Compaction bands due to grain crushing in porous rocks: A theoretical approach based on breakage...

Compaction bands due to grain crushing in porous rocks:A theoretical approach based on breakage mechanics

Arghya Das,1 Giang D. Nguyen,1 and Itai Einav1

Received 27 January 2011; revised 26 April 2011; accepted 3 May 2011; published 6 August 2011.

[1] Grain crushing and pore collapse are the principal micromechanisms controlling thephysics of compaction bands in porous rocks. Several constitutive models have beenpreviously used to predict the formation and propagation of these bands. However, they donot account directly for the physical processes of grain crushing and pore collapse. Theparameters of these previous models were mostly tuned to match the predictions ofcompaction localization; this was usually done without validating whether the assignedparameters agree with the full constitutive behavior of the material. In this study amicromechanics‐based constitutive model capable of tracking the evolving grain sizedistribution due to grain crushing is formulated and used for a theoretical analysis ofcompaction band formation in porous rocks. Linkage of the internal variables to graincrushing enables us to capture both the material behavior and the evolving grain sizedistribution. On this basis, we show that the model correctly predicts the formation andorientation of compaction bands experimentally observed in typical high‐porositysandstones. Furthermore, the connections between the internal variables and theirunderlying micromechanisms allow us to illustrate the significance of the grain sizedistribution and pore collapse on the formation of compaction bands.

Citation: Das, A., G. D. Nguyen, and I. Einav (2011), Compaction bands due to grain crushing in porous rocks: A theoreticalapproach based on breakage mechanics, J. Geophys. Res., 116, B08203, doi:10.1029/2011JB008265.

1. Introduction

[2] Localized narrow deformation zones are widelyobserved in porous rocks. Mollema and Antonellini [1996]discovered the existence of localized deformation bandscaused by pure compaction without being accompanied byany shear deformation within the Aeolian sandstone; theytermed it “compaction band.” Afterward the formation ofpure compaction bands were observed in many other in-vestigations through laboratory experiments on various kindsof rock samples [e.g., Olsson, 1999; Haimson, 2001; Tembeet al., 2008; Wong et al., 2001]. In different experimentalstudies compaction bands were also seen in nongeologicalmaterial, such as metal foam [Bastawros et al., 2000] andpolycarbonate honeycomb [Papka and Kyriakides, 1998].According to Rudnicki and Rice [1980], the formation ofsuch localization bands initiates either due to stress concen-tration in the place of local heterogeneities or any abruptchange in strength of material in a localized zone.[3] Menéndez et al. [1996] and Wu et al. [2000] provided

valuable experimental observations regarding the micro-mechanics driving the formation of planar deformationzones in porous rocks. Various mechanisms were identified,including: grain crushing, grain sliding, bond breaking andpore collapse. Shearing at low confining pressures facilitates

the fracture of grain bonding cement, allowing the grains torotate and slip, which could be followed by the flow ofgranulated materials. During this bond breaking process, themobilized shear strength of the material will obviouslyreduce [Menéndez et al., 1996; Wong et al., 1997]. Incontrast, high confining pressures frustrate grain rotationand slip when shear loading is applied. Instead, the con-tacting grains tend to crush under the pressure, leading tothe rearrangement of fragments, which further reduces theporosity and consequently hardens the material [Menéndezet al., 1996; Wong et al., 1997]. In both loading cases(under low and high pressures), pore collapse acts as a passivemechanism facilitated by either a bond breaking or a graincrushing event. At a macroscopic level these failures can beclassified as brittle failures followed by shear enhanceddilation at low confining stresses, or cataclastic flow withshear enhanced compaction in high‐pressure regimes [Wonget al., 1997].Generally, compaction bands and shear enhancedcompaction bands are observed in the transition zonebetween brittle and ductile behavior [Wong et al., 2001].[4] The theoretical prediction of compaction band for-

mation in porous rocks can be based on either continuum ordiscrete approaches. Discrete approaches [e.g.,Katsman et al.,2005; Katsman and Aharonov, 2006; Marketos and Bolton,2009;Wang et al., 2008] offer an excellent access to insightsinto the formation and propagation of compaction band atthe grain scale, but they critically depend on constitutiveassumptions at the microscale. Besides, they are very com-putationally demanding, tending to be subjected to strongnumerical noise when the number of grains is not suffi-

1School of Civil Engineering, University of Sydney, Sydney, Australia.

Copyright 2011 by the American Geophysical Union.0148‐0227/11/2011JB008265

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 116, B08203, doi:10.1029/2011JB008265, 2011

B08203 1 of 14

http://dx.doi.org/10.1029/2011JB008265

ciently high, and mostly suffering from numerical integra-tion inaccuracies due to the highly nonlinear nature of thiscomplex multibody system. On the other hand, continuumapproaches have also been widely used [Olsson, 1999; Issenand Rudnicki, 2000; Rudnicki, 2004; Challa and Issen,2004; Chemenda, 2009], and have the capacity to provideaccess to closed form solutions under particular conditions,that can clearly map the extent of the various effects. Issenand Rudnicki [2000] made a successful attempt toward theprediction of compaction band formation using the bifur-cation approach proposed by Rudnicki and Rice [1975] forshear localization. Challa and Issen [2004] further modifiedthis approach and proposed a constitutive model possessingtwo yield surfaces, one of which aims to capture the inelasticcompacting behavior during the formation of the compac-tion band. Grueschow and Rudnicki [2005] elaborated thisaspect of the model even further by introducing an ellipticcap yield surface. Their parametric study on the aspect ratioof the ellipse showed that for better prediction of localizationmode a cap is required which becomes steeper under con-tinued loading. We wish to stress that the parameters ofthese continuum models may be fitted to match the onsetand orientation of compaction bands, but this is only oneside of the story. Without readjusting those parameters, canthe same models predict experimental stress‐strain curves?This is not a trivial question, but one that we are willing toexplore in the current paper.[5] According to Aydin et al. [2006], a suitable constitu-

tive model should be able to track the evolving micro-structures of porous rocks due to grain crushing, poreenlargement and pore collapse, grain boundary sliding andpressure solution. Also the model needs to be capable ofcapturing the experimentally observed onset of compactionlocalization in the transitional regime between brittle frac-ture and ductile faulting [Wong et al., 1997]. In addition, asspecifically addressed by Wong et al. [2001], it is essentialfor constitutive models to have multiple distinct dissipativemechanisms in the transitional zone (between brittle frac-ture and ductile cataclastic flow) to faithfully describe thebehavior and the formation of localized deformation bandsin porous rocks. Given the above requirements, it can beseen that most of the current continuum models dedicated tothis issue are not adequate, as they are based on conven-tional plasticity theory [Olsson, 1999; Issen and Rudnicki,2000; Rudnicki, 2004; Challa and Issen, 2004; Chemenda,2009], or rely on the extension of fracture mechanics (theanticrack model by Sternlof et al. [2005]). Those modelscannot explain the physics dictating the porous rock behaviorand compaction band formation. This is due to the lack ofexplicit links between the only internal variable (plasticstrain) in these models and the two distinguishable under-lying micromechanical processes of grain crushing and porecollapse. As a consequence, these plasticity‐based modelsare unable to track the evolution of the grain size distributionduring grain crushing, resulting in difficulties in predictingcorrectly the permeability change due to such a microscopicprocess.[6] In this study, we approach the issue of constitutive

modeling of porous rocks and the formation of compactionbands from a rather more physical angle, which distin-guishes itself from elastoplasticity attempts. We use a con-

stitutive model based on the recently developed breakagemechanics theory for crushable granular material [Einav,2007a, 2007b]. The main feature of this theory is that itcan take into account the grain crushing phenomena throughthe evolving grain size distribution (gsd). This gsd is micro-mechanically linkedwith themacroscopic continuum behaviorthrough the breakage measure B, as an internal variable of thecontinuum model, so that the grain size distribution can becontinuously tracked during the deformation and associatedgrain crushing process. Moreover the theory is also able tocapture the effects of pore collapse on the macroscopicbehavior of the material. Models based on breakagemechanics theory possesses only a few physically identifiableparameters and were successfully used to predict the mechan-ical response and permeability reduction of porous rocks[Nguyen and Einav, 2009].[7] A model based on breakage mechanics theory is for-

mulated and used in this research to study the onset andorientation of compaction bands in sandstones. We note thatthese grain‐crushing‐induced compaction bands can onlybe formed in high‐pressure regimes, where we consider thecurrent breakagemodel to be adequate. Cement failure, whichaffects the cohesive strength, only dominates the materialbehavior in low‐pressure regime [Menéndez et al., 1996], andtherefore is expected to have insignificant contribution to themodel response and the prediction of compaction bands athigh pressures. We start the main parts of the paper with abrief introduction of breakage mechanics, followed by thedescription of a simple breakage model. The validity of thismodel is assessed against experimental data. The modelparameters are determined for a typical high‐porosity sand-stone, and the model is used to predict compaction localiza-tion observed in this rock. The localization analysis is basedon the loss of positiveness of the determinant of the strainlocalization tensor (or acoustic tensor) [Rudnicki and Rice,1975]. Using different loading conditions we determinefavorable conditions for compaction localization and linksbetween the material microstructures and the formation ofcompaction band. Agreements between numerical predic-tions and published experimental observations [Baud et al.,2006; Wong et al., 2001] for high‐porosity sandstone arefound. A parametric study is carried out to explore the effectof the variation of initial microstructures on the formation ofcompaction localization.

2. Breakage Mechanics

2.1. Fundamentals of Breakage Mechanics

[8] The breakage mechanics theory for crushable granularmaterials [Einav, 2007a, 2007b] is built on themicromechanicsof grains, using statistical homogenization for the upscalingprocedure. It relies on the following four assumptions:[9] 1. The elastic strain energy stored in the grains scales

with their surface areas [Einav, 2007a].[10] 2. The crushing process gradually shifts the initial

grain size distribution toward the ultimate one, which can takeany shape, but is conveniently assumed to follow a fractallaw. This assumption is often supported by field observationsof geological fault gouges [Sammis et al., 1987].[11] 3. The current grain size distribution (g(d,B)) is

determined by linear interpolation between the initial (g0(d))

DAS ET AL.: COMPACTION BANDS AND GRAIN CRUSHING B08203B08203

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and ultimate (gu(d)) grain size distribution (where d is thegrain diameter) [Einav, 2007a]:

g d;Bð Þ ¼ 1� Bð Þg0 dð Þ þ Bgu dð Þ ð1Þ

[12] 4. The dissipation due to grain crushing is equal tothe loss in the residual breakage energy [Einav, 2007a].[13] In equation (1), B is the internal variable of the

continuum breakage model, representing the degree of graincrushing. As can be seen, the current evolving gsd can betracked at any stage of the deformation process, given theinitial and ultimate ones. Assumptions 1–3 are used in astatistical homogenization scheme to upscale the grain‐scaleenergy potential to obtain the macro energy potential of thecontinuum model. The grading index # is a result of thisstatistical homogenization, and can be obtained from theinitial and ultimate gsd’s as:

# ¼ 1� J2u=J20 ð2Þ

where J20 and J2u are second‐order moments of the initialand final gsd’s [Einav, 2007a]. # admits values from 0 to 1,measuring the “distance” between the initial and ultimategsd’s. The energy balance postulate 4 enables us to derive ayield surface, on the basis of the loss of the energy ther-modynamically associated with the breakage internal vari-able, which reveals surprising analogies with Griffith’swell‐known theory in fracture mechanics [Einav, 2007d].

2.2. Constitutive Models Based on Breakage Mechanics

[14] A simple breakage model proposed by Einav [2007c]and later used and improved by Nguyen and Einav [2009] isadopted and further enhanced in this study. We use thefollowing standard notations: mean effective stress p, shearstress q, total volumetric strain "v and elastic volumetricstrain "v

e, total shear strain "s and elastic shear strain "se. As a

result of the statistical homogenization using assumptions1–3, the Helmholtz free energy potential takes the form:

Y ¼ 1� #Bð Þ 1

2K"e

2

v þ 3

2G"e

2

s

� �ð3Þ

For simplicity, we use linear elasticity for the presentbreakage models. However, the breakage formulation issufficiently general to accommodate any forms of elasticity,e.g., pressure‐dependent elasticity as illustrated by Einav[2007b] and Nguyen and Einav [2009]. From equation (3),the following stress‐strain‐breakage relationship andbreakage energy EB are obtained:

p ¼ @Y@"ev

¼ 1� #Bð ÞK"ev ð4Þ

q ¼ @Y@"es

¼ 3 1� #Bð ÞG"es ð5Þ

EB ¼ � @Y@B

¼ #

2 1� #Bð Þ2p2

Kþ q2

3G

� �ð6Þ

[15] Equations (4) and (5) can be written in tensorial formas

�ij ¼ 1� #Bð ÞDijkl "kl � "pkl� � ð7Þ

where sij (i, j = 1, 2, 3) is Cauchy stress tensor; "kl and "klp are

the total and plastic strain tensors, respectively; Dijkl is thelinear (isotropic) elastic tangent stiffness tensor;[16] On the basis of postulate 4 in section 2.1, the fol-

lowing breakage criterion for isotropic loading conditionscan be obtained [Einav, 2007b]:

yB ¼ 1� Bð Þ2EB

Ec� 1 � 0 ð8Þ

where Ec is the critical breakage energy (of stress dimen-sion). Apart from the breakage dissipation due to graincrushing, energy may be lost during shearing due to frictionaldissipation. To account for this shear‐related dissipation atthe macroscale, we can either adopt the generalized Hoek‐Brown failure criterion (equation (9)) (without consider-ing any cohesion) or the Mohr‐Coulomb failure criterion(equation (10)).

qf�� ¼ M*pr pf =pr

� �n ð9Þ

qf�� ¼ Mpf ð10Þ

where M and M* are friction‐related material constants;n is the Hoek‐Brown nonlinearity constant (0.5 to 1) andpr is reference pressure (1 KPa). Subscript f in equations(9) and (10) indicates stress state at failure. If n goes to 1, thecohesionless Hoek‐Brown failure criterion reduces to Mohr‐Coulomb type failure criterion which was used by Einav[2007c].[17] The combination of those failure criteria (equations (9)

or (10)) with breakage yield criterion (equation (8)) givesyield conditions (equations (11) or (12)) in mixed stress‐energy space (for models based on Hoek‐Brown and Mohr‐Coulomb criteria, respectively):

y ¼ 1� Bð Þ2EB

Ecþ q

M*p1�nr pn

� �2

�1 � 0 ð11Þ

y ¼ 1� Bð Þ2EB

Ecþ q

Mp

� �2

�1 � 0 ð12Þ

[18] The breakage model based on Hoek‐Brown failurecriterion is used for comparison purpose in some situations(wherever indicated), while most of the analysis in this studyis based on its Mohr‐Coulomb type counterpart. Figure 1depicts the breakage yield surfaces in p‐q‐B space withaccount given to equation (6).[19] We note that besides the energy loss due to frictional

shear, the dissipation from grain rearrangement/friction, fol-lowing a crushing event, can occur even in isotropic loadingconditions due to pore collapse [Einav, 2007b; Nguyen andEinav, 2009]. This kind of coupling between breakage andvolumetric plastic strain is accounted for through the intro-duction of the coupling angle w [Einav, 2007b] that governs


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the effects of pore collapse on the material behavior. Furtherdetails on w can be seen in section 3.2. The evolution lawsfor breakage and inelastic strains (for Mohr‐Coulomb andHoek‐Brown type shear failures, respectively) can be writ-ten as (dl is the common nonnegative multiplier; dij is theKronecker delta):

�B ¼ 2��1� Bð Þ2cos2 !

Ec; ð13Þ

�"pv ¼ 2��1� Bð Þ2EB sin

2 !

pEcand ð14Þ

�"ps ¼2��

q

M 2p2Mohr-Coulomb failure criterion

2��q

M*2 pr p=prð Þnð Þ2Hoek-Brown failure criterion

8>><>>:

ð15Þ

The flow rules above can be expressed in tensorial forms,which will later be needed for the derivation of the tangentstiffness tensor:

2.3. Formulation of Tangent Stiffness

[20] The formulation of the tangent stiffness tensor Lijkl,which will be used for the localization analysis, is brieflypresented in this section. Combining equation (13) and (16),we define:

Qij ¼�"pij�B

ð17Þ

[21] Using equations (7) and (17), the stress increment canbe expressed:

��ij ¼ 1� #Bð ÞDijkl�"kl � 1� #Bð ÞDijklQkl þ #�ij1� #Bð Þ

� ��B ð18Þ

From the consistency condition of the yield function(equations (11) or (12)) we have:

�y ¼ @y

@B�Bþ @y

@p

@p

@�ij��ij þ @y

@q

@q

@�ij��ij ¼ 0 ð19Þ

Substitution of dsij from equation (18) in (19) results in:

�y ¼ 1� #Bð ÞYijDijkl�"kl þ @y

@B� YijXij

�B ¼ 0 ð20Þ

where Xij ¼ 1� #Bð ÞDijklQkl þ #�ij1� #Bð Þ

and Yij ¼ @y

@p

@p

@�ijþ @y

@q

@q

@�ij:

The breakage increment is obtained from equation (20) as,

�B ¼ 1� #Bð ÞYijDijkl�"klYijXij � @y=@B

: ð21Þ

The stress increment can then be rewritten as

��ij ¼ Lijkl�"kl; ð22Þ

�"pij ¼�� 2

1� Bð Þ2EB sin2 !

pEc

�ij3þ 3sijM2p2

!Mohr-Coulomb failure criterion

�� 21� Bð Þ2EB sin

2 !

pEc

�ij3þ 3sij

M*2 pr p=prð Þnð Þ2 !

Hoek-Brown failure criterion:

8>>>><>>>>:

ð16Þ

Figure 1. Typical yield surface in q‐p‐B space for different failure criteria, (a) Mohr‐Coulomb;(b) Hoek‐Brown.


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where the elastic‐plastic‐breakage tangent stiffness tensorLijkl is,

Lijkl ¼ 1� #Bð ÞDijkl � 1� #Bð ÞXijYmnDmnkl

YijXij � @y=@B: ð23Þ

The modulus tensor (equation (23)) can be described interms of material shear and bulk modulus (G and K,respectively), slope of yield locus (m), dilatancy factor (bp)and stress state (p, q) at the crushing inception (B = 0) as,

�1 ¼M2p2=qEc for Mohr-Coulomb failure criterion

M*2p2r pf =pr� �2n

=qEc for Hoek-Brown failure criterion

8<:

and (see also the flow rules in section 2.2)

�p ¼ ��"pv=�"ps ð25Þ

[22] In the above expressions, the definitions of the yieldparameter m and dilatancy factor bp are identical to thoseused in previous works [e.g., Rudnicki and Rice, 1975;Wong et al., 1997; Issen and Rudnicki, 2000]. In the currentbreakage model these parameters also have links with otherphysical quantities governing the response of the model. Inparticular, m is governed by the shape of the yield surface,which through the breakage mechanics theory is a directconsequence of the assumption that the grain crushing dis-sipation is equal to the loss in the residual breakage energy;bp depends on the relative amount of dissipation due to graincrushing and pore collapse, through the coupling parameterw (see evolution rules for breakage and plastic strains insection 2.2).

3. Model Assessment

[23] In this section the model is assessed against availableexperimental data. This is essential to obtain meaningfulmodel parameters related to real material behavior, beforeexamining the localization features of the model. Becausewe focus on the formation of compaction bands in porousrocks, we selected the highly porous (23%) Bentheim

sandstone, which exhibits significant compactive behaviorand distinct compaction bands in experiments. Moreover,this rock is nearly monomineralic (95% quartz, 2% ortho-clase and 3% kaolinite). The model capability in capturingthe behavior of other sandstones and the evolving gsd due tograin crushing has been demonstrated by Nguyen and Einav[2009] and Nguyen et al. [2008]. The application of thecurrent analysis to these sandstones is therefore straight-forward, including the model assessment and localizationanalysis. Furthermore, the connections between the internalvariables and their underlying micromechanisms will beexplored in a parametric study.

3.1. Model Parameters and Response

[24] All parameters for the above breakage model wereobtained from the experimentally observed stress strainresponse and grain size distribution of Bentheim sandstone[Wong et al., 2001; Baud et al., 2004]. The stress‐strainresponse in elastic regime was used to calibrate the elasticmoduli (K and G), while the critical breakage energy Ec wasdetermined from the following formulation EC = Pcr

2 #/2K[Einav, 2007c], in which the (isotropic) crushing pressure

(Pcr) was obtained from the isotropic compression. Accord-ing to experimental findings [Wong et al., 2001; Baud et al.,2006] for Bentheim sandstone this crushing pressure Pcr

varies from 390 MPa to 420 MPa. It is important to note thatthe above range of Pcr were experimentally obtained fromboth dry [Wong et al., 2001; Baud et al., 2004] and wet[Baud et al., 2006; Tembe et al., 2008] Bentheim sandstonesamples. The parameter M, governing the frictional shear,was determined from the ratio of ultimate shear stress (qf) tomean stress (pf) from any conventional triaxial tests. For thegrading index #, equation (2) was used which requiresinformation on the initial and ultimate gsd’s. From experi-ments [Schutjens et al., 1995] we found that the grain sizedistribution is initially very narrow, having maximum graindiameter of 550 mm and minimum grain diameter of 50 mm.We assume that both the initial (g0(d)) and the ultimate(gu(d)) gsds are of power laws:

g0 dð Þ ¼ 3� �0ð Þd2��0

d3��0M � d3��0

m; gu dð Þ ¼ 3� �uð Þd2��u

d3��uM � d3��u

mð26Þ

where dm is the minimum grain size, dM is the maximumgrain size and a0 and au are the power law coefficients.We use a0 = −4 to reflect a narrow initial gsd; this is inagreement with the observation by Schutjens et al. [1995].For the ultimate fractal gsd the fractal dimension is taken asau = 2.7. This fractal dimension for the ultimate gsd is wellinside the proposed range of 2.5–2.8 [Sammis et al., 1986].The remaining parameter, the coupling angle w, was deter-mined from the slope of inelastic stress‐strain response. Allthe required parameters obtained for the breakage constitu-tive model in this study are listed in Table 1.

Lijkl ¼ Dijkl �3GNij þ �pK�ij þ #�ij�1 cos2 !� �

3GNkl þ �K�klð Þ# q� �pð Þ�1 cos2 !þ 6EBG�2

1 cos2 !

� �= #q�1 þ 6Gð Þ� �þ 3Gþ ��pK

where Nij ¼ sij=q; ð24Þ

Table 1. Constitutive Parameters of Bentheim Sandstone

Parametersa Mohr‐Coulomb Hoek‐Brown Type

G 7588 MPa 7588 MPaK 13833 MPa 13833 MPaM or M* 1.7 210Ec 4.67 MPa 5.41 MPan ‐ 0.6w 70° 70°

aFurthermore, # = 0.85. Distinguished from the above parameters, the #is a grading index, disconnected from stress‐strain data.


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[25] The model responses are plotted in Figure 2. Goodagreement between numerical predictions and experimentalobservations can be seen, thus confirming the above param-eter calibration. We can see that the breakage model basedon linear elasticity works well in this case for Bentheimsandstone, although in general pressure‐dependent elasticitymay be required [Nguyen and Einav, 2009].[26] The use of a limited number of physically identifiable

and determinable parameters and the agreement betweenexperimental observation and numerical prediction highlightthe importance of first principles in developing constitutivemodels. We note that the model was designed to tacklehigh‐pressure behavior due to grain crushing effects. Theincorporation of cementation effects and the modeling ofcement failure to deal with low‐pressure behavior within theframework of breakage mechanics would require resorting

to the micromechanics of bonded grains, rather than addingany fitting parameters. This enhancement of the theory isbeing carried out, with promising preliminary results. Asuccessful example of a further development of breakagemechanics theory to model grain crushing in unsaturatedgranular materials, based solely on first principles at thegrain scales, has been demonstrated in a recent work byBuscarnera and Einav [2011].

3.2. Effect of gsd and Pore Collapse on MaterialConstitutive Behavior

[27] In breakage mechanics theory, the grading index #(equation (2)) depends on the initial and ultimate gsd’s ofthe granular materials. The ultimate gsd can be convenientlyassumed to be fractal capped by an ultimate minimumparticle size and maximum particle size, and a reasonable

Figure 2. Numerical (using the model described in section 2) and experimental [Baud et al., 2004]comparisons of stress‐strain responses of Bentheim sandstone under drained triaxial loading at differentconfining pressures; (a) mean stress versus volumetric strain; (b) differential stress against axial strain.

Figure 3. Stress‐strain responses of Bentheim sandstone; (a) varying # with constant w; (b) varying wwith constant #.


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fractal dimension [Einav, 2007a]. It represents a fixed pointthat attracts the evolution of current gsd [Ben‐Nun et al.,2010]. On the other hand, the initial gsd depends on theinitial microstructure of the material, which is essentiallycontrolled by long‐term processes of healing and granula-tion [e.g., Marone et al., 1995; Renard et al., 2000]. If theinitial gsd is close to the ultimate gsd (i.e., smaller # for wellsorted gsd) higher external energy is required to crush thematerial compared to those having higher # [Einav, 2007b].This shows the important role of the grading index # ingoverning the crushing potential of the granular material.Figure 3a illustrates the effect of # on the volumetric stress‐strain behavior of a typical sandstone, while keeping the othermodel parameters fixed. It can be seen that the inelasticresponse is becoming softer with increasing #, consistentwith its notional role for crushing potential.[28] The effect of pore collapse arising passively from

fragment rearrangement, is presented in Figure 3b, simplyby varying the parameter w. We note that this parameteronly implicitly characterizes the role of the initial porosityand intergranular contact friction in governing the materialresponse: higher porosity (of both the grains and their rep-resentative volume element) and lower grain contact frictionwould result in higher stress drop following a crushingevent. As an example of this effect let us consider the caseof two materials with identical gsd, and hence same #. If,however, one material contains microporous grains but notthe other, it is straightforward to demonstrate that thematerials with microporous grains must suffer more porosityreduction than the other one. The volumetric plastic strain isthe internal variable representing these effects in the currentbreakage model. An explicit link between this plastic strain,its associated coupling parameter w, and the actual porosityreduction is not provided in this study. However, the readercan refer to a more sophisticated model in which the materialporosity acts as an internal variable of the continuum model[Rubin and Einav, 2011]. According to equations (13) and(16), decreasing w signifies the increase in breakage dissi-

pation, which indicates the dominance of dissipation due tograin crushing over the grain rearrangement and intergran-ular contact friction [Einav, 2007b]. On the other hand thevolumetric plastic dissipation dominates the failure processif the coupling angle w is higher. The maximum value of wis 90° which implies the limit of zero breakage increment(equation (13)) and total inelastic dissipation that is purelyplastic. Therefore themacroscopic material response is elasticperfectly plastic, as shown in Figure 3b. Any intermediatevalue of w between 0°–90° denotes a combined elasticbreakage plastic response.[29] The roles of the material initial microstructures in

governing the response of the material model can be clearlyseen in Figure 3, with reference to the changes in theirmicrostructural parameters # and w. This unique featuredistinguishes the present breakage model from many exist-ing plasticity‐based models in the literature. A hardeningparameter, as usually required in these plasticity‐like mod-els, is not necessarily derived here; this is because thehardening behavior of the model is directly governed by thegrading index # and pore collapse parameter w, besidesother elasticity properties.

4. Condition of Localization

[30] The presence of local heterogeneity/weak zone atmicroscopic level produces stress concentration during thedeformation process. Therefore the deformation behaviorof granular rock material under some circumstances canbifurcate from homogeneous to inhomogeneous localiza-tion. In such a case, a localization band is observed atmacroscopic level. We use the discontinuous bifurcationcondition [Rudnicki and Rice, 1975] to detect the formationof compaction band. The classical discontinuous bifurcationcondition is:

n � Li � Lo� �

: _eo þ Að Þ � _m ¼ 0: ð27Þ

where n is the band orientation vector (see Figure 4); Li

(equation (23)) is the tangent stiffness tensor inside thelocalization zone; Lo and _eo are the tangent stiffness andstrain rate tensor outside the localization zone, respectively;m is the orientation of relative velocity between oppositesides of the localization band; A = n · Li · n is the locali-zation tensor, also termed the acoustic tensor.[31] Equation (27) is usually simplified considering iden-

tical tangent modulus tensors inside and outside the locali-zation band [Rudnicki and Rice, 1975; Issen and Rudnicki,2001; Chemenda, 2007]. Therefore the modified bifurca-tion condition is,

n � Li � n�� ¼ Aj j ¼ 0 ð28Þ

[32] This assumption, however, only holds either forincrementally linear models [Chambon et al., 2000] or ifthere are constraints over continuously evolving strain rates[Neilsen and Schreyer, 1993]. Due to the fact that the tan-gent stiffnesses of the material inside and outside the bandare different in the case of our breakage model, the condition

Figure 4. Schematic representation of the orientation of alocalization band within a cylindrical sample.


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for discontinuous bifurcation is [Rudnicki and Rice, 1975;Chambon et al., 2000],

n � Li � n�� ¼ Aj j � 0: ð29Þ

[33] The normal vector n of the localization band isillustrated in Figure 4. This normal vector can be expressedas follows,

ni ¼n1n2n3

8<:

9=; ¼

cos� � cos sin�

cos� � sin

8<:

9=;: ð30Þ

For axisymmetric cases, equation (30) can be simplified tohave the following form:

ni ¼n1n2n3

8<:

9=; ¼

0sin�cos�

8<:

9=;: ð31Þ

5. Theoretical Analysis of Compaction BandFormation

[34] The breakage model described in section 2 and thelocalization condition (equation (29)) are used in a theoreticalanalysis of compaction band formation in porous rocks. Twodifferent loading conditions are taken into account: 1‐Dcompression and triaxial drained loading. The formation ofonly pure compaction band is assumed in the 1‐D com-pression case to make it analytically tractable. Neverthelessthis gives us an understanding of the micromechanics ofcompaction band formation in this simplest case. Similarassumption and analysis have also been used by Chemenda[2009]. On the other hand, drained triaxial loading conditionis chosen because it is the most widely used loading path formaterial characterization. Due to complicacy of the modelwe perform the entire analysis numerically for this case. Theresults of localization analysis using the above mentionedloading conditions along with discussions are presented insections 5.1 and 5.2.

5.1. The 1‐D Compression Test

[35] We consider only pure compaction band (a = ±p/2)in this case. Therefore the normal vector (equation (31))reduces to

ni ¼ 0 0 1f gT : ð32Þ

Substituting equation (32) into (29) yields the bifurcationcondition:

L3333 � 0: ð33Þ

The condition of compaction localization is determined at theonset of yielding/crushing (B = 0). The material is loaded in1‐D compression from zero stress state. The corresponding

stress and strain tensors for 1‐D compression are presentedin matrix form,

�ij ¼

�

1� 0 0

0�

1� 0

0 0 �

266664

377775; ð34Þ

"ij ¼

0 0 0

0 0 0

0 0�

K þ 4G=3

2666664

3777775 ð35Þ

where n is the Poisson’s ratio and s (= s33) is the principalcompressive stress at the onset of yielding. L3333 is deter-mined from equation (24) as

L3333 ¼ K þ 4G=3ð Þ

� Kþ4G=3ð Þ tan2 !þ2��2ð Þtan2 !þ 2

#

� �1��2ð Þþ�2 1� tan2 !

4G

3K

� �� þ �2

2 sec2 !

1��2ð Þ3K

4Gþ1

� �ð36Þ

where �2 ¼ 4G2=K2M2:

From equation (33) and (36) we obtain the critical gradingindex #cr for the onset of pure compaction:

#cr ¼ 2 2þ �2 tan2 !

1� �21þ 4G

3K

� �� 2

2 sec2 !

1� �2ð Þ2 1þ 3K

4G

� �" #�1

:

ð37Þ

[36] The variation in the critical grading index (#cr) isplotted in Figure 5 against the Poisson’s ratio, for differentvalues of M and w. It shows that the critical grading index(#cr) required for the compaction localization decreases withincreasing pore collapse parameter (w), and vice versa. Inother words, according to the current formulation the com-paction band formation arises from the combined effects ofgrain crushing and pore collapse. Any reduction in onecontributor’s effects indicates a different initial microstruc-ture, and hence requires the compensations from the other tomaintain the same probability for compaction band forma-tion. Our analysis shows that crushable granulated rockshaving larger #, i.e., those that initially consist of uniformlygraded particles, should be more susceptible to compactionlocalization. From the micromechanical point of view, thepotential for particle crushing in such a case is smaller thanthose having higher # [Einav, 2007a, 2007b]. This resultsin a stiffer, and less compacting, macroscopic response(Figure 3a). Therefore in such a case, the effects of porecollapse on the macroscopic behavior must be significantenough to maintain the same level of compacting response.Higher values of w are therefore required.


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5.2. Conventional Drained Triaxial Test

[37] The localization analysis at the onset of inelasticdeformation was also carried out under drained loading usingthe discontinuous bifurcation condition (equation (29)).During loading the principal stresses maintain the following

condition: s3 > s1 = s2. Figure 6 represents the initial yieldenvelopes obtained from our theoretical models (for dif-ferent failure criteria) and experiments [Baud et al., 2004].We determine the set of stress state during the initialyielding for which compaction bands most likely occur. The

Figure 5. Variation of #cr with Poisson’s ratio n, for various w andM in 1‐D compression. The line #cr =0.9 is added as an example to highlight a typical uniformly graded with narrow gsd. Any point below thisline would mark pure compaction for that material.

Figure 6. Initial yield envelopes and predicted stress states at the formation of compaction localizationfor Bentheim sandstone. (a) Mohr‐Coulomb; (b) Hoek‐Brown.


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thick black line over the initial yield surface in Figure 6denotes the numerically predicted zone where either shearenhanced or pure compaction bands may develop. The opencircular symbols in brittle‐ductile transitional regime denotethe experimentally observed stresses at the onset of shearenhanced compaction bands whereas the triangular solidsymbols are the peak stresses during brittle failure [Baudet al., 2004]. In very high pressure regime no localizationfailure is observed. This may be manifested due to homo-geneous deformation or diffuse compaction band formationover the entire sample under high confining stress. Similardamage distribution in delocalized manner in high‐pressureregime was also experimentally observed by Wong et al.[2001]. Furthermore, in our numerical test, shearing thematerial model beyond the initial yield surface and at rela-tively high pressure (e.g., up to 350 MPa), will eventuallylead to compaction localization. This pressure is, however,smaller than the experimental maximum pressure (450 MPa)at which compaction localization can be observed [Baudet al., 2004]. As also numerically experienced, the closer tothe isotropic compression line the loading path is, the easierthe deformation would evolve into cataclastic flow withoutany compaction localization.[38] One can also note in Figure 6 that the breakage model

with Hoek‐Brown type frictional criterion better fits the entireexperimental yield surface than the one based on Mohr‐Coulomb type failure criterion. This is because the Mohr‐Coulomb critical state line (equation (10)) does not allowthe breakage yield surface to match the experimental data atlow confining pressure. We may need a more sophisticatedmodel which gives the flexibility in critical state response,as suggested by Nguyen and Einav [2009]. The Hoek‐Brown type dissipation mechanism could be a better alter-native in this context due to the pressure‐dependent natureof the critical state line.[39] The experimental and theoretical inelastic compac-

tion factors DFp/D"ap (≈ D"v

p/D"ap, according to Wong et al.

[1997]) are compared in Figure 7 for Bentheim sandstone.Baud et al. [2006] also made a comparative study between

the experimental and theoretical inelastic compaction factorsusing DiMaggio and Sandler cap model [DiMaggio andSandler, 1971] and associative flow rules. However, thismodel predicts inelastic compaction factor higher than theobserved experimental data. We show here that the experi-mental inelastic volumetric strains are in good agreementwith the predictions by the breakage constitutive model.[40] Figure 8 highlights the contours of the determinant of

the acoustic tensor (or the strain localization tensor) againstthe band orientation angle (a) and the slope of yield surface(m) for Bentheim sandstone using the breakage modelassociated with Mohr‐Coulomb failure criterion. The colorbar in this plot indicates the values of the determinant ofthe acoustic tensor. In Figure 8 the innermost zone denotesnegative determinant of the acoustic tensor and it is boundedby ∣Aij∣ = 0. According to the discontinuous bifurcationcriterion (equation (29)) compaction localization is possibleonly within this zone. We already presented this similarzone in p‐q space through the thick black line in Figure 6. Itis clear from Figure 8 that for any stress state, in terms of m,we get a set of orientation angles which are physicallyequally favorable in forming compaction or shear enhancedcompaction band. In practice, localization band having ori-entation angle 0° ≤ a ≤ 10° can be classified as pure com-paction band whereas those with 10° ≤ a ≤ 45° are treated asshear enhanced compaction bands [Baud et al., 2004]. Ouranalysis predicts that the band orientation falls within therange of 0° to 40° which is quite close to the experimentalobservations [Baud et al., 2004].[41] The orientation angles corresponding to different stress

states at which compaction localization occur are plottedin Figure 9, showing different modes of localization. Thelocation and the size of the localization zone depicted on theinitial yield surface are completely dependent on the stressstate, grain size distribution and porosity of the respectivematerial. Shear bands are observed at the transition of brittleand ductile failure. On the other hand at the right boundary ofthe localization zone, pure compaction bands are observeddue to ductile nature of porous rocks under shearing at high

Figure 7. Prediction of the normality condition in Bentheim sandstone.


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confining pressure [Klein et al., 2001]. Beyond this zone nolocalization is observed, due to homogeneous deformationunder high confining pressure [Wong et al., 2001]. We notethat our model aims at tackling high‐pressure behavior withgrain crushing in porous rocks. Ongoing efforts are beingcarried out to equip the model with new capabilities tocapture the material behavior at low confining pressure,where the formation of extension/dilation bands [Bésuelle,2001] is expected.[42] We compare the predicted values of m and bp at the

onset of localization against the experimental results inFigure 10. Increase in the absolute value of m and bpindicates transition from compaction to shear band. Our

predictions fairly agree with experimental observation. It isclear (Figure 10) that the predicted range of m is almostidentical to its experimental counterpart. According to theexperimental observation byWong et al. [2001], m should begreater than −1.5 for compaction localization which is wellpredicted by our breakage model. The experimentallyobserved bp is higher than our prediction as shown inFigure 10. However, Figure 10 shows that enhancement tothe model behavior (e.g., using Hoek‐Brown criterion) leads

Figure 8. Contours of the determinant of the acoustic tensor for Bentheim sandstone at crushing incep-tion (B = 0).

Figure 9. Stress states and corresponding localization bandorientations over initial yield surface (Bentheim sandstone).

Figure 10. Possible zone of compaction localization in m −bp space for Bentheim sandstone. (The values of m and bp

calculated by some researchers [Issen and Rudnicki, 2000;Wong et al., 2001] were in p − � (� =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=2sijsij

p) space. Due

to the use of p–q (q =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3=2sijsij

p) stress space in this study,

appropriate adjustments were made for the presentation inp–q space.).


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to a better model prediction. Therefore improvements in themodel’s capabilities may be a hint for better predictions thatcan help explain the current mismatch. This is the topic ofour ongoing study.

5.3. Parametric Study

[43] Given the links between the model parameters (# andw) and the initial microstructures (gsd and porosity) of thematerial (section 3.2), a parametric study is carried out toexplore the effects of initial microstructures on the forma-tion of compaction localization. Figure 11 shows the effectsof initial gsd (through #) and pore collapse (through w) onthe formation of compaction localization. As can be seen,the size of compaction localization zone strongly dependson the initial gsd and pore collapse effects. It is quite clearthat the possible range of stress state (in terms of m) to triggercompaction localization gets narrower as w becomes smaller.We know that this parameter w reflects the effects of porecollapse on the material responses. In all cases, the dilatancyfactor (bp) decreases (in absolute value) with decrease in wfor any m, which indicates a reduction in inelastic volumechange due lower pore collapse effects. Taking a close lookat Figures 11a–11d, one can see the change in favorablestress state for compaction band formation due to the vari-ation in #. The range of m − bp associated with localization

zone reduces with the decrease in # and vice versa. Increasein # indicates that the initial gsd is poorly graded and faraway from ultimate gsd and therefore directly increases thechance of compaction localization. These numerical pre-dictions confirm the analytical results in 1‐D compressioncase.[44] It can be seen in Figure 11 that the compaction

localization regime extends to higher pressures (indicated bysmaller values of m) when the grain structure is poorlysorted (high #) and/or the effects of pore collapse are high(high w). Material with well‐sorted grain structure and lowporosity are therefore less likely to form compaction bands.The theoretical extreme cases totally precluding compactionlocalization are #→0 (initial and ultimate gsd’s coincide),and w→0 (very low internal porosity within the grains). Thecondition for no compaction localization, however, dependson the type of loading, and only in specific cases can ananalytical expression be derived (see section 5.1).We can seefrom the trend in Figure 5 that for certain values w <45° (low pore collapse effects) compaction localization cannever be observed regardless of what the initial gsd is.[45] The formation of compaction localization, as shown

in this paper, depends not only on the initial grain sizedistribution, but also on other features of the microstructure(e.g., initial porosity). From the experimental point of view,

Figure 11. Variation in zone of possible compaction localization for different # and w for Bentheimsandstone.


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another factor contributing to the formation of compactionlocalization could be the stress distribution in rock sam-ples, which is not always perfectly uniform, but dependson both the material microstructure and the specimengeometry. It is therefore possible that compaction locali-zation can develop in relatively homogeneous samples, asnumerically observed by Katsman et al. [2005] and Wanget al. [2008], and experimentally observed by Louis et al.[2007].

6. Concluding Remarks

[46] We have presented a novel continuum modelingapproach to capturing the micromechanics of compactionband formation in porous rocks. This model possessesexplicit links between the evolution of the gsd and themacromechanical behavior of porous rocks. Our predictionsare in agreement with experimental observations in terms ofstress strain behavior of porous rocks, yield locus, favorablestress states for compaction localization and compactionband orientations. It also explains the micromechanicsbehind the formation of compaction bands. The role of theinitial gsd, via the grading index #, is found to be a keyfactor for localization deformation. Any change to this ini-tial grading would affect the range of favorable stress statefor compaction band formation. On the other hand, theparameter w, as an indicator of pore collapse effects on thematerial response, controls both the favorable stress statesfor compaction localization and the inelastic volume changeduring localization deformation.[47] At this point we would like to emphasize the important

feature of the bifurcation analysis thus presented. We haveemployed the general bifurcation condition by Rudnicki andRice [1975] (R&R), applicable to any constitutive models.Our results have been presented using the usual yieldparameter m and dilatancy factor bp as suggested in R&R.However, in the proposed breakage model these parametersare strongly linked with the actual driving physical mechan-isms. In particular, the shape of the yield surface representedby m is here a direct consequence of the assumption that thegrain crushing dissipation is equal to the loss in the residualbreakage energy; b is linked with the competition betweenthe dissipations due to grain crushing and pore collapse.These two connections underpin equations (23)–(24) and(36)–(37). These features are usually missing in plasticity‐based models for geomaterials.[48] This simple approach provides a good understanding

of the underlying mechanisms governing compaction local-ization. It still requires further improvements and investiga-tions, which are an ongoing research in our group toward abetter understanding of the complex intrinsicmicromechanicsbehind compaction localization in porous rocks.

[49] Acknowledgments. Arghya Das wishes to thank the Universityof Sydney International Scholarship scheme. Giang Nguyen and Itai Einavwould like to acknowledge the Australian Research Council for the Discov-ery Projects funding scheme (projects DP110102645 and DP0986876).Support to Giang Nguyen through the University of Sydney PostdoctoralFellowship scheme is also gratefully acknowledged. Finally, the authorswould like to thank the two reviewers, Teng‐fong Wong and Patrick Baud,for their constructive comments that helped us improve the clarity and con-sistency of the paper.

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A. Das, I. Einav, and G. D. Nguyen, School of Civil Engineering,University of Sydney, Sydney NSW 2006, Australia. ([email protected]; [email protected]; [email protected])


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