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

The mechanical response of two-dimensional frictional granular materials under an oscillatory shear are numerically investigated. It is confirmed that the shear storage modulus $G'$ depends on the initial amplitude of the oscillation to prepare the system before the measurement. For sufficiently large initial strain amplitude, the shear jammed state satisfying $G'>0$ is observed even if the packing fraction is below the jamming point. The fragile state is also identified as a long lived metastable state where $G'$ depends on the phase of the oscillatory shear. The dynamic viscosity evaluated from the shear loss modulus $G''$ exhibits a sudden jump similar to the discontinuous shear thickening in the fragile state.