Emergence of linear isotropic elasticity in amorphous and polycrystalline materials

TL;DR

This paper explores how isotropic linear elasticity emerges in amorphous and polycrystalline materials through numerical simulations, revealing the role of a finite elastic length scale and its relation to structural correlations.

Contribution

It demonstrates the emergence of continuum isotropic elasticity from microscopic properties and identifies the elastic length scale's connection to structural and stress correlations.

Findings

Elastic properties are correlated over a finite length scale $\xi_E$.

Continuum elasticity emerges when specimen size exceeds $\xi_E$.

Elastic length scale influences stress and shear modulus correlations.

Abstract

We investigate the emergence of isotropic linear elasticity in amorphous and polycrystalline solids, via extensive numerical simulations. We show that the elastic properties are correlated over a finite length scale $\xi_E$, so that the central limit theorem dictates the emergence of continuum linear isotropic elasticity on increasing the specimen size. The stiffness matrix of systems of finite size $L > \xi_E$ is obtained adding to that predicted by linear isotropic elasticity a random one of spectral norm $(L/\xi_E)^{-3/2}$, in three spatial dimensions. We further demonstrate that the elastic length scale corresponds to that of structural correlations, which in polycrystals reflect the typical size of the grain boundaries and length scales characterizing correlations in the stress field. We finally demonstrate that the elastic length scale affects the decay of the anisotropic long-ranged correlations of locally defined shear modulus and shear stress.