A coordination-based approach to elasticity of floppy and stiff random networks

TL;DR

This paper investigates how the connectivity of random spring networks influences their elastic properties, revealing a unified critical point governing both floppy and stiff systems, with implications for understanding amorphous materials.

Contribution

It introduces a coordination-based framework to analyze the elasticity of random networks, combining numerical simulations with theoretical predictions to unify the behavior of floppy and stiff systems.

Findings

Critical point governs both floppy and stiff network responses

Derived exponents for shear modulus and non-affine displacements

Predictions applicable to glasses and fibrous networks

Abstract

We study the role of connectivity on the linear and nonlinear elastic behavior of amorphous systems using a two-dimensional random network of harmonic springs as a model system. A natural characterization of these systems arises in terms of the network coordination relative to that of an isostatic network $\delta z$; a floppy network has $\delta z<0$, while a stiff network has $\delta z>0$. Under the influence of an externally applied load we observe that the response of both floppy and rigid network are controlled by the same critical point, corresponding to the onset of rigidity. We use numerical simulations to compute the exponents which characterize the shear modulus, the amplitude of non-affine displacements, and the network stiffening as a function of $\delta z$, derive these theoretically and make predictions for the mechanical response of glasses and fibrous networks.