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Abstract |
Rigidity regulates the integrity and function of many physical and biological
systems. Depending on the type of system under study, different proxies are
suitable for measuring the rigidity of a system. Here, we propose that ``energetic
rigidity,'' in which all non-trivial deformations raise the energy of a structure,
is a more useful notion of rigidity in practice than two more commonly used tests:
Maxwell-Calladine constraint counting (first-order rigidity) and second-order
rigidity. We also show that there may be systems for which neither first nor
second-order rigidity imply energetic rigidity. We apply our formalism to examples
in two dimensions: random regular spring networks, vertex models, and jammed
packings of soft disks. Spring networks and vertex models are both highly under-
constrained and first-order constraint counting does not predict their rigidity,
but second-order rigidity does. In contrast, jammed packings are over-constrained
and thus first-order rigid, meaning that constraint counting is equivalent to
energetic rigidity as long as prestresses in the system are sufficiently small.
The formalism of energetic rigidity unifies our understanding of mechanical
stability and also suggests new avenues for material design.
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