The Effect of Local Kinematics on the Local and Global Deformations of Granular Systems

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2010 15: 3 originally published online 3 April 2008Mathematics and Mechanics of SolidsAntoinette Tordesillas, Stuart D.C. Walsh and Maya Muthuswamy

SystemsThe Effect of Local Kinematics on the Local and Global Deformations of Granular

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The Effect of Local Kinematics on the Local and GlobalDeformations of Granular Systems

ANTOINETTE TORDESILLAS

STUART D. C. WALSH

MAYA MUTHUSWAMYDepartment of Mathematics and Statistics, University of Melbourne, Victoria 3010,Australia

(Received 21 January 2008� accepted 29 January 2008)

Abstract: Despite the prevalent use of continuum mechanics for the modeling of granular materials, the con-troversy surrounding the relationship between the properties of the discrete medium and those of its equiv-alent continuum has far from abated. The concept of strain is especially problematic. In a continuum body,the strain represents the deformation of an infinitesimal region about a material point. In a discrete granularassembly, however, deformation is governed by the relative motions of the constituent grains. Herein, we in-troduce a new microstructural definition for the deformation of a granular material within the framework ofMicropolar Continuum Theory. The advantages of the new strain definition over existing formulations are:it accounts for particle rotations, it is relatively straightforward to calculate, and its global average matchesthe macroscopic strain of the assembly. The new definition leads to a patchwork strain field, the existence ofwhich is linked to the nonaffine strain at the particle scale. A key aspect of this study is the construction of aset of local micropolar strain and curvature measures on the scale of a particle and its first ring of neighbors.We dissect these local continuum quantities and, with the aid of discrete element simulations, examine themfor a specimen under biaxial compression. New insights are gained on the contributions of the relative parti-cle motions for specific types of contacts at different stages in the deformation history. Results are discussedin light of past experimental findings on shear banding, as well as Oda’s hypothesis on force chain buckling.

Key Words: granular media, micropolar, strain, nonaffine deformation

1. INTRODUCTION

Continuum mechanical models represent real material domains in terms of bodies that oc-cupy every point in space and, consequently, these have properties describable by continuousfunctions. While most solids and fluids are reasonably amenable to this idealization, largeconglomerations of discrete macroscopic particles that embody granular materials seem be-yond the realm of continuum mechanics. Indeed, alternatives to the continuum mechanicalapproach exist—the most popular of which is the discrete element method (DEM). Devel-

Mathematics and Mechanics of Solids 15: 3–41, 2010 DOI: 10.1177/1081286508089844

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4 A. TORDESILLAS ET AL.

oped in the 1970s, discrete element methods can reproduce the complex rheological behaviorof real assemblies by modeling the motion of individual grains [1, 2]. Until recently, DEMprovided the only complete source of data on the temporal and spatial evolution of particlescale properties. Even with the advent of new image analysis and computer aided imagingtechnologies [3–5], DEM simulations continue to provide invaluable insight into the particle-scale processes emerging in deforming granular systems [6–9].

The widespread success of DEM models has long raised the speculation that DEM maywell supplant continuum mechanics in the future modeling of granular media [10–11]. De-spite this, continuum mechanical models remain an integral tool for understanding the be-havior of these materials. One reason for the prevailing co-existence of these methodologies,albeit at the opposite extremes of the spectrum, is their complementary strengths. Whileindisputably useful for capturing observed behavior, DEM remains limited to a few millionparticles—roughly the number of particles in a handful of sand! Thus, many engineeringscale processes are still beyond the reach of DEM, while prototype scale problems requirethe use of oversized particles, which leads to scaling errors [12]. Continuum models, on theother hand, are generally powerful predictive tools that can be readily applied to large-scaleapplications. However, in the case of granular materials, the predictive capabilities of con-tinuum models remain impeded from two aspects: (a) insufficient knowledge of the physicsthat govern emergent mechanisms at the microscopic and mesoscopic length scales, and (b)ineffective homogenization techniques that fail to account for the essential physics in themodel formulation in the transition from particle to bulk.

A promising way forward for micromechanics is the complementary use of continuumtheory and DEM (e.g. [13, 14]). DEM, when used as a virtual laboratory for studies ofinternal structural evolution, not only provides the modeler with vital information on themost influential aspects of material behavior, but also facilitates an understanding of thecontinuumization or homogenization scheme. A key element of this scheme is a formulationthat can robustly relate particle properties to the continuum properties (e.g. stress and strain).To date, such relations for both stress and strain remain a matter of contention (e.g. [15–19]).In particular, recent attention has been paid to the definition of a micropolar continuum fordiscrete media [20–22]. The chief advantage of micropolar (or Cosserat) theory over othertypes of continuum formulations is that it embodies both rotational degrees of freedom andan internal length scale. Together, these elements permit the introduction of relevant physicsat the microscopic and mesoscopic scales [23–27].

This paper is focussed on one aspect of the homogenization scheme for granular ma-terials: the definition of deformation. Accordingly, the goal in this paper is to relate thedisplacements and rotations of the constituent particles in a granular assembly to the kine-matic variables, namely the strain and curvature, of the equivalent micropolar continuum.Currently, there are several different schools of thought as to how strain should be definedin granular media. Of these, the two most prominent are (a) the equivalent continuum ap-proach, and (b) the best-fit approach [28, 29]. In the first approach, an equivalent continuumis assigned to the assembly, where the displacement field is interpolated from the particledisplacements. The strain tensor is then given in terms of the volume averaged gradient ofthe displacement field in the same manner as a classical continuum. Examples of this ap-proach can be found in [16], [30] and [31]. In the second approach it is assumed that thestrain defines the mean displacement field which best fits the actual particle displacements.




THE EFFECT OF LOCAL KINEMATICS 5

This approach is adopted in [24]. Other commonly encountered definitions of the straininclude homogenization strain definitions with a weighting function that averages the particlemotion in a manner simular to smooth particle hydrodynamics methods [21, 32, 33] and thecontact-cell strain of [34, 35]. There are drawbacks to each of these aforementioned straindefinitions. In homogenization methods, the strain definition must be altered to account forthe presence of boundaries [21]. On the other hand, while the contact-cell method of Satakedoes not suffer from this problem, its calculation is prohibitively intensive from a numericalperspective [29]. Strains defined by best-fit methods are easier to compute, but often failto match the global strain measures [28]. In contrast, strains defined using the equivalentcontinuum methods generally show good agreement with the global strain experienced by theassembly as a whole. However, unlike the other strains listed above, equivalent continuummethods fail to capture the effects of particle rotations. An exception is the micropolarstrain definition given by [15], but again this strain definition disagrees with global straindefinitions under biaxial compression [29].

The objectives of this paper are threefold. The first is the derivation of a new definitionof strain that accounts for particle rotations, without any a priori assumption on its func-tional form. This definition must be relatively easy to compute, in good agreement with themacroscopic strain, and applicable to a particle cluster of arbitrary length scale. Secondly, byapplying this new definition of micropolar strain, and its associated relation for micropolarcurvature, we aim to develop a set of local micropolar strain and curvature relations for aparticle cluster comprising of a particle and its first ring of neighbors, in two dimensions.These relations are defined at the particle center. This is a necessary condition in existingelastoplastic micromechanical models (e.g. 13, 24, 36), the advantages and unique predictivecapabilities of which are discussed elsewhere (e.g. 13, 37–39). Thirdly, we dissect these lo-cal deformation quantities to gain new insights into the contributions of: (i) different modesof relative motion, as well as (ii) the relative particle motions from the different modes ofcontacts, at various stages in the deformation history. Discrete element simulations will beemployed to achieve this third objective, and the results discussed in light of past experimen-tal findings on shear banding. A key aspect of this investigation is the test on the validity ofOda’s hypothesis on force chain buckling, to be undertaken via a two-pronged analysis. Inthe first prong, the decompositions will be used to create a filter that is designed to capturethose particles chiefly responsible for strain-softening under dilatation. In the second prong,a new decomposition will be constructed which would distinguish the contributions fromcontacts between particles in buckling force chains and their laterally supporting first ringneighbors, from those in the complementary set of particles.

The paper is arranged as follows. In Section 2, the Delaunay triangulation of the granularassembly is modified to ensure that all contacting particles are included in the triangulationnetwork. In Section 3, the kinematic quantities of the equivalent continuum are establishedby interpolating the particle displacements and rotations over each element in the triangula-tion. The micropolar strain and curvature are then defined by imposing an energy equivalencebetween the discrete assembly and that of the equivalent continuum. Based on these, a set oflocal micropolar strain and curvature relations for a particle cluster comprising of a particleand its first ring of neighbors, and the corresponding decompositions of these relations ac-cording to the contributions from the relative motions of the different types of contacts, arepresented in Section 4. In Section 5, the new deformation measures are applied to generate





Figure 1. (a) In the Delaunay triangulation, nearest neighbors are joined by branch vectors. However,under certain conditions this definition excludes contacting particles from the triangulation network (e.g.,dashed line). (b) Instead a modified Delaunay triangulation is proposed, in which contacting particlesautomatically contribute an edge to the triangulation.

new insights on the local and global rheological behavior of a densely packed assembly ofcircular particles under quasistatic, biaxial compression using discrete element simulations.Conclusions are drawn in Section 6.

2. A TRIANGULATION OF THE GRANULAR ASSEMBLY

In certain cases, the Delaunay triangulation excludes contacting particles from the triangu-lation network. Here the relationship between the discrete assembly and the continuum isestablished by first introducing a modified Delaunay triangulation of the granular assemblysuch that contacting particles automatically contribute an edge to the triangulation.

A Delaunay triangulation of a set of points is defined such that, for each edge in the tri-angulation, a circle can be found that contains the edge’s endpoints, but not the other pointsin the set. In this case, the locations of the particle centers may be used to define a Delaunaytriangulation for the granular assembly. Thus each node in the triangulation represents a par-ticle, and each edge defines a branch vector that joins two neighboring particles (Figure 1(a)).Neighboring particles in the triangulation need not be in contact.

Thus it is possible for contacting particles to be excluded in the triangulation network.To avoid this possibility, a modified version of the Delaunay triangulation is considered, inwhich all contacting particles are automatically included in the network of branch vectors(Figure 1(b)). In three dimensions, an appropriate tessellation of the assembly can be foundusing a Delaunay triangulation that is dual to the Dirichlet tessellation [35].

For large two-dimensional assemblies, a simple equation relates the number of particlesto the number of branch vectors. From Euler’s Theorem for plane graphs, the number ofparticles N , branch vectors N b� and regions or elements N e satisfy





Figure 2. A zoomed in portion of an assembly undergoing a macroscopically homogeneous deformation,showing particles as well as the modified Delaunay triangulation. Each triangular element in thetriangulation is colored by its volumetric strain rate, ��vol, using equation (33). The displacement fieldobtained from the discrete granular assembly gives a nonaffine deformation of the continuum.

N � N b� � N e � 2� (1)

Each triangular element has 3 branch vectors that it shares with its neighbors. Thus, in alarge assembly where N � 1, N b� � 3N e�2 and

N b� � 3N� (2)

i.e., the number of branch vectors is always three times the number of particles. In otherwords, on average, each particle has six branch vectors.

The orientations of these branch vectors are described by a branch-vector density dis-tribution function �b� . The branch-vector density distribution function is defined such thatgiven a point on a particle’s surface with unit normal n, �b��n� dn describes the probabilityof finding a branch vector within a solid angle dn about n. From (2), in two dimensions

�

�b� �n� dn � 6� (3)

where the range of integration, , is taken over all orientations of the contact normal n.A definition for strain based on a triangular tessellation of the assembly provides more

information about the assembly’s deformation than a division of the area into polyhedra (e.g.[16, 35, 40]). This is because a strain tensor defined from a triangular tessellation uniquely





defines the deformation of each triangular region: see, for example, Figure 2. However, morethan one type of deformation can give rise to the same strain in a polygonal region. Considera polygon constructed from two triangles: for example, the polygon joining the centers ofparticles 1, 2, 3 and 4, can be constructed from the triangle joining the centers of particles1, 2 and 3, and the triangle joining the centers of particles 1, 3 and 4 in Figure 2. The totalstrain of this polygon is the weighted average of the relative deformation of the two triangles.Thus, different combinations of strains in the two triangular regions can contribute to thesame strain for the polygonal region. Moreover, a strain definition based on contacting andnon-contacting branch vectors can be extended easily to three dimensions by consideringtetrahedral regions. It is not clear how a definition based on polygonal regions could beextending to three dimensions without considering edges between non-contacting particles.In three dimensions, there is no guarantee that edges between contacting particles lie in thesame plane, so the volume in which the polyhedral strain would apply is not clearly defined.

3. CONTINUUM DESCRIPTION OF THE GRANULAR ASSEMBLY

The aim in this section is to develop new definitions for the micropolar strain and curvatureby imposing an energy equivalence between the discrete assembly and that of the equivalentcontinuum. We note that the functional form of the strain is not an a priori assumption in thefollowing formulation� the form of the strain arises as a result of invoking the condition thatthe free energy is constant under rigid body motion of the assembly. Since constituent parti-cles in a granular assembly have independent translational and rotational degrees of freedom,the deformation of the granular assembly is expressed in terms of two sets of vectors, namely,particle displacements, di , and rotations, ri . Similarly, as particles transmit both forces andmoments, the kinetics of the granular assembly are described by vectors giving the forces fi

and moments mi along each branch vector. Here it is assumed no long range forces act be-tween the particles, so that branch vectors between non-contacting particles carry zero forceand moment.

Consider a single element from the modified Delaunay triangulation defined in Sec-tion 2. The displacements of each vertex are given by dn

i and the rotations by rni , where

1 n � and � is the number of vertices in the element (i.e. � � 3 for triangular elementsin two dimensions, � � 4 for a tetrahedra in three dimensions). For a small variation in thenodal displacements and rotations, the work done on this element is given by

W ��

n�1

�Fn

i dni � Mn

i rni

�� (4)

where Fni is the net external force on the nth particle (i.e., the sum of the forces at each

node that arise from contacts that are not part of the triangular tessellation) and Mni is the net

external moment acting on that vertex of the element (i.e., the sum of the moments at eachnode that arise from contacts that are not part of the triangular tessellation).

Displacement and rotation fields are assigned to the element by assuming a linear inter-polation of the node (particle) displacements and rotations:





ui ��

n�1

N ndni � (5)

�i ��

n�1

N nrni � (6)

where ui is the displacement field, �i is the rotation field, and N n represent the shape func-tions for the linear interpolation. In two dimensions, N n can be expressed as

N 1 � 1� � � (7)

N 2 � � (8)

N 3 � � (9)

where � � [�y3� y1��x� x1�� x3� x1��y� y1�]�� , � [�x2� x1��y� y1�� y2� y1��x�x1�]�� . Here, � is twice the triangle area, and xi and yi are the coordinates of the 3 verticesof the triangle. The coordinates of the point inside the triangle where the shape function isbeing evaluated are given by x� y. Note that unlike the particle displacements and rotations,dn

i and rni , which are only defined for each particle, the fields ui and �i are defined at each

point within the element.The work done, W , on the element is equal to the change in the free energy, �, plus

the energy dissipated, D:

W � � � D� (10)

The change in free energy is equal to the integral of the variation of the free energydensity, � , over the element

� ��

V� dV � (11)

where V refers to the volume of the element (or area in two dimensions). The variation inthe free energy density can be expressed in terms of partial derivatives of the underlyingdeformation fields ui and �i :

� � ��ui� j � �i � �i� j � �� ui� j��

�ui� j� �i

��

��i� �i� j

��

��i� j� ��

�� (12)

where � are representative of internal variables related to plastic deformation. These variablesare assumed to be functions of the plastic contact displacements d p

i and plastic rotations r pi

arising from: sliding friction, rolling friction, or inelastic normal motion [41, 13].The dissipation function is given by

D ��c�B

f ci d

pci � mc

i rpci � (13)





where the sum is taken over the element’s branch vectors B, with contact forces f ci , plastic

contact displacements d pci , moments mc

i and plastic rotations r pci . Note again that branch

vectors between non-contacting particles do not register a dissipation.Equation (10) thus gives

��n�1

�Fn

i dni � Mn

i rni

� ��

ui� j��

�ui� j� �i

��

��i� �i� j

��

��i� j� ��

��

�dV

��

c

�f ci d

pci � mc

i rpci

�� (14)

As the above equation must hold for arbitrary variations in the displacement and rotations inthe nodes,

Fni �

�N n� j

��

�ui� jdV� (15)

Mni �

� �N n ��

��i� N n

� j

��

��i� j

�dV � (16)

Equations (15) and (16) imply

��

��dV � �

�c

�f ci d

pci �mc

i rpci

�� (17)

The relationship between the internal variables � and the plastic displacements d pci may

be further refined by specifying the law governing the contact forces and the functionaldependence of the internal variables on the plastic contact displacements and rotations. Thishowever lies outside the scope of the present paper. Although the detail of the dissipation isnot considered explicitly, the equations derived for the strain are purely kinematic relations,which are independent of the form of the contact forces.

For rigid body motions, the variation in the free energy must be zero:

� � 0� (18)

This condition constrains the free energy’s functional dependence on the particle dis-placements and rotations. For example under a rigid body rotation rn

i � �i , dni �

ei jk� j xnk , Equation (18) gives

��n�1

��

�rni

� ei jk xnk

��

�dnj

�� 0� (19)

where ei jk is the Levi-Civita tensor.





Hence the free energy may be written as

� � ��dni � ei jk xn

j rnk �� (20)

While the constraint on the free energy introduced in Equation (18) is sufficient to fixthe free energy’s functional dependence on the displacement and rotation, it is not enough toset the form of the strain. To do so, a slightly stronger assumption is introduced, namely thatthe variation in the free energy density is zero under rigid body rotations, i.e.

� � 0� (21)

Using Equation (21) and considering a rigid body rotation:

ui � ei jk� j xk (22)

�i � �i � (23)

leads to

��

��k� ei jk

��

�ui� j(24)

or

��ui� j � �i� � ��ui� j � ei jk�k� � ��i j�� (25)

i.e., the free energy density may be expressed as a function of the variable �i j � ui� j�ei jk�k ,the microstrain.

Using Equations (5) and (6), the strain and curvature are defined in terms of the particledisplacements and rotations as

�i j ��

n�1

�N n� j d

ni � N nei jkrn

k

�(26)

� i j ��

n�1

N n� j r

ni � (27)

This defines a mapping, D, that relates the particle displacements and rotations, u, to thestrain and curvature tensors, �, i.e.

u � D�� (28)

This definition also implies a connection between the stress and moments (represented by �)and the external forces and moments, f by virtue of the principle of virtual work [42]:





� � DT f� (29)

This relationship between the stress and the contact forces is similar to that found in finiteelement simulations [43].

The value of the strain and curvature for the element as a whole are obtained by takingthe volume average of the two fields:

��i j � 1

V

��n�1

� �N n� j d

ni � N nei jkr n

k

�dV (30)

� 1

V

��n�1

��N ndn

i s j dS ��

N nei jkr nk dV

�(31)

�� i j � 1

V

��n�1

�N nrn

i s j dS� (32)

where si is the outward normal vector of the surface S, bounding the volume V .From this point onwards, we specialize to the case of two dimensions. Hence, the equa-

tions above can be expressed as

��i j � 1

V

�ui s j dS � 1

Vei j3

�� dV � ui� j � ei j3 �r (33)

�� i � 1

V

��si dS � ��i (34)

where ui� j and ��i are the values of the displacement gradient and rotation gradient withinthe element, and �r is the average of the three particle rotations.

We note some commonalities and differences between the new definition in Equation(33) and past definitions of strain. Equation (33) is similar to the micropolar strain definitiongiven by [15], however, the discrete expression given in Equation (33) differs in that it doesnot employ vectors connecting the center of each element to its neighbors. The classicalstrain definition given by [16] is recovered from the symmetric part of ��i j . The functionalform of the strain in Equation (25) is not an a priori assumption, as is the case, for example,in [17, 44]� instead it arose naturally from the condition of invariance of the free energy un-der rigid body motion. In addition, while the above definition was derived for the smallestpossible meaningful area (i.e. that of a triangle in the modified Delaunay triangulation, seeFigure 2), by integrating over a larger area comprising of many such triangles, we may deriveaverage strain fields for any size cluster. In the next section, this is undertaken for a clustercomprising of a particle and its first ring of neighbors, in accordance with the homogeniza-tion scheme developed in earlier micromechanical continuum formulations [38, 25, 13].

For granular systems, the strain and curvature definitions given above in Equation (26)and Equation (27) result in a patchwork strain field, more akin to that defined in a finiteelement mesh than by a smooth continuum (Figure 2). Although the strain is given at everypoint within each element, it is not defined along branch vectors or at particle centers—an





indication of nonaffine motion. This is demonstrated by considering an infinitesimal linesegment overlapping the boundary between two triangular elements. The line segment be-comes kinked following deformation due to the separate displacement fields defined in thetwo elements, and thus there is no smooth mapping to describe the assembly’s deforma-tion. The patchwork nature of the continuum means that no single particle strain tensor canbe used to describe the relative grain motions. This is evident, for example, in the distinctrelative motions of the seven neighbors of particle 1 in Figure 2. Indeed, for any realisticcoordination number, the degrees of freedom of the particles in “first ring” cluster of Torde-sillas and co-workers [25, 38] exceed the four continuum strain and rotation variables ofthe equivalent two-dimensional Cosserat medium. On these grounds, the affine projectionscheme is inadequate even for an elastic response.

4. DEFORMATION ON THE SCALE OF A PARTICLE AND ITS FIRSTRING OF NEIGHBORS

Continuum mechanical models typically employ the Voigt hypothesis which is predicatedon the assumption that the deformation field is affine [24]. This assumption indicates that:(a) the fluctuations about the mean gradients of particle displacements and rotations can bereasonably neglected, (b) a single strain tensor is sufficient to describe the relative motionof the grains, (c) the relative displacement and rotations between a reference particle andits neighbor can be expressed as dot products of the strain and curvature with the branchvectors joining the particle centers [13, 36]. However, numerous studies have now shown,that quasistatic deformation of a densely packed granular material is strongly nonaffine andthe failure to account for this in constitutive modeling results in poor predictions of observedbehavior ([13] and references cited therein). Specifically, constitutive models which do notaccount for nonaffine deformation overpredict material stability to the extent that they failto reproduce defining behavior such as strain softening under dilatation and shear banding[13, 24, 45].

In what follows, we introduce an enriched projection scheme that accounts for nonaffinedeformation for a particle cluster comprising of a particle and its first ring of neighbors.We employ multiple strain tensors, each defined at the particle center, that are capable ofdescribing the underlying particle motions to greater accuracy than models that rely on asingle strain and curvature tensor. The new strain tensors represent the contributions from thedistinct modes of relative motion between the reference particle and its first ring of neighbors.

We begin by deriving an expression for the average strain for each particle in two dimen-sions, based on the relative displacements between a particle and its first ring of neighbors.Define a projection vector pc

i as

pci � uc

i � ui � ei j3lcj�� (35)

where ui and � describe the displacement and rotation of the reference particle, lci is the

branch vector joining the particle to its cth neighbor and uci is the displacement of the neigh-

boring particle. This vector describes the relative motion of the center of particle c from the





frame of the reference particle. The average strain assigned to each particle, � pi j , is given in

terms of the pci vectors associated with that particle as

�pi j �

1

V p

�S p

pi s j dS p � 1

2V p

�c�B

�pci � pc�1

i �e jk3�lc�1k � lc

k�� (36)

where the sum is taken anticlockwise over the set of branch vectors B associated with theparticle, V p is the area of the element formed by each particle’s first ring of neighbors, S p isthe surface bounding this area, and pi is a linear interpolation of the pc

i vectors. The averagestrain for a particle may therefore be expressed as a function of the strain and curvaturetensors given earlier in equations (26) and (27):

�pi j � 1

V p

�V p

��i j � ei j3�klk

�dV � (37)

The average curvature for a particle, � pi , is defined in a similar manner. The relative

rotation at a branch vector is given by

��c � �c � �� (38)

where �c is the rotation of the cth neighboring particle. Using this definition, the averagecurvature for a particle is given by

�pi �

1

V p

�S p

��si dS p � 1

2V p

�c�B

��c ��c�1�ei j3�lc�1j � lc

j�� (39)

where�� is a linear interpolation of the��c values. Although the above expressions for theaverage strain and curvature for a particle are functions of the relative motions of the particleand its immediate neighbors, these motions are poorly fitted by the dot product of the strainand curvature with the branch vectors. This discrepancy is due to the fact that there existdistinct modes of relative motion between the particles, which we identify in the followingsubsections. Thus we now construct multiple strain tensors—each of which represents thecontributions from a particular mode of relative motion between particles in differing modesof contact, and can be viewed as decompositions of the particle strain and curvature ten-sors. Examined here are decompositions that represent contributions from relative motionwith respect to: in-contact versus out-of-contact neighbors, the normal versus tangential di-rections, contacts with above versus below average contact force magnitude, contacts withabove versus below average contact moment magnitude, and elastic versus plastic contacts.

4.1. In-Contact versus Out-of-Contact

The average strain rate for a particle, �� pi j , can be decomposed into components describing

the relative motion at branch vectors between contacting particles, ��ini j , and that due to the

relative motion at branch vectors between particles that are out of contact, ��outi j , such that





�� pi j � ��in

i j � ��outi j � (40)

These two strain rate measures are given by

��ini j � 1

2V p

�c�B in

� �pci � �pc�1


k� (41)

��outi j � 1

2V p

�c�Bout

� �pci � �pc�1


k�� (42)

where the sums are over those branch vectors where the particles are in contact, B in, andthose branch vectors where the particles are out of contact, Bout, respectively. A similarstrain decomposition has been suggested by [31].

The average particle curvature rate, �� pi may be decomposed along the same lines, i.e.

�� pi � �� in

i � ��outi � (43)

where

�� ini � 1

2V p

�c�B in

� ��c � ��c�1�ei j3�l

c�1j � lc

j� (44)

��outi � 1

2V p

�c�Bout

� ��c � ��c�1�ei j3�l

c�1j � lc

j�� (45)

The above decomposition into contacting and non-contacting components enables an analy-sis of the relative contribution of the force-bearing contact network to the macroscopic strainand curvature.

4.2. Normal versus Tangential

In light of studies which show distinct trends in the strain evolution of the normal and tan-gential contact forces and displacements on the macroscopic level [7, 46], a second decom-position worth considering is that of

��ni j � 1

2V p

�c�B

� �pchnhni � �pc�1

h nhni�e jk3�lc�1k � lc

k� (46)

��ti j � 1

2V p

�c�B

� �pchthti � �pc�1

h thti�e jk3�lc�1k � lc

k�� (47)

where ��ni j and ��t

i j describe the contribution to the strain rate due to the relative normal andtangential motion respectively at branch vectors between particles, and ti is a unit vectortangential to the branch vector.





The successive application of the in-out and normal-tangential decompositions intro-duces four more strain rates: ��inn

i j and ��int

i j describing the contribution to the strain rate atbranch vectors between contacting particles due to the relative normal and tangential motionrespectively, and ��outn

i j and ��outti j describing the strain rate at branch vectors between particles

which are out of contact due to the relative normal and tangential motion respectively. Thesestrain rate measures are given by

��inn

i j � 1

2V p

�c�Bin



k� (48)

��int

i j � 1

2V p

�c�Bin



k� (49)

��outni j � 1

2V p

�c�Bout



k� (50)

��outti j � 1

2V p

�c�Bout



k�� (51)

As an illustration of this decomposition, in Figures 3(a)–(c) we show sample particlemotions that produce different contributions to the particle volumetric strain � p

vol � 12�i i . In

particular, Figure 3(a) shows how a dilative volumetric strain can arise from solely normalmotion from out-of-contact branch vectors (�outn

i j )� Figure 3(b) shows how dilative normalmotion and contractive tangential motion from out-of-contact branch vectors (�outn

i j � �outti j )

can combine to produce zero volume change in �i j � �outi j and �in

i j � and Figure 3(c) showshow a dilatation due to solely contacting branch vectors can arise. Finally, Figure 3(d) il-lustrates how contacting branch vectors contribute to the decomposition of � p

i (recall Equa-tions (43)–(45)).

4.3. Strong versus Weak Contact Forces

Radjai and co-workers characterized the contact force network into a “strong network” com-prising of all contacts that carry above average force, and a “weak network” comprising ofall contacts carrying below average force [47]. In light of this study, which found differ-ent trends in the contributions to stress from the strong and weak network, we also furtherdecompose the in-contact strain rate, ��in

i j , according to the magnitude of the contact force

��ini j � ��insCF

i j � ��inwCF

i j � (52)

where ��insCF

i j � ��inwCF

i j are the components describing the contributions due to the relative motionat contacts within the strong and weak networks respectively. These two strain rate measuresare given by





Figure 3. Schematics of a particle and its first ring of neighbors illustrating possible motions of particlesthat lead to different strain/curvature contributions. Particles that contribute to the strain/curvatureare colored dark gray, and their branch vectors are dotted lines� while those that do not contributeare colored light gray, with branch vectors as solid lines. (a) A dilatation resulting from solely normalmotion from out-of-contact branch vectors (� p

vol � �outnvol � 0�132� �outt

vol � 0�0). (b) A combinationof normal and tangential motion from out-of-contact branch vectors resulting in zero volume change(� p

vol � �outvol � 0� �outn

vol � ��outtvol � 0�016). (c) A dilatation resulting solely from in-contact branch

vectors (� pvol � �in

vol � 0�034). (d) A curvature resulting solely from rotation at contacting branch vectors(� in

2 � �4�78� � in1 � �2�38).

��insCF

i j � 1

2V p

�c�Bin�sCF

� �pci � �pc�1


k� (53)

��inwCF

i j � 1

2V p

�c�Bin�wCF

� �pci � �pc�1


k�� (54)

where the sums are over those in-contact branch vectors where the magnitude of the contactforce is above the average for the whole assembly, B in�sCF, and below the average for thewhole assembly, B in�wCF, respectively.

The average in-contact curvature rate, �� ini may be decomposed along the same lines,

i.e.

�� ini � �� insCF

i � �� inwCF

i � (55)

where

�� insCF

i � 1

2V p

�c�Bin�sCF

� ��c � ��c�1�ei j3�l

c�1j � lc

j� (56)





�� inwCF

i � 1

2V p

�c�Bin�wCF

� ��c � ��c�1�ei j3�l

c�1j � lc

j�� (57)

4.4. Strong versus Weak Contact Moment

In non-spherical granular materials, moments as well as forces can be transmitted at contacts.Hence, we also decompose the in-contact strain rate, ��in

i j , into components describing thecontribution due to relative motion at contacts carrying a strong (above average) contactmoment, ��insCM

i j , and due to the relative motion at contacts carrying a weak (below average)

contact moment, ��inwCM

i j , such that

��ini j � ��insCM

i j � ��inwCM

i j � (58)


��insCM

i j � 1

2V p

�c�Bin�sCM

� �pci � �pc�1


k� (59)

��inwCM

i j � 1

2V p

�c�Bin�wCM

� �pci � �pc�1


k�� (60)

where the sums are over those in-contact branch vectors where the magnitude of the contactmoment is above the average for the whole assembly, B in�sCM, and below the average for thewhole assembly, B in�wCM, respectively.

In a similar fashion, the in-contact curvature rate, �� ini may be decomposed such that

�� ini � �� insCM

i � �� inwCM

i � (61)

where

�� insCM

i � 1

2V p

�c�Bin�sCM

� ��c � ��c�1�ei j3�l

c�1j � lc

j� (62)

�� inwCM

i � 1

2V p

�c�Bin�wCM

� ��c � ��c�1�ei j3�l

c�1j � lc

j�� (63)

4.5. Elastic versus Plastic Contacts

Also, the in-contact strain rate, ��ini j , can be decomposed into components describing the con-

tribution due to relative motion at elastic contacts, ��ine

i j , and due to the relative motion atplastic contacts, ��inp

i j , such that





��ini j � ��ine

i j � ��inp

i j � (64)


��ine

i j � 1

2V p

�c�Bin�e

� �pci � �pc�1


k� (65)

��inp

i j � 1

2V p

�c�Bin�p

� �pci � �pc�1


k�� (66)

where the sums are over those in-contact branch vectors where the contact force is elastic,B in�e, and plastic, B in�p, respectively.

Similarly, we decompose the in-contact curvature rate, �� ini such that

�� ini � �� ine

i � �� inp

i � (67)

where

�� ine

i � 1

2V p

�c�Bin�e

� ��c � ��c�1�ei j3�l

c�1j � lc

j� (68)

�� inp

i � 1

2V p

�c�Bin�p

� ��c � ��c�1�ei j3�l

c�1j � lc

j�� (69)

In the simple case of a DEM simulation with frictional particles whose tangential forceis assumed to follow a spring-slider contact law, a plastic contact would be defined as onewhose tangential force is at the Coulomb threshold.

4.6. Nonaff ine Deformation

A fundamental principle of continuum mechanics is the axiom of locally affine motion [48].This axiom states that the deformation of each infinitesimal region in a material body is de-scribed by a linear mapping from the region’s undeformed state. Under an affine deformationof the assembly, the vector pc

i defined in Equation (35) would equal the dot product of theaverage particle strain along the cth branch vector, � p

i j lcj . However, granular materials exhibit

nonaffine motion. Consequently, the relative particle motion is only approximated by the dotproduct of the strain and curvature (Equations (36) and (39)) with the branch vectors.

To quantify this nonaffine motion, a vector � �pi is first defined such that

� �pi � �pci � �� p

i j l j � (70)

The vector � �pi measures the difference between the rate of change of the projection vectorand that implied by the particle strain. Normal and tangential contributions to the nonaffinemotion at each contact are then given by





� �pni � �� phnh�ni � (71)

� �pti � �� phth�ti � (72)

From these definitions, measures of the nonaffine motion experienced by each particle areintroduced, defined in a like manner to the strain given in Equation (36):

�ni j � 1

V p

�S p

� �pni s j dS p � 1

2V p

�c�B

� �pnc�1

i �� pnc

i

e jk3�l

c�1k � lc

k�� (73)

�ti j � 1

V p

�S p

� �pti s j dS p � 1

2V p

�c�B

� �ptc�1

i �� ptc

i

e jk3�l

c�1k � lc

k� (74)

and

� � 1

V p

�S p

��p� dS p � 1

2V p

�c�B

��pc�1� � ��pc��lc�1 � lc�

� 1

2V p

�c�B

��pc��lc�1 � lc� � �lc � lc�1�� (75)

The two tensors �ni j and �t

i j may be used to investigate directional dependance in the non-affine motion. The scalar � gives a measure of the total nonaffine motion experienced byeach particle. Although� is similar to the measure of nonaffine motion defined in both [24]and [49], there are important differences between the two. Firstly, the measure of nonaffinemotion given in Equations (75) is dimensionless, and therefore independent of the size of theparticle cluster. Secondly, here the strain is defined independently of the expression for thenonaffine motion. In contrast, both [24] and [49], define the strain such that the nonaffinemotion is minimised (i.e., the best-fit method). A full examination of nonaffine deformationin dense granular assemblies under biaxial compression has recently been undertaken in [50].

5. ANALYSIS OF PARTICLE STRAIN AND CURVATURE USING DEM

The definitions of particle strain and curvature and their respective decompositions presentedin Section 4 were applied to data taken from a discrete element simulation. The simulation ofa two-dimensional biaxial compression test was performed on a densely packed assembly ofpolydisperse circular disks. The DEM code used is derived from earlier work by [12]. Thecontact model for normal motion is based on Hooke’s law, while the model for tangentialmotion combines Hooke’s law with Coulomb’s law in order to model friction. In addition,resistance to relative rotation is introduced via a contact moment. The contact model used forthe contact moment is analogous to that for tangential motion, so that rolling friction may beincorporated. A more detailed description of the contact laws, sample preparation and sim-





Figure 4. (a) Boundary conditions and initial configuration of the particles in the discrete elementsimulation. (b) Evolution of the stress ratio, sin�, with the axial strain magnitude ��22�. In this and allfollowing figures that show evolution of a quantity with axial strain magnitude, the vertical dashed linesat ��22� � 0�018� 0�034 depict transitions between Stages 1 to 2, and 2 to 3 respectively.

ulation parameters is given in [41, 51]. Particle radii are taken from a uniform distribution,and gravitational effects are ignored.

To create the initial configuration, non-contacting particles were randomly distributedin a box and a confining stress was applied to all four walls of the box. The assembly wasthen allowed to settle to a state where the kinetic energy became negligible. This isotropicassembly was then subjected to boundary driven biaxial compression. Simulations wereperformed under conditions of zero gravity. The assembly was compressed at a constantstrain rate in the vertical direction, while allowed to expand in the horizontal direction un-der a continued constant confining stress. Figure 4(a) shows the boundary conditions andinitial configuration, and Table 1 lists parameters and material properties used. The rateof compression in the vertical direction was ��22 � �0�008 s�1. The dimensionless group,��22

pmass�� 11, where pmass is the average particle mass, used to quantify inertial effects, is ofthe order of 10�5: this is significantly less than that suggested to correspond to the transitionbetween the quasi-static regime and dense flow regime of 10�3 [52]. Additional simulationsperformed at even smaller rates of deformation confirmed that the strain rate is sufficientlysmall to ensure quasi-static conditions.

Figure 4(b) shows the evolution of the stress ratio, sin� � �� 11�� 22�� 11�� 22�, withaxial strain magnitude, ��22�. As is typical of real densely-packed granular assemblies, thematerial undergoes a period of strain-hardening up to a peak, followed by strain-softeningafter strain localization has occurred: see, for example, similar trends in [53].

The onset of a region of localized deformation, a shear band, occurs just before the peakstress ratio, with the shear band becoming fully developed and persistent shortly after thepeak. In real assemblies, these regions of strain localization typically measure 8–20 particlediameters in width [26, 54–56]. In the two dimensional simulation considered here, the shearband width varies between 5 and 15 particle diameters.





Table 1. DEM parameters and material properties used.

Parameter Value��22 �8� 10�3 s�1

Confining stress � 11 on side walls 7�035� 102 N m�1

Timestep increment 4� 10�6 sInitial height:width ratio 1 : 1Number of particles 5098Particle density 2�65� 103 kg m�3

Smallest radius 0�76� 10�3 mLargest radius 1�52� 10�3 mAverage radius (uniform distribution) 1�14� 10�3 mInter-particle friction coefficient 0�7Particle-wall friction coefficient (top, bottom) 0�7Particle-wall friction coefficient (sides) 0�0Rolling friction coefficient 0�02Normal spring stiffness constant 1�05� 105 N m�1

Tangential spring stiffness constant 5�25� 104 N m�1

Rotational spring stiffness constant 6�835� 10�2 Nm rad�1

5.1. Matching the Global Strain

A measure of the total strain of the assembly, �assemblyi j , is obtained by taking the average of the

individual particle strains (Equation (36)), weighted according to the volume of the elementformed by each particle’s first ring of neighbors:

�assemblyi j �

N p�p

V p�pi j�

N p�p

V p� (76)

where the sum runs over the N p particles in the assembly. The components of this strainmeasure are almost identical to those of the global strain given by the boundaries of theassembly, although small errors do arise due to openings between the straight confiningboundary and the irregular boundary of the granular assembly (Figure 5). In particular, theerror (expressed as a percentage of the total strain) for �22 is always less than 1.4%. While themaximum error is 7% for �11, this is only at the start of the deformation—for the majority ofthe deformation, this error is less than 3%. This level of agreement is not achieved by otherstrain measures defined in terms of particle contacts—in particular those given by “best fit”hypotheses ([28] and [29]). Bagi found that even the most accurate micropolar strain, that of[15], demonstrates errors of over 10% within the first 1% of strain [29].

5.2. Distinct Stages of Deformation and Associated Instabilities on the Mesoscale

In [41], three stages of deformation were identified for the specimen examined here: theseare indicated in Figure 4(b). In the discussions in this and subsequent sections, we will relate





Figure 5. Error in the strain components, �assembly11 and �assembly

22 , compared to those given by the boundariesof the assembly, �wall

11 and �wall22 .

temporal and spatial distributions of the local measures of strain and curvature in equations(36) and (39), along with their various decompositions, to these three distinct stages of thedeformation history. We briefly recall key aspects of rheological behavior during each ofthese stages. In Stage 1, the assembly deforms almost uniformly in accordance with themacroscopic strain rate dictated by the motion of the boundaries. This stage correspondsto the steep increase in the stress ratio and ends approximately at the axial strain value of��22� � 0�018. Rattlers are the only form of instability observed during this stage. A rattler isa particle that is free to move and rotate within the cage formed by its ring of neighbors [57].In Stage 2, a secondary displacement field emerges, that is characterized by the presenceof a new form of instability, i.e., microbands and vortex-like structures: here deformation islocally nonaffine. Near the peak stress ratio, the onset of what will later become the so-calledpersistent shear band, the precursor of material failure, can be observed. The developmentof this band continues through to the initial phase of the strain-softening regime, or Stage3. The band becomes fully formed at ��22� � 0�04. During the subsequent steady stateevolution of the band, i.e. persistent shear banding, the regions outside move collectivelyin essentially rigid-body motion. Throughout Stage 3, consecutive cycles of unjamming-jamming events (i.e. drop and rise in macroscopic stress ratio), reminiscent of the so-calledslipÐstick phenomenon observed in other granular systems [58–61], are evident.

We present spatial distributions of volumetric and deviatoric particle strains, obtainedfrom Equation (36). During Stage 1, the volumetric strain distribution is essentially affine,as the assembly compresses almost uniformly (Figure 6(a)), while during Stage 2, the rangeof particle volumetric strains starts to increase (Figure 6(b)). This trend continues in Stage 3,with particles registering the highest changes in volumetric strain (i.e. those coloured black





Figure 6. Spatial distributions of the volumetric strain, �vol, during (a) Stage 1 (��22� � 0�0073), (b)Stage 2 (��22� � 0�030) and (c) Stage 3 (��22� � 0�050). Spatial distributions of the deviatoric strain,� , during (a) Stage 1 (��22� � 0�0073), (b) Stage 2 (��22� � 0�030) and (c) Stage 3 (��22� � 0�050).Spatial distributions of the magnitude of curvature, ��, during (d) Stage 1 (��22� � 0�0073), (e) Stage 2(��22� � 0�030) and (f) Stage 3 (��22� � 0�050).

and white), concentrated in the shear band (Figure 6(c)). This simultaneous contractive anddilative local volumetric strain behavior in the shear band has been previously observed inexperiments [55] and DEM simulations [8, 62].

The spatial distributions of the deviatoric strains � � 12��11 � �22) exhibit a distinctive

pattern of behavior, as shown in Figure 6(d)–(f). During Stage 1, deviatoric strain is rela-tively small, as motion is essentially affine. In Stage 2, regularly spaced particle clusters thatexperience relatively little deviatoric strain emerge between lines of particle pairs experienc-ing intense shear. These fine bands of intense shear are a manifestation of the micro-bandsdescribed by Kuhn [30]. In Stage 3, a single shear band develops along one of the exist-





ing microband locations, consistent with findings from DEM simulations in [14]. Inside theshear band, deviatoric strain accumulates.

The evolution of the particle curvature magnitudes obtained from Equation (39) is pre-sented in Figure 6(g)–(i). Throughout all three stages, rattlers are evident. The large rotationof the rattler, and zero rotation of the outer cage particles results in the cage particles regis-tering large curvatures. In Stages 2 and 3, the greatest concentration of particle curvature isfound in microbands and the shear band, respectively, as these mesoscale structures involvelarge relative particle rotation. These results are in agreement with experimental observationsand results from other numerical studies [8, 63].

5.3. How Does Relative Motion from Distinct Modes of Contact Govern Strain and Curvature

In this section, we present and discuss results for the spatial and temporal evolution of thevarious decompositions of the local micropolar strain and curvature in Equations (36) and(39)). These decompositions represent the contributions from distinct modes of relative mo-tion, as well as relative motion from distinct modes of contact.

5.3.1. Decompositions of the volumetric strain

The total volumetric strain �vol undergoes a period of compression, following which the as-sembly begins to dilate or increase in volume with continuing deformation (Figure 7(a)).The sudden decrease in dilatation rate coincides with the appearance of the shear band. Thispattern of deformation is similar to that observed for other 2D DEM simulations of initiallydense assemblies of polygonal frictional particles [53]. Figure 7(a) also shows the contri-butions to �vol from in-contact branch vectors, �in

vol, and out-of-contact branch vectors, �outvol.

We reproduce here the trend observed by Dedecker and co-workers [31], who also inves-tigated this type of decomposition. Similar to their findings, the branch vectors betweenin-contact particles and those between out-of-contact particles do not contribute equally tothe volumetric strain experienced by the assembly as a whole. The decompositions evolvewith strain distinctly from each other and from the total macroscopic strain: specifically,the contribution from branch vectors between out-of-contact particles begin to dilate beforethe total volumetric strain, whereas that for in-contact particles continues to undergo com-pression even after the assembly has commenced dilatation. During Stage 3, �out

vol essentiallylevels off, while �in

vol� �vol continue to increase. The saturation point for �outvol may be tied to

a maximum average local void ratio possible for a particle and its first ring for a specificparticle size distribution. This warrants further study. On the other hand, during Stage 3, �in

vol

more closely follows the total volumetric strain, and is responsible for most of the dilatationhere. Moreover, we note that there is a correlation between decreases in sin� and increasesin �in

vol: recall the stress ratio presented in Figure 4. This is not surprising, since �invol is due to

the contacting particles, which are also those transmitting stress.We wish to probe further the types of motion responsible for the in-contact and out-

of-contact volumetric strain behavior. To do this, we investigate �inn

vol� �int

vol, �outnvol and �outt

vol , asshown in Figures 7(b) and (d). Assuming affine deformation of the displacement field, thenormal and tangential particle strains (�n

i j and �ti j , Equations (46) and (47)) can be related to

the total strain by





Figure 7. Decompositions of: (a) �vol, into �invol and �out

vol� (b) �invol, into �inn

vol and �intvol� (c) �in

vol, into �insCFvol , and

�inwCFvol � and (d) �out

vol, into �outnvol , and �outt

vol .

�ni j � 1

2ei j � 1

2i j�kk (77)

�ti j � 1

2ei j � ��i j� (78)

where ei j is the classical deviatoric strain tensor, i.e. ei j � 12��i j � � j i � i j�kk�, and where

subscripted curly brackets represent the skew symmetric part of a tensor, i.e. ��i j� � 12��i j �

� j i�. During Stages 1 and 2, relative displacements in the normal direction contribute almostall of both the in-contact and out-of-contact volumetric strain (Figure 7(b),(d)). This is inline with the predictions of the affine deformation of the assembly, where the volumetricstrain is solely a product of the normal displacements at the contacts. A possible situationshowing a dilatation from solely normal motion of out-of-contact branch vectors is shownin Figure 3(a). In Stage 3, however, the deformation of the assembly is highly nonaffineas the shear band has formed. Hence, tangential displacements now also contribute to thevolumetric strain (�int

vol� �outtvol �� 0).

As previously mentioned, the dilatation of �invol governs the global dilatation in Stage 3:

here we see that �invol is due predominantly to normal motion (Figure 7(b)). While the behavior

of �inn

vol is dilative on average over Stage 3, we note that during slow rises in stress ratio, �inn

vol

slowly compacts, while during rapid drops in stress ratio, �inn

vol has a rapid rate of dilatation�





compare the slopes of consecutive drops and rises in �inn

vol during Stage 3. In the regionsoutside the shear band, a scenario that could account for this behavior is the simple case ofparticles in force chains loading (compactions) or unloading (dilatations): this mechanismwas observed to be quite common in these outer regions. In this case, there would be little orno tangential movements that would lead to a change in structure of the force chain. Insidethe shear band, however, a different mechanism would explain this behavior: the buckling offorce chains. This mechanism has been suggested as intrinsic to the shear band [54, 56, 64].The schematic in Figure 3(c) shows a sample set of particle motions that could result in thisbehavior.

During Stage 3, when �outvol reaches essentially a steady state, the tangential motions are

mainly due to branch vectors that are out-of-contact (Figure 7(d)). During this stage, we seethe tangential relative motions to be solely contractive, while the normal relative motionsdilative� hence, on average, essentially no volume change can be observed from the out-of-contact branch vectors. One possible set of particle motions that results in this behavior isshown in Figure 3(b).

We now turn our attention to strong and weak contact networks as first examined byRadjai and co-workers [47]. They found that the strong contact network was responsiblefor the deviatoric stress, while the weak contact network contributed only to the averagepressure. Here, we wish to investigate contributions to the strain from these two networks,�insCF

vol � �inwCF

vol . In Stages 1 and 2, most of the in-contact volumetric strain is due to the strongcontact network, as force chains are formed and are then compressed, producing contractivevolumetric strains (Figure 7(c)). During sudden dilatations in �in

vol in Stage 3, both strong andweak contact networks contribute: the force chains buckle, while their surrounding weakcontact network rearranges. However, during the slow slight contractions or periods of nochange in �in

vol (e.g. at ��22� � 0�085 � 0�09), the strong contact network compacts as forcechains load up at the same time as the surrounding weak contact network dilates slightly.Continuing dilative motion of the weak contact network diminishes the lateral support forthe force chains and, in so doing, destabilizes them. The next sudden dilation in �in

vol (e.g.at ��22� � 0�093) results from the force chains buckling and collapsing. These results elu-cidate the mechanisms behind the well-known hypothesis by Oda and co-workers, based ontheir experiments and simulations on densely packed sand� they attributed the large dilata-tional strain in the shear band to the voids created between buckling force chains, whichis balanced by the large compressive strain due to these force chains collapsing [62, 64].From Figure 7(c), it is evident that the weak network particles between buckling force chainsare more responsible for dilatation than the strong network, during Stage 3. We would ex-pect force chains to collapse during sudden decreases in stress ratio. However, we observesmall dilative rather than contractive strains here for the strong network. This suggests thatmost of the force chains are in the initial stages of buckling (dilative motion) rather thanin the final stages where collapse (contractive motion) occurs. In addition, the behavior ofthe strong network outside the shear band (i.e. force chains unloading) contributes to thisthis dilatation. The decomposition of �in

vol into plastic and elastic contacts follows a similartrend to Figure 7(c) (data not shown), with the plastic contacts being mainly responsible fordilatation.





Figure 8. (a) Decomposition of � into � in and � out. (b) Proportion of branch vectors (N ) that are incontact (N in), as well as the proportion of � that is due to these branch vectors. (c) Decomposition of� into � n and � t . (d) Decomposition of � in into � inn

and � int. (e) Decomposition of � out into � outn and

� outt . (f) Decomposition of � in into � insCFand � inwCF

. (g) Proportion of in-contact branch vectors (N in)that correspond to an above average force (N insCF

), as well as the proportion of � in that is due to thesebranch vectors. (h) Decomposition of � in into � ine

and � inp. (i) Proportion of in-contact branch vectors

(N in) that correspond to a plastic contact force (N inp), as well as the proportion of � in that is due to these

branch vectors.

5.3.2. Decompositions of the deviatoric strain

Figure 8(a) compares contributions to the total deviatoric strain � from contacting (� in) andnon-contacting (� out) branch vectors, demonstrating that most of the deviatoric strain is dueto those particles out-of-contact. Here, the differences in the deviatoric strain values cannotbe attributed to the proportion of the branch vectors in-contact compared with those out-of-contact, since this ratio differs to the proportion of contacting deviatoric strain to totaldeviatoric strain, � in�� (Figure 8(b)). This result suggests that non-contacting branch vectorscontribute a disproportionately larger amount to the deviatoric strain per contact than theirin-contact counterparts. A possible explanation to this is that particles that are not touchingcan move more easily relative to one another.

Under an affine deformation of the granular assembly, normal displacements and tangen-tial displacements contribute equal amounts to the deviatoric strain (� t � � n). As shown inFigure 8(c), this is the case during Stages 1 and 2 when deformation is mainly affine. How-ever, during Stage 3, the tangential displacement components constitute a greater proportionof the deviatoric strain than the normal displacement components. In this study, the value





of � t exceeds that of of � n by as much as 30%. To investigate this difference further, welook at the proportion of � t to � n in the contacting and non-contacting strain contributions(Figure 8(d) and (e)).

Figure 8(d) demonstrates that the vast majority of the in-contact deviatoric strain arisesfrom motion tangential to the contact, � int

. This may be due to the influence of the physicalpresence of the reference particle and its neighbors. Contacting particles have less freedomto move normal to the branch vectors than they do tangentially in the form of slip. However,the out-of-contact deviatoric strain is equally due to normal and tangential motions at thecontact (� outn � � outt in Figure 8(e)). During Stage 1, the strong network contributes dispro-portionately more to � in than the weak network (Figure 8(f) and (g)). This could be due toparticles arranging themselves into force chains. However, this trend changes during Stage 2as the system forms microbands, which are not solely formed from the strong network. Dur-ing Stage 3, it is the weak network which contributes disproportionately more to � in, since itis much easier for these particles to rearrange as they are not held in strong force chains.

During Stage 1, the differences in the contributions to � in from plastic and elastic con-tacts, � inp

� � inerespectively, can be attributed to the proportion of the in-contact branch vec-

tors that are plastic compared with those that are elastic (Figure 8(h) and (i)), since theproportion of in-contact deviatoric strain due to plastic contacts, � inp

�� in closely tracks theratio of plastic contacting branch vectors to contacting branch vectors here (Figure 8(i)). Thissuggests that plastic and elastic contacts contribute equal amounts per contact to � in duringStage 1. During Stage 2, the proportion of plastic contacts N inp

�N in decreases from the peak,as rolling and sliding motions become increasingly confined to the microbands. However,the plastic contacting strain proportion � inp

�� in continues to increase, which suggests thatmost of the in-contact strain in this regime is due to the microbands. During Stage 3, theproportion of plastic contacts fluctuates about a mean value of 13.7%, while the proportion� inp

�� in levels off to a much higher value of 30%. Thus, most of the in-contact deviatoricstrain during shear banding is due to plastic contacts, as these are the contacts which wouldinduce the most relative motion and rearrangements.

5.3.3. Decompositions of the curvature

The contacting and non-contacting branch vectors also contribute differing amounts to theparticle curvature. Figure 9(a) shows that curvature is dominated by out-of-contact branchvectors. Since this simulation incorporates a resistance to relative rotations in the form ofa contact moment, this is to be expected: it is far easier for two out-of-contact particles torotate relative to one another than two particles that are in-contact and are thus resisted by thecontact moment. We also expect the contribution to curvature from in-contacts to decreasewith increasing resistance to relative rotations. We wish to probe now why and how particlesin contact rotate relative to one another during Stage 3 by examining decompositions of�� in�. To aid in our discussions, the inset to Figure 9(a) shows a scatterplot, taken at the startof Stage 3, where each data point represents a contact. This shows how the magnitude ofthe contact force relates to the magnitude of the contact moment. All contacts have forceand moment magnitudes that lie above a line through the origin. This is due to the factthat the law for the contact moment, m, in the DEM code is based on a Coulomb-like frictionthreshold which is a function of the magnitude of the normal contact force, � f n�. In particular,





Figure 9. (a) The magnitude of curvature due to the contributions from the branch vectors betweenparticles in, �� in

i �, and out, ��outi �, of contact. Inset shows normalized magnitude of contact force, fi ,

� fi �� fi �� , plotted against normalized magnitude of contact moment, m, �m�

��m�� , for all contact forces at the

start of Stage 3 (��22� � 0�034). (b) Ratio of the magnitude of the curvature due to � inpto that due to � ine

,and the corresponding ratio of the number of contacting branch vectors in plastic mode, N inp

, to thosein elastic mode, N ine

. (c) Ratio of the magnitude of the curvature due to � insCMto that due to � inwCM

,and the corresponding ratio of the number of contacting branch vectors that have above average contactmoment, N insCM

, to those with below average contact moment, N inwCM.

�m� � �r R� f n� if the contact moment is elastic, and �m� � �r R� f n� if the contact momentis plastic, where �r � R are the coefficient of rolling friction and particle radius respectively.The magnitude of the contact force � fi � at a contact is proportional to � f n�. Hence, a chosencontact force magnitude value � fi �� fi �� (normalized by the average contact force magnitudeof all contacts), will set the upper bound for the corresponding moments. Plastic contactmoments would thus correspond to those data points that are towards the right side of thedata for a chosen value of � fi �� fi ��.





Firstly, we see that it is the plastic contacts that contribute significantly more to �� in� thanthe elastic contacts (Figure 9(b)). Similar to deviatoric strain (recall Figures 8 (h) and (i)),we would expect this, since plastic contacts are those that would induce the most relativemotion. As explained earlier, those with a plastic contact moment are those situated at theright extreme for a given value of � fi �� fi � �: recall inset to Figure 9(a). Hence, theseplastic contacts constitute both strong and weak network particles.

Secondly, we see that contacts with strong contact moment contribute disproportionatelymore than those with weak moment (Figure 9(c)). This would be expected, since contactmoment is proportional to relative particle rotation. Also, this trend is seen in contributionsfrom the strong network of forces and weak network, with the strong network contributingdisproportionately more (data not shown). This could be explained by the large rotationstied to force chain buckling during shear banding. One such possible particle rotation isillustrated in Figure 3(d). In addition, this similarity between the strong contact forces andstrong contact moment can be explained by the fact that the overlap between the set ofcontacts with above average contact force (strong network) and the set of contacts with aboveaverage contact moment is quite large. This is evident in the inset to Figure 9(a): largemoments are necessarily limited by the contact force� on the other hand, if a contact hasa large force, it need not have a large moment. This could occur, for example, in forcechains that are in the outer regions of the assembly which have no relative rotation. Thereare a small number of contacts with above average moment but below average force—thesecorrespond to weak network particles with large relative rotations. Thus, it would seem thatit is the plastic contacts in the strong network that also have a strong contact moment, thatare mainly responsible for curvature. This supports the hypothesis of Oda, that buckling offorce chains is inevitably tied to particle rotation [56, 64].

By taking the rate form of the curvature magnitudes in Figure 9(a), we obtain � �� in� and� ��out�, shown in Figure 10. During Stage 1, particle rotation is confined to the rattlers, andhence curvature is negligible (recall Figure 6(g)). During Stage 2, particle rotations increaseslightly as microbands emerge� small peaks are evident in � �� in� and � ��out� (recall Figure 6(h)).In Stage 3, persistent shear banding is dominated by particle rotations, hence large peaks inboth � �� in� and � ��out� can be observed, coinciding with the sudden decreases in stress ratio.However, on closer examination of the two types of curvature, it seems that � �� in� correlatesbetter with decreases in stress ratio than � ��out� (e.g. note the absence of a corresponding dropin stress ratio in the peak in � ��out� at ��22� � 0�042). This is to be expected, since it is thein-contacts that carry the force, and thus are related to the macroscopic stress ratio.

5.3.4. Deformation patterns inside the shear band

Recent experiments by Rechenmacher on densely packed sand specimens revealed that highmagnitudes of local shear and rigid body rotation alternate in position along the length of ashear band [54]. This pattern was attributed to the mechanism of alternating build-up and col-lapse of force chains as the dominant mechanism of continuing deformation in shear bands,as suggested earlier by Oda and co-workers [56, 62]. To compare against these experimentsproperly, we note that sand particles are clearly not spherical and thus would exhibit a higherrolling resistance at interparticle contacts than is evident in the current simulation. Hence, toascertain whether our strain measures can reproduce these trends, we use data from another





Figure 10. The magnitude of curvature increment due to the contributions from the branch vectorsbetween particles in, � �� in

i �, and out, � ��outi �, of contact. The stress ratio, sin�, is also shown. Vertical

dashed lines in Stage 3 are a guide to the eye to indicate the start of the major drops in stress ratio.

DEM simulation of the biaxial compression test with a much higher rolling friction valueto achieve greater resistance to relative rotations at contacts. We refer the reader to [50] forfull details of the experimental setup, parameters and material properties used. This systemexhibits a V-shaped shear zone: see Figure 11(d). As in [54], for particles in one of the shearband regions, local shear strains and rigid body rotations � were computed respectivelyas

� �p��21 � � p��

12 � u�1�2 � u�2�1 (79)

� � 1

2��

p��21 � � p��

12 � 2�� 1

2�u�2�1 � u�1�2� (80)

where the starred variables indicate measurements with respect to axes oriented perpendicu-lar and parallel to the shear band (see Figure 11(d)), � p

i j is the particle strain tensor (Equation(36)), ui are particle displacements and � particle rotation. As before, we compute quantitiesover a strain interval where a decrease in stress ratio is observed, just after the peak stress.

We demonstrate using our new local strain measure that the patterns observed by Rechen-macher are reproduced in our simulations: see from Figure 11(a). Specifically, particle clus-ters in rigid body motion, i.e. large � , that are interlaced with regions of intense shear, i.e.large , are evident. Also shown are volumetric strains for the same strain interval: see Fig-ure 11(b). Although these were not discussed in [54], the following observations are relevant.As the global volumetric strain over this interval indicates considerable dilatation (data notshown), we expect and observe more particles within the shear band undergoing dilatationthan contraction. Interestingly, most of the particles that undergo large local dilatation are





Figure 11. (a) Particles in one of the shear band regions, where black particles are those whose localshear strain � 0�02, blue particles are those whose local rigid body rotation � � 0�015, and particlescolored black with a blue center satisfy both conditions. The strain interval used here is just after peakstress ratio. (b) Local particle volumetric strains ��p

vol for the same region and strain interval as in (a).Black particles have registered large dilatation (��p

vol � 0�005), and white particles large contraction(�� p

vol � �0�005). (c) Local particle volumetric strains�� pvol for the same region as in (a) and (b), except

the strain interval is during the persistent shear band regime. Black and white particles depict the sameconditions as in (b).

also those that experience large local shear strain � in contrast, those particles with largerigid body rotation � do not generally register large volume changes (compare spatial dis-tributions of particles in Figure 11(b) with (a)). In other words, these results show that theunidirectional rotation of particle clusters in rigid body motion induce simultaneously thecreation of large voids and intense shear in the regions between such clusters. When a straininterval is chosen within the persistent shear band region that registers no change in global�vol, local particle volumetric strains still vary widely along the band, consistent with Oda’sfindings from DEM simulations and experiments [62, 64].

5.4. A Test of Oda’s Hypothesis

Having shown above that the proposed strain and curvature measures (Equations (36) and(39)) can detect local kinematical patterns consistent with experiments, the key questionnow is: Can the new local measures of strain and curvature elucidate this mechanism, andthereby test the validity of Oda’s hypothesis [56, 64]? We now attempt to answer this ques-tion via a two-pronged analysis. In the first prong, we will use the decompositions in thepreceding sections to create a filter that would isolate the mechanism chiefly responsible formacroscopic strain-softening under dilatation in specimens undergoing shear banding. Fromexperiments, this mechanism was observed by Oda and co-workers to be restricted to theshear band and governed by relatively large particle rotations. These findings led Oda to the





well-known hypothesis that the mechanism in question was that of the confined buckling offorce chains [56, 64]. For the second prong of the analysis, we are to construct a new decom-position which would distinguish the contributions from branch vectors between particlesin the set of buckled force chains and their laterally supporting neighbors (later defined asCBFC particles), and the contributions from the complementary set (later denoted as non-CBFC particles), using the method of decompositions employed earlier for other modes ofcontact.

5.4.1. The first prong

The objectives are: (i) to construct a filter that would isolate particles which undergo strain-softening while sustaining significant levels of dilatation and relative rotations� and (ii) todetermine whether or not these particles are in fact buckling force chains and their later-ally confining neighbors. The filter will comprise of a combination of appropriate strainand curvature measures developed earlier in this paper. As the focus is on the process ofstrain-softening, the filter is to be applied during distinct drops in the stress ratio, or potentialenergy. These correspond to the strain intervals when confined buckling of force chains is ex-pected to prevail. To demonstrate, consider the strain interval ��22� � 0�038 to ��22� � 0�040,which corresponds to the largest drop in stress ratio (see Figure 4(b)). Coinciding with thisevent, we observe: large energy dissipation (evident in the peak in N inp

�N in in Figure 8(g)�see also [41]), large dilatation (see Figure 7(a)), and large curvature (see Figure 10).

In accordance with Oda’s experimental and DEM findings, we consider a measure ofvolumetric strain to identify particles that have undergone large dilatation, and a measureof curvature to identify those groups of particles with large relative rotations. In choosingan appropriate decomposition for the volumetric strain, we recall from Figure 7(a) that itis �in

vol that is responsible for most of the dilatation in �vol during the shear banding regime(i.e. Stage 3). Since we wish to focus on force chains, we choose that decomposition ofthe in-contact strain that is due to the strong network, i.e. �insCF

vol � Figure 7(c) shows thatthis registers dilatation during decreases in stress ratio. For the curvature decomposition,we note the following relevant observations from results earlier. Since force chains do notnecessarily buckle in one direction only, the appropriate decomposition measure is that of acurvature magnitude (recall Figures 9 and 10) instead of the vector components �1 and �2.We also recall observations from Figure 10 that the magnitude of the in-contact curvaturemore closely correlates with decreases in stress ratio, hence this measure forms a sensiblechoice of decomposition for the curvature.

In summary, the filter, Filter(�,�), comprises of the following steps: (1) at the start ofthe drop in stress ratio, eliminate all particles except those that belong to force chains and lieinside the shear band� (2) eliminate all particles that have not decreased in potential energyduring the drop in stress ratio� (3) eliminate those particles that have sustained an increasein ��insCF

vol and an increase in �� in� below a given threshold � and �, respectively, duringthe drop in stress ratio. Steps 1–2 ensure that only those particles that have undergone strainsoftening inside the shear band region during macroscopic softening are selected. Out ofthese, Step 3 then selects those particles that have undergone dilatation and curvature abovesome given threshold value.





Figure 12. The percentage of the filtered set, Filter(�� ), that are also part of buckling force chains (theset BFC), plotted against both � and �, the threshold values for ��insCF

vol and �� in� respectively.

To ascertain whether this group of particles corresponds to buckling force chains, we turnto an algorithm developed in [41], that identifies all three-particle buckled segments of forcechains—we call this set BFC. The reader is referred to [41] for a full discussion. In Figure 12,we plot the percentage of the filtered set Filter(�,�) that are also part of buckling force chains(i.e. the set BFC), as a function of both � and �. As either � or � (or both) are increased, thefilter captures those particles that have sustained increasing levels of dilatation and relativerotation, respectively. Further, as � or � (or both) are increased, the filter captures moreand more of the particles in the buckling force chains. This confirms that Oda’s hypothesisis indeed correct for this test. For the largest values of � and � plotted here, we note thatapproximately 20% of particles picked up by the filter are not part of the set BFC. On closerinspection of the data, these particles are either (i) at the ends of force chains, or (ii) at branchpoints, where two or more force chains meet (data not shown).

5.4.2. The second prong

Here we wish to investigate the kinematics of the set BFC, i.e. the set of all particles inbuckling force chains. In order to take into account the effect of the weak network particlesproviding lateral support to these particles, we define a new set, CBFC. This set consists ofthe set BFC plus all first ring neighbors of all particles in this set, obtained via the modified





Figure 13. (a) The total volumetric strain, �vol, decomposed into the contributions from the branch vectorsthat are between CBFC particles, �CBFC

vol , and those that are not from CBFC particles, �nonCBFCvol . (b)

Evolution of the ratio of CBFC branch vectors to the total number of branch vectors, NCBFC

N .

Delaunay triangulation: note these neighbors can be either in-contact or out-of-contact. Wenow consider a new decomposition of the local strain/curvature: that part of the local strainand curvature measures that is due to branch vectors between CBFC particles, �CBFC

i j , andthat part that is due to branch vectors between nonCBFC particles, �nonCBFC

i j , such that � pi j �

�CBFCi j � �nonCBFC

i j . Note that this decomposition is similar to that of the in-contact and out-of-contact in Section 4.1, except that in-contact branch vectors are replaced with CBFC branchvectors, and out-of-contact branch vectors with nonCBFC branch vectors. Further, in orderto investigate strain evolution of this decomposition, we need to know which particles arein the set CBFC at each timestep. Hence, we apply the algorithm for finding buckling forcechains in [41] over each individual timestep. Figure 13(a) shows the strain evolution of thevolumetric strain due to CBFC branch vectors, and that due to nonCBFC branch vectors.Prior to Stage 3, virtually all the volumetric strain is due to nonCBFC branch vectors, as theproportion of branch vectors that are between CBFC particles is near zero (Figure 13(b)).There are a few buckling force chains that are identified just before peak stress ratio inStage 2, at the onset of shear banding, but these register only small amounts of dilation.Remarkably, during Stage 3, half of the dilatation is due to CBFC branch vectors, and hencebuckling force chains, but the proportion of branch vectors that are between CBFC particlesnever exceeds 19%, and in fact mostly hovers between 0 and 10%. This lends further supportto Oda’s hypothesis that dilatation in shear banding is due to buckling force chains.

In a similar fashion to Figure 13, we show in Figure 14(a) the curvature magnitude that isdue to CBFC branch vectors, and that due to nonCBFC branch vectors. Figure 14(b) showsthe evolution of the ratio of these two curvature magnitudes, as well as the ratio of CBFCbranch vectors N CBFC to nonCBFC branch vectors N nonCBFC� the latter indicates the numbersof buckling force chains. During Stage 1, curvature is solely due to nonCBFC branch vectors(e.g. rattler branch vectors). During Stage 2, the ratio of the curvatures begins to increase justbefore the peak stress ratio as a few buckling force chains emerge� this corresponds to theonset of shear banding as was shown in [41]. These buckling force chains were identifiedas trigger events that cause the subsequent strain softening first observed inside the shear





Figure 14. (a) The magnitude of curvature due to the contributions from the branch vectors from CBFCparticles, ��CBFC

i �, and those that are not from CBFC particles, ��nonCBFCi �. (b) The evolution of the ratio of

CBFC branch vectors to nonCBFC branch vectors, NCBFC

NnonCBFC , and the ratio of the magnitude of curvature

due to CBFC branch vectors to nonCBFC branch vectors, ��CBFC��nonCBFC� .

band and elastic unloading of the two regions on either side of the band. During Stage 3, theCBFC contacts become the predominant contributor to curvature with the ratio ��CBFC�

��nonCBFC� risingto around 80%, while the ratio of CBFC branch vectors to nonCBFC branch vectors remainsbelow 25%. Therefore, despite CBFC branch vectors (buckling force chains) constituting arelatively small number of branch vectors (particles), they register disproportionately largeamounts of relative rotation, again adding credence to Oda’s hypothesis that force chainbuckling is tied to large relative rotations.

6. CONCLUSION

One of the core advantages of DEM over continuum models lies in the treatment of particlekinematics. As DEM models deal explicitly with the motion of each particle, determinedfrom its interactions with neighboring particles, particle kinematics arises from the analysisas an emergent behavior. By contrast, in continuum models, particle kinematics enters theformulation via the definition for strain—a predefined constraint or limitation on the degreesof freedom of the system that is imposed at the outset of the analysis. Not surprisingly, failureto account for the influential kinematics in the definition for strain has serious implicationswith respect to the model’s predictive capabilities.

By imposing an energy equivalence between the discrete assembly and that of the equiv-alent continuum, a new definition of micropolar strain and curvature for granular media hasbeen derived that has advantages over existing formulations: it accounts for particle rotations,is relatively straightforward to calculate, and its global average matches the macroscopicstrain of the assembly. Because the deformation of granular materials is generally nonaffine,the definition gives a patchwork strain and curvature field more akin to that defined by a





finite element mesh than a smooth continuum. Consequently, the strain and curvature arenot defined uniquely along branch vectors and at particle centers.

We have resolved this problem for a particle cluster comprising of a particle and its firstring of neighbors, in two dimensions, by introducing an enriched projection scheme. Thisscheme employs multiple strain tensors, each defined at the particle center, that are capableof describing the underlying particle motions to greater accuracy than models that rely ona single strain and curvature tensor. The various strain and curvature decompositions rep-resent the contributions from relative motions with respect to: in-contact and out-of-contactneighbors, the normal versus tangential directions, contacts with above and below averagenormal contact force, contacts with above and below average contact moment, and elasticversus plastic contacts. To gain a better understanding of local particle kinematics and howit governs local and global deformation, we have applied these decompositions to data fromDEM simulations of dense granular assemblies under biaxial compression. Whenever pos-sible, results were discussed in the light of past experimental findings, in particular, thosewhich focussed on specimens undergoing strain localization including shear bands. Resultswere found to be in agreement with past experimental findings on shear banding.

A key outcome of this investigation is the test on Oda’s hypothesis via a two-prongedanalysis. In the first prong of the analysis, the decompositions were used to create a filterthat would capture those particles which are chiefly responsible for strain-softening underdilatation. The great majority of the particles captured turned out to be those in CBFCs,i.e. the set of particles in buckled force chains and their laterally supporting neighbors.This method can be used to create other filters designed to identify specific local kinematicevents. For the second prong, we constructed a new decomposition which distinguished thecontributions from branch vectors between particles in CBFCs and those from the comple-mentary set of particles in nonCBFCs, using the method of decompositions employed earlierfor other modes of contact. The results of this study showed that the CBFCs had a dispro-portionately large impact on local and global rheology. Altogether, the findings from thistwo-pronged analysis provide further evidence of the validity of Oda’s hypothesis that forcechain buckling—a nonaffine mode of deformation—is indeed the governing mechanism forshear banding. In conclusion, the importance of nonaffine deformation cannot be ignored incontinuum models of dense granular assemblies, in which shear banding and other forms ofstrain localization are prevalent—a lesson that modelers will need to address in the futuredevelopment of micromechanical continuum theory.

Acknowledgments. We thank undergraduate scholars David Rafferty, Nana Liu, Sarah Traine and Christopher Hallalfor their assistance in the preparation of this paper. The support of the US Army Research Off ice (grant numberDAAD19-02-1-0216) and the Australian Research Council (DP0558808 and DP0772409) through a grant to AT isgratefully acknowledged, as is the University of Melbourne Advanced Research Computing section through a grant ofcomputer time.

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