Geometry and the onset of rigidity in a disordered network

TL;DR

This paper investigates how geometric factors induce rigidity in disordered spring networks under strain, revealing a topological transition marked by the emergence of a state of self stress at a critical shear.

Contribution

It introduces a topological analysis approach to predict and characterize rigidity transitions in disordered networks, beyond traditional degree-of-freedom methods.

Findings

Rigidity onset coincides with a single state of self stress.

Shear modulus exhibits critical scaling beyond the transition.

Singular value decomposition predicts rigidity emergence.

Abstract

Disordered spring networks that are undercoordinated may abruptly rigidify when sufficient strain is applied. Since the deformation in response to applied strain does not change the generic quantifiers of network architecture - the number of nodes and the number of bonds between them - this rigidity transition must have a geometric origin. Naive, degree-of-freedom based mechanical analyses such as the Maxwell-Calladine count or the pebble game algorithm overlook such geometric rigidity transitions and offer no means of predicting or characterizing them. We apply tools that were developed for the topological analysis of zero modes and states of self-stress on regular lattices to two-dimensional random spring networks, and demonstrate that the onset of rigidity, at a finite simple shear strain $\gamma^\star$, coincides with the appearance of a single state of self stress, accompanied by a single floppy mode. The process conserves the topologically invariant difference between the number of zero modes and the number of states of self stress, but imparts a finite shear modulus to the spring network. Beyond the critical shear, we confirm previously reported critical scaling of the modulus. In the sub-critical regime, a singular value decomposition of the network's compatibility matrix foreshadows the onset of rigidity by way of a continuously vanishing singular value corresponding to nascent state of self stress.