  The shear response of soft solids can be modeled with linear elasticity,
provided the forcing is slow and weak. Both of these approximations must break
down when the material loses rigidity, such as in foams and emulsions at their
(un)jamming point -- suggesting that the window of linear elastic response near
jamming is exceedingly narrow. Yet precisely when and how this breakdown occurs
remains unclear. To answer these questions, we perform computer simulations of
stress relaxation and shear startup experiments in athermal soft sphere
packings, the canonical model for jamming. By systematically varying the strain
amplitude, strain rate, distance to jamming, and system size, we identify
characteristic strain and time scales that quantify how and when the window of
linear elasticity closes, and relate these scales to changes in the microscopic
contact network. Our findings indicate that the mechanical response of jammed
solids are generically nonlinear and rate-dependent on experimentally
accessible strain and time scales.
