Hierarchical Elastic Network Mechanics

Structural hierarchy is ubiquitous in biological tissues, a long-used architectural strategy, and an increasingly common feature of mechanical metamaterials. All of these cases feature networks comprised of bonds and nodes. While the mechanics of frames with features on a single length scale has been studied for over a century, there exists a lingering need for general rules explaining the interplay of structures on disparate length scales. We present the results of normal mode analysis and uniaxial tensile simulations of ball-and-spring networks with a diluted hexatic lattice structure on two separate scales. Broadly, small-scale structure is found to govern the vibrational density of states, while large-scale structure is found to be important in frustrating low-energy bending modes. We additionally comment upon tolerance in hierarchical structures of random errors in assembly, a crucial attribute in biological systems.