JOURNAL OF GEOPHYSICAL RESEARCH: SOLID EARTH, VOL. 118, 2049–2066, doi:10.1002/jgrb.50193, 2013
The propagation of compaction bands in porous rocks based
on breakage mechanics
Arghya Das,1 Giang D. Nguyen,2 and Itai Einav3
Received 9 August 2012; revised 24 March 2013; accepted 16 April 2013; published 23 May 2013.
[1] 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. [2011] 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.
Citation: Das, A., G. D. Nguyen, and I. Einav (2013), The propagation of compaction bands in porous rocks based on
breakage mechanics, J. Geophys. Res. Solid Earth, 118, 2049–2066, doi:10.1002/jgrb.50193.
1.
Introduction
[2] 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.
[3] 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
1
School of Civil Engineering, The University of Sydney, Sydney NSW,
Australia.
2
School of Civil Engineering, The University of Sydney, Sydney NSW,
Australia.
3
School of Civil Engineering, The University of Sydney, Sydney NSW,
Australia.
Corresponding author: G. D. Nguyen, School of Civil Engineering,
The University of Sydney, Sydney NSW 2006, Australia. (giang.nguyen@
sydney.edu.au)
©2013. American Geophysical Union. All Rights Reserved.
2169-9313/13/10.1002/jgrb.50193
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. [2001] 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].
[4] 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).
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DAS ET AL.: PROPAGATION OF COMPACTION BANDS
[5] 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 [2001] 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 [2005] 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.
[2011] 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 [2006] and Katsman et al.
[2005]. 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
[1998], “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. [2011]). 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?
[6] 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.
[7] 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. [2011] 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.
[8] 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 rateindependent 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
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DAS ET AL.: PROPAGATION OF COMPACTION BANDS
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
[9] 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)):
pðx; BÞ ¼ ð1 BÞp0 ðxÞ þ Bpu ðxÞ;
2.2. A Constitutive Model Based on Breakage Mechanics
[10] 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.
[11] The stress-strain relationship is
tensor, eekl
# ¼ 1 J2u =J20
(3)
where J20 and J2u are second-order moments of initial and final
gsd [Einav, 2007a]. Physically # 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.
[12] 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:
y¼
2
EB ð1 BÞ2
q
þ
1≤0
Mp
Ec
(4)
where p = (1/3)sijdij is the mean stressp(positive
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiin compression; dij is Kronecker delta); q ¼ ð3=2Þsij sij is the
distortional stress (sij = sij + pdij 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:
EB ¼
#
2ð1 #BÞ2
p2 q2
þ
;
K 3G
(5)
(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].
sij ¼ ð1 #BÞDijkl eekl ;
homogenization (Einav, 2007a), can be obtained from the
initial and ultimate gsd’s as:
[13] 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 Ec ¼ Pc2 #=ð2K Þ
[Einav, 2007b].
[14] The evolution laws for breakage and plastic strain
are, respectively,
dB ¼ 2dlð1 BÞ2 cos2 o=EC ¼ Rdl and
depij
3sij
ð1 BÞ2 EB sin2 o dij
¼ dl 2
þ
pEc
3 M 2 p2
(6)
!
¼ Sij dl
(7)
[15] In the above expression, dl is the nonnegative breakage/plasticity multiplier, determinable from the consistency
condition for the yield function in equation (4); o is a parameter that couples the plastic volumetric deformation with
grain crushing [Einav, 2007b]. Physically, o represents the
pore collapse of the material, which is a consequence of
grain crushing and grain/fragment reorganization. Further
details on o and pore collapse can be found in Einav
[2007a,2007b] and Das et al., [2011].
(2)
where sij is Cauchy stress
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 #, which is a result of the statistical
3. Onset and Orientation of Compaction
Localization in Porous Rocks
[16] This section briefly summarizes the results by Das
et al. [2011] using the above breakage model, as an essential
basis for sections 4 and 5.
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DAS ET AL.: PROPAGATION OF COMPACTION BANDS
current breakage model [Chambon et al., 2000; Rudnicki
and Rice, 1975].
i
ni Lijkl nl ¼ Ajk ≤ 0
j j
Figure 1. Initial yield envelope and predicted stress states
at the formation of compaction localization for Bentheim
sandstone (results adapted from Das et al. [2011]).
3.1. Model Calibration
[17] 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
# ¼ 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 (o = 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. [2011]. 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
[18] We use the discontinuous bifurcation condition described in Rudnicki and Rice [1975]] 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
(8)
[19] In the above equation ni is the ith component of normal vector of the localization band, Liijkl 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. [2011].
[20] 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 [2009] and Das et al. [2011].
4. Rate-Dependent Regularization for the Current
Breakage Model
[21] 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 stressstrain 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 discretizationdependent 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
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DAS ET AL.: PROPAGATION OF COMPACTION BANDS
Figure 2. Effects of viscosity parameter on the mechanical behavior of a sandstone (see section 3.1
for other parameters) during the numerical simulations of drained triaxial loading at axial strain rate
e_ a ¼ 106 /s: (a) breakage evolution; (b) shear stress evolution.
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.
[22] 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:
dB ¼ 2
depij
h yi ð1 BÞ2 cos2 o
h yi
dt;
dt ¼ R
Ec
(9)
!
3sij
h yi
ð1 BÞ2 EB sin2 o dij
h yi
2
dt: (10)
¼
þ
dt ¼ Sij
pEc
3 M 2 p2
[23] In the above expressions, is the viscosity parameter,
hyi is a dimensionless overstress function derived from
the rate-independent breakage-yield function.
[24] The McCauley bracket implies that
h yi ¼
y; for y ≥ 0; i:e:; at inelastic material response
0; for y < 0; i:e:; at elastic material response
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.
[25] 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:
dl ¼
h yi
dt:
(11)
[26] 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 , 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 108/s and 104/s: thus, for the present
study, the effects of on the model response are presented
under constant axial strain rate e_ a ¼ 106 /s (via drained
triaxial tests). Figure 2 shows a comparison between rateindependent 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.
[27] 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
[28] 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 [1999],
the derivation of a tangent stiffness tensor consistent with
the implicit stress return algorithm is used here for the ratedependent breakage model. Details on the derivation are
given in Appendix A.
[29] 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
2053
DAS ET AL.: PROPAGATION OF COMPACTION BANDS
Figure 3. Normalized determinant of the acoustic tensor corresponding to stress state of p = 281 MPa and
q = 243 MPa (i.e., drained triaxial test at 200 MPa confining stress) at the onset of inelastic deformation: (a)
effect of viscosity and (b) effect of strain rate.
against the band orientation. It can be seen that for this ratedependent model, the determinant of the acoustic tensor can
drop below zero, even beyond rate-independent behavior for
certain combinations of the viscosity parameter 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. [2000] and Hickman and
Gutierrez [2005].
[30] 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 can be
deduced. Given the axial strain rate of 5 10 5/s [Wong
et al, 1997] that has been considered to give rateindependent behavior for the calibration of model parameters [Das et al. 2011], e_ a ¼ 106/s will be used in this study
as a starting point for rate-dependent effects. This is one way
to calibrate the viscosity parameter , highlighting the
simplicity of the employed regularization. It has been found
that the combination of axial strain rate e_ a ¼ 106 /s with
viscosity parameter = 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 rateindependent 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
[31] 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
Figure 4. Comparison of strain rate effect on mechanical behavior of Bentheim sandstone during the
numerical simulations of drained triaxial loading: (a) breakage evolution; (b) shear stress evolution.
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DAS ET AL.: PROPAGATION OF COMPACTION BANDS
400 elements
100 elements
1600 elements
6400 elements
Breakage, B
y
Weak element, 99.5% Pc
0.0 0.1 0.2 0.3 0.4
Symmetry axis
r
Figure 5. FE meshes and breakage contours showing the formation of localization bands under drained
triaxial loading (radial stress sr = 200MPa and axial strain ea = 2%).
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).
[32] 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
(a)
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
[33] 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
(b)
Reaction force (kN)
Reaction force (kN)
100
80
60
100 elements
40
400 elements
1600 elements
6400 elements
20
0
0
0.001
0.002
0.003
Axial displacement (m)
0.004
100
80
60
40
100 elements
400 elements
1600 elements
6400 elements
20
0
0
0.001
0.002
0.003
Axial displacement (m)
0.004
Figure 6. Reaction force against axial displacement in drained triaxial condition, (a) using the rateindependent model; and (b) using the rate-dependent model.
2055
DAS ET AL.: PROPAGATION OF COMPACTION BANDS
Breakage, B
y
z
Weak element
x
Figure 7. Localization bands in samples with different locations of a weak element, subjected to plane
strain compression. The localization in all the tests initiate when p = 207 MPa and q = 243.6 MPa, and the
orientation of the localization band is always 34 .
vertical compression with a constant strain rate (_ea ¼ 106 /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. [2011] 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.
[34] 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 (sx = sy = 120MPa, under ez = 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 ).
[35] 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.
Breakage, B
Weak element
1:1
1:2
1:4
Figure 8. Localization bands in samples with different aspect ratios, subjected to plane strain compression. The localization in all the tests initiates when p = 207 MPa and q = 243.6 MPa, and the orientation of
the localization band is always 34 .
2056
DAS ET AL.: PROPAGATION OF COMPACTION BANDS
Breakage, B
Weak element
Band angle
(FE analysis) =
Band angle for min. |A|
(Localization analysis) =
Stress state at
the onset of
localization
(see Fig. 1)
34°
33.7°
p = 207.0 MPa
q = 243.6 MPa
29°
25°
17°
0°
29.08°
24.8°
17.4°
0 .0 °
248.0 MPa
248.8 MPa
271.0 MPa
243.7 MPa
230.0 MPa
248.8 MPa
334.0 MPa
193.9 MPa
Figure 9. Variation of localization band pattern in numerical specimens due to variation is stress states at
the onset of localization.
[36] 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
[37] 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
Figure 10. Axial development of discrete compaction bands at different stages of drained triaxial loading (a) numerical simulation under sr = 270 MPa in a homogeneous specimen, with gray scale indicating
the predicted value of breakage. (b) Experimental observation under sr = 300 MPa with darker areas referring to denser materials [Baud et al., 2004]. (c) FE mesh and boundary conditions.
2057
DAS ET AL.: PROPAGATION OF COMPACTION BANDS
Figure 11. Propagation of pure compaction bands in numerical sample. (a–d) Stress-strain responses
at different material points (A, B, C, and D). (e) Breakage contours at different stages of loading. (f)
Incremental breakage contours at different stages of loading.
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).
[38] Figure 10 presents a comparison of compaction band
development during drained triaxial loading, seen through
experiments by Baud et al. [2004] (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. [2004], 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. [2011].
[39] 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.
[40] 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
2058
DAS ET AL.: PROPAGATION OF COMPACTION BANDS
Figure 12. Global stress-strain responses from numerical
prediction and experimental observation
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.
[41] 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. [2006] 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
[42] 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
105 mm
(a)
0.8 mm
5 mm
50 mm
2 mm
38.1 mm
(b)
2 mm
18.4 mm
Figure 13. Notched specimens: FE meshes and the enlarged view of notched area.
2059
DAS ET AL.: PROPAGATION OF COMPACTION BANDS
Figure 14. Breakage contour during shearing of notched
triaxial samples at different axial strain: (a) slow strain rate,
e_ ¼ 3:8 108 /s; (b) fast strain rate, e_ ¼ 3:8 104 /s.
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. [2009]: K = 16531 MPa; G = 9068 MPa).
[43] Two notched cylindrical specimens of different configurations, given in corresponding experiments in Stanchits
et al. [2009] and Vajdova and Wong [2003], 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.
[44] 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.
[45] 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. [2009]. 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. [2011] and
Stanchits et al. [2009] 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.
[46] The formation of pure compaction bands was also
observed in another similar study on notched Bentheim
(a)
(b)
(c)
(d)
Figure 15. Qualitative comparison of compaction band formation in notched samples of sandstones. (a–b)
Breakage contour obtained from FE analysis during slow loading rate and fast loading rate. (c–d) Corresponding
experimental observations of compaction band based on AE activity [Stanchits et al., 2009].
2060
DAS ET AL.: PROPAGATION OF COMPACTION BANDS
Figure 16. Comparison of experimental results [Vajdova et al., 2003] and our numerical predictions: (a)
optical microscopic images of experimental triaxial samples; (b) breakage contours obtained from FE
analysis; (c) stress-strain response.
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 [2003]. 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.
[2011] and the previous bifurcation analysis presented in
section 3.2.
[47] 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
[48] 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.
Figure 17. Propagation of localized breakage in sandstone specimen having different notch depths and
corresponding force-displacement plots.
2061
DAS ET AL.: PROPAGATION OF COMPACTION BANDS
Figure 18. Comparisons of compaction band propagation
rate in notch specimens under drained triaxial condition.
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.
[49] 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 (~1 107 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 1 106 m/s. This propagation velocity is found to be significantly higher, e.g., around 2
orders of magnitude, than the rate of axial deformation
(~3.8 108 m/s). In their experiments, Vajdova and Wong
[2003] 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.
[50] 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, e_ a ¼ 106 /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. [2011] 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
(a)
Breakage, B
130 MPa
(b)
150 MPa
175 MPa
130 MPa
150 MPa
175 MPa
200 MPa
270 MPa
Force (kN)
50
40
30
200 MPa
(c)
270 MPa
B.1
B
20
10
0
A
0
0.2
0.4
0.6
0.8
Displacement (mm)
Figure 19. (a) Propagation of localized breakage in sandstone specimen under different confining stresses.
(b) Corresponding force-displacement responses. (c) Enlarged view of localization band formation in notched
specimen while sheared at 175 MPa confining pressure (A = compaction band; B = compacted shear band).
2062
DAS ET AL.: PROPAGATION OF COMPACTION BANDS
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.
[51] 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.
7.
[55] The incremental form of stress-strain relation (equation
(2)) is
Δsij ¼ ð1 #BÞDijkl : Δekl e_ pkl Δt [52] 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.
[53] 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. [2011], 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.
nþ1
Δsnþ1
¼ ð1 ij
" #BÞDijkl :Δekl
ð1 #BÞDijkl :Qnþ1
kl þ
#
#snþ1
ij
ΔB;
ð1 #BÞ
ðA2Þ
where Qijnþ1 is the ratio between plastic strain increment and
breakage increment,
Δepij EBnþ1 tan2 o @pnþ1
¼
pnþ1
ΔB
@sijnþ1
þ
3snþ1
ij
Ec
M 2 ðpnþ1 Þ2 2ð1 Bnþ1 Þ2 cos2 o
:
ðA3Þ
[57] For deriving the analytical form of consistent tangent
operator, we establish the stress residual
ΔRijnþ1 ¼ ð1 " #BÞDijkl Δeklnþ1
ð1 #BÞDijkl Qklnþ1 þ
#
#sijnþ1
ΔB Δsnþ1
ij :
ð1 #BÞ
ðA4Þ
[58] Since the stress residual must be zero after each time
step, the root of ΔRijnþ1 ¼ 0 can be determined by NewtonRaphson iterative scheme. To do so, the stress residual is
expanded using the first-order Taylor expansion.
nþ1
nþ1
ΔRnþ1
ij jnew ¼ ΔRij jold þ dΔRij jold ;
(A5)
where
dΔRij ¼
@ΔRij
@ΔRij
@ΔR
dB;
dsij þ
deij þ
@B
@skl
@ekl
(A6)
@ΔRij
@Qkl
¼ ΔBð1 #BÞDijkl
@skl
@sij
#
Iijkl Iijkl ;
ΔB
ð1 #BÞ
ðA7Þ
where Iijkl = dikdjl
@ΔRij
¼ ð1 #BÞDijkl ;
@ekl
Appendix A: Formulation of Consistent Tangent
Stiffness for Rate-Dependent Breakage Model
(A1)
[56] The above form can be rewritten for any time increment Δt = tn + 1 tn considering the backward-Euler integration scheme
Qijnþ1 ¼
Conclusions
#sij _
BΔt:
ð1 #BÞ
(A8)
and
[54] 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 [1999] with modifications
for the multiple dissipation mechanisms in the current
breakage model.
@ΔRij
¼ #Dijkl Δekl
@B
"
2063
ΔB #Dijkl Qkl þ ð1 #BÞDijkl
ð1 #BÞDijkl Qkl þ
#sij
:
ð1 #BÞ
@Qkl
@B
þ
#
#2 sij
ð1 #BÞ2
ðA9Þ
DAS ET AL.: PROPAGATION OF COMPACTION BANDS
(a)
(b)
(c)
(e)
(d)
Figure B1. Propagation of compaction band under drained triaxial condition (only upper half portion of
the specimen is shown here). (a–b) Lateral propagation of compaction band 1. (c) Propagation of shear
bands from the edges of compaction band 1. (d) Development of compaction band 2 and propagation
weak shear band from the edge of both bands 1 and 2. (e) Propagation of compaction bands 3 and 4.
ΔB ¼ Δt
h yi
R:
(A10)
[60] The breakage increment during each step of NewtonRaphson iterative scheme can be obtained from equation
(A10) in the following manner
dB ¼
dB ¼
Δt @
@
ðyRÞdB þ
ðyRÞ ;
@B
@sij
@y
Δt
@s R
h
ij
i dsij ¼ xij dsij :
1 Δt @y R þ y @R
@B
(A11)
(A12)
@B
[61] Hence, we can write
dΔRij ¼
@ΔRij
@ΔRij
@ΔRij
dskl þ
dekl þ
ðxkl dskl Þ:
@skl
@ekl
@B
(A13)
[62] After iteration (when convergence criteria is met),
0¼
@ΔRij
@ΔRij
@ΔR
ðxkl :dskl Þ:
dsij þ
deij þ
@B
@skl
@ekl
(A14)
[63] The above results in
Lvp
ijkl ¼
1
@sij
@ΔRij
@ΔRij @ΔRij
:
¼
þ
:
xkl
@ekl
@ekl
@skl
@B
(A15)
[65] 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).
[66] 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).
[67] 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
Distortional stress, q (MPa)
[59] From the evolution rule for breakage (see equation
(9)), we get
Appendix B: Spacing of Discrete Compaction
Bands in a Homogeneous Specimen
[64] 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.
300
Global stress path
Local stress path outside
200
compaction band
Local stress path inside
compaction band
100
0
200
300
400
Mean stress, p (MPa)
Figure B2. Stress paths inside and outside the localization band.
2064
DAS ET AL.: PROPAGATION OF COMPACTION BANDS
(a)
(b)
(c)
Figure B3. Variation of compaction band spacing due to
the change in specimen aspect ratio (diameter/length): (a)
1:1; (b) 1:2; (c) 1:4.
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.
[68] Acknowledgments. 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.
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