CONTACT FORCE MODELS IN INELASTIC COLLISIONS

The Discrete Element Method (DEM) has become a valuable technique in modelling various industrial processes involving granular flows. There are a number of contact force models implemented in DEM that incorporate viscous or plastic dissipation but their validation is a difficult process given their performance is likely problem dependent. In this context, we examine the accuracy of two DEM viscous dissipation models (where the energy is dissipated through a dashpot mechanism) in simulating a simple inelastic oblique impact. We compare the rebound kinematics at various impact angles for a wide range of normal coefficients of restitutions for two types of dashpot models against previously validated plastic dissipation models. We discuss two important results. The first is that for high coefficients of restitutions both dashpot models behave similarly and compare well to plastic dissipation models. However as the coefficient of restitution decreases (and the energy dissipation increases) both dashpot models produce increasingly damped tangential forces and velocities. This is entirely different behaviour to the validated plastic dissipation models which show increasingly oscillatory tangential forces and velocities as the coefficient of restitution decreases. The final key result is the appropriate choice of the damping coefficient in the viscous dissipation models to ensure the normal coefficient of restitution is correctly reproduced. INTRODUCTION The Discrete Element Method (DEM), originally developed by Cundall and Strack (1979) for quasi-static deformation of compact particle systems is now used for a wide variety of problems ranging from highly collisional rapid granular flows to quasi-static deformation problems with enduring contacts, see for example Thornton (2009). The particle-particle interactions depend on how the contact normal and tangential forces are calculated, i.e the choice of contact force model. There are a number of different contact force models that have been implemented in DEM simulations, all of which incorporate some form of viscous or plastic energy dissipation. However, it is currently unclear what significance the choice of contact force model has on the resulting flow or its properties. Indeed the choice likely depends on whether the problem is highly collisional and/or has enduring contacts. So the validation of DEM contact models is challenging but worthwhile given the importance of DEM in modelling industrial processes. As part of this validation process we have recently completed a study comparing the performance of various contact models for a simple problem of a sphere impacting obliquely with a target wall, see Thornton et. al (2012). We studied normal and tangential contact force models that can be categorised as either viscous dissipation or plastic dissipation models for a broad range of normal coefficient of restitutions. It was found that existing experimental evidence and theoretical models supported the plastic dissipation model behaviour but did not support the viscous dissipation behaviour. The aim of this paper then is to expand further on the limitations of the viscous dissipations models. Specifically we will show how and why the behaviour of the viscous models diverges from the plastic models as the normal coefficient of restitution, en reduces. We will also discuss the choice of dashpot coefficient in these viscous models in order to reproduce the normal coefficient of restitution correctly. DESCRIPTION OF CONTACT FORCE MODELS The most common contact force model is the linear spring-dashpot (LS+D) model introduced by Walton (1983). LS+D models are widely used as they are robust and easy to implement. Non-linear variations of the LS+D model have also been implemented e.g. Tsuji et al. (1992). These dashpot models dissipate energy through viscous means; therefore we will label them viscous dissipation models. Other contact force models, e.g. Stronge (1994) and Thornton (1997), dissipate energy plastically by using different loading and unloading spring stiffnesses. We will call these interaction models plastic dissipation models. In the following sections we describe the viscous dissipation models to be examined and compare their performance to a validated plastic dissipation model. Viscous Dissipation Models In the following sections we present a linear spring dashpot and a non-linear spring dashpot model. Linear Spring Dashpot Model (LS+D) In the linear spring dashpot model (LS+D) the normal and tangential forces are calculated using the following equations below, n n n n v mk k F   2     t n te n te k F F   1 t t n te t v mk F F  2   if Ft  Fn then Ft=Fn (1) where Fn is the normal force, Ft is the tangential force, Fte is the elastic component of the tangential force, is the overlapm is the mass, kn is the normal spring stiffness, vn