Rigidity transitions in zero-temperature polygons

We study geometrical clues of a rigidity transition due to the emergence of a system-spanning state of self-stress in underconstrained 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 and estimate the transition strain using purely geometrical methods. These findings help determine the rigidity of an area-preserving polygon just by knowing its geometry. Since two-dimensional spring networks can be considered as a network of polygons, we look for similar geometric features in underconstrained spring networks under isotropic expansive strain. 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. Acyclic polygons in the network correlate with larger tensions, forming effective force chains.