Rigidity percolation in a random tensegrity via analytic graph theory

Functional structures from across the engineered and biological world combine rigid elements such as bones and columns with flexible ones such as cables, fibers, and membranes. These structures are known loosely as tensegrities, since these cable-like elements have the highly nonlinear property of supporting only extensile tension. Marginally rigid systems are of particular interest because the number of structural constraints permits both flexible deformation and the support of external loads. We present a model system in which tensegrity elements are added at random to a regular backbone. This system can be solved analytically via a directed graph theory, revealing a mechanical critical point generalizing that of Maxwell. We show that even the addition of a few cable-like elements fundamentally modifies the nature of this transition point, as well as the later transition to a fully rigid structure. Moreover, the tensegrity network displays a collective avalanche behavior, in which the addition of a single cable leads to the elimination of multiple floppy modes, a phenomenon that becomes dominant at the transition point. These phenomena have implications for systems with nonlinear mechanical constraints, from biopolymer networks to soft robots to jammed packings to origami sheets.