Constraints and vibrations in static packings of ellipsoidal particles

We numerically investigate the mechanical properties of static packings of ellipsoidal particles in 2D and 3D 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 $z_{\rm iso}$ required for them to be collectively jammed, amorphous packings of ellipsoidal particles generally possess fewer contacts than expected for collective jamming ($z < z_{\rm iso}$) 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 $N(z_{\rm iso} - z)$ eigenmodes ${\hat e}_0$ with zero eigenvalues even at finite compression, where $N$ is the number of particles. In addition, these modes ${\hat e}_0$ 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, ${\hat e}_0$. These findings illustrate the significant differences between static packings of spherical and ellipsoidal particles.