On inequalities of shear modulus contributions in disordered elastic bodies

We investigate generic inequalities of various contributions to the shear modulus $\mu$ in ensembles of amorphous elastic bodies. We focus first on a simple elastic network model with connectivity matrices (CMs) which are either annealed or quenched, at or out of equilibrium. The stress-fluctuation formalism relation for $\mu$ is rewritten as $\mu ={\mu }_1+{\mu }_a$ with ${\mu }_1\ge 0$ characterizing the variance of the quenched shear stresses and ${\mu }_a$ being a simple average over all states and CMs. For equilibrium CM-distributions ${\mu }_a$ becomes equivalent to the shear modulus of annealed systems, i.e. ${\mu }_a\ge 0$, while more generally ${\mu }_a$ may become strongly negative as shown by considering a temperature quench and a scalar active two-temperature model. Consistent relations are also found for glass-forming colloids where $\mu -{\mu }_1={\mu }_a=0$ for equilibrium ensembles, i.e. $\mu$ is set by the quenched shear stresses, while ${\mu }_a$ becomes again negative otherwise.