On inequalities of shear modulus contributions in disordered elastic bodies

\n\nWe 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 $\\mu = \\mu _{1} + \\mu _{\\textrm {a}}$ with $\\mu _{1} \\ge0$ characterizing the variance of the quenched shear stresses and $\\mu _{\\textrm {a}}$ being a simple average over all states and CMs. For equilibrium CM distributions $\\mu _{\\textrm {a}}$ becomes equivalent to the shear modulus of annealed systems, i.e., $\\mu _{\\textrm {a}} \\ge0$ , while more generally $\\mu _{\\textrm {a}}$ 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 $\\mu -\\mu _{1}=\\mu _{\\textrm {a}}=0$ for equilibrium ensembles, i.e., μ is set by the quenched shear stresses, while $\\mu _{\\textrm {a}}$ becomes again negative otherwise.\n