Constraints and vibrations in static packings of ellipsoidal particles
Carl F. Schreck, Mitch Mailman, Bulbul Chakraborty, Corey S. O'Hern
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
Δϕ
. While amorphous packings of spherical particles at jamming onset (
Δϕ=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
N(
z
iso
−z)
eigenmodes
e
^
0
with zero eigenvalues even at finite compression, where
N
is the number of particles. In addition, these modes
e
^
0
are nearly eigenvectors of the dynamical matrix with eigenvalues that scale as
Δϕ
, 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
δ
4
for perturbations by amplitude
δ
along these `quartic' modes,
e
^
0
. These findings illustrate the significant differences between static packings of spherical and ellipsoidal particles.