Branching toughens fibrous networks.

Fibrous collagenous networks are not only stiff but also tough, due to their complex microstructures. This stiff yet tough behavior is desirable for both medical and military applications but it is difficult to reproduce in engineering materials. While the nonlinear hyperelastic behavior of fibrous networks has been extensively studied, the understanding of toughness is still incomplete. Here, we identify a microstructure mimicking the branched bundles of a natural type I collagen network, in which partially cross-linked long fibers give rise to novel combinations of stiffness and toughness. Finite element analysis shows that the stiffness of fully cross-linked fibrous networks is amplified by increasing the fibril length and cross-link density. However, a trade-off of such stiff networks is reduced toughness. By having partially cross-linked networks with long fibrils, the networks have comparable stiffness and improved toughness as compared to the fully cross-linked networks. Further, the partially cross-linked networks avoid the formation of kinks, which cause fibril rupture during deformation. As a result, the branching allows the networks to have stiff yet tough behavior.