The interplay between anisotropy and strain localisation in granular soils:a multiscale insight

J. ZHAO� and N. GUO�

This paper presents a multiscale investigation on the interplay among inherent anisotropy, fabricevolution and strain localisation in granular soils, based on a hierarchical multiscale framework withrigorous coupling of the finite-element method (FEM) and discrete-element method (DEM). DEMassemblies with elongated particles are generated to simulate inherent anisotropy and are embedded tothe Gauss points of the FEMmesh to derive the required constitutive relation. Specimens preparedwithdifferent bedding plane angles are subjected to biaxial shear under either smooth or rough loadingplatens. Key factors and physical mechanisms contributing towards the occurrence and development ofstrain localisation are examined. The competing evolutions of two sources of anisotropy, one related toparticle orientations and the other related to contact normals, are found to underpin the developmentof the shear band. A single band pattern is observed under smooth boundary conditions, and itsorientation relative to the bedding plane depends critically on the relative dominance between the twoanisotropies. Under rough boundary conditions, the non-coaxial material response and the boundaryconstraint jointly lead to cross-shaped double shear bands. The multiscale simulations indicate that theDEM assemblies inside the shear band(s) undergo extensive shearing, fabric evolution and particlerotation, and may reach the critical state, while those located outside the shear band(s) experience mildloading followed by unloading. The particle-orientation-based fabric anisotropy needs significantlylarger shear and dilation for mobilisation than the contact-normal based one. The asynchrony inevolution of the two fabric anisotropies can cause non-coaxial responses for initially coaxial packings,which directly triggers strain localisation.

KEYWORDS: anisotropy; discrete-element modelling; finite-element modelling; numerical modelling;strain localisation

INTRODUCTIONMany intriguing phenomena observed in granular mediareflect non-trivial microstructural mechanisms that can bebetter understood from a multiscale perspective (Guo &Zhao, 2014). Strain localisation and anisotropy are amongthe representative ones. Strain localisation in sand is widelyregarded as an important precursor associated with thefailure of geomaterials and relevant geostructures, whileanisotropy underpins key mechanical responses of granularmedia including strength, deformation and failure. As mostin-situ granular soils are naturally inherently anisotropic,how anisotropy interplays with strain localisation during theinception and formation of the latter has drawn sustainedattention. Anisotropy has indeed been found among the keyfactors, along with density, boundary conditions, drainageconditions and specimen dimensions, affecting the occur-rence of strain localisation in granular soils (Vardoulakis,1996; Mokni & Desrues, 1999; Desrues & Viggiani, 2004;Rechenmacher, 2006). Experimental studies by Tatsuokaet al. (1990) and Lade et al. (2008) showed that the initiationand development of a shear band in sand can be greatlyinfluenced by the presence of anisotropy.

Numerous past studies, including those dealing with theeffect of anisotropy, have treated strain localisation as a

material instability based on bifurcation analysis (Rudnicki& Rice, 1975; Bigoni & Loret, 1999, among others). Therehave been attempts to simulate the phenomenon of strainlocalisation in anisotropic sand samples, based on eithercontinuum modelling, typically by finite-element method(FEM), or micromechanics using the discrete-elementmethod (DEM) (Bauer et al., 2004; Tejchman & Górski,2010; Fu & Dafalias, 2011b). Despite being successful inreproducing important aspects of strain localisation, such asthe correlation between shear band pattern and inclinationand inherent anisotropy, these studies are not withoutlimitations. DEM-based micromechanics approaches, forexample, typically have difficulties in handling the largenumber of particles required to simulate a real engineeringproblem. Continuum modelling often relies on constitutiverelations which are phenomenological in nature. To triggerstrain localisation, it is common to introduce imperfectionsin elements or random field distribution of materialproperties in a FEM simulation, which may add subjectivityand uncertainties to the simulations and distract from theidentification of key mechanisms of strain localisation. Thereis another popular class of micromechanics approachpioneered by Chang and co-workers, which has beenapplied to treat inherent anisotropy and instability in sand(Chang & Hicher, 2005; Hicher et al., 2008; Chang & Yin,2010; Chang et al., 2011). Notably, most studies haveneglected the evolution of fabric anisotropy and how itinteracts with the developing shear band. Indeed, anisotropymay evolve during the loading process, especially in regionswhere strain is localised. It is evident that an evolving fabricmay affect importantly the development of the shear band aswell as its final pattern (Gao & Zhao, 2013). To quantify theessential feature of fabric and its evolution, a

� Department of Civil and Environmental Engineering, Hong KongUniversity of Science and Technology, Clearwater Bay, Kowloon,Hong Kong.

Manuscript received 18 September 2014; revised manuscriptaccepted 15 April 2015.Discussion on this paper closes on 1 January 2016, for further detailssee p. ii.

Zhao, J. & Guo, N. (2015). Géotechnique 65, No. 8, 642–656 [http://dx.doi.org/10.1680=geot.14.P.184]

642

micromechanics approach has been proved more appropriate(Guo & Zhao, 2013b; Zhao & Guo, 2013), whereas acontinuum approach may be more efficient and straightfor-ward to simulate strain localisation as a boundary valueproblem (BVP). To capture the interplay between the two, acombination of the two approaches is desirable.In this study, the authors employ a hierarchical multiscale

approach coupling DEM and FEM (Guo & Zhao, 2013a,2014) to model the behaviour of anisotropy, fabric evolutionand strain localisation in a granular soil. The multiscaleframework is in line with early studies by Nitka et al. (2011)and Andrade et al. (2011). Specifically, the multiscaleapproach treats a continuum domain by FEM and embedsa DEM assembly as the representative volume element(RVE) at each FEM Gauss point to derive the materialconstitutive relation. It helps to circumvent the phenomen-ological assumptions and parameter fitting in conventionalcontinuum approaches, expedites a direct linkof macroscopicsoil behaviour with its microscopic origin and enablesmicromechanical insights to be gained into complex soilproperties such as history dependency, non-coaxiality,dilatancy and critical state (Schofield & Wroth, 1968;Gutierrez & Ishihara, 2000; Yu & Yuan, 2006; Tejchman &Wu, 2009; Muhlhaus et al., 2010), as well as the anisotropyand strain localisation to be considered here.

APPROACH AND FORMULATIONThe hierarchical multiscale framework is based on a

rigorous coupling of two open-source codes – Escript(FEM engine, Gross et al., 2007) and Yade (DEM engine,Šmilauer et al., 2010), which is briefly summarised below. Adetailed formulation and solution procedure of the schemecan be found in Guo & Zhao (2014).

Governing equation and the FEM modelThe governing equation of a BVP can be written in the

following strong form (considering quasi-static equilibriumin the absence of gravity)

σij;j ¼ 0 ð1Þwhere σij is the stress tensor. The FEM commonly employs avariational weak form of equation (1), which is transformedinto a system of linear equations upon discretisation

Ku ¼ f ð2Þwhere K is the tangent stiffness matrix, u is the FEM nodaldisplacement vector, and f is the nodal force lumped from theapplied boundary traction.A conventional FEM needs to assume an elasto-plastic (or

hypoplastic) relation to describe the material response and toassemble K from the elasto-plastic modulus D ep which ishighly non-linear and history-dependent for granular soils.The hierarchical multiscale approach abandons any assump-tion of phenomenological constitutive models, and incursseparate DEM simulations at each FEM Gauss point toprovide the material constitutive responses. For a givenglobal loading step, the typical solution procedure of themultiscale approach consists of the following substeps: (a)evaluating the tangent modulus Dijkl based on the DEMpacking at each Gauss point to form K; (b) interpolating thelocal deformation ui,j at each Gauss point from the (trial)FEM solution and applying them as boundary conditions forthe corresponding RVE packing; (c) solving the local RVE,updating the total stress at the Gauss point, and assemblingthe global tangent operator and stress tensor for the FEMdomain; (d ) performing Newton–Raphson iteration over

(a) to (c) to find a converged solution of equation (2) for thedisplacement field u.

DEM model and macroscopic definitionsTo simulate inherent anisotropy, the present authors have

modified the DEM model in Guo & Zhao (2014) byreplacing the circular particles therein with elongatedclumps consisting of two identical discs, as shown in Fig. 1(see also Evans & Frost, 2010; Li & Yu, 2010). The centroiddistance of the two clumped circular discs is set to be 1·25 r(r¼ radius of the constituent discs), which leads to an aspectratio of 1·625. The area Acl, the equivalent radius req and theinertia of moment (with regard to the mass centre) Icl of theclump can be computed as follows

Acl ¼ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 0�6252

p� 0�625þ π � arccosð0�625Þ

h i

� r2 � 5�4676r2req :¼

ffiffiffiffiffiffiffiffiffiffiffiffiAcl=π

p � 1�3192rIcl � 0�9994�mclr2

8>>>><>>>>:

ð3Þwhere mcl is the mass of the clump. Icl in equation (3) isestimated by discretising the clump boundary to a star-shaped polygon (see Fig. 1). The coefficient is determinedby approximating the clump boundary with a 3000-segment polygon (this leads to an accuracy up to fourdecimal places).A similar linear force–displacement contact law for the

DEM model as described in Guo & Zhao (2014) is followed,adopting a normal stiffness kn = req*¼455MPa and atangential stiffness kt¼0·3 kn, where req

* is the commonradius of the two contacted particles. The interparticlefriction angle governing the Coulomb friction criterion isset to be ϕ ¼0·5 rad. The density of the particles is set to ρcl¼2650 kg=m3. The DEM implementation of the clumps issimilar to Šmilauer et al. (2010).Two macroscopic quantities pivotal to the multiscale

computation, the tangent modulus and the Cauchy stresstensor, are homogenised from the RVE packing at eachGauss point. By adopting a uniform strain assumption(Wren & Borja, 1997; Kruyt & Rothenburg, 1998; Luding,2004), the tangent modulus used for iterative solution ofequation (2) is estimated as

Dijkl ¼ 1V

Xc[Nc

knnci dcj n

ckd

cl þ kttci d

cj t

ckd

cl

� �ð4Þ

where V is the total volume of the assembly, Nc is the totalcontact number in the assembly, n c is the outward contactnormal, d c is the branch vector connecting the centres of thetwo contacted clumps (see Fig. 1). t c is the unit vector alongthe tangential direction of a contact. The Cauchy stresstensor is evaluated from the Love’s formula:

σij ¼ 1V

Xc[Nc

dci f

cj ð5Þ

where f c is the interparticle contact force. From σij, themean effective stress p and the deviatoric stress q can becalculated (in two dimensions): p¼σii=2, q ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

sijs ji=2p

(sij¼σij� pδij, δij denoting the Kronecker delta). The localstrain is measured from the displacement gradient (obtainedfrom the FEM solution), namely εij¼�(ui,jþuj,i)=2.Accordingly, the volumetric and deviatoric strains can bedefined: εv¼ εii, εq ¼

ffiffiffiffiffiffiffiffiffiffiffiffi2eije ji

p, where eij¼ εij� εvδij=2. The

rigid body rotation (note that the rigid body rotation isspecifically referred to the overall rotation of the RVE

ANISOTROPYAND STRAIN LOCALISATION IN GRANULAR SOILS 643

packing here) ωij¼ (uj,i� ui,j)=2 is also considered for theRVE boundary condition to accommodate large defor-mations in strain localisation.

Quantification of fabric anisotropyAccording to Casagrande & Carillo (1944), granular soils

commonly possess two forms of anisotropy, namely, inherentanisotropy and induced anisotropy (see also Arthur &Menzies, 1972; Arthur et al., 1977; Wong & Arthur, 1985).Inherent anisotropy is formed due to preferentially orientedparticles, voids and contact normals during geologicaldeposition processes such as tectonic compaction andhydraulic=pneumatic transport or laboratory sample prep-aration through air=water pluviation. Induced anisotropy iscaused by the applied loading, which may alter thepre-existing inherent anisotropy and induce rearrangementsof the soil fabric. Both types of anisotropy affect the overallbehaviour of sand significantly (Oda, 1972a, 1972b; Abelev& Lade, 2003; Lade & Abelev, 2003; Hasan & Alshibli, 2010;Guo & Zhao, 2013b; Zhao & Guo, 2013). This paper adoptsthe fabric tensor to quantify both sources of soil anisotropy.Prior to loading, the fabric tensor represents the state ofinherent (or more accurately, initial) anisotropy in sand,whereas upon loading, it reflects the superposition ofinherent anisotropy and induced anisotropy. In particular,two different definitions of fabric tensor are compared. Onefollows the widely used definition proposed by Satake (1982)and Oda (1982) based on the contact normal distribution(ϕij

c), and the other is defined based on the particleorientation (ϕij

p)

ϕ�ij ¼

ðΘEðΘÞn�i n�j dΘ ¼ 1

N�

XN�

n�i n�j ð� ¼ c; pÞ ð6Þ

where the super=subscript * denotes either ‘c’ or ‘p’ for thetwo tensors (throughout this paper). np is a unit vectorpointing along the longest axis of the clump (see Fig. 1).Np isthe total number of clumped particles in the assembly. Thedistribution function E(Θ) can be approximated by asecond-order truncation of its Fourier expansion

EðΘÞ ¼ 12π

1þ F �ijn

�i n

�j

� �ð� ¼ c; pÞ ð7Þ

where the deviatoric tensors F cij and Fp

ij quantifying the fabricanisotropy can be calculated

F �ij ¼ 4 ϕ�

ij �12δij

� �ð� ¼ c; pÞ ð8Þ

A scalar Fc or Fp can be used to measure the anisotropicintensity

F� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12F �ijF

�ij

rð� ¼ c; pÞ ð9Þ

Preparation of RVEsA DEM assembly containing 400 particles proves to

be sufficient for the RVE (see also Guo & Zhao, 2013a,2014; Nguyen et al., 2013, 2014) and is used in the study.Each RVE contains clumps of equivalent radii rangingfrom 3mm to 7 mm with particle size distribution (PSD)according to Fig. 2(a). All particles (indeed rods) areassumed to have a thickness of 100 mm, that is about

Star-shaped polygon

2 r 1·25 r

f c

dc

np np

tc

nc kn

φkt

3·25 r

Fig. 1. Clump geometry and interparticle contact law

100P

erce

ntag

e fin

er b

y m

ass:

% 80

60

40

20

01 2 3

Equivalent radius: mm(a)

(b)

4 5 6 7 8 9 10

α = 0° α = 45°

α = 70° Random1

1

0 0

Fig. 2. (a) PSD of the RVE assembly; (b) snapshots of fourisotropically consolidated RVEs (partial) with different beddingplane angles

ZHAO AND GUO644

ten times the mean particle diameter. With the assumedthickness, the stress quantities in the current two-dimensional (2D) simulations are rendered in pressureunits (i.e. force over area). To prepare RVEs with differentinherent anisotropies, an assembly is generated with allparticles orientated in the same direction with no overlap.Their orientation with regard to the horizontal plane (the0� 0 axis) is denoted by an angle α, which is hereafterreferred to as the bedding plane angle. A two-stage isotropicconsolidation is then applied. First, the rotational degreesof freedom (RDOFs) of all particles are prohibited untila first-stage consolidation pressure p¼50 kPa is reached.A second stage of consolidation is then continued byrelaxing the RDOFs for all particles until the samplereaches a final designated pressure p0¼100 kPa. Duringboth stages, the interparticle friction angle is set small and isadjusted case by case to obtain different dense packings witha close initial void ratio of around e0¼0·13. A total of fiveRVE packings, with α¼0°, 22·5°, 45°, 70° and a randomlyorientated packing, respectively, are prepared as shown inFig. 2(b). During the multiscale simulations, periodicboundary conditions are imposed on each of the localRVE assemblies, which renders the Hill–Mandel-typecondition satisfied a priori (Miehe & Dettmar, 2004;Miehe et al., 2010). Note that in a pure DEM study,proper observation windows (see e.g. Meejun et al., 2008; Fu& Dafalias, 2011a) need to be defined for macroscopicinterpretations. They are not required in the current multi-scale approach since the mesoscale RVE packings serve abetter role in this regard.Figure 3 presents the homogenised responses of the

five RVEs under drained biaxial compression (note thatthe two vertical sides of the unit cell are kept withconstant confining pressure of 100 kPa, and the top=bottomis fixed against the vertical direction while allowinghorizontal movement when compression is applied fromthe top), mimicking the subsequent biaxial shear on thefull sample under smooth platen. The RVE responsesare consistent with laboratory observations on sand. Thepeak stress of the RVE decreases steadily with the increaseof α. The α¼0° RVE packing shows the most dilativeresponse accompanied with a drastic post-peak soften-ing behaviour, whereas the dilation and the softeningfor the α¼70° RVE are relatively mild. The randompacking gives rise to an intermediate strength and dilationbehaviour among all the RVEs. Note that the observedresponses in Fig. 3 are similar to those obtained by SeyediHosseininia (2012a, 2012b) using elongated polygonalparticles.

Model set-up for multiscale simulation of biaxial shearon sandThe influence of anisotropy on strain localisation is

investigated through multiscale modelling of biaxial com-pression tests on sand. Two boundary conditions arecompared, using either smooth or rough loading platenson the sand samples. A sand specimen with dimensions50 mm by 100 mm is discretised into 8�16 eight-nodequadrilateral elements, as shown in Fig. 4. A reducedintegration scheme using 2�2 Gauss points for eachelement is adopted, which provides good accuracy comparedto the full integration using 3�3 Gauss points to be shownin Fig. 6(a). The specimen is initially uniform by assigningidentical RVEs to all Gauss points. It is first isotropicallyconsolidated to a target initial mean effective pressure p0¼100 kPa, and is then compressed along the vertical directionwith a constant horizontal confining pressure σ00¼100 kPauntil it fails.

RESULTS AND DISCUSSIONGlobal stress responseFigure 5 provides the resultant global axial stresses

(compression is positive) plotted against axial strain fordifferent specimens under different loading conditions, in

7

6

5

4

Prin

cipa

l stre

ss ra

tio, σ

11/σ

00

Volu

met

ric s

train

ευ: %

3

2

0 2 4 6

(a)

(b)

8 10

Axial strain, ε11: %

Axial strain, ε11: %

0 2 4 6 8 10

1

–8

–6

–4

–2

0

2

α = 22·5°

Random

α = 0°

α = 70°α = 45°

α = 22·5°

Random

α = 0°

α = 70°

α = 45°

Fig. 3. Responses of different RVEs under drained loading condition:(a) stress–strain relations; (b) dilatancy curves

Smooth loading platen Rough loading platen

σ 100

= 1

00 k

Pa

100

mm

50 mm

Gausspoint

x1

x0

Fig. 4. FEM mesh containing 8×16 eight-node quadrilateralelements and boundary conditions for the biaxial compression tests

ANISOTROPYAND STRAIN LOCALISATION IN GRANULAR SOILS 645

comparison with pure RVE element test results. Fig. 6 furthercompares the global resultant stress of the five specimens inboth loading cases. Owing to the use of the rough loadingplaten, shear stresses are also resulted in the rough platen case(Fig. 6(c)). In both cases, the general trend of the peak axialstress of the five specimens is consistent with that of theRVEs; that is, the peak axial stress decreases with the increaseof α. However, the post-peak stress reduction in the multi-scale simulation is more dramatic than that in RVE elementtest, due to the formation of a shear band in the multiscaleBVP. Similarly to their respective RVE, the five specimensreach a comparable residual stress (around 200 kPa in thesmooth case and 300 kPa for the rough case). The pre-peakresponses of the multiscale simulation for both boundarycases coincide well with their corresponding RVE responsesfor all the samples, especially during the early elastic stage.For the cases of α¼0°, 45° and the randomly orientatedspecimens, the multiscale global peak axial stress is almostidentical to their respective RVE peak stress, suggesting arelatively uniform pre-peak deformation in these specimens.The other two samples (α¼22·5° and 75°) exhibit a smallermultiscale global peak axial stress that occurs earlier thantheir RVE peak stress under both loading conditions, whichindicates that considerable strain localisation may havedeveloped prior to the peak in these samples. Fig. 5 alsoshows that the peak stress is attained slightly earlier in therough loading condition than in the corresponding smoothcase.

The post-peak responses display a marked differencebetween the rough and smooth loading cases. The axialstress in the smooth case (Fig. 6(a)) first drops dramaticallyafter the peak and then becomes steady, while in the roughplaten condition (Fig. 6(b)), it first shows a dramatic dropafter peak before entering a quasi-steady state, and thenincreases to a mild second peak before dropping again to afinal steady value. These subtle post-peak changes inFig. 6(b) are intimately correlated with the development ofthe resultant shear stress as shown in Fig. 6(c). During theearly shearing stage, the resultant shear stress in mostspecimens (except the α¼70° sample) remains vanishinglysmall because the deformation in each specimen is stillsymmetric. The first dramatic drop in Fig. 6(b) indeedcorresponds to a rapid increase of the resultant shear stress,

while the second mild peak of the normal stress coincideswith the maximum shear stress. Towards the final steadystate, the resultant shear stress tends to vanish gradually. Themagnitude of the resultant shear stress is generally an ordersmaller than that of the axial stress.Figure 6(a) also shows for the smooth case of the α¼0°

sample, the predictions based on a reduced integrationscheme with 2�2 Gauss points (solid line) are almostidentical with those obtained from a full integrationscheme with 3�3 Gauss points (open circles). This indicatesthat the high-order eight-node element with reduced inte-gration is accurate enough while improving computationalefficiency.

Strain localisation and shear band patternsThe multiscale simulations show that a single shear band is

exclusively observed under the smooth loading condition,whereas a cross-shaped double band pattern is found for therough platen case. Fig. 7 shows contours of the accumulateddeviatoric strain εq and void ratio e at the peak stress state,and the final state for three specimens in both loading cases.Under smooth loading conditions (Figs 7(a) and 7(b)), thedeformation of the α¼45° sample at the peak stress stateremains relatively homogeneous (small range in εq and e),while the α¼22·5° specimen already shows a greatervariation of εq and e at the peak, suggesting that localisationhas developed at the peak, which results in a smaller globalpeak stress than the RVE element test (Fig. 5). An intenselylocalised shear band coinciding with the greatest void ratio isobserved in both samples at the final stage with accumulatedεq reaching 172~198%. The minimum εq at the final state inboth samples is also smaller than that in the peak stress state,indicating unloading in regions outside the shear band.Interestingly, a shear band inclining closely to the beddingplane direction (termed as a type-b band according toTatsuoka et al. (1990)) is observed in the α¼22·5° sample,while the one in the α¼45° sample is more to theperpendicular direction of the bedding plane (termed byTatsuoka et al. (1990) as a type-a band). Both types of shearband have been reported by Tatsuoka et al. (1986, 1990) intheir plane strain tests with lubricated loading platen. It wasargued that the type-a shear band is typically governed by the

700

600

500

400

300

200

100

700

600

500

400

300

200

1000 2 4 6 8 10 0 2 4 6 8 10

σ 11:

kP

a

α = 0°

α = 70° Random

α = 22·5° α = 45°

σ 11:

kP

a

700

600

500

400

300

200

100

700

600

500

400

300

200

100

σ 11:

kP

aσ 1

1: k

Pa

700

600

500

400

300

200

100

σ 11:

kP

a

ε11: %

0 2 4 6

RVE element test

Multiscale global,smooth boundary

Multiscale global,rough boundary

8 10ε11: %

0 2 4 6 8 10ε11: %

0 2 4 6 8 10ε11: %

ε11: %

Fig. 5. Global axial stress from the multiscale simulations with different boundary conditions in comparison with the RVE element responses

ZHAO AND GUO646

Coulomb’s failure condition, whereas the type-b band iscontrolled by the zero-extension failure mechanism(Tatsuoka et al., 1990; Gao & Zhao, 2013).Under rough loading conditions, a cross-shaped shear

band pattern is appreciable in the α¼45° sample at the peakaxial stress state, although the intensity of shear strainremains low. Shear stress with symmetric distribution isresulted along the top end of the sample by the loadingplaten constraint (Fig. 7(c)) and the net shear stress is zero(Fig. 6(c)). The type-b shear band develops dominantly afterthe peak axial stress, while the type-a band stays nearlystagnant until the resultant shear stress reaches its peak.After the peak shear stress state, the type-a shear bandrestarts to develop with the type-b band. At the final stage(ε11¼10%), the shear stress approaches zero and an almost

symmetric (in shape) crossed double shear band pattern isobserved, with the first developed type-b band showing moreintense shear strain concentration than the type-a band.Regarding the α¼70° sample, this exhibits an initialnon-coaxial material response from the beginning, leadingto a steady increase in the resultant shear stress and anasymmetric shear deformation (Fig. 7(d)). The type-b shearband is dominant over the type-a band until the peak shearstress state is reached (ε11¼3·6%). The subsequent develop-ment of the shear bands is similar to that of the α¼45° case,except that the final cross shear band pattern appears to beslightly asymmetric.

Fabric anisotropy and particle rotationIn addition to the accumulated shear strain and the void

ratio, it is also interesting to examine the distribution offabric anisotropy and the average particle rotation. Fig. 8presents the results at the final stage for two samples undersmooth loading conditions and two samples in the roughcase, wherein the average particle rotation within a DEMpacking is defined by

θ̄ ¼ 1Np

Xp[Np

θp ð10Þ

where θp is the accumulated rotation of an individual particle(anti-clockwise rotation is treated as positive).For the smooth case, it is evident from Figs 8(a) and 8(b)

that the contours of both Fp and θ̄ present rather similarpatterns to those of εq and e in Figs 7(a) and 7(b). Thelocalised θ̄ suggests substantial particle rotations inside theshear band. This is consistent with both laboratory exper-iments (Hall et al., 2010) and pure DEM simulations (Bardet& Proubet, 1991). Notably, θ̄ is anti-clockwise in the case ofα¼22·5° and is clockwise in the random case. This is notsurprising since the shear band develops from the upper-rightto the lower-left side in the α¼22·5° case, whereas it inclinesfrom the upper-left corner to the lower-right corner in therandom specimen. Despite these similar localised patterns,the two samples exhibit different behaviour in Fp. Theα¼22·5° specimen possesses a relatively large initial aniso-tropic intensity compared to the random case. Fp inside thelocalised band decreases with shearing and reaches aminimum as compared to the outside regions. In contrast,the random sample has a marginally small initial fabricanisotropy (Fp� 0), and its Fp gradually increases inside theshear band upon shearing and reaches a maximum at thefinal stage. Fig. 8(c) presents the contours of the deviatoricstress q and the contact normal based fabric anisotropy Fc,which indicate both are apparently unsuitable to identify theshear band (see also Guo & Zhao, 2014).Under rough boundary conditions, the contours of Fp and

θ̄ for the two specimens (Figs 8(d) and 8(e)) also depictsimilar localisation patterns as shown by εq and e. Theα¼70° specimen, with a higher initial fabric anisotropy,shows a continuously decreasing Fp inside the double shearbands upon shearing, whereas the Fp in the random specimen(with a nearly zero initial anisotropy) increases graduallyunder shear and reaches a maximum within the localisedbands. The contours of Fp show a similar pattern as observedbefore that one shear band dominates the other. The twosamples share similar contours of θ̄. Anti-clockwise rotation(positive θ̄) is found for particles inside the type-b band, andclockwise rotation (negative θ̄) is observed within the type-ashear band. At the intersection of the two bands, the particlespossess almost zero average accumulated rotation. Since thetwo shear bands have developed asynchronously, the par-ticles at this centre location are expected to rotate towards

700

600

500

400

300

200

1000 2 4 6

(a)

8 10

0 2 4 6(b)

8 10

Axial strain, ε11: %

0 2 4 6

(c)

8 10Axial strain, ε11: %

Axi

al s

tress

, σ11

: kP

a

700

600

500

400

300

200

100

50

40

30

20

10

–10

–20

–30

–40

–50

0

Axi

al s

tress

, σ11

: kP

aS

hear

stre

ss, σ

10: k

Pa

α = 22·5°

Random

α = 0°With full integration

α = 0°

α = 70°α = 45°

α = 22·5°

Random

α = 0°

α = 70°α = 45°

Fig. 6. (a) Resultant normal stress with smooth loading platen;resultant (b) normal and (c) shear stresses with rough loading platen

ANISOTROPYAND STRAIN LOCALISATION IN GRANULAR SOILS 647

one direction first, and then rotate inversely when the secondshear band becomes distinct.

Non-coaxialityAll simulations in this paper have been based on initially

uniform specimens with symmetric boundary conditions. Ina conventional FEM analysis, such samples are expected toundergo homogeneous deformation without observing anylocalisation. Most past studies have practically introduced

either imperfections or random fields for elements=materialparameters as triggers to simulate shear banding (Andrade &Borja, 2006; Andrade et al., 2008; Tejchman & Górski,2010). Despite being partially justified by the random natureof geomaterials, these treatments may appear to be subjectiveand potentially complicate the analysis of underlyingmechanisms with the random factors introduced. Indeed,the non-coaxial response of a granular material induced bythe presence of anisotropy can play a role of symmetrybreaker to trigger strain localisation, which has been

0·128

0·126

0·124

0·122

(a)

Peak normal stress state, ε11 = 3·2% Final state, ε11 = 10%

Peak normal stress state, ε11 = 3·1% Final state, ε11 = 10%(b)

(c)

(d)

α =

22·5

°α

= 45

°α

= 45

°α

= 70

°

εq εq0·0798 0·1291·98

0·0067

1·6 0·230·242

0·126

0·20

0·17

0·15

1·2

0·8

0·40·121

e

εq εqe

e

e

0·0473

0·07

0·06

0·0588

0·0584

0·0580

0·0576 0·0575

0·05940·0592

εq

0·06

0·04

0·02

0·08 0·0843

εq εq εq0·06 0·4

0·3

0·2

0·1

0·050·040·030·02

0·0116

0·0672

0·0074

0·459

0·00485

1·121·0

0·60·8

0·40·2

0·0158

Peak normal stress state,ε11: 2·6%

Peak normal stress state,ε11: 1·7%

Peak shear stress state,ε11: 3·6%

Peak shear stress state,ε11: 4·3%

Final state,ε11: 10%

Final state,ε11: 10%

εq εq

0·4

0·3

0·2

0·1

0·5 0·527

0·0079

0·1237

0·1236

0·1235

0·12380·124 1·72 0·215

0·126

1·60·2

0

0·18

0·16

0·14

1·2

0·8

0·4

0·0165

1·11·0

0·8

0·6

0·4

0·20·005

0·123

0·05

Fig. 7. Contours of the accumulated deviatoric strain and the void ratio for specimens under (a), (b) smooth and (c), (d) rough loading conditionsat different shearing levels. Note the difference in the shading ranges

ZHAO AND GUO648

demonstrated by Gao & Zhao (2013) and Guo & Zhao(2014). By virtue of its embedded DEM engine for eachGauss point, the multiscale modelling can naturally repro-duce the non-coaxial response of a granular soil, and helps tocapture strain localisation without resorting to introducingany artificial imperfection or randomness. The smoothplaten loading case is taken here as an example.Figure 9 presents the material responses of three RVEs

(α¼22·5°, 45° and 70°) based on their element test (notethat the element tests are performed on the unit cell).

Notwithstanding the fluctuated response curves due to alimited number of particles used, the non-coaxial behaviouris evident for each RVE. The non-coaxiality angle β (see theinset of Fig. 9) is defined as the angle between the majorprincipal stress direction and the major principal strainincrement direction. Notably, both α¼22·5° and α¼45°RVEs depict a roughly coaxial response (β� 0°) at the earlyshearing stage up to an axial strain of about 1·55% and 3%,respectively. After ε11¼1·55%, the α¼22·5° RVE graduallydevelops a negative β, which triggers a type-b shear band asobserved in Fig. 7(a), whereas the α¼45° RVE develops apositive β after ε11¼3%, leading to the formation of thetype-a shear band shown in Fig. 7(b). The α¼70° RVEdemonstrates a positive non-coaxiality from the very begin-ning of shearing, which leads to a type-a shear band. Thematerial non-coaxiality observed from the element tests isindeed consistent with the global responses in the BVPs.Shown in Fig. 10 are the accumulated displacement fields forthe three specimens at different shearing levels. For example,the displacement field in the α¼22·5° specimen remainssymmetric at ε11¼1·5% as the material response is coaxial(Fig. 9). When the axial strain reaches 2·5%, an appreciableleftwards-inclined displacement field is found, which triggersa type-b shear band shortly after.Clearly, the non-coaxiality observed in the RVEs is not

predetermined by the bedding plane orientation, but is aconsequence of the interplay among several key factorsincluding the particle orientation, the contact normaldistribution, the bulk density and the particle interlocking.Indeed, the two fabric tensors ϕij

p and ϕijc for soil are usually

orthogonal to each other. In the bedding plane direction,there are less contact normals (n c) but longer branch vectors(d c), while in the perpendicular direction of the beddingplane there are more nc but shorter dc. The number of contactnormals and the length of branch vectors can be identified astwo major competing factors in determining non-coaxiality,since they directly influence the principal directions of thetangent modulus Dijkl in equation (4), which predominantlygoverns the material non-coaxial response.

Local analysesThe hierarchical multiscale approach enables a direct

correlation of the macroscopic observations with theirunderlying microstructural mechanisms through local

1·48

0·662

1·41·21·00·8

0·791

0·0008

0·6

0·4

0·2

0·517

0·0456

0·50·40·30·20·1

0·00449

–0·671

0

–0·2

–0·4

–0·6

Random

110

820406080

100

0·3

0·8

0·6

0·4

1·55

0·496

1·41·21·00·80·6

0·351

–0·373

0·2

0

–0·2

0·45

0·0914

0·4

0·3

0·2

0·1

0·364

–0·378

0·2

0

–0·2

ACWCW

CW ACW

CWACW

CWACW

ACW

CW

ACW

CW

α = 22·5º

α = 22·5º

α = 70º

Fp

Fp

Fc

Fp

Fp

Random

q: kPa

(a)

(b)

(c)

(d)

(e)

Fig. 8. Contours of Fp, θ̄, q and Fc for specimens under (a)–(c)smooth and (d), (e) rough loading conditions at ε11=10%. For colourinterpretation of this figure, readers are referred to the web version ofthis paper (ACW: anti-clock wise; CW: clock wise)

20

15

10

β: d

eg 5

0

–5

–100 0·5 1·0 1·5 2·0 2·5 3·0 3·5

Axial strain, ε11: %

α = 45°

α = 22·5°

ε1σ1

σ3

β

αα = 70°

·

ε3·

Fig. 9. Non-coaxial responses of the RVEs. The inset illustrates thedefinition of the non-coaxiality angle, β

ANISOTROPYAND STRAIN LOCALISATION IN GRANULAR SOILS 649

analyses. Take the α¼0° sample in the smooth boundarycase as an example. Two typical local points, as shown inFig. 11(e), are selected for the analysis. Gauss point A islocated far away from the localisation zone, while the Gausspoint B is inside the shear band. Figs 11(a)–11(c) show thematerial responses of point A extracted from the correspond-ing RVE. Clearly, point A undergoes a small deformation,reaching a maximum shear strain εq¼6·4% followed byunloading. The loading path in Fig. 11(b) indicates the stressof point A almost returns to its initial isotropic state after theunloading. The overall volumetric change is also small(εv¼ �0·4%). During the process, Fp at point A stays

almost unchanged, while Fc first experiences a significantincrease during the loading stage and then decreases duringunloading. The force chain network of the embedded RVEpacking of point A at the final stage, shown in Fig. 11(d), isalmost isotropic. In contrast to point A, point B experiencesexcessively large deformation as shown in Figs 11(f)–11(i).Its deviatoric strain monotonically increases to around200%, reaching the critical state (Roscoe et al., 1958;Schofield & Wroth, 1968) when the stress, the volume andboth fabric anisotropies all become constant. The materialresponses at point B match reasonably well with the results ofa comparison pure DEM case under monotonic drained

α =

22·5

°α

= 45

°α

= 70

°

|u|: m0·00308

0

0·003

0·002

0·001

|u|: m0·000106

0·0001

7·5 × 10–5

5 × 10–5

2·5 × 10–5

0

|u|: m0·00354

0

0·003

0·002

0·001

0

|u|: m0·00262

0·002

0·001

|u|: m0·00151

(a)

(c)

(b)

0·0012

0·0018

0·0004

0

ε11 = 1·5% ε11 = 2·5%

ε11 = 3%

ε11 = 0·1%

ε11 = 3·2%

Fig. 10. Accumulated displacement field for: (a) α=22·5°; (b) α=45°; (c) α=70° specimens at different shearing levels

ZHAO AND GUO650

shear shown in Figs 11(f)–11(h) (open symbols). When bothFp and Fc evolve with shear strain, the zero-εv point (fromoverall contraction to dilation) at around εq¼9·5% can beregarded as a watershed point in observing their changes. Fcincreases steadily upon shearing, attains a peak of more thantwice its initial value at the watershed point, and then dropsgradually before reaching a steady value. Prior to thewatershed strain level, Fp only shows marginally smallincrease, and then decreases to a steady value of around40% of its peak.

The above observations suggest that Fp is generally harderto mobilise than Fc. In order for Fp to change, in addition toshearing, considerable dilation is needed to allow therelaxation of interlocking and reorientation of particles,which is in contrast to the immediate change of Fc uponshearing. The evolutions of the two sources of fabricanisotropy are hence not synchronous. This probably helpsto explain the previous observations on the α¼22·5° and theα¼45° samples in Fig. 9 that an initially coaxial packingmay later develop substantial non-coaxiality. Notably, it also

0

0·3

0·6

0·9

1·2

1·5

–1.0

–0·5

0

0·5

1.0

1·5

2.00

50

100

150

200

250

300

q: k

Pa

0

50

100

150

200

250

300

q: k

Pa

Loading pathslope = 1:1

q

p: kPa

(c)

(d)

(i)

(e)

AB

Pure DEM test resultsGauss point B responses

0

0·3

0·6

0·9

1·2

1·5

0 1 2 3 4 5 6 7

0 100 200

(b)

300 400

0 50 100 150 200

F p, F

cF p

, Fc

–12

–10

–8

–6

–4

–2

0

2

0

50

100

150

200

250

300

q: k

Pa

0

50

100

150

200

250

300

q: k

Pa

0 100 200 300 400p: kPa

Loading pathslope = 1:1

q

(f)

(h)

(g)

εq: %

(a)

0 1 2 3 4 5 6 7εq: %

εq: %

0 50 100 150 200εq: %

ε v: %

εv

ε v: %

Fp

Fc

Fp

Fc

εv

Fig. 11. Local Gauss points responses in the α=0° specimen: (a)–(c) Gauss point A responses; (d) RVE force chain of Gauss point A after shear; (e)locations of the two Gauss points in the deformed mesh; (f)–(h) Gauss point B responses; (i) RVE force chain of Gauss point B after shear. Note theforce chain width is proportional to the contact normal force and the width in (d) is scaled two times wider than that in (i) for better visualisation

ANISOTROPYAND STRAIN LOCALISATION IN GRANULAR SOILS 651

takes a significant shearing process for Fp to reach its criticalstate (e.g. εq. 150% for point B). This large deformationlevel is normally hard to attain in a uniform sample inroutine laboratory element tests, which partially explains whyOda (1972b) and Wong & Arthur (1985) observed in theirtests the survival of inherent anisotropy and limited particleorientation change (Wong &Arthur (1985) only sheared theirsoil samples to 6% of axial strain). Meanwhile, the aboveanalysis suggests that, although the stress and volumetricchanges can reach a steady state at a relatively early stage(e.g. εq� 60% in Fig. 12(f)), the critical state for fabricanisotropy (Li & Dafalias, 2012; Zhao & Guo, 2013; Gaoet al., 2014) can only be attained within the shear bandwherein much larger shear deformation occurs (e.g.εq. 100%) (Mooney et al., 1998; Fu & Dafalias, 2011a).Indeed, as shown in Fig. 11(i), the RVE packing at point Bexperiences severe distortion (c.f. Fig. 11(d) for point A) atthe final stage accompanied by a significant clockwise rigidbody rotation ωij. The force chain network in Fig. 11(i) alsocontains several penetrating strong force chains in the verticalcompression direction.

To further elucidate the natures of the two fabric tensors ϕijp

and ϕijc, in Fig. 12 their evolution is presented in terms of rose

diagrams extracted from a RVE inside the shear band formedin the α¼0° specimen and the random specimen undersmooth loading conditions. The RVE for the α¼0° case(Fig. 12(a)) possesses large initial anisotropies for bothmeasures. The major principal direction of the initial ϕij

p

aligns in the horizontal axis, which is almost perpendicular tothe major principal direction of the initial ϕij

c. After shear, themajor direction of ϕij

p rotates noticeably with a decreasedanisotropic intensity, but the rotation of ϕij

c and its change of

intensity is relatively small. For the randomly orientatedsample shown in Fig. 12(b), the RVE possesses nearlyisotropic fabric tensors at the initial state (both ϕij

p and ϕijc

depict minor anisotropies due to the limited number ofparticles used). After shear, the intensity of both anisotropymeasures increases. At the final stage, the major principaldirections of the two fabric tensors are roughly perpendicularto each other. A further scrutiny of the fabric tensors at thecritical state from both specimens suggests that, althoughdifferent in initial states, their critical state fabric anisotropiesare close in terms of both principal direction (the smalldeviation in the principal orientations of ϕij

p is attributable tothe different rigid body rotation ωij of the two RVE packings)and anisotropic intensity (the length of the major principalaxis observed from Fig. 12 is about 0·25 for ϕij

p and 0·23 forϕijc, noting that different radial length scales have been used

for ϕijp in the two cases).

A particularly interesting observation for the roughboundary case is the material response at the intersectionof two crossed shear bands. Fig. 13 shows the results of such apoint for the α¼45° specimen whose position is marked inFig. 7(c). The material responses at the Gauss point in Figs13(a)–13(c) match fairly well with the results of a mono-tonically drained pure DEM test as a comparison case. Thepoint appears to reach the critical state after the deviatoricstrain reaches 120%. Fig. 13(d) presents the force chainnetwork of the RVE packing at the peak shear stress (ε11¼4·3%, corresponding to the onset state of the second shearband), which depicts a noticeable anti-clockwise rigid bodyrotation ωij that is consistent with the first shear band(type-b) of the specimen. At the final state, the RVE packingis severely deformed, but the rigid body rotation becomes

Initial stateAfter shear

90°

135°

180°

225°

270°

315°

0°

45°

0·60·50·40·30·20·1

Initial stateAfter shear

90°

135°

180°

225°

270°

315°

0°

45°

0·30·250·2

0·150·10·05

Initial stateAfter shear

90°

135°

180°

225°

270°

315°

0°

45°

0·30·250·2

0·150·10·05

Initial stateAfter shear

90°

135°

180°

225°

270°

315°

0°

45°

0·30·250·2

0·150·10·05

Random

(a)

(b)

α = 0°

φ ijp

φ ijc

φ ijp

φ ijc

Fig. 12. Evolution of the two fabric tensors ϕijp and ϕij

c for RVEs inside shear band in (a) the α=0° and (b) the randomly orientated specimens. Notethe difference in the radial length scales

ZHAO AND GUO652

negligible again due to the balance of the second shear band(Fig. 13(e)). The observation is consistent with the observedzero θ̄ at this point in Fig. 8.Meanwhile, strong force chains areobserved for the RVE in Figs 13(d) and 13(e), with the majordirection slightly tilted from the vertical axis. The evolutionsof the two fabric tensors are also presented in Figs 13(f) and13(g). Interestingly, ϕij

p first flips anti-clockwise slightly untilthe peak shear stress state due to the rigid body rotation ofthe RVE. It then spins clockwise significantly to attain thecritical state. The anisotropy intensity of ϕij

p decreases steadilyto a critical state value (see also Fig. 13(c)), while ϕij

c rotatescontinuously clockwise to the critical state with a roughlyconstant anisotropic intensity in the course.

CONCLUSIONSThe interplays between anisotropy and strain localisation

in granular soils were investigated using a hierarchicalmultiscale approach based on rigorous coupling of FEMand DEM, wherein clumped particles were used in the DEMpart to simulate initial anisotropy. Biaxial compression tests

were simulated by the multiscale approach on samples undereither smooth or rough boundary conditions. Key findingsfrom the study are summarised as follows.

(a) The bedding plane orientation is not necessarily the‘weaker’ direction for the shear band to initiate. Eithertype-a or type-b shear bands can happen under smoothboundary conditions, which is consistent with theexperimental observations by Tatsuoka et al. (1990).Under rough boundary conditions, cross-shaped doublebands occur (type-a pairs with type-b), but they are notinitiated simultaneously. The inception of the first bandleads to increased resultant shear stress on the platen,while the later developing one offsets the shear stresstowards zero at the residual state. Significant particlerotation is observed inside shear bands except at theintersection of the cross-shaped double bands, wherevanishing average particle rotation is found.

(b) Non-coaxial material responses may serve as animportant direct (nominal) triggering mechanism for

(c)

(e)

90°

135°

180°

225°

270°

315°

0°

45°

Initial statePeak shear stressFinal state

0·60·50·40·30·20·1

90°

135°

180°

225°

270°

315°

0°

45°

Initial statePeak shear stressFinal state

0·30·250·2

0·150·10·05

(f) (g)

(d)

Pure DEM test resultsGauss point responses

0

0·3

0·6

0·9

1·2

1·5

0 50 100 150 200

–10

–8

–6

–4

–2

0

2

0

50

100

150

200

250

q: k

Pa

0

50

100

150

200

250

q: k

Pa

Loading pathslope = 1:1

q

0 100 200 300 400p: kPa

ε v: %

εq: %

(a)

0 50 100 150 200εq: %

(b)

φ ijp φ ij

c

F p, F

c

Fp

Fc

εv

Fig. 13. Local Gauss point (position marked in Fig. 7(c)) responses in the α=45° specimen: (a)–(c) evolution curves; (d) and (e) RVE force chains;(f) and (g) fabric tensors ϕ ij

p and ϕ ijc at different shearing levels respectively

ANISOTROPYAND STRAIN LOCALISATION IN GRANULAR SOILS 653

strain localisation, while the fundamental (physical)triggering factors may vary greatly in a BVP for sand.For example, initial anisotropy, imperfection, inhom-ogeneity, boundary constraints and drainage conditioncan also cause non-coaxial material responses andinduce strain localisation. Although non-coaxiality haslong been investigated in the context of constitutivemodelling of sand only, the present study is the first todiscuss its direct correlation with strain localisation byevidence from multiscale analysis.

(c) The two fabric anisotropy tensors ϕijc and ϕij

p, defined bythe distributions of the contact normals and the particleorientations, respectively, play different roles in trigger-ing strain localisation. ϕij

c begins to evolve immediatelyupon loading, while the evolution of ϕij

p needs to bemobilised with larger shear and dilation. The asynchro-nous evolution of the two anisotropies can cause aninitially coaxial packing to develop substantial non-coaxiality at small shearing levels to trigger strainlocalisation. The shearing strain level for ϕij

p to reach thecritical state is also much larger than that for ϕij

c.(d) The final shear bands feature a high concentration of

shear strain, large dilation and excessive particlerotations. Within the shear band, ϕij

p attains anextremum (which can be either a minimum for aninitially anisotropic specimen or a maximum for aninitially random specimen). The local RVEs inside theshear band may attain critical state after shear withconstant volume, constant stress and constant fabricanisotropies (in terms of both Fp and Fc), while thoseoutside the shear band may experience mild defor-mation followed by unloading.

The current study constitutes a first step towards bridging themicromechanics of a granular soil and the macroscopicphenomena such as strain localisation. It is apparent that themicro- and meso-scale models considered in such anapproach may influence the macroscopic responses andthus need careful calibrations to tackle specific problems.To account for inherent anisotropy, the observations and theconclusions drawn above have been based on a simple clumpDEMmodel, which may be improved by using more complexparticle shapes for better approximation of real sand particles(e.g. Mollon & Zhao, 2012, 2013, 2014). This multiscaleinvestigation highlights the complicated nature of non-coaxiality, which may involve multiple influential factorssuch as particle orientation and contact normal distributionsthat jointly contribute to the anisotropic soil behaviour. Thisdefinitely deserves further theoretical consideration in thecontext of continuum soil mechanics.

ACKNOWLEDGEMENTSThe simulations were performed on a high-performance

computing facility supported by the Science SchoolComputational Science Initiative, HKUST. The study wasfinancially supported by the Research Grants Council ofHong Kong through GRF 623211.

NOTATIONAcl, Icl,

ρclarea, inertia of moment and density of a clump particle

Dep elasto-plastic modulusDijkl homogenised tangent modulus of a DEM packingdc, di

c branch vector at a contactE(Θ) distribution function

e local void ratiof FEM nodal force vector

fc, fic contact force at a contactK FEM stiffness matrix

kn, kt contact normal stiffness and tangential stiffnessNp, Nc particle number and contact number within a DEM

packingn c, ni

c unit contact normal vectornp, ni

p unit particle orientation vectorp, q mean effective stress and deviatoric stress

p0, e0 initial mean effective stress and void ratio prior tobiaxial shear

r radius of the constituent disc of a clump particlereq equivalent radius of a clump particlereq* common radius of two contacting particles

t c, tic unit tangential vector at a contactu FEM nodal displacement vector

ui,j FEM deformation field and boundary condition forlocal RVEs

V total volume of a DEM packingα bedding plane angle of a RVE packingβ non-coaxiality angle in a RVE element testδij Kronecker delta

ε̇1; ε̇3 major and minor principal strain incrementεij, eij strain tensor and deviatoric strain tensorεv, εq volumetric strain and deviatoric strainθp; θ̄ individual particle rotation and average particle rotation

for a RVE Fpijpacking

σ1, σ3 major and minor principal stressσij, sij effective stress tensor and deviatoric stress tensor

ϕ ijc, F c

ij ,Fc

contact normal based fabric tensor, deviatoric tensorand anisotropic intensity

ϕ ijp, Fp

ij ,Fp

particle orientation based fabric tensor, deviatoric tensorand anisotropic intensity

φ interparticle friction angleωij rigid body rotation of a RVE packing

REFERENCESAbelev, A. V. & Lade, P. V. (2003). Effects of cross anisotropy on

three-dimensional behavior of sand. II: stress–strain behaviorand shear banding. J. Engng Mech. ASCE 129, No. 2, 160–166.

Andrade, J. E. & Borja, R. I. (2006). Capturing strain localization indense sands with random density. Int. J. Numer. Methods Engng67, No. 11, 1531–1564.

Andrade, J. E., Baker, J. W. & Ellison, K. C. (2008). Randomporosity fields and their influence on the stability of granularmedia. Int. J. Numer. Analyt. Methods Geomech. 32, No. 10,1147–1172.

Andrade, J. E., Avila, C. F., Hall, S. A., Lenoir, N. & Viggiani, G.(2011). Multiscale modeling and characterization of granularmatter: from grain kinematics to continuummechanics. J.Mech.Phys. Solids 59, No. 2, 237–250.

Arthur, J. R. F. & Menzies, B. K. (1972). Inherent anisotropy in asand. Géotechnique 22, No. 1, 115–128, http:==dx.doi.org=10.1680=geot.1972.22.1.115.

Arthur, J. R. F., Chua, K. S. & Dunstan, T. (1977). Inducedanisotropy in a sand. Géotechnique 27, No. 1, 13–30, http:==dx.doi.org=10.1680=geot.1977.27.1.13.

Bardet, J. P. & Proubet, J. (1991). A numerical investigation of thestructure of persistent shear bands in granular media.Géotechnique 41, No. 4, 599–613, http:==dx.doi.org=10.1680=geot.1991.41.4.599.

Bauer, E., Huang, W. & Wu, W. (2004). Investigations of shearbanding in an anisotropic hypoplastic material. Int. J. SolidsStructs 41, No. 21, 5903–5919.

Bigoni, D. & Loret, B. (1999). Effects of elastic anisotropy on strainlocalization and flutter instability in plastic solids. J. Mech.Phys. Solids 47, No. 7, 1409–1436.

Casagrande, A. & Carillo, N. (1944). Shear failure of anisotropicmaterials. J. Boston Soc. Civ. Engrs 31, No. 4, 74–87.

Chang, C. S. & Hicher, P. Y. (2005). An elasto-plastic model forgranular materials with microstructural consideration.Int. J. Solids Structs 42, No. 14, 4258–4277.

Chang, C. S. & Yin, Z.Y. (2010). Micromechanical modelling forinherent anisotropy in granular materials. J. Engng Mech.ASCE 136, No. 7, 830–839.

Chang, C. S., Yin, Z.Y. & Hicher, P. Y. (2011). Micromechanicalanalysis for inter-particle and assembly instability of sand.J. Engng Mech. ASCE 137, No. 3, 155–168.

ZHAO AND GUO654

Desrues, J. & Viggiani, G. (2004). Strain localization in sand: anoverview of the experimental results obtained in Grenoble usingstereophotogrammetry. Int. J. Numer. Analyt. MethodsGeomech. 28, No. 4, 279–321.

Evans, T. M. & Frost, J. D. (2010). Multiscale investigation of shearbands in sand: physical and numerical experiments. Int. J.Numer. Analyt. Methods Geomech. 34, No. 15, 1634–1650.

Fu, P. & Dafalias, Y. F. (2011a). Fabric evolution within shear bandsof granular materials and its relation to critical state theory.Int. J. Numer. Analyt. Methods Geomech. 35, No. 18,1918–1948.

Fu, P. & Dafalias, Y. F. (2011b). Study of anisotropic shear strengthof granular materials using DEM simulation. Int. J. Numer.Analyt. Methods Geomech. 35, No. 10, 1098–1126.

Gao, Z. & Zhao, J. (2013). Strain localization and fabric evolution insand. Int. J. Solids Structs 50, No. 22–23, 3634–3648.

Gao, Z., Zhao, J., Li, X.S. & Dafalias, Y. F. (2014). A critical statesand plasticity model accounting for fabric evolution.Int. J. Numer. Analyt. Methods Geomech. 38, No. 4, 370–390.

Gross, L., Bourgouin, L., Hale, A. J. & Mühlhaus, H.B. (2007).Interface modeling in incompressible media using level sets inEscript. Phys. Earth Planetary Interiors 163, No. 1–4, 23–34.

Guo, N. & Zhao, J. (2013a). A hierarchical model for cross-scale simulation of granular media. AIP Conf. Proc. 1542,1222–1225.

Guo, N. & Zhao, J. (2013b). The signature of shear-inducedanisotropy in granular media. Comput. Geotech. 47, 1–15.

Guo, N. & Zhao, J. (2014). A coupled FEM=DEM approach forhierarchical multiscale modelling of granular media.Int. J. Numer. Methods Engng 99, No. 11, 789–818.

Gutierrez, M. & Ishihara, K. (2000). Non-coaxiality and energydissipation in granular material. Soils Found. 40, No. 2, 49–59.

Hall, S. A., Bornert, M., Desrues, J., Pannier, Y., Lenoir, N.,Viggiani, G. & Bésuelle, P. (2010). Discrete and continuumanalysis of localised deformation in sand using X-ray μCT andvolumetric digital image correlation. Géotechnique 60, No. 5,315–322, http:==dx.doi.org=10.1680=geot.2010.60.5.315.

Hasan, A. & Alshibli, K. A. (2010). Experimental assessment of 3Dparticle-to-particle interaction within sheared sand using syn-chrotron microtomography. Géotechnique 60, No. 5, 369–379,http:==dx.doi.org=10.1680=geot.2010.60.5.369.

Hicher, P. Y., Chang, C. S. & Dano, C. (2008). A multi-scalemodeling of grouted sand. Int. J. Solids Structs 45, No. 16,4362–4374.

Kruyt, N. &Rothenburg, L. (1998). Statistical theories for the elasticmoduli of two-dimensional assemblies of granular materials.Int. J. Engng Sci. 36, No. 10, 1127–1142.

Lade, P. V. & Abelev, A. V. (2003). Effects of cross anisotropy onthree-dimensional behavior of sand. II: volume change behaviorand failure. J. Engng Mech. ASCE 129, No. 2, 167–174.

Lade, P. V., Nam, J. & Hong, W. P. (2008). Shear banding andcross-anisotropic behavior observed in laboratory sand tests withstress rotation. Can. Geotech. J. 45, No. 1, 74–84.

Li, X. & Yu, H.S. (2010). Numerical investigation of granularmaterial behaviour under rotational shear. Géotechnique60, No. 5, 381–394, http:==dx.doi.org=10.1680=geot.2010.60.5.381.

Li, X. S. & Dafalias, Y. F. (2012). Anisotropic critical statetheory: the role of fabric. J. Engng Mech. ASCE 138, No. 3,263–275.

Luding, S. (2004). Micro-macro transition for anisotropic, frictionalgranular packings. Int. J. Solids Structs 41, No. 21, 5821–5836.

Meejun, N., Skeldon, A. C., Tüzün, U. & O’Sullivan, C. (2008).Wavelet analysis of DEM simulations of samples under biaxialcompression. Granular Matter 10, No. 5, 389–398.

Miehe, C. & Dettmar, J. (2004). A framework for micro-macrotransitions in periodic particle aggregates of granularmaterials. Comput. Methods Appl. Mech. Engng 193, No. 3–5,225–256.

Miehe, C., Dettmar, J. & Zäh, D. (2010). Homogenization andtwo-scale simulations of granular materials for different micro-structural constraints. Int. J. Numer. Methods Engng 83, No. 8–9, 1206–1236.

Mokni, M. & Desrues, J. (1999). Strain localization measurementsin undrained plane-strain biaxial tests on Hostun RF sand.Mech. Cohesive-Frictional Mater. 4, No. 4, 419–441.

Mollon, G. & Zhao, J. (2012). Fourier-Voronoi-based generation ofrealistic samples for discrete modelling of granular materials.Granular Matter 14, No. 5, 621–638.

Mollon, G. & Zhao, J. (2013). Generating realistic 3D sandparticles using Fourier descriptors. Granular Matter 15, No. 1,95–108.

Mollon, G. & Zhao, J. (2014). 3D generation of realistic granularsamples based on random fields theory and fourier shapedescriptors. Comput. Methods Appl. Mech. Engng 279, 46–65.

Mooney, M. A., Finno, R. J. & Viggiani, M. G. (1998). A uniquecritical state for sand? J. Geotech. Geoenviron. Engng ASCE 124,No. 11, 1100–1108.

Muhlhaus, H., Moresi, L., Gross, L. & Grotowski, J. (2010). Theinfluence of non-coaxiality on shear banding in viscous-plasticmaterials. Granular Matter 12, No. 3, 229–238.

Nguyen, T. K., Combe, G., Caillerie, D. & Desrues, J. (2013).Modeling of cohesive granular materials by a multi-scaleapproach. AIP Conf. Proc. 1542, 1194–1197.

Nguyen, T. K., Combe, G., Caillerie, D. & Desrues, J. (2014). FEM�DEM modelling of cohesive granular materials: numericalhomogenization and multi-scale simulations. Acta Geophysica62, No. 5, 1109–1126.

Nitka, M., Combe, G., Dascalu, C. & Desrues, J. (2011). Two-scalemodeling of granular materials: a DEM-FEM approach.Granular Matter 13, No. 3, 277–281.

Oda, M. (1972a). Initial fabrics and their relations to mechanicalproperties of granular materials. Soils Found. 12, No. 1, 17–36.

Oda, M. (1972b). The mechanism of fabric changes duringcompressional deformation of sand. Soils Found. 12, No. 2,1–18.

Oda, M. (1982). Fabric tensor for discontinuous geologicalmaterials. Soils Found. 22, No. 4, 96–108.

Rechenmacher, A. L. (2006). Grain-scale processes governing shearband initiation and evolution in sands. J. Mech. Phys. Solids 54,No. 1, 22–45.

Roscoe, K. H., Schofield, A. N. & Wroth, C. P. (1958). On theyielding of soils. Géotechnique 8, No. 1, 22–53, http:==dx.doi.org=10.1680=geot.1958.8.1.22.

Rudnicki, J. W. & Rice, J. R. (1975). Conditions for the localizationof deformation in pressure-sensitive dilatant materials. J. Mech.Phys. Solids 23, No. 6, 371–394.

Satake, M. (1982). Fabric tensor in granular materials. InDeformation and failure of granular materials (edsP. A. Vermeer and H. J. Luger), pp. 63–68. Rotterdam, theNetherlands: Balkema.

Schofield, A. N. & Wroth, C. P. (1968). Critical state soil mechanics.London, UK: McGraw-Hill.

Seyedi Hosseininia, E. (2012a). Discrete element modeling ofinherently anisotropic granular assemblies with polygonalparticles. Particuology 10, No. 5, 542–552.

Seyedi Hosseininia, E. (2012b). Investigating the micromechanicalevolutions within inherently anisotropic granular materialsusing discrete element method. Granular Matter 14, No. 4,483–503.

Šmilauer, V., Catalano, E., Chareyre, B., Dorofeenko, S., Duriez, J.,Gladky, A., Kozicki, J., Modenese, C., Scholtès, L., Sibille, L.,Stránský, J. & Thoeni, K. (2010). Yade documentation, 1st edn.See https://www.yade-dem.org/doc/ (accessed 23/06/2015).

Tatsuoka, F., Sakamoto, M., Kawamura, T. & Fukushima, S.(1986). Strength and deformation characteristics of sand inplane strain compression at extremely low pressures. Soils Found.26, No. 1, 65–85.

Tatsuoka, F., Nakamura, S., Huang, C.C. & Tani, K. (1990).Strength anisotropy and shear band direction in plane straintests of sand. Soils Found. 30, No. 1, 35–54.

Tejchman, J. & Górski, J. (2010). Finite element study of patterns ofshear zones in granular bodies during plane strain compression.Acta Geotechnica 5, No. 2, 95–112.

Tejchman, J. & Wu, W. (2009). Non-coaxiality and stress-dilatancyrule in granular materials: FE investigation within micro-polarhypoplasticity. Int. J. Numer. Analyt. Methods Geomech. 33, No.1, 117–142.

Vardoulakis, I. (1996). Deformation of water-saturated sand:I. Uniform undrained deformation and shear band.Géotechnique 46, No. 3, 441–456, http:==dx.doi.org=10.1680=geot.1996.46.3.441.

ANISOTROPYAND STRAIN LOCALISATION IN GRANULAR SOILS 655

Wong, R. K. S. & Arthur, J. R. F. (1985). Induced and inherentanisotropy in sand. Géotechnique 35, No. 4, 471–481, http:==dx.doi.org=10.1680=geot.1985.35.4.471.

Wren, J. R. & Borja, R. I. (1997). Micromechanics of granularmedia Part II: overall tangential moduli and localization modelfor periodic assemblies of circular disks. Comput. Methods Appl.Mech. Engng 141, No. 3–4, 221–246.

Yu, H. S. & Yuan, X. (2006). On a class of non-coaxial plasticitymodels for granular soils. Proc. R. Soc. A 462, No. 2067,725–748.

Zhao, J. & Guo, N. (2013). Unique critical state characteristicsin granular media considering fabric anisotropy.Géotechnique 63, No. 8, 695–704, http:==dx.doi.org=10.1680=geot.12.P.040.

ZHAO AND GUO656