A minimal-length approach unifies rigidity in underconstrained materials

What do a guitar string and a balloon have in common? They are both floppy unless rigidified by geometric incompatibility. The same kind of rigidity transition in underconstrained materials has more recently been discussed in the context of disordered biopolymer networks and models for biological tissues. Here, we propose a general approach to quantitatively describe such transitions. Based on a minimal length function, which scales linearly with intrinsic fluctuations in the system and quadratically with shear strain, we make concrete predictions about the elastic response of these materials, which we verify numerically and which are consistent with previous experiments. Finally, our approach may help develop methods that connect macroscopic elastic properties of disordered materials to their microscopic structure.