  Rigidity regulates the integrity and function of many physical and biological
systems. In this work, 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 rigidiy tests:
Maxwell-Calladine constraint counting (first-order rigidity) and second-order
rigidity. We find that constraint counting robustly predicts energetic rigidity
only when the system has no states of self stress. When the system has states
of self stress, we show that second-order rigidity can imply energetic rigidity
in systems that are not considered rigid based on constraint counting, and is
even more reliable than shear modulus. 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.
