Dissipative anomalies of stresses in soft amorphous solids: footprints of density singularities

Abstract

In soft amorphous solids, localized irreversible (plastic) stress dissipation occurs as a response to external forcings. A crucial question is whether we can identify structural properties linked to a region's propensity to undergo a plastic stress drop when thermal effects are negligible. To address this question, I follow a theoretical framework provided by Onsager's ideal turbulence theory, representing a non-perturbative application of the renormalization group scale-invariance principle. First, I analyze the zero temperature limit for the fine-grained balance equation for the stress tensor corresponding to instanton realizations. I show that irreversible stress drops can occur if the density gradients diverge. I then derive a balance relation for the coarse-grained instantaneous stress tensor with arbitrary regularization scale $\ell$. From the latter, I obtain an expression for the local inter-scale stress flux in terms of moments of the density increments. By assuming that the density field is Besov regular, I determine the scaling of the stress flux with $\ell$. From this scaling, I show that distributional solutions of the noiseless Dean (NDE) equation can sustain stress dissipation due to a non-equilibrium inter-scale stress flux if the scaling exponents of the density structure functions are below a critical threshold. The athermal limit of fine-grained and coarse-grained descriptions must describe the same phenomenology, \textit{i.e.} the existence of stress dissipation must be independent of any regularization of the dynamics. Using this principle, I analyze the limit $\ell\rightarrow 0$ and argue that flow realizations of athermal disordered systems correspond to ultraviolet fixed-point solutions of the coarse-grained NDE equations with sufficiently low Besov regularity.