Granular contact dynamics with particle elasticity

Granular Matter (2012) 14:607–619DOI 10.1007/s10035-012-0360-1

ORIGINAL PAPER

Granular contact dynamics with particle elasticity

K. Krabbenhoft · J. Huang · M. Vicente da Silva ·A. V. Lyamin

Received: 22 November 2011 / Published online: 26 July 2012© Springer-Verlag 2012

Abstract A granular contact dynamics formulation forelastically deformable particles is detailed. The result-ing scheme bears some similarity to traditional moleculardynamics schemes in that the consideration of a finite elas-tic contact stiffness implies the possibility for inter-particlepenetration. However, in contrast to traditional moleculardynamics schemes, there are no algorithmic repercussionsfrom operating with a large or, in the extreme case infinite,contact stiffness. Indeed, the algorithm used—a standard sec-ond-order cone programming solver—is independent of theparticle scale model and is applicable to rigid as well as elas-tically deformable particles.

Keywords Contact dynamics · Discrete element method ·DEM · Elasticity

1 Introduction

The motion and interaction of discrete particles can be sim-ulated using one of two different methods. The most pop-ular method is the discrete (or distinct) element method(DEM) pioneered by Cundall [1]. In this method, which isoften referred to as a molecular dynamics (MD) method,

K. Krabbenhoft (B) · J. Huang · M. V. da Silva · A. V. LyaminCentre for Geotechnical Science and Engineering,University of Newcastle, Callaghan, NSW, Australiae-mail: [email protected];[email protected]

K. KrabbenhoftInstitute of Technology and Innovation,University of Southern Denmark, Odense, Denmark

M. V. da SilvaDepartment of Civil Engineering, Faculty of Sciences and Technology,Universidade Nova de Lisboa, Lisbon, Portugal

the positions of the particles are advanced in a fully explicitmanner using sufficiently small time steps. Interactionbetween grains is accounted for by relating the overlap thatmay occur between particles to forces via an appropriate con-stitutive relation. While the motivation for this approach tosome extent is algorithmic, the penetration that occurs atparticle contacts may be interpreted as the actual deforma-tion associated with particles of finite rigidity [2] and themicroscopic elastic constants (normal and tangential springstiffnesses) can be related to the equivalent macroscopicquantities (bulk and shear modulus) [3–5]. However, in manycases involving the flow and deformation of granular mate-rials, the macroscopic system stiffness is orders of magni-tude lower than the microscopic inter-particle stiffness. Assuch, the inter-particle stiffness is often chosen as a com-promise between the physics (large enough to not affectthe response significantly) and the computational perfor-mance (small enough to not require a prohibitively small timestep). A typical example of such a compromise is describedin [6].

The second approach to the modeling of granular assem-blies is the so-called non-smooth contact dynamics (CD)method originally developed by Moreau et al. [7–10]. In themost basic version of this method, the particles are consideredperfectly rigid and the contact forces are determined as thosethat exactly prevent inter-particle penetration. An implicittime discretization is usually employed, thus allowing forlarger time steps, and implying that collisions are ‘smeared’over a finite time interval. A key point often stressed in con-nection with the CD method is that assemblies consisting ofperfectly rigid particles do not allow for the propagation ofwaves at a finite speed. However, in many scenarios of prac-tical interest, wave propagation is of little significance, as iselastic particle deformation, and the two methods then leadto practically identical results (subject to identical tolerances

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608 K. Krabbenhoft et al.

being specified). Examples of application of the CD methodto the simulation of granular materials include [11–29].

Though aimed at simulating essentially the same phys-ics, the basic premises and the algorithmic implementationsof the MD and CD methods are so different that they areoften considered two distinct methods, each with a set ofinherent assumptions that cannot be easily circumvented. Inparticular, while the concept of particle deformability is atthe heart of the MD method, it is sometimes held that theassumption of perfectly rigid particles is fundamental to theCD method. However, as discussed by a number of authors,notably [30,31], the CD method is in fact quite general and inprincipe capable of accommodating any particle scale behav-iour. In this paper, the case of elastically deformable parti-cles is considered in detail. As in MD, the assumption ofa finite elastic contact stiffness leads to a situation whereparticles may penetrate each other. In contrast to MD, how-ever, the consideration of particle deformability is motivatedsolely by the physics rather than the numerics. Moreover, thebasic CD framework is maintained and algorithms originallydeveloped for the perfectly rigid case are applicable with littlemodification. In the present paper, we demonstrate this factwith respect to an optimization based solution scheme pre-viously developed for the perfectly rigid case [32].

The paper is organized as follows. In Sect. 2, the govern-ing equations for frictionless rigid particles are summarized.In the interest of clarity, the methodology is developed withrespect to two-dimensional circular particles. However, thebasic principles are straightforward to generalize to threedimensions. One possible extension is outlined in “Appen-dix 1”. Following [32] we make extensive use of variationalconcepts. This sets the scene for subsequent developments.In Sect. 3 the framework is extended to elastically deform-able particles and in Sect. 4 inter-particle friction is included.Finally, in Sect. 5, the effects of a finite elastic inter-particlestiffness are illustrated with respect to the common biaxialtest before conclusions are drawn in Sect. 6.

2 Frictionless rigid particles

2.1 Equations of motion

The equations of motion for a single frictionless rigid particleare given by

mv(t) = f ext (1)

where v(t) = (vx (t), vy(t))T are the linear velocities, m isthe mass, and f ext = ( fx , fy)

Text are external forces.

2.1.1 Time discretization

Relating position to velocity by x(t) = v(t), the equationsof motion can be written as

mv(t) = f ext

v(t) = x(t)(2)

These equations are discretized in time by the θ -method:

mv − v0

Δt= f ext

θv + (1 − θ)v0 = x − x0

Δt

(3)

where 0 ≤ θ ≤ 1 and x0 and v0 are the known positionand velocity at time t0 while x and v are the correspondingquantities at time t0 + Δt . We have here assumed that f extare constant, stemming, for example, from gravity. Straight-forward manipulations lead to the following expressions forthe displacements and velocities at time t0 + Δt :

mΔx = f 0

v = 1

θ

[ΔxΔt

− (1 − θ)v0

] (4)

where

m = 1

θΔt2 m, f 0 = f ext + mv0Δt (5)

and Δx = x − x0 are the displacements. The stability prop-erties of the θ -method are well known [33]: for θ = 1

2 thean unconditionally stable and energy preserving scheme isrecovered, for θ > 1

2 the scheme is unconditionally stableand dissipative, and for θ < 1

2 stability depends on the timestep. In the context of collisions, the algorithmic energy dis-sipation that occurs for θ > 1

2 can be related to the physicaldissipation associated with impact and thus to the restitutioncoefficient. Indeed, as shown in [32], a value of θ = 1

2 cor-responds to an elastic collision while θ = 1 reproduces aperfectly inelastic collision.

2.2 Non-penetration condition

Consider two circular particles as shown in Fig. 1. The posi-tions of the particles at time t0 are given by xi

0 and x j0. The

condition that the particles do not penetrate each other at timet0 + Δt can be stated as

||xi − x j || ≥ r i + r j (6)

This inequality constraint is non-convex for spatial dimen-sions greater than one and thus problematic to deal with.Consequently, throughout this paper, we consider the linearapproximation:

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Granular contact dynamics 609

Fig. 1 Frictionless contact geometry

(ni j

0

)T(Δxi − Δx j ) ≤ g0 (7)

where

ni j0 = x j

0 − xi0

||xi0 − x j

0||, g0 = ||xi

0 − x j0|| − (r i + r j ) (8)

are the initial normal and gap respectively (see Fig. 1). Thelinearized non-penetration can also be derived by expandingthe exact expression (6) in time and then using the θ -methodwith θ = 0 to express the velocities in terms of the displace-ments. As such, the linearized non-penetration conditionmay be viewed as being imposed in a fully explicit manner.In other words, it is assumed that the geometry does notchange over the time step. Although this could potentiallycompromise the stability of the system, practical experienceshows that stability for all but obviously unreasonable timesteps always is maintained.

2.3 Governing equations

We now consider the problem of frictionless contact involv-ing two particles, i and j . In this case the governing equationscomprise momentum balance for each particle (incorporat-ing contact forces), the linearized non-penetration conditionand conditions ensuring that the contact forces are positiveonly if the gap is closed and otherwise zero. These require-ments can be stated in terms of the following time discretegoverning equations:

mi Δxi = fi0 − ni j

0 p

m j Δx j = fj0 + ni j

0 p(ni j

0

)T(Δxi − Δx j ) ≤ g0

p ≥ 0

p

[(ni j

0

)T(Δxi − Δx j ) − g0

]= 0

(9)

where p is the contact force. The above governing equationsmay be viewed as the optimality conditions associated withone of four different optimization problems, or variationalprinciples. This is discussed in detail in the following sec-tions

2.4 Variational formulation

Before proceeding with a variational formulation of the gov-erning equations (9), it is convenient to introduce the follow-ing matrix quantities which cover general n-particle systems:

M = diag(m1, m1, . . . , mn, mn)

x = (x1, . . . , xn), v = (v1, . . . , vn)

g = (g1, . . . , gN ), p = (p1, . . . , pN )

(10)

where n is the number of particles and N is the number ofcontacts. Furthermore, collecting the normals associated withpotential contacts in a matrix N , the governing equations (9)can be written as

MΔx + N0 p = f 0

NT0 Δx ≤ g0

p ≥ 0

P(

NT0 Δx − g0

)= 0

(11)

where P = diag( p) and subscripts 0 again refer to theknown state. These equations constitute the first-orderKarush–Kuhn–Tucker (KKT) optimality conditions associ-ated with the following optimization problem (see “Appen-dix 3” for details):

minΔx

maxp

{ 12 ΔxT MΔx − ΔxT f 0

}+ {

ΔxT N0 p − gT0 p

}subject to p ≥ 0

(12)

The first term in the objective function is a time discreteform of the action integral associated with a collection ofnon-interacting particles while the second term accounts forthe effects of contact. This principle reproduces the govern-ing equations (11) after which the velocities at time t = t1 =t0 + Δt are calculated from the second equation of (4) andused to set up a new optimization problem to determine thedisplacements at time t2, etc.

Alternatively, the governing equations are recovered asthe solution to the following problem:

minΔx

maxt, p

− { 12 tT M−1 t − ΔxT(t − f 0)

}+ {

ΔxT N0 p − gT0 p

}subject to p ≥ 0

(13)

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Fig. 2 Potential contacts as given by Delaunay triangulation

where t are to be interpreted as dynamics forces (one ofthe optimality conditions associated with the above problemgives the relation t = MΔx).

2.4.1 Displacement based problem

Solving the max part of the problem (12) gives rise to thefollowing displacement based problem:

minimize 12 ΔxT MΔx − ΔxT f 0

subject to NT0 Δx − g0 ≤ 0

(14)

The contact forces are here recovered as part of the solu-tion, namely as the Lagrange multipliers associated with theinequality constraints.

2.4.2 Force based problem

Similarly, solving the min part of (13) gives rise to the fol-lowing force based problem:

maximize − 12 tT M−1 t − gT

0 p

subject to t + N0 p = f 0

p ≥ 0

(15)

The displacements are here recovered as the Lagrange mul-tipliers associated with the equality constraints.

In summary, the governing equations (11) can be cast interms of four different but equivalent variational statements:the mixed force-displacement problems (12) and (13), thedisplacement based problem (14), and the force based prob-lem (15). Each of these four problems have their merits interms of providing physical insights and forming the basisfor computations.

2.5 Potential contact specification

Following [32], potential contacts at t0 + Δt are defined bya Delaunay triangulation on the basis of the positions at t0.An example is shown in Fig. 2.

(a) (b)

Fig. 3 Contact between perfectly rigid (a) and elastic (b) particles

3 Frictionless elastic particles

The above model is now extended to cover elastically deform-able particles. Adopting an approach equivalent to that ofMD where the elastic contact forces are proportional to theinter-particle penetration, the only modification required con-cerns the measure of the gap g0. Thus, instead of requiringthe non-penetration condition satisfied exactly, we allow fora penetration equal to k−1 p where k is a constant charac-terizing the effective stiffness of the contact (see Fig. 3).The governing equations (9) for a binary system are thusgiven by

mi Δxi = fi0 − ni j

0 p

m j Δx j = fj0 + ni j

0 p(ni j

0

)T(Δxi − Δx j ) ≤ g0 + k−1 p

p ≥ 0

p

[(ni j

0

)T(Δxi − Δx j ) − g0 − k−1 p

]= 0

(16)

while, for a n-particle system, (11) generalizes to

MΔx + N0 p = f 0

NT0 Δx ≤ g0 + C p

p ≥ 0

P(NT0 Δx − g0 − C p) = 0

(17)

where C = diag(1/k1, . . . , 1/k N ) contains the contactspring compliances.

3.1 Variational formulation

The governing equations (17) may again be thought of asconstituting the optimality conditions associated with a num-ber of different optimization problems. One possibility is to

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extend min-max problem (13) to account for a finite particlestiffness. This gives rise to the following problem:

minΔx

maxp,t

− { 12 tT M−1 t − ΔxT(t − f 0)

}

+ {ΔxT N0 p − gT

0 p − 12 pTC p

}subject to p ≥ 0

(18)

We note the similarity of this problem to the classical Hel-linger–Reissner variational principle [34–37]. Similarly, theextension of (12) to account for particle elasticity can be car-ried out in two ways: either by introducing the elastic energyterm 1

2 pTC p as above or by introducing the elastic energyin terms of additional kinematic variables (see “Appendix 3”for details).

3.1.1 Force based problem

The force based version of (18) follows by solving its minpart to arrive at

maximize − 12 tT M−1 t − gT

0 p − 12 pTC p

subject to t + N0 p = f 0

p ≥ 0

(19)

where the only modification with respect to (15) is the addi-tion of the quadratic term accounting for particle elasticityto the objective function.

4 Frictional elastic particles

The most general case of frictional elastic particles is con-sidered next. Following [32] this problem appears by addingappropriate tangential forces and accounting for the momentsinduced by these tangential forces. The frictionless problem(19) thus generalizes to

minΔx,Δα

maxp,q,t,r

− { 12 tT M−1 t − ΔxT(t − f 0)

}

+{ΔxT N0 p − gT

0 p − 12 pTC N p

}− { 1

2 rT J−1r − ΔαT(r − m0)}

+{ΔxT N0q − ΔαT R0q − 1

2 ΔqTCT Δq}

subject to |q| − μ p ≤ 0

(20)

where the third and fourth terms in the objective functionaccount for sliding and rolling due to tangential forces.The tangential forces, aligned along ni j

0 (see Fig. 4), aregiven by q, while the angles of rotation are given by α.Elastic deformations are now accounted for by two contri-butions: one concerning normal deformations governed byeffective contact compliances C N = diag(1/k1

N , . . . , 1/k NN )

Fig. 4 Frictional contact geometry

and one concerning tangential deformations governed byCT = diag(1/k1

T , . . . , 1/k NT ). We note that while the nor-

mal elastic deformation can be accounted for on the basis ofthe known geometry at time t0, the tangential deformationrequires consideration of the shear forces both at time t0 andt0 + Δt .Regarding the additional rotational terms implied by the pres-ence of a shear force, the matrix J contains the scaled massmoments of inertia:

J = 1

θΔt2 J, m0 = Jω0Δt (21)

with J = diag(J 1, . . . , J n) being the mass moments of iner-tia. The vector m0, with ω0 being the rotational velocities, isthe rotational equivalent to the effective translational forcevector f 0. In the above, the rotational terms have been dis-cretized in time analogous to the translational terms and therotational velocities are thus calculated as

ω = 1

θ

[Δα

Δt− (1 − θ)ω0

](22)

The matrix R0 concerns the contribution of the total angularmomentum balance from the tangential forces and containsentries Ri I = r i , I ∈ Ci where Ci is the set of potentialcontacts associated with particle i .Finally, the Coulomb criterion is imposed with μ being theinter-particle friction coefficient. For further details on theabove formulation (for perfectly rigid particles) we refer to[32].

4.1 Optimality conditions

Following the procedure in “Appendix 3”, the KKT con-ditions associated with (20) can be shown to comprise the

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following sets of governing equations pertaining to linearmomentum balance:

t + N0 p + N0q = f 0,

t = MΔx(23)

angular momentum balance:

r − R0q = m0

r = JΔα(24)

sliding friction conditions:

|q| − μ p ≤ 0

diag(λ)(|q| − μ p) = 0, λ ≥ 0,(25)

and kinematics incorporating elastic deformation character-istics:

NT0 Δx + μλ = g0 + C N p

NT0 Δx − RT

0 Δα = CT Δq + Sgn(q)λ.(26)

where Sgn is the signum function. Regarding the kinemat-ics, we note that the variational formulation adopted leads toan apparent dilation proportional to μ. However, as arguedin [32] and following [27–29,38–40], this dilation may beviewed as an artifact of the time discretization which, with theexception of a few pathological cases, gradually is reducedas the time step is reduced. Moreover, in [32] it was shownthat the dilation, even for rather large time steps, is negligibleover a range of common conditions including both instancesof highly dynamic and relatively unconfined flows as well asconfined quasi-static deformation processes.

4.2 Force based problem

Finally, it is possible to cast (20) in terms of the followingforce based problem:

maximize − 12 tT M−1 t − 1

2 rT J−1r

−gT0 p − 1

2 pTC N p − 12 ΔqTCT Δq

subject to t + N0 p + N0q = f 0

r − R0q = m0

|q| − μ p ≤ 0

(27)

This is the problem actually solved in numerical calcula-tions. For this purpose, a general second-order cone program-ming solver, SONIC, recently developed by the authors isemployed. This solver is based on much the same principlesas the popular codes MOSEK [41] and SeDuMi [42]. Thoughoriginally designed with continuum plasticity applications inmind [37], it is ideally suited for the kinds of programs gen-erated by the present granular CD scheme.

4.3 Static limit

Omitting the dynamic forces t and r from the problem (27)gives rise to the following static problem which is valid inthe limit of Δt tending to infinity:

maximize −gT0 p − 1

2 pTC N p − 12 ΔqTCT Δq

subject to N0 p + N0q = f ext

R0q = 0

|q| − μ p ≤ 0

(28)

It is worth noting the similarity of this problem to those thatappear from finite element discretizations in continuum plas-ticity, in particular computational limit analysis [35–37,43].The above principle is useful for quasi-static problems gov-erned by an internal pseudo-time rather than physical time.Examples include common soil mechanics laboratory testssuch as triaxial tests, quasi-static soil-structure interactionproblems such as cone penetration, and various applicationsin the earth sciences where the time scales are such that thedeformations are of a quasi-static nature, e.g. [44,45].

The static problem (28) reveals a number of interestingproperties related to the indeterminacy of force networks ingranular media. It is well known that perfectly rigid particleslead to a situation where the force network solution is non-unique [46–49]. Setting C N = CT = 0 in (28) leads to alinear program where global optimality may be achieved bymore than one set of forces. Conversely, for finite values ofC N and CT , the solution is unique, i.e. there is a unique setof contact forces leading to the optimal value of the objectivefunction. This property that elasticity ‘regularizes’ the prob-lem has an interesting analogy in continuum plasticity whereit is well known that the assumption of a rigid-plastic mate-rial behaviour leads to a situation where the bearing capacityof a structure comprised of such material can be realized viamore than a single statically admissible stress field (the clas-sical example being the Hill and Prandtl solutions for therigid punch [50]), while the inclusion of elasticity eliminatesthis non-uniqueness.

5 Numerical examples

In the following, the effects of particle elasticity are stud-ied with reference to the common biaxial test shown inFig. 5. The initial dimensions of the sample are approx-imately 16 × 32 cm2. Two different initial packings wereconsidered, in the following referred to as loose (porosityof 0.22) and dense (porosity of 0.16). Each sample con-tains some 8,000 particles that are assumed weightless. Thesamples were generated by depositing rigid particles undergravity into an open rectangular container with a width of16 cm. The density of the samples was varied by varying the

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Rigid base

σa

σh= 100kN/m

H0

= 3

2 cm

B0 = 16 cm

Rigid movable walls, μw = tan15°

Weightless disks: d = 0.25cm ± 0.1cmμ = tan35°

σa

Fig. 5 Setup of biaxial test (actual samples contain approximately8,000 particles)

inter-particle friction coefficient from a low value to pro-duce the dense sample to higher values to produce the loosesample. Next, constraints specifying vertical and horizon-tal resultant forces corresponding to a uniform pressure of100 kN/m were imposed for a number of time steps untilthe displacement increments between subsequent steps were

sufficiently small. This typically required some 8–10 steps.The actual test was then carried out by progressively shiftingthe upper wall downwards while maintaining constant hor-izontal reactions. The Lagrange multipliers associated withthese latter constraints are the displacements of the walls.These were recorded in each time step, together with thedisplacement of the upper wall, to eventually infer strains.

5.1 Effects of normal stiffness

We first consider the case where kT = ∞. From a numeri-cal point of view, this case is particularly simple as the terminvolving Δq in (28) vanishes. This in turn means that noinformation about forces needs to be carried over from onetime step to the next. For each packing, three different valuesof the normal spring constant are used: kN = 10, 50 MN/m,and ∞. In all cases, a total of 400 increments of equal mag-nitude are used to induce an axial strain of εa = 0.2.

The results in terms of plots of deviatoric stress, σa − σh ,versus axial strain, εa = 1 − H/H0, and volumetric strain,εv = 1 − B H/B0 H0, versus axial strain are shown in Fig. 6.The tendency from these figures is—not surprisingly—thatthe inclusion of particle elasticity has a larger effect on theapparent macroscopic stiffness for the initially dense sample.Indeed, for the dense sample, particle stiffness constitutes themajor part of the contribution to the apparent macroscopic

0 0.05 0.1 0.15 0.20

100

200

300

400

Axial strain, a

Dev

iato

ric s

tres

s,

13

0 0.05 0.1 0.15 0.20.04

0.02

0

-0.02

-0.04

-0.06

Axial strain, a

Vol

umet

ric s

trai

n,v

0 0.05 0.1 0.15 0.20

50

100

150

Axial strain, a

Dev

iato

ric s

tres

s,

13

0 0.05 0.1 0.15 0.20.04

0.02

0

-0.02

-0.04

-0.06

Axial strain, a

Vol

umet

ric s

trai

n,v

Loose Dense

- -

kN = 10 MN/m kN = 50 MN/m kN =

Fig. 6 Biaxial test results for kT = ∞

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0 0.05 0.1 0.15 0.20

50

100

150

200

250

Axial strain, a

Dev

iato

ric s

tres

s,

13

0 0.05 0.1 0.15 0.20.01

0

-0.01

-0.02

-0.03

-0.04

Axial strain, a

Vol

umet

ric s

trai

n,v

-

kT /kN = 10 0.251

0.05 0.025 0.01

Fig. 7 Biaxial test results for kN = 25 MN/m (dense sample)

stiffness. Conversely, for the loose sample, particle rear-rangement governs the apparent macroscopic stiffness andparticle deformation plays an insignificant role. Regardingstrength properties, the results are also in qualitative agree-ment with what would be expected. That is, the dense sampledisplays an apparent peak shear stress before tending to theresidual level while the loose sample approaches the resid-ual—or critical—state in a monotonically increasing man-ner. The suppression of the peak by particle compliance isnot entirely unexpected either. As the particle stiffness, andthereby the system stiffness, decreases the attainment of thepeak stress will be delayed. And when it does occur, par-ticle rearrangements have brought the system into a statethat implies a smaller peak stress, if any at all. Finally, wenote that, for each of the samples, the macroscopic deviatoricstress reaches a constant value that is independent of the elas-tic properties while the rate of volumetric strain tends to zero,in agreement with standard continuum plasticity theories.

5.2 Effects of tangent stiffness

Next, we focus on the effects of a finite tangential stiff-ness. The dense sample is considered with a normal particlestiffness of kN = 25 MN/m. The effects of varying the tan-gential stiffness, kT , are illustrated in Fig. 7. Again, we

0.001 0.01 0.1 1 10 100

0.15

0.20

0.25

0.30

Nor

mal

ized

You

ng’s

mod

ulus

, Et/

k N

Particle stiffness ratio, kT/kN

Fig. 8 Dependence of initial tangent Young’s modulus on particlestiffness ratio kT /kN (dense sample)

0.001 0.01 0.1 1 10 1000

0.1

0.2

0.3

0.4

0.5

Poi

sson

’s r

atio

,

Particle stiffness ratio, kT/kN

Fig. 9 Dependence of initial tangent Poisson’s ratio on particlestiffness ratio kT /kN (dense sample)

observe that an decrease in kT leads to a more compliantsystem and thereby a gradual suppression of the peak stressas the stiffness ratio kT /kN decreases. The dependence of thesystem stiffness on the particle stiffness ratio can be charac-terized by the initial macroscopic tangent Poisson’s ratio,νt , and tangent Young’s modulus, Et . Assuming plane strainconditions, these are given by

νt = dεa − dεv

2dεa − dεv

(29)

and

Et = dσa − dσh

2dεa − dεv

(1 + νt ) (30)

The dependence of these parameters on kT /kN is shownin Figs. 8 and 9. As expected, the Young’s modulus increaseswith increasing kT /kN to ultimately reach an limiting valuecorresponding to kT = ∞. The Poisson’s ratio appears to

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Fig. 10 Biaxial test results for kN = kT = 25 MNm at εa = 0.2 (dense sample): particles colored according to their original position (left),particles colored according to relative magnitude of rotation rate (center), and distribution of porosity (right) (color figure online)

εa = 0.02(pre-peak)

εa = 0.033(peak)

εa = 0.04(post-peak)



εa = 0.10(residual)

εa = 0.20(residual)

Fig. 11 Evolution of shear bands as gauged by rotation rate for kN = kT = 25 MN/m (dense sample)

approach 0.5 (corresponding to the incompressible limit) forkT /kN tending to zero while, at the other extreme, a value of0 appears to be asymptotically approached as kT /kN tends toinfinity. These findings are qualitatively identical to those in[51]. Quantitatively, the present analyses reveals that both Eand ν are approximately linear functions of log kT /kN over alarge range of realistic particle stiffness ratio, approximately0.01 ≤ kT /kN ≤ 1. This provides a convenient means of cal-ibration. For example, to calibrate the particle assembly con-sidered in this example to a typical sand (0.1 ≤ ν ≤ 0.35),the relevant particle stiffness ratio would be approximatelyin the range of 0.1 ≤ kT /kN ≤ 1.

5.3 Shear banding

The load-displacement curves for the dense sample shownin Fig. 6 are quite typical of dense granular media such assands [52,53]. That is, a characteristic peak in shear stress isobserved followed by a decrease until a steady state whichis independent of the initial density. In terms of deforma-tions, the behaviour is characterized by an initial compac-tion followed by a dilation that reaches its maximum extentaround the peak stress, after which the rate of volumetric

straining eventually tends to zero as the steady, or critical,state is approached. This behaviour is again observed inFig. 7 although the peak, as already discussed, is suppressedby decreasing the kT /kN ratio.

In physical experiments, it is often observed that thepeak shear stress is accompanied by the formation of oneor more bands of localized deformation. These shear bandsmay form quite abruptly [52] or gradually become more pro-nounced as the deformation proceeds [53]. Figures 10 and11, which are concerned with the dense sample and a par-ticle stiffness ratio, kT /kN = 1, illustrate the latter typeof behaviour. Firstly, in Fig. 10, the state of deformationat an axial strain of εa = 0.2 is shown together with therate of rolling and the local porosities in the sample. Wehere observe a good correlation between the zones of intensedeformation and rate of particle rolling. Moreover, the localporosities are also rather indicative of the location of thesezones, with the porosity increasing quite substantially insidethe shear bands. Assuming a correlation between densityand shear strength, it is not difficult to appreciate that thedecrease in shear strength after the peak and the increasein porosity inside the shear bands are consequences of eachother.

123


Finally, in Fig. 11, the rate of rolling is shown at differencestages of the loading. From a relatively homogeneous distri-bution prior to the peak, the zones of particle rolling graduallylocalize after the peak to eventually define two quite distinctbands at the critical state.

6 Conclusions

A granular CD formulation for elastically deformable par-ticles has been detailed. The resulting scheme bears somesimilarity to traditional MD schemes in that the consider-ation of a finite contact stiffness implies the possibility for anelastically reversible inter-particle penetration. We show thatthe inclusion of particle elasticity reproduces the more basiccase of rigid particles in the limit of the inter-particle stiffnesstending to infinity. Moreover, in contrast to MD, there are noalgorithmic repercussions from operating with a large or, inthe extreme case infinite, stiffness. Indeed, the same algo-rithm is used regardless of the contact stiffness, with perfectrigidity being a limiting case that allows for certain simpli-fications. In the present paper we have only considered thecase of linear elasticity but extension to nonlinear elasticityis entirely possible as is the consideration of more complexcontact models incorporating hardening, viscous effects, etc.

Appendix 1: Three-dimensional formulation

The two-dimensional formulation discussed in the main partof the paper may be extended to three dimensions in a numberof ways. In the following, one possibility is outlined.

Compared to the two-dimensional case, the main com-plication is that the direction of the shear force is unknowna priori, the only requirement being that it is orthogonal tothe normal contact direction. The basic idea of the follow-ing three-dimensional formulation, sketched in Fig. 12, is toconsider a normal force directed along the particle normal asin the two-dimensional case. The shear forces are accounted

Fig. 12 Frictional contact geometry in three dimensions

for by separate force vector q I = (q Ix , q I

y , q Iz ) which must

satisfy(

nI0

)Tq I = 0 (31)

This condition is imposed explicitly in the final optimizationproblem.

With the above definition of forces, linear momentum bal-ance is given by

mi Δxi + pI nI0 + q I = f

i0

m j Δx j − pI nI0 − q I = f

j0

(32)

while the angular momentum balance equations read:

J i Δαi − r i0 × q I = mi

0

J j Δα j + r j0 × q I = m j

0

(33)

Similarly to the two-dimensional case, the frictional slidingcondition is given by

||q I || − μpI ≤ 0 (34)

Furthermore, for each contact, the quadratic term in theobjective function of (20), now reads

12 CN (pI )2 + 1

2 CT (Δq I )TΔq I (35)

where Δq = q − q0, with q0 being the known shear forceat the beginning of the time step as in the two-dimensionalcase. In summary, the three-dimensional formulation followsthe two-dimensional formulation closely, the only essen-tial modification being that the direction of the shear forceis unknown a priori, necessitating the orthogonality condi-tion (31). Finally, the optimization problems generated bythe above three-dimensional formulation can be solved usingthe same second-order cone programming algorithm as in thetwo-dimensional case.

Appendix 2: KKT optimality conditions

We consider the saddle-point problem representing the fric-tionless contact problem:

minΔx

maxp


}+ {

ΔxT N0 p − gT0 p

}subject to p ≥ 0

(36)

The first-order KKT optimality conditions associated withthis problem can be derived using the following procedure[35,54–56]. The inequality constraints are first converted intoequality constraints by subtraction of positively restrictedslack variables s. The objective function is then augmentedby a logarithmic barrier function which eliminates the need to

123


make explicit reference to the fact that s ≥ 0. The modified,equality constrained, problem is given by

minΔx

maxp,s


}+ {

ΔxT N0 p − gT0 p

} + β∑

I∈C ln s I

subject to p − s = 0

(37)

where β > 0 is an arbitrarily small constant and C is theset of potential contacts. The standard Lagrange multipliertechnique then applies, i.e. the solution to (37) is found byrequiring stationarity of the Lagrangian

L = 12 ΔxT MΔx + ΔxT(N0 p − f 0) − gT

0 p

+β∑

I∈C ln s I + λT( p − s)(38)

where λ are Lagrange multipliers. The stationary conditionsare given by

∂L

∂x= MΔx − f 0 + N0 p = 0

∂L

∂ p= NT

0 Δx − g0 + λ = 0

∂L

∂λ= p − s = 0

∂L

∂s I= β

s I− λI = 0 �⇒ s I λI = β, I ∈ C

(39)

By letting β tend to zero while bearing in mind that s ≥ 0,the KKT conditions (11) are recovered.

Appendix 3: Alternative variational principlesfor frictionless elastic particles

The governing equations (17) may be cast in a number of dif-ferent ways besides (18) and (19). The perhaps most directway is to include a quadratic term in the contact forces into(12). We thus have

minΔx

maxp


}+ {

ΔxT N0 p − gT0 p − 1

2 pTC p}

subject to p ≥ 0

(40)

While this problem reproduces the governing equations (17)directly, solving for the min part first does not produce a pureforce based problem. Similarly, solving first with respect tothe max part does not result in a pure displacement basedproblem.

To arrive at a pure displacement based problem, (12) isinstead extended by introducing a new set of variables, e,

such that we have

minΔx,e

maxp

12 ΔxT MΔx + 1

2 eT De − ΔxT f 0

+ pT(

NT0 Δx − g0 − e

)

subject to p ≥ 0

(41)

where D = C−1. The new variables e here represent the totalelastic normal deformation (see Fig. 3). We note the similar-ity of this problem to the Hu–Washizu variational principle[34]. Finally, a problem containing only kinematic variablesas unknowns follows by solving the max part of the aboveproblem:

minimize 12 ΔxT MΔx + 1

2 eT De − ΔxT f 0

subject to NT0 Δx ≤ g0 + e

(42)

We note that the objective function is here the time discreteaction integral associated with a collection of linear elasti-cally deformable particles. The KKT conditions are givenby

MΔx + N0 p = f 0

NT0 Δx ≤ g0 + e

p = De

p ≥ 0

P(NT0 Δx − g0 − e) = 0

(43)

which are equivalent to those of (17).

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Granular contact dynamics with particle elasticity