The emergence of an unusual stiffness profile in hierarchical biological tissues

Biological tissues usually exhibit complex multiscale structural architectures. In many of these, and particularly in mineralized tissues, the basic building block is a staggered array-a composite material made of soft matrix and stiff reinforcing elements. Here we study the stiffness of non-overlapping staggered arrays, a case that has not previously been considered in the literature, and introduce closed-form analytical expressions for its Young's modulus. These expressions are then used to estimate the stiffness of natural staggered biocomposites such as low-mineralized collagen fibril and mineralized tendon. We then consider a two-scale composite scheme for evaluating the modulus of a specific hierarchical structure, the compact bone tissue, which is made of mineralized collagen fibrils with weakly overlapping staggered architecture. It is found that small variations in the staggered structure induce significant differences in the macroscopic stiffness, and, in particular, provide a possible explanation for the as yet unexplained stiffening effects observed in medium-mineralized tissues.