Rigidity transitions in zero-temperature polygons

TL;DR

This paper investigates the geometric conditions leading to rigidity transitions in polygonal and spring network systems, highlighting the role of convexity, cyclic configurations, and geometric features in predicting system rigidity.

Contribution

It establishes geometric criteria for rigidity transitions in polygons and spring networks, linking convexity and cyclicity to the emergence of self-stress and rigidity.

Findings

Convexity of polygons is necessary for self stress during strain.

Cyclic configuration of polygons guarantees the presence of self stress.

Convex polygons in networks signal the onset of rigidity.

Abstract

We study geometrical clues of a rigidity transition due to the emergence of a system-spanning state of self stress in under-constrained systems of individual polygons and spring networks constructed from such polygons. When a polygon with harmonic bond edges and an area spring constraint is subject to an expansive strain, we observe that convexity of the polygon is a necessary condition for such a self stress. We prove that the cyclic configuration of the polygon is a sufficient condition for the self stress. This correspondence of geometry and rigidity is akin to the straightening of a one dimensional chain of springs to rigidify it. We predict the onset of the rigidity transition using a purely geometrical method. We also estimate the transition strain for a given initial configuration by approximating irregular polygons as regular polygons. These findings help determine the rigidity of an area-preserving polygon just by looking at it. Since two-dimensional spring networks can be considered as a network of polygons, we look for similar geometric features in under-constrained spring networks under isotropic expansive strain. In particular, we observe that all polygons attain convexity at the rigidity transition such that the fraction of convex, but not cyclic, polygons predicts the onset of the rigidity transition. Interestingly, acyclic polygons in the network correlate with larger tensions, thus, forming effective force chains.