Generic elasticity of thermal, under-constrained systems

Athermal (i.e. zero-temperature) under-constrained systems are typically floppy, but they can be rigidified by the application of external strain, which is theoretically well understood. Here and in the companion paper, we extend this theory to finite temperatures for a very broad class of under-constrained systems. In the vicinity of the athermal transition point, we derive from first principles expressions for elastic properties such as isotropic tension $t$ and shear modulus $G$ on temperature $T$, isotropic strain $\varepsilon$, and shear strain $\gamma$, which we confirm numerically. These expressions contain only three parameters, entropic rigidity $\kappa_S$, energetic rigidity $\kappa_E$, and a parameter $b_\varepsilon$ describing the interaction between isotropic and shear strain, which can be determined from the microstructure of the system. Our results imply that in under-constrained systems, entropic and energetic rigidity interact like two springs in series. This also allows for a simple explanation of the previously numerically observed scaling relation $t\sim G\sim T^{1/2}$ at $\varepsilon=\gamma=0$. Our work unifies the physics of systems as diverse as polymer fibers & networks, membranes, and vertex models for biological tissues.