Constraints and vibrations in static packings of ellipsoidal particles

Abstract

We numerically investigate the mechanical properties of static packings of ellipsoidal particles in 2D and 3D over a range of aspect ratio and compression $\Delta \varphi$. While amorphous packings of spherical particles at jamming onset ($\Delta \varphi =0$) are isostatic and possess the minimum contact number $z_{\mathrm{iso}}$ required for them to be collectively jammed, amorphous packings of ellipsoidal particles generally possess fewer contacts than expected for collective jamming ($z<z_{\mathrm{iso}}$) 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 $N(z_{\mathrm{iso}}-z)$ 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 $\Delta \varphi$, 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.

18 pages, 21 figures

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