Rigidity of generic random tensegrity structures

Abstract: Many mechanical structures, both engineered and biological, combine heavy rigid elements such as bones and beams with lightweight flexible ones such as cables and membranes. These are referred to as tensegrities, reflecting that cables can only support extensile tension. We model such systems via simulations of depleted triangular lattices in which we minimize the energies of tensegrities subject to strained boundary conditions. When there are equal numbers of cables and struts (which support only compressive tension), a cable and a strut together each contribute as much toward rigidity as a rod, with the two contributions being equal in the case of shear strain. Due to the highly nonaffine deformations at the rigidity transitions, the contribution of a cable (strut) can be significant even under global compression (dilation) despite a cable's inability to resist local compression. Further, we find that when neighboring elements tend to point away from one another, as is common in real systems, cables interact significantly more strongly with other cables than do cables with struts in supporting stress. These phenomena shed new light on a variety of realistic, disordered systems at the threshold of mechanical stability.