Second Order Rigidity in Small Underconstrained Networks

Rigidity is an important feature of many biological materials, including extra-cellular fiber networks and cellularized tissues, which can transition between a floppy, fluid-like state and a rigidified solid-like state. In some systems this transition is driven by changes in network connectivity and can be studied with constraint counting arguments (first-order rigidity). However, even underconstrained systems can be rigidified with sufficient deformation, in which case the transition is driven by the geometry and is described by second-order rigidity. We investigate the nature of this second-order rigidity transition in disordered sub-isostatic spring networks. By studying small toy networks with only a single junction, we attempt to develop a mean-field description of the geometry near the critical point. We then study the relationship between geometry and rigidity numerically by removing a single junction from a simulated rigid spring network with very low tensions so the system is close to the critical point. This causes the rigid configuration to become floppy, and we characterize the space of allowed motions and attempt to map them to analytic results in small networks. A better understanding of this rigidity transition and the space of configurations at the critical point could lead to new ideas in the design of mechanical metamaterials.