Stiffening of under-constrained spring networks under isotropic strain

TL;DR

This study demonstrates that under-constrained disordered spring networks become rigid under isotropic strain, with shear modulus linearly scaling with stress near the transition, confirmed across various network types.

Contribution

The paper validates analytical predictions of rigidity transition in diverse spring networks and clarifies the stress-shear modulus relationship near the transition.

Findings

Shear modulus scales linearly with isotropic stress near the transition.

Analytical predictions apply broadly across different network classes.

Finite-size effects are minimal or weak depending on the network class.

Abstract

Disordered spring networks are a useful paradigm to examine macroscopic mechanical properties of amorphous materials. Here, we study the elastic behavior of under-constrained spring networks, i.e.\ networks with more degrees of freedom than springs. While such networks are usually floppy, they can be rigidified by applying external strain. Recently, an analytical formalism has been developed to predict the mechanical network properties close to this rigidity transition. Here we numerically show that these predictions apply to many different classes of spring networks, including phantom triangular, Delaunay, Voronoi, and honeycomb networks. The analytical predictions further imply that the shear modulus $G$ scales linearly with isotropic stress $T$ close to the rigidity transition; however, this seems to be at odds with recent numerical studies suggesting an exponent between $G$ and $T$ that is smaller than one for some network classes. Using increased numerical precision and shear stabilization, we demonstrate here that close to the transition linear scaling, $G\sim T$, holds independent of the network class. Finally, we show that our results are not or only weakly affected by finite-size effects, depending on the network class.