Simulation of localized compaction in high-porosity calcarenitesubjected to boundary constraints

Arghya Das, Giuseppe Buscarnera n

Northwestern University, Evanston, IL, USA

a r t i c l e i n f o

Article history:Received 9 November 2013Received in revised form1 July 2014Accepted 2 July 2014

Keywords:High-porosity rocksConstitutive modelingFinite element analysesStrain localizationCompaction bands

a b s t r a c t

This paper studies the mechanics of localized compaction in porous rocks subjected to axisymmetricdeformation. The material selected for the study is Gravina calcarenite, a soft rock prone to pore collapseand compaction banding. The stress–strain response has been simulated through a plasticity modelcapturing inelastic processes in the brittle–ductile transition, while the bifurcation theory has been usedto calibrate the model constants and identify stress paths able to generate heterogeneous compaction.The onset of strain localization upon application of the selected paths has been assessed numerically,simulating the global response of calcarenite specimens via the Finite Element Method. Simulatedtriaxial compression tests have been compared with published experiments, showing good agreement interms of both macroscopic response and localization patterns. In addition, oedometric compression hasbeen simulated to inspect the role of material heterogeneity, kinematic constraints and boundary effects.The results show that the interplay between these factors has important implications for the resultinglocalization process. In particular, heterogeneity and boundary conditions have been found to control theformation of unexpected strain patterns, such as compactive shear zones that do not reflect thesymmetry of the imposed constraints. Moreover, the simulations have suggested that such forms ofheterogeneous compaction may not be easily identifiable from global measurements, as they tend todisappear with further strains and the averaging of the quantities measured at the boundary tends togenerate global signatures not easily distinguishable from those associated with pure (horizontal)compaction banding.

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1. Introduction

High-porosity rocks display a wide range of localizationmechanisms controlled by microstructural attributes and stressconditions [1]. In particular, when such rocks are loaded at highconfinements they tend to exhibit compaction bands orthogonalto the maximum compressive stress [2–5]. These modes oflocalization have attracted the interest of different communities,such as structural geology, geophysics, geotechnical and petroleumengineering. This interest is motivated by the ability of thesestructures to reduce the permeability of reservoir rocks andthreaten the stability of deep boreholes [6,7]. After being identifiedin the field by Mollema and Antonellini [8], the mechanismsleading to the formation of compaction bands have been studiedvia laboratory tests [2–4,9–14], theoretical investigations [15–20]and numerical analyses [21–25]. These studies, together withmicrostructural inspections, identified grain crushing and porecollapse as the key micromechanical processes that control the

formation of compaction bands. While these processes induce alocal loss of strength, they also increase the frequency of inter-granular contacts, thus promoting the rearrangement of thecrushed fragments, the reduction of the local porosity and there-hardening of the post-localization regime. As a result, unlikesingle shear bands observed at low confinements, the brittle–ductile regime promotes multiple compaction zones that propa-gate across the sample until a complete re-hardening of thespecimen [26].

The systematic observation that compaction bands tend to form insuch a peculiar regime of deformation has inspired testing proceduresbased on pre-selected triaxial compression paths, i.e., stress pathsdesigned to pass through the transitional regime of the tested rock[2,3,13]. While these methods have disclosed important characteristicsof compaction banding, the conditions imposed in the laboratorymay significantly differ from those occurring in the field [27,28]. Forexample, recent experiments have pointed out the importanceof anisotropic stress paths for the onset of strain inhomogenei-ties [29,30], while theoretical studies have suggested that non-axisymmetric loading [31] and kinematic constraints [32] may hinderthe formation of compaction bands. Insights on this matter have beenrecently provided by Soliva et al. [33], who discussed the effect of

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International Journal ofRock Mechanics & Mining Sciences

http://dx.doi.org/10.1016/j.ijrmms.2014.07.0041365-1609/& 2014 Elsevier Ltd. All rights reserved.

n Corresponding author. Tel.: þ1 847 491 5884; fax: þ1 847 491 4011.E-mail address: g-buscarnera@northwestern.edu (G. Buscarnera).

International Journal of Rock Mechanics & Mining Sciences 71 (2014) 91–104

factors such as burial history and local tectonics on stress paths andconsequent strain localization mechanisms.

It is therefore arguable that the theoretical/numerical inspectionof different kinematic conditions and stress paths is a valuable toolto improve the understanding of compaction bands and identifythe key processes that control their formation in the field. For thesereasons, here we study the relation between the axisymmetricstress paths that derive from imposed kinematic constraints andthe onset of compaction banding. In particular, numerical modelingwill be used to explore the influence of kinematic conditions,boundary restraints and material heterogeneity. The referencematerial selected for the study is Gravina calcarenite, a porousrock from Southern Italy prone to pore collapse, degradation of itscement matrix and strain inhomogeneities [14,34–36].

The study consists of two distinct parts. In the first one, theconstitutive response of the selected soft-rock is simulatedthrough an elasto-plastic model developed by Nova and co-workers [35,37,38], which has been chosen for its ability tocapture inelastic mechanisms in the brittle–ductile transition.The model predictions are then inspected via a bifurcationcriterion for strain localization [39], identifying a set of stresspaths able to generate localized compaction. In the second part,numerical simulations are used to model the response of calcar-enite specimens as boundary value problems. The objective of thenumerical study is to inspect onset and propagation of localizedcompaction and elucidate the role of imposed kinematic con-straints. For this purpose, a Finite Element model enriched with arate-dependent regularization scheme [40,41] has been used toreproduce triaxial compression and oedometric compaction inpresence of different boundary conditions. The computedresponse is finally compared with experimental data, discussingthe interplay between the imposed conditions and the predictedpatterns of heterogeneous compaction.

2. Constitutive analysis of localized compaction

2.1. Constitutive model for porous rocks

An essential component for compaction band analyses is a con-stitutive law able to replicate the rheological response of specimensloaded in the brittle–ductile transition zone, as well as to capture theassociated strain localization processes. A popular strategy involvesthe use of cap plasticity [42], which allows the simulation of inelasticcompaction during compression. While parabolic cap models [43] andelliptic caps [42] are typical choices, numerous enhancements havebeen proposed in recent years to better fit the data [44,45]. Anotheroption for constitutive analyses is critical state plasticity [46,47], whichlinks the stress–strain response to the evolving plastic volumetricstrains. Despite these features, cap plasticity and critical state theorieshave been typically used to assess the potential for localized compac-tion at yielding [45,48,49], and only few studies have considered theinterplay between stress–strain response and patterns of localizedcompaction [22,24,50]. In addition, most of the existing works do notinclude a description of the irreversible processes that control themechanics of compaction, such as bond/grain breakage and porecollapse. A notable exception is the work by Das et al. [16,21], inwhichthe explicit incorporation of grain crushing allowed the authors topoint out the importance of the rheological response on the simula-tion of both onset and propagation of compaction bands.

Here we use a similar logic, in that we use a constitutive lawable to reproduce realistically the rheology of porous rocksobserved in experiments. The selected model derives from a seriesof contributions by Nova and co-workers [35,38,51,52], and is herechosen for its ability to capture the transition from brittle toductile behavior in a wide range of porous geomaterials [14,32,35].

At variance with typical models, this law includes multiple inter-nal variables mimicking a number of inelastic processes involvedin compaction banding (i.e., rupture of microstructural compoundsand subsequent pore collapse, as well as compaction hardening).A major advantage of this strategy is the possibility to naturallyreproduce the porosity loss generated by the degradation of thecement, which is captured by the plastic compensatory mechan-ism created by the competition between an internal variable thatincreases with compaction (thus mimicking a denser skeletonpacking) and a second term that decreases (thus reflecting astrain-induced degradation of the microstructure, [35]). Experi-mental and theoretical studies have demonstrated the ability ofthis model to capture the macroscopic signatures of compactionbanding upon oedometric compression [53], while a recent studyby Buscarnera and Laverack [32] has discussed the possibility touse it for capturing localized compaction in both porous sand-stones and carbonate rocks subjected to triaxial compression.

Hereafter we provide a brief description of the constitutiveformulation, focusing on the functions that allow the incorpora-tion of non-normality. In particular, the yield function and theplastic potential are assumed to be given by the expressionsproposed by Lagioia et al. [37]:

f

g

)¼ AK1h=Ch

h B�K2h=Chh pn�p'hs ¼ 0 ð1Þ

Ah ¼ 1þ 1K1hMh

qpn

ð2Þ

Bh ¼ 1þ 1K2hMh

qpn

ð3Þ

K1h=2h ¼μhð1�αhÞ2ð1�μhÞ

17

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�4αhð1�μhÞ

μhð1�αhÞ2

s !ð4Þ

Ch ¼ ð1�μhÞðK1h�K2hÞ ð5Þ

Mh ¼2MchcMh

1þcMh �ð1�cMh Þ sin ð3ϑÞ ð6Þ

where pn ¼ pnþrpm ¼ σijδij=3þrpm is a modified mean effectivestress; q¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið3=2Þsijsij

pis the deviatoric stress ðsij ¼ σij�p0δijÞ and

the subscript h indicates either the yield function (h¼ f) or theplastic potential (h¼g). The size of the elastic domain ðpf s'Þ ismeasured along the hydrostatic axis (Fig. 1c) and is defined by alinear combination of independent internal variables, herereferred to as ps and pm (i.e., by p0f s ¼ psþpm þrpm , where theterm rpm represents the hydrostatic yield threshold in the tensilestress regime).

As summarized by Eqs. (1)–(5), independent parameters areneeded to reproduce non-associated plastic flow. In particular, theshape of yield surface and plastic potential is defined by two setsof four parameters (αh, μh, Mch and Meh). The constants αh and μhcontrol the shape of the surfaces in meridian sections of the stressspace (Fig. 1c), while Mch and Meh control the geometry of thesurfaces in the region of compression and extension loading,respectively. In particular, the ratio cMh ¼Mch=Meh defines theshape of the deviatoric section (Fig. 1b), which is expressed as afunction of the Lode angle, ϑ, according to Eq. (6) [54]. These twosets of parameter must be calibrated based on experiments, thusdefining the shape of the yield surface from observed yieldingpoints, and that of the plastic potential from the irreversible strainincrements measured in the post-yielding regime [32]. The typicalshape of the initial yield envelope in the principal stress space anddeviatoric plane is presented in Fig. 1.

The most notable features of the model are embedded in thehardening laws [51,52]. Indeed, unlike conventional critical state

A. Das, G. Buscarnera / International Journal of Rock Mechanics & Mining Sciences 71 (2014) 91–10492

models, size and location of the yielding locus are controlled by theevolution of the internal variables, ps and pm (Fig. 1c), which simulatethe hardening/softening response driven by compactive/dilativeplastic strains and the strain-softening driven by the loss ofstructure due to the breakage of the microstructural compounds(e.g., cement bridges, particles, etc.), respectively. These twointernal variables are coupled, in that they evolve simultaneouslyas a function of the plastic strains:

_ps ¼psBp

_εpv ð7Þ

_pm ¼ �ρmpmðj_εpv jþξm _εps Þ ð8Þ

where Bp is a material constant that controls the rate of compac-tion hardening, as well as the plastic compressibility under hydrostaticcompression, while ρm and ξm govern the rate of material degradationand introduce strain-softening terms into the formulation.

The plastic strains are eventually obtained by incorporating theabove mentioned constitutive functions in the plastic flow rule, asfollows

_εpij ¼ Λ∂g∂σij

; ð9Þ

where Λ is a non-negative plastic multiplier determined from theconsistency condition.

The model is completed by defining the elastic response insidethe yield envelope. For the sake of simplicity, a linear elasticrelationship is used hereafter

σij ¼Dijklεekl; ð10Þ

where σij is the Cauchy stress tensor; εekl is the elastic strain tensor;Dijkl is the conventional linear elastic tangent stiffness tensor whichcan be expressed in term of shear (G) and bulk (K) modulus,Dijkl ¼ GðδikδjlþδilδjkÞþðK�ð2G=3ÞÞδijδkl and δij is the Kronecker delta.

2.2. Calibration of the constitutive parameters

The constitutive law has been used to capture the response ofGravina calcarenite, a soft carbonate rock from Southern Italy thathas been found to exhibit localized compaction [14,36]. Theadvantage of studying this material derives from the large numberof triaxial compression experiments enabling the quantification ofthe model parameters [35,14], as well as from the availability ofoedometric compression tests displaying global signatures ofcompaction banding [14,34,53]. As a result, this porous rockprovides unique opportunities for theoretical/numerical studiesfocused on stress-path effects.

Although most parameters derive from a recent constitutivestudy by Buscarnera and Laverack [32] (Table 1), some material

constants are here re-discussed in light of bifurcation analyses.Indeed, in presence of strain localization the post-localizationregime is controlled by strain inhomogeneities, and constitutiveanalyses should be regarded only as approximations of the realsample response [55]. This aspect is crucial to calibrate thedegradation parameter ξm, which mimics inelastic phenomena inthe brittle–ductile regime, where localized compaction is likely tooccur. As noted by Buscarnera and Laverack [32], this modelconstant affects both the stress–strain response (Fig. 2b) and thedomain of strain localization in the stress space (Fig. 2a). Thisproperty has allowed us to obtain a first-order estimate for ξm,which has been evaluated by identifying the strain localizationdomain through the classical bifurcation condition by Rudnickiand Rice [39]:

jAjkj ¼ jniLijklnljr0; ð11Þ

where jAijj is the determinant of the so-called acoustic tensor.Condition (10) can also be used to define the orientation vector ni ofpossible deformation bands, which have to be determined based onthe characteristics of the fourth order tangent stiffness tensor Lijkl

Lijkl ¼Dijkl�Dijklð∂g=∂σklÞð∂f =∂σmnÞDmnkl

Hþð∂g=∂σpqÞDpqrsð∂f =∂σrsÞð12Þ

Table 1Constitutive parameters for Gravina calcarenite.

Parameters Definition Calcarenite

K (kPa) Elastic bulk modulus 80,000G (kPa) Elastic shear modulus 75,000ρm Parameter governing the volumetric degradation 4.5Bp Isotropic plastic compressibility 0.034μf Shape parameters of the yield surface 1.2αf 0.45Mcf 0.9Mef 0.7μg Shape parameters of the plastic potential 1.3αg 0.05Mcg 1.5Meg 1.167ξm Parameter governing the deviatoric degradation 0.75a

R Size of the yield surface in the tensile stress domain 0.05pm0 (kPa) Expansion in size of the elastic domain in cemented

media2200b

ps0 (kPa) Size of the initial elastic domain for cohesionlessmedia

200b

η (s) Viscosity parameter 131c

a Initial assumption based on data from triaxial tests.b Initial values of the plastic internal variables.c Used only in finite element analyses to include rate-dependent regularization.

p (kPa)

q (k

Pa)

r.pm0 ps0 pm0

Fig. 1. Typical shape of the yield envelope: (a) principal stress space; (b) deviatoric stress plane; (c) q–p meridian plane (the dotted line represents the yield surface with nocementation effects; the solid line represents the expanded yield surface of a cemented material).

A. Das, G. Buscarnera / International Journal of Rock Mechanics & Mining Sciences 71 (2014) 91–104 93

H¼ � ∂f∂ps

∂ps∂εpij

∂g∂σij

þ 1þr�r∂f∂pn

� �∂f∂pm

∂pm∂εpij

∂g∂σij

!ð13Þ

By performing a parametric study it is possible to show thevariation of the localization domain for different values of ξm(Fig. 2a). The domain is found to expand with increasing values ofξm: Similarly, Fig. 2b indicates the variation of the predicted stress–strain response for varying ξm in two triaxial compression testssimulated at 1100 kPa and 2000 kPa confining pressure. It isapparent that the predicted constitutive response does not varysignificantly with ξm, while the extent of the bifurcation zone does.This confirms the convenient choice to assess such degradationparameter via the localization zone rather than through a directcomparison with the measured stress–strain behavior of thecalcarenite.

2.3. Triaxial compression paths causing localized compaction

The simulations in Fig. 2 display a stress plateau, which can beassociated with a pore collapse enhanced by structural degrada-tion. As is typical for porous rocks sheared at high confinement,this stage is followed by a re-hardening trend [1]. Since thesefeatures of the stress–strain response are usually interpreted asindications of shear-enhanced compaction and/or pure compac-tion banding [15], it is convenient to assess the potential for strainlocalization along these paths. In the case of Gravina calcarenite, infact, post-mortem observation of specimens sheared from aninitial confinement of 2 MPa has disclosed the formation ofpseudo-horizontal localization bands [14]. Similarly, a reinterpre-tation of earlier experiments performed by Lagioia and Nova [35]on the same rock has suggested the potential for formation of non-homogeneous shear-enhanced compaction in samples sheared at1.1 MPa confinement [36].

The two stress paths in Fig. 2 can therefore be used to constrainthe extent of the strain-localization zone and define a tentativevalue for the parameter ξm. The analysis suggests that a value of0:75 is sufficient to generate a localization zone large enough toencompass both paths. For this reason, ξm ¼ 0:75 is used in Fig. 3 to

define potential orientations of the deformation bands by plottingfor different stress states the determinant of the normalizedacoustic tensor against the band angle. Angles corresponding tozero or negative determinant suggest the possible formation ofstrain localization zones at the considered stress. Band orienta-tions are expressed in terms of the angle between the normal tothe plane of strain localization and the horizontal axis (Fig. 3). As aresult, the stress domain associated with the potential formationof pure compaction bands is found to be located in the high-pressure portion of the yield locus, where horizontal bands (i.e.,α¼901) provide the minimum value of the acoustic tensor deter-minant. By contrast, at lower confinements, the inclination of thepredicted localization zone changes, with high-angled shear bandsbecoming the preferential form of localization. To compare ouranalyses with previous results based on different conventions ofstress invariants [3,31,56–58], we have computed a modified slopeindex for the yield surface ðμ¼ ðdq=dpÞ=

ffiffiffi3

pÞ and the plastic

potential ðβ¼ ðdεpv=dεps Þ=ffiffiffi3

pÞ. Both indices have been evaluated at

the predicted onset of localization (see table inset in Fig. 3b), andthe computed values are consistent with previous theoreticalpredictions [31], as well as with evidences of compaction bandingand/or shear enhanced compaction available for various porousrocks [3].

This strategy can also be used to monitor the evolving potentialfor strain localization during deformation. This is shown in Fig. 4,in which the constitutive response predicted for the triaxialcompression paths in Fig. 3 is analyzed. The localization analysisbased on the bifurcation condition has been performed at severalpost-yielding states (open circle symbols in Fig. 4) by plotting thenormalized determinant of the acoustic tensor against theband angle. As expected, both stress paths fulfill the bifurcationcriterion at the onset of yielding, as the acoustic tensor determi-nant drops to zero or to negative values. Again, the formation of ashear-enhanced compaction band is suggested for the test simu-lated at lower confinement (1.1 MPa), while pure (horizontal)compaction bands are predicted for the stress path starting froma higher confinement level (2 MPa). It is interesting to note thatthe bifurcation indices tend to increase with further deformation,reflecting a reduced tendency of the material to undergo strain

0

1000

2000

3000

4000

5000

0.00 0.10 0.20

0.00.7524

0

500

1000

1500

-500 0 500 1000 1500 2000 2500

ξm =0, 0.75, 2.0, 4.0

q(k

Pa)

p (kPa)

p0= 1100 kPa

p0 =2000 kPa

εa

q(k

Pa)

ξm

Fig. 2. Parametric study of the effect of the degradation parameter ξm: (a) yield surface with localization domain; (b) simulated stress–strain response for triaxialcompression.

Fig. 3. Localization analysis at different yield points (here function of the initial confining pressure prior to triaxial compression).

A. Das, G. Buscarnera / International Journal of Rock Mechanics & Mining Sciences 71 (2014) 91–10494

localization. This feature of the predicted response can be attrib-uted to the evolution of the internal variable pm (Eq. (8)), whichtends to decrease upon post-yielding shearing because of strain-softening. In other words, with decreasing values of pm the effectsassociated with the loss of structure tend to vanish, promotinga progressive re-hardening. These effects are controlled by thedegradation parameters, and hence by ξm. As a result, the calibra-tion of ξm is found to exert an important control on the predictedpatterns of localized compaction.

2.4. Effect of kinematic constraints: oedometric compression

As suggested by Buscarnera and Laverack [32], refinements ofthe calibration can be achieved by cross-correlating data fromdifferent loading paths. Oedometric tests, for example, can bea useful resource for compaction band analyses [38]. Indeed, thedeformations associated with oedometric loading coincide withthe kinematic conditions inside a compaction band. This idea iscorroborated by data provided by Olsson [15] and Baud et al. [2],who report deviations of the measured strain path at the onset ofcompaction banding during triaxial compression tests performedat constant radial confinement. The authors reported indeed axialshortening taking place with negligible radial strains (i.e., adeformation mode enforced directly by oedometer tests). Alongthese lines, recent studies by Arroyo et al. [14] and Castellanzaet al. [53] have indicated the occurrence of localized compactionduring oedometric tests, showing that the onset of strain inho-mogeneity was linked to specific global signatures (e.g., a peak ofthe vertical stress and/or a curl of the averaged stress pathobtained from radial stress measurements). The K0 consolidationresponse reported by Lagioia and Nova [14] for Gravina calcarenitedisplays similar features (see data reported in Fig. 11), thussuggesting that also this material is susceptible to localizationprocesses upon radially constrained deformation. This evidencecan be used to further constrain the value of the parameter ξm,thus obtaining a more accurate calibration of the model. This isshown in Fig. 5, which shows the stress paths and the associatedlocalization analysis for two simulated oedometric compressiontests characterized by different values of ξm. In the case ofξm ¼ 0:75 (i.e., the value previously estimated from triaxial tests)the bifurcation analysis rules out localization processes, as the

computed determinant is always positive (Fig. 5a and b). On thecontrary, the use of ξm ¼ 4:0 modifies the stress path (Fig. 5c and d)and increases the potential for mechanical instability [32].

This result can be explained with the fact that increasing valuesof ξm cause the expansion of the localization domain (Fig. 2), thusfavoring the intersection between the evolving zone of potentiallocalization and the stress path [32]. This is confirmed by theinspection of the stress–strain response via bifurcation analyses,which provide negative values of the bifurcation indices (Fig. 5d).By inspecting the range of possible band angles, it can be noticedthat inclinations of 901 are possible at a certain point of thedeformation process (point iv). Despite this, horizontal bands arenot the preferential localization mode, as inclined bands of shear/compaction are activated first (at iii) and are featured by thelowest value of determinant even at strains for which thehorizontal bands become theoretically possible. In other words,regardless of the value of ξm, the stress path generated by theoedometric constraint tends to cross the localization zone in astress domain in which compactive shear bands are more likelyto occur.

While these analyses highlight the interaction between kine-matic constraints, stress path and localization mode [32,33], theycannot be considered to be conclusive, as the patterns of localizedcompaction in actual specimens eventually depend on materialheterogeneity and boundary effects. This raises two major ques-tions that will be addressed hereafter via numerical modeling:(i) is the adopted calibration procedure based on bifurcationanalyses adequate to capture the global behavior of rock samplesand the consequent localization patterns? And (ii) how do bound-ary constraints and material heterogeneity interact with thegeometric patterns of localized compaction?

3. Finite element analysis of localized compaction

3.1. Regularization scheme

Reliable numerical modeling of strain localization in plasticsolids requires regularization schemes to remove possible discre-tization dependent solutions [40,41,60,61]. Indeed, the energydissipation during localization is controlled by the size of the

-0.05-0.03-0.010.010.030.050.070.090.11

0.0320.0150.0080.004

-0.05

0.00

0.05

0.10

0.15

30 60 90 120 150

30 60 90 120 150

0.0070.0180.0330.056

0

1000

2000

3000

4000

det(A

)/det

(Ae)

band orientation angle, (°)a

q(k

Pa)

a

0

1000

2000

3000

4000

0.00 0.05 0.10 0.15 0.20

0 0.05 0.1 0.15 0.2

q(k

Pa)

a

a

det(A

)/det

(Ae)

band orientation angle, (°)

Fig. 4. Constitutive response for triaxial compression and associated localization analysis: (a and b) simulated triaxial compression test at 1100 kPa initial confinement;(c and d) simulated triaxial compression test at 2000 kPa initial confinement.

A. Das, G. Buscarnera / International Journal of Rock Mechanics & Mining Sciences 71 (2014) 91–104 95

process zone. Given the lack of an internal length in traditionalelasto-plastic models, such zone may depend on the mesh size[62], thus producing a pathological mesh dependency of simula-tions performed in the post-localization regime.

To overcome this problem, we have implemented a computa-tionally inexpensive rate-dependent regularization of the Perzyna-type [63], which has been successfully used for several types ofgeomaterials [21,41,64,65]. The enhancement of the constitutivemodel has been carried out by modifying the plastic flow rule inthe following manner:

_εpij ¼⟨ϕðf Þ⟩η

∂g∂σij

ð14Þ

where ⟨…⟩ represents the McCauley brackets; ϕ is the overstressfunction (here assumed to be given by f =ðps0þð1þrÞpm0Þ); f is thecurrent value of the yield function; η is the viscosity parameter anddt is the time increment during which an axial strain rate isimposed ð_εa ¼ dεa=dtÞ. A value of axial strain rate εa ¼ 10�6=s hasbeen used, as is typical for quasi-static loading. Moreover, thevalue of the viscosity parameter ðη¼ 131 sÞ has been chosen toobtain a rate-dependent response as close as possible to that of therate-independent version of the model, while preserving at thesame time the mesh independence of the computed solution.

Details about the calibration procedure of the selected regulariza-tion scheme can be found in [21].

The model has been implemented in the ABAQUS Finite Elementcode [66] through a UMAT subroutine. The constitutive relation hasbeen integrated though an implicit algorithm, while the rate depen-dent regularization has been constructed through the algorithmichierarchy proposed by Wang et al. [40]. The effectiveness of theregularization strategy has been assessed through a simple numericalexercise. Two plane strain simulations have been performed usingrectangular specimens (40 mm�20 mm) made with eight nodequadrilateral finite elements and two mesh densities (800 and 2200elements, respectively). The bottom boundary of the specimens hasbeen subjected to roller boundary conditions, while the horizontaldisplacement at the bottom-left corner has been constrained toprevent lateral movements. Finally, both specimens have been sub-jected to a constant rate of displacement at the top boundary. Theexpected outcome from this numerical exercise is the activation ofa shear band in specimens subjected to unconfined compression.To trigger the band, a slight weakness has been assigned at one of thebottom elements of each model (i.e. 1% reduction in pm0, marked witha white box in Fig. 6a and b).

The effect of the mesh density is presented in Fig. 6, where theformation of a shear band is captured through the contours of

-0.05

0.05

0.15

0.25

0.35

0.45

viviiiiii

-0.05

0.05

0.15

0.25

0.35

0.45

viviiiiii

0

500

1000

1500

q(k

Pa)

p (kPa)

det(A

)/det

(Ae)

band orientation angle, (°)

0

500

1000

1500

30 60 90 120 150

30 60 90 120 150-500 500 1500 2500

-500 500 1500 2500q

(kPa

)

p (kPa)

det(A

)/det

(Ae)

band orientation angle, (°)

(i) (ii) (iii)(iv)

(v)

(i)(ii)

(iii)

(iv)

(v)

m =0.75m =0.75

m =4.0m =4.0

Fig. 5. Predicted stress path for oedometric compression test and localization analysis at five selected stress states (marked by i, ii, …). Results obtained by using ξm¼0.75 (a, b)and ξm¼4.0 (c, d).

0

0.005

0.01

0.015

0.02

0 0.2 0.4 0.6 0.8

800 elements (a)2200 elements (b)

0.800.720.640.560.480.400.320.240.160.080.00

U (mm)

F(k

N)

Weak element

Fig. 6. Simulated response of rate-dependent rock specimens subjected to plane strain conditions: (a) coarse mesh specimen; (b) fine mesh specimen; (c) force–displacement plot.

A. Das, G. Buscarnera / International Journal of Rock Mechanics & Mining Sciences 71 (2014) 91–10496

plastic volumetric strain. At a certain value of axial strain bothspecimens form shear bands with identical features (i.e., thecomputed orientation, thickness and progression of the bandsare identical). Most importantly the predicted axial stress–strainresponse (Fig. 6) is very similar, which ensures almost identicaldissipation irrespective of the mesh size. The numerical test clearlyindicates that the rate-dependent regularization scheme effi-ciently overcomes the pathological mesh dependency arising frombifurcation instability.

3.2. Finite element model of calcarenite specimens

The response of calcarenite samples has been simulated through3D finite element models both for the case of triaxial compressionand oedometric compression. For this purpose, two different typesof specimen have been used. Triaxial tests have been simulatedthrough a cylindrical specimen of size of 38 mm�76 mm (Fig. 7a)discretized with 32650 linear brick elements (8-nodes). Oedometerspecimens have been assigned a size of 55 mm�22 mm anddiscretized with 52835 linear tetrahedron elements (4-nodes)(Fig. 7b). Vertical movements (Y) and rotations about the verticalaxis have been prevented using appropriate restraints at thebottom boundary of each specimen. The horizontal (X–Z) move-ments of the bottom central node have also been restricted to avoidlateral instability. To ensure oedometric boundary conditions,additional constraints have been imposed in the oedometer speci-men by restricting the radial displacements of the lateral boundary.

The loading process has been applied in two steps; (a) initiallyboth specimens have been subjected to isotropic compression ðp0Þ;and then (b) displacement control ð_εa ¼ 10�6=sÞ has been appliedat the top boundary. All the parameters for calcarenite given inTable 1 have been used, except ξm which has been assumed to beequal to 4.0, as indicated by bifurcation studies. To initiateheterogeneous deformation, a stress concentration is needed tocreate a hotspot of inelastic restructuration. This aspect has beenincluded in the model through a simulated material heterogeneity(Fig. 7), here imposed via random spatial variation of pm0 with

71% from the average calibrated value and standard deviation of0.014. This strategy generates varying yielding points at differentlocations within the specimen, thus allowing the spontaneousgeneration of heterogeneous compaction zones.

3.3. Simulation of specimen response upon drained triaxialcompression

Drained triaxial compression has been simulated to trackformation and propagation of compaction zones, as well as toreproduce the global response of calcarenite specimens. To locatethe areas of the samples subjected to concentrated inelasticity andloss of strength, the variation of second-order work has also beenmonitored. The first simulation is a compression test starting froma hydrostatic confinement of 2 MPa, i.e. a condition that inexperiments has originated heterogeneous deformations in theform of horizontal compaction zones [14]. Fig. 8 reports thecontours of cumulative and incremental plastic volumetric strainsin the simulated sample, as well as the local second-order work atdifferent stages of the process (open circles in Fig. 9). Compactionbands are mostly accumulated at the boundaries (see εa¼0.008–0.016 in Fig. 8) and propagate towards the center of the specimen.Two mechanisms of band development are found: (i) bands thatderive from stress concentration and extend across the width ofthe specimen and (ii) progressive formation of new bands alignedat different locations along the axis of the specimen. Negativevalues of computed second-order work are concentrated insidethe bands, thus defining the active zone of inelastic strainlocalization. By contrast, positive values are found outside thebands, within portions of the sample undergoing elastic unloading.As discussed with reference to material point analyses, thepotential for strain localization decreases once the mechanicaldegradation stage is complete, with densely packed conditionscausing the re-hardening of the specimen. Once these conditionsare attained, a homogeneous response is restored in the strain-hardening regime (Fig. 8).

2.2112.2092.2062.2042.2022.2002.1982.1952.1932.1912.189

pm0 (MPa)

76 m

m

38 mm

22 mm

55 mm

Fig. 7. Finite element mesh for (a) triaxial compression tests; (b) oedometer tests with random spatial heterogeneity in numerical specimens of Gravina calcarenite.

A. Das, G. Buscarnera / International Journal of Rock Mechanics & Mining Sciences 71 (2014) 91–104 97

The computed solution clearly supports the predictionsobtained through material point simulations, as indeed theband inclination coincides with that derived from a localizationanalysis (Fig. 4) corresponding to the minimum acoustic tensordeterminant. As discussed by Das et al. [21], this feature can be

seen as an outcome of the maximum dissipation principle inthe localization plane, according to which the band inclinationassociated with the minimum value of determinant of theacoustic tensor can be seen as the natural weakest directionof deformation.

0.0400.0360.0320.0280.0240.0200.0160.0120.0080.0040.000

6.05.44.84.23.63.02.41.81.20.60.0

1.00.50.0-0.5-1.0-1.5-2.0-2.5-3.0-3.5-4.0

Fig. 8. Simulated distribution and propagation of compaction bands in triaxial compression tests (2000 kPa initial confinement) on Gravina calcarenite specimen.

0.6

0.7

0.8

0.9

1

1.1

1.2

BVP response

Experiment

0

1000

2000

3000

4000

5000

0 1000 2000 3000 40000.00 0.05 0.10 0.15 0.20

BVP responseExperiment

q(k

Pa)

a

Selected points refers to Fig. 8

p (kPa)

e

Fig. 9. Stress–strain response for simulated triaxial compression tests at 2000 kPa initial confinement. A comparison with measured data is reported (data after [35]).

A. Das, G. Buscarnera / International Journal of Rock Mechanics & Mining Sciences 71 (2014) 91–10498

The accuracy of these simulations can also be assessed bycomparing the simulated global response with measured data forcalcarenite specimens. This comparison is provided in Fig. 9 in termsof both stress–strain behavior and volumetric response. The figureillustrates a satisfactory agreement between computations and experi-ments, thus corroborating the accuracy of our calibration strategy.

4. Simulation of the effect of boundary constraints

Triaxial compression produces highly controlled stress paths thatcross the bifurcation domain in predefined locations. This feature islikely not to reflect conditions in the field, where the stress pathresults from local burial history and complex tectonic processes [33].Oedometric tests can therefore be a useful scheme to study realisticfeatures of compaction banding [38], as they mimic a simpledeposition history (i.e., K0 consolidation) and impose boundaryconstraints able to generate the same strain kinematics taking placein pure compaction bands. Gravina calcarenite offers an interestingopportunity to meet this goal, as its K0-compression obtained fromtriaxial testing exhibits considerable loss of structure and porecollapse. Although the original calcarenite samples tested by Lagioiaand Nova [35] were discarded without subjecting them to post-mortem inspection, subsequent studies by Arroyo et al. [14] andCastellanza et al. [53] on similar materials have reported horizontalcompaction zones. Most of these studies were based on a modifiedoedometer with deformable boundaries, which was able to produce astress–strain response similar to that observed in K0-compressiontests done in a triaxial cell [34,35]. For this reason, we have

reproduced numerically the conditions imposed in both types ofexperiments, with the aim to elucidate the processes that may haveoriginated localized compaction in the tested samples.

4.1. Rigid boundary constraints

To compare numerical predictions and measured data, a finiteelement model of a slender specimen (Fig. 7a and b) has been usedto mimic the K0-consolidation originally performed in a triaxialcell (where average radial strains were prevented through anautomated servo-control, [35]). In this case, the K0 stress pathhas been replicated by applying varying horizontal nodal forces atthe lateral boundary. The values of such varying radial stressesrequired to prevent average lateral strains have been inferred froma prior material point analysis. Moreover, to understand themechanisms that may have caused localized compaction in oedo-meter tests done in subsequent studies [53], a second numericalmodel has been used, which is closer to the sample geometry usedin a typical oedometer apparatus (Fig. 7c and d). In this case, theexperiment has been simulated in a simpler way, i.e. by fixing thehorizontal displacements of the lateral boundary, thus directlypreventing the occurrence of radial strains. To replicate the sameinitial conditions of the experiment, both models have beenhydrostatically compressed up to a pressure of 210 kPa and thendeformed axially by imposing the vertical displacements at the topboundary. Fig. 10 shows the computed deformation patterns in thenumerical specimens, showing that in both cases shear zonesdevelop inside the domain.

Fig. 10. Snapshots of deformation patterns within the numerical specimens during K0-consolidation; (a, b) servo-controlled triaxial configuration at selected axial strains;(c, d) conventional oedometric configuration.

A. Das, G. Buscarnera / International Journal of Rock Mechanics & Mining Sciences 71 (2014) 91–104 99

Positive plastic volumetric strains within the band highlight thecompactive nature of the localized process. Despite horizontalbands may be seen as a natural mode of strain localization for thistype of loading, the analysis shows that the stress path generatedby the imposed constraints induces localized shear zones insidethe specimen. While such non-trivial mechanisms do not matchthe symmetry imposed by the boundary conditions (especially inthe case with rigid lateral constraints), the possibility of activatinga shear zone was already predicted by the bifurcation analysis inSection 2. In other words, the complex deformation pattern inFig. 10 can be interpreted as the spontaneous outcome of localstress paths that cross the bifurcation domain in the zone ofcompactive shear bands rather than in correspondence of thedomain of pure compaction banding.

Fig. 11 superposes the computed global behavior with the datafor K0 consolidation reported by Lagioia and Nova [35]. Althoughthe three curves can be compared only qualitatively, it is readilyapparent that their stress paths share similar attributes. On theone hand, such result confirms the interpretation of earlierexperiments [14], in which these attributes were interpreted as asignature of strain localization. On the other one, they point outthat the interpretation of these global signatures as pure horizon-tal compaction bands might be misleading, as it can overlookcomplex localization processes that are not easily detectablethrough boundary measurements.

4.2. Flexible boundary constraints

A factor of interest for the onset of localized compaction duringconstrained deformation is the flexibility of the boundaries thatimpose the kinematic restraints. While the use of rigid boundariesis typical for oedometer testing, flexible boundaries can be

adopted to measure the radial stress originated by a constraineddeformation. This configuration was adopted by Arroyo et al. [14]and Castellanza et al. [53] in the apparatus used to detecthorizontal compaction bands in specimens of highly porousgeomaterials.

To simulate such conditions, the numerical model has beenenriched by adding a flexible ring around the specimen. Brass wasused to simulate the ring (G¼37.59 GPa and K¼100 GPa), whichwas eventually discretized via the same finite elements used forthe calcarenite sample. Given the deformability of the lateralboundaries, the specimen is allowed to slightly expand duringvertical compression. The compressive load has been applied usinga displacement boundary condition for the rigid plate on the top ofthe specimen, while a second rigid plate has been placed at thebase of the specimen to restrict vertical movements. Steel hasbeen used to simulate the top and bottom plates (G¼78.74 GPaand K¼144.9 GPa), with the purpose to create sufficiently rigidboundaries, and a small friction coefficient of 0.1 has beenimposed at the interface between the specimen and the rigidboundaries. This arrangement is sketched in Fig. 12a, and has beenused in combination with a calcarenite model identical to thespecimen shown in Fig. 7 (i.e., again characterized by a randomspatial heterogeneity).

Fig. 12b shows the computed patterns of deformation interms of cumulative and incremental plastic volumetric strains.It can be readily observed that flexible lateral boundaries andsmall amounts of friction promote the formation of horizontalcompaction bands. These bands are originated at the top andbottom of the specimen and propagate towards the center, thusreproducing a response very similar to that observed in thepreviously mentioned experimental studies. In addition, thesimulations show that minor alterations to the boundary

5.04.03.02.01.00.0

0.100.080.060.040.020.00

Fig. 12. Numerical simulation of K0 consolidation with flexible ring oedometer, showing the propagation of compaction bands.

0

500

1000

1500

0

500

1000

1500

0 500 1000 1500 2000 25000 0.05 0.1 0.15a

q(k

Pa)

q(k

Pa)

p (kPa)

ExperimentBVP response (K0 in triaxial setup)BVP response (K0 in oedometric setup)Selected points refer to Fig. 10

ExperimentBVP response (K0 in triaxial setup)BVP response (K0 in oedometric setup)Selected points refer to Fig. 10

Fig. 11. Comparison between computed response and measured data for oedometric compression of Gravina calcarenite; (a) stress–strain response; (b) stress path (dataafter [35]).

A. Das, G. Buscarnera / International Journal of Rock Mechanics & Mining Sciences 71 (2014) 91–104100

conditions are sufficient to alter the localization response, thusconsiderably deviating from the idealized conclusions obtainedfrom a bifurcation analysis of the constitutive predictions.

4.3. Effect of patterned material heterogeneity

Here we study the role of the spatial pattern of heterogeneity,passing from a random distribution of the internal variablecontrolling the yielding threshold (pm) to a layered heterogeneity.In other words, a spatially varying pm has been assigned along thevertical axis (Y), while the material parameters have been assumedto be homogeneous across horizontal planes (X–Z), as shown inFig. 13. This scheme mimics the natural heterogeneity of sedimen-tary formations, in which depositional layers of lithified materialform under high pressures, creating patterns of spatially varyingdensity and strength. Similar to kinematic conditions, such geo-metric features constrain the mechanical response and may forcethe system to deform in accordance with the symmetry require-ments. As a result, it is arguable that a spatially patterned structuremay influence also the formation of compaction bands in layeredrock masses, as it has been found both in field and laboratoryinvestigations by Aydin and Ahmadov [67] and Townend et al.[68], respectively.

Once a vertical layering has been assigned, the localizationprocess is initiated at the weakest zone, affecting the entire layerthat has reached yielding conditions. This process generateshorizontal compaction bands that propagate towards the bound-ary of the specimen with further deformation. At variance withthe previous cases, no lateral extension of these bands is obtained,as each band forms across an entire layer. In addition, as a resultof boundary effects the band alignment tends to change as

the localization process approaches the top boundary of thespecimens.

Fig. 14 compares the stress–strain response and the stress pathassociated with the three numerical simulations of oedometersamples. As is readily apparent, the macroscopic averaging of theexternal quantities generates global responses that are not easilydistinguishable one from another, thus indicating that suchheterogeneous deformation mechanisms generated by oedometriccompression (e.g., non-symmetric shear zones or pure compactionbands) are associated with similar macroscopic signatures, and aretherefore not easily distinguishable via external force–displace-ment measurements.

These results (Fig. 14) prompted additional inspections of thelocal response within the localization zone, with the goal toelucidate the mechanisms underlying the patterns of localizedcompaction predicted in the oedometer simulations. Local stresspaths at three selected integration points (marked with a whitetriangle in Fig. 15a, d, g) have been plotted in Fig. 15(b, e, h). All theselected points are located inside the process zone of heteroge-neous compaction, thus providing a snapshot of the active defor-mation mechanisms inside the bands. Furthermore, localizationanalyses similar to those discussed in Section 2.4 have been usedto track the characteristics of the localization events at specificstress states along the computed paths (states i and ii). Suchanalyses have been performed in a qualitative manner, by exploit-ing the similarity between the response provided by rate-independent and rate-dependent formulations (as indeed thelatter cannot be rigorously analyzed via the localization conditionin Eq. (11)). In other words, rate-independent localization analyseshave been carried out for stress states very close to thosecomputed through the regularized finite element model. For thispurpose, the elasto-plastic constitutive tensor was re-computed by

0

1000

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3000

0 0.05 0.1 0.15

Rigid ring oedometerFlexible ring oedometerPatterned heterogeneity

0

500

1000

1500

0 500 1000 1500 2000 2500

Rigid ring oedometerFlexible ring oedometerPatterned heterogeneity

q(k

Pa)

p (kPa)a

q(k

Pa)

Fig. 14. Comparison of the response computed for the three simulated oedometer specimens.

Fig. 13. Numerical simulation of K0 consolidation in calcarenite specimen with pattern heterogeneity, showing the propagation of compaction bands.

A. Das, G. Buscarnera / International Journal of Rock Mechanics & Mining Sciences 71 (2014) 91–104 101

using the same hardening parameters (ps, pm) and modifiedstress-ratios (ηn¼q/(pþr.pm)) obtained from the numerical simu-lations at the selected points. This strategy has allowed us toquantify the localization potential, thus identifying the preferentiallocalization modes at the selected points and providing a mechan-istic interpretation for the predicted patterns of heterogeneouscompaction.

Fig. 15 displays the evolution of the normalized acoustic tensorat the three selected points. In all cases the discontinuousbifurcation is predicted to be initiated in the form of an inclinedshear band, eventually turning into pure compaction localizationwith horizontal band angles. Nevertheless, Fig. 15b indicates thatthe simulation characterized by rigid boundaries and randomheterogeneity deviates significantly from the ideal stress pathassociated with oedometric conditions (Fig. 5c). In addition,stresses prone to pure compaction banding are achieved onlywhen the values of the acoustic tensor determinant computed forα¼901 are just below zero (i.e., there is no longer significantpotential for localization when a horizontal band becomes thepreferential mode of heterogeneous compaction, Fig. 15c). Suchcharacteristics are clearly an outcome of the prior shear bandmechanism, which deteriorates the potential for further strainlocalization and reduces dramatically the likelihood of purecompaction banding right at the beginning of the stage of inelasticloading (Fig. 10). By contrast, the use of flexible boundary con-straints and layered heterogeneity tends to promote local stresspaths that maintain a significant potential for pure compactionbanding up to stress states located in proximity of the plastic cap(Fig. 15e, h), thus favoring the formation and propagation ofhorizontal compaction bands (Fig. 15f, i).

5. Conclusions

We have studied the patterns of localized compaction emer-ging from constrained and unconstrained axisymmetric loading onsimulated specimens of porous calcarenite. In the first part of thepaper we have discussed a strategy to refine the calibration of themodel parameters by means of the bifurcation theory. It has beenshown that the cross-correlation of data from different stresspaths can be used to quantify material properties and achieveimproved predictions. In addition, the dependence of strainlocalization on the confinement stress has been captured correctly,reproducing transitions from high-angle shear bands to pure(horizontal) compaction bands.

In the second part of the paper, finite element analyses havebeen used to investigate the role of loading paths and kinematicconstraints. Rate-dependent regularization and spatial randomi-zation of the rock properties have allowed us to reproduce realis-tically the response of calcarenite specimens, obtaining a satisfac-tory agreement with the measured data. Such numerical modelhas been used to inspect the effect of heterogeneity and boundaryeffects on oedometric compression, a loading mode useful tounderstand the mechanics of compaction in the natural environ-ment. It has been shown that certain types of boundary conditions(e.g., flexible lateral boundaries with friction) tend to divert thestress paths towards the domain of pure compaction banding,while other conditions (e.g., rigid lateral boundaries) exacerbatethe tendency to develop shear zones. In all cases, however, thedevelopment of heterogeneous deformation zones has been pre-dicted to disappear with increasing strains. In addition, theaveraging of the quantities computed at the boundaries has been

Fig. 15. Zones of localized compaction mapped through incremental volumetric plastic strains, local stress paths and rate-independent localization analyses for oedometriccompression simulated with (a–c) rigid boundary constraints; (d–f) flexible boundary constraints; (g–i) layered heterogeneity.

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found to generate global signatures that are not easily distinguish-able from those of pure compaction banding, suggesting that theidentification of specific patterns of localization from force–displacement measurements can be particularly elusive if notcomplemented by local measurements at selected stress/strainlevels.

The numerical results reported in this work provide guidanceto identify compaction bands at both laboratory and field scale,especially in case of unconventional stress paths mimicking theeffect of natural processes. In particular, they disclose two majorfeatures of localized compaction: (i) the structures generated bythese mechanisms tend to disappear upon further straining, and(ii) their geometric and mechanical characteristics may not benecessarily regarded as a pure product of the constitutive responseof the parent rock, but rather as the outcome of complex interac-tions between structural effects, boundary conditions, local het-erogeneities and material properties. These findings may providea justification for the elusive characteristics of pure compactionbands, which until now have been found in the field only undervery particular circumstances.

Acknowledgments

This research has been initiated with a Booster Award from theInstitute for Sustainability and Energy at Northwestern (ISEN) andhas been partially supported by the U.S. National Science Founda-tion, Geomechanics and Geomaterials Program under GrantCMMI-1351534.

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