Second Order Rigidity: A Unifying Theory of Rigidity and its Origins

From jammed packings to biological tissues, rigidity plays a crucial role in the integrity and functionality of systems. Typically, Maxwell constraint counting (CC) is used to predict whether a system will be rigid or not based on how constrained its individual components are; however, there are cases where CC fails. For instance, epithelial tissues have been known to exhibit fluid-solid transitions in development or due to underlying disease. Similarly, biopolymer networks can show rigidity transitions under applied strain. In both cases, the transition can happen without any changes to the number of constraints, demonstrating the limitations of CC. Using energy expansion, we show that rigidity is an inherently nonlinear phenomenon and that CC is only guaranteed to work for specific systems. We introduce the framework of second order rigidity to classify systems based on their internal prestresses and make general predictions about their rigidity. We show for example that prestresses can rigidify under-constrained systems such as spring networks and vertex models of epithelia. This formalism unifies our understanding of the origin of rigidity and can be useful in material design by predicting the stability of rigid systems under external loads.