Generic Elasticity of Thermal, Under-constrained Disordered Systems

Under-constrained disordered systems, where there are more degrees of freedom than constraints, are frequently applied to understand rigidity of disordered biological materials, such as fiber/spring networks and vertex models for biological tissues. These systems are typically floppy based on the Maxwell counting at zero temperature. However, they can be rigidified by the application of external strain or at a finite temperature. Strain-induced athermal rigidification was shown to be caused by the emergence of a state of self-stress (SSS), where non-zero tensions give net zero force at local nodes. While the role of SSS is well studied in this case, it is unclear for the thermal rigidification. We extend our previous SSS theory for athermal rigidity of disordered networks to finite temperatures. From first principles, we derive expressions for elastic properties such as isotropic tension t and shear modulus G on temperature T, isotropic strain and shear strain, which we confirm numerically. These expressions contain only three parameters: entropic rigidity, energetic rigidity, and a parameter for the interaction between isotropic and shear strain. Our results imply that in under-constrained systems, entropic and energetic rigidity interacts like two springs in series. This also allows a simple explanation for the nontrivial scaling t ∼ G ∼ T^1/2 , previously found in simualtions.