Breakdown of continuum elasticity in amorphous solids

We show numerically that the response of simple amorphous solids (elastic networks and particle packings) to a local force dipole is characterized by a lengthscale l(c) that diverges as unjamming is approached as l(c) similar to (z - 2d)(-1/2), where z >= 2d is the mean coordination, and d is the spatial dimension, at odds with previous numerical claims. We also show how the magnitude of the lengthscale l(c) is amplified by the presence of internal stresses in the disordered solid. Our data suggests a divergence of l(c) similar to (p(c) - p)(-1/4) with proximity to a critical internal stress p(c) at which soft elastic modes become unstable.