Finite temperature mechanical instability in disordered lattices

TL;DR

This paper investigates how thermal fluctuations influence mechanical instability in disordered lattices, revealing that temperature stabilizes these systems and alters their shear modulus scaling near critical points.

Contribution

It provides an analytic theory demonstrating the stabilizing effect of thermal fluctuations on disordered lattices and characterizes the different scaling behaviors of shear modulus.

Findings

Thermal fluctuations stabilize disordered lattices near mechanical instability.

Triangular lattice shear modulus scales as G ~ T^{1/2}.

Square lattice shear modulus scales as G ~ T^{2/3}.

Abstract

Mechanical instability takes different forms in various ordered and disordered systems. We study the effect of thermal fluctuations in two disordered central-force lattice models near mechanical instability: randomly diluted triangular lattice and randomly braced square lattice. These two lattices exhibit different scalings for the emergence of rigidity at $T=0$ due to their different patterns of self stress at the transition. Using analytic theory we show that thermal fluctuations stabilize both lattices. In particular, the triangular lattice displays a critical regime in which the shear modulus scales as $G \sim T^{1/2}$, whereas the square lattice shows $G \sim T^{2/3}$.