Stress correlations in glasses

Abstract

We rigorously establish that, in disordered three-dimensional isotropic solids, the stress autocorrelation function presents anisotropic terms that decay as 1/ at long-range, with being the distance, as soon as local stress fluctuations are normal, by which 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/ 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 with the increasing sphere volume. Our conclusion is reached because, in view of (ii), the normal decay of stress fluctuations is only compatible with both the pressure autocorrelation and the trace of the autocorrelation of stress deviators being integrable; in turn, due to the precise analytic relation (i) fixed by isotropy and mechanical balance, this condition demands the spatially anisotropic stress correlation terms to decay as 1/ at long-range.

Connecting shear localization with the long-range correlated polarized stress fields in granular materials

“…Nonetheless, a power-law r −n fit of h C τ 1 ðrÞi, yields an exponent n = 1.98 ± 0.03 as shown in Fig. 3e, which indicates that an azimuthally averaged harmonic projection of the correlation function does show the scaling consistent with the field theoretical predictions of n = 2 in 2D systems for the description of stress chains in the continuum limit [23][24][25][26] . A finite-size analysis of h C τ 1 ðrÞi is given in the Supplementary Fig.…”

“…decays as a power law of r −2 , which is still consistent with the continuum description of the stress chains in the theories [23][24][25][26] . Finally, we verify that the autocorrelations of inherent shear stress τ 2 show a quadrupole-like pattern with a power law decay of r −2 , providing the experimental evidence of the theoretical predictions [23][24][25][26] .…”

“…It is a great challenge to understand the emergence of those self-organized, cooperative particle rearrangements in the seemingly disordered granular material. Recall that recent simulations and theories show the exhibition of long-range correlated shear stress of inherent structures in amorphous solids, including granular materials [23][24][25][26] , though no experimental evidence has been found yet, in particular, not in granular materials. If it does exist, such as in the force-chain network shown in Fig.1b, it is urgent to unravel the possible connection between the force network structure and the particle dynamics.…”

“…Compared to the polarized stress τ 1,i (α), in literature [23][24][25][26] , much attention has been paid to the quantity τ 2,i (α), the so-called inherent shear stress, due to the anisotropy and long-range characteristics in its spatial autocorrelation despite that no experimental evidence has been reported thus far [23][24][25][26] . First note that the τ 2,i (α) is directly related to the polarized stress τ 1,i (α) at individual particles by τ 2,i (α) = τ 1,i (α) − τ 1,i (α + (π/2)).…”

Green-Kubo stress correlation function at the atomic scale and a long-range bond-orientational ordering in a model liquid

“…In our view, it is worth making here a comment related to the history of application of the Green-Kubo expression. As far as we understand, the original derivations of the Green-Kubo expression [42][43][44] are actually microscopic and thus microscopic considerations adopted relatively recently [11,14,15,[22][23][24][33][34][35][36][37][38][39][40][41] are much closer in spirit to the derivations of the Green-Kubo expression than the macroscopic view of the Green-Kubo expression usually used. Our research of the literature suggests that the macroscopic view of the microscopically-derived Green-Kubo expression has been adopted in one of the first papers on viscosity calculations in computer simulations [70] (see also Ref.…”