On inequalities of shear modulus contributions in disordered elastic bodies

Abstract

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 $μ= μ_1 + μ_a$ with $μ_1 \ge 0$ characterizing the variance of the quenched shear stresses and $μ_a$ being a simple average over all states and CMs. For equilibrium CM-distributions $μ_a$ becomes equivalent to the shear modulus of annealed systems, i.e. $μ_a \ge 0$, while more generally $μ_a$ 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 $μ-μ_1=μ_a=0$ for equilibrium ensembles, i.e. $μ$ is set by the quenched shear stresses, while $μ_a$ becomes again negative otherwise.