Mechanics of hierarchical, filamentous tissues
Michel, Jonathan A.
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Structural hierarchy, the property of possessing spatial organization on multiple, distinct length scales, is omnipresent in biological tissues, and is increasingly popular as a means of pursuing designer properties of human-made materials. Hierarchy can offer economy of material, resilience against fracture, and novel mechanical response; however, the apparent opportunity for errors in assembly at multiple stages na\"ively seems to present an imposing obstacle to the evolution of hierarchical tissue. Nonetheless, many organisms, from many evolutionary lineages, exhibit structural hierarchy. In this work, we build upon previous efforts to model tissue as spring networks. We create networks with a nested, self-similar structure, whose geometrical attributes can be independently varied at each scale. Following previous researchers, we focus upon the mean coordination number, which gives the typical number of nearest neighbors to which a vertex in a network is connected, as a parameter for controlling the elastic properties of structures. We extend this idea, defining separate coordination numbers for the network architecture, and find a simple scaling law relating a material's stiffness to its structural attributes at each length scale. We validate this scaling law with simulations, and find it to hold for structures derived from crystalline lattices and triangulations of random point sets. From this scaling law, we predict that the variability in the stiffness of a network resulting from variability in its structural attributes at each length scale diminishes with increasing levels of hierarchy, up to some threshold. Our results suggest that robustness to errors in assembly may be a generic benefit of a modular assembly process. Finally, we elucidate the role of large-scale and small-scale structural attributes. We find the small scale structure sets the vibrational density of states of our model systems at large frequency, while the large-scale structure is important in coordinating a system-wide, percolating force network to stiffen the material.