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Particuology 8 (2010) 119–126

Contents lists available at ScienceDirect

Particuology

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uasi-static simulations of compact polydisperse particle systems�

olin Thorntonchool of Civil Engineering, University of Birmingham, Birmingham, UK

r t i c l e i n f o

rticle history:eceived 30 March 2009ccepted 3 July 2009

a b s t r a c t

The Discrete Element Method (DEM) was originally devised by Cundall and Strack (1979), as a techniqueto examine the micromechanics of granular media with the anticipation that this would lead to more

eywords:iscrete Element Methoduasi-static deformationranular media

physically reliable continuum theories to describe the quasi-static deformation of granular material suchas sand. However, the methodology models the evolution of a system of particles as a dynamic process.Consequently there have been numerous publications of the application of DEM to an increasingly widervariety of problems in many areas of engineering and science. This paper, however, focuses on the orig-inal motivation for DEM and attempts to provide a state-of-the-art understanding of the quasi-staticdeformation of granular media.

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© 2009 Chinese So

. Introduction

Sands, grains and powders are granular materials whoseacroscopic behaviour during quasi-static deformation is related,

n some way, to the spatial and size distributions of the constituentarticles and the load–displacement behaviour at the interparticleontacts. Traditional theoretical and experimental investigationsf the mechanical behaviour of granular materials are restricted byhe limited quantitative information about what actually happensnternally. Laboratory experiments on real materials rely on esti-

ates of the macroscopic stress and strain states from boundaryeasurements, which themselves depend on assumptions made

bout the material behaviour. Information about the internalicromechanics is rare, since attempts at direct observation andeasurement intrude upon the material response. In addition,

omparisons between sets of data are uncertain due to the inabil-ty to prepare exact replicas of the physical system. Traditionalttempts to mathematically model the mechanical behaviour ofranular media are normally based on intuitive speculation as toow the observed experimental behaviour might best be modelledy modifying existing continuum mechanics theories. However,

he resulting theories invariably include new parameters, therecise physical meaning of which is relatively obscure. This leadso difficulties in selecting appropriate experiments to rigorouslyest a theory.

� Conference papers selected from International Symposium on Discrete Elementethods and Numerical Modelling of Discontinuum Mechanics, September 2008,

eijing.E-mail address: C.Thornton@bham.ac.uk.

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674-2001/$ – see front matter © 2009 Chinese Society of Particuology and Institute of Process Eoi:10.1016/j.partic.2009.07.007

of Particuology and Institute of Process Engineering, Chinese Academy ofSciences. Published by Elsevier B.V. All rights reserved.

An alternative approach is to perform computer-simulatedxperiments using the Discrete Element Method (DEM), origi-ally devised by Cundall and Strack (1979), in which the particle

nteractions are modelled as a dynamic process and the time evo-ution of the system is advanced using a simple explicit finiteifference scheme. The technique offers perfect control over exper-

ments, non-intrusive measurement and, with computer graphics,ermits visualisation of the internal micromechanical processes,hereby providing additional information not readily accessiblen real physical experiments. Perhaps, most significantly, manyifferent tests can be simulated on exactly the same initially pre-ared sample, avoiding the main source of uncertainty in laboratoryxperiments.

Most of the early applications of DEM to quasi-static deforma-ion were restricted to two-dimensional systems (Cundall & Strack,983; Thornton & Barnes, 1986; Rothenburg & Bathurst, 1989). Dueo the kinematic constraints on particle movement in two dimen-ions and the restricted loading paths available, comparisons witheal experiments were of limited value. Following the develop-ent of the three-dimensional simulation code TRUBAL by Cundall

1988) it is possible to explore the ability of DEM to simulateuasi-static shear deformation of idealised granular media undereneral 3D states of stress and strain and to compare the resultsbtained, at least qualitatively, with laboratory experiments,ncluding results obtained from true triaxial apparatuses. In thisaper, results are presented of 3D DEM simulations of polydisperse

ystems of elastic frictional spheres subjected to (i) axisymmet-ic compression and (ii) radial deviatoric strain paths. The workeported is, in effect, a review of the author’s previous workresented at conferences and published in journals (Thornton,

ngineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.

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997, 2000; Thornton & Antony, 1998; Thornton & Zhang,006).

. Simulation details

In soil mechanics, the traditional laboratory test is the so-calledstandard triaxial test’ in which a cylindrical specimen of soil isompressed axially between two rigid platens. The specimen isnclosed by a flexible rubber membrane that permits the horizon-al normal stresses to be controlled by fluid pressure. This ensureshat the principal stresses are �1 > �2 = �3. The objective of suchests is to apply uniform strain conditions and, assuming homo-eneous stress states, identify the material behaviour from whichontinuum models can be developed. This cannot be achieved dueo boundary influences. In DEM simulations one may choose to

imic laboratory experiments but, if the objective is to relatehe micromechanics to the meso-scale constitutive behaviour, its necessary to use a periodic cell. In this paper, all the simulationseported were performed using a representative volume element,ith periodic boundaries, subjected to uniform strain fields. In thisay, ‘perfect’ experiments are created free from boundary effects.

n order to control the deformation of the system, a strain-rate ten-or εij is specified, according to which the centres of the spheres inhe periodic cell move, as though they are points in a continuum,o satisfy the equation

xi = εijxj�t, (1)

n which xi are the coordinates of the sphere centre and �t ishe small timestep used to advance the evolution of the system.dditional incremental displacements occur as a result of the inter-ctions between contiguous spheres.

In order to permit quasi-static (ε = 10−5) simulations to be con-inued to large strains (ε = 0.5) within a reasonable timescale it isecessary to use density scaling. In the simulations reported thearticle density is scaled up by a factor of 1012. This increases theimestep from microseconds to seconds and does not affect theorces, displacements, work or energy (Thornton, 2000).

In order to follow desired stress paths, servo-control algorithmsre required and take the general form

˙ = g(�∗ − �) or ε = ε + g(�∗ − �), (2)

here �* is the desired value of stress, � is the calculated value andis a gain parameter whose appropriate value is obtained by trialnd error. The first equation is used to bring the system to equilib-ium at a desired stress state, e.g. isotropic compression. The secondquation is used to adjust the strain-rate to minimise the differenceetween the desired and calculated stress states when following aesired stress path, e.g. when performing shear deformation withhe mean stress held constant. Eq. (2) may be expressed in termsf individual components or combinations of components of thetrain-rate and stress tensors depending on the desired loadingath to be followed. More than one servo-control algorithm maye used but care must be taken to avoid conflicting adjustments tohe strain-rate tensor.

The ensemble average stress tensor is calculated from

ij = 1V

C∑

1

(R1 + R2)Nninj + 1V

C∑

1

(R1 + R2)Tnitj,

here n t = 0, (3)

i i

here the summations are over the C contacts in the volume V,1 and R2 are the radii of the two spheres in contact, N and T arehe magnitudes of the normal and tangential contact forces for theontact orientation defined by the unit vector normal to the contact

saFtb

8 (2010) 119–126

lane ni and ti defines the unit vector parallel to the contact plane.he structural anisotropy is characterised by a fabric tensor (Satake,982) defined as follows

ij = 〈ninj〉 = 1C

C∑

1

ninj. (4)

The average number of contacts per particle, the coordinationumber, is normally defined by Z = 2C/Np where Np is the numberf particles. However, in all simulations in a periodic cell with zeroravity it is found that there are always a number of particles thatave only one or no contacts and such particles do not contributeo the force transmission through the system (Thornton & Antony,998). Consequently, we define a mechanical coordination numbers

m = 2C − N1

Np − N0 − N1, (5)

here N1 and N0 are the number of particles with only one or noontacts, respectively.

To start a simulation, spheres are randomly generated within auboidal periodic cell sufficiently large to provide an initial solidraction of about 0.5 with no interparticle contacts. After gener-tion, the system is subjected to isotropic compression using atrain-rate of 10−4 s−1 until the mean stress has reached a valuef about 10 kPa. Isotropic compression is then continued using therst servo-control algorithm given in Eq. (2) to incrementally raisehe mean stress to the desired final value. If the value of inter-article friction to be used in the shear stage is introduced at thetart of isotropic compression then a relatively loose sample will bebtained. In order to obtain a dense sample the interparticle frictions set to zero during isotropic compression until the mean stress haseached 90% of the desired final value and then the desired value ofnterparticle friction is introduced for the final increment of meantress. A very loose sample can be obtained by using the desiredalue of interparticle friction and switching off any particle rota-ion until the mean stress has reached 90% of the desired final value.sing the above procedures ensures that, with a sufficiently largeumber of particles, the prepared sample is isotropic in terms ofoth the stress tensor and the fabric tensor prior to shearing.

. Axisymmetric compression

Figs. 1–4 show results obtained from simulations of axisymmet-ic compression (�1 > �2 = �3) on a polydisperse system of 3620lastic spheres during which the mean stress (�1 + �2 + �3)/3 waseld constant at 100 kPa using the second of the servo-control algo-ithms given in Eq. (2). At the start of shear, the solid fractions are.660 (dense) and 0.618 (loose) and the mechanical coordinationumbers are 6.07 (dense) and 5.02 (loose). Further details can be

ound in Thornton (2000).Fig. 1 shows the evolution of deviator stress (�1 − �3) with devi-

tor strain (ε1 − ε3) for both the dense and loose systems. Theorresponding evolution of volumetric strains is shown in Fig. 2.he figures show that, qualitatively, the stress–strain-dilationesponses obtained for both the dense and loose systems are typicalf those obtained in laboratory experiments. The initial shear mod-lus is much higher for the dense system, which exhibits a peak

n the stress–strain curve at about 5% strain followed by strain-oftening behaviour. The loose system does not exhibit any strain

oftening: the deviator stress increases at a decreasing rate untiln essentially constant value is reached at about 15% strain (seeig. 1). The volumetric strain responses show that the dense sys-em expands and the loose system contracts (Fig. 2). At large strains,oth systems deform at constant volume and this is associated with

C. Thornton / Particuology 8 (2010) 119–126 121

Fig. 1. Evolution of deviator stress during axisymmetric compression (constantmean stress = 100 kPa).

Fig. 2. Evolution of volumetric strain during axisymmetric compression (constantmean stress = 100 kPa).

Fig. 3. Evolution of deviator fabric during axisymmetric compression (constantmean stress = 100 kPa).

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ig. 4. Evolution of average mechanical coordination number during axisymmetricompression (constant mean stress = 100 kPa).

constant deviator stress that is independent of the initial solidraction.

In addition to the macroscopic behaviour shown in Figs. 1 and 2,t is also possible with DEM to examine various internal variablesnd how they evolve. It is now well known that shear deformationf granular media produces an induced structural anisotropy thats developed primarily as a result of contact separation occurringn directions that are approximately orthogonal to the direction ofhe major principal stress �1 (Thornton & Barnes, 1986). Structuralnisotropy is defined by the distribution of contact orientations andharacterised by the fabric tensor �ij defined by Eq. (4). Fig. 3 showshe evolution of the induced structural anisotropy, defined by theeviator fabric (�1 − �3), for both dense and loose systems. The fig-re shows that the structural anisotropy increases at a decreasingate to a maximum value that is dependent on the initial pack-ng density. The dense system exhibits a decrease in structuralnisotropy at strains above 10% until, at large strains, the degreef structural anisotropy is the same for both systems. It may alsoe noted that the shape of the curves are somewhat similar tohe deviator stress curves shown in Fig. 1. However, there is noseful correlation between the deviator stress and the deviatorabric.

The evolution of the mechanical coordination number, definedy Eq. (5), is shown in Fig. 4. During the initial 3% deviator strainhere is a rapid change in the mechanical coordination numberntil a critical value is attained that remains essentially constanthereafter, irrespective of whether the system is expanding or con-racting. Other simulations (Zhang & Thornton, 2005) have shownhat, if the mean stress is increased, this ‘critical’ coordination num-er increases and the deviator fabric at large strains decreases.

Eq. (1) demonstrates that the state of stress is a function of theistribution of contact forces. From both photoelastic studies ofD arrays of discs (Dantu, 1957; de Josselin de Jong & Verruijt,969; Drescher & de Josselin de Jong, 1972; Oda & Konishi, 1974;akabayshi, 1957) and both 2D and 3D DEM simulations (CundallStrack, 1979; Thornton & Barnes, 1986; Thornton, 1997) it has

een shown that strong forces are transmitted by relatively rigid,eavily stressed chains of particles forming a relatively sparse

etwork of above-average contact forces. Even when both theicrostructure and the stress state are isotropic, some contacts

ransmit forces several times those of others but with no pre-erred direction for the larger contact forces. During shear the

122 C. Thornton / Particuology 8 (2010) 119–126

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taselects favourable contact orientations within the overall struc-ture. Consequently, a knowledge of the overall induced structuralanisotropy is necessary but not sufficient.

ig. 5. Evolution of the contributions of the weak (f < 1) and strong (f > 1) sub-etworks to the deviator stress.

arge forces are orientated in the direction of the major princi-al stress �1. Radjai, Jean, Moreau, and Roux (1996) suggestedhat the contact network may be partitioned into two complemen-ary sub-networks: a ‘strong’ percolating sub-network of contactsransmitting above-average contact forces; and a ‘weak’ sub-etwork of contacts carrying below-average contact forces. FromD simulations of quasi-static biaxial compression on a polydis-erse system of about 4000 rigid discs, they concluded that thetrong sub-network accounts for all the deviator stress, whereashe weak sub-network contributes only to the isotropic stress. Thismportant observation was verified from results of 3D DEM simu-ations of axisymmetric compression on a polydisperse system of000 ‘soft’ spheres by Thornton and Antony (1998).

Fig. 5 illustrates the evolution of the deviator stress and itsecomposition into the contributions due to contacts transmitting

ess than average forces (f < 1) and contacts transmitting larger thanverage forces (f > 1). The figure clearly shows that the contributionf the weak sub-network of contacts is small, never deviating fromn isotropic stress state by more than ±4 kPa. Inspection of the dataeveals that the weak sub-network is composed of about 67% of allontacts and their contribution to the isotropic stress is about 35%Thornton & Antony, 1998).

The above figure demonstrates that the shear strength of granu-ar media is entirely due to the transmission of larger than averageontact forces in the sub-network of heavily stressed chains ofarticles within the system, which are aligned with the majorrincipal stress direction. The conventional view that higher sheartrengths are due to increased frictional energy dissipation andxtra work done in dilating the system does not agree with thehysics observed at the grain scale. Only 10% of contacts are slidingt any point in the test and they are always in the weak sub-etwork. If the interparticle friction is increased the percentage ofliding contacts decreases (Thornton, 2000) and there is no signif-cant increase in the amount of energy dissipation but the stabilityf the particles in the strong force chains is enhanced. The resultsuggest that shear strength is an elastic buckling problem and thatuckling of the strong force chains leads to strain localisation andhear band formation.

The fabric tensor, defined by Eq. (4), may be partitioned as fol-ows

ij = (1 − q)�wij + q�s

ij, (6)

ig. 6. Evolution of the contributions of the weak (f < 1) and strong (f > 1) sub-etworks to the induced structural anisotropy.

here the superscripts w and s indicate the weak and strongub-networks and q is the proportion of contacts in the strongub-network. Consequently we can calculate the separate contri-utions of the strong and weak sub-networks to the fabric tensor.ig. 6 shows the evolution of the overall deviator fabric with theontributions of the weak and strong sub-networks superimposed.he figure shows that, although the overall induced structuralnisotropy is relatively small, the distribution of contact orienta-ions in the strong sub-network is strongly anisotropic. The weakub-network is only slightly anisotropic and, in agreement with theD simulation data for rigid discs presented in (Radjai et al., 1996),here are more contacts oriented orthogonally to the major prin-ipal stress direction since the weak sub-network provides lateralupport to the forces in the strong sub-network.

Because of the way in which the forces are transmitted throughhe system it is not necessary for the system to develop a stronglynisotropic microstructure since the strong force transmission

Fig. 7. Initial stress states for radial strain probes.

C. Thornton / Particuology 8 (2010) 119–126 123

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iHoproperties of the particles are: Young’s Modulus = 70 GPa, Poisson’sRatio = 0.3, interparticle friction coefficient = 0.5 and sizes ranging

ig. 8. Deviatoric radial strain probing for different values of deviatoric strain.

. General stress and strain space

The vast majority of current experimental research in soilechanics is focused on axisymmetric compression tests. This isstate of stress that is rarely encountered in real boundary valueroblems. Even though plane strain deformation is a close approx-

mation to many geotechnical engineering problems there is veryittle work devoted to examining the plane strain response of gran-lar media. Although the general case in geotechnical engineeringroblems is one in which �1 /= �2 /= �3 there are probably lesshan ten research groups with facilities to examine general states

f stress, i.e. have ‘true’ triaxial apparatuses in their laboratory.lthough the tradition is that theory is tested by experiment it is notlear whether such complicated ‘true’ triaxial apparatuses provideeliable results. An alternative is to simulate, using DEM, com-

ig. 9. Stress response envelopes for different deviatoric strain amplitudes whenrobing from the initial isotropic stress state O.

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ig. 10. Fabric response envelopes for different deviatoric strain amplitudes whenrobing from the initial isotropic stress state O.

lex loading paths in general 3D stress space. The big advantagef DEM is that many tests may be performed on identical sampleshereby eliminating a major source of uncertainty in laboratoryxperiments.

Three-dimensional DEM simulations of deviatoric strain prob-ng have been performed on a dense polydisperse system of 3600ertz-Mindlin elastic spheres in a periodic cell. The initial void ratiof the assembly was 0.567 at an isotropic stress of 100 kPa. The

rom 0.25 to 0.33 mm.Five sets of deviatoric strain probing were carried out on the

eviatoric plane of principal strain space from different stress states

ig. 11. Contributions to the stress response envelope by the strong (f > 1) and weakf < 1) force sub-networks.

124 C. Thornton / Particuology 8 (2010) 119–126

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Fig. 12. Response envelopes for different deviatoric strain amplitu

t O, A, B, C and D respectively, as illustrated in Fig. 7. The initialtress state O is the isotropic state. The sample was then sub-ected to axisymmetric compression, axisymmetric extension andlane strain conditions to achieve the three stress states A, B and, respectively. From stress state C, the sample was loaded usingservo-control to maintain the deviatoric stress in the 1-direction

onstant until the deviatoric strain vector had rotated 30◦ and theorresponding stress state is referred to as D, which coincides withtress state A.

In each test the deviatoric strain increment �εd was kept con-tant for values ranging from 0.0001 to 0.05, in which

εd = 1√3

[(�ε1 − �ε2)2 + (�ε1 − �ε3)2 + (�ε2 − �ε3)2]1/2

.

(7)

The strain probes were carried out for 10◦ increments in the Lodengle from 0◦ to 360◦, as shown in Fig. 8. For each value of devi-

toric strain (indicated in the key to Fig. 8) the correspondingtress response envelope was determined and plotted on the devia-oric plane of principal stress space. The stress response envelopesbtained for probing from the isotropic stress state O are shownn Fig. 9 in which S1/p, S2/p and S3/p are the principal deviatoric

a

fef

Fig. 13. Response envelopes for different deviatoric strain amplitudes f

llowing preloading in axisymmetric compression to stress state A.

tresses normalised by dividing by the mean stress p. The corre-ponding fabric response envelopes are shown in Fig. 10. It can beeen from Fig. 9 that for all values of deviatoric strain the form ofhe stress response envelopes corresponds to the shape defined byhe failure criterion proposed by Lade and Duncan (1975) whichakes the form

I31

I3− 27 = �, (8)

here I1 = �1 + �2 + �3 is the first stress invariant and I3 = �1�2�3 ishe third stress invariant.

Interestingly, as shown in Fig. 10, the fabric response envelopesan be fitted by an inverted Lade surface where

I31

2I1I2 − 3I3= �f, (9)

here, in this case, I1, I2 and I3 are the first, second and third invari-

nts of the fabric tensor defined by Eq. (4).

The stress contributions from the strong (f > 1) and weak (f < 1)orce sub-networks are compared with the total stress responsenvelope in Fig. 11, which shows that the contribution of the weakorce sub-network to the stress response envelope is negligible,

ollowing preloading in axisymmetric extension to stress state B.

C. Thornton / Particuology 8 (2010) 119–126 125

Fig. 14. Response envelopes for different deviatoric strain amplitudes following preloading in plane strain to stress state C.

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rrespective of the strain probing direction. This demonstrates that,or general states of stress, the mobilized shear strength (the devi-toric stress) is entirely due to the contact forces in the strong forceetwork.

The stress and fabric response envelopes obtained after preload-ng in axisymmetric compression to stress state A are shown inig. 12. It is clear from the figure that for small deviatoric strainrobes, e.g. εd ≤ 0.005, both the stress and the fabric envelopesre very sensitive to the previous loading history. The spacingsetween each envelope are wide towards the bottom and nar-ow towards the top which means that the system remembers itas preloaded in axisymmetric compression. At larger deviatoric

trains, as peak strength is approached, the memory is graduallyiped out and the stress envelopes correspond to Lade surfaces,

entred on the isotropic state.Fig. 13 shows the response envelopes obtained after preloading

n axisymmetric extension to stress state B. Both stress and fabric

esponse envelopes are initially ‘pulled’ towards the isotropic stateor small strain probes but become symmetric about the isotropictate as the peak shear strength is approached. The same trendan be found in Fig. 14 for probing from stress state C followingreloading in plane strain.

iatsm

s for preloading paths OA and OCD.

Comparisons between the stress and fabric response envelopesbtained for preloading paths OA and OCD are shown in Fig. 15.lthough, in both cases, the two sets of envelopes are quite similar it

s observed that those obtained after the more complex preloadingath OCD indicate an initial memory of having visited stress state.

All the above figures show that at large deviatoric strains, aseak strength is approached, the memory of the preloading historyas been erased and the stress response envelopes are Lade surfacesnd the fabric response envelopes are inverted Lade surfaces, botheing centred at the isotropic state as was obtained from probingrom the isotropic state, as shown in Figs. 9 and 10.

. Concluding remarks

Constitutive theories for quasi-static deformation of granularedia are needed to model complex three-dimensional load-

ng histories involving general changes in the principal stressesnd their direction. The results reported in the paper illustratehat, at least in a qualitative sense, realistic three-dimensionaltress–strain-dilation behaviour can be generated by discrete ele-ent simulations. DEM simulations can, therefore, be used to

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26 C. Thornton / Partic

alidate continuum formulations of the constitutive behaviour ofranular materials subjected to complex loading histories.

The requirements of a good theory are that it needs to be math-matically consistent and reflects the physics of the problem. Toate, continuum approaches to the constitutive modelling of gran-lar materials are based on intuitive speculation about what may oray not happen inside granular systems, as inferred from observa-

ions of the macroscopic response of element tests conducted in theaboratory. DEM simulations can provide a better understandingf how granular systems behave at the grain scale, including newnformation, and this should lead to improved continuum models.he challenge is to identify how to modify continuum models toeflect more reliably the microscopic behaviour at the grain scale.

cknowledgments

The author is indebted to Drs. Gexin Sun, Joseph Antony and Linghang who performed the actual DEM simulations. All the workeported in the paper was funded by the Engineering and Physi-al Sciences Research Council (Grants Nos. GR/H14427, GR/K05832nd GR/R91588).

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undall, P. A., & Strack, O. D. L. (1983). Modeling of microscopic mechanisms ingranular materials. In J. T. Jenkins, & M. Satake (Eds.), Mechanics of granular

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akabayshi, T. (1957). Photoelastic method for determination of stress in powderedmass. In Proceedings of the 7th Japan national congress on applied mechanics (pp.153–158).

hang, L., & Thornton, C. (2005). Characteristics of granular media at the ‘criticalstate’. In R. García-Rojo, H. J. Herrmann, & S. McNamara (Eds.), Powders andgrains 2005 (pp. 267–270). Leiden: Balkema.