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.