Ultralight Fractal Structures from Hollow Tubes

Abstract

A fractal design is shown to be highly efficient both as a load bearing structure and as a general metamaterial. Through changing the hierarchical order of the structure, the scaling of material required for stability against loading can be manipulated. We show that the transition from solid to hollow beams changes the scaling in a manner analogous to increasing the hierarchical order by one. An example second order solid beam frame is constructed using rapid prototyping techniques. The optimal hierarchical order of the structure is found for different values of loading. Possible fabrication methods and applications are then discussed.

Honeycomb phononic crystals with self-similar hierarchy

“…This geometrical change results in an increase in multiple scattering of the propagating waves at the cell walls and consequently opening up the Bragg-type band gaps [31]. The alterations of the band structure indicate a hierarchy-dependent transition, which parallels the effect of hierarchy on mechanical behavior in other contexts [10][11][12][13][14].…”

“…Examples include collagen [1], bone [2,3], tooth [2], tendon [3], wood [3,4], nacre [5], gecko foot pads [6], Asteriscus plant [7], Euplectella sponge [8], and water-repellent biological systems [9]. The purely structural role of hierarchy in boosting mechanical performance is now well known [10][11][12][13][14]. In addition to hierarchy, periodic organizations aimed at influencing the wave-propagation behavior, for instance in structural colorations, can also be found in nature [15][16][17].…”

Imperfections in a two-dimensional hierarchical structure

“…Higher-order structures can be obtained through the same replacement procedure. The notation here will follow that used in related works [8,9]: X G,i represents the parameter X at the ith hierarchical level (i = 0 being the smallest and i = G being the largest) of a structure with G + 1 characteristic length scales. …”

“…Recent theoretical works assessed the benefits that can be obtained from self-similar hierarchical designs [6][7][8][9][10]. However, in attempting to create structures of high efficiency, inevitable initial imperfections can result in problems of reliability [11].…”

“…The technique of naive optimization (used in Refs. [6][7][8][9][10]) is a useful tool in creating theoretical bounds for efficiency of structures. It is often the case, however, that through its implementation, high imperfection sensitivity is introduced [11].…”