  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.
