Applied Numerical Modeling in Geomechanics – 2020 – Billaux, Hazzard, Nelson & Schöpfer (eds.) Paper: 10-05
©2020 Itasca International, Inc., Minneapolis, ISBN 978-0-9767577-5-7
Microscopic calibration of rolling friction to mimic particle shape effects
Riccardo Rorato1, Marcos Arroyo1, Antonio Gens1 & Edward Andò2
1 Universitat Politécnica de Catalunya (UPC), Barcelona, Spain
2 Université Grenoble Alpes, CNRS, Grenoble INP, Laboratoire 3SR, Grenoble, France
Much work has been done to characterize granular shape and to understand its influence on overall soil
behavior. Thus, Wadell (Wadell 1932) introduced the concept of “sphericity” that quantifies how a particle
differs from a sphere, in terms of surface area. Krumbein (Krumbein 1941) presents the first chart to visually
estimate shape from the grain lengths ratios.
There is much evidence showing that particle shape is relevant for mechanical responses of soils. Andò
(Andò 2013, Andò et al. 2012) tested in triaxial conditions different sands with shape ranging from very
angular to rounded. Using Digital Image Correlation, he showed that angular sands exhibited a larger shear
band thickness compared to rounded sands. Rorato (Rorato et al. 2019b) demonstrated that a rounded sand
(Caicos ooids) exhibits higher grains rotations compared to an angular sand (Hostun sand).
In this work, we propose a new procedure for an optimal calibration of the DEM contact model parameters
that is able to mimic the effect of particle shape without dramatically increase the computational time. In
particular, our approach aims to (1) limit the number of free parameters requested, (2) respect the mechan-
ical and kinematic triaxial responses of the sheared granular materials and (3) maintain low the computa-
tional time. The Particle Flow Code (PFC3D) developed by Itasca Consulting Group, Inc. (Itasca 2014) has
been used in this work.
2 DESIGN AND ANALYSIS
The most widely used shape used in DEM is the sphere, because it allows straightforward and computa-
tionally efficient contact detection. Unfortunately, soil particles are not spheres. Some researchers has tried
to tackle this challenge by introducing non-spherical elements, like clumps (e.g., Katagiri et al. 2010, Lu
& McDowell 2007), polyhedrons (e.g., Elias 2013, Langston et al. 2013) or grain-shape-inspired particles
(e.g., Jerves et al. 2016, Kawamoto et al. 2018), at the price of increasing dramatically the complexity of
the contact detection and computational time. Other researchers (Iwashita & Oda 1998, Jiang et al. 2005,
Sakaguchi et al. 1993) have proposed the introduction of a resisting moment (i.e., rolling resistance) into
the contact law, beside normal and shear forces, in order to consider the influence of flat (i.e., not punctual)
contacts between real grains.
In this work, a simplified version - as implemented in the PFC software - of the Iwashita & Oda contact
model has been used under the following assumptions:
(1) The rolling stiffness (kr) is defined as the Iwashita & Oda’s original contact model:
where ks is the contact shear stiffness and R the effective radius defined as
being R1 and R2 the radii of the two particles in contact.
(2) The moment-rotational contact law is implemented as an elastic-perfectly plastic model with the yield-
ing moment (M*) defined as:
M* = μ
where μr is defined as rolling friction coefficient and Fn is the normal contact force.
This paper exploits a novel approach to relate the particle shape with the rolling resistance applied at the
contacts, extending the model that was originally proposed in (Rorato et al. 2018). In particular, it is hy-
pothesized that the degree of true sphericity 1 (ψ), of one particle is univocally related with its coefficient
of rolling friction, through a relation
valid for all the spherical particles participating in the DEM simulation. Therefore, if the statistical distri-
bution of sphericity is known for one particular sand, it is possible to extract infinite values so that one
measure of ψ can be assigned to each sphere of the numerical specimen, and therefore the rolling friction
coefficients can be distributed through all the discrete elements. The histograms of true sphericity for three
different sands (Hostun, Caicos and Ticino sands), computed as in (Rorato et al. 2019a), are showed in
Figure 1. Statistical distributions of 3D true sphericity for Hostun, Caicos and Ticino sands.
The question then is what shape function (ψ) might take. We tried to find the equation of (ψ) that could
best match the experimental triaxial tests performed on Hostun sand (specimen “HNEA01”) and Caicos
ooids (specimen “COEA01”). The calibration procedure here proposed aims to fit the conventional macro-
mechanical responses together with kinematic measures. In particular, the histories of the cumulated grain
rotations are known for each grain from the experiments have been measured and the particles rolling fric-
tions have been adjusted to reproduce similar kinematic responses inside the shear bands of the numerical
specimens. It is indeed well known from past DEM studies (Cheng et al. 2017, Estrada et al. 2008, Wensrich
et al. 2014, Wensrich & Katterfeld 2012) that the same macroscopic friction angle can be obtained from
1 Defined by Wadell (1932) as the ratio between the surface area (Sn) of the equivalent sphere (i.e., same volume as the grain) and
the surface area (S) of the particle.
several couples of sliding friction coefficient () and rolling friction coefficient (μr). Both parameters con-
tribute to the shear resistance of the numerical sample, and their influence is coupled. However, the rota-
tional information - from the experimental measures of grains rotations - provides a unique numerical so-
3 RESULTS AND DISCUSSION
The equation of (ψ) has been finally chosen, after an iterative procedure, according to a power law written
= 0.1963(ψ). (5)
with an upper bound fixed at ψ= 1 (perfect sphere).
It is extensively shown in the full paper that it is able to reproduce the macro-mechanical responses (i.e.,
stress-volumetric-strain) of HNEA01 and COEA04 sand specimens and the mean rotations inside the shear
bands (i.e., the kinematics at failure) throughout the execution of the triaxial test (Fig. 2).
Mean rotations inside the shear band
Mean rotations inside the shear band
Figure 2. Histories of mean particle rotations for the grains located inside the shear bands for both the experimental
and numerical samples, during triaxial shearing. The good fit ensures the kinematics at failure is respected.
The proposed approach has been then tested for validation in three different situations, achieving successful
results, (1) at higher confining pressures, (2) testing a third type of sand (Ottawa sand) for which the statis-
tical distribution of 3D sphericity was known and (3) testing a fourth type of sand (Ticino sand) for which
the distribution of 3D sphericity was not known.
Regarding the third case, an innovative method is exploited to determine the statistical distribution of the
degree of true sphericity (3D shape parameter) from 2D measures, as originally proposed by (Rorato et al.
2019a). In particular, a table scanner has been used to obtain an “oriented” projection of thousands of sand
grains laying on their plane of greatest stability. The 2D outlines of all the particles thus obtained, can be
then studies by image analysis techniques in order to extrapolate2 the statistical distribution of ψ, and there-
fore of μr, according to Eq. 5.
2 It is known from (Rorato et al. 2019a) that the degree of true sphericity (ψ) is highly correlated with the perimeter
sphericity, 2D shape parameter, after “oriented” particle projection (i.e., perpendicularly to the minor particle length).
This paper presents an innovative technique to relate univocally the degree of true sphericity of each grain
contained in a sand sample with the coefficient of rolling friction to apply to its numerical avatar of spherical
shape. The main advantage of the proposed model is that it reduces the number of free parameters to set by
trial-and-error procedures when performing DEM simulations, albeit respecting the grains kinematics at
failure. Indeed, if the statistical distribution of sphericity is known, either from experiments either from the
literature, the resisting rolling moment is entirely determined since all the parameters involved in the contact
model are known or predictable.
Therefore, if the initial numerical sample reproduces the experimental void ratio (matched by adjusting the
initial friction coefficient) and the PSD, the only crucial free parameter that must be determined for the
shearing phase by trial-and-error procedures is the inter-particle sliding friction coefficient. Moreover, the
contact detection remains economical and advanced algorithms are not required, maintaining low the com-
putational time. This will open new frontiers to the use DEM for studying engineering applications at larger
scales, especially in geotechnical problems in which the 3D particulate nature of the soil cannot be ignored.
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