A class of variational formulations for discrete element analysis of granular media is presented. These formulations lead naturally to convex mathematical programs that can be solved using standard and readily available tools. In contrast to traditional discrete element analysis, the present granular contact dynamics formulation uses an implicit time discretization, thus allowing for large time steps. Moreover, in the limit of an infinite time step, the general dynamic formulation reduces to a static formulation that is useful in simulating common quasi-static problems such as triaxial tests and similar laboratory experiments. A significant portion of the paper is dedicated to exploring the consequences of the associated frictional sliding rule implied by the variational formulation adopted. In this connection, a new interior-point algorithm for general linear complementarity problems is developed and it is concluded that the associated sliding rule, in the context of granular contact dynamics, may be viewed as an artifact of the time discretization and that the use of an associated flow rule at the particle scale level generally is physically acceptable.

Keywords: Contact Dynamics, Discrete Element Method (DEM), Mathematical Programming, Optimization, Second-Order Cone Programming

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