Rigidity percolation in a random tensegrity via analytic graph theory

TL;DR

This paper introduces an analytical model for tensegrity structures with randomly added elements, revealing a new critical point and collective avalanche behavior that influence the transition to rigidity in complex systems.

Contribution

It presents a novel analytical graph theory approach to model rigidity percolation in tensegrity networks, uncovering new critical phenomena and collective behaviors.

Findings

Identification of a new mechanical critical point extending Maxwell's theory.

Discovery of a collective avalanche phenomenon where single cable additions eliminate multiple floppy modes.

Analysis of how cable-like elements alter the rigidity transition in tensegrity structures.

Abstract

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 novel 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 fundamentally new 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.