On the mechanical behaviour of dry cohesionless soils by DEM simulations

Dorival M. Pedroso, Sergio Galindo-Torres & David J. WilliamsThe University of Queensland, Brisbane, Australia

This paper presents a study on the macroscopic mechanical characteristics of granular assemblies (dry cohe-sionless soils) with three dimensional complex shaped (non-spherical) particles. The study is conducted usingthe discrete element method (DEM) with the so-called sphero-polyhedra particles and simulations of mechan-ical true triaxial tests for a range of Lode angles. Focus is given to the strength characteristics. The observedmathematical failure envelopes are investigated in the Haigh-Westergaard stress space. It is verified that theDEM simulations with complex particles are more stable than with spheres, either considering virtual rollingresistance or not. It is also verified that the discrete element method with non-spherical particles produces re-sults qualitatively similar to experimental data on the Toyoura sand. The simulations reproduce quite well thestrength of assemblies of granular media such as higher strength during compression than during extension.Finally, it is observed that the Matsuoka-Nakai failure envelope averages very well the simulated data and thatthe macroscopic friction angle can be considered constant for the range of mean pressures investigated, both forspheres and sphero-polyhedra.

1 INTRODUCTION

The mechanical properties of assemblies of naturalgrains, such as dry cohesionless soils, are determinedprimarily by particle size, surface texture, size distri-bution, and shape of grains. Another important prop-erty is the structure of the packing, such as cubic,pyramidal, or tetrahedral. The packing also affects theporosity and density. These properties are basicallyof physical nature and have great influence on themacroscopic behaviour such as strength of the assem-bly. In this paper, attention is given to the capabilitiesof a numerical model in predicting the strength of as-semblies of non-spherical particles. Dense packingsof polyhedron-shaped particles with rounded corners,the so-called sphero-polyhedra, are considered. Nu-merical simulations are carried out using the discreteelement method (DEM).

Previous works on the two dimensional proper-ties of assemblies of polygonal shaped particles areavailable (Alonso-Marroquın and Herrmann 2002;Alonso-Marroquın and Herrmann 2005; Galindo-Torres et al. 2010) and some works consider alsothree-dimensional particles of non-spherical shapes(Galindo-Torres et al. 2009; Wang et al. 2010;Galindo-Torres and Pedroso 2010), although not alldiscuss the mechanical behaviour and strength prop-erties of cubic assemblies, in particular the shape ofthe failure envelopes in the Haigh-Westergaard space.

Experimentally, the macroscopic mechanical be-haviour of granular assemblies can be investigated us-ing triaxial cells, as is customary in Soil Mechanics.Cylindrical cells are usually employed; however, withthese cells, only a two dimensional (axis-symmetric)stress field can be generated. On the other hand, byusing cubical, or true triaxial cells, the investigationof the three principal components of stress and strainscan be carried out; hence, allowing the observationof the influence of the Lode angle on the octahedralplane.

Usually, macroscopic phenomenological criteriasuch as the Mohr-Coulomb are fitted to the observeddata collected through triaxial tests. These criteria arethen used to predict the strength of the granular as-sembly. Loading tests with shearing and stress pathsin which the mean stress is kept constant are conve-nient for the investigation of the pattern of the peakstresses on the octahedral plane. Therefore, the shapeof the failure envelope on this plane can be assessedas well.

Matsuoka and Nakai (Matsuoka and Nakai 1974)presented a failure criterion that predicts the samestrength for compression and extension as the Mohr-Coulomb criterion; however its predictions for in-termediate stresses (influence of the second stresseigenvalue) exhibit higher strength than the Mohr-Coulomb criterion – this is often indicated by tests

on soils and other granular media. Moreover, theMatsuoka-Nakai criterion has a smooth envelope onthe octahedral plane – a great convenience for numer-ical models – and is product of intensive research onthe mechanical properties of soils, leading to the con-cept of the Spatially Mobilized Plane (SMP) (Mat-suoka and Nakai 1977; Nakai 1980; Matsuoka andNakai 1982; Nakai and Matsuoka 1983; Matsuokaand Nakai 1985). The spatially mobilized plane formsa convenient and rational framework for the definitionof continuum models for granular matter.

Other criteria that define failure envelopes withsimilar shape to that of the Matsuoka-Nakai criterionwere proposed in the literature (Argyris et al. 1974;von Wolffersdorff 1996). However, the lack of con-vexity of the failure envelopes of some of these cri-terion may cause problems to numerical simulations(Pedroso et al. 2008). The main algebraic differenceis that while the Matsuoka-Nakai failure criterion isdefined directly as a function of the characteristic in-variants Ik of the stress tensor, the other criteria aredefined by functions of the ratio between the devia-toric (q) and mean (p) stress invariants with respectto the Lode angle θ; the so-called M = q/p := M(θ)methods.

A goal of this research is to observe the shape of thefailure envelopes that best fit the results of DEM sim-ulations; therefore, virtual true triaxial tests are car-ried out. In the following, the methods are briefly ex-plained but further details can be found in (Galindo-Torres et al. 2009; Galindo-Torres and Pedroso 2010;Galindo-Torres et al. 2011). One key step for the truetriaxial test is the packing generation, which is care-fully explained here. Afterwards, the modelling of thevirtual true triaxial apparatus is discussed and a briefreview of some common failure criteria in Soil Me-chanics is presented. Results with spheres considering(and not) the so-called rolling resistance are presentedas well.

2 METHODS

Cubic ensembles of particles are mechanically loadedand the deformations are measured. The kinematicsand dynamics of each single particle are computa-tionally represented and the macroscopic behaviouris observed. The problem is solved using the dis-crete element method (DEM) as originally presentedby Cundall and Strack (Cundall and Strack 1979) ex-cept that some modifications are adopted in order toaccount for multi-contact and particles of complex(quasi-general) shape, including non-convex particles(Galindo-Torres et al. 2009; Galindo-Torres and Pe-droso 2010; Galindo-Torres et al. 2011).

A main innovation of the method employed hereis the use of a technique for the smoothing of theparticles in which all edges have actually the shape

Figure 1: Some sphero-polyhedral particles, includ-ing convex and non-convex particles. All edges areof capsule-shape due to the smoothing with a givensphero-radius. Particles can be made of just one ver-tex, or one edge, or one face

of pharmaceutical pills or capsules (see examplesin Fig. 1). This technique names the particles assphero-polyhedra (Alonso-Marroquın 2008; Galindo-Torres et al. 2009; Galindo-Torres and Pedroso 2010;Galindo-Torres et al. 2011). Basically, all edges areswept by spheres, turning them into capsules, wherethe radius R of the sweeping sphere defines thesphero-radius of the particle. The faces will then havea thickness equal to 2R and smooth corners. The ge-ometry of each particle is defined by a set of vertices,edges, and faces. Different shaped particles can bepresent in the same simulation; thus a complicate mixof particles of quasi-general shape can be simulatedat the same time (Fig. 1). This technique largely sim-plifies the numerical method, since now the contactforces have a unique definition, contrary to schemesemploying polygons or polyhedra.

The mass properties of the sphero-polyhedra par-ticles are a little more difficult to be calculated thanthe mass properties of conventional polyhedra. Themain reason is the rounding (smoothing) of the edgesby applying the sweeping method. Therefore, closedform equations are unavailable, except, perhaps, forsimple geometries, such as a rounded cube or a tetra-hedron. To solve this problem, a numerical integrationwith the Monte-Carlo method is employed. The inte-gration finds the volume (hence the mass), the tensorof inertia, and the centre of mass of the particles byMonte-Carlo approximations. These procedures alsowork for non-convex particles as the ones illustratedin Fig. 1.

2.1 Particle generation

Dense cubic packings of irregular particles are gen-erated using the method discussed in (Galindo-Torresand Pedroso 2010). Due to its importance for the truetriaxial simulations in this paper, the method is furtherexplained here. The method is named Voronoi-erosion

Figure 2: Voronoi tessellation generated by Voro++(extracted from its online documentation). The yellowspheres are shown in order to help the visualizationof the 3D lattice only and are not used as particleshere. Particles are created by the erosion algorithmdescribed in the text, considering the “Voronoi-cells”defined by the blue edges.

since it is based on the Voronoi tessellation (Voronoi1907) and requires an algorithm for the erosion of theinitial assembly of particles. The erosion is requiredmainly because the particles need to be converted intosphero-polyhedra, i.e. have all edges smoothed, andno overlapping should be present at the beginning ofthe simulations, otherwise the system would not be atequilibrium.

A three-dimensional grid is first generated with thedimensions of the cubic sample, then random pointsare sampled in the interior of each grid cell. The num-ber and position of the points will control the final par-ticle size and distribution. With these points, anotherlibrary, named Voro++ (Rycroft 2009), is called in or-der to generate the Voronoi tessellation. One exampleof the output of Voro++ is shown in Fig. 2, extractedfrom its user manual (Rycroft 2009).

The Voronoi tessellation partitions the domain intocells as illustrated in Fig. 2. In this figure, the yel-low spheres are shown to help the visualisation ofthe 3D volume only; they are not used as DEM parti-cles in this research. Afterwards, with the 3D Voronoitessellation (the blue edges in Fig. 2 defining theVoronoi-cells), particles are created by eroding thecells by an amount equal to the sphero-radii of thesphero-polyhedra. Basically, the faces and edges of aVoronoi-cell are displaced to the interior of the celland all intersections are computed. A control for the

displacements must be applied, because if the dis-placements are too large, the solution may not existand the cell may become degenerated. Therefore, thesphero-radii of the particles must be limited and theirvalues are generally found by trial-and-error.

With this technique, the packing is ready for thetrue triaxial simulation, since the algorithm guaran-tees that no overlaps exist, i.e. the particles are al-ready in perfect contact one with each other but notoverlapping. Clearly, this technique can only createdense packings. Another method to generate the pack-ing would be to run a preliminary DEM simulationwith the application of gravity, in order to move thegrains until a stable condition is found. Nonetheless,this method would create a geo-static situation wherecontact forces do exist; this is not the isotropic sit-uation desired for the investigation in this research,especially at the first stage of the true triaxial testing– zero gravity situation.

By controlling the position and number of pointsrandomly distributed inside the grid, the size andshape of the Voronoi particles can be indirectly con-trolled. However, in all analyses presented in this pa-per, the number was chosen in such a way that the as-sembly is reasonably isotropic with particles not tooelongated or with small angles. This can be done bydistributing equal, but random, spaced points. In ad-dition, because of this construction, the size of theparticles is approximately uniform. Further details arefound in (Galindo-Torres et al. 2009; Galindo-Torresand Pedroso 2010).

Fig. 3 illustrates a cubic packing of 1000 sphero-polyhedra created by the Voronoi-erosion techniqueand which is adopted as the experimental specimenfor the numerical simulations of true triaxial tests.

3 TRUE TRIAXIAL TEST

The shear properties of granular assemblies can beobtained with samples subjected to either compres-sion or extension stress paths. These tests allow forthe investigation of the relationship between devia-toric stresses and mean stresses at failure, and hencethe macroscopic friction properties. Shearing testswith stress paths in which the mean stress is kept con-stant allow for the investigation of the pattern of theultimate stress points on the octahedral plane. There-fore, the shape of the failure envelope on this planecan be investigated as well. The true triaxial is a con-venient apparatus to obtain these information.

The true triaxial apparatus is composed by a systemof six rigid plates forming a parallelepiped. Becausethe DEM code employed in this work can handlecomplex (quasi-general) shaped particles, the loadingplates of the apparatus can also be defined as DEMparticles. This is very convenient since no changeto the code is necessary in order to implement the

Figure 3: Cubic packing with 1000 quasi-general(sphero-polyhedra) particles used in the true triaxialtests, generated by means of the Voronoi-erosion tech-nique.

contact between the loading plates and the granularmaterial representing the sample. The friction coeffi-cient between the plates and the particles can easilybe controlled – here, the friction coefficient is set tozero simulating lubricated walls. In addition, the in-teraction between the plates one with respect to eachother is switched off; therefore, the plates can easilycross each other in order to confine the particles (seeFig. 4). Clearly, problems such as the interaction be-tween plates do not exist in the virtual apparatus.

Forces are applied to the plates of the virtual ap-paratus according to a pre-defined stress path. To cal-culate the stresses, the cross-sectional areas for eachdirection are updated as the plates move. Stressesare defined as negative during compression (ClassicalMechanics’ convention). Principal strain componentsare calculated by the change on the distance betweenopposite plates divided by their initial distance. Thefollowing Continuum Mechanics’ quantities are cal-culated:

pcam = −σ1 + σ2 + σ3

3(1)

qcam =

√

(σ1 − σ2)2 + (σ2 − σ3)2 + (σ3 − σ1)2√2

(2)

εv = ε1 + ε2 + ε3 (3)

εd =

√2

3

√

(ε1 − ε2)2 + (ε2 − ε3)2 + (ε3 − ε1)2 (4)

Therein, σi are the principal stress components and εiare the principal strain components, pcam is the nega-tive of the Cambridge mean stress invariant (Schofield

Figure 4: Cubic ensemble of spheres representing asample for the true triaxial test. There are 1000 parti-cles randomly generated. The same packing is usedfor simulations with and without rolling resistance.The loading plates of the apparatus can freely crosseach other. There is no friction between the plates andthe particles.

and Wroth 1968), qcam is the Cambridge deviatoricstress invariant, εv is the volumetric strain, and εd isthe deviatoric strain.

To verify the strength properties of cubic assem-blies of granular media subject to the combination ofthree stress components, the stress paths illustrated inFig. 5 can be applied. These allow the construction ofthe failure envelopes in the Haigh-Westergaard space.Each path is described as follows:

• (1) Initial confinement of the specimen by meansof isotropic stresses;

• (2a) Conventional (cylindrical) compressiontests with constant lateral stresses and increasingvertical stress;

• (2b) Conventional (cylindrical) extension testswith constant vertical stress and increasing lat-eral stresses;

• (3) p-constant tests with a combination ofstresses such that a pre-defined constant Lodeangle (θ) can be reproduced on the octahedralplane (see Fig. 6). These paths vary from exten-sion (θ = −30◦) to compression (θ = +30◦).

The Lode angle is illustrated in Fig. 6 and is defined

failure envelopes compress

ion

extension

(1)

(2a)

(2b)

(3)

Figure 5: Applied stress paths.

Figure 6: Octahedral plane and definition of Lode an-gle θ.

according to the following expression:

θ =1

3asin

(

−27s1s2s32q3cam

)

(5)

where si are the principal deviatoric stress compo-nents, calculated by means of: si = σi + pcam.

4 FAILURE CRITERIA

Stress-based failure criteria can be defined by observ-ing either the peak or the residual stresses attained inmechanical tests, such as triaxial tests. For instance,the ultimate stresses measured in triaxial tests withincreasing deviatoric stresses can be used for this def-inition. Results from triaxial tests can be used to de-fine failure criteria regarding either the macroscopicfriction angle at compression (θ = +30◦) or the fric-tion angle at extension (θ = −30◦). Only one thesetwo angles is necessary for a mathematical model. Inthis paper, the results of friction angles at compressionφcomp are thus considered. After obtaining this singlematerial parameter, the following criteria are fitted tothe simulated data: Mohr-Coulomb, Matsuoka-Nakai,and Lade-Duncan (Lade and Duncan 1973).

The Mohr-Coulomb failure criterion predicts a lin-ear relationship between qcam and pcam for the stressstates at failure and can be mathematically expressedaccording to (disregarding cohesion):

σ∗

1− σ∗

3

σ∗

1+ σ∗

3

= sinφcomp (6)

Therein, σ∗

i are the sorted (increasing) principal stressvalues. With this expression, the strength at extensionwill be smaller than for compression. Therefore, theshape of the Mohr-Coulomb failure envelope is ofa deformed hexagon when viewed in the octahedralplane.

The Matsuoka-Nakai (Matsuoka and Nakai 1974)failure criterion predicts the same strength as theMohr-Coulomb failure criterion for both compressionand extension. For plane strain or when the Lode an-gle is in the range −30◦ < θ < +30◦, the Matsuoka-Nakai failure criterion predicts higher strength thanthe Mohr-Coulomb criterion. The Matsuoka-Nakaifailure criterion is directly defined based on the threecharacteristic invariants Ii of the stress tensor accord-ing to

I1I2I3

= 9+ 8tan2 φcomp (7)

Another failure criterion similar to the Matsuoka-Nakai failure criterion is the Lade-Duncan (Lade andDuncan 1973). This criterion predicts higher strengthat extension than that predicted by the Matsuoka-Nakai or Mohr-Coulomb failure criteria. The Lade-Duncan criterion is given by

I31

I3=

(3− sinφcomp)3

(1 + sinφcomp)(1− sinφcomp)2(8)

5 RESULTS

To observe the strength properties of cubic assembliesof granular media with the DEM, virtual true triax-ial tests are carried out. The applied stress path is theone numbered (3) in Fig. 5, which is enforced afteran initial isotropic compression is applied (path (1) inFig. 5).

When using spherical particles, in order to accountfor the effect of eventual non-spherical shapes thatare typically observed in natural grains, for instancein soils, the rolling resistance technique can be em-ployed as a convenient alternative (Iwashita and Oda1998). However, some artificial parameters have to beintroduced. For the sake of comparison, simulationswith spheres and rolling resistance are carried out andthe scheme presented in (Luding 2008) for modellingthe rolling resistance is adopted in this study.

Simulations of true triaxial tests are carried out withassemblies of spheres (Fig. 4) and sphero polyhedra(Fig. 3). The packings are randomly generated as de-scribed earlier in this paper. With spheres, simulationsare carried out with and without rolling resistance.Each simulation is repeated with a different numberof particles in order to also investigate the influenceof the number of particles. The microscopic constantsadopted in all analyses are organized in Table 1. In thefollowing, pcte means constant pcam.

5.1 Linear model

Tests with spheres, spheres with rolling resistance,and sphero-polyhedra are carried out in order to verifywhether the macroscopic strength of a cubic packingof grains can be represented by a linear model or not.

Table 1: Microscopic constants for the DEM analyses.

Constant Description

Kn = 10000.0 Contact normal stiffnessKt = 5000.0 Contact tangential stiffnessµ = 0.4 Microscopic friction coefficientgn = 8.0 Normal viscous coefficientgt = 0.0 Tangential viscous coefficientβ = 0.12 Rolling resistance stiffness coeffi-

cient (only for spheres with rolling

resistance)

η = 1.0 Plastic moment coefficient (only for

spheres with rolling resistance)

0 10 20 30 40 50pcam [kPa]

0

5

10

15

20

25

q cam

[kPa

]

DEM datafit: �=12.9

Figure 7: Tests with 1000 spheres under compression(path (3) with constant pcam).

The linear model here applies to the relationship be-tween deviatoric and mean stresses. The compression(θ =+30◦) path with p-constant is employed for suchtask.

Results with 1000 spheres and 10000 spheres aregiven in Fig. 7 and Fig. 8, respectively. It is observedthat the results with 1000 spheres are more scattered(noisy) than those obtained with 10000 spheres. Forthe latter, the boundary conditions have less influenceand the representative volume (RVE) is better definedbecause of the higher number of particles.

To illustrate the difference on material response dueto the number of particles, the stress-strain curves foreach one of the packings with 1000 spheres and 10000spheres are given in Fig. 9 and Fig. 10, respectively,where it can be seen that the increase on the number ofparticles makes the simulation more stable with qual-itatively better representation when compared withreal materials; Toyoura sand for instance.

The stability situation and the quality of represen-tation are improved by adding rolling resistance to

0 10 20 30 40 50pcam [kPa]

0

5

10

15

20

25

q cam

[kPa

]

DEM datafit: �=12.0

Figure 8: Tests with 10000 spheres under compres-sion (path (3) with constant pcam).

0 2 4 6 8 10 12

εd [%]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

q cam/p

cam

θ=+30 ∘

θ=+20 ∘

θ=+10 ∘

θ=0 ∘

θ=−10 ∘

θ=−20 ∘

θ=−30 ∘

Figure 9: Stress-strain behaviour of a sample with1000 spheres (pcte = 20kPa).

0 2 4 6 8 10 12

εd [%]

0.0

0.1

0.2

0.3

0.4

0.5q c

am/p

cam

θ=+30 ∘

θ=+20 ∘

θ=+10 ∘

θ=0 ∘

θ=−10 ∘

θ=−20 ∘

θ=−30 ∘

Figure 10: Stress-strain behaviour of a sample with10000 spheres (pcte = 20kPa).

the assembly of spheres. This makes sense as longas the DEM has to reproduce Nature materials suchas sands. The linear fitting in this case is illustratedin Fig. 11 and Fig. 12 for 1000 spheres and 10000spheres, respectively. The results with 1000 particlesare again more scattered than with 10000 particles.

Even though the results in Fig. 12 weakly suggesta nonlinear fitting, a linear model is adopted, sincethe selected points representing the peak stresses arenot necessarily the most robust indication of failurestresses. It is important to note that these may varysomewhat from simulation to simulation and dependon the boundary conditions, REV size, and stress

0 10 20 30 40 50pcam [kPa]

0

10

20

30

40

50

60

q cam

[kPa

]

DEM datafit: �=25.8

Figure 11: Tests with 1000 spheres with rolling re-sistance under compression (path (3) with constantpcam).

0 10 20 30 40 50pcam [kPa]

0

5

10

15

20

25

30

35

40

45

q cam

[kPa

]

DEM datafit: �=20.9

Figure 12: Tests with 10000 spheres with rolling re-sistance under compression (path (3) with constantpcam).

path. Therefore, it is reasonable to adopt a linear fit-ting.

To illustrate the difficulty on selecting the pointsrepresenting the failure states, stress-strain curves areplotted as in Figs. 13 and 14 (see also Figs. 9 and 10)for assemblies with 1000 particles and 10000 parti-cles (spheres with rolling resistance), respectively. Inthese figures, the selected points are indicated by up-ward triangles (the same is done for all other stress-strain curves). With 10000 particles, there is less noiseand the predicted curves are more similar to the onesobtained with real sand, as shown in a next section,i.e. the 10000 particles cube is a better REV.

Simulations of true triaxial tests are also car-ried out with assemblies of complex shaped parti-cles (sphero-polyhedra). Different number of parti-cles, from 1000 to 10000 are considered. It is ob-served that the stress-strain response does not changeradically for these numbers of particles (see Fig. 15).This is due to the already high number of contacts ina sphero-polyhedra packing, therefore, well reproduc-ing a dense granular packing. In Fig. 15, the macro-scopic strength properties (ultimate stresses ratio) ofthe packing with complex particles are similar foreach number of particles. For instance, the ultimatestress ratio at failure is about qcam/pcam = 1.2. Be-cause of this similarity, only the tests with 1000 parti-cles are further discussed in the following text.

The deviatoric-mean stresses relationship for thespecimen of 1000 sphero-polyhedra (assembly illus-trated in Fig. 3) is given in Fig. 16, for compres-sion stress paths with constant mean pressure. It isobserved that the data can be very well fitted by astraight line (linear model), contrary to the simulation

0 2 4 6 8 10 12

εd [%]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

q cam/p

cam

θ=+30 ∘

θ=+20 ∘

θ=+10 ∘

θ=0 ∘

θ=−10 ∘

θ=−20 ∘

θ=−30 ∘

Figure 13: Stress-strain behaviour of a sample with1000 spheres with rolling resistance (pcte = 20kPa).

0 2 4 6 8 10 12

εd [%]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

q cam/p

cam

θ=+30 ∘

θ=+20 ∘

θ=+10 ∘

θ=0 ∘

θ=−10 ∘

θ=−20 ∘

θ=−30 ∘

Figure 14: Stress-strain behaviour of a sample with10000 spheres with rolling resistance (pcte = 20kPa).

0 1 2 3 4 5 6 7 8 9�d [%]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

q cam/pcam

1000 particles2000 particles5000 particles10000 particles

Figure 15: Stress-strain behaviour of samples of com-plex (sphero-polyhedra) particles (pcte = 5kPa).

with spheres with rolling resistance or not.

The stress-strain curves obtained with the 1000sphero-polyhedra packing are also more smooth thanthat obtained with 1000 spheres with rolling resis-tance or not. This is mainly due to the higher numberof contacts (vertices, edges, faces) and the interlock-ing provided by the complex shapes. The curves ob-tained with 1000 sphero-polyhedra can hence be com-pared with those obtained with 10000 spheres. Later,it is shown that the sphero-polyhedra packing qualita-tively represents very well the Toyoura sand.

The stress-strain behaviour in this case is illustratedin Fig. 17 for a constant mean pressure of 20kPa.The reason for a less chaotic behaviour of the sphero-polyhedra is related to the dissipation of the kinetic

0 10 20 30 40 50pcam [kPa]

0

20

40

60

80

100

q cam

[kPa

]

DEM datafit: �=44.4

Figure 16: Tests with 1000 sphero-polyhedra particlesunder compression (path (3) with constant pcam).

0 2 4 6 8 10 12

εd [%]

0.0

0.5

1.0

1.5

2.0q c

am/p

cam

θ=+30 ∘

θ=+20 ∘

θ=+10 ∘

θ=0 ∘

θ=−10 ∘

θ=−20 ∘

θ=−30 ∘

Figure 17: Stress-strain behaviour of a sample with1000 sphero-polyhedra (pcte = 20kPa).

energy. The noise observed with spheres is indeedof chaotic nature and its effect is mitigated by con-straining particles movement (as with rolling resis-tance). When spheres are free to roll, the kinetic en-ergy of the system is not as quickly dissipated as inthe other constrained cases, hence producing a noisyresponse. On the other hand, the angularity of thesphero-polyhedra, and eventual multi-contact (non-convex particles), will produce a strong constraint forthe particles movement in addition to a larger numberof collisions for the same number of particles. Thishigher frequency of inelastic collisions will then re-duce the kinetic energy and the noise.

5.2 Octahedral plane

The data obtained with DEM simulations of true tri-axial tests are now plotted in the Haigh-Westergaard3D space of principal stresses. In particular, a viewalong the hydrostatic axis is considered, with focus onthe so-called octahedral plane. This allows the inves-tigation of an appropriated macroscopic failure enve-lope for isotropic materials. Although the failure en-velope is a 3D surface in the Haigh-Westergaard stressspace, only its cross-section on the octahedral plane isdrawn, for a fixed pcam value.

For the mathematical definition of a particularfailure criteria with linear deviatoric-mean stressrelationship (e.g. Mohr-Coulomb, Matsuoka-Nakai,Lade-Duncan), a constant macroscopic friction anglemust be defined beforehand. Either the angle at com-pression or extension can be employed for such task.Here φcomp at compression is considered.

For spheres, as shown in Figs. 7, 8, 11, and 12, aconstant friction angle may not fit all sets of data forall range of mean stresses. For the case of pcam =20kPa, the fitting seems to be fairly accurate. This

−σ1 ,θ=+30 ∘

−σ3 −σ2

θ=0 ∘

θ=−30 ∘

ϕcomp=12.9 ∘

Mohr/Coulomb

Matsuoka/Nakai

Lade/Duncan

Figure 18: Octahedral view of data obtained with1000 spheres (pcte = 20kPa).

case is then selected to illustrate the DEM results onthe octahedral plane, including the shape of some fail-ure criteria. These results are given in Figs. 18, 19, 20,21.

For the packing with 1000 sphero-polyhedra, theconstant friction angle fits quite well all data obtainedwith different values of pcam. The results for pcam =20kPa are given in Fig. 22.

For spheres, spheres with rolling resistance, andsphero-polyhedra, it is observed that most data lie inbetween the Mohr-Coulomb criterion (lower bound)and the Lade-Duncan criterion (upper bound), with

−σ1 ,θ=+30 ∘

−σ3 −σ2

θ=0 ∘

θ=−30 ∘

ϕcomp=12 ∘

Mohr/Coulomb

Matsuoka/Nakai

Lade/Duncan

Figure 19: Octahedral view of data obtained with10000 spheres (pcte = 20kPa).

−σ1 ,θ=+30 ∘

−σ3 −σ2

θ=0 ∘

θ=−30 ∘

ϕcomp=25.8 ∘

Mohr/Coulomb

Matsuoka/Nakai

Lade/Duncan

Figure 20: Octahedral view of data obtained with1000 spheres with rolling resistance (pcte = 20kPa).

−σ1 ,θ=+30 ∘

−σ3 −σ2

θ=0 ∘

θ=−30 ∘

ϕcomp=20.9 ∘

Mohr/Coulomb

Matsuoka/Nakai

Lade/Duncan

Figure 21: Octahedral view of data obtained with10000 spheres with rolling resistance (pcte = 20kPa).

−σ1 ,θ=+30 ∘

−σ3 −σ2

θ=0 ∘

θ=−30 ∘

ϕcomp=44.4 ∘

Mohr/Coulomb

Matsuoka/Nakai

Lade/Duncan

Figure 22: Octahedral view of data obtained with1000 sphero-polyhedra (pcte = 20kPa).

the Matsuoka-Nakai criterion being a good average.Regarding the sphero-polyhedra, it is interesting tonote that the (isotropic) geometry of the (complex)particles does not affect much the shape of the fail-ure envelope, apart from increasing the macroscopicfriction angle, when compared with spheres.

5.3 Qualitative observation

To asses the capabilities of the DEM on representingassemblies of complex shaped grains, real experimen-tal data from true triaxial tests on the Toyoura sandof Japan (Nakai 1989), illustrated in Fig. 23(a,c,e),are qualitatively compared with the numerical resultsobtained with 1000 sphero-polyhedra, as given inFig. 23(b,d,f). It can be observed that both the stress-strain behaviour (Fig. 23(a,b)) and dilatancy curves(Figs. 23(c,d)) of experimental data and simulationsare quite similar, indicating that the 1000 particlespacking of sphero-polyhedra is a reasonable repre-sentation of this particular sand, regarding the macro-scopic mechanical behaviour.

It can also be observed that the influence of theLode angle is similar when comparing experimentswith simulations. In particular, it can be seen that thestrength at extension is smaller than at compression(Figs. 23(e,f)). This also illustrates the great impor-tance of considering the stress path on the strengthcharacteristics of granular materials. Finally, for bothsets of data and predictions, ideal failure envelopeshave similar shapes.

0 1 2 3 4 5 6

εd [%]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

q cam/p

cam

θ=+30 ∘

θ=+20 ∘

θ=+10 ∘

θ=0 ∘

θ=−10 ∘

(a) Toyoura sand

0 2 4 6 8 10 12

εd [%]

0.0

0.5

1.0

1.5

2.0

q cam/p

cam

θ=+30 ∘

θ=+20 ∘

θ=+10 ∘

θ=0 ∘

θ=−10 ∘

θ=−20 ∘

θ=−30 ∘

(b) Sphero-polyhedra

0 1 2 3 4 5 6

εd [%]

−0.5

0.0

0.5

1.0

1.5

2.0

ε v [

%]

(c) Toyoura sand

0 2 4 6 8 10 12

εd [%]

0

2

4

6

8

10

ε v [%]

(d) Sphero-polyhedra

−σ1 ,θ=+30 ∘

−σ3 −σ2

θ=0 ∘

θ=−30 ∘

ϕcomp=40 ∘

Mohr/Coulomb

Matsuoka/Nakai

Lade/Duncan

(e) Toyoura sand

−σ1 ,θ=+30 ∘

−σ3 −σ2

θ=0 ∘

θ=−30 ∘

ϕcomp=44.4 ∘

Mohr/Coulomb

Matsuoka/Nakai

Lade/Duncan

(f) Sphero-polyhedra

Figure 23: Comparison of experimental data on Toyoura sand (left, pcte = 196kPa, data after Nakai, 1989) withDEM results with 1000 sphero-polyhedra (right, pcte = 50kPa).

6 CONCLUSIONS

A DEM code that considers grains of non-spherical(quasi-general) shapes is employed for simulations ofcubic packings representing true triaxial tests. Atten-tion is given to the strength properties of assembliesof granular media. The pattern of the failure data bothon the octahedral and on the deviatoric-mean pressureplanes are observed.

The generation of dense packings of complex par-ticles is carried out employing a three-dimensionalVoronoi tessellation, after which an erosion algorithmis applied in order to build sphero-polyhedral parti-cles, i.e. smooth particles with rounded edges.

Non-physical corrections, such as the virtualrolling resistance technique, necessary for the simu-lations of Nature grains, such as sands or gravels, arecompletely avoided by the method employed in thisresearch thanks to the consideration of particles ofquasi-general (complex) shapes.

It is verified that the DEM simulations with sphero-polyhedra are more stable than with spheres, eitherwith or without rolling resistance. This is mainly dueto the interlocking of the grains that may cause moredissipation of energy, and, therefore, more damping,in addition to provide a higher frequency for con-tacts. Because the number of contacts in assembliesof sphero-polyhedra may be higher than in assem-blies of spheres, the number of particles in packingsof sphero-polyhedra may be smaller than in packingsof spheres.

The instabilities of DEM simulations with spheresare a reported problem in the DEM literature re-lated to simulations of natural grains and one of thesolutions is precisely the adoption of rolling resis-tance. However, the rolling resistance introduces vir-tual (non-physical) parameters. The instabilities aremainly due to the rolling of grains that do not de-pend on the friction angle. Controlling the rolling pre-vents this chaotic behaviour. The non-spherical shapeof sphero-polyhedra is a much more natural solution,allowing a sound physical representation of naturalmaterials.

It is also verified that, for all particles studied here,spherical or sphero-polyhedral, the relationship be-tween deviatoric and mean stresses at failure (for peakstresses) is best fitted with a straight line. This im-plies that the best phenomenological model for thismaterial is a linear model with constant macroscopicfriction coefficient. In addition, it is verified that thestress values at failure, when plotted on the octahe-dral plane, usually lie between the Mohr-Coulomband the Lade-Duncan envelopes with the Matsuoka-Nakai model being a good average.

The numerical results qualitatively agree with a setof experimental data on Toyoura sand. Therefore, theDEM, especially with particles more similar to real

Nature grains (non-spherical), can describe quite wellthe macroscopic properties of granular assemblies, in-cluding higher strength at compression than at exten-sion and an intermediate strength from compressionto extension.

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