  Rigidity regulates the integrity and function of many physical and biological
systems. This is the first of two papers on the origin of rigidity, wherein 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 rigidity 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. The formalism of energetic
rigidity unifies our understanding of mechanical stability and also suggests
new avenues for material design.
