On inequalities of shear modulus contributions in disordered elastic bodies

We investigate generic inequalities of various contributions to the shear modulus μ 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 μ is rewritten as with characterizing the variance of the quenched shear stresses and being a simple average over all states and CMs. For equilibrium CM distributions becomes equivalent to the shear modulus of annealed systems, i.e., , while more generally 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 for equilibrium ensembles, i.e., μ is set by the quenched shear stresses, while becomes again negative otherwise.