Shear band emergence in granular materials - A numerical study

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICSInt. J. Numer. Anal. Meth. Geomech., 2007; 31:373–393Published online 3 January 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nag.590

Shear band emergence in granularmaterials}A numerical study

Alison Ord1,*,y, Bruce Hobbs1 and Klaus Regenauer-Lieb1,2

1CSIRO Exploration and Mining, P.O. Box 1130, Bentley, WA 6102, Australia2University of Western Australia, Australia

SUMMARY

In contrast to continuum systems where localization or shear banding arises through a bifurcation in apredefined system of differential equations, shear bands emerge in numerical simulations of deforminggranular systems with no prescribed mathematical relations other than simple contact forces betweenparticles. Shear bands emerge from the self-organization of large numbers of particles with long-rangegeometrical interactions playing a dominant role; both translation and rotation of particles are important.Granular media therefore deform more like materials with non-local constitutive relations than materialswhere only first-order interactions are relevant. In this paper we adopt a thermo-mechanical approach andexplore the fluxes of energy in the evolving granular system (that has cohesion as well as friction betweenthe particles) as it is loaded through the unstable localization regime, and track the evolution of energydissipation. As in continua, the sign of the second-order work defines the emergence of instability in thesystem. Initially, these instabilities decay into stable configurations of particles but with continued loading,force chains collapse locally to generate geometrically necessary fractures. These zones then propagate togenerate localization zones. When these fractures form a continuous network, the system is at thepercolation thresh-hold for broken bonds. However, long before this stage, the second-order workfluctuates in bursts weakly correlated with bursts in kinetic energy as damage accumulates. This behavioursuggests that any continuum constitutive description of granular media must be (i) non-local inan anisotropic manner, (ii) micro-polar, and (iii) involve damage evolution. Copyright # 2006 John Wiley& Sons, Ltd.

Received 26 October 2006; Accepted 27 October 2006

KEY WORDS: shear bands; emergence; granular materials

1. INTRODUCTION

An understanding of shear band emergence in granular media is important because thedevelopment of fracture systems, shear zones, zoned mineral alteration systems (including ore

*Correspondence to: Alison Ord, CSIRO Exploration & Mining, P.O. Box 1130, Bentley, WA 6102, Australia.yE-mail: [email protected]

Copyright # 2006 John Wiley & Sons, Ltd.

bodies), and fluid percolation networks in deforming rock systems can all be thoughtof as emergent phenomena in geo-materials. However, prediction of the initiation, development,and evolution of fracturing and shear zone formation, and of associated properties suchas dilatancy and critical percolation thresh-holds, is remarkably difficult. This is a functionpartly of the heterogeneity of rock properties, and partly because we cannot see into thespecimen to see how it behaves during physical deformation experiments. This iswhere computer experiments can provide such valuable information. In this case we look atthe opportunities provided by a particle code for exploring the emergence of micro-structuresduring deformation [1].

Emergent phenomena are characteristic of large systems, in which energy is being dissipated,and which are being driven far from equilibrium. In such systems, wonderful patterning andorder can be generated, both in time and space, in systems that otherwise one would expect tobehave in uniform, or perhaps random, manners [2]. This emergence can commonly berepresented as a symmetry breaking phase transition (see [3]). A characteristic of such systems isthat there are first-order feedback relationships between several processes. Another importantcharacteristic is that the resultant patterning occurs on a number of scales so that something of afractal structure evolves; this is a fundamental characteristic of natural systems.

As Cundall [4] has pointed out, there is a basic difference between continuum codes andparticle codes such as PFC [5]. In continuum codes the (commonly imprecisely known, butcomplicated) physics has to be incorporated explicitly in the form of partial differentialequations, or perhaps empirical relationships, whereas relatively simple interactions betweenparticles in particle codes commonly suffice to produce quite complicated and emergentphenomena. Such emergent phenomena do not arise in continuum codes unless the basic physicsbehind the phenomena is incorporated a priori into the constitutive relations. Thus, asan example, localization arises in plasticity continuum codes only if non-associated plasticity,corners on a yield surface or strain softening are specifically incorporated in the constitutiverelations [6, 7]. By contrast, localization occurs in particle codes with the assumptions only ofNewton’s Laws of Motion, together with simple elastic and frictional interactions betweenparticles.

The continuum approach to simulation is a classical Aristotelian one: the problem is firstbroken down into its constituent parts and then the relationships between the parts are definedby means of mathematical expressions of physical and/or chemical laws such as constitutiverelations, Darcy’s Law or the Law of Mass Action. The assumption then is that the behaviour ofthe total system can be arrived at by combining the constituent parts in simple, or sometimesquite complicated non-linear ways. Normally this is done by writing a coupled set of partialdifferential equations that is then solved numerically by some differential equation solvingengine, such as FLAC [8].

The ultimate result of such a process is deterministic and arises solely from the proposedconstitutive behaviour together with assumed initial conditions and/or boundary conditions.This is the classical Aristotelian–Galilean–Newtonian way of doing science. It is deterministicand the results follow once the physical and chemical laws are prescribed along with initial and/or boundary conditions. As indicated above, we now know that in these kinds of systems,emergent behaviour will not appear unless bifurcation phenomena are explicitly built into themathematics that describe the system behaviour. The behaviour of particle codes, on the otherhand, is such that constitutive relations capable of exhibiting bifurcation behaviour do not needto be specifically incorporated in order for localized or persistent patterning to arise.

A. ORD, B. HOBBS AND K. REGENAUER-LIEB374

Copyright # 2006 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2007; 31:373–393DOI: 10.1002/nag

One important class of constitutive relations is the so-called non-local constitutive relations inwhich the mechanical behaviour at a point depends on the behaviour at points far removed fromthat point. The gradient class (see, for instance, [9, 10]), where the spatial gradients of thedeformation rate or stress rate are included within the constitutive relation, is a special case ofnon-local constitutive behaviour. Such constitutive relations automatically include a length scaleand, in addition to having the advantage that mesh sensitivity is no longer an issue atbifurcation points in continuum codes, these constitutive relations also do not posemathematical problems such as loss of ellipticity at bifurcation points. These materials exhibita large range of emergent behaviour in continuum calculations. The physical significance of non-local constitutive relations is that since each node in a continuum mesh is influenced by theevolution of nodes one or more removed from itself, long-range feedback interactions areautomatically incorporated into the mechanics of the system.

Another, related, important class of constitutive relations are the micro-polar relations thatinvolve rotations and/or force moments at the micro-granular scale [11]. Here, the stress tensoris asymmetric. Again, particle codes involve particle rotation as an intrinsic part of the micro-deformation processes and so one would expect any continuum representation to be micro-polarin nature.

Both these types of behaviour are intrinsic features of particle codes in that commonly, themotion of an individual particle is strongly influenced by the motion of a particle many timesremoved from itself and rotations of grains are fundamental aspects of the deformation. This isparticularly true for tightly packed assemblies with cohesion but less true for loosely packedmaterials without cohesion. Thus, a form of non-local, micro-polar interaction is an intrinsicfeature automatically incorporated within particle codes. In particular, Kuhn [12] has shownnumerically that the shearing stress in granular media depends upon the first two gradients ofshear strain. That is, we see behaviour that had not been considered feasible in continua withgradient plasticity. In such gradient theories, the case had been made that a dependence onthe odd order gradients of shear strain was not physically realistic [13–15] because of therequirement that the constitutive relation should be invariant with respect to a reversal in thesense of shearing.

2. BACKGROUND-OBSERVATIONS ON GRANULAR MEDIA

We are concerned here with the micro-mechanical processes that lead to the emergence of shearbands in deforming granular media. As such we explore the interaction and rearrangement ofparticles and groups of particles, and how these motions are expressed in the form of commonlyused kinematic and dynamic quantities that are routinely related to instability and bifurcationduring deformation, albeit normally in continua (see [16]). There are three important aspects ofthe problem:

a. Criteria that define the stability of the deformation history with respect to an infinitesimalperturbation of the system.

b. Criteria that define the uniqueness of the deformation path and whether bifurcation in thedeformation path will occur.

c. Criteria that define whether the bifurcation mode is one of localization or of some otherkind.

SHEAR BAND EMERGENCE 375


A deforming body is said to be stable if a vanishingly small perturbation in the deformationremains vanishingly small [17, 18]. The criterion proposed by Hill [19] for stability of a volumeV is

Z

V’sij ’eij dV > 0 ð1Þ

where ’sij is the stress rate tensor and ’eij is the total (elastic plus plastic) strain rate tensor.Equation (1) is true if

’sij ’eij > 0 ð2Þ

for all kinematically admissible but otherwise arbitrary ’eij : Drucker [20–22] proposed a stabilitycriterion ’sij ’e

pij > 0 where ’epij is the plastic strain rate tensor. Following tradition, we refer to ’sij ’eij

as the second-order work.The quantities we are interested in include the time rate of change of kinetic energy and the

second-order work in particular. We are also concerned with measures of the total dissipation ofenergy and how these are related to instability and the subsequent evolution of instability. Forinstance, just as Kuhn [12] has asked the question: ‘Are granular materials simple materials?’one could ask the question: ‘Are granular materials standard dissipative materials in the sense ofNguyen [16]?’ This would mean that the stability of the system could be expressed as a functionof only the Helmholtz Free Energy and the Intrinsic Dissipation Function and its evolutionwould be described by Biot’s Equation [16]. If this were to be true we would need to know thedetailed micro-mechanisms of deformation so that these could be built into the Helmholtz FreeEnergy and the Intrinsic Dissipation Function.

The situation in granular media is made rich because contributions to energy arise fromrotations of grains together with translations, and it is of interest to understand the relativecontributions of rotation and translation to instability development and evolution.

A number of different micro-structural processes have been observed in numericalexperiments on granular systems, contributing to dissipation of energy. These are brieflysummarized below:

(i) The formation and evolution of strong force chains between grains [23]. This processdivides the system at a particular instant into ‘strong force chain columns’ and ‘weakregions’ between these columns. Dissipation occurs when the strong force chains collapse,often due to the motion of a single grain in a particular chain to occupy a neighbouringvoid. Such a collapse only occurs at high local stresses. Both translation and rotation ofparticles are involved in this process, however, this motion of a single grain influences thesubsequent motion of grains perhaps 10 diameters removed from that grain.

(ii) Breakage of bonds between grains with subsequent frictional sliding. Wolf et al. [24] haveshown that dissipation arising from frictional sliding occurs dominantly in the weakregions between strong force chains.

(iii) The development of micro-bands approximately parallel to the direction of shearing ininhomogeneous simple shearing experiments [12]. These represent local areas of dilationand compaction that develop from the onset of deformation, that is, well within what onemight commonly think of as the ‘elastic’ range of deformation. These structures are notpersistent and tend to grow, interact, and disappear. They appear to be unrelated to thestrong force chains at any one time.

(iv) Non-persistent micro-slip bands that form early in the deformation [25] and that areoriented at 40–608 to the maximum compression direction in a pure shearing deformation.



(v) Persistent shear bands that tend to form relatively late in the deformation history [25, 26].These are the classical shear bands that would be interpreted as the result of bifurcationin classical continua. Oda and Konishi [27, 28], Oda and Kazama [29] and Walsh et al.[30] have interpreted these as forming as a result of buckling or kinking of the strongforce chains.

These observations are somewhat reminiscent of what has been observed in deformingcontinua: typically at various points on a loading curve for a continuum, starting at a criticaldisplacement, bifurcation modes may develop in the deformation. These modes may besymmetrical or asymmetrical but commonly are transient in the sense that they do not growwith continued deformation (see [31, 32]). If we test for the stability of the deformation thencommonly ’sij ’eij > 0 and the subsequent deformation remains homogeneous. If an asymmetricbifurcation does begin to grow then localization results and ’sij ’eij becomes zero or negative [33].The important distinction between continua and granular media is that unstable phenomenaseem to develop much earlier in the deformation history in granular media than in continua [12].

3. EXPERIMENTAL PROCEDURE

One objective of this paper is to explore the behaviour of the energy fluxes in a system that isloaded beyond the stage characterized by persistent localization of deformation. A secondobjective is to explore the relationship of this localization to the development of gradients in thedisplacement field. We also need to explore the micro-mechanics of shear band formation andthe further development of the shear band through to a discontinuity. This process is followedto see whether or not precursors to localization can be found. Further development of this workis designed to yield an improved micro-structural understanding of the energies and behavioursat the granular scale that could be incorporated within a continuum formulation for scaling-upof the micro-structural understanding into macrostructural descriptions of shear bandbehaviours through to the 100’s of kilometre scale.

The computational or virtual experiments described below are based on the numericalexperiments of Ord [34] which themselves explore the physical experiments of Edmond andPaterson [35]. The material in this case is based on the results for Gosford Sandstone (see also[1, 36]).

3.1. The virtual rock

The experiments reported here are two dimensional and are performed in plane strain; thevirtual material is 0.2m square on edge. It is comprised of particles with a range of radii from2 to 3mm, uniformly distributed, and packed so as to approach a porosity of 0.13 (seeFigure 1(a)), and a density of 2226 kg m#3: A process analogous to physical compaction isfollowed [5]. Particles of artificially small radii are created within the specimen space at randomco-ordinates. The particles are then expanded to radii within the desired range until the desiredporosity is obtained, which in this case required 1772 particles.

The initial normal and shear stiffnesses of the particle contacts, using a linear contact model,are both 1:16$ 1010 Nm#1; as is the wall normal stiffness, for wall–ball interaction. Parallelbonds, providing the attribute of cohesion to the material, are created for all contacts. The



normal and shear stiffnesses of the parallel bonds are both 1:16$ 1010 Nm#3; and the parallelbond normal and shear strengths are 3:5$ 106 Nm#2: The frictional coefficient assigned to theparticles is 1.0 (see [37], for a discussion of the role of interparticle friction in a spherical particlesystem). The material is pre-stressed to attain 10MPa in both the vertical and horizontaldirections, with a standard deviation of 0.005.

3.2. Boundary conditions

The two horizontal boundaries are rigid, and are moved at specified velocities, while a confiningpressure of 10MPa is maintained on the vertical walls. The maximum total bulk shorteningattained is 8%. A few experiments were run with velocities applied to all four walls in which case

(a) (b)

(d)(c)

Figure 1. (a) 1772 particles representing the virtual rock; (b) localized shearing, represented by populations ofcracks. Vertical axial strain ¼ 0:0196; (c) velocity field, axial strain ¼ 0:004; maximum velocity vector ¼0:025m s#1; and (d) contact force chains, axial strain ¼ 0:0196; maximum contact force ¼ 0:0675MN:



the applied velocities were changed during shortening of the specimen to produce an isochoricpure shearing deformation, based on an incremental shortening formulation.

The horizontal boundaries are accelerated gradually from zero velocity to the specifiedvelocity before the formal experiment takes place; some loading of the specimen takes place atthis stage. The loading rate during the formal experiment is sufficiently slow that the specimenremains in quasi-static equilibrium throughout the test. This was checked by monitoring thework done by the platens, and by comparing this to the strain energy stored in the particleassembly ([5]; PFC allows direct access to the data required to calculate these energies). Thesetwo energy quantities are required to be equal during the elastic portion of the experiment forthe conditions to be quasi-static. Otherwise the loading conditions are dynamic. It was foundthat this process of ensuring quasi-static deformation is very important. Quite differentbehaviours of force chains and patterns of ball motion develop if dynamic loading is imposedrather than quasi-static loading. Most of the published studies do not make this distinction.

Figure 2 shows the stress–strain curves (Figure 2(a) and (b)) as well as the comparisonbetween the energies of the system (Figure 2(c) and (d)) for both damped (Figure 2(a) and (c))and undamped (Figure 2(b) and (d)) conditions. The total strain energy of the entire assemblyacquired during formation of the virtual rock has been subtracted so as to allow for easiercomparison of these data. Damping is shown to affect the results most strongly in the post-yield

(a)

(c) (d)

(b)

Figure 2. Comparison of axial stress, and of the energies of the system, versus axial strain fordamped and undamped conditions: (a) axial stress versus axial strain, damped; (b) axial stress versusaxial strain, undamped; (c) the energies of the system versus axial strain, damped; and (d) the

energies of the system versus axial strain, undamped.



regime (Figure 2(a) and (b)). The boundary energy or total accumulated work done by all wallson the assembly represents the work done by the platens. The total strain energy represents thetotal strain energy of the entire assembly stored at all contacts. They are seen to be closelysimilar up to a strain of about 0.0025 for damped and undamped runs. Figure 2(c), undamped,and Figure 2(d), damped, show also the strain energy of the assembly stored in the parallelbonds, the frictional work or energy dissipated by frictional sliding at all contacts, the kineticenergy of all particles accounting for both translational and rotational motion, and the sum ofall the energies in the system. The sum of all the energies in the system tracks the boundaryenergy until a strain of about 0.008. There is a direct correlation between the rate of increase oftotal strain energy starting to decrease at a strain of about 0.0025, and an increase in the parallelbond strain energy, the frictional work, and the kinetic energy.

3.3. Mechanical behaviour

The bulk mechanical behaviour of the virtual specimen is represented by the stress–strain curveshown in Figure 3(a). The beginning of the formal experiment is assumed to be that step whenthe velocity specified for the loading platens for the experiment is attained, and this isrepresented by zero axial strain in this figure. The peak or maximum deviatoric stress attainedby the virtual specimen is about 22MPa.

Figure 3(b) represents the change in bulk volume with axial strain; regions below thehorizontal zero line represent bulk volume decrease, with bulk volume increase beingrepresented by regions above the horizontal line. The former may be attributed to settlingand compaction of the specimen, including elastic or reversible compaction. Volume increase ordilation may be associated with shearing of regions over each other, and locally, the opening upof holes in the structure (see [7] for a discussion of dilation and associated physical processes).

That section of the curve prior to peak stress, that is, that section for which the axial strain isless than 0.014, was analysed as described in [38]. The initial structure of the specimen is shownin Figure 1(a) and the development of localized zones of shearing and fracturing at a strain of0.0196 is shown in Figure 1(b). The vertical loading throughout the specimen, represented by thevelocity field for a strain of 0.004 (Figure 1(c)), is demonstrated for a strain of 0.0196 by thevertical and continuous lines of contact forces, as shown in Figure 1(d).

Figure 3. (a) Behaviour of axial deviatoric stress versus axial strain for the virtual rock; and (b) behaviourof volumetric strain versus axial strain for the virtual rock.



The unloading behaviour of the material is represented in Figure 4. Although there is minorhysteresis of the unloading/loading curve at low strains (Figure 4(a)), the elastic modulus staysvery much the same until after obvious yield in the material at a strain of about 0.008.Thereafter, hysteresis is marked, and the gradient of the unloading curve decreases withincreasing strain. The volume strain/axial strain behaviour (Figure 4(b)) shows convincingly theelastic response at low strains, and strong hysteresis in the volume strain at higher strains.

4. EXPERIMENTAL RESULTS

4.1. Micro-structural results

The micro-structural changes associated with various sections of the stress–strain curve areshown in Figure 5 by plots of broken parallel bonds existing between particles in the assembly,and by tracking the numbers of contacts within the assembly during loading. At the loweststrain, in the ‘elastic’ (or initially linear) section of the curve, almost all the bonds are intact.Prior to peak stress, a number of bonds are broken; post peak stress, there is no continuity ofunbroken bonds through the specimen; the specimen is seen to be composed of clusters ofbonded regions [38].

During the pure shearing, particles move, both by translation and by rotation. As a resultof such movements, contacts are broken resulting in the formation of shear- and normal-displacement cracks.

In Figure 5(a) and in subsequent Figures 7, 9, 10 and 11, three arrows are presented toindicate (i) the strain corresponding to the first burst of frictional dissipation (see Figure 11(d)),(ii) the strain corresponding to the first major burst of kinetic energy (see Figure 9(g)), and(iii) the strain corresponding to the critical percolation thresh-hold for bond breakage inthe specimen, that is, when the broken bonds form a continuous network across the specimen(see Figure 5(d) and (e)).

The details of the micro-structure are exhibited best in a model with many more grains. Sohere are shown micro-structural results for a model for which all the parameters are for the‘virtual rock’ described above, except that the grains are 10 times smaller, with a range of radii

Figure 4. Unloading behaviour of the system: (a) axial stress versus axial strain; and (b) volumetricstrain versus axial strain.



from 0.2 to 0.3mm (see [1], Figure 3). Force chains for the whole specimen at a strain of 0.017are shown in Figure 6(a) and the associated cracks are shown in Figure 6(b). The distribution offorce chains and cracks is shown at a higher magnification in Figure 6(c) and (d). The crackstend to occupy regions in the system where the strong force chains are less developed, in keepingwith the observation of Wolf et al. [24]. In Figure 6(d) the displacement field associated with thezone of localization of cracking in Figure 6(b) is shown together with the cracks at that strain.Clearly, the zone of localization corresponds to a discontinuity in the displacement field

Figure 5. Association of the evolution of the micro-structure, represented by broken bonds, of the virtualrock with axial strain. The snapshots are: (a) 508 200 steps (strain of 0.004); (b) 1 008 200 steps (strain of0.008); (c) 1 508 200 steps (strain of 0.012); (d) 2 008 200 steps (strain of 0.016); and (e) 2 508 200 steps

(strain of 0.020); and (f) number of contacts versus axial strain.



although the cracks are distributed on a much wider scale than simply the displacementdiscontinuity.

4.2. Energy fluxes in the system

Let us now explore the energies in the system. We begin with the kinetic energy. What is neededis not the total kinetic energy for the entire system, but the difference between the observedkinetic energies and the predicted kinetic energies. By ‘predicted energies’ here we mean thoseenergies that would have arisen if the deformation throughout the specimen had remainedhomogeneous in accordance with the imposed boundary conditions. In order to obtain this,

(a)

(c) (d)

(b)

Figure 6. Micro-structural behaviours for material with grain sizes of 200–300 mm (177 234)particles: (a) contact force chains, maximum contact force ¼ 0:25MN; (b) cracks; (c) contact forcechains and cracks (zoom), maximum contact force ¼ 0:10MN; and (d) cracks and displacement

field (zoom), maximum displacement vector ¼ 8:55$ 10#4 m:



deviations from homogeneous deformation are determined by subtracting the velocity field forhomogeneous isochoric pure shearing from the observed velocity field. The evolution of theperturbed velocity field is shown in Figure 7. Note that the cellular patterns of the perturbedvelocity field are not nearly as well developed here (except perhaps in Figure 7(b)) as has beenreported by other workers [25]. We note that the perturbed velocity field becomes closer tohomogeneous as quasi-static conditions are approached. For comparison, Figure 8 compares

(a) (b) (c)

(d)

(e)

Figure 7. Association of the evolution of the velocity perturbation field of the virtual rock with axialstrain, against deviatoric stress. The snapshots are: (a) 508 200 steps (strain of 0.004), maximum velocity of0:015m s#1; (b) 1 008 200 steps (strain of 0.008), maximum velocity of 0:015 m s#1; (c) 1 508 200 steps(strain of 0.012), maximum velocity of 0:1m s#1; (d) 2 008 200 steps (strain of 0.016), maximum velocity of

0:1m s#1; and (e) 2 508 200 steps (strain of 0.020), maximum velocity of 0:05 m s#1:



perturbed velocity fields for approximately 5% shortening for dynamic conditions (Figure 8(a))and for a (quasi-static) strain rate 100 times slower (Figure 8(b); 4% shortening).

For a given grain, the kinetic energy ec is

ec ¼ 12mv2 þ 1

2 Io2 ð3Þ

where m is the grain mass, v the speed, I the inertia moment, and o the rotation speed.The local kinetic energy is determined from the difference velocity at each node.

The translational part and the rotational part of the kinetic energy are calculated separatelyso as to allow exploration of the differences in the processes by which the micro-structuralrearrangements occur during loading. Although shedding some light on the spatial andtemporal variations of kinetic energy during loading, the local results do not provide any furtherprecursor information than exploring the aggregation of particles described earlier.

Let us now explore the system globally rather than locally. First, we examine the deviatoricstress versus strain and its gradient or first derivative. As might be expected, the gradient of thedeviatoric stress shows a marked deviation from linear behaviour during the hardening section(Figure 9(a) and (b)). Second, let us now look at the global kinetic energy, still calculated usingthe difference velocity field. The global kinetic energy sums the kinetic energy per grain overN, the total number of grains in the assembly:

Ec ¼XN

k¼1

ekc ð4Þ

The global kinetic energies are shown in Figure 9, including the translational (Figure 9(c)),rotational (Figure 9(e)), and total kinetic energies (Figure 9(g)). The first major burst of energyoccurs during the hardening section of the stress–strain curve, where the gradient of thedeviatoric stress shows its first marked deviation. Note that both translational and rotationalkinetic energies contribute strongly to the total kinetic energy. Comparison of Figures 5 and 7

(a) (b)

Figure 8. Perturbed velocity field: (a) experiments for which the applied velocity is 100 timesfaster than most of the experiments reported here, maximum velocity ¼ 5:0 m s#1; axial strain¼ 0:049; and (b) applied velocity the same as most of the experiments reported here,

maximum velocity ¼ 0:05 m s#1; axial strain ¼ 0:040:



Figure 9. Kinetic energy diagrams: (a) axial stress versus axial strain; (b) gradient of the stress–straincurve; (c) translational kinetic energy; (d) gradient of the translational kinetic energy; (e) rotationalkinetic energy; (f) gradient of the rotational kinetic energy; (g) total kinetic energy; and (h) gradient

of the total kinetic energy.



indicate that at the first burst of kinetic energy there is little in the way of bond breakagealthough cellular patterns for the perturbation velocity field are well developed.

We now explore the second-order work, ’s ' ’e; which, using the Hill criterion, is positive forstability, and non-positive for instability [33].

In the discrete case, the second-order work is suggested [39] to be represented by

d2W ¼ ds de ð5Þ

Following [39] we calculate

d2W ¼ dF ' dlþ dC dO ð6Þ

where dF is the force increment on a given grain, dl is the displacement of the grain under thisforce increment, dC is the resultant couple, and dO is the rotation about the particle axis. Forthe assembly of grains, the global work is the local work summed over N, the total number ofgrains in the assembly:

D2W ¼XN

k¼1

d2Wk ð7Þ

We access the absolute magnitudes of the out-of-balance force for each particle, and the out-of-balance moment, together with the perturbed displacement and the rotational velocitymultiplied by the time step. We calculate the change in work for each time step, and divideby the time step to obtain the second-order work. The results are shown in Figure 10.

This global second-order work mimics the first burst in global kinetic energy, during the strainhardening section of the stress–strain curve. However, there are minor bursts of rotationalinstability quite early in the loading history, prior to this, that are not accompanied by bursts inkinetic energy. Figure 10(a) shows the rotational part of the second-order work (cf., Equation(6)), Figure 10(b) shows the translational part and Figure 10(c) the total second-orderwork, D2W : Again, both translational and rotational parts contribute to the total work. InFigure 10(d) the translational and rotational parts are compared. Initially, the particles onlyrotate; translational movement only begins at a strain of about 0.002. At higher strains, the twoare not strongly correlated. In Figure 10(e) the particle rotations are plotted against theirmoments. As expected, there is a strong positive correlation. Comparison of Figures 5, 7 and 9show that the main bursts of second-order work occur quite late in the loading history,associated with the onset of the main bursts of kinetic energy (although in detail not stronglycorrelated) and not particularly associated with the percolation thresh-hold for bond breakagein the specimen.

A different representation of the second-order work comes through consideration of the workassociated with the particle contacts (Cundall, Lambert, pers. comm., 2006) rather than withtranslation and rotation of the grains. We access one of the six energy partitions in the entireparticle assembly, the total energy, Ef ; dissipated by frictional sliding at all contacts [5]:

Ef Ef #X

Nc

ððF si ÞðDU

si Þ

slipÞ ð8Þ

where Nc is the number of contacts; and F si and Us

i slip are the average shear force and theincrement of slip displacement, respectively, at the contact for the current timestep. The work iscalculated through division by the time step. The most obvious difference between the workderived from frictional sliding at all contacts (Figure 10(f)) and the work derived from particle



translations and rotations (Figure 10(c)) is the cumulative nature of the work associated with thecontacts. Also, the magnitudes reached (Figure 10(f)) are on average about five times greaterthan for the work associated with the particle movement (Figure 10(c)). The first perturbation isseen at a strain slightly less than 0.004, detectable in both cases (see Figure 10(c) and (f)). Theperturbation at a strain of about 0.005 is detectable in Figure 10(a)–(c) and (f), being at afractionally smaller strain in the latter case.

Figure 11 presents the evolution of the total strain energy (Figure 11(c)), the gradient ofthe frictional dissipation (Figure 11(d)) and the evolution of the total energy of the system(Figure 11(e)) compared to the stress–strain curve (Figure 11(a)) and its gradient (Figure 11(b)).The essential feature here is that bursts of frictional dissipation begin early in the loading historywith no particular relationship to features of the stress–strain curve but more or less matchingbursts of kinetic energy. The first burst in frictional dissipation also occurs before the first

Figure 10. Second-order work diagrams: (a) rotational second-order work (particles); (b) translationalsecond-order work (particles); (c) total second-order work (particles); (d) correlation of rotational (vertical)with translational (horizontal) second-order work (particles); (e) rotation (vertical) versus moment

(horizontal) (particles); and (f) total second-order work (contacts).



departure in linearity of the stress–strain curve. The essential picture painted here is one ofprogressive evolution of damage and associated frictional dissipation rather than the emergenceof a discrete ‘bifurcation’ point.

5. DISCUSSION

Sporadic bursts of the kinetic energy of the system occur during the hardening section of thestress–strain curve, post- first yield, and pre- peak stress. These bursts of kinetic energy occurfrom very early in the loading history and correspond to local re-arrangements of grains thatpresumably correspond to the collapse of strong force chains and perhaps the development ofthe micro-compaction and micro-shear bands described by Kuhn [25]. Both translational androtational micro-structural re-arrangements give rise to these bursts of energy. Spatially, these

Figure 11. (a) Axial stress versus axial strain; (b) gradient of the stress–strain curve; (c) totalstrain energy; (d) gradient of the frictional dissipation energy; and (e) total energy ¼elastic energyþ frictional dissipation energyþ total kinetic energyþ parallel bond energy:



re-arrangements emerge spontaneously and apparently, at least initially, randomly within thespecimen. During this initial loading a strong anisotropy develops [23–25, 27, 28, 30, 40, 41]which seems to be fundamental to future localization development. One of the chief expressionsof this anisotropy is the formation of strong force chains that isolate regions of very low averagestress. Many but not all bursts of kinetic energy are associated with non-positive values of thesecond-order work. This presumably indicates unstable behaviour but these instabilities do notgrow in the early stages of loading and the perturbation generated by the kinetic energy burstis absorbed into stable new particle arrangements. Still early in the loading history, grainre-arrangements result in breakage of bonds but these are isolated and not connected spatially.With increased deformation, as these re-arrangements increase in number, they coalesce, into amacro-structural arrangement spatially controlled by the global boundary conditions.Ultimately, these broken bonds form a continuous network across the specimen and thispercolation thresh-hold corresponds approximately, but not exactly, with peak stress. Thisthresh-hold corresponds to intense bursts of kinetic energy but not the first large non-positivevalues of the second-order work. At this stage, the deformation is not only unstable but theinstability amplifies with continued deformation. Throughout this process, the bond-breakingactivity is concentrated initially at large gradients in the velocity field. Such gradients seem todevelop through local collapse of strong force chains. These gradients are preserved throughoutthe deformation and seem to spread from initial, local sites of heterogeneity. These brokenbonds can be thought of as ‘geometrically necessary’ fractures. The curvature associated withthese initial heterogeneities spreads along with the development of other geometrically necessaryfractures as the deformation proceeds until a percolation thresh-hold is reached. This part of theprocess corresponds to the propagation of the newly nucleated localization zone. From then on,new fractures are added to the array as the deformation proceeds until eventually a wide zone ofdamage is generated which is identified macroscopically as a zone of localization. Thedevelopment of such fracture systems is associated with strong, random bursts of kinetic energyand of non-positive second-order work.

6. CONCLUDING REMARKS

PFC is capable of producing self-organized or emergent behaviour with the initial incorporationof only the simplest of contact relations between particles [4].

One of the most interesting challenges in mechanics at present is to develop the continuumconstitutive relationship for such granular media. This presents all the challenges of cross-scaleprojections and of understanding the micro-mechanical behaviour of granular media at manyscales and of how to formulate this behaviour in classical, non-local constitutive behaviour or innon-local ways that capture the physics of what actually occurs at the granular (sensu stricto)or multi-grain scale. One approach is to insist that the proposed constitutive relation iscompatible with the Second Law of Thermodynamics, namely that the dissipation of energy benon-negative. This is the approach followed by Walsh and Tordesillas [42] and by Walsh et al.[30]. However, this approach also carries with it other necessities: How do we use thisthermodynamic approach to understand if the current configuration of the granular assembly isstable? Is it such that local instabilities dissipate and do not grow? Or do unstable bifurcationprocesses dominate and ultimately grow to express themselves as localized shear bands? This iswhy it is important to make connection with the work of Nguyen [16].



These questions are still quite open as to the important criteria we are trying to address whenit comes to: (i) instability in granular media; and (ii) the criteria for bifurcation and when is thisnecessarily equivalent to instability? For example, Cherukuri and Shawki [43, 44] have shownthat the rate of change of kinetic energy is a parameter that tracks instability nucleation andsubsequent evolution in viscous continua with thermal–mechanical feedback; is this criterionimportant for granular media? Is the second-order work of relevance? Is maximization of thedissipation important? As indicated, the relevant criteria are quite open at the moment. Thechallenge is to integrate the micro-mechanical behaviour with the assembled array of knowledgethat is available on the micro-mechanical behaviour of granular media and the thermodynamicbehaviour of materials together with associated instability criteria.

The present study shows that granular media do not behave in the ‘clean’ manners thatcontinua do, even non-local continua. In granular media there is no clear bifurcation pointmarked by a clear non-positive value for the second-order work or by a clear peak in the kineticenergy rate (see [43, 44]). Instead, there is a gradual evolution of instabilities some of whichdecay back to stable assemblies of particles and some of which begin to grow. The progressionto persistent localization is gradual and marked by bursts of non-positive second-order workand weakly correlated bursts of kinetic energy. It is this gradual evolution of damage that needsto be incorporated into a continuum formulation. It would appear that the important aspects ofa thermo-mechanical approach to developing a continuum constitutive relation for granularmedia involve:

(i) a gradient approach that accounts for long range (10 particle diameter) interactions; suchinteractions are strongly anisotropic in that interactions commonly occur along thestrong force chains;

(ii) a micro-polar approach that accounts for grain rotations; and(iii) the progressive evolution of damage.

ACKNOWLEDGEMENTS

Congratulations to ITASCA on their 25th anniversary. Without Peter Cundall and his, the ITASCA,codes, this work would not have been pursued. We acknowledge Yanhua Zhang’s contribution ofsupporting models. Alison thanks the CSIRO Complex Systems Science group, particularly John Finnigan,for support of this research area. We thank Ioannis Vardoulakis, George Exadaktylos, attendees, andsponsors of the 7th International Workshop on Bifurcation, Instabilities and Degradation inGeomechanics for an invigorating meeting in Crete. Felix Darve’s critique resulted in much new work.The recommendation of David Schaeffer to discuss this work with Antoinette Tordesillas has resulted in amuch improved manuscript as a result of stimulating discussions with her and her students.

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Shear band emergence in granular materials - A numerical study

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