# Theory of elastic constants of athermal amorphous solids with internal stresses

## TL;DR

This paper develops a microscopic theory for the elastic constants of athermal amorphous solids, incorporating internal stresses, and shows how internal stresses modify the classical rigidity threshold and shear modulus.

## Contribution

It introduces a novel framework using nonaffine lattice dynamics to include internal stresses in the elastic properties of amorphous solids, extending the classical Maxwell criterion.

## Key findings

- Shear modulus at jamming matches simulation results.
- Internal stresses alter the rigidity threshold below the Maxwell limit.
- Analytical expression for the stress-dependent factor f.

## Abstract

A new microscopic derivation of the elastic constants of amorphous solids is presented within the framework of nonaffine lattice dynamics, which makes use of a perturbative form of the low-frequency eigenvectors of the dynamical matrix introduced in [V. Mazzacurati, G. Ruocco, M. Sampoli EPL 34, 681 (1996)]. The theory correctly recovers the shear modulus at jamming, $\mu \sim (z-2d)$, including prefactors in quantitative agreement with simulations. Furthermore, this framework allows us, for the first time, to include the effect of internal stresses. The theory shows that the Maxwell rigidity criterion $z=2d$ is violated with internal stress. In particular, $\mu \sim (z-2df)$ where $f<1$ if the bonds are, on average, stretched, and the solid is thus rigid below the Maxwell isostatic limit, while $f>1$ if the bonds are, on average, compressed. The coefficient $f$ is derived in analytical form and depends only on $d$ and on the average particle displacement from the interaction energy minimum.