Stress-dependent normal mode frequencies from the effective mass of granular matter

A zero-temperature critical point has been invoked to control the anomalous behavior of granular matter as it approaches jamming or mechanical arrest. Criticality manifests itself in an anomalous spectrum of low-frequency normal modes and scaling behavior near the jamming transition. The critical point may explain the peculiar mechanical properties of dissimilar systems such as glasses and granular materials. Here, we study the critical scenario via an experimental measurement of the normal modes frequencies of granular matter under stress from a pole decomposition analysis of the effective mass. We extract a complex-valued characteristic frequency which displays scaling $|\omega^*(\sigma)|\sim\sigma^{\Omega'}$ with vanishing stress $\sigma$ for a variety of granular systems. The critical exponent is smaller than that predicted by mean-field theory opening new challenges to explain the exponent for frictional and dissipative granular matter. Our results shed light on the anomalous behavior of stress-dependent acoustics and attenuation in granular materials near the jamming transition.