Shear jamming, discontinuous shear thickening, and fragile states in dry granular materials under oscillatory shear

We numerically study the linear response of two-dimensional frictional granular materials under oscillatory shear. The storage modulus $G'$ and the loss modulus $G''$ in the zero strain rate limit depend on the initial strain amplitude of the oscillatory shear before measurement. The shear jammed state (satisfying $G'>0$) can be observed at an amplitude greater than a critical initial strain amplitude. The fragile state is defined by the emergence of liquid-like and solid-like states depending on the form of the initial shear. In this state, the observed $G'$ after the reduction of the strain amplitude depends on the phase of the external shear strain. The loss modulus $G''$ exhibits a discontinuous jump corresponding to discontinuous shear thickening in the fragile state.