# Vibrations of jammed disk packings with Hertzian interactions

## Abstract

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 interact 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 harmonic approximation 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 near jamming by contact breaking as found for systems with repulsive linear spring interactions.

### Keywords

Granular materials Acoustics Jamming Vibrations## Notes

### Acknowledgments

We acknowledge support from NSF Grant No. CBET-0968013 (MS), DTRA Grant No. 1-10-1-0021 (CO), and Yale University (CS). This work also benefited from the facilities and staff of the Yale University Faculty of Arts and Sciences High Performance Computing Center and the NSF (Grant No. CNS-0821132) that in part funded acquisition of the computational facilities. We thank Bob Behringer for his kind mentorship and honest science over the past 25 years.

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