Shear-strain and shear-stress fluctuations in generalized Gaussian ensemble simulations of isotropic elastic networks

TL;DR

This paper explores shear-fluctuations and correlations in isotropic elastic networks using generalized Gaussian ensembles, revealing how shear-stress and strain fluctuations depend on ensemble parameters and connecting stress autocorrelation to relaxation modulus.

Contribution

It introduces a generalized Gaussian ensemble framework to tune shear-fluctuations and derives explicit relations between stress fluctuations, autocorrelations, and the shear modulus in elastic networks.

Findings

Stress fluctuations depend linearly on the ensemble parameter λ.

Stress autocorrelation functions relate directly to the shear relaxation modulus.

Theoretical predictions are supported by numerical simulations of spring networks.

Abstract

Shear-strain and shear-stress correlations in isotropic elastic bodies are investigated both theoretically and numerically at either imposed mean shear-stress $\tau$ ($\lambda=0$) or shear-strain $\gamma$ ($\lambda=1$) and for more general values of a dimensionless parameter $\lambda$ characterizing the generalized Gaussian ensemble. It allows to tune the strain fluctuations $\mu_{\gamma\gamma} \equiv \beta V \la \delta \gamma^2 \ra = (1-\lambda)/G_{eq}$ with $\beta$ being the inverse temperature, $V$ the volume, $\gamma$ the instantaneous strain and $G_{eq}$ the equilibrium shear modulus. Focusing on spring networks in two dimensions we show, e.g., for the stress fluctuations $\mu_{\tau\tau} \equiv \beta V \la \delta\tau^2 \ra$ ($\tau$ being the instantaneous stress) that $\mu_{\tau\tau} = \mu_{A} - \lambda G_{eq}$ with $\mu_{A} = \mu_{\tau\tau}|_{\lambda=0}$ being the affine shear-elasticity. For the stress autocorrelation function $c_{\tau\tau}(t) \equiv \beta V \la \delta \tau(t) \delta \tau(0) \ra$ this result is then seen (assuming a sufficiently slow shear-stress barostat) to generalize to $c_{\tau\tau}(t) = G(t) - \lambda \Geq$ with $G(t)$ being the shear-stress relaxation modulus.