Shear stress relaxation and ensemble transformation of shear stress autocorrelation functions revisited

TL;DR

This paper clarifies the relationship between shear stress relaxation modulus and autocorrelation functions in networks, showing they differ in solids and cannot be used interchangeably to find the equilibrium shear modulus.

Contribution

It provides a theoretical and computational analysis of shear stress autocorrelation functions and their relation to the shear relaxation modulus in isotropic spring networks.

Findings

$G(t)$ equals $C(t)|_{ au}$ and differs from $C(t)|_{ ext{ extgamma}}$ in solids.

$G_{eq}$ cannot be obtained solely from $C(t)|_{ ext{ extgamma}}$ in solids.

An intermediate plateau in autocorrelation functions reflects the shear modulus of the quenched network.

Abstract

We revisit the relation between the shear stress relaxation modulus $G(t)$, computed at finite shear strain $0 < \gamma \ll 1$, and the shear stress autocorrelation functions $C(t)|_{\gamma}$ and $C(t)|_{\tau}$ computed, respectively, at imposed strain $\gamma$ and mean stress $\tau$. Focusing on permanent isotropic spring networks it is shown theoretically and computationally that in general $G(t) = C(t)|_{\tau} = C(t)|_{\gamma} + G_{eq}$ for $t > 0$ with $G_{eq}$ being the static equilibrium shear modulus. $G(t)$ and $C(t)|_{\gamma}$ thus must become different for solids and it is impossible to obtain $G_{eq}$ alone from $C(t)|_{\gamma}$ as often assumed. We comment briefly on self-assembled transient networks where $G_{eq}(f)$ must vanish for a finite scission-recombination frequency $f$. We argue that $G(t) = C(t)|_{\tau} = C(t)|_{\gamma}$ should reveal an intermediate plateau set by the shear modulus $G_{eq}(f=0)$ of the quenched network.