Mean-field description of aging linear response in athermal amorphous solids

Abstract

We study the linear response to strain in a mean-field elastoplastic model for athermal amorphous solids, incorporating the power-law mechanical noise spectrum arising from plastic events. In the ``jammed'' regime of the model, where the plastic activity exhibits a nontrivial slow relaxation referred to as aging, we find that the stress relaxes incompletely to an age-dependent plateau, on a timescale which grows with material age. We determine the scaling behavior of this aging linear response analytically, finding that key scaling exponents are universal and independent of the noise exponent $\ensuremath{\mu}$. For $\ensuremath{\mu}>1$, we find simple aging, where the stress relaxation timescale scales linearly with the age ${t}_{\mathrm{w}}$ of the material. At $\ensuremath{\mu}=1$, which corresponds to interactions mediated by the physical elastic propagator, we find instead a ${t}_{\mathrm{w}}^{1/2}$ scaling arising from the stretched exponential decay of the plastic activity. We compare these predictions with measurements of the linear response in computer simulations of a model jammed system of repulsive soft athermal particles, during its slow dissipative relaxation towards mechanical equilibrium, and find good agreement with the theory.