Stability and anomalous entropic elasticity of sub isostatic random-bond networks

We study the elasticity of thermalized spring networks under an applied bulk strain. The networks considered are sub-isostatic random-bond networks that, in the athermal limit, are known to have vanishing bulk and linear shear moduli at zero bulk strain. Above a bulk strain threshold, however, these networks become rigid, although surprisingly the shear modulus remains zero until a second, higher, strain threshold. We find that thermal fluctuations stabilize all networks below the rigidity transition, resulting in systems with both finite bulk and shear moduli. Our results show a $T^{0.66}$ temperature dependence of the moduli in the region below the bulk strain threshold, resulting in networks with anomalously high rigidity as compared to ordinary entropic elasticity. Furthermore we find a second regime of anomalous temperature scaling for the shear modulus at its zero-temperature rigidity point, where it scales as $T^{0.5}$, behavior that is absent for the bulk modulus since its athermal rigidity transition is discontinuous.