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.