Conference PaperPDF Available

Microscopic calibration of rolling friction to mimic particle shape effects in DEM

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
kr=ksR2 (1)
Microscopic calibration of rolling friction to mimic particle shape effects
in DEM
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.
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
μr=(ψ) =
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.
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-
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).
Caicos (COEA04)
Mean rotations inside the shear band
Hostun (HNEA01)
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.
Andò, E. 2013. Experimental investigation of microstructural changes in deforming granular media using x-ray to-
mography. PhD Thesis. Université de Grenoble.
Andò, E., Hall, S.A., Viggiani, G., Desrues, J. & Bésuelle, P. 2012. Grain-scale experimental investigation of localised
deformation in sand: A discrete particle tracking approach. Acta Geotech. 7, 113.
Cheng, K., Wang, Y., Yang, Q., Mo, Y. & Guo, Y. 2017. Determination of microscopic parameters of quartz sand
through tri-axial test using the discrete element method. Comput. Geotech. 92, 2240.
Elias, J. 2013. DEM simulation of railway ballast using polyhedral elemental shapes, in: PARTICLES 2013 - III In-
ternational Conference on Particle-Based Methods Fundamentals and Applications. Barcelona, pp. 110.
Estrada, N., Taboada, A. & Radjaï, F. 2008. Shear strength and force transmission in granular media with rolling
resistance. Phys. Rev. E 78, 1–11.
Itasca Consulting Group, Inc. 2014. PFC3D - Particle Flow Code in 3 Dimensions, Ver. 5.0 User's Manual. Minne-
apolis: Itasca.
Iwashita, K. & Oda, M. 1998. Rolling resistance at contacts in simulation of shear band development by DEM. J. Eng.
Mech. 124, 285292.
Jerves, A.X., Kawamoto, R.Y. & Andrade, J.E. 2016. Effects of grain morphology on critical state: A computational
analysis. Acta Geotech. 11, 493503.
Jiang, M.J.J., Yu, H.-S. & Harris, D. 2005. A novel discrete model for granular material incorporating rolling re-
sistance. Comput. Geotech. 32, 340357.
Katagiri, J., Matsushima, T., Yamada, Y., Katagiri, J., Matsushima, T. & Yamada, Y. 2010. Simple shear simulation
of 3D irregularly-shaped particles by image-based DEM. Granul. Matter 12, 491497.
Kawamoto, R., Andò, E., Viggiani, G. & Andrade, J.E. 2018. All you need is shape: Predicting shear banding in sand
with LS-DEM. J. Mech. Phys. Solids 111, 375392.
Krumbein, W.C. 1941. Measurement and Geological significance of shape and roundness of sedimentary particles. J.
Sediment. Petrol. 11, 6472.
Langston, P., Ai, J. & Yu, H.-S. 2013. Simple shear in 3D DEM polyhedral particles and in a simplified 2D continuum
model. Granul. Matter 15, 595606.
Lu, M. & McDowell, G.R. 2007. The importance of modelling ballast particle shape in the discrete element method.
Granul. Matter 9, 6980.
Rorato, R., Arroyo, M., Andò, E. & Gens, A. 2019a. Sphericity measures of sand grains. Eng. Geol. 254, 4353.
Rorato, R., Arroyo, M., Andò, E., Gens, A. & Viggiani, G. 2019b. Linking shape and rotation of grains during triaxial
compression of sand. Geotechnique (submitted).
Rorato, R., Arroyo, M., Gens, A., Andò, E. & Viggiani, G. 2018. Particle shape distribution effects on the triaxial
response of sands: a DEM study, in: Giovine, P., et al. (Eds.), Micro to MACRO Mathematical Modelling in Soil
Mechanics, Trends in Mathematics. Reggio Calabria (Italy), pp. 277286.
Sakaguchi, H., Ozaki, E. & Igarashi, T. 1993. Plugging of the Flow of Granular Materials during the Discharge from
a Silo. Int. J. Mod. Phys. B 7, 19491963.
Wadell, H. 1932. Volume, Shape, and Roundness of Rock Particles. J. Geol. 40, 443451.
Wensrich, C.M. & Katterfeld, A. 2012. Rolling friction as a technique for modelling particle shape in DEM. Powder
Technol. 217, 409417.
Wensrich, C.M., Katterfeld, A. & Sugo, D. 2014. Characterisation of the effects of particle shape using a normalised
contact eccentricity. Granul. Matter 16, 327337.
ResearchGate has not been able to resolve any citations for this publication.
Full-text available
Particle shape has a strong effect on the mechanical response of coarse soils. This has been usually observed examining specimen-scale or engineering-scale responses, which are the sum of many microscale interactions. In this work we observe the effects of particle shape directly at the microscale level. X-ray tomography (μ-CT) of two sand specimens is exploited to measure three-dimensional particle shape descriptors but also to track individual particle motions during triaxial compression. A discrete digital volume correlation algorithm is employed to track the motion of individual grains (around 50,000 for each sand specimen) during the test and measure, with good precision, their cumulated displacements and rotations. The specimens examined failed in a clearly localised shear mode. Advantage is taken of this to obtain data relevant for very different kinematical regimes: one uniform and more constrained and the other close to critical state. A direct comparison between the shape and kinematic databases shows to what degree particle shape descriptors are related to observed kinematics. It appears that true sphericity is a good predictor of upper bound rotational restraint.
Full-text available
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.
Full-text available
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.
Full-text available
We introduce a new DEM scheme (LS-DEM) that takes advantage of level sets to enable the inclusion of real grain shapes into a classical discrete element method. Then, LS-DEM is validated and calibrated with respect to real experimental results. Finally, we exploit part of LS-DEM potentiality by using it to study the dependency of critical state (CS) parameters such as critical state line (CSL) slope (Formula presented.), CSL intercept (Formula presented.), and CS friction angle (Formula presented.) on the grain’s morphology, i.e., sphericity, roundness, and regularity. This study is carried out in three steps. First, LS-DEM is used to capture and simulate the shape of five different two-dimensional cross sections of real grains, which have been previously classified according to the aforementioned morphological features. Second, the same LS-DEM simulations are carried out for idealized/simplified grains, which are morphologically equivalent to their real counterparts. Third, the results of real and idealized grains are compared, so the effect of “imperfections” on real particles is isolated. Finally, trends for the CS parameters (CSP) dependency on sphericity, roundness, and regularity are obtained as well as analyzed. The main observations and remarks connecting particle’s morphology, particle’s idealization, and CSP are summarized in a table that is attempted to help in keeping a general picture of the analysis, results, and corresponding implications.
Full-text available
This doctoral thesis presents an experimental investigation into the mechanics of granular media.The novelty that this work brings is that the specimens of sand tested in this work are systematicallyand non-destructively imaged using x-ray tomography. Sample size is considerably reducedfrom standard (specimens measure approximately 22 mm height by 11 mm diameter), allowingentire specimens to be scanned at a sufficiently high resolution to identify all the grains (morethan fifty thousand) in each specimen.A campaign of triaxial compression tests has been run on a series of three different naturalsands with different grain shapes (Hostun sand, Ottawa sand and Caicos ooids – all prepared atrelatively dense initial states), and tested at 100 or 300 kPa cell pressure. In each test around 15x-ray scans are performed. In the 3D images resulting from the reconstruction of the x-ray scansperformed, grains are identified each state using a standard watershed algorithm. Starting fromthese discretised data, techniques are developed in order characterise grain-to-grain contacts,as well as to measure the kinematics of all the identified grains between imaged states. Grainkinematics are measured with two specifically-developed tools: “ID-Track” to track grains yieldingtheir displacements, and a discrete image correlation technique to measure grain rotations.Grain-scale measurements are reported in detail for one test, and are then compared to testsin different conditions, in order to highlight the micro-mechanisms responsible for the observedmacroscopic behaviour. This comparison highlights some important micro-scale mechanisms suchas the increasing rotational frustration of more angular grains when the sample’s deformation isconcentrated in a fully developed shear band; this is used to explain to some extent the highervalue of their residual stress for these materials. Signs of localised deformation are seen to occurwell before the peak in many samples, and complex patterns of rotating grains (which match alocal, grain-based measurement of strain) are noticed around the peak of each sample’s response.
Full-text available
Recent developments in the application of x-ray micro-tomography in laboratory geomechanics have allowed all the individual grains of sand in a test sample to be seen and identified uniquely in 3D. Combining such imaging capabilities with experiments carried out “in situ” within an imaging set-up has led to the possibility of directly observing the mechanisms of deformation as they happen. The challenge has thus become extracting pertinent, quantified information from these rich time-lapse 3D images to elucidate the mechanics at play. This paper presents a new approach (ID-Track) for the quantification of individual grain kinematics (displacements and rotations) of large quantities of sand grains (tens of thousands) in a test sample undergoing loading. With ID-Track, grains are tracked between images based on some geometrical feature(s) that allow their unique identification and matching between images. This differs from Digital Image Correlation (DIC), which makes measurements by recognising patterns between images. Since ID-Track does not use the image of a grain for tracking, it is significantly faster than DIC. The technique is detailed in the paper, and is shown to be fast and simple, giving good measurements of displacements, but suffering in the measurement of rotations when compared with Discrete DIC. Subsequently, results are presented from successful applications of ID-track to triaxial tests on two quite different sands: the angular Hostun sand and the rounded Caicos Ooids. This reveals details on the performance of the technique for different grain shapes and insight into the differences in the grain-scale mechanisms occurring in these two sands as they exhibit strain localisation under triaxial loading.
This paper presents discrete element method (DEM) simulations with experimental comparisons at multiple length scales—underscoring the crucial role of particle shape. The simulations build on technological advances in the DEM furnished by level sets (LS-DEM), which enable the mathematical representation of the surface of arbitrarily-shaped particles such as sands. We show that this ability to model shape enables unprecedented capture of the mechanics of granular materials across scales ranging from macroscopic behavior to local behavior to particle behavior. Specifically, the model is able to predict the onset and evolution of shear banding in sands, replicating the most advanced high-fidelity experiments in triaxial compression equipped with sequential X-ray tomography imaging. We present comparisons of the model and experiment at an unprecedented level of quantitative agreement—building a one-to-one model where every particle in the more than 53,000-particle array has its own avatar or numerical twin. Furthermore, the boundary conditions of the experiment are faithfully captured by modeling the membrane effect, as well as the platen displacement and tilting. The results show a computational tool that can give insight into the physics and mechanics of granular materials undergoing shear deformation and failure, with computational times comparable to those of the experiment. One quantitative measure that is extracted from the LS-DEM simulations that is currently not available experimentally is the evolution of three dimensional force chains inside and outside of the shear band. We show that the rotations on the force chains are correlated to the rotations in stress principal directions.
In discrete element modelling it is quite common to employ rolling friction models to mimic the effects of particle shape. This paper presents an investigation of the mechanisms at play when using this technique and compares the behaviour of a rolling friction model with various non-spherical particle systems. The motivation behind this work revolves around forming a theoretical framework behind the selection of a coefficient of rolling friction. As a part of this study, we describe an approach where the normalised average contact eccentricity of non-spherical particles (in this case multispheres) is used to characterise the effects of shape. This description is found to capture some aspects of material behaviour reasonably well. When compared to the behaviour of a common rolling friction model, it was found that similar behaviour could be approximated by spheres with a coefficient of rolling friction equal to one half of the normalised eccentricity of non-spherical material. This is approximately in-line with previous studies involving 2D polyhedral particles (Estrada et al. in Phys Rev E 84:011306, 2011).
Laboratory experiments and numerical simulations were performed to find the mechanism of the plugging during the gravitationally emptying a silo. Some interesting results were given by this investigation: 1) Two major slip lines can be seen symmetrically about a vertical midline in the continuous flow, but particles flow only on a one-side slip line was found at each moment; 2) At transition of the flow from one-side to the other, two-side flows collided each other on a vertical midline. And this collision possibly resulted in the formation of the arches of granular materials; 3) In the special case for 2) when arches were stable enough under certain geometrical condition the plugging of the flow occurred. For computer simulation on this issue, the Discrete Element Method specifically taking into account of rolling friction effect was used. Results of numerical simulations were in good agreement with the experimental measures not only in the flow pattern but also in the occurrence of the plugging.