On inequalities of shear modulus contributions in disordered elastic bodies

TL;DR

This paper explores inequalities in shear modulus contributions in disordered elastic bodies, revealing conditions under which these contributions are positive or negative, with implications for understanding amorphous materials.

Contribution

It introduces a formal decomposition of shear modulus into components and analyzes their inequalities across different ensemble conditions, including equilibrium and out-of-equilibrium states.

Findings

In equilibrium, shear modulus contributions are non-negative.

Out-of-equilibrium conditions can lead to negative shear modulus contributions.

The formalism applies to both elastic networks and colloidal glasses.

Abstract

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.