Coefficient of restitution for elastic disks

We compute the coefficient of restitution ε starting from a microscopic model of elastic disks. Collisions are found to be inelastic, in general, with a finite fraction of translational energy being transferred to elastic vibrations. The coefficient of restitution depends on the relative velocity of the colliding disks, such that collisions become increasingly more inelastic for larger relative velocities. The predictions of the model are in agreement with the approach of Hertz [J. Reine Angew. Math. 92, 156 (1882)] in the quasistatic limit. The coefficient of restitution depends on the elastic constants of the material via Poisson's number. The elastic vibrations absorb kinetic energy more effectively for soft materials with low ratios of shear to bulk modulus.