Avalanche contribution to shear modulus of granular materials

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 $\varphi$, ${\varphi }_J$, 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 $\varphi -{\varphi }_J$ can be understood by slip avalanches.