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

TL;DR

The paper develops a correlated random-matrix framework to analyze nonaffine elastic responses in strongly disordered amorphous solids. By representing the force-constant matrix as a correlated Wishart ensemble and employing Dyson-Schwinger equations for the four-point resolvent, it derives the covariance of nonaffine displacements as a sum of ladder and twisted contributions, revealing both a power-law and an exponential decay with a disorder--controlled length scale $ξ$. Near isostaticity, where $κ = 1 - N^{\prime}_{\rm dof}/N_{\rm b}$ is small, $ξ$ diverges as $ξ \sim κ^{-1/2}$ and governs large-scale correlations in the divergence, with rotor correlations following a similar scaling except under volumetric strain. Numerical tests in rigidity percolation and MD simulations of amorphous polystyrene confirm exponential tails with the predicted $ξ$, and show that rotor correlations can exhibit a shorter length scale, especially for volumetric deformations. The work provides a quantitative link between disorder strength, heterogeneity length scales, and the spatial structure of nonaffine elasticity, with implications for nanocomposites and jammed/rigid systems near the isostatic point.

Abstract

The correlation properties of nonaffine elastic response in strongly disordered materials are investigated using the theory of correlated random matrices and numerical models. The random matrix theory shows that the divergence of the nonaffine displacement field has large-scale exponentially decaying correlations. The corresponding length scale $ξ$ is determined by the strength of the disorder and can be indefinitely large, significantly exceeding the correlation length of the disorder. The rotor of the nonaffine displacement field has the same length scale $ξ$ except the case of the volumetric deformation. The main theoretical dependencies are confirmed by the numerical investigation of the rigidity percolation model and the molecular dynamics simulations of a model polystyrene in the amorphous state.

Figures (5)

• Figure 1: Schematic illustration of the bands $θ_n(\boldsymbol{p})$ (color lines) and discrete eigenvalues $θ_v$ for the finite system (color points) for $\,\tilde{\!\varkappa} = 0.03$. Vertical gray lines show the position of $pξ\sim 1$. The vertical dotted lines show the position of $pξ_0\sim 1$.
• Figure 2: Correlation function of the nonaffine displacement field $K_{αα}(r)$ in the rigidity percolation model system under volumetric (a) and shear (b) strain. Dashed lines represent the dependence $1/r$ as a visual guide.
• Figure 3: Correlation function of the divergence (a, c) and the rotor (b, d) of the nonaffine displacement field in the rigidity percolation model under volumetric (a, b) and shear (c, d) strain. Shaded areas represent two standard deviations of the obtained data. Dashed and dotted lines represent the dependency $\exp(-r/ξ)$. (e) Dependence of the heterogeneity length scale $ξ$ on the vicinity of the percolation threshold $p - p_c$.
• Figure 4: Correlation function of the nonaffine displacement field $K_{αα}(r)$ in model polystyrene system under volumetric strain (a) and shear strain (b) for different values of the smoothing parameter $w$. Shaded areas represent two standard deviations of the obtained data. Dashed lines represent the dependence $1/r$ as a visual guide.
• Figure 5: Correlation function of the divergence (a, c) and the rotor (b, d) of the nonaffine displacement field in model polystyrene under volumetric (a, b) and shear (c, d) strain for different values of the smoothing parameter $w$. The normalization factor $n_{\rm at}r/ε^2$ is used to plot the data. Shaded areas represent two standard deviations of the obtained data. Dashed and dash-dotted lines represent the dependencies $\exp(-r/ξ)$ and $\exp(-r/ξ_0)$, respectively.