣUCS; Σd ¼ ð ffiffiffi

3p

=3ÞΣUCS

into the strength criterion [Eq. (39)]. The matrix-inclusion andgranular microstructures, modeled by the Mori-Tanaka andself-consistent schemes, are implemented in the comparisons.The contour plots display the absolute difference between the nor-malized predictions of the models. The comparisons in Fig. 2 areestablished only for the elliptical strength regime, which is in accor-dance with the applicability of the effective strain-rate solution.Relatively small differences between the two homogenizationmodels are observed in Fig. 2, where the largest differences corre-spond to the granular microstructure.

Solid Reinforced by Rigid Inclusions

Solutions for the case of a Drucker-Prager solid intermixedwith rigid inclusions have been presented by Lemarchand et al.(2002) and Barthélémy and Dormieux (2004). The strength cri-terion [Eq. (51)] for Level II is adapted for the case of a puresolid phase at Level I, which implies that BI

hom ¼ �α2AIhom;

ΣIhom ¼ cs=α; AI

hom → 0. The resulting macroscopic criterion is

F IIðΣÞ ¼ Σd þ αhom

�Σm � cs

α

�≤ 0 ð61Þ

and can be understood as a Drucker-Prager-type criterion with ahomogenized friction coefficient of the form

αhom ¼ α

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� f incÞMII

2� α2KIIð1� f incÞ

sð62Þ

The solutions of Barthélémy and Dormieux for the homogenizedfriction coefficient considering perfectly adherent (A) and slip-type(S) interface conditions read

αMT;Ahom ¼ α

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 3

2 finc

1� 43α

2f inc

s;

αMT;Shom ¼ α

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� f incÞð1þ 3

5 fincÞ

ð1� 25 f

incÞð1� 43α

2f incÞ

s ð63Þ

whereas the solution by Lemarchand et al. (2002) based on amixed-secant formulation and perfectly bonded interfaces is

αMT;Ahom ¼ α

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 3

2f inc

rð64Þ

These micromechanics solutions were derived based on a matrix-inclusion morphology (modeled by the Mori-Tanaka scheme).Fig. 3(a) displays an example of the predicted homogenizedfriction coefficient by the effective strain-rate and LCC models,showing adequate comparisons between them. In fact, the solutionsin Eq. (63) remain a good approximation to the order of Oðα3Þ. Astrict comparison between the model of Lemarchand et al. (2002)and the two remaining models cannot be formulated, since the

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.0 0.2 0.4 0.6

Inclusion volume fraction, f inc [1]

Hom

ogen

ized

fri

ctio

nco

effi

cien

t, a

[1]

Lemarchand et al., 2002Barthelemy & Dormieux, 2004Model

(a) (b)

Adherent interface

Slip interface

0.0

0.2

0.4

0.6

0.0 0.2 0.4 0.6 0.8

Friction coefficient, α [1]

Cri

tica

l inc

. vol

ume

frac

tion

,fin

c [

1]

Adherent inferfaceSlip interface

crit

MT

hom

α

Fig. 3. (a) Predictions of the homogenized friction coefficient αhom for a Drucker-Prager solid reinforced with rigid inclusions. The microstructure ismodeled by the Mori-Tanaka scheme and considers perfectly adherent (A) and slip-type (S) interface conditions. (b) Critical inclusion volume fractionf inccrit as a function of solid friction α for a solid-rigid inclusion composite modeled by the self-consistent scheme

Packing density, η [1]

Fric

tion

coef

fici

ent,

α [

1]

0.00 0.25 0.50 0.75 1.000.0

0.2

0.4

0.6

0.8

0.01

0.02

0.03

0.04

Packing density, [1]

Fric

tion

coef

fici

ent,

[1]

0.6 0.7 0.8 0.90

0.2

0.4

0.6

0.8

0.02

0.04

0.06

0.08

(a) (b)

MT SC

η

α

050.max =∆ 100.max =∆

Fig. 2. Comparisons between predictions of the effective strain rate (ESR) and LCC models for unconfined compressive strength. The microstructureof the porous solid is modeled by (a) the Mori-Tanaka (MT) and (b) self-consistent (SC) schemes. The contour plots for the absolute differencebetween predictions jΔj ¼ ð1=csÞjΣUCS

ESR � ΣUCSLCCj are generated as functions of the friction coefficient α and the solid packing density η

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mixed-secant method does not consider the local flow rule of theDrucker-Prager solid (Barthélémy and Dormieux 2004). The LCCresult in Eq. (62) for granular composites predicts homogenizedfriction coefficients for a restricted range of inclusion-volume frac-tion values. Fig. 3(b) displays the critical inclusion-volume fractionvalues f inccrit as functions of the solid friction α at which the homog-enized friction coefficient αhom is unbounded. Hence, inclusion-volume fraction values below f inccrit ensure finite predictions ofstrength properties for the solid–rigid inclusion composite withgranular microstructure.

Analysis of Model Strength Predictions

This section analyzes the predictive capabilities of the LCChomogenization model following the multiscale structure dis-played in Fig. 1. The composite strengths for the porous solid(Level I) and for the porous solid reinforced with rigid inclusions(Level II) are investigated with regard to the types of predictedstrength domains, the effects of rigid inclusions and theirinterface conditions, and the consideration of differentmicrostructures.

Strength Predictions for Level I

The strength criterion for the porous solid composite determined bythe LCC method supports two different domain forms: ellipticaland hyperbolic. From inspecting Eq. (39), the particular form ofthe predicted domain is determined from the sign of the homogeni-zation factor BI

hom, which is a function of the solid packing density,the friction coefficient, and the modeled microstructure. It is con-venient to define a critical solid packing density ηcrit marking thetransition between elliptical and hyperbolic strength domains, suchthat

η � ηcritðα;KIJ¼MT;SCÞ

8<:> 0 Hyperbolic criterion¼ 0 Limit parabola< 0 Elliptical criterion

ð65Þ

Substituting the inclusion morphology factors for theMori-Tanaka [Eq. (40)] and self-consistent [Eq. (41)] estimatesin Eq. (37) delivers the critical packing densities for Level I:

0 ≤ ηMTcrit ¼ 1� 4

3α2 ≤ 1 ð66a Þ

23≤ ηSCcrit ¼ 1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1216α4 þ 432α2 þ 81

p� ð16α2 þ 9Þ

2ð20α2 þ 3Þ ≤ 1

ð66b ÞThese critical packing densities are displayed in Fig. 4(a), showingthat the effect of microstructure is more significant for higherfriction values of the solid phase. To illustrate the transitionbetween elliptical and hyperbolic regimes for a particular stressstate, the predictions of unconfined compressive strengths (referto the section “Comparison with Micromechanics Solutions”)are developed as a function of solid packing density. Fig. 4(b) dis-plays the UCS predictions for porous solids with matrix-inclusionand granular microstructures. The evolution of the UCS strength forthe granular microstructure modeled by the self-consistent schemedisplays a solid percolation threshold at η0 ¼ 0:5, which is relatedto spherical morphologies assumed for the different hetero-geneities. In contrast, the Mori-Tanaka strength estimates coverthe entire range of packing densities. As shown in Fig. 4(b),a smooth transition exists between the strengths associated withthe elliptical and hyperbolic regimes close to the critical packingdensity. The ability of the LCC-based strength criterion[Eq. (39)] to transition from the elliptical to the hyperbolic domainrepresents an improvement over the present form of the effectivestrain-rate solution, which is restricted to the elliptical regime.Thus, the LCC model allows consideration of the entire rangeof Drucker-Prager frictional behaviors in the strength predictionsfor the porous solid.

Strength Predictions for Level II

In addition to the response of the pressure-dependent porous solid,the underlying microstructure, the interface conditions betweenphases, and characteristic amounts of rigid inclusions modifythe homogenized strength predicted by the LCC method for themacroscopic composite. The reinforcing effect attributable tothe presence of rigid inclusions that are perfectly bonded to the(homogenized) porous solid is shown graphically in Fig. 5(a).The macroscopic composite is modeled by the Mori-Tanaka andself-consistent schemes. The strength domains in Fig. 5(a) showthat a granular microstructure amplifies the reinforcing effect attrib-utable to rigid inclusions compared with the predictions for amatrix-inclusion morphology (the self-consistent estimates forthe Level II strength predictions are limited to specific ranges ofinclusion volume-fraction values, as noted in the section “Inclusion

0.0

0.5

1.0

1.5

2.0

2.5

0.00 0.25 0.50 0.75 1.00

Packing density, η [1]

UC

S / c

s [

1]

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8

Friction coefficient, α [1]

Cri

tical

pac

king

den

sity

, η

crit

[1]

MTSC

Ellipticalregime

Hyperbolicregime

(a) (b)

MTcritηSCcritη

MT SC

3.0=α

Fig. 4. (a) Critical packing density ηcrit as a function of solid friction coefficient α for microstructures modeled by the Mori-Tanaka and self-consistentschemes. (b) Predictions of unconfined compressive strength (UCS) for Level I as functions of packing density η for different microstructures. Thesolid lines correspond to the predictions associated with the elliptical strength regime (η < ηcrit), whereas the dashed lines correspond to predictionsassociated with the hyperbolic strength regime (η > ηcrit)

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Morphology Factors for Level II”). A similar set of comparisons isdeveloped in Fig. 5(b) for the case of slip (nonfrictional) interfaceconditions. The predictions for the strength domains in the tensilemean stress regime (Σm > 0) are not included in the figure, giventhat the micromechanics expressions [Eqs. (57) and (58)] assumethe transmission of loads only through normal stresses (Barthélémyand Dormieux 2004). As expected, the strength domains for slipinterface conditions are smaller than those related to perfectlybonding interfaces. The pronounced strength enhancement forthe granular microstructure compared with the matrix-inclusionmorphology is also observed for slip interface conditions. However,a particular behavior is observed for the matrix-inclusion morphol-ogy, for which an increase in inclusion-volume fraction results indecreased deviatoric strength capacities within a certain range ofmean stress states. In turn, the application of the self-consistentscheme translates into an overall enlargement of the predictedstrength domain with increasing inclusion-volume fraction values.

Compared with the smooth transitions observed between ellip-tical and hyperbolic strength domains at Level I for moderate meanstress states, the presence of rigid inclusions reinforcing the poroussolid response yields distinctive homogenized behaviors at Level II.By way of example, Fig. 6(a) displays UCS predictions for Level IIas a function of packing density for a fixed inclusion volume frac-tion and different microstructures. The figure reveals a nonsmoothtransition between predictions associated with the elliptical andhyperbolic strength regimes. From a yield design perspective,

the behaviors of the predicted UCS strengths displayed inFig. 6(a) respect the expected dissipation capacities of themacroscopic material system close to the critical packing densityseparating the elliptical and hyperbolic domains, that is, ~ΠII

homðη <ηcrÞ < ~ΠII

homðη > ηcrÞ for a fixed f inc value. The latter is equivalentto increasing (UCS) strengths with increasing packing densities fora fixed amount of rigid inclusions. However, it was determinedfrom a series of parametric studies that modeled macroscopicstrengths related to a granular microstructure at Level II could vio-late the expected dissipation capacities in the hyperbolic strengthregime for characteristic inclusion volume fractions. Fig. 6(b) dis-plays the critical inclusion volume-fraction values corresponding toa characteristic strength measure (unconfined compressivestrength) as functions of the friction coefficient α, which definesthe separation between elliptical and hyperbolic domains [seeEq. (66)]. Similar calculations can be performed for other stressstates (e.g., hydrostatic tension, pure deviatoric strength). The criti-cal inclusion volume fraction f inccrit in Fig. 6(b) establishes theamount of rigid inclusions below which strength predictions remainvalid for the entire range of packing densities. Strength predictionsat Level II are otherwise restricted to packing densities below thecritical packing density η < ηcr (i.e., elliptical strength domain).The analysis of the predictive capabilities at Levels I and II ofthe multiscale structure thought model completes the implementa-tion of the LCC methodology for strength homogenization.

0.0

0.5

1.0

1.5

-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.50.0

0.5

1.0

1.5

-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5s

m c/Σ sm c/Σ

f inc = 0

f inc = 0.4, MT-A

f inc = 0.4, SC-A

f inc = 0

f inc = 0.4, MT-S

f inc = 0.4, SC-S

(a) (b)

sd

c/Σ

sd

c/Σ

Fig. 5. Effect of rigid inclusions with (a) perfectly adherent (A) and (b) slip (S) interface conditions on the homogenized strength at Level II. Theporous clay at Level I (f inc ¼ 0) is modeled by the self-consistent scheme with η ¼ 0:7, α ¼ 0:3. The Level II properties are predicted forMori-Tanaka and self-consistent microstructures

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.0 0.2 0.4 0.6 0.8

Friction coefficient, α [1]

Cri

tical

inc.

vol

ume

frac

tion,

fin

c [

1]

0

1

2

3

4

0.5 0.6 0.7 0.8 0.9 1.0

Packing density, η [1]

30

280

.

.f inc

==

α

SC-SC

UC

S / c

s[1

]

MT-SC-A

crit

SC-SC-A

MT-SC-S

SC-SC-S

UCS

(a) (b)

SC-MT

Fig. 6. (a) Predictions of UCS strength at Level II as functions of solid packing density η for a fixed inclusion volume fraction f inc value. The poroussolid is modeled by the self-consistent scheme and α ¼ 0:3. The predicted UCS values for the elliptical regime are displayed as solid lines, whereasthose corresponding to the hyperbolic regime are displayed as dashed lines. (b) Critical inclusion volume fraction f inccrit as a function of the solid frictioncoefficient α for MT-SC and SC-SC configurations at Levels I and II, respectively, and for perfectly adherent (A) and slip (S) interfaces

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Application to Shale

A first application of the LCC strength homogenization focuses onthe investigation of strength properties of shale, a common type ofsedimentary rock that serves as a geological cap for most hydro-carbon reservoirs. As for most rock materials, the characterizationof the strength behavior of shale has been mainly derived frommacroscopic techniques such as conventional triaxial experiments.In turn, little quantitative understanding exists on the local strengthin the load-bearing clay phase. This section studies the claystrength properties in shale through the use of the multiscalestrength model. In fact, the structure thought model of Fig. 1 iswell suited for the strength modeling of shale, as it has been sat-isfactorily implemented in predicting poroelastic properties (e.g.,Ulm and Abousleiman 2006; Ortega et al. 2009). The compositionof shale can be broadly described as the intermix of the porous clayfabric (clay particles and porosity at the submicrometer scale) andsilt-size inclusions (typically quartz of micrometer sizes). Thedifference in scales between the heterogeneities at the level ofthe porous clay composite and the rigidlike quartz grains justifiesapplication of the structure thought model.

The investigation of clay properties follows an inverse approach,in which a series of triaxial test results obtained for core samplesfrom a particular data set of shale materials is used to back-calculatestrength attributes for the assumed Drucker-Prager type solid. Thecohesion cs and friction angle α for the clay phase in a shalematerial are inferred by minimizing the difference between n ex-perimental values and modeled strengths for the macroscopic shale

mincs;α

�Xnj¼1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½Σexp

m;j � Σmodm;opt�2 þ ½Σexp

d;j � Σmodd ðΣmod

m;opt�2q �

ð67Þ

where Σmodd is the deviatoric strength derived from Eq. (51) and the

homogenization factors [Eq. (52) or Eq. (53)] and Σmodm;opt is the

solution to the equation

ddΣmod

mf½Σexp

m;j � Σmodm �2 þ ½Σexp

d;j � Σmodd ðΣmod

m Þ�2g ¼ 0 ð68Þ

The previous expressions correspond to the best fit of triaxialstrength data by the modeled strength criterion for Level II ofthe structure thought model. The experimental data set is obtainedfrom the work of Jizba (1991), which compiles results of conven-tional triaxial compression experiments for several shale samples(Travis Peak formation, East Texas). In addition to strength data,

porosity and mineralogy information (in terms of clay content)were documented. These data are required data for estimatingthe model input parameters: the clay packing density η andthe inclusion volume fraction f inc [see Eqs. (15) and (16)].Four shale materials with similar clay packing density values(η ¼ 0:91� 0:03) and varying amounts of inclusion volume frac-tion (f inc ¼ 0:3–0:57) are considered for the inverse analysis. Thesematerials were retrieved from different wells, and four core sampleswere tested for each shale material under drained conditions andvarying degrees of confinement. In addition to the computationof input volume fractions ðη; f incÞ, the microstructural descriptionof shale at Levels I and II must be defined for model implementa-tion. For this, the porous clay has been characterized through nano-indentation experiments as a nanogranular composite, exhibitinga solid percolation threshold at approximately η0 ¼ 0:5 (Ulmand Abousleiman 2006; Bobko and Ulm 2008). The granularresponse is captured in micromechanics by applying the self-consistent scheme for spherical morphologies, which attests tothe random orientations of contact surfaces between ensemblesof clay grains (Ortega et al. 2010). The mechanical interactionsbetween the porous clay and silt (rigid) inclusions are modeledby the Mori-Tanaka scheme and perfectly bonded interfaces, whichallows the consideration of large inclusion-volume fractions.

Fig. 7(a) shows the results of the inverse analysis for shale sam-ple 9898. The triaxial data plotted in equivalent mean and devia-toric stresses are captured by the strength model with solid clayproperties α ¼ 0:64, cs ¼ 27 MPa. These strength parametersfor the solid clay phase in conjunction with the input values forthe clay packing density and inclusion-volume fraction deliver ahyperbolic strength domain for the macroscopic composite. Theresults for the inverse analysis for the four shale materials are sum-marized in Fig. 7(b). The frictional properties for the four materialsare relatively similar, α ¼ 0:54–0:64, whereas the clay cohesionvaries between cs ¼ 21–40 MPa. The volumetric propertiesðη; f incÞ for each shale material are also provided in Fig. 7(b).A particular trend is observed between the cohesion and theinclusion-volume fraction, in which increasing cs values are relatedto decreasing f inc values. Analogously, the increase in cohesioncan be related to increasing clay volume fractions f s ¼ηð1� f incÞ, given the similar packing density for the four shalesconsidered. In these materials, the presence of larger volume frac-tions of (rigidlike) quartz inclusions dominates the overall cohesiveresponse, which translates into decreased cohesions for the effec-tive solid clay phase. The relatively constant clay friction may

0.0

0.2

0.4

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0.8

10 20 30 40 50

Clay cohesion, c s [MPa]

Cla

y fr

ictio

n co

effi

cien

t, α

[1]

0.0

0.2

0.4

0.6

0.8

1.0

Vol

ume

frac

tions

[1]

Jizba

eta

finc

0

100

200

300

-300 -200 -100 0 100

Σm [MPa]

Σ d [

MPa

]

LCC modelShale 9898 data

Inverse analysis

ηf inc

7053 9898 8675 6275

(a) (b)

Fig. 7. (a) Fitting of triaxial experimental data for shale (Jizba 1991) using the multiscale LCC strength homogenization model. (b) Results of theinverse analysis for four shale materials in terms of cohesion and friction coefficient for the effective solid clay phase. The volume fractions η , f inc forthe different shale materials are also displayed

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relate in turn to the characteristic clay packing density for this set ofshale data.

Conclusions

The modeling and prediction of strength properties for complexheterogeneous media such as geomaterials require explicit consid-erations of local strength properties and microstructural features. Tothis end, this work developed a novel micromechanics method forstrength homogenization within a yield design framework that isapplicable to cohesive-frictional porous composites. The inherentlynonlinear problem of strength upscaling is treated through the LCCtheory, which reformulates the nonlinear homogenization in termsof modeling a linear comparison composite with suitably chosenproperties and similar microstructure. For a porous solid micro-structure, the LCC formulation allows consideration of the com-plete range of frictional behaviors related to the Drucker-Pragersolid matrix. Such improvement translates into predictions of bothelliptical and hyperbolic strength regimes that are naturally definedby the LCC solutions. In contrast, effective strain-rate solutionshandle only elliptical strength criteria, although extensionsof the theory for delivering hyperbolic domains can be pursued(Cariou et al. 2008). The definitions of mechanical interactionsbetween phases, captured by so-called inclusion morphologyfactors, are also refined in the LCC approach by lifting therestriction on the incompressible character of the solid matrixinherent to the effective strain-rate approach. Nevertheless, thestrength criteria for porous solids under drained conditions pre-dicted by both approaches compare adequately within theircommon domains of application, in particular in the tensile andmoderate compressive mean stress regimes. Adequate agreementsare also observed between the two approaches for a solid reinforcedby rigid inclusions.

The proposed micromechanics-based upscaling of strengthproperties exhibits some of the limitations that are intrinsic to con-tinuum mechanics and yield-design approaches. The principle ofmaximum plastic dissipation, which is at the core of the limit the-orems of yield design, does not consider the nonassociated plasticflow and hardening or softening behaviors that are characteristic ofcertain material systems. In addition, the multiscale strengthhomogenization method cannot capture scale effects commonlylinked to other dissipation mechanisms at grain scales (e.g., frac-ture). Therefore, adequately defining the length scales to be mod-eled with a micromechanics methodology is crucial. Within theaforementioned limitations, the LCC method represents a robustframework for developing micromechanics solutions of the multi-scale strength response of cohesive-frictional materials. Thestrength criteria here derived through the LCC method can bereadily implemented in the interpretation of hardness measure-ments on porous materials from instrumented indentation experi-ments (see Ulm et al. 2007; Bobko et al. 2010). With theincreased deployment of experimental techniques such as nanoin-dentation and scratch testing, properties derived from grain scalescan become key inputs to strength homogenization models forsedimentary rocks and cement-based materials.

Acknowledgments

We are grateful for the financial support of this study by the MIT-OU GeoGenome Industry Consortium (G2IC) directed by Prof.Younane Abousleiman of the University of Oklahoma at Norman.

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