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

TL;DR

This paper investigates how density singularities in soft amorphous solids lead to irreversible stress dissipation, linking structural properties to plastic events through a theoretical framework inspired by turbulence and renormalization group concepts.

Contribution

It introduces a novel theoretical approach connecting density singularities to stress dissipation in amorphous solids using turbulence theory and regularity analysis.

Findings

Irreversible stress drops occur when density gradients diverge.

Stress dissipation is linked to the scaling of density structure functions.

Flow realizations correspond to fixed-point solutions with low Besov regularity.

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.