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Journal of the Mechanics and Physics of Solids

Journal of the Mechanics and Physics of Solids 57 (2009) 706–727

0022-50

doi:10.1

� Cor

E-m

journal homepage: www.elsevier.com/locate/jmps

On the modeling of confined buckling of force chains

Antoinette Tordesillas �, Maya Muthuswamy

Department of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia

a r t i c l e i n f o

Article history:

Received 11 February 2008

Received in revised form

3 November 2008

Accepted 1 January 2009

This paper is dedicated to Professor Robert

P. Behringer on the occasion of his 60th

birfthday.

Keywords:

Force chains

Buckling

Granular materials

Energy methods

Microstructure

96/$ - see front matter & 2009 Elsevier Ltd. A

016/j.jmps.2009.01.005

responding author. Tel.: +613 8344 9685; fax:

ail address: [email protected] (A. T

a b s t r a c t

Buckling of force chains, laterally confined by weak network particles, has long been

held as the underpinning mechanism for key instabilities arising in dense, cohesionless

granular assemblies, e.g. shear banding and slip-stick phenomena. Despite the

demonstrated significance of this mechanism from numerous experimental and discrete

element studies, there is as yet no model for the confined buckling of force chains. We

present herein the first structural mechanical model of this mechanism. Good

agreement is found between model predictions and confined force chain buckling

events in discrete element simulations. A complete parametric analysis is undertaken to

determine the effect of various particle-scale properties on the stability and failure of

force chains. Transparency across scales is achieved, as the mechanisms on the

microscopic and mesoscopic domains, which drive well-known macroscopic trends in

biaxial compression tests, are elucidated.

& 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Today, 300 years on after the birth of Euler, the mechanism of buckling not only remains at the forefront of mechanics,but its relevance has spread beyond the realm of traditional structural mechanics—perhaps none more notable than to theburgeoning areas of micromechanics and nanomechanics (e.g. Chaudhuri et al., 2007; Majmudar and Behringer, 2005). Astechnological advances provide new and more powerful tools to probe materials at ever-diminishing length scales, wholenew worlds of structures are continuing to be discovered. Structures in naturally occurring and man-made materialsabound from the mesoscale to the nanoscale, and a common mode of failure of many of these structures is that of buckling;examples can be found in biological materials, foams, polymers, micellar systems, DNA and carbon nanotubes (e.g. Ji et al.,2004; Gioia et al., 2001; Gibson and Ashby, 1988; Goodman et al., 2005; Falvo et al., 1997).

The focus of this paper is on buckling mechanisms occurring in deforming dense granular materials. Considered as theultimate paradigm of a complex system, granular materials exhibit behavior that has eluded scientists for centuries. Todate, this class of materials, despite its ubiquity in everyday life, possesses no constitutive model of the same level ofreliability as the Navier–Stokes equation for fluids (Duran, 2000). Consequently, systems and processes involving granularmaterials rarely reach 60% of their design performance—a far cry from fluid processing which operates on average at 96%(Knowlton et al., 1994).

Numerous experimental studies, from those by Oda et al. (2004) and references cited therein, to the more recentdevelopments in soil mechanics and physics (Rechenmacher, 2006; Corwin et al., 2005; Majmudar and Behringer, 2005),

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A. Tordesillas, M. Muthuswamy / J. Mech. Phys. Solids 57 (2009) 706–727 707

suggest that a missing piece of the puzzle of constitutive theory for granular materials lies in the manner of forcetransmission and associated kinematics. Specifically, in a deforming granular material, instabilities emerge at multiplelength scales, and a prevalent source of instability is known to be the buckling of so-called force chains (e.g. Tordesillaset al., 2009; Rechenmacher, 2006; Oda and Kazama, 1998). Physical experiments have shown that force chains—

quasi-linear, chain-like particle groups through which above average contact forces are transmitted—form to resistdeformation, and that these ‘‘columnar structures’’ of particles preferentially align in the direction of maximumcompressive stress in the system. Under continued axial compression, force chain columns ultimately buckle. Mountingevidence suggests that this event is the underlying mechanism for shear banding and slip in slip-stick phenomena (e.g. Odaand Kazama, 1998; Rechenmacher, 2006; Thornton and Zhang, 2006; Aharonov and Sparks, 2004; Alonso-Marroquin et al.,2006).

Although extensive experimental and theoretical effort has been devoted to the study of force chains (e.g.Rechenmacher, 2006; Majmudar and Behringer, 2005; Oda et al., 2004; Aharonov and Sparks, 2004; Cates et al., 1998;Radjai et al., 1998), a proper analysis of the structural stability and associated kinematics of the process of force chain failure

by buckling is still lacking. This is surprising given the fundamental role of force chains in force transmission, energy storageand macroscopic strength. More recently, the role of force chain failure as an underpinning mechanism for energydissipation and macroscopic failure, has been assessed in the context of constitutive theory for dense, cohesionlessgranular systems, in the absence of particle damage (Tordesillas, 2007a, b; Walsh et al., 2007; Tordesillas and Walsh, 2005).For these systems, the two dissipative mechanisms that generally operate under quasi-static loading conditions arefrictional slip and buckling of force chains. Constitutive models which account only for plastic slip over-predict stability,and fail to capture the defining behavior of these materials: i.e. strain-softening under dilatation (see Tordesillas, 2007a;Walsh et al., 2007; Chang and Hicher, 2004 and references cited therein). Specifically, although these models predict loss ofcontacts in the direction of extension in a biaxial test, the normal contact force continues to grow in the direction of mostcompressive principal stress. In other words, the amount of energy dissipated through slip at the contacts is much less thanthe stored energy that is accumulated from the steady growth of the normal contact forces as loading proceeds.Consequently, the stress ratio increases monotonically, even in the presence of plastic slip and loss of contacts in thedirection of extension. Viewed from the standpoint of the force chain network, this result signifies that the force chains,which initially align themselves with the major principal stress axis, continue to sustain a steady increase in load evenunder continuing loss of lateral supporting contacts. What essential physics then is missing in these models? The answer istwofold. First, there is nothing in the formalism of plastic slip that limits the buildup of the normal forces at the contacts.Second, slip only limits the tangential force at the contacts but in the absence of softening. To resolve this, one might betempted to adopt a more sophisticated contact law that allows for plastic softening in the normal contact force. However,this would not necessarily resolve the problem, unless the softening is tied to dilatation—in which case, one is thenconfronted with a dissipative mechanism that is beyond the particle scale. The mesoscopic mechanism of force chainbuckling is one event in which plastic softening seems inextricably linked to dilatation. The recent constitutive formulationin Tordesillas and Muthuswamy (2008) and Walsh et al. (2007) presents one approach to account for this mechanism viathe kinematics—specifically, from the standpoint of nonaffine deformation. A rigorous analysis of both the statics andkinematics of this event is thus key to the advancement of constitutive theory for granular materials—and to bridging thegap between fundamental advances in the physics of force chains and the development of constitutive models that areof core importance to a broad range of applied settings: geomechanics, in which such models are used in mining,exploration and construction; chemical engineering for control of handling and processing of powders, pharmaceuticalsand food products; and agriculture for design of cutting tools, off-road vehicles and machines—to name a few examples(Duran, 2000).

In an attempt to fill this gap in the current state of knowledge in the physics and mechanics of granular media, thisstudy seeks to establish the first structural mechanical model of confined force chain buckling. Model predictions are to bevalidated against data drawn from an analysis of local buckling events from two-dimensional DEM simulations. Next, weundertake two further investigations. First, we use this model to: demonstrate the influence of various resistances to forcechain buckling and their interplay on the progression of buckling; and elucidate the mesoscopic origin of strain-softeningunder dilatation (Rechenmacher, 2006), and other well-established trends from soil mechanics (e.g. effects of confiningpressure and particle interlocking on macroscopic shear strength, as recently underlined by Kuhn and Chang, 2006).Second, we test the veracity of conclusions drawn from earlier studies on the importance of confined force chain bucklingin constitutive development (Tordesillas, 2007a, b; Walsh et al., 2007). The first investigation is performed in this paper.The second is the subject of a companion paper (Tordesillas and Muthuswamy, 2008). Therein, we employ the bucklingmodel to derive a thermomicromechanical constitutive law and assess this law’s predictive capabilities for strain-softeningunder dilatation and shear banding. The findings in this companion paper will be summarized in a later section.

In preparation for the development of the buckling model, a series of preliminary studies on the evolution of forcechains and its implications for constitutive modeling have been undertaken. These studies were aimed at the quantitativecharacterization of force chain particles and their supporting weak network neighbors, their associated kinematics andstability, and their role in unjamming transitions and shear banding (Tordesillas et al., 2008a, b, 2004; Tordesillas, 2007a;Muthuswamy and Tordesillas, 2006; Peters et al., 2005). By unjamming transition, we refer to the transition from solid-liketo fluid-like behavior, characterized by a decrease in macroscopic stress (also known as the slip phase in stick-slipphenomena, e.g. Aharonov and Sparks, 2004; Alonso-Marroquin et al., 2006). In the first of these preliminary studies, we

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Step 3Step 2Step 1

Fig. 1. Snapshots from DEM simulations showing result of applying successive filters in the algorithm for determining buckled force chain segments over

a single unjamming transition. (a)–(c) Biaxial compression test with 5098 particles. (d)–(f) Rigid flat punch indentation test with 38 849 particles

(zoomed in region shown). (a) and (d) All particles in force chains. (b) and (e) All particles that have decreased in potential energy, out of (a) and (d),

respectively. (c) and (f) All particles in buckled segments, out of (b) and (e), respectively.

A. Tordesillas, M. Muthuswamy / J. Mech. Phys. Solids 57 (2009) 706–727708

developed an algorithm to distinguish force chain particles from the complement set termed the weak network particles(Peters et al., 2005). Using discrete element analysis, we then performed a detailed analysis of buckling events insimulations of two-dimensional, densely packed, cohesionless granular assembly subject to quasi-static, boundary drivenbiaxial compression (Tordesillas, 2007a). This involved the development of another algorithm, aimed at identifying parts ofthe force chain particle network that have undergone buckling, i.e. buckled force chain segments. A strain interval ½�A; �B� ischosen, usually an unjamming transition, and a set of three filters applied, as shown in Fig. 1: (a) eliminate all particles notin force chains at �A; (b) out of those remaining, eliminate those which have not decreased in potential energy; (c) out ofthose remaining, identify all three-particle segments which have buckled. It was shown that these buckled force chainsegments are a key mechanism responsible for nonaffine modes of deformation, dilatation and energy dissipation(Tordesillas et al., 2008a, 2009; Tordesillas, 2007a). Moreover, results confirm the well-known hypothesis of Oda and co-workers that these buckling events are confined to and represent the governing mechanism behind the formation of shearbands. The implications of these findings for continuum modeling are significant, since these formulations are generallybased on the assumption of affine deformation and thus ignore this important source of instability. Indeed, recent studieshave shown that this assumption, which ignores the inherently nonaffine mode of deformation of force chain buckling, isthe main reason why such models fail to reproduce the defining behavior of densely packed granular systems, e.g. strain-softening under dilatation and shear banding (Walsh et al., 2007; Agnolin et al., 2006; Tordesillas and Walsh, 2005; Changand Hicher, 2004).

In summary, the specific problem to be addressed in this paper is the development of a two-dimensional analyticalmodel for the elastic and plastic buckling behavior of a force chain, under an applied axial compression, withlateral support from weak network particles. Several system properties will be drawn from this model, including: bucklingmodes, progression of plastic contacts during buckling, evolution of the load-carrying capacity of the force chain,and evolution of particle kinematics in the cluster comprising the force chain and its supporting weak network neighbors.To achieve this, we proceed in two steps. In the first step, presented in Section 2, we conceive the physical modelfrom a suitable choice of initial configuration, degrees of freedom (DOFs), boundary conditions, and contact models, inaccordance with discrete element method (DEM) simulations. The second step, presented in Section 3, involves thedevelopment of the mathematical model for buckling, using structural stability theory for elastic buckling, andthermodynamical principles for plastic buckling. Next, the kinematics of confined force chain buckling and its implicationsfor constitutive modeling are addressed in Section 4. Input parameters and calibration of the model with DEM simulationsare discussed in Section 5. Results from model validation and application are presented in Section 6. Conclusions are drawnin Section 7.

2. Development of the physical model

The aim of this section is to establish the physical model of force chain buckling under lateral support, accounting forthe effect of particle shape irregularities. Here key results from a recent analysis of buckling events in DEM simulations andphotoelastic disk experiments (Tordesillas, 2007a; Tordesillas et al., 2009), as well as past observations on force chainevolution (Taboada et al., 2005; Aharonov and Sparks, 2004; Radjai et al., 1998), are used to guide model development. Inwhat follows, particular attention is paid to the initial configuration, DOFs, boundary conditions, and the contact laws thatgovern particle interactions.

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N

α

α+1

N

Δ F

θ(α)

1

α

α+1

1

ks

μs

ks

μs

Undeformed DeformedNormal

Rotational

Δn(�)knα+1

αTangential

k tΔ t(α)α+1

α

(α+1)

(α)

Enlargedviews invicinity ofparticles α

Δ s(α)

Δ s(α+1)

kr

μr

α+1

α

(α+1)

(α)

and α+1

FC

x2

x1

Fig. 2. Boundary conditions, degrees of freedom, and contact models for an N-particle force chain: (a) prior to buckling; (b) after buckling; (c) enlarged

views of (b).


2.1. Initial configuration, DOFs and boundary conditions

As with any structural mechanics analysis, the initial configuration of the structure under study must be properlydefined. Given the vast number of possible initial force chain configurations, a judicious choice must be made to achieveprogress in the analysis. Thus, to ensure the formulation is sufficiently broad in scope, we consider two initialconfigurations comprising (i) a perfect force chain column, and (ii) an imperfect force chain column. In the analysispresented here and in the next section, the initial configuration involves a perfectly straight columnar structure, consistingof N deformable particles of radius R, as illustrated in Fig. 2. Particle shape irregularities, common in real granularassemblies, will be accounted for later in the analysis via the contact model. Initially, the particles are assumed to be inpoint contact, i.e. deformation at contacts is zero. This initial configuration leads to an analysis of the complete bucklingprocess, and provides an upper bound on the strength and stability of the column. A second initial configuration, whichembodies a force chain with an imperfection or misalignment, is considered in the Appendix. This configuration is commonin real systems and DEM simulations, and results in a force chain column with a weaker load-carrying capacity, as shown ina later section.

Numerical experiments have revealed that lateral support is provided to force chains by particles which belong to theweak network (Taboada et al., 2005; Aharonov and Sparks, 2004; Radjai et al., 1998). Thus, we model their contribution bymeans of forces acting laterally on the N � 2 central particles. Consider the system to be loaded under a controlled verticaldisplacement D, leading to a reaction force F which forms part of the model output. As shown in the figure, the boundaryconditions are such that: (a) all particles may rotate, wð1Þ;wð2Þ; . . . ;wðNÞ, where positive rotation is anticlockwise; (b) all butparticle 1 may translate in the x2 direction; and (c) all but particles 1 and N may translate in the x1 direction. Thus, thedeformation of the system of N particles can be characterized by, at most, 3N � 3 independent DOFs. These boundaryconditions are akin to a pin-ended solid column subject to an axial load, which lead to specific buckling modes (Bazant andCedolin, 2003).

There are a number of ways that the 2N � 3 independent translational DOFs for the system can be specified, such thatthe kinematics of the system is completely defined. However, in preparation for the treatment of contact interactionsbetween the particles, it is more convenient to express these in terms of relative normal contact displacements andbuckling angles: Dnð1Þ;Dnð2Þ; . . . ;DnðN�1Þ where DnðaÞ is the relative normal displacement of the contact between particles aand aþ 1; and yð1Þ;yð2Þ; . . . ; yðN�1Þ where yðaÞ is the angle representing the deflection of particle aþ 1 from the vertical,measured with respect to the center of particle a, i.e. the ‘‘buckling angle’’. Note that the relation sin yð1Þ þ sin yð2Þ þ � � � þsin yðN�1Þ

¼ 0 must hold from geometrical considerations. This results in the elimination of yðN�1Þ and hence one less DOF.While the above DOFs are sufficient and completely define the system, we nonetheless introduce additional relative

displacements that depend on the above DOFs in order to express the contact laws in a more accessible form:Dtð1Þ;Dtð2Þ; . . . ;DtðN�1Þ where DtðaÞ is the relative horizontal displacement of the contact between particles a and aþ 1, i.e.,the displacement of the contact from particle aþ 1’s reference frame; Dsð2Þ;Dsð3Þ; . . . ;DsðN�1Þ where DsðaÞ is the relativedisplacement of the lateral supporting spring-slider acting on particle a. These quantities are interim variables, and will beeliminated in Section 3.1 when the buckling analysis is performed. They are introduced simply to make the exposition ofthe fundamental hypotheses more transparent.

All output quantities, predicted by the model, can be expressed in terms of the control variable D, or, the buckling anglesyðaÞ for a specific buckling mode of the N-particle force chain. Later, for the system with N ¼ 3, in which there is only one

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buckling mode possible, the model’s output quantities can be expressed with respect to one internal kinematicvariable—the buckling angle.

2.2. Contact models

To establish an appropriate set of contact laws for the buckling model, we use results from a local analysis of bucklingevents in two-dimensional, DEM simulations (Tordesillas and Muthuswamy, 2008; Tordesillas et al., 2008a, b; Tordesillas,2007a, b) and an experiment involving photoelastic disks (Tordesillas et al., 2009). Accordingly, in what follows, we firstdiscuss the key findings from these studies, focussing on: the nature of interactions at contacts within force chains andtheir laterally supporting neighbors, the key resistances to buckling and the interplay between these resistances. Ourultimate objective here is to keep the contact laws in the buckling model as simple as possible, but sufficiently robust tocapture the essential interactions and resistances observed in these buckling events.

One contact mechanism that has been emphasized to play a key role in microstructural evolution inside shear bands,especially in force chain buckling, is the resistance to relative particle rotations at contacts (e.g. Iwashita and Oda, 2000;Bardet, 1994). This resistance is due to particle interlocking, an effect borne out of particle shape irregularities inherent inreal granular materials. To take this effect into account but still retain the simplest particle geometry, Sakaguchi et al.(1993) and Iwashita and Oda (2000) introduced a contact moment, the so-called rolling resistance, to their DEM simulationsof idealized assemblies comprising perfectly round particles. It is important to keep in mind that, in these simulationsinvolving near-rigid circular particles, the actual contact moment is negligible owing to the contact zone being very small.Thus the contact moment introduced in these models is merely an artificial, albeit simple, means of capturing the effect ofparticle shape irregularities. In our DEM simulations (Tordesillas and Muthuswamy, 2008; Tordesillas et al., 2008a,b, 2009;Tordesillas, 2007a, b), the contact moment is regarded identically as an effective property by which resistance to relativerotation at contacts—irrespective of its source—can be accounted for in the mathematical formulation, withoutexplicit representation of particle shape irregularities. Consequently, apart from the contact forces between the particles inthe force chain column, there are essentially two main resistances to buckling events arising in these simulations: thecontact moment between the force chain particles, and the lateral support to the force chain from the surrounding weaknetwork particles.

Past results from DEM simulations suggest a synergetic interplay between the various resistances to buckling(Tordesillas and Muthuswamy, 2008; Tordesillas, 2007a). Accordingly, we probed the nature of this interplay in threedifferent DEM simulations. The results are presented in Table 1. Here we analyzed three different systems, differing only inthe coefficient of rolling friction, as shown in table rows. Using the algorithm for identifying buckling force chains(Tordesillas, 2007a), we classified buckled force chain segments according to the behavior of the aforementionedresistances during unjamming transitions. The data in Table 1 represent averaged values, taken over the first threeunjamming transitions. Note that the trends uncovered here are representative of those found in subsequent unjammingtransitions. We identified four classes of buckled force chain segments, as shown in table columns. These are distinguishedaccording to the modes of contact and the progression of plastic contacts realized in the buckling process. Class 1comprises buckled force chain segments in which the contact moment and laterally supporting contacts remained elasticthroughout unjamming. Here the laterally supporting contacts encompass those contacts with and between the laterallysupporting weak network of particles. Force chain segments in classes 2–4 exhibited elastic–plastic buckling duringunjamming. Class 2 comprises force chain segments in which a contact moment along the force chain and a laterallysupporting contact reached plastic threshold at the same time. Classes 3 and 4 comprise force chain segments in which thefirst contact to reach its plastic threshold belongs, respectively, to the laterally supporting contacts and to the contactmoment along the force chain.

Within the force chain column, relative rotations dominate contact kinematics. Hence, the contact moment is the keysource of buckling resistance within force chains. Specifically, for all three systems summarized in Table 1, less than 5% offorce chain segments buckled due to sliding at contacts, i.e. the tangential force reaching its plastic threshold. This is inaccordance with earlier observations that sliding between particles is rarely realized in contacts carrying above averagecontact forces (Radjai et al., 1998). The predominant mode of plastic contact developed along force chains, in buckling

Table 1Percentages of confined buckled segments for four classes, distinguished according to plastic behavior of contacts with weak network particles (WN) and

contact moment within force chain (CM): (1) neither WN nor CM reaches plastic threshold, (2) WN and CM reach plastic threshold at the same time, (3)

WN reaches plastic threshold before CM, (4) WN reaches plastic threshold after CM.

mr (1) Both elastic (%) (2) WN same as CM (%) (3) WN before CM (%) (4) WN after CM (%)

0:2 7.1 14.8 44.1 34.0

0.05 1.2 39.2 21.7 37.9

0.02 0 52.7 18.8 28.5

Percentages, shown for three DEM simulations of boundary driven biaxial compression, differing only in the contact moment rolling friction coefficient mr

(m ¼ 0:7), represent averaged values over the first three unjamming transitions.

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segments or otherwise, is via the contact moment reaching its plastic threshold. The same was found in Tordesillas andMuthuswamy (2008) and Tordesillas et al. (2009). As shown in the combined sum of those in classes 2 and 4 of Table 1, theforce chain segments in which the contact moment reached its plastic threshold before and/or at the same time as thelateral support during unjamming consistently represent the majority, despite an order of magnitude increase in rollingfriction. These results are indicative of the governing influence of the contact moment in the buckling of force chaincolumns, and thus corroborate earlier observations of Oda and co-workers (Oda et al., 2004; Oda and Iwashita, 2000;Iwashita and Oda, 2000, 1998).

We now pay attention to the resistance to force chain buckling from lateral support. The weak network contacts arethose that provide lateral support and stability to force chains (e.g. Aharonov and Sparks, 2004; Radjai et al., 1998).Consequently, even the slightest rearrangements via sliding and rolling in such contacts may induce buckling of forcechains. We examined these important interactions in DEM simulations and confirm this finding. In Table 1, we show thepercentages of force chain segments in class 3 in which the plastic threshold was first reached at a contact with thelaterally supporting weak network neighbors. For high rolling friction, this class dominates. Thus, in essence, lateralconfinement of force chains from weak network particles is governed by the Coulomb frictional limit for the tangentialforce as well as the analogous limit for the contact moment. These limiting values depend on the normal forces actingat these contacts. By the definition of a weak network contact (Radjai et al., 1998), the upper bound on the normal forceat these contacts is given by the global average normal contact force which, in turn, is proportional to the applied confiningpressure.

In what follows, we employ a set of contact laws that capture the key resistances described above. The interactionsbetween the particles in our physical model of confined force chain buckling are assumed to be governed by a combinationof spring and spring-slider contact models, in common with our DEM simulations and those by others (e.g. Kruyt andAntony, 2007; Luding, 2004). In particular, the resistance to relative rotations between particles due to particle interlockingis similarly accounted for in the model, via the introduction of a contact moment.

2.2.1. Contact force between force chain particles

The normal and tangential forces between particles in the force chain are, respectively, given by

f nðaÞ¼ knDnðaÞ; f tðaÞ

¼ ktFCD

tðaÞ, (1)

where kn; ktFC are spring stiffnesses. For the range of buckling angles considered here, the relative displacement in the

horizontal direction, DtðaÞ, is a good approximation to the relative tangential contact displacement. The absence of a sliderin the tangential contact force model is in accordance with DEM simulations which show a negligibly small number ofsliding contacts developed between force chain particles, as discussed above.

2.2.2. Contact moment between force chain particles

In common with our DEM simulations, we use a spring-slider model for the contact moment mðaÞ along the force chain,with a plastic threshold that is analogous to Coulomb’s law:

jmðaÞj ¼krR2jwðaþ1Þ �wðaÞj; krR2

jwðaþ1Þ �wðaÞjomrRf nðaÞ;

mrRf nðaÞ; krR2jwðaþ1Þ �wðaÞjXmrRf nðaÞ;

((2)

where kr is the rotational spring stiffness, mr the coefficient of rolling friction, and R the radius. An important question ofhow the buckling mechanism can occur in the majority of DEM simulations using circular particles with no rollingresistance (e.g. Thornton and Zhang, 2006) will be addressed later in Section 6.3.

2.2.3. Lateral supporting force from weak network particles

Here, we model all possible sources of support from the surrounding weak network particles to the force chain as oneeffective lateral supporting force, f sðaÞ. These sources encompass the applied confining pressure, normal forces, tangentialforces and moments operating at contacts between weak network particles and between weak network and force chainparticles. The force f sðaÞ acts in the x1 direction, and is governed by a spring-slider contact model:

f sðaÞ¼

ksDsðaÞ; jksDsðaÞjoRmsss;

Rmsss; jksDsðaÞjXRmsss;

((3)

where ks is the spring stiffness, ms the friction coefficient for side support, and ss a maximum stress that the lateralsupporting particles can exert. We consider ss as a fixed parameter; henceforth, msss is taken as a single parameter.Although sliding and rolling friction at contacts do not appear explicitly in the above formulation, these clearly contributeto the effective plastic threshold Rmsss. The explicit influence on force chain buckling of the particular details of thisconfinement, viz. the distribution of the lateral supporting forces along the sides of the force chain and of the sliding androlling friction at the supporting contacts, is the subject of ongoing studies.

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3. Mathematical model of confined force chain buckling

The aim of this section is to present a mathematical model of confined force chain buckling, by employing structuralstability theory for elastic behavior, and thermodynamical principles for plastic behavior. Structural stability theory hasbeen applied to simple rigid-link mechanical models (e.g. Bazant and Cedolin, 2003), and to N-link systems in the contextof buckling of highly oriented polymer fibres in kink band formation (DeTeresa et al., 1985). However, several keydifferences exist between our approach and those adopted in these models. The model developed here comprises discreteparticles with full DOFs. On the other hand, multi-link polymer chain models have connectivity which constrains themotion of the individual links. Thus, in the modeling of evolving particulate force chain structures, the key challengeresides in the many DOFs that must be captured during the course of the deformation. In particular, it is now wellestablished that rotations hold the key to the rich complexity that emerges within these materials, including that of forcechain buckling (Bardet, 1994; Kuhn, 1999; Kuhn and Bagi, 2002; Oda and Iwashita, 2000; Oda et al., 2004). This aspectraises certain unique challenges in the modeling that have not been addressed in the aforementioned N-link models, sincethese are confined to only one or two translational DOFs for each link.

Our analysis also differs from much of traditional structural mechanics in that the focus is not just on the elastic bucklingprocess but also—and more importantly—the plastic buckling regime. For this, we adopt the thermodynamical approach thatwas previously employed by one of us in elastoplastic constitutive formulations (Tordesillas and Walsh, 2005; Walsh et al.,2007). As underlined by Bazant and Cedolin (2003), the use of thermodynamics has generally been limited to constitutiveformulations, while the thermodynamical aspects of structural stability have not received proper attention. Indeed, a thoroughexposition of thermodynamical concepts for structural stability studies can be found in their book. We emphasize, however,that the applications they focus on are confined to solid as opposed to discrete structures. Moreover, the techniques used bythem and by DeTeresa et al. (1985) are different from the particular thermodynamic treatment employed in this study.

Traditional studies of buckling do not generally concern themselves with the plastic rearrangements of constituentstructural elements (e.g. Croll and Walker, 1972; DeTeresa et al., 1985; Hunt et al., 1997; Bazant and Cedolin, 2003). Incontrast, one of the main objectives of the present effort is the derivation of kinematical relations for the plastic motions ofparticles in the cluster comprising the force chain column and its laterally supporting neighbors. These relations are vitalfor micromechanical constitutive modeling, as discussed further in Section 4. Plastic deformation from frictional particlerearrangements dominates rheological behavior of granular systems. Dissipation of energy is strongly correlated to thenonaffine deformation; numerical and experimental results show that these become significant during force chain buckling(e.g. Tordesillas and Muthuswamy, 2008; Tordesillas et al., 2009; Tordesillas, 2007a). In the plastic buckling regime, bothdissipation and deformation are governed by a complex interplay between the resistances to buckling operating at thevarious contacts between force chains, as well as the contacts with and between laterally supporting neighbors.Understanding this synergetic interplay is crucial to establishing the interconnections in material behavior across lengthscales, from the particle to bulk (Kuhn and Chang, 2006).

In the analysis below, we proceed in two steps. First, we focus on the elastic regime in Section 3.1. Here the formulationis initially employed for an N-particle force chain, and then specialized to the case N ¼ 3 for reasons that will be discussedlater in this section. In the second study in Section 3.2, we focus on the entire elastic–plastic loading history, as well as theelastic unloading regime, using thermodynamical principles.

3.1. Confined elastic buckling of an N-particle force chain

As recently highlighted by Thornton and Zhang (2006), ‘‘the initiation of shear bands is an elastic buckling problem’’.The quantitative examination of force chain buckling in Tordesillas (2007a) supports this. Thus, the aim of this section is touse structural stability theory to develop a model for elastic buckling behavior. The energy criterion of stability is used tocalculate equilibrium paths and buckling behavior for the initial elastic loading regime.

In this regime, neither the contact moment nor the side supporting force has reached their respective Coulomb failurecriteria. Thus, the change in the total potential energy V of the structure after being compressed by a distance D (defining adatum of zero stored energy in the uncompressed, unbuckled state) is given by

V ¼ c�W ¼XN�1

i¼1

1

2knðDnðiÞÞ2þ

1

2kt

FCðDtðiÞÞ2þ

1

2krR2ðwðiþ1Þ �wðiÞÞ2

� �þXN�1

i¼2

½1

2ksðDsðiÞÞ2� � FD, (4)

where c is the stored energy, and W the work done by the axial force F, defining compressive force as positive. Fromgeometrical considerations, DtðaÞ;DsðaÞ;D can be written in terms of yð1Þ; . . . ; yðaÞ;Dnð1Þ; . . . ;DnðN�1Þ;wðaÞ;wðaþ1Þ,

DtðaÞðwðaÞ;wðaþ1Þ; yðaÞÞ ¼ ð2R� DnðaÞ

Þ sin yðaÞ � RðwðaÞ þwðaþ1ÞÞ (5)

� 2R sin yðaÞ � RðwðaÞ þwðaþ1ÞÞ, (6)

DsðaÞðyð1Þ; . . . ; yða�1Þ

Þ ¼Xa�1

k¼1

ð2R� DnðkÞÞ sin yðkÞ � 2R

Xa�1

k¼1

sin yðkÞ, (7)

DðDnð1Þ; . . . ;DnðN�1Þ; yð1Þ; . . . ; yðN�1ÞÞ ¼ 2ðN � 1ÞR� ð2R�Dnð1Þ

Þ cosyð1Þ � � � � � ð2R�DnðN�1ÞÞ cos yðN�1Þ, (8)

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where we have used the fact that DnðaÞ5R; 8a. Note that this assumption also allows the decoupling of the normal and

tangential relative displacements.To calculate the equilibrium paths, we invoke the energy criterion for equilibrium, which requires that the partial derivative

of V with respect to each DOF is zero. One equilibrium path that is always satisfied is when yðaÞ ¼ wðaÞ ¼ 0; a 2 ½1;N � 1�. Here,the structure remains straight and does not buckle. In particular, qV=qDnðaÞ

¼ 0; 8a yields

F ¼ knDnðaÞ; 8a; ‘ F ¼kn

N � 1D. (9)

The reaction force F and its corresponding displacement D result from the top and bottom particles compressing the centralparticles via normal interactions only. For other equilibrium paths, where yðaÞa0, solving the partial derivatives of V for F isconsiderably more complex.

In what follows, we consider the special case of N ¼ 3. We note that the technique outlined below similarly applies forlarger N. We have, for example, undertaken an analysis of N ¼ 4;5 using the same formulation; however, this lies outsidethe scope of this paper.

3.1.1. A three-particle force chain and its lateral support

There are several reasons for considering the particle cluster comprising a three-particle force chain segment and itslaterally supporting neighbors, i.e. the N ¼ 3 model. First, this is the smallest confined force chain structure that couldphysically buckle. Second, the underlying interactions in this particle cluster lend themselves to direct physicalinterpretation, as their relative contributions are still reasonably tractable. Third, an N-particle force chain can be analyzedsegment-by-segment whereby each segment is made up of three contacting particles in the force chain, as was shown inTordesillas (2007a): an N-particle force chain would comprise N � 2 such segments. Fourth, current homogenizationschemes focus on a representative volume element comprising a particle and its first ring of neighbors, which enablepredictions to be made of emergent behavior down to this length scale (Tordesillas and Muthuswamy, 2008; Walsh andTordesillas, 2004; Tordesillas et al., 2004; Tordesillas and Walsh, 2002). Results from this cluster model are used to derivesuitable internal variables and their evolution laws in the companion paper (Tordesillas and Muthuswamy, 2008), asdiscussed in more detail later in Section 4.

The geometrical relation sin yð1Þ þ sin yð2Þ ¼ 0 results in �yð2Þ ¼ yð1Þ ¼ y. Hence, by symmetry, Dnð1Þ¼ Dnð2Þ

¼ Dn. Aspreviously mentioned, output quantities can be expressed in terms of the internal kinematic variable, the buckling angle y,instead of the control variable D. Determining F, Dn;wð1Þ;wð2Þ and wð3Þ, and then using Eq. (8) to obtain D, yields Eq. (10)

FðyÞ ¼Rkn

cos y1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�

2 cos2 yð2ktFCkrþ kskt

FC þ kskrÞ

knðkt

FC þ krÞ

s !, (10)

DðyÞ ¼ 4R� 2R 1þ



FC þ kskrÞ

knðkt

FC þ krÞ

s !cos y, (11)

DnðyÞ ¼ R 1�



FC þ kskrÞ

knðkt

FC þ krÞ

s !, (12)

wð2ÞðyÞ ¼ 0; wð1ÞðyÞ ¼ �wð3ÞðyÞ ¼ wðyÞ ¼2kt

FC sin ykt

FC þ kr . (13)

This is the only buckling mode, and is that which corresponds to the top and bottom particles rotating in oppositedirections while the central one does not rotate and only translates. As in classical stable symmetric bifurcation problems(Bazant and Cedolin, 2003), this second equilibrium path does not emanate from the origin. Rather, this path starts fromforce and displacement values of

Fð0Þ ¼ Fcrit ¼ Rkn 1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�

2ð2ktFCkrþ kskt

FC þ kskrÞ

knðkt

FC þ krÞ

s !, (14)

Dð0Þ ¼ Dcrit ¼ 2R 1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�

2ð2ktFCkrþ kskt

FC þ kskrÞ

knðkt

FC þ krÞ

s !. (15)

These correspond to the bifurcation point or critical buckling load/displacement of a perfectly straight particle column.

3.2. Confined elastic–plastic buckling and unloading regimes

The aim of this section is to extend the N ¼ 3 model in Section 3.1.1 to the plastic regime using the lawsof thermodynamics. The general equations applicable for any regime are first developed in rate form, and then specialized

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for four stages of buckling, thus accounting for the entire loading history of confined elastic–plastic buckling and elasticunloading.

3.2.1. General formulation

In the plastic regime, we can no longer use energy methods, since the total potential energy function V does not exist.For a given displacement D, nonunique loads exist, depending on the deformation history. Hence, it is necessary to turn tothermodynamical principles to account for dissipative behavior. For this, we employ the approach previously taken in theconstruction of an elastoplastic constitutive law (Tordesillas and Walsh, 2005; Walsh et al., 2007).

To model the plastic behavior of the side supporting force and contact moment, wp and tp are introduced, where wp isthe plastic rotation, and tp a scaled form of plastic displacement for the side support (2Rtp is the actual plasticdisplacement). Eqs. (2) and (3) now become

mð1Þ ¼ �mð2Þ ¼krR2jw�wpj; krR2

jw�wpjomrRjfnj;

mrRjfnj; krR2

jw�wpjXmrRjfnj;

((16)

f sð2Þ¼ f s¼

2Rksðsin y� tpÞ; 2Rks

ðsin y� tpÞoRmsss;

Rmsss; 2Rksðsin y� tpÞXRmsss:

((17)

Noting that (a) the wð1Þ;wð2Þ;wð3Þ DOFs have been replaced with the single DOF of w; (b) Dnð1Þ¼ Dnð2Þ

¼ Dn hencef nð1Þ¼ f nð2Þ

¼ f n; and (c) yð1Þ ¼ �yð2Þ ¼ y, due to the symmetry of the only stable buckled mode, known from Section 3.1.From the first law of thermodynamics, the increment of work done on a structure, dW , must equal the change in stored

energy dc plus the energy dissipated D, due to this work

dW ¼ dcþ D. (18)

The increment of work done on the structure is

dW ¼ FdD ¼ FqDqDn dDn

þqDqy

dy� �

¼ 2F cos ydDnþ 2Fð2R� Dn

Þ sin ydy, (19)

where D ¼ DðDnð1Þ;Dnð2Þ; yð1Þ; yð2ÞÞ ¼ DðDn;yÞ (Eq. (8)). The stored energy of the structure in the buckled configuration is

cðDn; y; tp;w;wpÞ ¼ knðDnÞ2þ kt

FCR2ð2 sin y�wÞ2 þ krR2

ðw�wpÞ2þ 2ksR2

ðsin y� tpÞ2. (20)

The energy dissipated during the increment of work is

D ¼

0; dwp ¼ 0; dtp ¼ 0;

2R2msssdtp; dwp ¼ 0; dtpa0;

2mrRf ndwp ¼ 2mrRknDndwp; dwpa0; dtp ¼ 0;

2R2msssdtp þ 2mrRknDndwp; dwpa0; dtpa0;

8>>>><>>>>:

(21)

where the factor of two for dwp arises due to the symmetry of contact moments (Eq. (16)). Substituting Eqs. (19), (20) andthe total differential of Eq. (20) into Eq. (18) yields

2F cos ydDnþ 2Fð2R� Dn

Þ sin ydy ¼ 2knDndDnþ 4R2

½ktFC cos yð2 sin y�wÞ þ ks cos yðsin y� tpÞ�dy

� 4R2ksðsin y� tpÞdtp þ 2R2

½krðw�wpÞ � kt

FCð2 sin y�wÞ�dw� 2krR2ðw�wpÞdwp

þ

0; dwp ¼ 0; dtp ¼ 0;

2R2msss dtp; dwp ¼ 0; dtpa0;

2mrRknDn dwp; dwpa0; dtp ¼ 0;

2R2msss dtp þ 2mrRknDn dwp; dwpa0; dtpa0:

8>>>>><>>>>>:

(22)

We now specialize the above equation to obtain F;D;Dn;w as a function of y for four different stages of buckling, whichreflect the evolution of the side supporting force and contact moment. As previously discussed in Section 2.2, the modes ofcontact and the progression of plastic contacts during buckling is dependent on the values chosen for the parameters:recall Table 1. In what follows we have chosen to show equations for class 3 from Table 1, since we focus on the effect ofstrong grain interlocking between particles that belong to a force chain. Nonetheless, we have also derived the equationsfor class 4. However, for brevity, we forgo the derivations since the steps are identical to class 3 but we do include themodel predictions for this class later in Section 6.3.

Stage 1: This stage of buckling represents the elastic loading regime. Neither the laterally supporting force nor contactmoment have reached their plastic thresholds. Note that we recover the equations previously derived in Section 3.1. Wedenote the period prior to buckling as Stage 1a and the period of elastic buckling as Stage 1b.

Stage 2: This first stage of plastic loading commences when the lateral supporting force reaches its plastic threshold,whilst the contact moment remains elastic.

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Stage 3: This second stage of plastic loading is when both the lateral supporting force and contact moment have reachedtheir respective Coulomb thresholds.

Stage 4: This unloading stage is when the side supporting force as well as the contact moment return to elastic behavior.

3.2.2. Stage 1: elastic loading regime

Initially, all behavior is elastic, that is, dwp ¼ wp ¼ dtp ¼ tp ¼ 0. Hence, Eq. (22) becomes

2F cos ydDnþ 2Fð2R� Dn

Þ sin ydy ¼ 2knDn dDnþ 4R2

½ktFC cos yð2 sin y�wÞ þ ks cos y sin y�dy

þ 2R2½krw� kt

FCð2 sin y�wÞ�dw.

Since this must hold for arbitrary variations dDn;dy;dw, the following must hold:

2F cos y ¼ 2knDn,

2Fð2R� DnÞ sin y ¼ 4R2

ðktFC cos yð2 sin y�wÞ þ ks cos y sin yÞ,

2R2ðkrw� kt

FCð2 sin y�wÞÞ ¼ 0.

These equations are the same as the partial derivatives of V from Section 3.1, after substitution of w ¼ wð1Þ ¼ �wð3Þ;wð2Þ ¼ 0.Thus we recover Eqs. (9)–(13).

3.2.3. Stage 2: plastic loading regime 1

At the point y ¼ ys, the side supporting force reaches its plastic threshold, where ys is given by ys¼ sin�1

ðmsss=ð2ksÞÞ

using Eq. (17). During this first regime of plastic loading, dtpa0; tpa0 while dwp ¼ wp ¼ 0. Thus it follows fromEq. (22) that

2F cosydDnþ 2Fð2R� Dn

Þ sin ydy ¼ 2knDn dDnþ 4R2

½ktFC cos yð2 sin y�wÞ þ ks cos yðsiny� tpÞ�dy� 4R2ks

ðsin y� tpÞdtp

þ 2R2½krw� kt

FCð2 sin y�wÞ�dwþ 2R2msss dtp.

Since this must hold for arbitrary variations dDn;dy;dw;dtp, the following equations must be true:

2F cos y ¼ 2knDn,

2Fð2R� DnÞ sin y ¼ 4R2

½ktFC cos yð2 sin y�wÞ þ ks cosyðsin y� tpÞ�,

2R2msss � 4R2ksðsin y� tpÞ ¼ 0,

2R2½krw� kt

FCð2 sin y�wÞ� ¼ 0.

Solving these and using Eq. (8) to obtain D yields

FðyÞ ¼Rkn

cos y1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�

2 cos2 ykn

2ktFCkr

ktFC þ kr þ

msss

2 sin y

!vuut0@

1A, (23)

DðyÞ ¼ 4R� 2R 1þ


2 cos2 ykn

2ktFCkr

ktFC þ kr þ

msss

2 sin y

!vuut0@

1A cos y, (24)

DnðyÞ ¼ R 1�


2 cos2 ykn

2ktFCkr

ktFC þ kr þ

msss

2 sin y

!vuut0@

1A; wðyÞ ¼

2ktFC sin y

krþ kt

FC

, (25)

noting that wðyÞ is the same as that in Eq. (13), since the rotations are still elastic.

3.2.4. Stage 3: plastic loading regime 2

At the point y ¼ y�, where we assume ysoy�, the contact moment reaches its plastic threshold. Here, the angle y� is thesolution to the following equation:

2ktFCkr sin y

ktFC þ kr ¼ mrkn 1�


2 cos2 ykn

2ktFCkr

ktFC þ kr þ

msss

2 sin y

!vuut0@

1A,

using Eqs. (16) and (25). During this second regime of plastic loading, the following holds: dtpa0; tpa0;dwpa0;wpa0. Wenow follow the same procedure as for Stage 2, and substitute these relations into Eq. (22). As Eq. (22) must hold forarbitrary variations dDn;dy;dw;dtp;dwp, we obtain the following equations:

FðyÞ ¼Rkn

sin y cos y�mr cos2 yþ sin y�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðmr cos2 y� sin yÞ2 �

msss sin y cos2 ykn

s0@

1A, (26)

ARTICLE IN PRESS


DðyÞ ¼ 2R 2� cos y 2�1

sin y�mr cos2 yþ sin y�


msss sin y cos2 ykn

s0@

1A

0@

1A

0@

1A,

DnðyÞ ¼

R

sin y�mr cos2 yþ sin y�


msss sin y cos2 ykn

s0@

1A,

wðyÞ ¼ �mrk

n

RktFC

DnðyÞ þ 2 sin y.

As expected, there is no dependence of the force on either the tangential or rotational springs (ktFC ; k

r) during this plasticregime, since rotations are now linked with the normal displacements.

3.2.5. Stage 4: elastic unloading regime

Finally, suppose that at some chosen point in Stage 3, yu4y�, the structure unloads elastically (dDo0). Now,dwp ¼ dtp ¼ 0, but wpa0; tpa0: these plastic variables remain at their value at yu from Stage 3 (that is, wpðyÞ ¼wpðyu

Þ; tpðyÞ ¼ tpðyuÞ). Again we follow the procedure employed for Stage 2, resulting in the following equations:

FðyÞ ¼Rkn

cosy1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�

2 cos2 ykn sin y

ktFCkrð2 sin y�wpðyu

ÞÞ

ktFC þ kr þ ks

ðsin y� tpðyuÞÞ

" #vuut0@

1A,

DðyÞ ¼ 2R 2� cos y 1þ


2 cos2 ykn sin y


ÞÞ

ktFC þ kr þ ks


" #vuut0@

1A

0@

1A,

DnðyÞ ¼ R 1�


2 cos2 ykn sin y


ÞÞ

ktFC þ kr þ ks


" #vuut0@

1A,

wðyÞ ¼2kt

FC sin yþ krwpðyuÞ

ktFC þ kr .

4. Kinematics of confined force chain buckling

Knowledge of the kinematics of key rearrangement events is a critical ingredient in resolving many open problems inthe constitutive modeling of dissipative media. For dense granular systems, two challenges come to the fore. The first is theidentification of the underlying origins of strain-softening under dilatation—a hallmark phenomenon exhibited by thesesystems on the macroscale. The second concerns the constitutive formulation, specifically, the establishment of a physicalmeaning for internal variables and a rigorous method by which evolution laws for these variables can be derived incompliance with thermodynamical principles (Valanis, 1995; Deboeuf et al., 2005; Tordesillas, 2007a, b; Walsh et al., 2007).These challenges require full information on the statics and kinematics of not only elastic buckling, but also and moreimportantly, the plastic buckling regime. Indeed, as discussed earlier in Section 3, this is one of the key differences betweenthe analysis presented here and much of traditional structural mechanics where buckling models are usually confined toelastic regimes only (e.g. Bazant and Cedolin, 2003; Croll and Walker, 1972; DeTeresa et al., 1985). The task in this section isto determine expressions for the motion of the four side supporting particles which, together with the displacement earlierderived for the particles within the force chain column, yield a set of kinematic relations that quantify the motions withinthe N ¼ 3 model, i.e. the particle cluster comprising a force chain and its laterally supporting particles. We draw attentionto two specific applications of these kinematical relations, before proceeding with their derivation.

Our studies of force chain evolution in DEM simulations and, more recently, in photoelastic disk experimentscorroborate the hypothesis of Oda and co-workers: force chain buckling is the underpinning mechanism for shear bandingand related phenomena of strain-softening and dilatancy observed for dense granular systems (Tordesillas andMuthuswamy, 2008; Tordesillas et al., 2009; Tordesillas, 2007a; Oda et al., 2004; Oda and Iwashita, 2000). Thus, later inSection 6.2, we will use these kinematic relations to demonstrate the dilatation of the particle cluster or N ¼ 3 model,under a weakening load-carrying capacity of the force chain column.

In addition, we have shown in the companion paper, Tordesillas and Muthuswamy (2008), that these kinematicrelations can be used to determine the micropolar strain and curvature of the particle cluster, as well as the cluster’snonaffine deformation throughout the buckling process, in accordance with Tordesillas et al., (2008a, b). The nonaffinedeformation is of specific interest in continuum modeling, as there is compelling evidence to suggest that these assume therole of the so-called internal variables in plasticity theory (Lubliner, 1990; Valanis, 1995, 1996). In thermomicromechanicalconstitutive formulations, these internal variables represent the kinematics of mechanisms chiefly responsible for the loss

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of stored free energy in the system (Tordesillas, 2007a, b; Walsh et al., 2007). A key challenge in constitutive developmentwas reiterated recently by Deboeuf et al. (2005): ‘‘the underlying question of a physically based identification of therelevant internal variables and of their evolution laws remains open’’. In Tordesillas and Muthuswamy (2008), we showedhow the kinematical relations derived in this paper deliver an answer to this question for dense, cohesionless granularmaterials by providing: (i) internal variables with clear physical meaning, i.e. the nonaffine deformation of confinedbuckling of a force chain, on the scale of a particle and its first ring of neighbors, and (ii) a rigorous method for derivingtheir corresponding evolution laws in compliance with the 2nd Law of Thermodynamics. The resulting constitutive lawderived therein can reliably predict not only the defining behavior of strain-softening under dilatation on the macroscopicscale, but also the formation and evolution of shear bands. The thickness and angle of the shear band, the distribution ofparticle rotations and the evolution of the normal contact force distribution inside the band, are consistent with thoseobserved in two-dimensional, discrete element simulations and physical experiments (e.g. Calvetti et al., 1997; Bardet andProubet, 1991; Thornton and Zhang, 2006). For a detailed discussion of the relevance of this analysis to constitutivedevelopment we refer to Tordesillas and Muthuswamy (2008), and to Tordesillas (2007a, b).

4.1. Kinematics for a cluster in an initially hexagonal close packing

A quantitative characterization of the kinematics requires a more detailed treatment of the laterally supporting weaknetwork particles than that presented in Sections 2.2.3 and 3.2 in the form of a single effective lateral force f s (Eqs. (3) and(17)). There are a number of strategies for modeling this lateral support. In this section, we discuss one simple method fordefining the kinematical behavior of the laterally supporting weak network particles for the specific case of N ¼ 3, asillustrated in Fig. 3.

We assume the simplest initial packing configuration: hexagonal close packing (Fig. 3(a)). Once buckling has occurred inone direction, a void is created. From DEM, the changes in the force and hence relative displacement of the nonbucklingside particles encircling the void (dashed particles in Figs. 3(b) and (c)) are relatively small compared to those for the sideparticles being ‘‘pushed out’’ by the buckling motion. Therefore, we neglect the motion of the nonbuckling side particles,and assume that they remain stationary over the whole buckling process. We only take into account interactions betweenthe central particle and the side particles that are pushed out during buckling. We further ignore all side particles’ rotation,since results from DEM show that their rotations are small when compared to the rotations of the three particles of theforce chain column. In Table 2, we identify which kinematics are predicted vs. assumed for the three sets of particles in thecluster (left side supporting particles, right side supporting particles, and force chain column). For each predicted quantity,we indicate the relevant section where the derivation for this quantity is presented.

The effective side force of f sðyÞ, imposed by the weak network particles on the force chain, is then decomposed into two

equal forces acting horizontally, initially at 30� above and below the horizontal (Fig. 3(c)). The motion of the side particles,as illustrated in Fig. 3(b). may be computed from their positions xðyÞ; yðyÞ as follows.

Each side force f sðyÞ=2 is decomposed into normal (f n

ðyÞ) and tangential (f tðyÞ) components. Assuming the side contact

angle remains close to 30�, the contact normal and tangential vectors are n ¼ ð�ffiffi3p

2 ;12Þ; t ¼ ð12 ;

ffiffi3p

2 Þ: Thus, the decomposition

4R

x2

x1

2 3R

f s(θ)

θ

f s(θ)/2

f s(θ)/2θ

4R-Δ

(-x,y)

(-2Rsinθ,0)

θ

3 R+xF

(-x,-y)

Fig. 3. Schematics of (a) initial configuration; (b) deformed configuration, with dark arrows showing movement of central particle and left side particles;

(c) two side supporting particles, each with side force acting horizontally on central particle. Particles in nonbuckling side shown by dashed lines.

Table 2Predicted vs. assumed kinematics of the particles in the cluster in Fig. 3.

Two left side particles Two right side particles Three force chain particles

Displacements (x1 ; x2) Predicted (Section 4.1) Assumed zero Predicted (Section 3)

Rotations Assumed zero Assumed zero Predicted (Section 3)

The relevant section that contains the derivation of each of the predicted kinematical quantity is indicated.

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of f sðyÞ=2 becomes jf n

ðyÞj ¼ffiffiffi3p

f sðyÞ=4; jf t

ðyÞj ¼ f sðyÞ=4; and hence relative normal (DunðyÞ) and tangential (DutðyÞ)

displacements between central and top left side particles are

jDunðyÞj ¼jf nðyÞj

kn ¼

ffiffiffi3p

f sðyÞ

4kn ; jDutðyÞj ¼jf tðyÞjkt ¼

f sðyÞ

4kt . (27)

The spring constant kt is associated with the amount of relative tangential motion between a weak network particle and aforce chain particle: note that this is distinct from kt

FC , which governs the relative motion between force chain particles, andks, the stiffness parameter for the effective lateral supporting force f s. The need for this parameter kt, in addition to kt

FC andks, will be explained in Section 5.2.

We can now compute the positions of the particles after buckling: see Fig. 3(b), where x40; y40. The relativedisplacement vector DuðyÞ is

DuðyÞ ¼ us � uc ¼ ð�xþffiffiffi3p

Rþ 2R sin y; y� RÞ, (28)

where us;uc are the displacement vectors of the top left side, and central particle, respectively. From the unit contactvectors n; t above, and Eq. (28), jDutðyÞj; jDunðyÞj can be calculated, and combined with Eqs. (27) to solve for x, y:

xðyÞ ¼ffiffiffi3p

Rþ 2R sin y�f sðyÞðkn

þ 3ktÞ

8knkt ; yðyÞ ¼ Rþ

ffiffiffi3p

f sðyÞðkn

� ktÞ

8knkt . (29)

A more realistic representation of the confinement due to lateral particles than that delineated above requires a moreextensive and quantitative examination of the interactions within the laterally supporting weak network. This study isongoing and will be reported on in a future paper.

5. Procedure for model calibration and validation

Before we can apply the buckling model, make predictions and assess its performance, all input parameters must first bedefined, so that a comparison can be made with DEM simulations. Thus, the aim of this section is twofold: (i) to identify themethod employed for the acquisition of data for the calibration and validation of the model, and (ii) to properlycharacterize the system under study by establishing all input parameters to the model. Recall that the effect of particleshape irregularities is accounted for in both the DEM simulation and the buckling model via the introduction of a contactmoment. There are a total of eight input parameters to the model: R, kn, kr , mr , msss, kt

FC , ks, kt . The first four of theseparameters (i.e. R; kn; kr ;mr) are identical to those in the DEM simulation: see Table 3. The remaining four parameters (i.e.the maximum lateral supporting force msss and the stiffness parameters kt

FC , ks; kt) are calibrated to the DEM simulation.

5.1. Collection of ‘‘DEM buckling data’’

Data for the calibration of the model, as well as those used for the validation in the next section, are obtained from alocal analysis of buckling events in a DEM simulation of a two-dimensional assembly of 5098 bidisperse circular disksunder biaxial compression. For completeness and consistency, the complementary analysis in the companion paper wasalso undertaken for this particular DEM simulation (Tordesillas and Muthuswamy, 2008). This system is described in thefirst row of Table 1 (i.e. mr ¼ 0:2). The complete list of all the parameters used for the model and this simulation is shown,

Table 3Model parameters with typical constants used in DEM simulations of stiff particles (i.e. small overlap).

Model Our DEM Other DEM simulations

R̄ (m) 1:14� 10�3 1:14� 10�3 1� 10�3 (Luding, 2004);

5� 10�3 (Iwashita and Oda, 1998)

kn (N/m) 1:05� 105 1:05� 105 1:05� 105 (Luding, 2004);

6� 107 (Iwashita and Oda, 1998)

kt=kn 0.5 0.5 0.1 (Iwashita and Oda, 2000);

0.5 (Kruyt and Antony, 2007);

0.67 (Iwashita and Oda, 1998)

kr=kn 0.5 0.5 0.47 (Iwashita and Oda, 1998);

0.28 (Iwashita and Oda, 2000);

8.3�10�4 to 0.21 (Jiang et al., 2005)

mr 0.2 0.2 � 0:08, � 0:4(Iwashita and Oda, 2000);

0.02 to 0.27 (Jiang et al., 2005)

ktFC=kn ; ks=kn ; 0.014, 0.06,

msss=kn 0.024 Calibrated to DEM buckling data

kr is normalized by R̄2, for consistency of dimensions with kt ; kn .

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respectively, in columns 2 and 3 of Table 3. Additionally, the input parameters used in similar DEM simulations by otherworkers are included for comparison in column 4 of Table 3. The initially isotropic state of the system in our DEMsimulation (column 3) has a density of 0.858, and an average coordination number of 4.3. Peak stress ratio is 0.66 and isattained at an axial strain of 0.05. We use the two algorithms developed in Peters et al. (2005) and Tordesillas (2007a) forfinding buckling segments of force chains: recall these were summarized in Section 1 and also in Fig. 1. Once bucklingsegments of force chains are identified, we then establish properties for such segments that can be used to either calibratethe model (e.g. f s

ðyÞ), or validate model predictions (e.g. F;Dn;w).In this study, buckling force chain segments are identified over three unjamming transitions or drops in stress ratio.

These correspond to the axial strain intervals of 0.046–0.053, 0.059–0.066, and 0.080–0.087. The change in strain is thesame over these three intervals, i.e. 0.007. Thus each of these can be divided uniformly into i ¼ 1;2; . . . ;X small strainincrements. The properties of force chain segments in the DEM simulation are then measured at each of these strainincrements. This allows us to quantify various aspects of the structural evolution of each buckling force chain segment overeach of the three drops in stress ratio. We implement the following procedure. For each drop, we define an initial strainstate. The initial strain state is the strain at the start of the drop. Based on this, the change in the properties of each bucklingforce chain segment at the ii strain increment can be quantified relative to the initial reference configuration, i.e. theconfiguration of the segment at the initial state or start of the drop. Once the properties of all the buckled force chainsegments are established for all the X strain increments, we then record the ‘‘global’’ average value for each property at theiTtH strain increment, i.e. the average value of a given property over all the buckled force chain segments that were foundfrom all three drops or unjamming transitions at the ith strain increment. Hereafter, this data is referred to as the ‘‘DEMbuckling data’’.

5.1.1. Accounting for initial imperfections in force chains in DEM

In real assemblies and in DEM simulations, there is no clearly identifiable initial or reference configuration for theprocess of force chain buckling. Even the formation of a force chain is a gradual process commencing in the early stages ofcompression, but with no clearly defined starting point. Therefore, typically, force chains in DEM have already sustained acertain degree of imperfection and deformation at the start of each unjamming event, i.e. ya0 and Da0. By contrast, theinitial reference configuration in the model involves a perfectly straight and undeformed force chain such that y ¼ D ¼ 0.Thus, the nonzero average values for the initial imperfection and initial displacement of force chains in DEM will need to bedetermined in order to establish a reference force chain configuration—a state from which the DEM buckling data can beappropriately compared with model predictions.

The buckling angle of a force chain segment at a given strain state is defined as the average of the two acute anglesbetween the axial line connecting the centers of the two outer particles of the force chain segment and the lines connectingthe centers of the middle particle and the two outer particles. It may be useful to refer back to Fig. 3(b) which shows thebuckling angle for the model; however, keep in mind that, in DEM, the axial line need not be vertical and the two acuteangles need not be identical as in the model. Thus, the initial imperfection or deviation from the perfectly straightconfiguration of each force chain segment at the start of each unjamming event is simply the buckling angle at the initialstrain state. The initial average imperfection is then the average value of this angle over all buckled force chain segments forthe three unjamming transitions. This was found to be 14�. Thus, when comparing model predictions to DEM, we reconcilethis difference by translating the DEM data to the buckling angle equal to the initial average imperfection of 14�. Similarly,to compare displacement data, we need a nonzero reference value for D=R. A sensible value for D=R is that whichcorresponds to the initial imperfection of 14�. This was found to be D=R � 0:25.

5.1.2. Computing properties of buckling force chains in DEM

In this section, we delineate the procedure for computing three properties of buckling force chain segments from DEMsimulations, to be used subsequently in the calibration and validation of the model. The properties are: the average axialforce F, the average displacement D, and the average rotation w.

To compute the average axial force F for all three-particle force chain segments at the ith strain increment, we proceedas follows. For each such force chain segment, we let the bottom particle of the force chain column be particle 1; themiddle, particle 2; and the top, particle 3. We define the axial line to be that which connects the centers of particles 1 and 3.We then define two parallel lines, L1 and L3, which are perpendicular to the axial line and pass through the centers ofparticles 1 and 3, respectively. To compute the axial force F1 on particle 1, a scalar quantity, we first compute the total forcevector as a sum of the force vectors from all the particles which: (i) are in contact with particle 1, and (ii) whose centers lieon the opposite side of line L1 to particles 2 and 3. In general, this force vector may not be parallel to the axial line, so F1 issimply the component of this total force vector along the axial line. To determine the axial force on particle 3, F3, weemploy the same procedure except we compute the force from the particles that are in contact with particle 3 whosecenters lie on the opposite side of line L3 to particles 2 and 1. The total axial force for the segment Fseg is simply the averageof the two scalar axial forces F1 and F3. The average axial force F is then the average Fseg over all force chain segments fromthe three unjamming transitions at the ith strain increment.

To compute the average displacement D at the ith strain increment, we use the same axial line that was used for theaxial force. We compute the amount by which the line joining the centers of particles 1 and 3 for each segment hasdecreased from the initial state to the ith strain increment to obtain Dseg . We then average Dseg over all force chain

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segments to obtain D. To compute Dn, we first compute f nseg which represents the average of the normal forces at the two

force chain contacts for each segment, i.e. between particles 1 and 2 and between particles 2 and 3. We then take theaverage of this over all segments and then divide by kn.

Recall from Section 3 that the top and bottom particles of the force chain column in the model rotate with equaland opposite rotations (e.g. Eq. (13) for the elastic regime). We wish to compare these rotations with an equivalentrotation measure w, obtained from the DEM buckling segments. First, we define the initial strain state of each unjammingtransition as zero rotation. At the ith strain increment, we measure the amount by which each particle has rotatedsince the initial strain state. At this point, we have three rotation values. Note that in the DEM, rigid body rotation ofwhole parts of the assembly is possible. To remove this effect, we now compute relative rotations at the ith strainincrement: in particular, we compute the relative rotation w1;2 between particles 1 and 2, and between particles 3 and 2,w3;2. For the ith strain increment, we have two (relative) rotation values. Since the model predicts zero rotation ofparticle 2, and opposite rotations of particles 1 and 3, we calculate the rotation for each segment, wseg , as: jw1;2 �w3;2j=2.The negative sign arises here because, in general, we expect w1;2 and w3;2 to be of opposite signs. The average rotation w isthen the average of the wseg values, over all force chain segments from the three unjamming transitions at the ith strainincrement.

5.2. Calibration of the effective lateral supporting force

Recall from Section 2.2.3 that f s is an ‘‘effective’’, or average, lateral supporting force on a force chain particle. Thus, thequantity msss multiplied by R, as shown earlier in Eq. (3), represents the limiting or maximum lateral force the weaknetwork particles can exert on the force chain particle. This lateral force is related to the confining pressure and can becalibrated to the DEM buckling data. Specifically, we measured the average total lateral (horizontal) supporting force fromthe weak network particles acting on the central particle of each buckled segment, and then averaged these as describedabove in the procedure for collecting the DEM buckling data. This force was found to be essentially constant over all thestrain increments. This is to be expected, since the confining pressure applied to the boundaries of the assembly is constant.Dividing this quantity by the average radius R̄, we obtain msss (Eq. (3)). For the DEM simulation considered here, a confiningpressure of 7:0� 102 N=m results in an average f s of 2.9 N, and hence msss ¼ 2:9=R̄ ¼ 0:024kn (using R̄; kn from Table 3).Clearly, we would expect that as confining pressure increases, msss would also increase.

5.3. Calibration of stiffness parameters

Three of the input stiffness constants, i.e. kt ; ktFC ; k

s, must be calibrated to the DEM buckling data. Recall from Section 4that there is a need for a spring constant kt , which governs the amount of relative tangential motion between a weaknetwork particle and a force chain particle, in addition to kt

FC , which governs the relative motion between force chainparticles. Why is this so, given that this distinction is entirely unnecessary in DEM? As discussed earlier in Section 2.2.1,sliding between particles is rarely observed in contacts carrying above average contact forces (Radjai et al., 1998). Indeed, inall our DEM simulations, less than 5% of force chains buckle due to plastic slip in contacts within the force chain(Tordesillas et al., 2008a, 2009; Tordesillas, 2007a). Thus, in these simulations, the tangential contact forces and relativetangential displacements developed between force chain particles during the chain’s lifetime are generally much smallerthan those in the buckling model if kt

¼ ktFC . To understand this disparity, a useful quantity to focus on is the change in

buckling angle during an unjamming transition; if this angle is small, then so are the relative tangential displacementsdeveloped before the chain’s collapse, bearing in mind nearly all buckling force chains collapse by the end of an unjammingtransition. There are two possible sources of this disparity. The first lies in the many more DOFs available in the DEM thanthe six DOFs allowed for in the model. A key lesson learned from DEM simulations is that the complex rearrangements inthe force chain neighborhood have a strong influence on the force chain’s stability. Recall the data from the first row ofTable 1 in Section 2.2 which show that, for the simulation used to calibrate and validate the model, the buckling segmentsthat dominate are from class 3 in which the first contact to reach plastic threshold belongs to the laterally supportingneighbors. In particular, we observed that the whole gamut of such local and nonlocal plastic rearrangements leads to arelatively small average change in buckling angle during an unjamming transition. On average, this change in bucklingangle is only of the order of 10� in the DEM simulation used here. By contrast, the buckling model reaches a change inbuckling angle of around 20� even before the onset of Stage 3. The second source of disparity lies in the initial referenceconfiguration of buckled force chain segments in DEM simulations. This configuration typically bears an imperfection, i.e. amisalignment from the perfectly straight column that embodies the reference state in the model. As discussed earlier inSection 5.1.1, this imperfection is approximately 14� in the DEM simulation. Such imperfection renders the force chain weakand vulnerable to collapse (we show quantitative evidence of this in the next section).

To resolve the aforementioned inconsistency in a simple yet systematic way, we exploit previous assumptions madeearlier in the construction of the model. That is, we reduce the value of kt

FC relative to kt—while limiting the particle

overlaps to lie in the range consistent with the linear spring contact laws assumed earlier for the deformation of theparticles in Section 2. For stiff grains, our DEM simulations have shown that particle overlaps for force chain particlesgenerally lie below 10% of the average particle radius at the onset of buckling (Tordesillas, 2007a). Hence, by enforcing that

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the maximum relative normal overlap 12Dcrit (Eq. (15)) must not exceed lR; l51, we get

2ktFCkrþ kskt

FC þ kskr

knðkt

FC þ krÞ

o2l� l2

2. (30)

As an example, by choosing a typical maximum particle overlap of lR ¼ 0:1R, the values of ktFC ¼ 0:014kn, and ks

¼ 0:06kn,together with the five fixed parameters in Table 3, satisfy inequality 30.

Clearly, a model that would entirely obviate the need to calibrate the above input stiffness parameters and that of thelateral force discussed in Section 5.2 to local DEM buckling data presents an obvious extension of this study. Thisdevelopment is underway but is work that lies outside the scope of the present investigation.

6. Results of model validation and application

In this section, we validate model predictions in three studies. In the first (Section 6.1) we are to compare the bucklingproperties as predicted by the model against the DEM buckling data, while accounting for the initial imperfection andinitial displacement that are inherent in the latter and in real granular assemblies. The question we wish to answer is: canthe structural mechanical model for N ¼ 3 reasonably reproduce trends observed in buckling force chains in DEMsimulations? In the second (Section 6.2), we are to establish whether this cluster model predicts loss of load-carryingcapacity of the force chain column under dilatation. In the third (Section 6.3), we examine the effects of various particle-scale properties on the variation of the axial force vs. displacement. Here the objective is to determine whether theseeffects show qualitative concordance with well established trends observed on the macroscale from both DEM simulationsand physical experiments. Recall that buckling force chains occur spatially in the regions of greatest dissipation, and are agoverning mechanism that drives macroscale behavior (Tordesillas, 2007a). It seems reasonable to expect, then, thatvarying parameters in the cluster model would have the same qualitative effect on the axial force F, as varying parametersin large-scale DEM simulations of biaxial compression tests would have on the macroscopic shear stress. If so, then thisstudy elucidates possible mesoscopic origins of these important trends observed on the macroscale.

6.1. Model predictions against DEM buckling data

Fig. 4 presents a comparison of model predictions with DEM buckling data for the variation of the axial force andkinematical properties with either buckling angle (in graphs (a)–(d)) or D=R (in graph (e)). The stages of buckling, namely1a, 1b, 2, 3, and 4, as identified earlier in Section 3.2.1, are indicated on each of the curves. Comparison of the DEM bucklingdata with model predictions commences from the reference force chain configuration. As discussed previously in Section5.1.1, this occurs at a buckling angle equal to the initial imperfection of 14� and corresponding displacement value ofD=R � 0:25.

6

12

10 20

1b 2

3

4

10 20

1a

1b

2 3

4

5

10

0.2 0.4

1a

1b

2 3

4

5

10

0.2

0.4

10 20

1a

1b

2

3

4

0.08

0.04

10 20

1a

1b

23

4

Model (imperfection of 2°)Model (loading)Model (unloading)DEM buckling data

KEY:

F [

N]

θ [°]

Δ/R

θ [°]

w [°

]

θ [°]

Δn /R

θ [°]

F [

N]

Δ/R

Fig. 4. (a) Bifurcation plot of the axial force F vs. the buckling angle y for the model with an initially perfectly straight force chain and that with an initial

imperfection (Appendix A); (b) normalized vertical displacement D=R vs. y; (c) total rotation w vs. y; (d) normalized relative normal displacement Dn=R vs.

y; and (e) force–displacement curve obtained by combining (a) and (b). Constants used for the DEM simulation and buckling model are given in Table 3.

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The model predicts F;Dn;D as a function of the degree of buckling reasonably well. As expected, loss of load-carryingcapacity commences with buckling, as is evident in the decrease in the axial force, measured along the axis of force chain,for both DEM buckling data and the model (Figs. 4(a) and (e)). Good agreement between model predictions and DEM dataon buckling segments can also be seen in Figs. 4(b) and (d) for the vertical displacement D and the normal relativedisplacement Dn, respectively.

A monotonic increase of w with y is reproduced by the model, although due to the large ratio of kr to ktFC , significant

rotation only begins once the analogous Coulomb plastic threshold for the contact moment has been reached, as expected(Fig. 4(c)). Also in agreement with DEM are the model’s predictions on relative rotation: the relative rotation of the centralparticle is predicted to be wð2Þ � 0, and the rotations and directions of rotation of the top and bottom particles satisfywð1Þ � �wð3Þ (data not shown).

6.2. Model predictions on strain-softening under dilatation

The F vs. D=R curve in Fig. 4(e) exhibits post-peak softening. The tie between this and local dilatancy (i.e. the growth ofassociated void space) during buckling is crucial in understanding the origins of strain-softening under dilatancy at themacroscopic level. To see this connection, we turn to the corresponding structural evolution of the confined buckling ofthe force chain in Fig. 5. Here the particle motions were obtained from the kinematics derived and discussed in Sections 3and 4, and summarized in Table 2. Comparing Figs. 4 and 5 yields new insights into the mechanism of buckling indeforming, dense granular assemblies. Prior to buckling, Stage 1a, the force sustained by the force chain increases linearlyuntil the critical buckling load is reached (Fig. 4(e)). The column remains perfectly straight: no dilatation occurs (left imagein Fig. 5). The critical buckling load marks the initial point of instability, which corresponds to the onset of strain-softening.Following this is elastic buckling, Stage 1b, which manifests in a slight decrease in Dn and F (Figs. 4(d–e)). Dilatation of theparticle cluster commences, and slight rotations of the two outer force chain particles occur: see Fig. 4(c) and second fromleft image in Fig. 5. In Stages 2 and 3, the interplay between the weak supporting particles and the contact moment withinthe force chain becomes evident. Specifically, the lateral force becomes plastic in Stage 2, at which point the weaksupporting particles are no longer able to prevent further buckling, consistent with the findings in Radjai et al. (1998). Thisplastic behavior leads to a further reduction in F and Dn in Fig. 4(e). The weak and now plastic support also induces thecontact moments to reach their plastic threshold even faster in Stage 3, at which point particle rotation starts to increasesharply Fig. 4(c). Loss of load-carrying capacity of the force chain column under local dilatation is clearly evident in Figs. 4and 5 during Stages 1b–3. From the start of unloading through to the end of Stage 4 at which F ¼ Dn

¼ 0, the plasticbehavior of the contact moment and lateral supporting force results in a residual rotation of the particles and plastic strainof the entire mesoscale structure (Fig. 4(c) and right image in Fig. 5).

6.3. Consistency with macroscopic trends from experiments and DEM

We now present the results of a parametric study which serves a dual purpose. The first is to address a problem recentlyunderlined in Kuhn and Chang (2006). This problem concerns the lack of a comprehensive micromechanical explanationfor key trends derived from soil mechanics experiments: e.g. effects of particle interlocking and confining pressure on thestrain at peak shear stress, the friction angle at the peak stress, the rate of softening at post-peak strains and the steadycritical-state shear stress (e.g. Desrues and Viggiani, 2004). The second is to determine concordance between the effects ofparticle-scale properties on force chain evolution as derived from the model, and the effects of these same properties onmacroscopic strength as derived from DEM simulations and physical experiments (Oda et al., 2004; Oda and Kazama, 1998;Kolymbas and Wu, 1990; Desrues and Viggiani, 2004). As was shown earlier in Figs. 1(c) and (f), buckling of force chainsoccurs spatially in the regions of greatest dissipation (e.g. shear band), and is a mechanism that governs macroscalebehavior (Tordesillas, 2007a). Consequently, we expect that the various parameters would influence the load-carryingcapacity of the force chain—in the same manner as these would affect the macroscopic shear stress in biaxial compressionexperiments and DEM simulations.

To understand the dominant contributors to stability and strain-softening on the mesoscale and hence give insight intowell-known trends observed for these two aspects of material behavior on the macroscale, we present in Fig. 6 the effect on

FStage 1a FStage 1b FStage 4FStage 3Stage 2 F

Fig. 5. Structural evolution from model predictions on particle kinematics, with displacements of side particles computed from Eq. (29). Rotations of the

top and bottom force chain particles are indicated in both magnitude and direction via the rotation of the initially vertical solid lines through the centers

of these particles.

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0.2 0.4 0.6 0.2 0.4 0.6

0.2 0.4 0.6

0.2 0.4 0.6

10

5

10

5

10

5

10

5

kn ktFC kr

μr

0.2 0.4 0.6

10

5ks

0.2 0.4 0.6

10

5μsσs

F [

N]

Δ/R

F [

N]

Δ/R

F [

N]

Δ/R

F [

N]

Δ/R

F [

N]

Δ/R

F [

N]

Δ/R

Fig. 6. Effect on FðDÞ curve of varying each parameter. Arrows point in direction of increasing parameter value. Dotted lines in (d) show model with initial

imperfection angle of 2� . Varied parameters: kn¼ 7:00� 104; 1:05� 105; 1:40� 105; kt

FC=kn¼ 0:009;0:014;0:03; kr=kn

¼ 0:0;0:01;0:1;0:5; mr ¼

0:02;0:2;0:3;0:45;msss=kn¼ 0:01;0:024;0:04;0:06; ks=kn

¼ 0:04;0:06;0:09.


FðDÞ of varying each parameter apart from two quantities: R which is simply a multiplicative factor that scales FðDÞvertically, and kt which only affects the kinematics of weak network particles as discussed earlier in Section 4. Aspreviously mentioned, although we confined the derivations in Section 3 to the case where Stage 2 precedes Stage 3, wehave also undertaken the analysis for the reverse buckling progression. For completeness, we include the results for thislatter progression which may arise, for example, in cases of large msss or small mr . Recall that all manner of bucklingprogression was presented in the data in Table 1. As shown therein, the plastic behavior changes as mr is reduced from 0.2 to0.02: the percentage of buckled segments in class 4 switches from being less than to greater than that for class 3.

The F vs. D=R curves in Fig. 6 all exhibit post-peak softening. The phenomenon known as ‘‘snapdown’’ can be observed insome cases, indicating a loss of stability despite the system being displacement controlled. Snapdown can be observed forsmall mr , large ks and small msss in Figs. 6(d), (e) and (f), respectively: in these cases, nonunique F values can be found forcertain values of D=R. Bearing in mind that each point on the F vs. D=R curve represents an equilibrium state, portions of theforce–displacement curves where D=R decreases during softening represent unstable and thus unattainable equilibriumstates. In other words, under a controlled displacement where D=R is increased gradually from zero, the predicted force F

may follow a discontinuous path—as the model snaps down to the smaller post-critical force value at the next increasedvalue of D=R (Bazant and Cedolin, 2003).

6.3.1. Effects of kn and ktFC

Similar to R, the normal stiffness kn is a multiplicative factor that scales FðDÞ vertically, as shown in Fig. 6(a). Hence, thedisplacement corresponding to the critical buckling load is unaffected by kn. The quantity kn is a mechanical stiffness thatarises from the deformability of the particles. By contrast, kt

FC reflects the combined effect of geometric and mechanicalstiffness of the particles in the force chain, where the geometric stiffness is due to particle interlocking, an effect of particleshape irregularities (discussed in Section 2.2). As shown in Fig. 6(b), unlike kn, kt

FC bears an influence on the displacement atwhich buckling commences, and hence the stability of the force chain, in agreement with Kuhn and Chang (2006).

6.3.2. Effects of kr and mr

It is now well-established that the macroscopic strength and stability of the system is governed by the microstructuralmechanism of particle interlocking due to irregularities in particle shape (e.g. Jiang et al., 2005; Kuhn and Chang, 2006).Peak shear stress, strain at peak shear stress and the steady critical-state stress all increase as the degree of interlockingamong constituent particles increases. Experimental observations also suggest that a higher degree of interlocking leads tostronger and more stable force chains (Oda et al., 2004; Oda and Kazama, 1998).

In the model, the effect of particle interlocking is governed by kr and mr . The more irregular the shape and the moreinterlocking present, the larger are kr and mr . The coefficient of rolling friction mr also reflects the resistance to relativerotations between contacting particles. We observed four key trends in the load-carrying capacity of the force chaincolumn, as shown in Figs. 6(c) and (d), all of which are qualitatively consistent with those observed in the macroscopicshear stress of a specimen under biaxial compression (e.g. Tordesillas, 2007a). First, just as the peak macroscopic shear

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stress and the strain at which this is attained increase with increasing kr and mr in biaxial tests, so do the criticalbuckling load and its corresponding displacement for the force chain column. The effect of mr may not be immediatelyevident, as the peak (or critical buckling) load shows no dependence on mr for the initially perfectly straight structure.Note, however, that for a small mr , i.e. mr ¼ 0:02, the slightest amount of elastic buckling triggers the contact momentto become plastic (Stage 3), followed closely by further plastic behavior of the weak network (Stage 2), with high ratesof softening. The model for mr ¼ 0:02 displays an instability, under displacement controlled conditions, as evident inthe sudden snapdown behavior of the force-deflection curve, when the contact moment becomes plastic. This instabilitysuggests that for real granular systems, force chains that ever did actually reach their critical buckling load would suddenlydecrease in load at the slightest amount of further compression. Thus, we would expect that the likelihood of observingthis sharp discontinuity in the force behavior of force chains in real assemblies would be quite small. Hence, to gain a betterunderstanding of what would be a more likely buckling behavior, we investigated the effect of an initial imperfection in themodel (see Appendix A). The example we show in Fig. 6(d) is for an initial buckling angle of 2�. Although the concept of acritical buckling load has no meaning now, we see that the peak or critical buckling load as well as the strain at which thepeak is attained do indeed increase with increasing values of mr .

Second, as noted previously, the progression of the plastic regime, i.e. whether Stage 2 is followed by Stage 3 or viceversa, is dependent on the model parameters (see Table 3). For small mr , e.g. mr ¼ 0:02 in Fig. 6(d), the contact moment isweak, and thus becomes plastic before the lateral supporting force: Stage 3 precedes Stage 2. For higher mr values,e.g. mr ¼ 0:2;0:3;0:45 in Fig. 6(d), Stage 2 precedes Stage 3. These findings are consistent with the DEM results shown inTable 1: there, the percentage of buckled segments in class 4 can be seen to switch from being less than to greater than thatfor class 3, as mr is decreased from 0.2 to 0.02.

Third, in biaxial tests, different specimens can be characterized by their eventual steady, critical-state shear stress.Although there is no such steady state for buckling, we can nonetheless probe the fully plastic buckling regime (i.e. Stage 3)by comparing the predicted axial load values taken at a fixed D=R value, for different values of mr . Well within this regime,e.g. D=R ¼ 0:5, we see from Fig. 6(d) that as mr increases, so does the corresponding value of the axial load at D=R ¼ 0:5. Asmall mr means hardly any elastic rotation occurs before the plastic regime is reached. Thus, for small mr , the axial loadmust decrease from a much lower initial value (i.e. critical buckling load), resulting in generally lower post-critical loadvalues in the fully plastic regime. Fourth, the model predicts that the peak load saturates at high values of rolling friction.This is consistent with DEM simulations of biaxial compression tests which show that peak shear stress saturates for highvalues of rolling friction (Tordesillas, 2007a).

Finally, setting kr¼ mr ¼ 0 corresponds to the system considered in the majority of DEM models for circular

particles which do not include any resistance to rolling: see bottom curve in Fig. 6(c). However, buckling still occurs,as noted in Thornton and Zhang (2006). In this system, the resistance to buckling is provided by weak network particlesand tangential forces within the force chain only; no extra support is provided by the contact moment along thechain. Thus, we shed light on the somewhat controversial issue of the use of rolling resistance in DEM simulations:the introduction of rolling resistance increases the strength of the structure. This reinforces the fact that particle rotations,or the inhibition of these, are an important mechanism in the stability of force chains (Oda et al., 2004; Oda andKazama, 1998).

6.3.3. Effects of ks and msss

The last major contributor to force chain stability is the lateral confining force from the supporting weak networkparticles. Here model predictions are consistent with several well-known trends that result from increasing confiningpressure in biaxial compression experiments: the peak shear stress, the strain at peak shear stress, and the steady critical-state shear stress all increase, while the rate of softening decreases (e.g. Desrues and Viggiani, 2004; Kolymbas and Wu,1990). First recall the earlier discussion in Section 6.3.2 on the effect of an imperfection in interpreting the trends for mr:this similarly applies in interpreting the results in Fig. 6(f) for changing msss. Bearing this in mind, the model predicts thatthe peak or critical buckling load and the displacement at which this load is achieved both increase as ks and msss areincreased in Figs. 6(e) and (f). For post-peak behavior, for a given ks, the larger the msss, the less plastic displacement in thelateral support, and hence less dissipation. Thus, as the limiting lateral force msss increases, the post-critical loads increaseand the rate of softening decreases, as is evident in Fig. 6(f).

7. Conclusions

The direct modeling of self-organized emergent phenomena in complex media, especially of those in the technologicallyimportant and prevalent class of granular materials, has obvious intrinsic value. In this paper, we focussed on force chains,in particular, their structural evolution and failure—an underlying physical cause of the loss of macroscopic strength underdilatation of dense granular materials. Developed herein was a structural mechanical model for the mechanism of bucklingof a force chain, subject to lateral support from weak network particles. The effect of particle shape irregularities, i.e.interlocking, was accounted for by introducing a contact moment or rolling resistance. Validation of the model wasperformed against results from a quantitative analysis of force chain buckling events in deforming granular systems usingDEM. Good agreement was found between model predictions and data from DEM simulations.

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By varying particle-scale properties, we shed new light on the interconnections between material behavior at three lengthscales: the microscopic or particle scale, the mesoscopic event of confined buckling of force chains, and the observed behavioron the macroscale. Using the buckling model, we have shown explicitly and quantitatively the effect of resistance to relativeparticle rotations on macroscopic strength. Specifically, it was found that the introduction of rolling resistance serves tostabilize force chains, both directly and indirectly. It increases the stability of the force chain structure directly, by providinggreater resistance to relative rotation which is an underpinning mechanism of buckling. Thus, it enables force chains towithstand more load and store more energy, before the onset of instability or critical buckling load. Although the effect ofrolling resistance on the effective lateral supporting force was not considered explicitly in this study, it is reasonable to expectthat a greater rolling resistance in contacts involving weak network particles would stabilize the lateral support to force chainsand thereby indirectly enhance force chain stability. For very small rolling resistance, the plastic buckling regime is reachedrapidly, resulting in a force–displacement response exhibiting a snapdown path. However, the addition of an initialimperfection smooths this path, and results in the peak stress increasing with increasing rolling friction. These findingselucidate why, as observed in DEM simulations of biaxial tests where force chains are rarely if ever perfectly straight, both themacroscopic peak shear stress and the steady critical-state shear stress increase with increasing rolling friction. Moreover, themodel can reliably reproduce the trends observed in DEM on the progression of plastic buckling. As rolling friction is increased,the origin of plastic buckling switches from the force chain column to the laterally supporting weak network contacts. This isconsistent with DEM data which show that, as rolling friction is increased, so does the population of buckling segments wherethe first contact to reach plastic threshold belongs to the laterally supporting weak network.

The model has shed light on the controversial issue of the use of rolling resistance or contact moment in DEMsimulations involving circular or spherical particles. In the absence of rolling resistance in DEM simulations, the tangentialforces within the force chain and surrounding weak network particles, along with the confining pressure, are the elementsthat provide the resistance to force chain buckling. This explains why shear bands can still form in these systems.

This study has quantified the tremendous influence that the lateral support, provided by the weak network particles,has on the stability of force chains. The degree of lateral support determines the extent to which force chains can persistbefore buckling: the greater this support, the higher are the critical buckling load and the displacement at the criticalbuckling load. Moreover, a higher limiting lateral stress from the weak network particles means less plastic displacementand dissipation, and hence a relatively higher post-critical load-carrying capacity. These results thus yield new insights intothe underlying causes of several well-known trends observed under increasing confining pressure: peak shear stress, strainat peak shear stress, steady critical-state shear stress all increase while the rate of softening decreases.

Above and beyond the establishment of fundamental interconnections in material behavior across multiple length scales,the analysis presented here has yielded two missing elements in micromechanical constitutive development. The first is theset of relations describing the evolution of particle kinematics in the confined buckling of a force chain. As discussed inearlier studies (e.g. Tordesillas, 2007a; Walsh et al., 2007 and references cited therein), compelling experimental andnumerical evidence suggests that the nonaffine deformation of this dissipative mechanism and its evolution with strain canbe appropriately tied to internal variables and their evolution laws. Thus, by following this connection, the formulationpresented here provides clear physical meaning to and quantifies these internal variables and their evolution laws. Second,as the kinematical relations are entirely expressed in terms of particle-scale properties, these relations complete theingredients required to construct a thermomicromechanical constitutive model (Tordesillas, 2007a; Walsh et al., 2007), withfull physical transparency across length scales—from the particle to bulk. A first generation constitutive model of this kindhas been developed and presented in the companion paper (Tordesillas and Muthuswamy, 2008).

Ongoing studies are focused on three aspects of force chain buckling. The first concerns a sensitivity analysis to theinitial configuration that is assumed for the structural mechanics analysis of the particle cluster comprising the force chainand its lateral supporting neighbors. The second deals with a detailed characterization of the lateral support in an attemptto establish a connection between it and the input spring stiffnesses, rolling and sliding friction coefficients and the appliedconfining pressure. This would enable us to construct a more accurate model of the lateral support and would obviate theneed to conduct local force chain buckling analysis in order to calibrate the parameters associated with this support. Thethird deals with the process by which buckling propagates within the force chain network, in particular, how the bucklingof one force chain triggers the buckling of nearby force chains as well as the interaction between buckling columns. Drivingthis third study is the motivation to understand the lateral and longitudinal propagation of shear bands.

Acknowledgments

We thank our reviewers whose constructive comments and insights have helped us improve this paper. We acknowledgethe support of the Australian Research Council (Discovery Grants DP0558808 and DP0772409) and the US Army ResearchOffice (Grant W911NF-07-1-0370) to AT and the Pratt Foundation Scholarship for postgraduate support to MM.

Appendix A. Model of a force chain column with an imperfection

Consider the case of an imperfect force chain column in the N ¼ 3 model. We introduce an initial deflection of thecentral force chain particle from the vertical axis of yimp, such that the initial length of the undeformed column becomes

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4R cos yimp. Noting that �yð2Þ ¼ yð1Þ ¼ y and Dnð1Þ¼ Dnð2Þ

¼ Dn still hold (as in Section 3.1), Eqs. (5)–(8) become

Dtð1Þðwð1Þ;wð2Þ; yÞ ¼ 2Rðsin y� sin yimpÞ � Rðwð1Þ þwð2ÞÞ,

Dtð2Þðwð2Þ;wð3Þ; yÞ ¼ �2Rðsin y� sin yimpÞ � Rðwð2Þ þwð3ÞÞ,

Dsð2ÞðyÞ ¼ 2Rðsin y� sin yimpÞ; DðD

n; yÞ ¼ 4R cos yimp � 2ð2R� DnÞ cos y.

Using the same method in Section 3.1, the only equilibrium path for the elastic regime is

FðyÞ ¼Rkn

cosy1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�


FC þ kskrÞðsin y� sin yimpÞ

knðkt

FC þ krÞ sin y

vuut0@

1A; y4yimp. (A.1)

For Stage 2, we use the same method to obtain Eq. (23) in Section 3.2.3 to get

FðyÞ ¼Rkn

cosy1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�

2 cos2 ykn sin y

2ktFCkrðsin y� sin yimpÞ

ktFC þ kr þ

msss

2

!vuut0@

1A. (A.2)

For Stage 3, FðyÞ is identical to the case with no imperfection (Eq. (26)).

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