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, μ ZFC , is associated with the linear response of a single energy minimum. The second, μ 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 μ FC / μ 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.