Theoretical approach to and numerical simulation of instantaneous collisions in granular media using the A-CD$^2$ method

Abstract

This paper presents a model for the description of instantaneous collisions and a computational method for the simulation of multiparticle systems’ evolution. The description of the behavior of a collection of discrete bodies is based on the consideration that the global system is deformable even if particles are rigid. Making use of the principle of virtual work, the equations describing the regular (that is, smooth) as well as the discontinuous (that is, the collisions) evolutions of the motion system are obtained. For an instantaneous collision involving several rigid particles, the existence and the uniqueness of the solution as well as its satisfaction of a Clausius–Duhem inequality (proving that the evolution is dissipative) are proved. In this approach, forces are replaced by a succession of percussions (that is, forces concentrated in time). The approach is therefore named Atomized stress Contact Dynamics respecting the Clausius–Duhem inequality (A-CD${}^2$). This paper focuses also on nonassociated behaviors, and in particular on Coulomb’s friction law. The use of this constitutive law represents a further theoretical and numerical enhancement of the model. The theory is finally illustrated by some numerical examples, using the associated constitutive laws and Coulomb’s (nonassociated) friction law.