Micro-mechanical lengthscales in soft elastic solids

We provide numerical evidence and supporting scaling arguments that the response of soft elastic solids to a local force dipole is characterized by a lengthscale $\ell_c$ that diverges as unjamming is approached as $\ell_c \sim (z - 2d)^{-1/2}$, where $z \ge 2d$ is the mean coordination, and $d$ is the spatial dimension, at odds with previous claims based on numerics. We also show how the magnitude of the lengthscale $\ell_c$ is amplified by the presence of internal stresses in the disordered solid. Our data raise the possibility of a divergence of $\ell_c$ with proximity to a critical internal stress at which a buckling instability takes place.