Fluctuation-dissipation relation between shear stress relaxation modulus and shear stress autocorrelation function revisited

Abstract

The shear stress relaxation modulus $G(t)$ may be determined from the shear stress $\tau (t)$ after switching on a tiny step strain $\gamma$ or by inverse Fourier transformation of the storage modulus $G^{\prime }(\omega )$ or the loss modulus $G^{\prime \prime }(\omega )$ obtained in a standard oscillatory shear experiment at angular frequency $\omega$. It is widely assumed that $G(t)$ is equivalent in general to the equilibrium stress autocorrelation function $C(t)=\beta V⟨\delta \tau (t)\delta \tau (0)⟩$ which may be readily computed in computer simulations ($\beta$ being the inverse temperature and $V$ the volume). Focusing on isotropic solids formed by permanent spring networks we show theoretically by means of the fluctuation-dissipation theorem and computationally by molecular dynamics simulation that in general $G(t)=G_{eq}+C(t)$ for $t>0$ with $G_{eq}$ being the static equilibrium shear modulus. A similar relation holds for $G^{\prime }(\omega )$. $G(t)$ and $C(t)$ must thus become different for a solid body and it is impossible to obtain $G_{eq}$ directly from $C(t)$.

13 pages, 15 figures