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

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.