Fractality in optimal selfsimilar elastic structures

In this paper we study the diffusely observed occurrence of fractality in mechanical systems. We analytically show that these phenomena can be viewed as the result of the contemporary attainment of mass minimization and global stability in elastic systems. In recent years the experimental evidence of fractal systems has increasingly interested many technological and theoretical fields of research [2]. Nowadays, fractality is recognized as a paradigm of material and structure morphological optimization. Indeed, through billions years evolution, nature developed complex, hierarchical multiscale structures delivering performances unreached by human technologies [1]. Typical examples of natural hierarchical systems are represented by spider silks, geckoes pads, and keratin materials that attain their incredible properties based on the creation of multiscale structure morphologies, often characterized by self-similarity. The study of the physical mechanisms underlying the diffuse experimental observation of fractal systems is then fundamental to understand the features of many nature and biological phenomena and is also crucial for the design of new efficient bioinspired materials and structures. To this end we adopt a very simple prototypical, tensegrity type device, aimed at the transmission of compression loads. More precisely, starting from the elementary Euler column, we consider a geometrical complexification by means of self-similar tensegrity substructures that favors the mass minimization whence the global stability condition is imposed. As we analytically show, in the limit cases of decreasing load or increasing slenderness, the requirement of both mass optimization and global stability delivers a fractal like morphology, characterized by power laws behavior. Furthermore, the limit optimal structure exhibits a contemporary attainment of critical states at all involved scales and we are able to determine the dependence of the fractal dimension D of the limit optimal structure by the geometrical and constitutive parameters. Of course, as usual in the context of fractal systems this (‘infinite’) refinement, leading to a fractal morphology, is ideal, but clarifies why so many systems in nature exhibit self-similar character. In conclusion, we deem that the analytical evidence of the proposed results clarifies that mass minimization and global stability may be fundamental for interpreting the scale-free geometries of many natural systems.