Large-scale exponential correlations of nonaffine elastic response of strongly disordered materials

TL;DR

This paper investigates the correlation properties of nonaffine elastic responses in strongly disordered materials, revealing large-scale exponential decay governed by a heterogeneity length scale, supported by numerical simulations.

Contribution

It introduces a theory linking heterogeneity length scale to correlation decay in nonaffine responses, supported by numerical and molecular dynamics evidence.

Findings

Correlation functions of divergence and rotor exhibit exponential decay with length scale ξ.

Heterogeneity length scale ξ depends on disorder strength and can be very large.

Numerical simulations confirm exponential decay and power-law tails in rotor correlations.

Abstract

The correlation properties of the nonaffine elastic response in strongly disordered materials are investigated using the theory of correlated random matrices and supported by numerical models. While the nonaffine displacement field itself predominantly exhibits power-law decay, we demonstrate that its spatial derivatives reveal large-scale exponentially decaying correlations. Specifically, the correlation functions of the divergence and (for most deformations) the rotor of the nonaffine field are governed by a heterogeneity length scale $\xi$. This length scale is set by the disorder strength and can become indefinitely large, far exceeding the structural correlation length. A notable exception occurs under volumetric deformation, where the rotor correlations lack the exponential tail with the length scale $\xi$. The theory also predicts that the rotor correlations may have small power-law tails. We directly observe the exponential decay, characterized by $\xi$, in numerical studies of a rigidity percolation model and in molecular dynamics simulations of amorphous polystyrene and the Lennard-Jones glass. The latter example also confirms the existence of the power-law tail in the rotor correlation function at large distances.