Reentrant Rigidity Percolation in Correlated Elastic Networks*

Tissues made of semiflexible biopolymers are often modeled as a dilute network of fibers that resist bending and stretching, with the network elastic moduli depending critically on the fraction of bonds. While homogenous dilution is applicable for some biopolymer networks, our interest in networks with pronounced spatial variations, such as collagen in articular cartilage, motivates a network construction protocol that produces structural correlation. We model structurally correlated, dilute Kagome lattice-based networks and compute their linear response to simple shear. We find that the threshold bond fraction needed to produce a rigid network nonmonotonically depends on the structural correlation, with a global minimum for an optimal correlation strength. We also find a mechanical length scale, inferred from correlations in non-affine deformation, that grows monotonically with correlation.  This suggests the reentrance of rigidity percolation arises from a weak coupling between large, stiff domains. We further discuss extending our model to address composite tissues by coupling our correlated networks to a continuous elastic background.