Protocol-dependent shear modulus of amorphous solids

We investigate the linear elastic response of amorphous solids to a shear strain at zero temperature. We find that the response is characterized by at least two distinct shear moduli. The first one, mu(ZFC), is associated with the linear response of a single energy minimum. The second, mu(FC), is related to sampling, through plastic events, an ensemble of distinct energy minima. We provide examples of protocols that allow one to measure both shear moduli. In agreement with a theoretical prediction based on the exact solution in infinite spatial dimensions, the ratio mu(FC)/mu(ZFC) is found to vanish proportionally to the square root of pressure at the jamming transition. Our results establish that amorphous solids are characterized by a rugged energy landscape, which has a deep impact on their elastic response, as suggested by the infinite-dimensional solution.