A study on tangential force laws applicable to the discrete element method (DEM) for materials with viscoelastic or plastic behavior

H. Kruggel-Emden S. Wirtz, V. Scherer
Department of Energy Plant Technology,
Ruhr-Universitaet Bochum, Universitaetsstrasse 150,
D-44780 Bochum,
Germany
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Abstract

The discrete element method is a widely used particle orientated simulation approach for modeling granular systems. It is based on tracking each particle’s movement and its interactions with the surroundings over time. The motion of a particle is given by a system of coupled ordinary differential equations which are solved numerically. Therefore, models for the forces acting between particles in contact need to be specifed. In the past, detailed investigations dealing with the accuracy of tangential force–displacement models have been very limited, with sparse experimental data considered and the frequent restriction of including only fully elastic materials. In large scale discrete element simulations, on the other hand, viscoelastic or plastic material behavior is often assumed for normal contacts and combined with arbitrary tangential models. To address this situation a number of tangential force–displacement models are reviewed including linear models by Cundall and Strack [1979. A discrete numerical model for granular assemblies, Geotechnique 29, 47–65], Di Maio and Di Renzo [2004. Analytical solution for the problem of frictional-elastic collisions of spherical particles using the linear model. Chemical Engineering Science 59(16), 3461–3475], Brendel and Dippel [1998. Lasting contacts in molecular dynamics simulations. In: Herrmann, H.J., Hovi, J.-P., Luding, S. (Eds.), Physics of Dry Granular Media, Dordrecht. Kluwer Academic Publishers, pp. 313], Walton and Braun [1986. Viscosity, granular temperature and stress calculations for shearing assemblies of inelastic, frictional disks. Journal of Rheology 30, 949] and simple non-linear models by Brilliantov et al. [1996. Model for collisions in granular gases. Physical Review E 53(5), 5382–5392], Tsuji et al. [1992. Lagrangian numerical simulation of plug row of cohesionless particles in a horizontal pipe. Powder Technology 71, 239–250] and Di Renzo and Di Maio [2005. An improved integral non-linear model for the contact of particles in distinct element simulations. Chemical Engineering Science 60(5), 1303–1312]. Whereas for fully elastic materials the parameters of the tangential force–displacement models can be derived directly from mechanical properties a scaling approach is proposed for the estimation of the parameters in the non-elastic case. The effect of different normal force–displacement models is analyzed. For all model combinations macroscopic collision properties are derived and compared to experimental results by Foerster et al. [1994. Measurements of the collision properties of small spheres. Physics of Fluids 6(3), 1108–1115], Lorenz et al. [1997. Measurements of impact properties of small, nearly spherical particles. Experimental Mechanics 37(3), 292–298], Gorham and Kharaz [2000. The measurement of particle rebound characteristics. Powder Technology 112(3), 193–202] and Dong and Moys [2003. Measurement of impact behaviour between balls and walls in grinding mills. Minerals Engineering 16(6), 543–550; 2006. Experimental study of oblique impacts with initial spin. Powder Technology 161(1), 22–31]. 2007 Elsevier Ltd. All rights reserved.

Introduction

The discrete element method as introduced by Cundall and Strack (1979) is a numerical technique for modeling particle systems by tracking each particle’s movement and its interaction with its surroundings individually over time. The method provides information on each particle’s position, orientation and translational and angular velocity. These properties are obtained by numerically integrating a set of fully deterministic equations describing the motion of each individual particle. In order to do so, the forces acting between and on the particles are required in an explicit form. In general it is possible to calculate these forces directly from the deformation the particles experience during contact (Johnson, 1989). However, it is difficult to select the appropriate models and model parameters for such an approach. Furthermore it is far too time consuming from the computational viewpoint to calculate the detailed particle deformations, even if only small numbers of particles are considered. Therefore, the overlap of the particles is utilized to represent deformation and to estimate the resulting forces. For the integration of the Newton’s equations of the translational motion and the Eulerian equations of the rotational motion several numerical integration schemes are available (Allen and Tildesley, 1987), whereas the most simple are based on truncated Taylor expansions. With the steady increase of available computer power the discrete element method has established itself as an important simulation technique for engineering applications involving granular systems. To give some examples, Silo discharge was studied by Yang and Hsiau (2001), Goda and Ebert (2005), Langston et al. (1994) and Kruggel-Emden et al. (2006), particle mixing on grates and in drums was examined by Peters et al. (2005), Kruggel-Emden et al. (2007a) and Finnie et al. (2005), fluidized beds were studied by Tsuji et al. (1993), Zhong et al. (2006), Tatemoto et al. (2005) and Limtrakul et al. (2004) and simulations dealing with plug flow were performed by Tsuji et al. (1992).



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