Nonaffine Correlations in Random Elastic Media

TL;DR

This paper investigates how spatially varying elastic properties in amorphous materials cause nonaffine deformations under stress, revealing universal scaling laws and the influence of different types of elastic heterogeneity.

Contribution

It provides analytical and numerical analysis of nonaffine responses in random elastic media, establishing universal scaling behavior and clarifying the role of various sources of elastic disorder.

Findings

Nonaffine displacement correlations scale as |x|^{-(d-2)}.

Variance in local elastic moduli determines nonaffinity amplitude.

Random stress alone does not induce nonaffine displacements.

Abstract

Materials characterized by spatially homogeneous elastic moduli undergo affine distortions when subjected to external stress at their boundaries, i.e., their displacements $\uv (\xv)$ from a uniform reference state grow linearly with position $\xv$, and their strains are spatially constant. Many materials, including all macroscopically isotropic amorphous ones, have elastic moduli that vary randomly with position, and they necessarily undergo nonaffine distortions in response to external stress. We study general aspects of nonaffine response and correlation using analytic calculations and numerical simulations. We define nonaffine displacements $\uv' (\xv)$ as the difference between $\uv (\xv)$ and affine displacements, and we investigate the nonaffinity correlation function $\mathcal{G} = < [\uv'(\xv) - \uv' (0)]^2>$ and related functions. We introduce four model random systems with random elastic moduli induced by locally random spring constants, by random coordination number, by random stress, or by any combination of these. We show analytically and numerically that $\mathcal{G}$ scales as $A |\xv|^{-(d-2)}$ where the amplitude $A$ is proportional to the variance of local elastic moduli regardless of the origin of their randomness. We show that the driving force for nonaffine displacements is a spatial derivative of the random elastic constant tensor times the constant affine strain. Random stress by itself does not drive nonaffine response, though the randomness in elastic moduli it may generate does. We study models with both short and long-range correlations in random elastic moduli.