Contact breaking and Hertzian interactions between grains can both give rise to nonlinear vibrational response of static granular packings. We perform molecular dynamics simulations at constant energy in 2D of frictionless bidisperse disks that inter
act via Hertzian spring potentials as a function of energy and measure directly the vibrational response from the Fourier transform of the velocity autocorrelation function. We compare the measured vibrational response of static packings near jamming onset to that obtained from the eigenvalues of the dynamical matrix to determine the temperature above which the linear response breaks down. We compare packings that interact via single-sided (purely repulsive) and double-sided Hertzian spring interactions to disentangle the effects of the shape of the potential from contact breaking. Our studies show that while Hertzian interactions lead to weak nonlinearities in the vibrational behavior (e.g. the generation of harmonics of the eigenfrequencies of the dynamical matrix), the vibrational response of static packings with Hertzian contact interactions is dominated by contact breaking as found for systems with repulsive linear spring interactions.
We propose a `phase diagram for particulate systems that interact via purely repulsive contact forces, such as granular media and colloidal suspensions. We identify and characterize two distinct classes of behavior as a function of the input kinetic
energy per degree of freedom $T_0$ and packing fraction deviation above and below jamming onset $Delta phi=phi - phi_J$ using numerical simulations of purely repulsive frictionless disks. Iso-coordinated solids (ICS) only occur above jamming for $Delta phi > Delta phi_c(T_0)$; they possess average coordination number equal to the isostatic value ($< z> = z_{rm iso}$) required for mechanically stable packings. ICS display harmonic vibrational response, where the density of vibrational modes from the Fourier transform of the velocity autocorrelation function is a set of sharp peaks at eigenfrequencies $omega_k^d$ of the dynamical matrix evaluated at $T_0=0$. Hypo-coordinated solids (HCS) occur both above and below jamming onset within the region defined by $Delta phi > Delta phi^*_-(T_0)$, $Delta phi < Delta phi^*_+(T_0)$, and $Delta phi > Delta phi_{cb}(T_0)$. In this region, the network of interparticle contacts fluctuates with $< z> approx z_{rm iso}/2$, but cage-breaking particle rearrangements do not occur. The HCS vibrational response is nonharmonic, {it i.e} the density of vibrational modes $D(omega)$ is not a collection of sharp peaks at $omega_k^d$, and its precise form depends on the measurement method. For $Delta phi > Delta phi_{cb}(T_0)$ and $Delta phi < Delta phi^*_{-}(T_0)$, the system behaves as a hard-particle liquid.
This is a response to the comment on our manuscript Repulsive contact interactions make jammed particulate systems inherently nonharmonic (Physical Review Letters 107 (2011) 078301) by C. P. Goodrich, A. J. Liu, and S. R. Nagel.
