Rigidity percolation in a random tensegrity via analytic graph theory
Rocklin, D. Zeb
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Tensegrities are mechanical structures that include cable-like elements that are strong and lightweight relative to rigid rods yet support only extensile stress. From suspension bridges to the musculoskeletal system to individual biological cells, humanity makes excellent use of tensegrities, yet the sharply nonlinear response of cables presents serious challenges to analytical theory. Here we consider large tensegrity structures with randomly placed cables (and struts) overlaid on a regular rigid backbone whose corresponding system of inequalities is reduced via analytic theory to an exact graph theory. We identify a novel coordination number that controls two rigidity percolation transitions: one in which global interactions between cables first support external loads and one in which the structure becomes fully rigid. We show that even the addition of a few cables strongly modifies conventional rigidity percolation, both by modifying the sharpness of the transition and by introducing avalanche effects in which a single constraint can eliminate multiple floppy modes.