Absence of inelastic collapse for a 1D gas of grains with an internal degree of freedom

In a cooling gas of rigid particles interacting with a constant coefficient of restitution, groups of particles within the gas may experience an infinite number of collisions in a finite time. This singularity, named inelastic collapse, is a shortcoming of the mathematical model, and it hampers the efforts of simulating a freely evolving, cooling granular system. After a brief review of previous works addressing the problem, we propose a one-dimensional model where a grain is seen as a pair of point masses joined by a massless, dissipative spring. We show that binary interactions of such grains are described as impacts with a constant restitution coefficient, whose expression is given in terms of the spring parameters. However, the impact is not instantaneous, but it requires a finite time. We show that in situations that would lead to inelastic collapse, multiple interactions among grains transfer kinetic energy into potential energy associated with the deformation of the springs, rather than dissipate it. This effectively avoids the collapse. Finally, we discuss the results of the simulations of a cooling granular system in comparison with other models.