Content uploaded by Marcos Arroyo
Author content
All content in this area was uploaded by Marcos Arroyo on Jan 10, 2021
Content may be subject to copyright.
1
Imagebased calibration of rolling resistance in discrete element 1
models of sand 2
Rorato R., Arroyo M., Gens A., Andò E., Viggiani, G. 3
Riccardo Rorato, Dr. 4
Universitat Politécnica de Catalunya (UPC), Barcelona (Spain)  Department of Civil and 5
Environmental Engineering. 6
riccardo.rorato@upc.edu 7
ORCID ID: 0000000241893058 8
UPC  Barcelonatech 9
Moduli D2 Campus Nord UPC – Office 212 10
C Jordi Girona 13 11
Barcelona 08034 12
13
Marcos Arroyo Alvarez de Toledo, Prof. 14
Universitat Politécnica de Catalunya (UPC), Barcelona (Spain)  Department of Civil and Envi15
ronmental Engineering 16
marcos.arroyo@upc.edu 17
ORCID ID: 0000000193849107 18
19
Edward Carlo Giorgio Andò, Prof. 20
Univ. Grenoble Alpes, CNRS, Grenoble INP, 3SR, F38000 Grenoble (France) 21
edward.ando@3srgrenoble.fr 22
ORCID ID: 0000000155095287 23
24
Antonio Gens Solé, Prof. 25
Universitat Politécnica de Catalunya (UPC), Barcelona (Spain)  Department of Civil and Envi26
ronmental Engineering 27
antonio.gens@upc.edu 28
ORCID ID: 0000000175887054 29
30
Gioacchino Viggiani, Prof. 31
Univ. Grenoble Alpes, CNRS, Grenoble INP, 3SR, F38000 Grenoble (France) 32
cino.viggiani@3srgrenoble.fr 33
ORCID ID: 0000000226096077 34
PRE_PRINT
For the version of record see DOI: 10.1016/j.compgeo.2020.103929
FREE (until 24/02/21) at: https://authors.elsevier.com/c/1cMfU,63b~s4GS
2
Abstract 35
Models that introduce rolling resistance at the contact are widely employed in simulations using 36
the discrete element method (DEM) to indirectly represent particle shape effects. This approach 37
offers substantial computational benefits at the price of increased calibration complexity. This 38
work proposes a method to simplify calibration of rolling resistance. The key element is an em39
pirical relation between a contact parameter (rolling friction) and a 3D grain shape descriptor (true 40
sphericity). Values of true sphericity can be obtained by image analysis of the grains, either directly 41
by 3D acquisition or by correlation with simplertoobtain 2D shape measures. Evaluation of roll42
ing friction is thus made independent from that of other model parameters. As an extra benefit, 43
the variability of grain shape in natural sands can be directly mapped into the discrete model. A 44
mapping between rolling friction and true sphericity is calibrated using specimenscale and grain 45
scale results from two triaxial compression tests on Hostun sand and Caicos ooids. The mapping 46
is validated using different triaxial tests from the same sands and from other reference sands 47
(Ottawa, Ticino). In the case of Ticino grainshape acquisition is made in 2D, using an ordinary 48
table scanner. The results obtained support this direct calibration procedure. 49
50
Keywords: Discrete Element Method; Rolling Resistance; Particle Shape; Xrays micro tomogra51
phy; Triaxial Test; Shear Resistance. 52
Introduction 53
Powered by increased computational performance, the discrete element method (DEM) has 54
gained much relevance in geomechanics since originally proposed by Cundall & Strack (1979). 55
DEM models at specimen scale are now a basic tool of research to study and illuminate many 56
features of soil mechanics observed in the laboratory (Ciantia et al. 2019a; Li et al. 2018; Hosn et 57
3
al. 2018). There is also a growing trend to use DEM models to analyse large scale problems of 58
direct engineering relevance (Zhang et al 2019; Zhang & Evans, 2019; Butlanska et al. 2018; Ka59
wano et al. 2018). As in other numerical modelling approaches, there is always an underlying 60
conflict between model resolution and computational efficiency. This tension is particularly vivid 61
in the consideration of particle shapes in DEM. 62
63
The most widely used shape in 3D DEM is the sphere (as is the disk in 2D DEM). The reason is 64
pragmatic: spheres allow straightforward and computationally efficient contact detection, which 65
is a large part of the computational cost of every step. Unfortunately soil particles are not gener66
ally spherical but have instead very varied shapes. In coarse soils, research has clearly identified 67
large particle shape effects for several important properties such as extreme void ratios (Cho et 68
al. 2006), critical state friction (Yang and Luo 2015) or dilatancy and peak friction (Xiao et al. 2019). 69
Particle shape also affects responses of major engineering significance, like liquefaction resistance 70
(Vaid et al. 1985) or cone tip resistance (Liu & Lehane 2013). 71
72
Direct experimental evidence for the role of shape in soils was reinforced by DEM models in 73
which grain shape was directly controlled. For instance, just switching from disks to ellipses 74
(Rothemburg & Bathurst, 1992) or from spheres to ellipsoids (Lin & Ng, 1998) raised numerical 75
shear strength and dilatancy values to within the range observed in soils. 76
77
Ellipses and ellipsoids are still far from the shapes observed in most soil particles. Several tech78
niques have been developed to incorporate more realism into element shapes: they use clumps 79
or aggregates of spheres (Matsushima 2002, Lu and McDowell 2007, Katagiri et al. 2010); polyhe80
drons (Zhao et al. 2006; Boon et al. 2012); superquadrics (Williams & Pentland, 1992; Zhao et al. 81
2018) or level sets (Jerves et al. 2016, Kawamoto et al. 2018). Increased morphological realism has 82
advantages and disadvantages. One advantage is that it may be thus possible to represent the 83
4
nonnegligible variability in shape that is observed in granular soils. A significant disadvantage 84
is added computational cost. Indeed, for the same problem dimensions, orders of magnitude in85
creases in computational time with respect to spherebased models are typically reported (Lu et 86
al. 2015; Irazabal et al. 2017). 87
88
The micromechanics underlying the effect of element shape on shear strength was clarified by 89
Bardet (1994), who noted that disks showed a high concentration of rotations in shear bands and 90
that, if rotation was blocked, realistic values of friction and dilatancy ensued. Based on this and 91
similar observations, several researchers (Sakaguchi et al. 1993, Iwashita & Oda 1998, Jiang et al. 92
2005; Mohamed & Gutierrez, 2010) proposed the introduction of a resisting moment (i.e., rolling 93
resistance) at particle contacts (see Figure 1). The moment applied is typically dependent on rela94
tive particle rotation, opposing it through an elastoplastic mechanism analogous to that acting 95
for contact forces. Sometimes a viscous component is also added to the contact formulation (see 96
Ai et al. 2011, for a review). 97
98
Figure 1: Origin of rolling resistance at contact (Iwashita and Oda 1998) 99
Several DEM studies (Zhou et al. 2013, Wensrich et al. 2014) have compared the results obtained 100
using aggregates of particles (i.e., clumps) or adding rotational constrains, showing that both ap101
proaches result in very similar behaviour, at least for the quasistatic conditions relevant in most 102
soil mechanics problems. The main advantage of the rolling resistance model is that contact de103
tection remains efficient; the calibration of contact properties is, however, far from trivial. 104
5
The majority of the previous studies (Iwashita and Oda 1998, 2000, Jiang et al. 2005, Belheine et 105
al. 2009, Zhou et al. 2013) calibrate rolling resistance through an empirical macroscopic approach. 106
Specimenscale responses of “identical” numerical and experimental test are matched by trial107
anderror. The process is difficult because the effects of rolling resistance in macroresponse are 108
coupled with those of other parameters (the coefficient of sliding friction in particular), and mul109
tiple solutions are possible to match key experimental results, such as dilatancy or peak mobilised 110
friction (Estrada et al. 2008, Wensrich and Katterfeld 2012, Cheng et al. 2017). Calibration can thus 111
become a very time consuming and somewhat subjective process. Alternatives based on 112
statistically driven semiautomated calibration have been proposed, (Cheng et al. 2018) but they 113
appear computationally intensive. 114
115
Some researchers (Calvetti et al. 2003, Arroyo et al. 2011, Ciantia et al. 2015) have simplified 116
radically the calibration process by directly assuming very large values of moment resistance and 117
stiffness, so as to inhibit relative rotation at the contact. This assumption does not limit the ability 118
of the resulting discrete model to match and predict largescale soil responses, but it does lack 119
some subtlety. 120
121
A different approach to simplify the calibration problem would be to give some specific physical 122
base to rolling resistance. Little work has been done to explore this possibility. Wensrich & Kat123
terfeld (2012) proposed a definition of average contact particle eccentricity as such basis. Rorato 124
et al. (2018) suggested instead that a physical measure of grain shape such as sphericity would 125
offer a good basis to calibrate rolling resistance. Herein, this latter idea is developed in detail and 126
tested with several sands. 127
6
Methodology 128
2.1 Contact rolling resistance model 129
This work is based on the Iwashita & Oda contact model (1998) as implemented in the commercial 130
DEM software PFC3D V5 (Itasca Consulting Group Inc. 2014), which has been used for all the 131
simulations presented here. The model is schematically illustrated in Figure 2a, it includes a con132
ventional linear elastic – frictional contact model for particle relative displacement at the contact 133
plus an additional set of elastic spring notensional joint and slider for the rolling motion. 134
(a)
(b)
Figure 2: Rolling resistance contact model (a) and elasticperfectly plastic model accounting for rolling 135
resistance at contact (b). 136
The contact normal and shear stiffness are defined as 137
=
; =
(1)
where and are material parameters to be calibrated, A is the diameter of the smallest 138
contacting sphere and L is the distance between grain centres. This formulation guarantees scale139
invariance of the interaction law (Feng & Owen, 2014). 140
141
7
For the moment rotation law (Figure 2b) the following assumptions are used 142
(1) The rolling stiffness () is defined as: 143
=k R2 (2)
where is the contact shear stiffness and R the effective radius defined as 144
=1
R1
+1
R2
(3)
with R1 and R2 being the radii of the two particles in contact. The proportionality of rolling and 145
shear stiffness was derived by Iwashita & Oda (1998) to make identical the elastic moment due 146
to shear and that due to rotation. Wensrich & Katterfield (2012) compared this formulation of 147
rolling stiffness with available alternatives and noted that the IwashitaOda approach has some 148
numerical advantages as a) it dampens elastic oscillations without the need to introduce extra 149
parameters and b) simplifies the computation of critical time steps. 150
(2) The momentrotational contact law is implemented as an elasticperfectly plastic model with 151
the yielding moment (M*) defined as: 152
= μrFnR (4)
where µr is defined as rolling friction coefficient and Fn is the normal contact force. The rolling 153
resistance part of the contact model used in this study is illustrated in Figure 2b. 154
155
The Iwashita and Oda (1998) original formulation also includes viscous dissipation at the contact. 156
Wensrich and Katterfeld (2012) showed that the effect of rotational contact viscosity on simula157
tion outcomes is negligible for the quasistatic conditions with low inertial numbers which are of 158
interest here. Therefore, viscous dissipation at the contact was not included in the models for 159
simplicity. 160
8
2.2 Shape description 161
The degree of true sphericity, ψ (Wadell 1932) is employed to describe grain shape. ψ is defined as 162
ψ = sn
S =
36πV2
3
S (5)
where (S) is the particle surface area and (sn) is the surface area of a sphere with the same volume 163
(V) as the particle. As argued by Rorato et al. (2019a), ψ offers a compact, easy to interpret, and 164
conceptually sound measure of how similar a given particle is to a sphere. 165
Despite its conceptual simplicity, this shape descriptor had seen relatively little use because meas166
uring the surface area of irregular sand grains is difficult. This has changed in recent years, as 167
computerbased 3D image analysis techniques made such measurements possible. Still, access to 168
3D imaging equipment is sometimes limited, and 2D images are much easier to acquire and pro169
cess. For this reason a number of 2D proxy measures of sphericity have been proposed over the 170
years (Rorato et al. 2019a) In this work we use 2D perimeter sphericity . which is defined as 171
the ratio of the perimeter of the circle with area equal to that projected by the particle to the pe172
rimeter of the actual particle projection. Note that we use oriented particle projection – i.e., the 173
projection is made against the plane of maximum particle stability. 174
175
2.3 Relating rolling resistance and particle shape 176
Rorato et al. (2018) hypothesized that the degree of true sphericity may be univocally related with a 177
coefficient of rolling friction, through a relation such as 178
μr=(ψ) (6)
This kind of relation maps a physical measured sand property into a discrete element property. 179
Such mapping may be made just on the average value of sphericity, to obtain a single value of 180
9
rolling resistance to apply for all particles in a DEM model of such sand. However, when an 181
experimental distribution of ψ such as those in Figure 3 is available, the process can be also made 182
element by element, assigning to each one a sampled value from the measured distribution of ψ 183
and then applying the mapping function to initialize its rolling friction coefficient. In this way, 184
the variability in grain shape distribution is directly reflected in the numerical model through a 185
distribution of particle rolling friction coefficient. 186
187
Because rolling friction is a contact property, an extra rule is necessary to assign rolling resistance 188
to a contact between two particles. The solution to avoid this ambiguity is to select the minimum, 189
as 190
μr= (μr,1 , μr,2 ) (7)
where µr,1 and µr,2 are the rolling friction coefficients of the two contacting spheres. This is the 191
same rule that the PFC code applies to the sliding friction coefficient when two bodies of different 192
materials contact. Thus, the rolling resisting yielding moment () varies at each contact depend193
ing on (1) the radii of the contacting spheres, that is the effective radius, , (2) the normal contact 194
force F and (3) the coefficient of rolling friction, different for each contact (from Eqs. 67). 195
Model calibration 196
3.1 Target experimental data 197
Two natural sands with very different particle shape were selected for calibration: Hostun sand, 198
very angular, and Caicos ooids, very spherical (see Table 1, where physical properties for these 199
sands are reported alongside those of sands later used in validation). A triaxial test campaign on 200
various specimens of Hostun sand and Caicos ooids was performed by Andò (Andò 2013) at 201
10
Laboratoire 3SR (Grenoble). Systematic tomographic acquisition was carried out throughout. 202
Two tests on dense Hostun sand (specimen “HNEA01”) and Caicos ooids (specimen “COEA04”) 203
under 100 kPa confining pressure were selected for the calibration. The macroscopic stress strain 204
and volumetric responses recorded in these tests are shown in Figure 4. 205
Mineralogy
Hostun
338
1.41
0.95
2.65
0.605
0.927
Quartz
Caicos 420 1.39 1.09 2.80*  
Aragonite (96%)
Calcite (3%)
Ottawa
310
1.31
0.95
2.65
0.499
0.850
Quartz
Ticino 540 1.60 2.32 2.68 0.582 0.934
Feldspar (65%)
Quartz (30%)
Table 1: Physical properties of the different sands used for calibration and/or validation [ = mean grain 206
size, = Coefficient of uniformity, = Coefficient of curvature, = Specific gravity, / = Min207
imum/maximum void ratio]. Data for Hostun from (Combe 1998), data for Caicos from (Andò 2013), data 208
for Ottawa from (Lee et al. 2007), data for Ticino from (Jamiolkowski et al. 2003). * = typical value for 209
carbonate sands. 210
Rorato et al. (2019a) examined the tomographic images of the Hostun HNEA01 and Caicos 211
COEA04 specimens to acquire threedimensional shape properties (e.g., volume, surface area, 212
lengths, etc.) of every grain. One of the results thus obtained were statistical distributions of ψ 213
for Hostun and Caicos sands (Figure 3). Another important finding from that work was that, in 214
the Hostun and Caicos grains, 3D true sphericity () showed good linear correlation with the 215
much easier to measure 2D perimeter sphericity . The correlation obtained is given by 216
= 1.075()0.067 (8)
217
11
218
Figure 3: Statistical distribution of the degree of true sphericity of Caicos (COEA04) and Hostun 219
(HNEA01) sands (Rorato et al. 2019a). 220
221
222
Figure 4: Triaxial stressvolumetricstrain responses of Hostun sand (specimen HNEA01) and Caicos sand 223
(specimen COEA04) 224
A different set of analyses of the scanning data from these two specimens was made to obtain a 225
database of grain motions. In that work (see Rorato et al. 2020; Rorato, 2019b) Discrete Digital 226
Volume Correlation (DDVC, see Hall et al. 2010) is used to obtain the kinematical history of each 227
sand grain in these triaxial specimens. Averaging that grain scale result for a selection of grains 228
it is possible to obtain the average grain kinematics in a particular zone of the specimen. Because 229
12
the specimens failed in a localized shear mode, there was interest in separating the behaviour 230
inside and outside the shear bands. To individuate the grains belonging to the shear band, a nom231
inal strain (called “microstrain”) was assigned to each grain. That was done – following Catalano 232
et al. (2014) – by means of a Voronoibased allocation of spatial domains centred around each 233
particle. Once microstrains are computed, a threshold shear strain value (0.1 in this study) is 234
used to separate the particles that belong to the shear band from those that are outside of it. As a 235
result of this work the average cumulative rotation for the grains in these specimens can be plot236
ted (Figure 5). 237
238
Figure 5: Average cumulative rotation of sand grains inside and outside the shear band (Rorato, 2019b). 239
A further result form that previous work of interest here involves correlations between individual 240
grain shape descriptors and grain rotations . The study (Rorato et al. 2020; Rorato, 2019b) showed 241
that ψ is one of the shape descriptors that best correlated with cumulative grain rotation, partic242
ularly for grains that are inside the shear bands. 243
3.2 Mapping function 244
A monotonically increasing mapping function, (ψ), seems reasonable, as it provides low values 245
of rolling friction when grain sphericity is high, and viceversa. Rolling friction values used in 246
13
previous studies usually range between 0 and 1, although some researchers (e.g., Hosn et al. 2017) 247
have explored higher values. True sphericity ψ has a relatively narrow range in practice. A cube, 248
for instance, has a value of ψ=0.81; detailed examination shows that grain ψ values below 0.6 in 249
Figure 3 likely result from image segmentation errors (Rorato et al., 2019a). 250
251
In a first approximation it may seem tempting to assume a zero value of µr for a sphere (ψ=1). 252
However, such assumption has a serious limitation, as the rolling resistance of spherical particles 253
may be significant due to mechanisms such as contact deformation (Jiang et al. 2005) and/or sur254
face interlocking due to contact roughness (Huang et al. 2017). It is thus preferable to allow for a 255
finite value of µr at the upper limit of sphericity. The mapping function selected for calibration 256
takes then an exponential form 257
μr= (ψ) (9)
3.3 Calibration procedure 258
The parameters of the contact model available for fitting are thus , and contact sliding 259
friction . Apart from that, the two parameters of the sphericity to rolling resistance mapping 260
function, A and b also require calibration. 261
262
This set of parameters was calibrated, by trial and error, to adjust not only the specimen scale 263
macroscopic response illustrated in Figure 4, but also the observed evolution of average rotation 264
within the shear bands reported in Figure 5. This rotation evolution was included as the macro265
scopic stressstrain information does not offer enough information on the particle rolling behav266
iour that is directly related to rotational resistance in DEM. In principle, the contact model pa267
rameters may be different for each sand, as they are presumed to reflect grain properties not 268
explicitly accounted for in the model, such as mineralogy or roughness. On the other hand, the 269
14
sphericityrolling resistance mapping function parameters are presumed to be unique for all 270
sands as the function already incorporates the effect of different sand grain shapes. 271
DEM cylindrical specimens were prepared to be tested in triaxial compression. The specimens 272
matched the particle size distribution (PSD) of the actual specimens. The small size of the tested 273
specimens (10mm diameter and 20mm height) made scaling unnecessary, and the numerical 274
specimens maintained the same scale as the experiments. Doing so, the initial models contain 275
about 60.000 particles, close to the number of sand grains identified inside specimens HNEA01 276
and COEA04. To attain prescribed initial conditions of density and pressure arbitrarily low initial 277
friction coefficients (µ0) were used to facilitate specimen formation after seeding. The specimens 278
were then isotropically compressed up to 100kPa. 279
280
The DEM specimens were limited by a cylindrical wall element on the outer periphery and two 281
horizontal walls at the top and bottom. The radius of the horizontal wall was servo controlled 282
during loading to maintain a constant pressure. During shearing a constant vertical velocity is 283
applied to the top and bottom walls; this velocity was selected to maintain a low inertial number, 284
, which is defined as (Da Cruz et al. 2005): 285
=
/ (10)
where is the shearing rate, is the pressure level (confining pressure) and is the particles 286
density. 287
The same Voronoicell based procedure (Catalano et al. 2014) employed to assign microstrain to 288
grains in the experimental specimens was also applied to the numerical specimens. Grains be289
longing to a shear band were identified by the assigned shear strain value attained towards the 290
end of the test. The same microstrain shear threshold value used to analyse the experiments (0.1) 291
was also applied here. For all elements assigned to the shear band, their kinematic history was 292
15
then analysed to extract individual particle rotations, which were then averaged to compare with 293
the equivalent experimental data. 294
295
The trial and error parameter calibration procedure followed well established heuristics for the 296
linear elastoplastic model (Butlanska, 2014), with parameter mostly selected to match initial 297
stiffness, to match initial dilatancy and to match peak strength. The mapping function 298
parameters were mostly adjusted to match the average rotation vs strain curves, although they 299
also affected postpeak stressstrain behaviour. After a few rounds of iterations, the responses 300
illustrated in Figure 6, Figure 7 and Figure 8 were considered to offer a satisfactory match to the 301
experiments. These results were obtained using the parameters reported in Table 2 where the 302
parameters of other sands used in later validation simulations are also included. 303
Parameter Symbol Hostun Caicos Ottawa Ticino
Specimen sizes
(height, diameter)
()
()
20
10
20
10
20
10
20
10
Effective normal
contact stiffness
(10)
2.0 2.0 1.5 4.0
Normaltoshear
stiffness ratio
2.0 2.0 2.0 2.0
Interparticle
friction coefficient
0.575 0.575 0.450 0.600
Degree of true sphericity ψ (Fig. 13) (Fig. 13) (Fig. 13) (Eq. 8)
Rolling friction coefficients µr (Eq. 11) (Eq. 11) (Eq. 11) (Eq. 11)
Rolling stiffness (Eq. 2) (Eq. 2) (Eq. 2) (Eq. 2)
Local damping () 0.7 0.7 0.7 0.7
16
Parameter Symbol Hostun Caicos Ottawa Ticino
Ball density (
) 2500 2500 2500 2500
Ball scaling factor () 1 1 1 1
Confining pressures ()
• 100
• 300
• 100
• 300
• 100
• 100
• 200
• 300
Inertial number (10) 4.00 4.74 3.67 6.28
Table 2: Parameters and input variables employed in the DEM simulations 304
Hostun sand at 100kPa
Figure 6: Comparison between the triaxial responses (100kPa confining pressure) of the experiments (spec305
imens HNEA01 and HNEA03) and the numerical model (DEM) replicating Hostun sand. 306
17
Caicos ooids at 100kPa
Figure 7: Comparison between the triaxial responses (100kPa confining pressure) of the experiments (spec307
imens COEA03 and COEA04) and the numerical model (DEM) replicating Caicos ooids. 308
Caicos (COEA04)
Mean rotations inside the shear band
Hostun (HNEA01)
Mean rotations inside the shear band
Figure 8: Mean particle rotations for the grains located inside the shear bands (the black grains of Figure 309
10) for both the experimental and numerical samples, throughout the execution of the triaxial test. The good 310
fit ensures the kinematics at failure is respected. 311
18
Somewhat surprisingly, the adjusted values of the sliding friction coefficients (0.575) and the stiff312
ness parameters (effective normal stiffness, stiffness ratio) were identical for both sands. The 313
parameters fitted for the mapping function result in 314
μr= 0.1963(ψ). (11)
and this power function is plotted in Figure 9. The calibration result assigns a minimum limiting 315
rolling friction coefficient of about 0.2 to perfectly spherical particles (ψ=1). 316
317
Figure 9: Calibrated matching function between particle true sphericity and rolling friction coefficient. The 318
shaded area indicates the inadmissible values of true sphericity. 319
The match obtained for the axial stressstrain behaviour is rather good. Figure 6 and Figure 7 also 320
include results from two experimental replicas of the tests used in calibration, test HNEA03 for 321
Hostun and test COEA03 for Caicos. It can be seen that the numerical results fit well within the 322
baseline experimental variability given by the test replicas. 323
A slight discrepancy is noted in the volumetric vs. axial strain curves. The numerical specimens 324
keep on dilating towards the end when the experimental curves are becoming flat. This is most 325
likely an effect of the simplified model used to represent the cylindrical membrane employed in 326
the physical experiments. The servo controlled external rigid wall employed in the numerical 327
model forces a uniform radial expansion of the specimen, that is particularly unrealistic after 328
19
shear localization takes place. This effect of radial rigid walls on apparent (post localization) di329
latancy was recently demonstrated by Khoubany & Evans (2018). 330
Figure 8 shows that an excellent match was attained for the mean particle rotation history inside 331
the shear bands. The overall aspect of the shear bands identified in the numerical simulations is 332
compared with the experimental results in Figure 10. The numerical sample is clearly able to 333
localise the strain, although due to the rigid radial boundary condition applied, the bands are 334
thicker and extend further to the corners. However, the fact that the shear band of Hostun is 335
thicker than that of Caicos (due to increased interlocking effects) indicates that shear band thick336
ness variation is qualitatively reproduced in these DEM simulations. 337
Caicos Hostun
Experiment DEM Experiment DEM
(a)
(b)
(c)
(d)
Figure 10: Shear band identification for the experiments and the DEM simulation for both sands (specimens 338
HNEA01 and COEA04). Physical (ac) and numerical (bd) particles are coloured black if they belong to 339
the shear band. The same threshold separates the grains from both sands and both physical and numerical 340
samples. 341
20
Model validations 342
4.1 Further tests on Hostun and Caicos sands 343
The experimental dataset used for calibration was part of a larger triaxial testing campaign, as 344
described by (Andò 2013). That campaign included other triaxial tests on dense Hostun and Cai345
cos specimens at higher confining pressures (300 kPa). Such tests offered a first suitable target for 346
validation. 347
348
Numerical specimens identical to those described before were created, trying to approximate the 349
initial porosities (before shearing) of the physical specimens as much as possible (Table 3). The 350
specimens were compressed isotropically to 300kPa and then sheared in triaxial condition until 351
failure was attained. For each sand, all contact properties remain the same as in the 100kPa con352
fining case. The mapping function applied to assign rolling resistance to the elements is also the 353
same. 354
21
Sand Specimen
Relative
density
(EXP)
Confining
pressure
Initial
porosity
(EXP)
Initial
porosity
(DEM)
  (%) () ,(%) ,(%)
Hostun
HNEA01
83
100
39.7
39.0
Hostun
HNEA02
95
300
38.2
38.6
Caicos
COEA04

100
31.9
33.2 (*)
Caicos
COEA02

300
33.2
34.3 (*)
Ottawa
OUEA04
112
100
31.4
34.1 (*)
Ticino
TC1
47
109
43.5
44.1
Ticino
TC2
46
200
43.7
43.0
Ticino
TC3
41
300
44.1
43.8
Ticino
TC4
72
100
40.5
39.8
Ticino
TC5
74
200
40.3
39.7
Ticino
TC6
75
300
40.1
40.2
Ticino
TC7
90
100
38.2
38.2
Ticino
TC8
93
200
37.8
38.4
Ticino
TC9
93
300
37.8
37.6
Table 3: Drained triaxial compression tests performed in this study. The relative density and porosity of 355
each experimental/numerical test are reported. The symbol (*) means that a denser specimen could not be 356
generated for the DEM simulation. 357
Figure 11 and Figure 12 show the stressstrainvolumetric response of the numerical and experi358
mental tests. The numerical curves compare well with the experiments, except for the volumetric 359
dilation of the Caicos specimen, which is again overestimated. This may be partly due to the 360
22
boundary effects induced by the rigid lateral wall on the postlocalization response. Another ef361
fect at play may be breakage and/or particle abrasion at this higher stress level. Even small 362
amounts of particle breakage can have a significant effect on dilatancy (Ciantia et al. 2019b). 363
Hostun sand at 300kPa
Figure 11: Comparison between the triaxial responses (300kPa confining pressure) of the experiments 364
(specimens HNEA02 and HNEA04) and the numerical model (DEM) replicating Hostun sand. 365
23
Caicos ooids at 300kPa
Figure 12: Comparison between the triaxial responses (300kPa confining pressure) of the experiment (spec366
imen COEA02) and the numerical model (DEM) replicating Caicos ooids. 367
4.2 Triaxial test on Ottawa sand 368
The experimental campaign by Andò (2013) also included tests in a different silica sand: Ottawa 369
50/70, which is also a material frequently used in geotechnical research (Table 1). The tests 370
performed at 100 kPa (OUEA04) and 300kPa (OUEA02) were selected for validation. The physical 371
specimen OUEA04 had been scanned using the same tomographic procedures applied for the 372
case of Hostun & Caicos, therefore 3D grain images are available. True sphericity values were 373
computed for all the grains in the specimen (110.000), obtaining the distribution illustrated in 374
Figure 13. The sphericity values of Ottawa peak between those of Caicos and Hostun. 375
24
376
Figure 13: Statistical distributions of 3D true sphericity for Hostun, Caicos, Ottawa and Ticino sands. 377
The DEM specimens were then prepared matching the experimental PSD and approximating as 378
much as possible the initial porosity (Table 3). The specimens were prepared without scaling, 379
including about 102.000 spherical elements, slightly below the number of grains contained in the 380
physical sample. Specimen preparation and triaxial testing followed identical procedures as those 381
previously described. 382
383
In principle, only the contact parameters and (i.e., and ) and coefficient of sliding 384
friction () were free to adjust, as the sphericity to rolling resistance mapping function applied 385
was the same. In practice only and were adjusted  using the test at 100 kPa  to values 386
of 0.15 GPa and 0.45, respectively. The value was maintained as 2.0, as in the other sands 387
(Table 2). 388
389
Figure 14 and Figure 15 show the comparison between the experimental and numerical results at 390
100kPa and 300kPa confinement, respectively. The figures include two experimental replicas of 391
the tests, which were also available (OUEA06 and OUEA03, respectively for the 100kPa and 392
300kPa confinements). The simulations provide a good fit to the experiments; in this case even 393
the postlocalization volumetric mismatch is small. 394
25
Ottawa sand at 100kPa
Figure 14: Comparison between the triaxial responses (100kPa confining pressure) of the experiments 395
(specimens OUEA04 and OUEA06) and the numerical model (DEM) replicating Ottawa sand. 396
Ottawa sand at 300kPa
26
Figure 15: Comparison between the triaxial responses (300kPa confining pressure) of the experiments 397
(specimens OUEA02 and OUEA03) and the numerical model (DEM) replicating Ottawa sand. 398
4.3 Triaxial tests on Ticino sand 399
Ticino sand (Table 1) is a poorly graded mediumsized sand, with grains of medium angularity. 400
The macroscopic responses of Ticino sand have been well reproduced using DEM by several re401
searchers either using clumps (Gotteland et al. 2009) or by inhibiting rotations (Calvetti 2008, 402
Arroyo et al. 2011, Butlanska et al. 2014). Ticino sand will be here modelled using the rolling 403
resistance model with rolling resistance values assigned through the mapping function (Eq. 11). 404
405
There were no 3D tomographic images of Ticino sand readily available to establish the sphericity 406
distribution to be input in the mapping function. As an alternative, a table scanner (CanoScan 407
LiDE 25) was employed to acquire 2D images of about 4000 grains of Ticino sand. Rorato et al. 408
(2019a) showed that this sample size is enough to obtain a good definition of the sphericity sta409
tistics. The table scanner used had 1200 dpi, or equivalently about 20µm/pixel, which is only 410
slightly above the 15µm voxel side used in the 3D µCT image acquisition for the other sands. 411
The parallel projection of each grain on the scan surface avoids parallax errors. The scanned im412
age was binarised, segmented and labelled using the opensource python package SPAM (Andò 413
et al. 2017), as shown in Figure 16. 414
27
Original image (from scanner) Labelled image
(a)
(b)
Figure 16: Zoom on (a) the original scan and (b) the labelled image of Ticino sand. 415
Using a dedicated python script 2D perimeter sphericity was calculated for each grain identified 416
in the image. Equation 8 relating the known 2D perimeter sphericity () and the 3D true sphe417
ricity () was applied to obtain a statistical distribution of 3D true sphericity. This is plotted in 418
Figure 13: as for Ottawa sand, the mean sphericity of Ticino sand was located between those of 419
Hostun and Caicos sands. It can be noted that the dispersion is higher than for the other sands, 420
with a wide tail of low sphericities. This is likely a side effect of the approximations involved in 421
the 2D procedure. 422
423
DEM triaxial simulations were carried out for a total of nine triaxial tests, including specimens at 424
dense (90%), medium (75%) and loose (50%) states, at variable confining pres425
sures (100, 200 and 300kPa). The triaxial chamber used for the DEM simulations kept the same 426
geometry as that employed for Caicos, Hostun and Ottawa (Table 2), which – in this case – was 427
much smaller than that employed in the physical experiments. Again, no particle scaling was 428
applied and, due to the larger grain size of Ticino sand, about 16.000 elements were employed in 429
each numerical specimen. 430
431
28
One specimen (100kPa confining, 75%) was selected to calibrate contact parameters 432
and ( the value was kept at 2, as in the other sands). After a few iterations the values 433
finally selected were, respectively, 0.4 GPa and 0.60 (Table 2). 434
435
The mechanical responses (stressstrainvolumetric) of the nine DEM simulations are shown 436
alongside the corresponding experimental results in Figure 17 (50%), Figure 18 (75%) 437
and Figure 19 (90%). It is evident that the triaxial response is well reproduced under all 438
stress and state conditions used in the tests. Small discrepancies can be seen for the DEM simula439
tions at 300kPa but it must be mentioned that some particle crushing  not modelled here  was 440
noted in the physical samples at the end of those tests. 441
This case strongly suggests that there is no need to have a full threedimensional tomographic 442
identification of sand grain shapes. Thanks to the good correlation between true sphericity and 443
2D perimeter sphericity (equation 8), the required rolling resistance for DEM analysis can be 444
readily determined based on the observations of 2D images. 445
29
Loose state (50%)
Figure 17: Triaxial responses of loose Ticino sand at 100200300kPa 446
30
Medium Dense state (75%)
Figure 18: Triaxial responses of medium Ticino sand at 100200300kPa 447
31
Dense state (90%)
Figure 19: Triaxial responses of dense Ticino sand at 100200300kPa 448
Discussion 449
A comparison can be made with the approach proposed by Wensrich & Katterfeld (2012), who, 450
based on geometric deductive reasoning, suggested that rolling resistance could be evaluated as 451
the ratio of grain surface – averaged contact eccentricity and the equivalent grain radius (i.e., 452
the radius of a sphere with equal volume as the grain), 453
μr=
(12)
32
The rolling friction values evaluated with Eq. 12 are compared with those predicted by Equation 454
11 in Figure 20. The rolling friction values obtained are similar for high sphericity values. Despite 455
this coincidence, it is noted that further work with the average contact eccentricity concept 456
(Wensrich et al. 2014) concluded that values given by Equation 12, should be halved to obtain 457
good matches with clumpbased simulations, which would separate further the eccentricity458
based rolling friction values from those derived from our proposal. This difference may reflect 459
the decision to represent in our model not just structural rolling resistance  i.e., the one reflected 460
by eccentricity  but also contactlevel sources or rolling resistance, allowing nonzero rolling re461
sistance for spheres. 462
463
Figure 20: Comparison of rolling friction values derived from average eccentricity following Wensrich & 464
Katterfeld (2012) and those given by the calibrated mapping function 465
Another interesting comparison is with the work of Kawamoto (Kawamoto et al. 2018) who built 466
a DEMbased “avatar” of specimen of HNEA01 in which a much higher level of detail of each 467
grain shape was represented, via level sets. The “avatar” approach does not individuate single 468
grains for validation, but, as done here with average shear band rotation, relies on emerging en469
semble measures (like shear band orientation) for validation. The simulation results obtained by 470
Kawamoto et al (2018) for Hostun using a level set method to incorporate explicitly grain shape 471
33
are also included in Figure 6 for comparison. The results are not very different from those ob472
tained here, with discrepancies in dilatancy attributable to the much more realistic numerical 473
representation of boundary conditions (loading platens and flexible membrane surrounding the 474
specimen) in the work of Kawamoto et al. (2018). It is also interesting that the sliding friction 475
value that was calibrated (0.575) by Kawamoto et al. (2018) was very close to the one calibrated 476
here (0.55). Of course, the computational cost is very different: while simulating specimen 477
HNEA01 through level set approach took 17 h in the 480 cores of the San Diego Supercomputer, 478
using the model calibrated here a simulation of the same specimen lasts 20h in a fourcore desktop 479
computer. 480
Conclusions 481
This paper presents a novel technique to relate univocally the degree of true sphericity of each grain 482
contained in a sand sample with the coefficient of rolling friction to apply to its numerical counter483
part of spherical shape. This approach greatly simplifies the complex calibration procedure of 484
rolling resistance contact models, easily incorporating information on natural shape variability in 485
the numerical discrete model. 486
487
It has been also shown that easilyacquired 2D proxy measures of sphericity can be used instead 488
of the more difficult to acquire direct 3D measurements. A relatively cheap table scanner may be 489
all that is required to evaluate the coefficient of rolling friction. 490
491
For the four different sands examined, the approach appeared to work successfully using a 492
unique mapping function between true sphericity and rolling friction. However, the validity of 493
the approach has only been tested for triaxial compression paths. with different stress paths 494
34
seems necessary in this respect. Future work will also explore if the proposed mapping of sphe495
ricity into rolling friction holds true when other modelling features are modified, such as the 496
allocation rule for rolling friction at the contact, or the definition of contact stiffness (e.g. by using 497
a Hertzian contact model instead of linear stiffness). 498
Acknowledgments 499
The support of EU through 645665  GEORAMP  MSCARISE and of the Ministry of Economy 500
of Spain through research Grant BIA201459467R is gratefully acknowledged. Laboratoire 3SR 501
is part of the LabEx Tec 21 (Investissements d’Avenir  grant agreement n_ANR11LABX0030). 502
References 503
Ai, J., Chen, J.F., Rotter, J.M., and Ooi, J.Y. 2011. Assessment of rolling resistance models in 504
discrete element simulations. Powder Technology, 206(3): 269–282. Elsevier B.V. 505
doi:10.1016/j.powtec.2010.09.030. 506
Alshibli, K.A., Druckrey, A.M., AlRaoush, R.I., Weiskittel, T., and Lavrik, N. V. 2015. Quantifying 507
Morphology of Sands Using 3D Imaging. Journal of Materials in Civil Engineering, 27(10). 508
doi:10.1061/(ASCE)MT.19435533.0001246. 509
Alshibli, K.A., Jarrar, M.F., Druckrey, A.M., and AlRaoush, R.I. 2017. Influence of Particle Mor510
phology on 3D Kinematic Behavior and Strain Localization of Sheared Sand. Journal of Ge511
otechnical and Geoenvironmental Engineering, 143(2): 04016097. 512
doi:10.1061/(ASCE)GT.19435606.0001601. 513
Andò, E. 2013. Experimental investigation of microstructural changes in deforming granular 514
media using xray tomography. PhD Thesis. Université de Grenoble. 515
35
Andò, E., Hall, S.A., Viggiani, G., Desrues, J., and Bésuelle, P. 2012. Grainscale experimental in516
vestigation of localised deformation in sand: A discrete particle tracking approach. Acta 517
Geotechnica, 7(1): 1–13. doi:10.1007/s1144001101516. 518
Andò, E., Cailletaud, R., Roubin, E., Stamati, O., and the spam contributors. 2017. SPAM: The 519
Software for the Practical Analysis of Materials. 520
Arroyo, M., Butlanska, J., Gens, A., Calvetti, F., and Jamiolkowski, M. 2011. Cone penetration tests 521
in a virtual calibration chamber. Geotechnique, 61(6): 525–531. doi:10.1680/geot.9.P.067. 522
Bardet, J. P. (1994). Observations on the effects of particle rotations on the failure of idealized 523
granular materials. Mechanics of materials, 18(2), 159182 524
Barret, P.J. 1980. The shape of rock particle, a critical review. Sedimentology, 27: 291–303. 525
doi:10.1111/j.13653091.1980.tb01179.x. 526
Bathurst, R.J., and Rothenburg, L. 1992. Micromechanical features of granular assemblies with 527
planar elliptical particles. Géotechnique, 42(1): 79–95. doi:10.1680/geot.1992.42.1.79. 528
Belheine, N., Plassiard, J.P., Donzé, F. V., Darve, F., and Seridi, A. 2009. Numerical simulation of 529
drained triaxial test using 3D discrete element modeling. Computers and Geotechnics, 36(1–530
2): 320–331. doi:10.1016/j.compgeo.2008.02.003. 531
Boon, C. W., Houlsby, G. T., & Utili, S. (2012). A new algorithm for contact detection between 532
convex polygonal and polyhedral particles in the discrete element method. Computers and 533
Geotechnics, 44, 7382. 534
Butlanska, J., Arroyo, M., Gens, A., and O’Sullivan, C. 2014. Multiscale analysis of cone 535
penetration test (CPT) in a virtual calibration chamber. Canadian Geotechnical Journal, 536
51(1): 51–66. doi:10.1139/cgj20120476. 537
Butlanska, J. (2014). Cone penetration test in a virtual calibration chamber (Doctoral dissertation, 538
Universitat Politècnica de Catalunya (UPC)). 539
Calvetti, F. 2008. Discrete modelling of granular materials and geotechnical problems. European 540
Journal of Environmental and Civil Engineering, 12(7–8): 951–965. 541
36
doi:10.3166/EJECE.12.951965. 542
Calvetti, F., Viggiani, G., and Tamagnini, C. 2003. A numerical investigation of the incremental 543
behavior of granular soils. Rivista Italiana di Geotecnica, 37(3): 11–29. Available from 544
http://www.associazionegeotecnica.it/sites/default/files/rig/RIG_2003_3_11.pdf 545
Catalano, E., Chareyre, B., and Barthélémy, E. 2014. Porescale modeling of fluidparticles inter546
action and emerging poromechanical effects. International Journal for Numerical and Ana547
lytical Methods in Geomechanics, 38(1): 51–71. John Wiley & Sons, Ltd. 548
doi:10.1002/nag.2198. 549
Cavarretta, I., O’sullivan, C., Ibraim, E., Lings, M., Hamlin, S., and Wood, D.M. 2012. Characteri550
zation of artificial spherical particles for DEM validation studies. Particuology, 10: 209–220. 551
doi:10.1016/j.partic.2011.10.007. 552
Cheng, H., Shuku, T., Thoeni, K., & Yamamoto, H. (2018). Probabilistic calibration of discrete 553
element simulations using the sequential quasiMonte Carlo filter. Granular matter, 20(1), 554
11. 555
Cheng, K., Wang, Y., Yang, Q., Mo, Y., and Guo, Y. 2017. Determination of microscopic parame556
ters of quartz sand through triaxial test using the discrete element method. Computers and 557
Geotechnics, 92(August): 22–40. Elsevier Ltd. doi:10.1016/j.compgeo.2017.07.017. 558
Ciantia, M.O., Arroyo, M., Calvetti, F., and Gens, A. 2015. An approach to enhance efficiency of 559
DEM modelling of soils with crushable grains. Géotechnique, 65(2): 91–110. 560
doi:10.1680/geot.13.P.218. 561
Ciantia, M.O., Arroyo, M., O’Sullivan, C., Gens, A., and Liu, T. 2019a. Grading evolution and 562
critical state in a discrete numerical model of Fontainebleau sand. Géotechnique, 69(1): 1–563
15. Thomas Telford Ltd. doi:10.1680/jgeot.17.P.023. 564
Ciantia, M. O., Arroyo, M., O’Sullivan, C., & Gens, A. (2019b). Micromechanical inspection of 565
incremental behaviour of crushable soils. Acta Geotechnica, 14(5), 13371356. 566
Combe, A.L. 1998. Comportement du sable d’Hostun S28 au triaxial axisymétrique. 567
37
Comparaison avec le sable d’Hostun RF. Université Joseph Fourier, Grenoble. 568
Combe, G., and Roux, J.N. 2003. Discrete numerical simulation, quasistatic deformation and the 569
origins of strain in granular materials. Discrete numerical simulation, quasistatic 570
deformation and the origins of strain in granular materials. In 3ème Symposium 571
International sur le Comportement des sols et des roches tendres. Lyon. pp. 1071–1078. 572
Available from https://hal.archivesouvertes.fr/hal00354754. 573
Cui, L., and O’Sullivan, C. 2006. Exploring the macro and microscale response of an idealised 574
granular material in the direct shear apparatus. Géotechnique, 56(7): 455–468. 575
doi:10.1680/geot.56.7.455. 576
Cundall, P.A., and Strack, O.D.L. 1979. A discrete numerical model for granular assemblies. 577
Géotechnique, 29(1): 47–65. doi:10.1680/geot.1979.29.1.47. 578
Cho, A.G., Dodds, J., and Santamarina, J.C. 2006. Particle Shape Effects on Packing Density , 579
Stiffness and Strength – Natural and Crushed Sands. Journal of Geotechnical and 580
Geoenvironmental Engineering, 132(5): 591–602. doi:10.1061/(ASCE)1090581
0241(2006)132:5(591). 582
Da Cruz, F., Emam, S., Prochnow, M., Roux, J.N., and Chevoir, F. 2005. Rheophysics of dense 583
granular materials: Discrete simulation of plane shear flows. Physical Review E  Statistical, 584
Nonlinear, and Soft Matter Physics, 72(2): 1–17. doi:10.1103/PhysRevE.72.021309. 585
Elias, J. 2013. DEM simulation of railway ballast using polyhedral elemental shapes. In 586
PARTICLES 2013  III International Conference on Particlebased Methods – Fundamentals 587
and Applications. Barcelona. pp. 1–10. 588
Estrada, N., Taboada, A., and Radjaï, F. 2008. Shear strength and force transmission in granular 589
media with rolling resistance. Physical Review E  Statistical, Nonlinear, and Soft Matter 590
Physics, 78(2): 1–11. doi:10.1103/PhysRevE.78.021301. 591
Fonseca, J., O’Sullivan, C., Coop, M.R., and Lee, P.D. 2012. Noninvasive characterization of par592
ticle morphology of natural sands. Soils and Foundations, 52(4): 712–722. Elsevier. 593
38
doi:10.1016/j.sandf.2012.07.011. 594
Feng, Y. T., & Owen, D. R. J. (2014). Discrete element modelling of large scale particle systems—595
I: exact scaling laws. Computational Particle Mechanics, 1(2), 159168. 596
Gotteland, P., Villard, P., Salot, C., Nakagawa, M., and Luding, S. 2009. Using Nonconvex 597
Discrete Elements to Predict Experimental Behaviour of Granular Materials. In Proceedings 598
of the 6th International Conference on Micromechanics of Granular Media. AIP. pp. 361–599
364. doi:10.1063/1.3179934. 600
Hall, S.A., Bornert, M., Desrues, J., Pannier, Y., Lenoir, N., Viggiani, G. & Bésuelle, P. (2010) Dis601
crete and continuum analysis of localised deformation in sand using Xray μCT and volu602
metric digital image correlation. Géotechnique. 60 (5), 315–322. 603
Hosn, R. A., Sibille, L., Benahmed, N., & Chareyre, B. (2018). A discrete numerical model involv604
ing partial fluidsolid coupling to describe suffusion effects in soils. Computers and Ge605
otechnics, 95, 3039. 606
Huang, X., Hanley, K. J., O’Sullivan, C., & Kwok, C. Y. (2017). Implementation of rotational re607
sistance models: a critical appraisal. Particuology, 34, 1423. 608
Itasca Consulting Group Inc. 2014. PFC — Particle Flow Code, Ver. 5.0. Minneapolis: Itasca. 609
Available from https://www.itascacg.com/softwarefaq. 610
Irazábal, J., Salazar, F., & Oñate, E. (2017). Numerical modelling of granular materials with spher611
ical discrete particles and the bounded rolling friction model. Application to railway ballast. 612
Computers and Geotechnics, 85, 220229. 613
Iwashita, K., and Oda, M. 1998. Rolling resistance at contacts in simulation of shear band 614
development by DEM. Journal of Engineering Mechanics, 124(3): 285–292. 615
doi:10.1061/(ASCE)07339399(1998)124:3(285). 616
Iwashita, K., and Oda, M. 2000. Microdeformation mechanism of shear banding process based 617
on modified distinct element method. Powder Technology, 109(1–3): 192–205. 618
doi:10.1016/S00325910(99)002363. 619
39
Jamiolkowski, M., Lo Presti, D.C.F., and Manassero, M. 2003. Evaluation of Relative Density and 620
Shear Strength of Sands from CPT and DMT. In Symposium on Soil Behavior and Soft 621
Ground Construction Honoring Charles C. “Chuck” Ladd. Cambridge, USA. 622
doi:10.1061/40659(2003)7. 623
Jerves, A.X., Kawamoto, R.Y., and Andrade, J.E. 2016. Effects of grain morphology on critical 624
state: A computational analysis. Acta Geotechnica, 11(3): 493–503. doi:10.1007/s11440015625
04228. 626
Jiang, M.J.J., Yu, H.S., and Harris, D. 2005. A novel discrete model for granular material 627
incorporating rolling resistance. Computers and Geotechnics, 32(5): 340–357. 628
doi:10.1016/j.compgeo.2005.05.001. 629
Katagiri, J., Matsushima, T., Yamada, Y., Katagiri, J., Matsushima, T., and Yamada, Y. 2010. 630
Simple shear simulation of 3D irregularlyshaped particles by imagebased DEM. Granular 631
Matter, 12: 491–497. doi:10.1007/s1003501002076. 632
Kawamoto, R., Andò, E., Viggiani, G., and Andrade, J.E. 2018. All you need is shape: Predicting 633
shear banding in sand with LSDEM. Journal of the Mechanics and Physics of Solids, 111: 634
375–392. Elsevier Ltd. doi:10.1016/j.jmps.2017.10.003. 635
Kawano, K., Shire, T., & O'Sullivan, C. (2018). Coupled particlefluid simulations of the initiation 636
of suffusion. Soils and foundations, 58(4), 972985. 637
Khoubani, A., & Evans, T. M. (2018). An efficient flexible membrane boundary condition for DEM 638
simulation of axisymmetric element tests. International Journal for Numerical and Analyti639
cal Methods in Geomechanics, 42(4), 694715. 640
Kong, D., and Fonseca, J. 2018. Quantification of the morphology of shelly carbonate sands using 641
3D images. Géotechnique, 68(3): 249–261. doi:10.1680/jgeot.16.P.278. 642
Krumbein, W.C. 1941. Measurement and Geological significance of shape and roundness of sed643
imentary particles. Journal of Sedimentary Petrology, 11(2): 64–72. 644
Langston, P., Ai, J., and Yu, H.S. 2013. Simple shear in 3D DEM polyhedral particles and in a 645
40
simplified 2D continuum model. Granular Matter, 15: 595–606. doi:10.1007/s10035013646
04210. 647
Lee, J.S., Dodds, J., and Santamarina, J.C. 2007. Behavior of RigidSoft Particle Mixtures. Journal 648
of Materials in Civil Engineering, 19(2): 179–184. doi:10.1061/(ASCE)08991561(2007)19. 649
Li, T., Jiang, M., & Thornton, C. (2018). Threedimensional discrete element analysis of triaxial 650
tests and wetting tests on unsaturated compacted silt. Computers and Geotechnics, 97, 90651
102. 652
Lin, X., & Ng, T. T. (1997). A threedimensional discrete element model using arrays of ellipsoids. 653
Geotechnique, 47(2), 319329. 654
Liu, Q.B., and Lehane, B.M. 2013. The influence of particle shape on the (centrifuge) cone 655
penetration test (CPT) end resistance in uniformly graded granular soils. Géotechnique, 656
60(2): 111–121. doi:10.1139/t06037. 657
Lu, M., and McDowell, G.R. 2007. The importance of modelling ballast particle shape in the 658
discrete element method. Granular Matter, 9(1–2): 69–80. doi:10.1007/s1003500600213. 659
Lu, G., Third, J. R., & Müller, C. R. (2015). Discrete element models for nonspherical particle 660
systems: from theoretical developments to applications. Chemical Engineering Science, 127, 661
425465. 662
Matsushima. 2002. Discrete Element Modeling for IrregularlyShaped Sand Grains. La 663
Modélisation aux Éléments Discrets d’un sable a grains de forme irrégulière (1998): 239–246. 664
Midi, G. 2004. On dense granular flows. Eur. Phys. J. E, 14: 341–365. doi:10.1140/epje/i200310153665
0. 666
Mohamed, A., & Gutierrez, M. (2010). Comprehensive study of the effects of rolling resistance on 667
the stress–strain and strain localization behavior of granular materials. Granular Matter, 668
12(5), 527541. 669
Ng, T.T. 1994. Numerical simulations of granular soil using elliptical particles. Computers and 670
Geotechnics, 16(2): 153–169. doi:10.1016/0266352X(94)900191. 671
41
O’Sullivan, C., Riemer, M.F., Bray, J.D., and Riemer, M. 2004. Examination of the Response of 672
Regularly Packed Specimens of Spherical Particles Using Physical Tests and Discrete Ele673
ment Simulations. Journal of Geotechnical and Geoenvironmental Engineering. 674
doi:10.1061/(ASCE)07339399(2004)130:10(1140). 675
Plassiard, J.P., Belheine, N., and Donzé, F.V. 2009. A spherical discrete element model: calibra676
tion procedure and incremental response. Granular Matter, 11(5): 293–306. 677
doi:10.1007/s100350090130x. 678
Rorato, R., Arroyo, M., Andó, E., Gens, A., & Viggianni, G. (2020) Linking shape and rotation of 679
grains during triaxial compression of sand, under review, Granular Matter 680
Rorato, R., Arroyo, M., Andò, E., and Gens, A. 2019a. Sphericity measures of sand grains. 681
Engineering Geology, 254(4): 43–53. 682
Rorato, R., 2019b. Imaging and discrete modelling of sand shape. PhD Thesis. Universitat 683
Politecnica de Catalunya (UPC). 684
Rorato, R., Arroyo, M., Gens, A., Andò, E., and Viggiani, G. 2018. Particle Shape Distribution 685
Effects on the Triaxial Response of Sands: A DEM Study. In micro to MACRO Mathematical 686
Modelling in Soil Mechanics, Trends in Mathematics. Edited by P. Giovine and et al. Reggio 687
Calabria (Italy). pp. 277–286. doi:10.1007/9783319994741_28. 688
Rothenburg, L., & Bathurst, R. J. (1992). Micromechanical features of granular assemblies with 689
planar elliptical particles. Géotechnique, 42(1), 7995. 690
Sakaguchi, H., Ozaki, E., and Igarashi, T. 1993. Plugging of the Flow of Granular Materials during 691
the Discharge from a Silo. International Journal of Modern Physics B, 7(09–10): 1949–1963. 692
doi:https://doi.org/10.1142/S0217979293002705. 693
Shire, T., & O’Sullivan, C. (2013). Micromechanical assessment of an internal stability criterion. 694
Acta Geotechnica, 8(1), 8190. 695
Skinner, A.E. 1969. A note on the influence of interparticle friction on the shearing strength of a 696
random assembly of spherical particles. Géotechnique, 19(1): 150–157. 697
42
Vaid, Y., Chern, J., and Tumi, H. 1985. Confining pressure, Grain angularity and Liquefaction. 698
Journal of Geotechnical Engineering, 111(10): 1229–1235. 699
Wadell, H. 1932. Volume, Shape, and Roundness of Rock Particles. The Journal of Geology, 40(5): 700
443–451. 701
Wensrich, C.M., and Katterfeld, A. 2012. Rolling friction as a technique for modelling particle 702
shape in DEM. Powder Technology, 217: 409–417. Elsevier B.V. doi:10.1016/j.pow703
tec.2011.10.057. 704
Wensrich, C.M., Katterfeld, A., and Sugo, D. 2014. Characterisation of the effects of particle shape 705
using a normalised contact eccentricity. Granular Matter, 16(3): 327–337. doi:10.1007/s10035706
01304651. 707
Williams, J. R., & Pentland, A. P. (1992). Superquadrics and modal dynamics for discrete elements 708
in interactive design. Engineering Computations, 9(2), 115127. 709
Xiao, Y., Long, L., Matthew Evans, T., Zhou, H., Liu, H., and Stuedlein, A.W. 2019. Effect of 710
Particle Shape on StressDilatancy Responses of MediumDense Sands. Journal of 711
Geotechnical and Geoenvironmental Engineering, 145(2). doi:10.1061/(ASCE)GT.1943712
5606.0001994. 713
Yang, J., and Luo, X.D. 2015. Exploring the relationship between critical state and particle shape 714
for granular materials. Journal of the Mechanics and Physics of Solids, 84: 196–213. 715
doi:10.1016/j.jmps.2015.08.001. 716
Zhang, N., Arroyo, M., Ciantia, M. O., Gens, A., & Butlanska, J. (2019). Standard penetration test717
ing in a virtual calibration chamber. Computers and Geotechnics, 111, 277289. 718
Zhang, N., & Evans, T. M. (2019). Discrete numerical simulations of torpedo anchor installation 719
in granular soils. Computers and Geotechnics, 108, 4052. 720
Zhao, D., Nezami, E. G., Hashash, Y. M., & Ghaboussi, J. (2006). Threedimensional discrete ele721
ment simulation for granular materials. Engineering Computations, 23(7), 749770. 722
Zhao, B., and Wang, J. 2016. 3D quantitative shape analysis on form, roundness, and compactness 723
43
with microCT. Powder Technology, 291: 262–275. Elsevier B.V. doi:10.1016/j.pow724
tec.2015.12.029. 725
Zhao, S., Evans, T., & Zhou, X. (2018). Effects of curvaturerelated DEM contact model on the 726
macroand micromechanical behaviours of granular soils. Géotechnique, 17(158), 21. 727
Zhou, B., Huang, R., Wang, H., and Wang, J. 2013. DEM investigation of particle antirotation 728
effects on the micromechanical response of granular materials. Granular Matter, 15(3): 315–729
326. doi:10.1007/s1003501304099. 730
Zhou, Y.C., Wright, B.D., Yang, R.Y., Xu, B.H., and Yu, A.B. 1999. Rolling friction in the dynamic 731
simulation of sandpile formation". Physica A: Statistical Mechanics and its Applications, 732
269: 536–553. doi:10.1016/j.physa.2005.01.019. 733
734