Elasticity of highly cross-linked random networks

TL;DR

This paper derives a universal elastic energy expression for highly cross-linked random networks, showing that the shear modulus is independent of microscopic details and characterizes the amorphous solid state.

Contribution

It provides a microscopic derivation of the elastic properties of randomly cross-linked networks, revealing the universality of the shear modulus.

Findings

Shear modulus is universal and independent of microscopic length-scales.

Elastic energy form matches that of an isotropic amorphous solid.

Derived from a microscopic model with quenched disorder.

Abstract

Starting from a microscopic model of randomly cross-linked particles with quenched disorder, we calculate the Laudau-Wilson free energy S for arbitrary cross-link densities. Considering pure shear deformations, S takes the form of the elastic energy of an isotropic amorphous solid state, from which the shear modulus can be identified. It is found to be an universal quantity, not depending on any microscopic length-scales of the model.