Ultralight Fractal Structures from Hollow Tubes

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.

Figure 1

Generation-1 and generation-2 frames shown labeled (a) and (b) respectively. Images are stereographic: to view a single three-dimensional structure hold the figure 20 cm away and focus ‘through‘ the page until both images merge.

Figure 2

The end tetrahedron and octahedron of a generation-2 structure constructed through a rapid prototyping procedure; this structure is constructed using solid beams. In principle, this could be used as a scaffold [13] to construct a hollow, metallic structure of similar geometry. Inset shows the layered nature of the material in a single beam, with a layer thickness of $25\mu \mathrm{m}$.

Figure 3

Volume required for a stable structure against loading for which the structure is optimized, showing generations 0–4.

Figure 4

The material saving for a given $F$ and $L$ over a solid beam assuming material properties close to those of steel. Also depicting the progression of optimality from simple structures (chair legs) to structures with many levels of hierarchy. Also shown is the point $F=10\mathrm{kN}$, $L=200\mathrm{m}$ ($\times$) and the regions appropriate for construction of a solar sail compression beam and the boom of a crane [20].