Rigidity of random tensegrities

Tensegrities are mechanical structures that include cable-like (strut-like) elements that are strong and lightweight relative to rigid rods yet support only extensile (contractile) stress. Such structures abound in nature, combining structural hierarchy, high strength and toughness, light weight, and flexibility to efficiently manage complex loads. However, the intrinsically nonlinear mechanical response prevents the analysis that has characterized the rigidity percolation transition in model systems of rigid elements. Here we consider random tensegrities on generic triangulated lattices with random bond dilution and identify a modified rigidity transition. We identify the number of cables/struts (two) that compensate for a single rod in achieving rigidity under symmetric conditions. Finally, we delve into the role that microstructure and non-affine deformations play in responding to different strains, leading to counterintuitive effects such as cables strongly resisting compression. These results hint at new ways of understanding disordered biological structures at all scales, as well as design principles for novel artificial mechanical assemblies.