Chapter 6 . Criticality and Isostaticity in Fiber Networks

It has been known since Maxwell that collections of particles interacting via central forces only become rigid above the isostatic threshold, where the constraints and internal degrees of freedom just balance [1,2]. However, such networks can be stabilized below this threshold by additional interactions [3–5]. Here we elucidate the relative roles of bending versus central force interactions in stabilizing fibrous networks [6–16]. We study disordered networks with variable connectivity that exhibit both bending rigidity and central-force thresholds. Although the former determines the true onset of rigidity, the latter controls a cross-over between various mechanical regimes exhibiting rich critical behavior, including an anomalous power-law dependence of the shear modulus on both stretching and bending rigidities, as well as a breakdown of mean field theory. At the central force isostatic point, we also find divergent strain fluctuations together with a divergent correlation length ξ, implying a violation of continuum elasticity in this simple mechanical system. These results may apply to systems ranging from bond-bending network glasses [2,17–19] to the cellular cytoskeleton [21,22].