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Corresponding author: G. D. Nguyen, School of Civil Engineering, The University of Sydney, Sydney NSW 2006, Australia. (email@example.com)
 We analyze the propagation of compaction bands in high porosity sandstones using a constitutive model based on breakage mechanics theory. This analysis follows the work by Das et al.  on the initiation of compaction bands employing the same theory. In both studies, the theory exploits the links between the stresses and strains, and the micromechanics of grain crushing and pore collapse, giving the derived constitutive models advantages over previous models. In the current post localization analysis, the bifurcation instability of the continuum model is suppressed by the use of a rate-dependent regularization. This allows us to perform a series of finite element analyses of drained triaxial tests on porous sandstone specimens. The obtained numerical results compare well with experimental counterparts, in terms of both the initiation and propagation of compaction bands, besides the macroscopic stress-strain responses. On this basis, a parametric study is carried out to explore the effects of loading rate, degree of structural imperfections, and confining pressure on the propagation of compaction bands.
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 Compaction bands are narrow, tabular deformation zones that are oriented at high angles to the maximum compressive stress in high porosity rocks. Both field [Mollema and Antonellini, 1996; Sternlof et al., 2005] and experimental observations [Baud et al., 2004; Charalampidou et al., 2011; Haimson and Kovacich, 2003; Olsson, 1999; Stanchits et al., 2009; Vajdova and Wong, 2003; Wong et al., 2001] showed that the initiation and propagation of compaction bands depend on grain crushing, pore collapse, and grain sliding. The ways these mechanisms control the initiation and propagation of compaction bands as a function of the material properties are still an open question, essentially for advancing theoretical models of faulting in porous rocks and the associated structural changes.
 Experimental studies usually focus on the development of compaction bands in small sandstone samples in triaxial tests under different loading and geometrical conditions. Observations using Acoustic Emission (AE) [Baud et al., 2004; Digiovanni et al., 2000; Fortin et al., 2006; Tembe et al., 2008] and X-ray tomography [Charalampidou et al., 2011; Wolf et al., 2003] demonstrate that compaction bands develop at discrete locations and are perpendicular to the major principal stress. Microstructural analysis of deformed specimens confirmed a spatial correspondence of AE hypocenter distribution and compaction bands [Baud et al., 2004]. These bands seem to propagate very quickly, as they are usually seen across the full width of the specimens, from the two ends toward the mid height of the specimen [Digiovanni et al., 2000]. At first, intergranular bond breaking takes place and diffuse compaction develops, which is then overprinted by more pronounced localized compaction bands accompanied by grain crushing [Digiovanni et al. 2000]. Klein et al.  observed the propagation and development of discrete compaction bands in brittle to ductile transition regime using drained triaxial compression tests in Bentheim sandstone (with porosity of about 22%). The observed episodic stress drops corresponding to accumulated AE hypocenters are the evidences of discrete compaction band developed axially within the sample, and the cumulative number of stress drops is approximately equal to the number of compaction band formations [Klein et al., 2001].
 The lateral propagation of a single compaction band has been studied by introducing strong imperfections in forms of notches at mid height of the cylindrical specimens in triaxial tests [Stanchits et al., 2009; Tembe et al., 2006; Vajdova and Wong, 2003]. In such cases, a single compaction band is forced to propagate in transverse/lateral direction from the tips toward the center of the specimen. The effects of loading rate on the orientation of the band can be carefully studied in such cases. The propagation speed in transverse direction is found to be significantly higher than the axial shortening of the entire rock sample [Stanchits et al., 2009; Vajdova and Wong, 2003]. This explains why these compaction bands usually occur nearly instantaneously across the width (in both notched and unnotched specimens).
 While the onset of compaction localization has been widely studied and discussed in the literature [see Das et al., 2011] for a complete review), their propagations seem to have received less attention. Olsson  developed analytical solution to the propagation of compaction front in porous sandstone by balancing the mass and energy inside and outside compactive zone. In line with the experimental work on the propagation of a single compaction band in notched specimen, theoretical work by Rudnicki and Sternlof  provides information on the energy release due to the advancement of an existing compaction band. Alternatively, continuum models based on plasticity theory have been used by Chemenda [2009, 2011] and Oka et al.  to numerically study the mechanisms of compaction band initiation and propagation. Similar studies using lattice element method and simplified elastoplastic models appear in Katsman and Aharonov  and Katsman et al. . Despite the valuable gains from previous studies, it is clear that theoretical works are still far from being able to reliably explain and predict the formation and propagation of compaction bands in porous rocks. The key ingredient that is missing in current continuum models is the link between the observable localization bands and the true origins of compaction localization. None of the existing constitutive models possess an explicit link between the only internal variable, plastic strain, and the evolving grain size distribution (gsd), or the pore collapse during the propagation of compaction bands. In the words of Krajcinovic , “an internal variable inferred from the phenomenological evidence and selected to fit a particular stress-strain curve may provide a result that pleases the eye but seldom contributes to the understanding of the processes represented by the fitted curve.” Alternatively, while localization patterns that match the experimentally observable counterparts have been reproduced in several numerical works [e.g., Chemenda, 2009, 2011; Katsman and Aharonov, 2006; Katsman et al., 2005], the stress-strain responses corresponding to such patterns are left untouched. In contrast, in other studies only the stress-strain behavior is focused on and verified with experimental data, while the validation of the localization pattern is not experimentally confirmed (e.g., Oka et al. ). All these issues put forward a question for future research: are we able to capture all observable features related to the initiation and propagation of a compaction band, while still being able to keep track of the evolving physics behind it?
 We are going to provide a possible answer to this question with a model that encapsulates the underlying physics of compactive failure in crushable granular rocks. Such a model, based on the breakage mechanics theory [Einav, 2007a, 2007b], has been successfully developed and used for quantifying the microstructural effects such as evolving gsd and pore collapse on the initiation of compaction bands [Das et al., 2011]. It is also important to address the fact that the model relies on only a few parameters determined directly from experiments [Das et al., 2011]. Both the experimentally observed stress-strain responses and onset of compaction bands under high confining pressures are predicted well using the same set of parameters. More importantly, as shown in our previous study [Das et al., 2011], this model provides a map showing the effects of both material microstructures and stress conditions on the failure of the material in diffuse compactive mode, discrete (localized) compactive and shear modes or their combinations. This is an important feature of the model that, to our knowledge, any other existing constitutive models for porous rocks do not possess.
 It has been experimentally known that compaction localization observed at the macro (continuum) scale of porous granular rocks is a result of grain crushing and pore collapse at the micro (grain) scale [Digiovanni et al., 2000; Fossen et al., 2007; Menéndez et al., 1996]. In continuum modeling, this compaction localization is usually assumed to associate with the bifurcation conditions of continuum mechanics [Rudnicki and Rice, 1975]. Although these bifurcation conditions have been widely used to predict the onset and orientation of localization bands [Borja and Aydin, 2004; Chemenda, 2009; Das et al., 2011; Issen and Rudnicki, 2000, 2001; Olsson, 1999], the prediction of their propagation (post-localization) using solutions of boundary value problems (BVP) requires the enrichment of continuum models to overcome the instability caused by this bifurcation [Etse and Willam, 1999; Needleman, 1988; Schreyer and Neilsen, 1996; Wang et al., 1997]. This is because models based on conventional continuum mechanics lack a certain length scale, such that their employment in the numerical analysis of BVPs usually leads to the dependency of the solutions on the spatial discretization. The breakage mechanics model used in Das et al.  is not an exception. The introduction of material rate dependency in the constitutive models is one of the possible ways to overcome pathological mesh sensitivity of ill-posed BVPs [Needleman, 1988], as this implicitly incorporates a length scale that ensures the positive definiteness of the localization (acoustic) tensor. Although for very slow loading, the macroscopic behavior of sandstone samples is usually thought to be rate-independent, the onset and propagation of compaction bands is associated with microstructural changes at the microscopic scale due to grain crushing and pore collapse that may be rate-sensitive [Yamamuro and Lade, 1993]. In this sense, the introduction of rate-dependent evolutions of internal variables (breakage and plastic strain) representing the grain crushing and pore collapse can also be seen as a natural extension of the current breakage model. In this study, the Perzyna's type [Perzyna, 1966] viscous regularization is employed, with its viscosity parameters determined from experiments.
 This paper is organized as follows. Section 2 provides a brief representation of a rate-independent constitutive model based on breakage mechanics theory. This is then followed in section 3 by a recap of our previous analysis of the onset of compaction localization in high porosity sandstones [Das et al., 2011], as a basis for the post-localization analysis in the current study. A rate-dependent breakage model is developed in section 4 along with a localization analysis. Toward this analysis, a tangent stiffness tensor consistent with the implicit stress return algorithm is derived, and the rate-independent and rate-dependent responses of the breakage model are investigated. Thereafter, finite element (FE) analyses of drained triaxial and plane strain compression tests are performed in section 5 to study the orientation, formation and propagation of both compactive shear bands and pure compaction bands. Particular attention is paid to the formation of compaction bands in specimens with structural imperfections under different loading rates. The obtained numerical results are validated against experimental observations. On this basis, a parametric study is carried out in section 6 to observe the effects of confining pressure, strain rate variation, and degree of structural imperfections on the orientation and propagation of compaction bands in notched sandstone specimens.
2 A Model Based on Breakage Mechanics Theory
2.1 Breakage Mechanics Theory
 Grain crushing followed by grain sliding and pore collapse are micromechanical processes governing the deformation of crushable granular materials under high confining pressures. The breakage mechanics theory [Einav, 2007a, 2007b] was developed considering these basic mechanisms and the associated energy transformation during the crushing process. A proper energy scaling law was established [Einav, 2007a, 2007b] to link the macroscopic and microscopic stored elastic energy, whereby the total stored elastic energy of the material is assumed to distribute among the individual grains proportional to their surface areas. During the process of fragmentation, part of this stored energy is released as strictly positive dissipation rate that controls the evolution of an internal state variable called Breakage, B. The evolving gsd can be determined in a simplest way by employing B as a scaler for linear interpolation between the initial gsd and ultimate gsd (equation ((1))):
where x is the grain diameter, p0 (x) is the initial gsd and pu (x) is the ultimate gsd, which can be conveniently assumed to be of fractal type [Sammis et al., 1987; Turcotte, 1986]. The evolution law of B is derived from the postulation that the dissipation due to grain crushing is equal to the loss in the residual breakage energy [Einav, 2007a]. The details on the fundamentals of breakage mechanics theory, along with a series of model developments and various applications in geotechnical engineering and geophysics, can be found in earlier works [Buscarnera and Einav, 2012; Einav and Valdes, 2008; Nguyen and Einav, 2009; Zhang et al., 2012].
2.2 A Constitutive Model Based on Breakage Mechanics
 A brief outline of a breakage constitutive model used in this study is next presented. The essential constitutive equations include the stress-strain-breakage relationship, a breakage-yield function, and corresponding evolution rules, all of which have been derived within a rigorous thermomechanical framework.
 The stress-strain relationship is
where σij is Cauchy stress tensor, the elastic strain tensors, and Dijkl the linear (isotropic) elastic stiffness tensor expressed in terms of the shear (G) and bulk (K) modulii. The grading index ϑ, which is a result of the statistical homogenization (Einav, 2007a), can be obtained from the initial and ultimate gsd's as:
where J20 and J2u are second-order moments of initial and final gsd [Einav, 2007a]. Physically ϑ indicates the distance between the initial and ultimate gsd's and is thus related to the crushing potential of the material; its value lies within 0 and 1.
Einav [2007c] derived an elastic-plastic-breakage yield criterion based on an energy balance between the rates of change in dissipation and in breakage energy to describe comminution during pure compression; it is also based on a Coulomb type failure law to describe frictional shear deformations. The resulting breakage-yield criterion in true stress and breakage energy space takes the following form:
where p = − (1/3)σijδij is the mean stress (positive in compression; δij is Kronecker delta); is the distortional stress (sij = σij + pδij is the deviatoric stress), M the slope of the critical state line in p − q space, and EB is the breakage energy, the thermodynamical conjugate to the breakage internal variable, which in this particular model has the following form:
 In the above expression, Ec is the critical breakage energy which can be determined directly from the pressure marking the onset of comminution during isotropic loading conditions (Pc) through the relationship [Einav, 2007b].
 The evolution laws for breakage and plastic strain are, respectively,
 In the above expression, dλ is the nonnegative breakage/plasticity multiplier, determinable from the consistency condition for the yield function in equation ((4)); ω is a parameter that couples the plastic volumetric deformation with grain crushing [Einav, 2007b]. Physically, ω represents the pore collapse of the material, which is a consequence of grain crushing and grain/fragment reorganization. Further details on ω and pore collapse can be found in Einav [2007a,2007b] and Das et al., .
3 Onset and Orientation of Compaction Localization in Porous Rocks
 This section briefly summarizes the results by Das et al.  using the above breakage model, as an essential basis for sections 4 and 5.
3.1 Model Calibration
 The model parameters for a typical high porosity (23%) sandstone, the Bentheim sandstone, are determined from available experimental data. Parameters, such as the shear and bulk stiffness moduli (G = 7588 MPa and K = 13833 MPa), critical state parameter (M = 1.7), and critical breakage energy (Ec = 4.65 MPa), are obtained from published experimental stress-strain responses [Baud et al., 2004; Wong et al., 2001]. It is worth to address that the pressure dependency of the elastic material properties observed in porous rocks is not taken into account in the current model. This allows the use of experimental stress-strain response [Baud et al., 2004; Wong et al., 2001] without unloading paths for the determination of shear and bulk stiffness moduli. Although our analysis [Das et al., 2011] shows that this simplification does not significantly affect the numerical predictions, a more advanced constitutive model [Das et al., 2012] has been developed to take into account the effects of pressure on the elastic properties of porous rocks. The grading index ϑ = 0.85 is determined from basic gsd [Schutjens et al., 1995] information and the assumption of a power law distribution for the ultimate gsd. The coupling angle (ω = 70°) is chosen by matching the inelastic stress-strain response with experimental results in isotropic compression. The details of this model calibration can be found in Das et al. . It is worth to mention that this breakage mechanics model only depends on few physically identifiable material parameters, which represents a major advantage over conventional plasticity based constitutive models.
3.2 Numerical Prediction of Compaction Localization
 We use the discontinuous bifurcation condition described in Rudnicki and Rice ] to detect the onset and orientation of localization bands in porous rocks under shearing at high confining pressure. Equation ((8)) represents the simplified form of this discontinuous bifurcation condition, considering the fact that the tangent stiffnesses of the material inside and outside the band are different in the case of the current breakage model [Chambon et al., 2000; Rudnicki and Rice, 1975].
 In the above equation ni is the ith component of normal vector of the localization band, is the fourth-order tangent stiffness tensor of the material inside the localization zone, and Aij the strain localization tensor, also termed the acoustic tensor [Rice and Rudnicki, 1980]. The detailed formulations of this fourth-order stiffness tensor were already given in Das et al. .
 The model described in the previous sections is capable of capturing the experimentally observed localization features of porous rocks, besides its capability in describing the material behavior. Figure 1 highlights (the thick black line) the set of favorable stress states for the formation of compaction localization at the onset of yielding. In addition, the range of possible band orientation angles at different stress states and the orientation angles corresponding to minimum acoustic tensor determinant (within bracket) is also highlighted in Figure 1. The results compare well with their experimental counterpart [Baud et al., 2004] in terms of both the onset and orientation of compaction localization [see Das et al., 2011 for details]. At much higher-pressure regime, no localization failure is observed at the onset of inelastic deformation. As also numerically experienced, the closer to the isotropic compression line the stress path is the easier the deformation would evolve into cataclastic flow without any compaction localization. Further details on the model behavior can be found in Nguyen and Einav  and Das et al. .
4 Rate-Dependent Regularization for the Current Breakage Model
 The bifurcation condition in equation ((8)) has been successfully used to determine the onset and orientation of compaction localization [Borja and Aydin, 2004; Chemenda, 2009; Das et al., 2011; Issen and Rudnicki, 2000, 2001; Olsson, 1999]. Beyond the onset of localization, the stress-strain measure loses its physical interpretation, since this measure can only be defined over a certain homogeneously deformed volume. Any attempt to use conventional continuum models in the BVP analysis of solids/structures made of materials exhibiting softening or localization features will run into nonmeaningful results in the sense of discretization-dependent solutions. The meaningfulness of this stress-strain measure usually requires the enrichment of the continuum model with a length or temporal scale [Etse and Willam, 1999; Needleman, 1988; Schreyer and Neilsen, 1996; Wang et al., 1997]. The use of a length scale related to the microstructure of the material (e.g., mean grain size), via a nonlocal regularization, has been incorporated into the current breakage model [Nguyen and Einav, 2010]. However, the application of such a regularization scheme in practice is still computationally expensive, due to the fact that the size of finite elements must be much smaller than the width of a localization band, the location and orientation of which are unknown in advance. Given the physical width of the localization band in the order of 13–30 d50 [Vardoulakis and Sulem, 1995] (d50 is the median diameter for which half of the sample is finer), a very fine finite element mesh for the whole computational domain must be used. In the current paper, the regularization of the breakage model will be based on the development of a simple and computationally cheaper rate-dependent enhancement.
 For this purpose, the strain rate effects on the model response are incorporated into the current breakage constitutive model. Through the use of the Perzyna type regularization [Perzyna, 1966], the model enhancement is carried out by modifying the evolution laws of breakage and plastic strain (see equations ((6))–(7)) in the following manner:
 In the above expressions, η is the viscosity parameter, 〈y〉 is a dimensionless overstress function derived from the rate-independent breakage-yield function.
 The McCauley bracket implies that
It can be noted that, unlike the conventional viscosity parameter, the viscosity parameter within the present model contains an inversed stress dimension, as the breakage-yield function in the current breakage model is dimensionless.
 Comparing the viscoplastic flow condition with conventional rate-independent evolution laws in equations (6)–(7), the nonnegative multiplier for this rate-dependent mode is written as:
 The above expression indicates that unlike the consistency condition of rate-independent models, Perzyna type models provide an explicit form of nonnegative multiplier. In plasticity-based models, it can be proven that for any positive values of viscosity parameter η, the magnitude of inelastic strain is always smaller than its corresponding counterpart in a rate-independent model at any stress state. However, similar analytical proof for the present breakage model is not trivial. In experimental practice [Stanchits et al., 2009; Vajdova and Wong, 2003], the range of axial strain rates used for triaxial tests on sandstone specimens varies between 10−8/s and 10−4/s: thus, for the present study, the effects of η on the model response are presented under constant axial strain rate /s (via drained triaxial tests). Figure 2 shows a comparison between rate-independent and rate-dependent behavior for different viscosity parameters. It shows that, with the increase in the viscosity parameter, the model response approaches pure elastic behavior. On the other hand, the model response collapses to rate-independent behavior for a low-viscosity parameter.
 It may be questioned whether the above introduction of rate dependency into the breakage model is a reasonable enrichment to account for the difference in the time scales related to loading rate and grain crushing. Nevertheless, while the current rate-dependent model involves an implicit, but not necessarily physical length scale related to the strain rate and viscous parameter, for all the studied cases (Figures 15 and 16b), it does actually predict realistic localization bandwidths.
4.1 Strain Rate Effect on Constitutive Response
 The enhanced breakage model would not exhibit any bifurcation instability as long as the acoustic tensor (section 3) is positive definite. Proving this requires the formulation of a tangent stiffness tensor for the localization analysis [Carosio et al., 2000; Etse and Willam, 1999; Heeres et al., 2002]. Based on the work by Etse and Willam , the derivation of a tangent stiffness tensor consistent with the implicit stress return algorithm is used here for the rate-dependent breakage model. Details on the derivation are given in Appendix A.
 The rate-dependent effects on the determinant of the acoustic tensor are shown in Figure 3, where the normalized determinant of this acoustic tensor (with respect to that of the elastic acoustic tensor) at the onset of yielding is plotted against the band orientation. It can be seen that for this rate-dependent model, the determinant of the acoustic tensor can drop below zero, even beyond rate-independent behavior for certain combinations of the viscosity parameter η and strain rate. This is due to the difference in the way the acoustic tensors are derived in the two rate-independent and viscous cases. Continuum tangent stiffness is used in the former [Das et al., 2011] and consistent tangent stiffness in the latter (this study), which generally yields a smaller determinant of the stiffness tensor. Nevertheless, as seen in Figure 3, the use of consistent stiffness tensor does not lead to change in critical angle corresponding to the minimum determinant of the acoustic tensor, compared to the use of continuum stiffness tensor for the rate-independent model. Similar observations for plasticity type models have also been documented by Carosio et al.  and Hickman and Gutierrez .
 Numerical analyses are carried out to observe the effect of strain rate on the material behavior in drained triaxial compression. We aim to match both the strain rate and model response with the corresponding experimental counterparts [Stanchits et al., 2009; Vajdova and Wong, 2003], from which an estimate of viscosity parameter η can be deduced. Given the axial strain rate of 5 × 10− 5/s [Wong et al, 1997] that has been considered to give rate-independent behavior for the calibration of model parameters [Das et al. 2011], /s will be used in this study as a starting point for rate-dependent effects. This is one way to calibrate the viscosity parameter η, highlighting the simplicity of the employed regularization. It has been found that the combination of axial strain rate /s with viscosity parameter η = 7.05 × 10− 5 s/Pa gives a response identical to that of the rate-independent breakage model for parameters given in section 3.1, and also the determinant of acoustic tensor is just greater than zero (see Figure 3). This is the basis for adjusting the loading rate in the rest of this study. Figure 4b indicates that the ultimate stress increases with axial strain rate, and the transition from elastic to inelastic zone is smoother. Alternatively, for slow loading, the model response approaches that of rate-independent behavior. The rate of breakage growth also reduces with the increase in strain rate (Figure 4a). From a micromechanical point of view, this happens because under high strain rates, there is no sufficient time for fragments to rearrange [Yamamuro and Lade, 1993]. In such cases, the material becomes stronger, as reflected in the macroscopic stress-strain response of the proposed model. We notice that the above calibration of rate-dependent parameters has been carried out only for the stress state corresponding to yielding under drained triaxial loading at lateral stress of 200 MPa. A more general scheme may require a state-dependent calibration which is computationally expensive. Ongoing work [Nguyen et al., 2012] has been initiated toward a physically better regularization scheme.
4.2 Strain Rate Effects on the Propagation of Compaction Bands
 The stability of the above rate-dependent breakage constitutive model is illustrated through the numerical analysis of a sandstone specimen under a drained triaxial loading condition. an UMAT (user defined material subroutine for ABAQUS standard) subroutine is developed for the implementation of the current breakage constitutive model in ABAQUS (ABAQUS standard 6.8). We construct a cylindrical specimen (38.1 mm × 18.4 mm) using linear quadrilateral finite elements (Figure 5) and impose axisymmetric conditions for the purpose of the current evaluation. The entire loading arrangement follows a two-stage process, where initially isotropic pressure is applied and the material deforms isotropically, and then shearing process is initiated through prescribed vertical displacement under a constant axial strain rate while the confining (radial) stress remains constant. The vertical movement of the bottom boundary is restricted throughout the entire analysis, while the incremental vertical displacement at the top boundary is kept constant during shearing. To trigger off the localization, we introduce a local defect via a weak element at the bottom of the axis of symmetry, having lower critical comminution pressure (99.5% of Pc).
 The effect of the spatial discretization on the numerical solutions is presented in Figure 5 using four FE meshes employing 100 elements, 400 elements, 1600 elements, and 6400 elements, respectively. The contours in Figure 5 depict the spatial breakage state after subjecting an initially isotropically compressed sample to 2% axial strain. As can be seen in Figures 5 and 6b, both structural response and localization pattern converge upon mesh refinement. In contrast, the rate-independent model (Figure 6a) gives nonconverged solutions upon mesh refinement. In the analysis using this model, for the fine discretization, it is impossible (under direct displacement control) to trace the post-localization behavior due to severe snap back, as a consequence of localizing the inelastic behavior onto narrow bands with widths equal to the element size.
5 Numerical Analysis of Compaction Band Propagation
5.1 Band Orientation
 We start by presenting results from numerical 2D FE analysis (based on eight-noded rectangular elements) of sandstone specimens under plane strain condition. This plane strain condition is used as a computationally cheap alternative to the more expensive full 3D simulations, since the 2D axisymmetric condition is lost once an inclined localization band has occurred. As only the band orientation is concerned in this section, this plane strain condition is an appropriate choice. In section 5.2, when both pure compaction band development and specimen response are studied, the use of 2D axisymmetric elements can be justified. To trigger localization, a local material imperfection is introduced by a weak element, having 99.5% critical comminution pressure (Pc) relative to the rest of the elements. The samples were initially subjected to isotropic confining pressure and then sheared through displacement controlled vertical compression with a constant strain rate (/s) prescribed at the top nodes. Zero vertical displacement is assigned at the bottom nodes of the specimens throughout the entire test. During the shearing, the nodes along vertical boundaries are allowed to displace freely in all directions while maintaining the lateral stress constant. Also, the node at the middle of the bottom boundary is restricted from any horizontal movement to avoid lateral instability. Using this analysis, we confirm our theoretical expectations in Das et al.  for the onset and orientation of compaction localization based on the elementary discontinuous bifurcation condition (equation ((8))). Note that for any favorable stress state at the onset of yielding, this bifurcation condition implies a set of mathematically plausible solutions for the band orientations (Figure 1), corresponding to the range of conditions satisfying the inequality requirement of equation ((8)). However, an important observation is that at the structural scale, the propagation of compactive shear bands and compaction bands always seems to follow a unique orientation depending on the stress state at the onset of localization; the unique orientation appears to be predictable, as demonstrated in the following section through different sets of numerical analysis.
 First, we study the effects of the location of material defect (a weak element) on the orientation of the localization bands. Three rectangular numerical specimens (aspect ratio of 1:2) are vertically compressed under an initial confinement of 120 MPa (σx = σy = 120MPa, under εz = 0) which eventually reach a stress state defined by p = 207 MPa and q = 243.6 MPa on the initial yield surface as shown in Figure 1. In all three samples, we vary the location of weak element while maintaining the other parameters and conditions unchanged. Despite the difference in the relative positions of the bands in various tests, Figure 7 highlights that the orientations of the bands are always the same (34°).
 Next, additional numerical tests were carried out to explore the sensitivity of the band orientations to the sample's aspect ratio, under identical plane strain compression conditions. To achieve a prefixed aspect ratio, the widths of the specimens are varied while their heights are kept constant. Figure 8 shows some typical results for which the stress state at the onset of localization is p = 207 MPa and q = 243.6. The obtained shear band orientations (34°) are found to be the same in all specimens irrespective of their aspect ratio.
 In addition, the effects of variation in stress states at the onset of localization on the alignment of (compactive shear) compaction bands is studied through another series of numerical analyses, while the aspect ratio (1:2) of the specimen is fixed and the weak element is centrally located. We controlled the stress path by varying the initial isotropic stress, such that during shearing, different stress states will be reached on the initial yield surface. At this structural scale, seen in Figure 9, it is observed that this propagation always follows closely the orientation given by the minimum determinant of the acoustic tensor in the localization analysis [see Das et al., 2011]. This should not be entirely unexpected, as the acoustic tensor over a plane with normal n actually dictates the constitutive behavior across this plane. The smaller the determinant of the acoustic tensor is, the weaker the constitutive response becomes in terms of traction and displacement jump across this plane. Therefore, the structural response just naturally “finds” the weakest plane to localize the behavior on, which is consistent with the maximum principle of dissipation at the structural scale.
5.2 Compactive Failure of an Unnotched Specimen
 The development of multiple compaction bands is next studied. A cylindrical sandstone specimen is simulated using an axisymmetric FE mesh consisting of 1600 elements (Figures 5 and 10c). Eight-noded rectangular elements are used for this simulation. The boundary conditions remain the same as those described in section 4.2. A drained shear test under confining pressure of 270 MPa is carried out. This is the confining pressure that initiates a pure compaction band at shearing stage, as expected in the preceding sections on localization features. The theoretical analysis suggests that for these conditions, pure compaction bands appear (i.e., horizontal planes) and, consistent with the theory, the FE numerical analysis does reveal horizontal localized planes. From laboratory experiments, it has been observed and reported [Olsson, 2001] that a compaction band initiates from the two ends of the sample due to the stiffness mismatch between the material and the cap of the testing device. Therefore, to realistically simulate the experimental conditions, such a stiffness mismatch is reproduced here in the numerical model through the use of stiff loading plates (see Figure 10c) at the top and bottom of the numerical specimen (instead of using any weak element to trigger localization). In addition, the roughness between the plates and the sample surfaces is modeled by introducing interface friction (coefficient of friction = 0.2).
 Figure 10 presents a comparison of compaction band development during drained triaxial loading, seen through experiments by Baud et al.  (Figure 10b) and current numerical analysis (Figure 10a). The numerical results in Figure 10a show the amount of grain crushing in terms of breakage, with the most intense activity occurring inside the compaction bands. Initially, the sample undergoes homogeneous deformation until 270 MPa confining pressure, which is the maximum pressure that can trigger pure compaction localization (Figure 1) [see also Das et al., 2011]. Thereafter, localized breakage occurs at the two ends of the specimen followed by the development of discrete compaction bands toward the center of the specimen (Figure 10a). In the experiments of Baud et al. , it was observed that compaction bands also developed from the boundaries. The onset of localization in the experiments corresponds to 300 MPa confining pressure, where the mismatch with the numerical pressure of 270 MPa has been explained at length in Das et al. .
 Despite the similarities in the evolution trend, a regular and uniform band formation pattern is noticed in the initially homogeneous numerical specimen, while in the experimental specimens (intrinsically heterogeneous), the bands develop at random locations (Figure 10b). In the numerical analysis, the band spacing depends on various factors such as specimen geometry, localization characteristics, stress conditions, and material heterogeneity. For example, Appendix B reveals the role of localization characteristics and the specimen's aspect ratio on the compaction band spacing.
 The variation of distortional stress against axial strain is plotted in Figures 11a–11d for four material points to highlight important events during the propagation of compaction bands in a drained triaxial test. In Figures 11a–11d, the stress-strain responses at four points (A, B, C, D) are observed during the deformation of the specimen. Three of the points (A, B, and D) were chosen, such that they will reside inside corresponding compaction bands, while C is purposely chosen to be outside any compaction localization zones. Also plotted are the contours of breakage and its increment for three distinct times (t1, t2, and t3). It is shown that the behavior of all the material points along the specimen switches between hardening, softening, and elastic unloading. These material points take turns in the crushing process (Figure 11). Due to this process of simultaneous loading and unloading, parallel compaction bands are formed at discrete locations along the specimen height, from the two ends of the sample toward its center. This effect is also visible through the oscillating nature of the global stress-strain response during shearing (Figure 12). This structural response can be seen to be in good agreement with its experimental counterpart [Baud et al., 2004]. The stress drops in the inelastic branch of the episodic stress-strain response indicate the development of (already formed) compaction localization zones, whereas the peaks indicate formation of new compaction bands. In between is the hardening branch where the whole specimen is loading, after the evolving states of stress and breakage, when they can no longer favor localized deformation. The thickening effects of compaction bands can also be observed, and gradually, the entire sample becomes densely crushed, as portrayed for larger strains in Figure 10.
 While the development of discrete compaction bands in Figure 10, explained in Figure 11, is responsible for the jagged plateau of inelastic stress-strain response in Figure 12, Fortin et al.  observed a similar discrete compaction pattern but a smooth inelastic plateau at very low strain rates. In experiment, the oscillation in the structural response in Figure 12 may also be affected by both strain rate and degree of heterogeneity of the material, as factors controlling the formation of discrete/diffuse compactive failure.
5.3 Compaction Band Propagation in Notched Specimen
 We know that strain localization can occur due to heterogeneities at different spatial scales. While experimental tests on structurally homogeneous (e.g., unnotched) specimens are usually performed in the laboratory, field observations mostly show the formation of compaction bands induced by strong heterogeneity [Mollema and Antonellini, 1996]. This strong heterogeneity effect has been experimentally addressed in the literature, with the use of notched specimens to trigger compaction localization at known locations [Charalampidou et al., 2011; Stanchits et al., 2009; Tembe et al., 2006; Vajdova and Wong, 2003]. We perform a triaxial drained test using axisymmetric FE analysis on a cylindrical sample of Bentheim sandstone having a circumferential notch at the center (Figure 13). We adopted similar boundary conditions as those given in section 4.2. The commercial FE software package ABAQUS was used for this analysis along with a VUMAT (user defined material subroutine for ABAQUS explicit) subroutine for the breakage constitutive model. Explicit dynamics in combination with the Arbitrary Lagrangian Eulerian type remeshing technique in ABAQUS were employed to adequately deal with the dynamic effects and mesh distortion issues locally at the notches. All model parameters (sections 3.1 and 4.1) for the unnotched specimen are used for this study (the stiffness parameters are slightly adjusted to be in accordance with Stanchits et al. : K = 16531 MPa; G = 9068 MPa).
 Two notched cylindrical specimens of different configurations, given in corresponding experiments in Stanchits et al.  and Vajdova and Wong , were used in the numerical analysis (Figure 13). In the experiments [Stanchits et al., 2009], the specimens were loaded isotropically up to a pressure of 185 MPa, where shearing was applied. This is the pressure that triggers compactive shear band in unnotched specimens, as can be seen in the preceding sections. However, it may not be the case in this notched specimen, as the local stress fields at the notches may deviate from the macro one.
 Numerical analysis of drained triaxial loading on the notched sample shows that the compaction band initiates near the notched area and propagates laterally toward the center of the specimen. There are no other bands observed apart from that in the middle of the specimen. Two different strain rates were applied for the first example during shearing. Figures 14a and 14b show the compaction band propagation via the breakage contour for both fast and slow loading cases.
 It is observed that the completion of compaction band propagation across the specimen width takes place earlier in slow strain rate loading with the production of a thin and straight band (Figures 14a and 14b). However, irrespective of strain rate, the significant growth of compaction band takes place under the axial strain difference of 0.1%–0.2% (difference between total axial strain at the initiation (~0.8%) and completion (~0.9%–1.0%)). Similar band propagation was reported in the experiments by Stanchits et al. . In the present analysis, it is also important to note that compaction band orientations are not perfectly horizontal, particularly at higher loading rates. Charalampidou et al.  and Stanchits et al.  describe this type of pattern as coalescence of individual defect clusters. These competing effects between strain rate and strong heterogeneities will be further explored in the next section. The qualitative comparisons of compaction band formation and orientation between numerical and experimental results are presented in Figure 15. A good agreement in the trend can be seen.
 The formation of pure compaction bands was also observed in another similar study on notched Bentheim sandstone at relatively high confining stress (300 MPa) [Vajdova and Wong, 2003; Vajdova et al., 2003]. In this study, we also reproduce similar band formation in a numerical sample having similar geometry and boundary condition as that used by Vajdova and Wong . However, it is important to note that no localization was observed at 300 MPa confining stress, and thus 270 MPa confining stress was applied. This observation is consistent with the analysis of Das et al.  and the previous bifurcation analysis presented in section 3.2.
 Due to the strong heterogeneity, localized breakage takes place near the notch areas and further propagates toward the center of the specimen (Figure 16b). The rest of the sample (apart from the central zone) remains uncrushed during shearing. The axial strain corresponding to completion of the central compaction band is similar to that observed experimentally [Vajdova et al., 2003]. Figures 16a and16b show striking similarities between experimental and numerical analysis.
6 Parametric Study
 On the ground of the validated model, a parametric study is performed to observe the effect of notch depth and strain rate on the development of compaction bands in porous sandstone during drained triaxial tests. The notched specimen is used to mimic field observations where compaction bands are formed from existing cracks. Figure 17 shows the counter effects of strain rate and strong structural imperfection on the formation and propagation of compaction bands. While a high strain rate favors the development of a diffuse compaction zone, stronger imperfection facilitates the formation and propagation of localized compaction. In particular, a diffuse compaction zone dominates the behavior at high strain rate irrespective of how strong the imperfection is. At low strain rates, sharper compaction zones are created for deeper notches. At the structural scale, as expected, the strength of the sample in terms of maximum load carrying capacity reduces with increasing notch depth. Yielding takes place earlier in samples having deeper notch, and as a consequence, the compaction band initiates at a lower external load than in the case of shallow notched specimens. The force-displacement responses suggest that, due to higher strain rate, the sandstone specimen can sustain higher force even for deeper notches.
 A comparison of axial loading velocity and propagation velocity of compaction band is shown in Figure 18 for notched Bentheim specimen. The initial propagation velocity is found to be relatively slow (~1 × 10−7 m/s) with a diffuse breakage around the notched area of the specimens. At higher axial strain, and more localized breakage, the band propagates quickly at velocity in the order of 1 × 10−6 m/s. This propagation velocity is found to be significantly higher, e.g., around 2 orders of magnitude, than the rate of axial deformation (~3.8 × 10−8 m/s). In their experiments, Vajdova and Wong  also observed similar relationship between the axial deformation rate and band propagation rate, which they termed as “dynamic runway.” Alternatively, when the strain rate is higher, diffuse compaction is observed (Figure 17c) and the axial deformation rate in roughly of the same order of magnitude of lateral compaction band propagation rate. This feature indicates lateral band propagation rate could be a possible factor for the formation of either discrete or diffuse compaction localization in different sandstones. If the lateral band propagation rate is very slow, there is a possibility that the material might exhibit diffuse compaction localization even under quasistatic loading condition. However, further experimental evidences (e.g., various sandstones subjected to wide range of strain rates) are required to justify this argument.
 Another parametric study is carried out to examine the effect of confining pressure on compaction band propagation in notched specimens. In this case, the axial strain rate is kept unchanged throughout the study, /s. It can be observed from Figure 18 that the orientation and the compaction patterns are largely dominated by the confining stresses. This is consistent with the elementary localization analysis in Das et al.  at the local scale and the numerical study at the structural scale in section 4. At lower confining stresses, the localization band is shear induced, and hence compaction bands are not purely horizontal with respect to the maximum compressive stress. Alternatively, pure compaction bands are formed at high confining stress. Furthermore, the introduction of circumferential notch induces strong structural heterogeneity which seems to facilitate the formation of pure compaction bands (with respect to the macro stress field) at low pressures due to the difference in the local (around the notch) and macro stress fields. While the results from our localization analysis apply well to the macro stress field, as mentioned above, the deviation of the local stress field from the macro one leads to the formation of pure compaction localization emanating from the two notches. In all cases (of different pressures), we can see the competition between two distinct localization bands: a pure compaction band (type A) emanating from the two notches and a second localization band (type B) above it (see Figure 19a). Due to these competing mechanisms, the band inclinations in notched specimens are smaller than those in unnotched samples given the same loading condition. For example, at 200 MPa confining pressure, the unnotched sample produces inclined compactive shear bands (Figure 5), whereas pure compaction bands are found in the notched samples due to the strong imperfection. This finding is also true for other stress conditions.
 The localization patterns corresponding to confining pressures of 150 and 175 MPa in Figure 19a show internal mesh-dependent localization bands (Figure 19, type B.1) within the main inclined compactive shear band (type B). Since this occurred only during a small time period over the course of the entire failure process, the bigger bands (A and B) and structural responses are independent of the discretization. Ongoing work [Nguyen et al., 2012] has been initiated toward a physically better regularization scheme which shows promising mesh-independent results.
 We have improved the capability of a breakage model to deal with post-localization issues in the analysis of boundary value problems. The link between localization characteristics of the model and its predicted compactive localization pattern has been demonstrated. In particular, our study explains numerically how compaction bands develop in both structurally homogeneous specimens and heterogeneous specimens with strong imperfections under different loading rates. More importantly, we have answered the question put forward in section 1 on how a model can give reliable predictions in all aspects of failure in crushable granular rocks: from observable features such as compaction patterns and stress-strain responses under different geometrical and loading conditions to the internal physics of failure. In short, it is the micromechanics of crushable failure embedded in the model that drives its compaction localization patterns, stress-strain response, and the evolution of the gsd under various loading conditions. To our knowledge, no such model and study were ever presented.
 This study should be considered as a preliminary step toward the development of better micromechanically enriched constitutive models that can give reliable predictions in all aspects of localized failure of crushable granular rocks. As has been addressed at length in Das et al. , a major shortcoming in the predictive capability of the current model occurs at low-pressure regimes, where cement failure and dilation are the dominant mechanism. A new model that takes into account cementation effects has shown very promising outcomes [Das et al., 2012], and further results from it will come in a near future.
Appendix A: Formulation of Consistent Tangent Stiffness for Rate-Dependent Breakage Model
 The formulation of a consistent tangent stiffness tensor and associated acoustic tensor is presented for the localization analysis, as it is hard to obtain the continuum tangent stiffness tensor in this case. We follow the general approach described in Etse and Willam  with modifications for the multiple dissipation mechanisms in the current breakage model.
 The incremental form of stress-strain relation (equation ((2))) is
 The above form can be rewritten for any time increment Δt = tn + 1 − tn considering the backward-Euler integration scheme
where is the ratio between plastic strain increment and breakage increment,
 For deriving the analytical form of consistent tangent operator, we establish the stress residual
 Since the stress residual must be zero after each time step, the root of can be determined by Newton-Raphson iterative scheme. To do so, the stress residual is expanded using the first-order Taylor expansion.
where Iijkl = δikδjl
 From the evolution rule for breakage (see equation ((9))), we get
 The breakage increment during each step of Newton-Raphson iterative scheme can be obtained from equation ((A10)) in the following manner
 Hence, we can write
 After iteration (when convergence criteria is met),
 The above results in
Appendix B: Spacing of Discrete Compaction Bands in a Homogeneous Specimen
 Numerical study of drained triaxial tests is performed to analyze the propagation of compaction bands under axisymmetric and slow loading conditions. The loading path that induces the formation of pure (horizontal) compaction band is used, based on strain localization analysis (Figure 1). Homogeneous numerical specimens are used for this purpose with minor imperfection (friction between loading plate and specimen) to trigger off the localization. The numerical analysis shows that the development of compaction bands within cylindrical specimen follows a regular spacing.
 As seen in Figure B1, localized deformation initiates from the interface between the specimen and the loading plate, with compaction band 1 propagating in the lateral direction (Figure B1a–b). This lateral propagation is governed by the globally imposed stress field and stopped at the edge/boundary of the specimen. However, locally inside the compaction band, material hardening takes place along with pore collapse which causes a drop in the mean stress (p) and rise in deviatoric stress (q). Therefore, the local stress path inside the band deviates (Figure B2) from the global one and favors the shear band formation (see Figure 1 for band orientations corresponding to low-pressure shearing).
 In this situation, the materials at the edge of the compaction band tries to form compactive shear bands (Figure B1c), governed by the local stress field. Due to the axisymmetric loading arrangements, these shear bands create stress concentration at the center of the specimen and trigger off the next compaction band. It is also interesting to note that sometimes the combined effects of two parallel compaction bands create stress concentration away from the center of the specimen (Figure B1d–e for compaction band 3) and activate another one. Therefore, the orientation of these weak shear bands at the edge of the specimen governs the spacing between compaction bands, as there is no effect of material heterogeneity present. The spacing can also be changed by making the specimen slender or shorter under similar loading condition (Figure B3).
 According to our numerical analyses, viscosity and/or strain rate have little impact on the spacing, despite their strong effects on the width of localization bands. In reality, there are several aspects (e.g., geometry, boundary conditions or imposed stress field, presence of structural or material heterogeneity) which could be reasonable for the spacing of compaction bands within porous rock specimen. A thorough investigation is therefore beyond the scope of this study.
 Arghya Das wishes to thank the University of Sydney International Scholarship scheme. Giang Nguyen and Itai Einav would like to acknowledge the Australian Research Council for the Discovery Projects funding scheme (projects DP110102645 and DP120104926). The authors wish to thank Professor Cino Viggiani for the fruitful discussions along with the associate editor and two anonymous reviewers for their constructive comments, which helped to improve the clarity and consistency of the paper.