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Models that introduce rolling resistance at the contact are widely employed in simulations using the discrete element method (DEM) to indirectly represent particle shape effects. This approach offers substantial computational benefits at the price of increased calibration complexity. This work proposes a method to simplify calibration of rolling resistance. The key element is an empirical relation between a contact parameter (rolling friction) and a 3D grain shape descriptor (true sphericity). Values of true sphericity can be obtained by image analysis of the grains, either directly by 3D acquisition or by correlation with simpler-to-obtain 2D shape measures. Evaluation of rolling friction is thus made independent from that of other model parameters. As an extra benefit, the variability of grain shape in natural sands can be directly mapped into the discrete model. A mapping between rolling friction and true sphericity is calibrated using specimen-scale and grain scale results from two triaxial compression tests on Hostun sand and Caicos ooids. The mapping is validated using different triaxial tests from the same sands and from other reference sands (Ottawa, Ticino). In the case of Ticino grain-shape acquisition is made in 2D, using an ordinary table scanner. The results obtained support this direct calibration procedure.
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1
Image-based 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: 0000-0002-4189-3058 8
UPC - Barcelonatech 9
Moduli D2 Campus Nord UPC Office 212 10
C Jordi Girona 1-3 11
Barcelona 08034 12
13
Marcos Arroyo Alvarez de Toledo, Prof. 14
Universitat Politécnica de Catalunya (UPC), Barcelona (Spain) - Department of Civil and Envi-15
ronmental Engineering 16
marcos.arroyo@upc.edu 17
ORCID ID: 0000-0001-9384-9107 18
19
Edward Carlo Giorgio Andò, Prof. 20
Univ. Grenoble Alpes, CNRS, Grenoble INP, 3SR, F-38000 Grenoble (France) 21
edward.ando@3sr-grenoble.fr 22
ORCID ID: 0000-0001-5509-5287 23
24
Antonio Gens Solé, Prof. 25
Universitat Politécnica de Catalunya (UPC), Barcelona (Spain) - Department of Civil and Envi-26
ronmental Engineering 27
antonio.gens@upc.edu 28
ORCID ID: 0000-0001-7588-7054 29
30
Gioacchino Viggiani, Prof. 31
Univ. Grenoble Alpes, CNRS, Grenoble INP, 3SR, F-38000 Grenoble (France) 32
cino.viggiani@3sr-grenoble.fr 33
ORCID ID: 0000-0002-2609-6077 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 em-39
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 simpler-to-obtain 2D shape measures. Evaluation of roll-42
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 specimen-scale 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 grain-shape 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; X-rays micro tomogra-51
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; Ka-59
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 gener-66
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 tech-78
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); polyhe-80
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
non-negligible 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 in-85
creases in computational time with respect to sphere-based 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 rela-94
tive particle rotation, opposing it through an elasto-plastic 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 ap-101
proaches result in very similar behaviour, at least for the quasi-static conditions relevant in most 102
soil mechanics problems. The main advantage of the rolling resistance model is that contact de-103
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
Specimen-scale responses of identicalnumerical and experimental test are matched by trial-107
and-error. The process is difficult because the effects of rolling resistance in macro-response are 108
coupled with those of other parameters (the coefficient of sliding friction in particular), and mul-109
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 semi-automated 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 large-scale 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 & Kat-123
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 con-132
ventional linear elastic frictional contact model for particle relative displacement at the contact 133
plus an additional set of elastic spring no-tensional joint and slider for the rolling motion. 134
(a)
(b)
Figure 2: Rolling resistance contact model (a) and elastic-perfectly 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 scale-139
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 Iwashita-Oda 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 moment-rotational contact law is implemented as an elastic-perfectly 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 simula-157
tion outcomes is negligible for the quasi-static 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 meas-166
uring the surface area of irregular sand grains is difficult. This has changed in recent years, as 167
computer-based 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 pro-169
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 pe-172
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 depend-193
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. 6-7). 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, /  = Min-207
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 three-dimensional 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 stress-volumetric-strain 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 (D-DVC, 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 nom-231
inal strain (called “micro-strain”) was assigned to each grain. That was done following Catalano 232
et al. (2014) by means of a Voronoi-based allocation of spatial domains centred around each 233
particle. Once micro-strains 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 plot-236
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, partic-242
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 vice-versa. 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 sur-254
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 macro-265
scopic stress-strain information does not offer enough information on the particle rolling behav-266
iour that is directly related to rotational resistance in DEM. In principle, the contact model pa-267
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
sphericity-rolling 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 Voronoi-cell 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 be-289
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 elasto-plastic 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 post-peak stress-strain 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
Normal-to-shear
stiffness ratio
 2.0 2.0 2.0 2.0
Inter-particle
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 (spec-305
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 (spec-307
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 stiff-312
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 stress-strain 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) di-329
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 thick-336
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 (a-c) and numerical (b-d) 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 Cai-345
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 con-352
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 stress-strain-volumetric response of the numerical and experi-358
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 post-localization response. Another ef-361
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 (spec-366
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 post-localization 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 medium-sized sand, with grains of medium angularity. 400
The macroscopic responses of Ticino sand have been well reproduced using DEM by several re-401
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 sta-409
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 µ-CT image acquisition for the other sands. 411
The parallel projection of each grain on the scan surface avoids parallax errors. The scanned im-412
age was binarised, segmented and labelled using the open-source 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 sphe-417
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 pres-425
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 casewas 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 (stress-strain-volumetric) 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 simula-439
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 three-dimensional 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 100-200-300kPa 446
30
Medium Dense state (75%)
Figure 18: Triaxial responses of medium Ticino sand at 100-200-300kPa 447
31
Dense state (90%)
Figure 19: Triaxial responses of dense Ticino sand at 100-200-300kPa 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 clump-based simulations, which would separate further the eccentricity-458
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 contact-level sources or rolling resistance, allowing non-zero rolling re-461
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 DEM-based avatarof 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 en-469
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 ob-472
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 four-core 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 counter-483
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 easily-acquired 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 sphe-495
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 - GEO-RAMP - MSCA-RISE and of the Ministry of Economy 500
of Spain through research Grant BIA2014-59467-R is gratefully acknowledged. Laboratoire 3SR 501
is part of the LabEx Tec 21 (Investissements d’Avenir - grant agreement n_ANR-11-LABX-0030). 502
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... The drawback of this approach is that new micro contact parameters are introduced, without any other calibration means that observation of their effect on mesoscale response, a procedure that may be time consuming and difficult (Cheng et al. 2017). Rorato et al., (2021) proposed a technique to calibrate this model through 2D image analysis, which is quick and inexpensive. Herein, the basics of this approach are explained and applied to characterize a sand from the river Ticino. ...
... Where and are two fitting parameters. These parameters were calibrated by (Rorato et al., 2021) using a experimental database of particle rotation within shear bands (Rorato et al, 2020). The same calibration appeared to work for all tested sands (Hostun, Caicos, Ottawa, Ticino). ...
... For the DEM two cases were considered; a cylindrical sample with rigid walls and a cubic model confined by periodic boundary conditions. The contact parameters are taken from Rorato et al., (2021). The cylindrical DEM model, allowing for shear band formation, captures the experiment better whilst the periodic boundary REV shows some deviation from the result. ...
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Contact rolling resistance is the most widely used method to incorporate particle shape effects in the discrete element method (DEM). The main reason for this is that such approach allows for using spherical particles hence offering substantial computational benefits compared to non-spherical DEM models. This paper shows how rolling resistance parameters for 3D DEM models can be easily calibrated with 2D sand grain images.
... Some input parameters in the DEM simulations and d ed calculation. (Qu et al., 2020a;2020b;Rorato et al., 2021). Hennes (1953) has found experimentally that the shear strength of uniform-sized granular soil is related to particle size, showing a clear semi-log linear relationship as shown in Fig. 2. To demonstrate the role of material property between multi-sized particles in continuously graded coarse-grained soils, each class of particles was assigned with exclusive contact parameters. ...
... The selection of μ r is related to realistic particle shape. As mentioned by Rorato et al. (2021), a quantitative relation between μ r and the 2D perimeter sphericity S p exists as follows: Altuhafi et al. (2013) have revealed the distribution range of S p in different shape types. To emphasize the strength difference between coarse and fine particles, we assumed that coarse particles are richly angular while fine particles are more rounded. ...
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Particle size distribution (PSD) affects the strength behavior of continuously graded coarse-grained soils by modifying the granular skeleton state. However, due to the lack of a proper indicator linking the skeleton state to PSD, the quantitative relation between shear strength and granular skeleton state remains unexplored. Based on a concept of effective dominant skeleton size ded in multi-sized particle packing model, this study attempted to link granular skeleton state with friction angles of continuously graded coarse-grained soils. Using the discrete element method (DEM) and an improved PSD model, dense packing specimens with 36 PSDs were generated and conducted for biaxial tests. The results indicate that ded has a strong correlation with the size and stress distribution of active particles, i.e., ded can reflect the overall size level of skeleton particles in continuously graded granular soils. A definite semi-log linear correlation between φp and ded was revealed, wherein fitted intercept A and slope B were found to be related to particle-scale strength properties. According to the similarity of φp-di relation in uniform-sized particles and φp-ded relation in continuously graded soils, a method for estimating φp of coarse-grained soils with any PSD was proposed and validated by a series of experimental direct shear tests.
... And hence a probabilistic selection of input properties might be more appropriate in the discrete element method (DEM) simulations for the materials showing high discrepancy in their properties (Ren et al., 2021). Almost at the same time, Rorato et al. (2021) proposed a novel method to simplify the calibration of rolling resistance by relating the rolling friction and particle true sphericity, which enabled that the variability of grain shape could be directly mapped into the DEM. These novel works naturally necessitate the studies with regard to the effect of the heterogeneity of particle properties on the variability of soil properties using the DEM. ...
... where μ r p1 and μ r p2 are the particle rolling friction of two contacting particles, respectively. The above introduced simple linear rolling resistance contact model in PFC 3D has also been adopted by many researchers to efficiently study the sandy soil behaviors considering the effect of particle shape (e.g., Gu et al., 2020;Rorato et al., 2021). ...
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Many experimental results have presented that soil deformation and strength properties for the same type of soil samples are with a certain variability at the same density and stress states, and even at the identical laboratory conditions. Micro mechanical interpretations of this phenomenon are the random spatial arrangement of constituent particles and heterogeneity of particle geometry and mechanical properties. This paper mainly focuses on the influence of the latter, which is still an open issue for the community of granular materials, by the random discrete element method. The random field model is firstly incorporated into the discrete element method to characterize the heterogeneity of equivalent particle properties for a simple linear rolling resistance contact model. Two RFM parameters named coefficient of variation and scale of fluctuation are adopted to represent the variability and spatial correlation pattern of particle properties, respectively. Monte Carlo simulations are then performed by repeatedly conducting DEM simulations of drained triaxial compression tests on a series of dense heterogeneous specimens. The macroscopic soil deformation and strength parameters are analyzed statistically. The contact network characteristics of three representative heterogeneous specimens are discussed and compared with those of the homogenous specimen to explore the underlying microscopic mechanisms. Moreover, the variability of soil properties due to the random spatial arrangement of constituent particles is compared with that due to the heterogeneity of particle properties. The study helps to provide more insights into the cross-scale understanding of the variability of sandy soil properties in the laboratory.
... If the particle shape is irregular and the particle shape is required, the triaxial size measurement and modeling of the particle are adopted [23,24]. Particles' three-dimensional shape and size can be determined by obtaining point cloud data [25]. It is also possible to use the method of image recognition and edge extraction to obtain the outline of the particles and carry out 3D modeling [26]. ...
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The discrete element method and simulation analysis of the interaction between granular materials and implements provide a convenient and effective method for the optimal design of farming machinery. However, the parameter differences between different materials make discrete element simulation impossible to carry out directly. It is necessary to obtain the specific material parameters and contact parameters through parameter calibration of the simulation object, so as to make the simulation results more reliable. Parameter calibration mainly includes intrinsic parameter measurement, contact model selection, contact parameter selection, and parameter calibration. The test methods of the calibration test include the Plackett–Burman test and other methods of screening parameters with significant influence, and then selecting the optimal parameters through the climbing test, response surface analysis method, etc., and finally carrying out the regression analysis. This paper will describe the existing parameter measurement methods and parameter calibration methods and provide a reference for the scholars who study parameter calibration to carry out parameter calibration.
... Several contact models with rolling resistance have been proposed and implemented in DEM code [1,29,31,64]. Although the rolling resistance contact models have been widely used in DEM studies to consider the particle inter-locking effects [19,56,66], Zhao et al. [71] recently pointed out that the rolling resistance models have limitations in considering irregular particle shapes. Similar results have been found by the authors [79]. ...
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The shear strength of granular materials has been found to increase nonlinearly with particle asphericity before reaching a steady value independent of particle asphericity. Although the origin of shear strength has been extensively studied, the underlying mechanism of its nonlinear dependency on particle shape remains unclear. In this study, we present a microscopic investigation of shape-dependent shear strength from the perspective of particle dynamics. A series of numerical simple shear tests on assemblies of ellipsoids with different aspect ratios are performed using the discrete element method. It is confirmed that the power-law scaling in nonlocal granular rheology is still valid for granular materials composed of non-spherical particles, such that the macroscopic shear strength and microscopic dynamics can be bridged using granular temperature. Analogous to other amorphous solids, granular materials with higher granular temperature are much softer and exhibiting less resistance to shear. The statistics of the clusters of the particles with higher granular temperature indicate that granular systems with different particle shapes showing different collective motion patterns. We further explore the coupling between rotational and translational particle dynamics and their long-range correlations. The macroscopic shear strength shows a clear monotonic relation with the intrinsic length scale reflecting long-range dynamic correlations. Finally, we propose a picture illustrating the negative feedback mechanism between particle rolling and sliding, which leads to the nonlinear increase and steady value of shear strength with particle asphericity. Our finding may shed light not only on the particle shape effects, but also on the fundamental understanding of the microscopic origin of shear strength of granular materials.
... (3) According to the inversion results, the influence of particle breakage on the macroscopic properties of broken rock granules can be studied [7,10,13,34,36,70]. This method has been widely applied to simulate the mechanical behaviour of soil particle assemblies [9,23,37]. The impact of factors such as the particle size, particle strength, particle shape, and stress paths on the particle breakage of the backfilling gangue was studied by using a similar discrete element method (DEM) [21,26,51,65]. ...
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An accurate understanding of the breakage mechanism of broken coal and rock mass and its coupling relationship with stress and porosity is important for achieving efficient and safe production in coal mines, storage and utilisation of gas and water resources in goafs, and environmental ecological protection. In this study, a novel 3D simulation method is proposed for broken rock and coal granule compaction and breakage. This method can simulate the re-breakage characteristics of broken rock and coal granules during laterally confined compression (LCS). On this basis, numerical simulations combined with laboratory tests are conducted to quantitatively analyse the stress, porosity, and breakage rate evolution characteristics of broken rock granules during LCS. The entire loading process of broken rock granules is divided into three stages: self-adjustment, broken, and elastic. The stress evolution and breakage evolution characteristics of the broken rock granules during each loading stage are delineated. The breakage characteristics of broken rock granules are the main reasons for the evolution of stress, porosity, and breaking rate. In the elastic stage, only uniform compressive stress acts on the broken rock granule, inhibiting further breakage of the sample. When the loading stress reaches the tensile strength of the broken rock granules, the breakage rate of the models increases the fastest. The effects of the broken sample strength and sample size on the breakage characteristics and stress evolution law of the broken models during loading are further discussed. The secant modulus of the broken models in the elastic stages is approximately equal to the elastic modulus of the coal and rock samples. The coordination number evolution law of the broken granules during loading is the main factor affecting its breakage.
... The fundamental issue that prevents DEM from being a practical tool for geotechnical engineers is the challenge in measuring particle-scale properties, kinematics and contact forces at the particle scale (e.g. Fuchs et al., 2014;Hurley et al., 2016;Rorato et al., 2021). Because the micro-mechanical parameters are not always measurable in an experiment, their estimation is mostly formulated as an inverse problem, using statistical inference and optimisation techniques, such as gradient descent (e.g. ...
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This work presents an efficient probabilistic framework for the Bayesian calibration of micro-mechanical parameters for Discrete Element Method (DEM) modelling. Firstly, the superior behaviour of the iterative Bayesian filter over the sequential Monte Carlo filter for calibrating micro-mechanical parameters is shown. The linear contact model with rolling resistance is used for simulating the triaxial responses of Toyoura sand under different confining pressures. Secondly, synthetic data from DEM simulations of triaxial compression are used to assess the reliability of iterative Bayesian filtering with respect to the user-defined parameters, such as the number of samples and predefined parameter ranges. Excellent calibration results with errors between 1 and 2% are obtained when the number of samples is chosen high enough. It is crucial that the sample size is representative for the distribution of individual parameters within the predefined parameter ranges. The wider the ranges, the more samples are required. The investigation also shows the necessity of including both stress and strain histories, at certain confidence levels, for estimation of the correct mechanical responses, especially the correct fabric responses. Finally, based on the findings of this work a fully-automated open-source calibration tool is developed and demonstrated for selected stress paths.
... Based on the findings of Oda and Iwashita (2000) and Goddard (2008), contact couples are taken into consideration in some of the simulations. According to Oda and Iwashita (2000) and Rorato et al. (2021), the rolling friction coefficient that limits the contact couples should reflect the particle shape. Parametric studies, using two-dimensional DEM simulations of biaxial tests and three-dimensional DEM simulations of triaxial compression tests, on the influence of the rolling friction coefficient and particle shape on macroscopic behavior are given in Zhang et al. (2013) and Zhao et al. (2018). ...
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For granular materials, the kinematic degrees of freedom at the microscale of particles are the particles' displacements and rotations. In classical continuum mechanics, the kinematic degree of freedom at the macroscale is the (local) displacement field. The rotation of a material element is not independent but is determined by the antisymmetric part of the displacement gradient. The objective of this study is to investigate, mainly by means of discrete-element method simulations, whether the average particle rotation is equal to the continuum rotation determined from the average displacement gradient. In the three-dimensional discrete-element method simulations of shear tests (with nonzero average continuum rotation), simulations with and without contact couples have been analyzed. The simulation results show that the average particle rotation is effectively equal to the continuum rotation, over the whole range of strains. Additionally, the results of an X-ray tomography test of a rounded granular soil under triaxial compression are analyzed. The average rotation of soil particles inside shear bands agrees well with the average continuum rotation determined from the particle displacements. Comparison of simulation results with contact couples to those where contact couples were not considered reveals that the presence of contact couples has a significant effect on the stress ratio and on the volumetric strain. The stress tensor is symmetric, even when contact couples are included.
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Accurate modeling and simulation of irregular shape sand particle fluidization are challenging as particle shape-resolved simulation is complex and time-consuming. In this study, the coarse-grained discrete element model (CG-DEM)-computational fluid dynamics (CFD) was adopted to improve the computational feasibility of traditional CFD–DEM and simplify the mathematical modeling by lumping a certain amount of physical particles into a numerical parcel. 3D CGDEM–CFD simulated the fluidization behavior of irregular shape sand particles in a fluidized bed, and several key parameters or model settings were assessed to identify critical model factors. Moreover, the impact of the model parameters on hydrodynamics was quantified by comparing the simulation results with the experimental data. The results demonstrate that applying a rolling friction model can increase the simulation accuracy of the fluidization behavior of non-spherical sand particles.
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Two different DEM models are proposed for quantitatively simulating Toyoura sand macroscopic response along various monotonous loading paths and for a wide range of initial densities. The first model adopts spherical particles and compensates for the irregular shapes of Toyoura sand grains by adding an additional rolling resistance stiffness to the classical linear contact model. The second model follows a different strategy whereby rolling stiffness is abandoned in favor of more complex shapes in the form of a few different 3D polyhedrons defined from a 2D micrograph of Toyoura particles. After a preliminary analysis of the number of particles for optimal REV simulations, the two different modeling approaches are calibrated using triaxial compression in so-called drained conditions, adopting a common contact friction angle for the two models. Similar predictive abilities are then obtained along so-called undrained (constant volume) triaxial compression and extension paths. Although it leads to 9-times longer simulations, the polyhedral approach is easier to calibrate regarding the contact parameters. It also enables a more precise description of the microstructure in terms of particle shapes and initial fabric anisotropy, whose crucial role is evidenced in a parametric analysis.
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In granular soils grain crushing reduces dilatancy and stress obliquity enhances crushability. These are well-supported specimen-scale experimental observations. In principle, those observations should reflect some peculiar micromechanism associated with crushing, but which is it? To answer that question the nature of crushing-induced particle-scale interactions is here investigated using an efficient DEM model of crushable soil. Microstructural measures such as the mechanical coordination number and fabric are examined while performing systematic stress probing on the triaxial plane. Numerical techniques such as parallel and the newly introduced sequential probing enable clear separation of the micromechanical mechanisms associated with crushing. Particle crushing is shown to reduce fabric anisotropy during incremental loading and to slow fabric change during continuous shearing. On the other hand, increased fabric anisotropy does take more particles closer to breakage. Shear-enhanced breakage appears then to be a natural consequence of shear-enhanced fabric anisotropy. The particle crushing model employed here makes crushing dependent only on particle and contact properties, without any pre-established influence of particle connectivity. That influence does not emerge, and it is shown how particle connectivity, per se, is not a good indicator of crushing likelihood.
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It is widely recognised that particle shape influences the mechanical response of granular materials. Rolling resistance elasto-plastic contact models are frequently used to approximate particle shape effects in simulations using the Discrete Element Method (DEM). Such contact models require calibration of several micro-parameters, most importantly a rolling resistance coefficient. In this work, the value of rolling resistance is directly linked to true sphericity, a basic measure of grain shape. When shape measurements are performed, this link enables independent evaluation of the rolling resistance coefficient. It does also allow the characteristic shape variability of natural soils to be easily taken into account. In this work, we explore the effect of shape variability on the triaxial response of sand. It is shown, using realistic values of shape distributions, that shape variability significantly affects observed triaxial strength.
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The virtual calibration chamber technique, based on the discrete element method, is here applied to study the standard penetration test (SPT). A macro-element approach is used to represent a rod driven with an impact like those applied to perform SPT. The rod is driven into a chamber filled with a scaled discrete analogue of a quartz sand. The contact properties of the discrete analogue are calibrated simulating two low-pressure triaxial tests. The rod is driven changing input energy and controlling initial density and confinement stress. Energy-based blowcount normalization is shown to be effective. Results obtained are in good quantitative agreement with well-accepted experimentally-based relations between blowcount, density and overburden. It is also shown that the tip resistance measured under impact dynamic penetration conditions is close to that under constant velocity conditions, hence supporting recent proposals to relate CPT and SPT results.
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The sphericity of a grain should measure the similitude of its shape with that of a sphere. Sphericity is a shape descriptor of long-standing interest for sedimentology. Now it has gained also interest to facilitate discrete element modelling of granular materials. True sphericity was initially defined by a surface ratio that requires three-dimensional (3D) grain surface measurement. That kind of measurement has been pactically impossible until recently and, as a consequence, a number of alternative 3D measures and 2D proxies were proposed. In this work we present results from a study of grain shape based on x-ray tomography of two different sand specimens, containing more than 110.000 particles altogether. Sphericity measures were systematically obtained for all grains. 2D proxy measures were also obtained in samples of oriented and not-oriented grains. It is shown that the 2D proxy best correlated with true sphericity is perimeter sphericity, whereas the traditional Krumbein-Sloss chart proxy is poorly correlated. 2D measures acquired through minor axis projection are more closely related to 3D measures than those acquired using random projections.
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The effect of particle shape on the strength, dilatancy, and stress-dilatancy relationship was systematically investigated through a series of drained triaxial compression tests on sands mixed with angular and rounded glass beads of different proportions (0%, 25%, 50%, 75%, and 100%). A distinct overall regularity parameter was used to define the particle shape of these mixtures, which ranged from 0.844 to 0.971. The test results showed that all of the samples at an initial relative density of 0.6 exhibited strain-softening and volume-expansion behavior. It was found that the peak-state deviatoric stress, peak-state axial strain, and peak-state friction angle at a given confining pressure decreased with increasing overall regularity. The maximum differences in the peak-state deviator stress, peak-state axial strain, peak-state friction angle, excess friction angle, and maximum dilation angle due to changes in particle shape could be as much as 0.61 MPa, 5.4%, 8.6, 1.5, and 3° at a given confining pressure of 0.4 MPa. In addition, it was found that the slope of the relationship between the peak-state friction angle and maximum dilation angle was independent of the particle shape, whereas the intercept (i.e., the critical-state friction angle) was significantly influenced by the particle shape. A stress-dilatancy equation incorporating the effect of overall regularity was proposed and provided a good estimate of the observed response accounting for the different particle shapes investigated.
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Torpedo anchors are a viable approach for mooring marine hydrokinetic (MHK) energy devices to the seafloor. These anchors can serve to maintain station and to provide the reaction force for an MHK device. The ability of the anchor to perform these duties is a strong function of its penetration depth during installation. This is a large-strain problem not amenable to typical continuum numerical approaches. In the current work, we propose that the discrete element method (DEM) is a more appropriate tool to investigate the shallow penetration of torpedo anchors in sands. The effects of anchor mass, impact velocity, and soil interparticle friction are considered in the DEM simulations. The relative maximum penetration depths for different penetration conditions are quantified and presented. Granular material response at the microscale during penetration are used to provide insight into system response. Energy dissipation in the assembly by both friction and collision at the particle scale are considered. Results show that anchor penetration increases approximately linearly with an increase in impact velocity or anchor weight. Penetration decreases with an increase in interparticle friction (i.e., soil strength). Observations of microscale behaviors and energy calculations are used to provide insight into overall system response.
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Suffusion or internal instability is a major form of internal erosion in cohesionless soils. Hypotheses based on experimental work suggest complex micro-scale influences of stress and material fabric on suffusion. This study presents the results of coupled discrete element method – computational fluid dynamics (DEM-CFD) simulations of permeameter tests. These simulations were carried out to study the influence of micro-scale variables on the initiation of particle transport in gap-graded cohesionless soils with varying fines contents and relative densities. The results highlight the importance of both particle stress and connectivity on the initiation of suffusion. Particles which do not transfer externally applied stress and those with low initial connectivity are shown to be particularly susceptible to suffusion.
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Calibration chambers are frequently used to verify, adapt, or both verify and adapt empirical relations between different state variables and in situ test results. Virtual calibration chambers (VCC) built with 3D discrete element models may be used to extend and partially substitute costly physical testing series. VCC are used here to explore the mechanics of flat dilatometer penetration and expansion. Results obtained for a simulation of physical tests in Ticino sand are presented. Blade tip resistance during penetration is in good agreement with the experiments. A piston-like design is used for the blade so that larger displacements may be applied than it is possible with a membrane. Initial piston pressures in the expansion curves are very low, strongly affected by the scaled-up grain sizes. Despite that difficulty, expansion curves may be easily interpreted to recover dilatometer moduli ED close to those observed in the physical experiments. Particle-scale examination of the results allows a firmer understanding of the current limitations and future potential of the technique
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Granular materials reach critical states upon shearing. The position and shape of a critical state line (CSL) in the compression plane are important for constitutive models, interpretation of in situ tests and liquefaction analysis. It is not fully clear how grain crushing may affect the identification and uniqueness of the CSL in granular soils. Discrete-element simulations are used here to establish the relation between breakage-induced grading evolution and the critical state line position in the compression plane. An efficient model of particle breakage is applied to perform a large number of tests, in which grading evolution is continuously tracked using a grading index. Using both previous and new experimental results, the discrete element model is calibrated and validated to represent Fontainebleau sand, a quartz sand. The results obtained show that, when breakage is present, the inclusion of a grading index in the description of critical states is advantageous. This can be simply done using the critical state plane concept. A critical state plane is obtained for Fontainebleau sand.