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 ${\mathrm{\ell }}_c$ that diverges as unjamming is approached as $\ell_c \sim (z - 2d)^{-1/2}$,where$z \ge 2d$isthemeancoordination,and$d$ is the spatial dimension, at odds with previous numerical claims. We also show how the magnitude of the lengthscale ${\mathrm{\ell }}_c$ is amplified by the presence of internal stresses in the disordered solid. Our data suggests a divergence of $\ell_c\sim (p_c-p)^{-1/4}$withproximitytoacriticalinternalstress$p_c$ at which soft elastic modes become unstable.