Soft random solids and their heterogeneous elasticity

TL;DR

This paper uses vulcanization theory to analyze the spatial heterogeneity of elastic properties in soft random solids, revealing large residual stresses and universal disorder characteristics linked to their formation process.

Contribution

It introduces a theoretical framework connecting vulcanization theory with the heterogeneous elasticity of soft random solids, highlighting the role of quenched disorder and residual stresses.

Findings

Large residual stresses in equilibrium states

Disorder correlators are long-ranged and universal

Mean shear modulus is governed by a universal parameter

Abstract

Spatial heterogeneity in the elastic properties of soft random solids is examined via vulcanization theory. The spatial heterogeneity in the \emph{structure} of soft random solids is a result of the fluctuations locked-in at their synthesis, which also brings heterogeneity in their \emph{elastic properties}. Vulcanization theory studies semi-microscopic models of random-solid-forming systems, and applies replica field theory to deal with their quenched disorder and thermal fluctuations. The elastic deformations of soft random solids are argued to be described by the Goldstone sector of fluctuations contained in vulcanization theory, associated with a subtle form of spontaneous symmetry breaking that is associated with the liquid-to-random-solid transition. The resulting free energy of this Goldstone sector can be reinterpreted as arising from a phenomenological description of an elastic medium with quenched disorder. Through this comparison, we arrive at the statistics of the quenched disorder of the elasticity of soft random solids, in terms of residual stress and Lam\'e-coefficient fields. In particular, there are large residual stresses in the equilibrium reference state, and the disorder correlators involving the residual stress are found to be long-ranged and governed by a universal parameter that also gives the mean shear modulus.