Energetic rigidity II: Applications in examples of biological and underconstrained materials

TL;DR

This paper extends the formalism of energetic rigidity to two-dimensional biological and underconstrained materials, demonstrating how second-order rigidity predicts stability in underconstrained networks and how constraint counting applies to overconstrained systems.

Contribution

It applies energetic rigidity formalism to biological and underconstrained materials, highlighting the importance of second-order rigidity and modified constraint counting.

Findings

Second-order rigidity predicts stability in underconstrained networks.

Constraint counting aligns with energetic rigidity in overconstrained jammed packings.

Modified constraint counting is needed for hypostatic systems.

Abstract

This is the second paper devoted to energetic rigidity, in which we apply our formalism to examples in two dimensions: underconstrained random regular spring networks, vertex models, and jammed packings of soft particles. Spring networks and vertex models are both highly underconstrained, and first-order constraint counting does not predict their rigidity, but second-order rigidity does. In contrast, spherical jammed packings are overconstrained and thus first-order rigid, meaning that constraint counting is equivalent to energetic rigidity as long as prestresses in the system are sufficiently small. Aspherical jammed packings on the other hand have been shown to be jammed at hypostaticity, which we use to argue for a modified constraint counting for systems that are energetically rigid at quartic order.