Oscillatory instabilities in three-dimensional frictional granular matter

The dynamics of amorphous granular matter with frictional interactions cannot be derived in general from a Hamiltonian and therefore displays oscillatory instabilities stemming from the onset of complex eigenvalues in the stability matrix. These instabilities were discovered in the context of one- and two-dimensional systems, while the three-dimensional case was never studied in detail. Here we fill this gap by deriving and demonstrating the presence of oscillatory instabilities in a three-dimensional granular packing. We study binary assemblies of spheres of two sizes interacting via classical Hertz and Mindlin force laws for the longitudinal and tangent interactions, respectively. We formulate analytically the stability matrix in three dimensions and observe that a couple of complex eigenvalues emerge at the onset of the instability as in the case of frictional disks in two dimensions. The dynamics then shows oscillatory exponential growth in the mean-square displacement, followed by a catastrophic event in which macroscopic portions of mechanical stress and energy are lost. The generality of these results for any choice of forces that break the symplectic Hamiltonian symmetry is discussed.