Avalanche contribution to shear modulus of granular materials

Abstract: Shear modulus of frictionless granular materials near the jamming transition under oscillatory shear is numerically investigated. It is found that the shear modulus $G$ satisfies a scaling law to interpolate between $G \sim (\phi - \phi_J)^{1/2}$ and $G \sim \gamma_0^{-1/2}(\phi-\phi_J)$ for a linear spring model of the elastic interaction between contacting grains, where $\phi$, $\phi_J$, and $\gamma_0$ are, respectively, the volume fraction of grains, the fraction at the jamming point, and the amplitude of the oscillatory shear. The linear relation between the shear modulus and $\phi - \phi_J$ can be understood by slip avalanches.