Programming rigidity transitions and multifunctionality in disordered underconstrained spring networks

Our goal is to design multifunctional materials that are tuned to be near a rigidity transition and exhibit other desired behaviors. We focus on underconstrained central force networks near a second-order rigidity transition. Unlike the jamming transition seen in granular systems, which is governed by first-order perturbations and constraint counting, a second-order rigidity transition occurs when geometric incompatibility drives the system to a critical configuration that possesses a state of self-stress. The space of all critical configurations forms a hypersurface in configuration space -- the critical manifold. If a network is strained, it will eventually intersect this critical manifold, and the resulting self-stress is an emergent property. To program features into the self-stress, we treat the self-stresses as the degrees of freedom and develop an expression for all configurations on the critical manifold. Applying a force-density method first developed by engineers, we can calculate derivatives of any objective function with respect to the self-stresses on this manifold. This allows the critical manifold to be leveraged as a design space. As an example, we compare features of random disordered networks that have been strained to the rigidity transition to those in networks that optimize a stiffness metric.