Constraints and vibrations in static packings of ellipsoidal particles

We numerically investigate the mechanical properties of static packings of frictionless ellipsoidal particles in two and three dimensions over a range of aspect ratio and compression Delta phi. While amorphous packings of spherical particles at jamming onset (Delta phi = 0) are isostatic and possess the minimum contact number z(iso) required for them to be collectively jammed, amorphous packings of ellipsoidal particles generally possess fewer contacts than expected for collective jamming (z < z(iso)) from naive counting arguments, which assume that all contacts give rise to linearly independent constraints on interparticle separations. To understand this behavior, we decompose the dynamical matrix M = H - S for static packings of ellipsoidal particles into two important components: the stiffness H and stress S matrices. We find that the stiffness matrix possesses 2N(z(iso) - z) eigenmodes <(e)over cap>(0) with zero eigenvalues even at finite compression, where N is the number of particles. In addition, these modes (e) over cap (0) are nearly eigenvectors of the dynamical matrix with eigenvalues that scale as Delta phi, and thus finite compression stabilizes packings of ellipsoidal particles. At jamming onset, the harmonic response of static packings of ellipsoidal particles vanishes, and the total potential energy scales as delta(4) for perturbations by amplitude delta along these quartic modes, (e) over cap (0). These findings illustrate the significant differences between static packings of spherical and ellipsoidal particles.