Stress correlations in glasses

TL;DR

This paper proves that in 3D glasses with normal stress fluctuations, stress correlations decay as 1/r^3 at long distances, indicating a universal anisotropic tail in these disordered solids.

Contribution

The paper provides a rigorous analytical proof that stress autocorrelation functions in isotropic 3D glasses exhibit a universal 1/r^3 decay at long-range under normal stress fluctuations.

Findings

Stress correlations decay as 1/r^3 at long-range.

Anisotropic stress correlation tails are universal in glasses.

Normal stress fluctuations are key to the decay behavior.

Abstract

We rigorously establish that, in disordered three-dimensional (3D) isotropic solids, the stress autocorrelation function presents anisotropic terms that decay as $1/r^3$ at long-range, with $r$ the distance, as soon as either pressure or shear stress fluctuations are normal. By normal, we mean that the fluctuations of stress, as averaged over spherical domains, decay as the inverse domain volume. Since this property is required for macroscopic stress to be self-averaging, it is expected to hold generically in all glasses and we thus conclude that the presence of $1/r^3$ stress correlation tails is the rule in these systems. Our proof follows from the observation that, in an infinite medium, when both material isotropy and mechanical balance hold, (i) the stress autocorrelation matrix is completely fixed by just two radial functions: the pressure autocorrelation and the trace of the autocorrelation of stress deviators; furthermore, these two functions (ii) fix the decay of the fluctuations of sphere-averaged pressure and deviatoric stresses for windows of increasing volume. Our conclusion is reached because, due to the precise analytic relation (i) fixed by isotropy and mechanical balance, the constraints arising via (ii) from the normality of stress fluctuations demand the spatially anisotropic stress correlation terms to decay as $1/r^3$ at long-range. For the sake of generality, we also examine situations when stress fluctuations are not normal.