Approximate analytical description of the nonaffine response of amorphous solids

TL;DR

This paper introduces an analytical approximation scheme for disordered solids that accurately predicts elastic constants, including the vanishing shear modulus at the isostatic point, accounting for atomic displacement inhomogeneity.

Contribution

It provides a fully analytical method to evaluate elastic properties of amorphous solids considering nonaffine displacements, aligning well with simulations.

Findings

Quantitative agreement with simulations for central-force systems.

Predicts shear modulus vanishing at the isostatic point with linear law.

Shows rigidity loss due to cancellation of affine and nonaffine terms.

Abstract

An approximation scheme for model disordered solids is proposed that leads to the fully analytical evaluation of the elastic constants under explicit account of the inhomogeneity (nonaffinity) of the atomic displacements. The theory is in quantitative agreement with simulations for central-force systems and predicts the vanishing of the shear modulus at the isostatic point with the linear law {\mu} ~ (z - 2d), where z is the coordination number. The vanishing of rigidity at the isostatic point is shown to be a consequence of the canceling out of positive affine and negative nonaffine terms.