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

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.

Automated summary

This paper discusses the calculation method of energetic rigidity and applies it to examples in two dimensions. Underconstrained spring networks and vertex models require second-order rigidity to predict their rigidity, while overconstrained spherical jammed packings can be calculated using first-order constraint counting. Aspherical jammed packings are jammed at hypostaticity and require a modified constraint counting.