INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICSInt. J. Numer. Anal. Meth. Geomech. 2011; 35:293–308Published online 11 March 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nag.895

Rate effects in dense granular materials: Linear stability analysisand the fall of granular columns

V. Lemiale1,2,∗,†, H.-B. Muhlhaus3, C. Meriaux2,‡, L. Moresi2 and L. Hodkinson4

1CSIRO Process Science and Engineering, Melbourne, Victoria 3168, Australia2School of Mathematical Sciences, Monash University, Melbourne, Victoria 3800, Australia

3Earth Systems Science Computational Centre (ESSCC), The University of Queensland, St Lucia,QLD 4072, Australia

4Victorian Partnership for Advanced Computing, Victoria, Australia

SUMMARY

In this paper, the suitability of rate-dependent constitutive relationships to model the rheology of granularmaterials is investigated. In particular, the formation of shear bands as predicted by this approach isstudied. First, a rate-dependent model is investigated in terms of a linear stability analysis. It turns outthat at low to moderate strain rates, the orientation of shear bands tends to vary from the so-called Roscoeand Coulomb solutions towards a unique admissible orientation with an increase of the so-called inertialnumber I . This effect is confirmed by numerical simulations of a compression test performed with aparticle in cell finite element program. To further assess the validity of continuum approaches for thesimulation of dense granular flows, a quasi-static fall of a granular column is studied numerically and theresults are confronted to available experimental data. It is shown that a satisfying agreement is obtained atdifferent aspect ratios and for the two materials investigated in this paper, i.e. sand and glass beads. Theresults reported in the present paper demonstrate the relevance of continuum approaches in the modellingof dense granular flows. Copyright q 2010 John Wiley & Sons, Ltd.

Received 12 February 2009; Revised 16 September 2009; Accepted 4 January 2010

KEY WORDS: granular columns; finite element; shear bands; linear stability analysis

1. INTRODUCTION

While important advances have been made in our understanding of granular physics [1], thereare still many aspects yet to be elucidated. For example, the description of the transition from asolid-like to a fluid-like behaviour in dense granular media remains a difficult problem. This is ofprimary importance considering the number of applications involving the flow of dense granularmaterials, ranging from natural geophysical hazards to industrial processes. A consequence of thisinherent complexity is that there is no accepted unified theory for the mechanical description ofdense granular flows [2, 3]. At the continuum level, a model based on a rate-dependent constitutiverelationship has been proposed in Jop [4], Jop et al. [5] and Pouliquen et al. [6]. These authorsdemonstrated that quantitative agreement could be achieved under different flow configurations.However, it is unclear how the occurrence of instabilities in the form of shear bands would be

∗Correspondence to: V. Lemiale, CSIRO Process Science and Engineering, Private bag 33, Clayton South MDC,Vic. 3169, Australia.

†E-mail: Vincent.Lemiale@csiro.au‡Now at the Institute Dom Luiz, Universidade de Lisboa, Lisboa, Portugal.

Contract/grant sponsor: ARC grant; contract/grant numbers: DP0663258, DP0985662Contract/grant sponsor: Merit Allocation Scheme on the NCI National Facility at the ANUContract/grant sponsor: Auscope infrastructure grant

Copyright q 2010 John Wiley & Sons, Ltd.

294 V. LEMIALE ET AL.

captured by this approach. This question is discussed in this paper. A linear stability analysis isapplied to the model developed by Jop et al. [5]. The characteristics of incipient shear bands forthe bifurcation problem are derived and discussed. The theoretical findings are compared againstfinite element simulations on a compression test. Numerically, the moving integration point schemeproposed by Moresi et al. [7] has been used in the present work. The fluid dynamics-orientedformulation of this finite element program makes it an ideal candidate for testing the particular classof macroscopic constitutive relationships discussed here. Moreover, the suitability of the so-calledmaterial point method for the simulation of dense granular flows has been demonstrated [8, 9]. Ourresults, carried out at low to moderate applied velocities, indicate that the shear band orientationas predicted by the rate-dependent rheology of Jop et al. [5] varies as a function of strain rate.

Recently, laboratory experiments have been developed in various forms to analyse the quasi-static fall or the collapse of granular cylinders or columns [10–13]. The relative simplicity of thesetests provides a unique opportunity to assess the validity of constitutive laws over experimentaldata. Based on the experimental work of Meriaux [12], a numerical model has been developed tostudy the deformation of granular columns and is presented in this paper. Our results show that theessential features of the flow in the quasi-static regime are reproduced for different aspect ratiosand two different materials, namely sand and glass beads.

The paper is organized as follows. In the next section, the rheology investigated in this work isintroduced and its main characteristics are discussed. Then the occurrence of bifurcation modes inthe form of shear bands is studied by means of a linear stability analysis. The numerical frameworkused in the simulations is subsequently described. In particular, a numerical algorithm that hasbeen specifically developed for the treatment of friction boundary conditions between the granularmedium and the walls is detailed in that section. In the remaining sections of the paper, numericalsimulations are presented. First, results obtained from a compression test are compared againstthe theoretical findings of the linear stability analysis. The effect of strain rate on the orientationof shear bands is highlighted. Finally, a numerical model of the sliding of granular columns ispresented. A quasi-static test is developed and compared against available experimental data. Theagreement between simulations and experiments is an encouraging result in the further developmentof continuum models for dense granular flows.

2. CONSTITUTIVE BEHAVIOUR WITH STRAIN RATE AND PRESSURE DEPENDENCE

As highlighted in the introduction, there is no available continuum model capable of describingthe mechanics of granular flows. However, in the intermediate dense regime, a rate-dependentconstitutive relationship has recently been proposed by Jop et al. [5]. This model is briefly discussedin this section.

The Cauchy stress tensor r is written as

r=2�D− p1 (1)

where � is an effective viscosity, D is the strain rate tensor, p is the pressure and 1 represents theidentity tensor.

In this rheology, the effective viscosity is given as a function of the strain rate and pressure, asfollows:

�= �(I )p

�(2)

In this expression, � is the equivalent strain rate defined as �=√2DijDij.The friction coefficient �(I ) depends on the inertial number I , a dimensionless parameter

defined as

I = �d√p/�

(3)

where d represents the particle diameter and � is the particle density.

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RATE EFFECTS IN DENSE GRANULAR MATERIALS 295

This number can be seen as the ratio between two timescales, namely a macroscopic deformationtimescale 1/� and a confinement timescale (d2�/p)0.5 [2]. In the quasi-static limit, I thereforeconverges to 0, whereas for infinite strain rates, I becomes infinitely large.

Having introduced the inertial number I , the friction coefficient can be expressed as

�(I )=�s+�2−�sI0I

+1(4)

where I0 is a constant. �s and �2 are limiting values of the friction coefficient at zero and high I ,respectively.

With the incompressibility constraint imposed on the viscous and plastic parts of D, the set ofequations (1)–(4) defines the rheology of an incompressible dense granular medium. Thus, thematerial may be seen as a viscous-plastic material with the effects on both strain rates and pressurebeing incorporated via the introduction of an effective viscosity.

In particular, for quasi-static regimes, the friction coefficient reduces to �(I )=�s, so that theformulation becomes equivalent to a viscous-plastic rheology with a Drucker–Prager yieldingcriterion discussed by the present authors in [14]. In the latter paper, we have shown by means of alinear stability analysis that under these conditions the so-called Roscoe and Coulomb orientationsfor the shear bands are admissible solutions of the bifurcation problem. In the next section, asimilar analysis is performed using the rheology described above, in order to assess the effects ofstrain rate on the characteristics of incipients shear bands.

3. LINEAR STABILITY ANALYSIS

In this section, the occurrence of shear bands is investigated by a linear stability analysis. A two-dimensional plane strain configuration is assumed. A (x1, x2) coordinate system is adopted, suchthat the shear band is parallel to the x1 axis. The principal axes of the stress tensor are inclined atan angle of � with respect to the band axis, see Figure 1.

Initially, the state of deformation is assumed to be homogeneous. The strain rate tensor thereforereduces to

D=D0II

[−cos2� sin2�

sin2� cos2�

](5)

where D0II corresponds to the initial strain rate in the principal coordinate system.

Figure 1. Orientation of principal stresses with respect to the incipient shear band.

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2011; 35:293–308DOI: 10.1002/nag

296 V. LEMIALE ET AL.

The strain rate is subsequently incremented and the possibility of an inhomogeneous deformationmode in the form of a shear band is investigated. Hence, the incremental strain rate tensor has thefollowing form:

�D=⎡⎣ 0 �D12= �v1,2

2

�D12 0

⎤⎦ (6)

where �v1 is the velocity associated with the development of a shear band.Furthermore, in the coordinate system adopted here, the incremental equilibrium conditions may

be written as follows:

��12,2 = 0

��22,2 = 0(7)

It is thus necessary to determine the expression of the incremental stress tensor.The differentiation of (1), using (2), leads to (see Appendix A of [14] for details of the derivation)

��ij=−�p�ij+2�p0

�

(1

2(�il�jk+�ik� jl)− 2

�2Dkl Dij

)�Dkl+2

��p+��p0

�Dij (8)

where p0 is the initial pressure in the homogeneous state.From (4), the following expression can be derived:

��= �2−�s(I0I

+1

)2

I0I

(��

2|D0II|

− 1

2

�p

p0

)(9)

Moreover, the following equation holds:

��= 2Dij�Dij

�(10)

which can be further expressed as follows:

��= 2D0II sin2�

12�v1,2

|D0II|

(11)

Inserting this expression into (9), one obtains

��= �2−�s(I0I

+1

)2

I0I

(D0II sin2��v1,2

2|D0II|2

− 1

2

�p

p0

)(12)

Using this last expression, (8) can be rearranged in a matrix form as follows:

�r= −[

�p 0

0 �p

]+ �p0�v1,2

2|D0II|

[0 1

1 0

]+ −p0 sin2�(�− I�,I )�v1,2

2|D0II|

[−cos2� sin2�

sin2� cos2�

]

+2D0

II

(�− I

2�,I

)�p

2|D0II|

[−cos2� sin2�

sin2� cos2�

](13)

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RATE EFFECTS IN DENSE GRANULAR MATERIALS 297

where �,I represents the derivative of � with respect to I :

�,I = �2−�s(I0I

+1

)2

I0I 2

(14)

It is now possible to insert the expression of the incremental stress tensor (13) into the equilibriumconditions (7). This leads to the following system of equations:⎡⎢⎢⎢⎢⎢⎢⎢⎣

�p0

2|D0II|

(1−sin2 2�)

(1− I�,I

�

)D0II�

|D0II|(1− I�,I

2�

)sin2�

− �p0

2|D0II|

sin2�cos2�

(1− I�,I

�

)−(1− D0

II�

|D0II|(1− I�,I

2�

)cos2�

)

⎤⎥⎥⎥⎥⎥⎥⎥⎦[

�v1,22

�p,2

]=[0

0

](15)

Non-trivial solutions exist if the determinant of (15) vanishes, which is equivalent to(1−sin2�

(1− I�,I

�

))(1+�

(1− I�,I

2�

)cos2�

)

+�sin2�cos2�

(1− I�,I

�

)(1− I�,I

2�

)sin2�=0 (16)

This equation can be further rearranged as

1+�

(1− I�,I

2�

)−2

(�

(1− I�,I

2�

)+2

(1− I�,I

�

))sin2�+4

(1− I�,I

�

)sin4�=0 (17)

Equation (17) represents a quadratic equation for sin2�. Thus, we finally obtain the followinggeneral result for the orientations of shear bands:

sin2�1/2 = 1

8

(1− I�,I

�

)⎛⎝2

(�

(1− I�,I

2�

)+2

(1− I�,I

�

))

±√4�2

(1− I�,I

2�

)2

+16

(−1+ I�,I

�

)I�,I

�

⎞⎠ (18)

This result can be simplified when the condition I�,I /��1 is met. In that case, we obtain

sin2�1 = 1

2

((1+�)−8

I�,I

�

)

sin2�2 = 1

2+8

I�,I

�

(19)

In the limit case I =0, the shear band orientations are therefore �1=/4+/2 and �2=/4, respectively, which is consistent with the result obtained previously under equivalentconditions [14].

In summary, according to the present linear stability analysis, the angle of shear bands dependson the dimensionless number I through the relationship (18). In the limit case of low I , the solutionconverges to the classical Roscoe and Coulomb angles as admissible shear band orientations.

It should be noted that the theory applied here is of course not valid for all possible flow regimesof granular materials. The slow deformation regime is very well described by the engineering

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298 V. LEMIALE ET AL.

Figure 2. Shear band angles as a function of I calculated using (18).

plasticity and hypo-plasticity theories [15]. As the deformation rates increases, the fraction ofthe internal momentum transfer associated with inter particle collisions increases as well. Themomentum transfer in the limit for very fast deformations is governed purely by inter particlecollisions [16], whereas the slow flow regime is characterized by permanent particle contactsgoverned by the Coulomb friction. The present theory describes an intermediate regime.

Figure 2 shows the evolution of the two shear band orientations as predicted by Equation (18).Material parameters of glass beads given in Jop [4] have been used here. Following our previousremark on the domain of applicability of the present model, the solution is only plotted for moderatevalues of I . Above a critical value Ic (about 0.019 in the numerical application considered here),the square root term in (18) becomes negative; therefore, the linear stability analysis is no longerapplicable at higher values of I . It can be seen that the shear band angles vary from the so-calledRoscoe and Coulomb solutions at zero strain rate towards a unique admissible angle at the criticalIc value. This result will be compared with finite element predictions in Section 5.

4. NUMERICAL FORMULATION

4.1. Finite element particle in cell modelling

The rheology discussed in the previous sections has been implemented in our finite element particle-in-cell code Underworld [17]. The formulation of this code has already been discussed elsewhere[7]. The most noticeable feature lies in the use of material points (or particles) to discretize materialdomains and to carry material properties that are history dependent (in our case, the plastic historyof the material). Therefore, the quadrature points usually used in the integration scheme are nowreplaced by the (arbitrarily distributed) material points. This approach combines several advantagesof both the Lagrangian and Eulerian formulations, by allowing large deformations at the same timeas being capable of efficient solution of the underlying momentum equation on the static mesh.

In the present study the polynomial interpolation is piecewise linear in velocity, and constantin pressure (calculated at the centre of each finite element). In addition, we found it advantageousto consider a gradient recovery method to obtain a continuous pressure/strain-rate field from theindividual element values [18].

From (1), it is apparent that this rheology is well suited for a fluid dynamic-oriented programsuch as Underworld. Indeed the momentum equation to be solved can be written as follows:

2�(�Dij)

�x j− p,i = fi (20)

where the term fi represents the volume forces, i.e. gravity in our case.

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RATE EFFECTS IN DENSE GRANULAR MATERIALS 299

Hence from (20) it is seen that the set of global equations to be solved is equivalent to that ofan incompressible viscous flow, the viscosity being replaced by an effective viscosity determinedto satisfy the yielding constraints.

Theoretically, the relations (1)–(4) imply that a yield criterion exists and can be written asfollows [5]:

|�|>�s p (21)

where |�|=√(�ij�ij)/2 and s is the deviatoric stress tensor.Therefore, the algorithm discussed in [14] in the context of a Drucker–Prager-type rheology

still applies here, the only difference lying in the definition of the friction coefficient.Another consequence of (1)–(4) is that the effective viscosity varies from infinity at zero strain

rate to zero at infinitely large strain rates. However, numerically, lower and upper limits must beenforced to avoid any difficulties. It is therefore necessary to determine an appropriate viscosityrange so that the experimental time scale is recovered in the simulations, as will be more apparenton the models of granular columns.

4.2. Numerical algorithm for friction boundary conditions

In the presence of rigid boundaries, friction against these boundaries need to be considered.For that purpose, a special algorithm has been developed that is specifically designed for thefluid dynamics-oriented formulation of Underworld. The friction forces are implemented using a‘meshless element’ scheme. During the assembly of the stiffness matrix, a set of extra elements istemporarily created adjacent to any frictional boundaries involved in the system. The mechanicalbehaviour of these elements is similar to those representing the continuum, following the sameviscous flow laws, but in addition a special constitutive rule is used to satisfy the friction conditions.Here a Coulomb-type model is assumed as follows:

�s=�w�n+cw (22)

where �s and �n are the tangential and normal forces to the boundary, respectively. �w is thecoefficient of friction associated with the wall and cw is a coefficient that represents the adhesionbetween the continuum and the wall. To prevent the continuum material from moving through theboundary, velocity values are constrained with u(c)

n =u(w)n , where u(c)

n and u(w)n are the velocities

normal to the boundary at the continuum and the wall, respectively.In order to estimate the friction forces, an initial solve for the stresses in the continuum must

first be performed. The tangential and normal forces �s and �n can then be evaluated. From thesevalues, the viscosity of the frictional elements is modified so as to satisfy Coulomb type frictionconditions for the current stress solution. This process is repeated until a prescribed convergencecriterion is met.

5. STRAIN RATE EFFECT IN COMPRESSION

In this first test, the evolution of the shear bands orientation as a function of strain rate is numericallyinvestigated. The results are discussed in the context of the theoretical findings obtained by theprevious linear stability analysis.

5.1. Initial setup

The initial setup of the compression model is shown schematically in Figure 3. A box of dimensions4cm×1cm is considered. A velocity V is prescribed on the right and left edges of the box. Free slipboundary conditions are applied to the other edges of the box. To accommodate the volume change,a compressible background layer is included on top of the viscous-plastic material. To facilitatethe shear band measurements, a small notch of dimensions 0.04cm×0.04cm is introduced in themiddle of the box, next to the bottom wall, to encourage the formation of the bands there.

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300 V. LEMIALE ET AL.

Figure 3. Compression test model and corresponding boundary conditions. Also shown on this figure arethe point locations defined for output purposes.

Table I. Material parameters for glass beads used in the compression test.

Parameter Value

Initial viscosity (Pa s) 107

Final viscosity (Pa s) 1Background viscosity (Pa s) 0.1�s 0.38�2 0.64�(kgm−3, used in (3)) 2450I0 0.279d(�m) 530

Gravity is neglected in this section, so that the only parameter influencing the characteristics ofthe shear bands is the strain rate. The material parameters used in this test are given in Table I.Apart from the viscosities, these values correspond to those of glass beads as given in Jop [4].The initial and final viscosities correspond to the upper and lower numerical limits of the effectiveviscosity, respectively.

5.2. Shear band evolution as a function of I

Since gravity is neglected, the pressure is reduced to its dynamic component and is directlyproportional to the applied velocity in the case considered here. To estimate the correspondingvalue of I at a specific applied velocity, the following procedure was adopted. Two particle tracerswere introduced in initial locations indicated by points P1 and P2 in Figure 3. More precisely,P1 and P2 were situated at (0.5,0.2) and (3.5,0.3), respectively, i.e. outside the region of shearband development. At the end of each increment, the numerical values of pressure, strain rate andstress tensors calculated on these points were monitored. I was estimated from the first increments,before the shear bands had fully developed, by taking the average of the two values calculated atP1 and P2.Figure 4 illustrates the formation of shear bands as obtained numerically with Underworld under

two different applied velocities. The strain rate invariant is plotted on this figure. Assuming that theprincipal directions of stress remain constant at the onset of shear banding, the shear band anglecan be estimated from these figures. Practically, in each case the average value was computed fromthe orientation of the two incipient shear bands. It is apparent in Figure 4 that the shear bandsare more pronounced at lower strain rate. Indeed, it was found that the shear bands more easilydeveloped under smaller prescribed velocity conditions. It is interesting to note that at velocitiescorresponding to a value of I higher than the critical Ic discussed in Section 3, the shear bandshardly developed. The band thickness was also found to slightly increase with the strain rate inour simulations. This may be attributed to the fact that the band contours get more diffuse as thepressure sensitivity increases with the increasing strain rate. However, in the present study we onlyfocused on the orientation of shear bands as a function of strain rate or equivalently here of I .It was shown in [14] that the shear band angles could be accurately captured by our numericalapproach provided that the mesh was sufficiently refined.

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RATE EFFECTS IN DENSE GRANULAR MATERIALS 301

Figure 4. Strain rate invariant at the onset of shear banding for an applied velocity of(a) 10−5 cms−1 and (b) 10cms−1.

Figure 5. Comparison between the shear band angle obtained numerically and the theoretical orientations�1 and �2 defined by (18). Error bars of ±1◦ are added to the numerical angles.

The evolution of shear bands angles as a function of I is compared in Figure 5 with the theoreticalcurves discussed in Section 3. As explained in that section, only a low to moderate values of Iinterval is plotted on this figure, corresponding to a velocity range of [10−5;24]cms−1.

The numerical solution closely follows one of the orientations obtained by the linear stabilityanalysis presented in Section 3. In the quasi-static limit, the angle converges to the Coulombsolution, which is consistent with our previous findings in [14]. As the strain rate increases, thereis a small but clear decrease of the shear band angle. The rate-dependent rheology proposed byJop et al. [5] therefore implies that the orientations of shear bands depend on the dimensionlessnumber I for small to moderate values of this number. Experiments reported in [19] did notshow a dependence of the shear bands inclination on the rate of loading. However, these resultswere obtained under different conditions, in particular the specimens were loaded in extension.Therefore, additional experimental data would be required in order to assess the suitability of thepresent material model for the prediction of shear band formation.

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302 V. LEMIALE ET AL.

6. QUASI-STATIC TEST ON GRANULAR COLUMNS

Having considered the effects of strain rate associated with the rheology proposed by Jop et al. [5],we now turn our attention to the simulation of the quasi-static fall of granular columns. Meriaux[12] has developed a set of experiments to analyse the flow of dense granular materials. Thoseexperimental results are compared with numerical simulations in the present section.

6.1. Test description

Experimentally, a column of sand was deposited between two walls. The right wall was fixedthroughout the test, whereas the left wall was constrained to move at a specified velocity of2cmh−1. Pictures of the granular column were recorded at regular intervals. To visualize theinternal dynamic of the flow, two different colours of otherwise identical layers of sand or glassbeads were used.

The numerical model developed to reproduce the experimental setup is presented in Figure 6.A compressible viscous layer was added on top of the box to accommodate the volume change. Itsviscosity was set to 10−3 Pas, a value sufficiently small to ensure that the compressible material didnot influence the overall results. Friction boundary conditions were applied to the right and bottomwalls, using the method highlighted in Section 4.2. Both sand and glass beads were used. Jop [4]has shown that while the mechanical behaviour of glass beads subjected to intermediate strain rateloading can be captured by the simple rheology used in the present work, the situation is morecomplex with sand. However, in the quasi-static case considered here, the rheology will convergeto a Drucker Prager yield criterion that appears to be sufficiently accurate both for sand and glassbeads materials. Consequently, the rheology described in the previous sections was retained here.The corresponding mechanical properties of sand and glass beads are listed in Table II. The grainsize diameter and density were determined from experimental data provided in [12]. The othermaterial parameters I0, �s and �2 were determined as follows. I0 and �2 were chosen in accordancewith [5], bearing in mind that these parameters will not play a significant role under the presentquasi-static conditions. �s was set to a representative value of the internal friction angle for sandand glass beads, respectively.

The other relevant parameters for this test are given in Table III. The numerical values of thefriction parameters were determined from [12]. The aspect ratio is defined as the initial heightto length ratio of the column. The upper and lower limits of the viscosity were determined sothat the time of the simulation matched the actual duration of the experiments. Contrary to thecompression test of the previous section, it is obvious that gravity must be included here.

6.2. Angle of repose

Figure 7 shows the final shape of the deposit obtained numerically with sand and glass beads. Ascan be seen, the model correctly predicts a greater angle of repose in the case of sand. From

Figure 6. Initial set up for the quasi-static test. Friction boundary conditions are applied on the right andbottom edges of the box. A velocity is prescribed on the left edge of the box.

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RATE EFFECTS IN DENSE GRANULAR MATERIALS 303

Table II. Material parameters for sand and glass beads used in the quasi-static test.

Parameter Sand Glass beads

Grain size diameter (�m) 700 800�(kgm−3) 1490 1790�s 0.577 0.40�2 0.64 0.64I0 0.279 0.279

Table III. Parameters defined for the quasi-static test.

Parameter Value

Aspect ratios 1.5, 3 and 5.5Applied velocity (cmh−1) 2�w 0.45 (sand), 0.31 (glass beads)cw 0Maximum viscosity (Pa s) 107

Minimum viscosity (Pa s) 104

Background viscosity (Pa s) 10−3

g (ms−2) 9.81

Figure 7. Final deposit as obtained with our numerical model for sand (left) and glass beads (right).

these figures, was measured as 21.3◦ and 28.6◦ for glass beads and sand, respectively. Thesevalues are consistent with the relation tan=sin�, with sin�=�s in our case. They also agreewell with those that were measured in the experiments [12] with 22◦±0.5 and 27.5◦±0.5 forglass beads and sand, respectively. Numerical results obtained with a Drucker Prager rheologygave similar angles of repose and internal material flow, which demonstrates that the effect ofstrain rate on the rheology can be neglected under the quasi-static conditions of the present test.

6.3. Numerical prediction of the internal flow in sandpile

In this section, the internal flow of sandpile as obtained in our computation is compared with theexperimental results from [12]. Because experimentally the internal flow was only analysed forsand through the use of two different colours, here the comparison will be made on this material.

A representative sequence of the sliding of a sand column is shown in Figure 8 for an initialaspect ratio of 1.5. A shear band is initiated from the lower left-hand corner of the pile along whichthe material begins to slide. With further motion of the wall, new shear bands are continuouslyactivated, as shown in Figure 9. This figure illustrates the shifts that operate between a previouslyactivated shear band and a new one, oriented at a higher angle with respect to the horizontaldirection. Eventually, the shear band is aligned with the inclined surface of the pile. Finally, theright foot of the column remains static throughout the simulation.

The qualitative motion of the column as described here agrees well with experimental obser-vations reported in [12]. Quantitatively, the final shape of the pile is recovered in our numerical

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304 V. LEMIALE ET AL.

Figure 8. Sequence of sand column deformation for an initial aspect ratio of 1.5, as obtainednumerically (left images) and experimentally (right images): (a) at t=0 (simulation and exper-iments); (b) at t=36’ (simulation) and t=35’ (experiments); and (c) at t=2h38’ (simulation)and t=2h30’ (experiments). Experimental figures reprinted with permission from [12]. Copyright

2006, American Institute of Physics.

Figure 9. Illustration of a shift between an old shear band and a newly activated, steeper one with furthermotion of the left wall. Contour plot represents the strain rate invariant, given in s−1.

simulations, as will be discussed later in Section 6.4. However, the internal flow of the granularlayers is not perfectly reproduced by our model. Indeed, from the experimental data provided in[12], the slip planes observed experimentally seem to form at a higher angle with respect to thehorizontal plane. Moreover, the most appreciable difference is observed on the left-hand side ofthe box, at the wall interface, see Figure 8. This can be attributed to the treatment of boundary

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RATE EFFECTS IN DENSE GRANULAR MATERIALS 305

Figure 10. Sequence of sand column deformation for an initial aspect ratio of 3, as obtained numerically(left images) and experimentally (right images): (a) at t=0 (simulation and experiments); (b) at t=41’(simulation) and t=40’ (experiments); (c) at t=1h53’ (simulation) and t=1h50’ (experiments); and (d)at t=4h8’ (simulation) and t=4h10’ (experiments). Experimental figures reprinted with permission from

[12]. Copyright 2006, American Institute of Physics.

conditions associated with the moving wall. Numerically, the mesh is modified to accommodatethe motion of the wall, which results in a spurious deformation mode in extension applied to thematerial. In other words, the displacement of the material points is not fully decoupled from thewall motion in our model. This could be eliminated by introducing an additional layer between thewall and the sandpile, although this solution has not been tested in the present study. Nevertheless,the agreement between our numerical prediction and experimental data appears to be satisfactory.

A similar sequence of deformation is shown in Figure 10, but this time for an aspect ratio of 3.Most aspects of the observed experimental flow are captured by our model when compared withdata presented in [12]. In particular, in the first part of the deformation the material flows alongtwo main slip planes intersecting at the right wall. This results in one surface sloping downwardto the fixed wall (phase b of Figure 10). Subsequently the material slides along one main slipplane, resulting in the transient shape observed in phase c of Figure 10, before the final deposit isobtained. Similarly to the preceding case, the main difference between experimental observationand numerical prediction occurs at the interface of the moving wall. This is apparent already onphase b of Figure 10. This boundary condition effect largely explains the overall differences inthe final deposition of sand layers.

It is interesting to note that inside the sandpile, the shape of the inhomogeneously deformed layersis correctly captured in our simulations, as can be seen in Figure 10(d). Various simulations withdifferent mesh sizes have been conducted. They demonstrate the necessity of using a sufficientlyrefined mesh to capture this level of details inside the pile. Figure 11 is an illustration of this fact.With a coarser mesh, the deformation of the layers appear smoother than in the actual experiments.

The sequence of deformation for an initial aspect ratio of 5.5 is shown in Figure 12 forcompleteness. All the previous remarks formulated on the analysis of ratio 1.5 and ratio 3 stillapply here.

6.4. Final deposits characterization

Following [12], the final deposit may be characterized by two dimensionless numbers definedfrom the ratios between final and initial lengths and heights of the columns. The numerical values

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2011; 35:293–308DOI: 10.1002/nag

306 V. LEMIALE ET AL.

Figure 11. Comparison of the final deposit for an initial aspect ratio of 3 calculated on a (a) 64×64elements mesh size and (b) 128×128 elements mesh size.

Figure 12. Sequence of sand column deformation for an initial aspect ratio of 5.5, as obtained numerically(left images) and experimentally (right images). (a) at t=0 (simulation and experiments); (b) at t=46’(simulation and experiments); (c) at t=1h39’ (simulation) and t=1h41’ (experiments); and (d) at t=3h33’(simulation) and t=3h31’ (experiments). Experimental figures reprinted with permission from [12].

Copyright 2006, American Institute of Physics.

obtained in the present study are given in Table IV, along with their corresponding experimentalvalues. These values have been determined from the experimental work reported in [12].

Overall, the agreement is better for glass beads than for sand, the relative difference betweensimulation and experiments being in most cases smaller than 5% for glass beads. The differenceobtained with sand is larger, especially at smaller ratios. This may be partly explained by the factthat at small ratios a slight variation in measurement will result in an appreciable relative differencebetween the quantities at play. Therefore, the measurement precision has a larger influence atsmaller ratios. More generally, it is not entirely surprising that the prediction is more accurate forglass beads, since as has been discussed previously the rheology used in the present study has beenspecifically developed for this class of materials. For sand, the mechanical behaviour is likely tobe more complex. Nonetheless, it can be concluded from the results reported here that our modelnot only reproduces the flow of granular materials qualitatively, but a quantitative agreement isalso found.

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2011; 35:293–308DOI: 10.1002/nag

RATE EFFECTS IN DENSE GRANULAR MATERIALS 307

Table IV. Estimated initial to final height and initial to final length ratios for several simulations. Alsoindicated is the corresponding experimental value as determined from [12], and the relative difference

between experimental and numerical values.

Dimensionless Calculated Experimental RelativeMaterial Ratio number value value difference (%)

Glass beads 1.5 HiH f

1.45 1.4 3.5

Glass beads 1.5L f −Li

Li1.7 1.7 0

Glass beads 3 HiH f

2.12 2.2 −3.6

Glass beads 3L f −Li

Li2.86 3 −4.7

Glass beads 5.5 HiH f

2.84 2.7 5.2

Glass beads 5.5L f −Li

Li4.05 4.1 −1.22

Sand 1.5 HiH f

1.24 1.13 9.7

Sand 1.5L f −Li

Li1.34 1.23 8.9

Sand 3 HiH f

1.8 1.67 7.8

Sand 3L f −Li

Li2.29 2.23 2.7

Sand 5.5 HiH f

2.42 2.28 6.1

Sand 5.5L f −Li

Li3.53 3.56 −0.8

7. CONCLUSION: TOWARDS AN UNDERSTANDING OF DENSE GRANULAR MEDIAAT THE CONTINUUM LEVEL

A rate-dependent constitutive relationship proposed by Jop et al. [5] to describe the rheology ofdense granular materials has been investigated in terms of shear band formation. It has been shownthat this model produces shear bands oriented at an angle dependent on the so-called inertialnumber I . For small to moderate values of I , the orientation of shear bands varies from the Roscoeand the Coulomb solutions to a unique admissible angle. This result was confirmed by a numericalanalysis of a compression test carried out with a particle in cell finite element program. Thisdependence of the shear band angle on the inertial number I should be confronted to experimentaltests in order to assess the validity of the strain rate-dependent rheology investigated in this paper.In the last part of the paper, a numerical model of a quasi-static fall of granular columns wascompared against experimental data. It was shown that a quantitative agreement could be achievedfor different aspect ratios of the columns and for two different materials, namely sand and glassbeads. In order to further develop continuum models of dense granular flows, it would be essentialto compare numerical predictions with the collapse of granular columns, which occurs under muchhigher strain rates than the quasi-static case considered here. Meriaux and Triantafillou [20] havedeveloped the dynamic equivalent of the quasi-static test used in the present paper. Since theformulation of our code Underworld does not incorporate inertial effects yet, it was not possibleto perform a meaningful comparison between our numerical results and the experiments reportedin [20]. However, this type of dynamic testing should be of interest to further assess the validityof mechanical models of dense granular media.

ACKNOWLEDGEMENTS

This research was supported under Australian Research Council’s Discovery Projects funding scheme(project number DP0663258 and DP0985662). This work was supported by an award under the MeritAllocation Scheme on the NCI National Facility at the ANU. The second author wishes to acknowledgefinancial support from the Auscope infrastructure grant, an NCRIS DISR initiative.

The first author would like to thank Julian Giordani for his technical assistance with the programUnderworld.

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Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2011; 35:293–308DOI: 10.1002/nag