Fractal space frames and metamaterials for high mechanical efficiency

We resort to variational methods to evaluate the asymptotic behavior of fine metamaterials as a function of cell size. To zeroth order, the metamaterial behaves as a micropolar continuum with both displacement and rotation degrees of freedom, but exhibits linear-elastic fracture mechanics scaling and therefore no size effect. To higher order, the overall energetics of the metastructure can be characterized explicitly in terms of the solution of the zeroth-order continuum prob...

In an extension of speculations that physical space-time is a fractal which might itself be embedded in a high-dimensional continuum, it is hypothesized to "compensate" for local variations of the fractal dimension by instead varying the metric in such a way that the intrinsic (as seen from an embedded observer) dimensionality remains an integer. Thereby, an extrinsic fractal continuum is intrinsically perceived as a classical continuum. Conversely, it is suggested that any v...

Rapid progress in additive manufacturing methods has created a new class of ultralight and strong architected metamaterials that resemble periodic truss structures. The mechanical performance of these metamaterials with a very large number of unit cells is ultimately limited by their tolerance to damage and defects, but an understanding of this sensitivity has remained elusive. Using a stretching-dominated micro-architecture and metamaterial specimens comprising millions of u...

This work shows that fractals can be obtained from Mechanical Laws without being forced by any algorithm, closing the gap between the Platonic world of Mathematics and Nature. Fractal tree crown directly emerges when applying elasticity theory to branching stresses in a binary tree. Vertical displacements of nodes are given by the Takagi curve, while the horizontal ones are given by a linear combination of inverses of $\beta$-Cantor functions. In addition, both fractal dimens...

The quest for novel designs for lightweight phononic crystals and elastic metamaterials with wide lowfrequency band gaps has proven to be a significant challenge in recent years. In this context, lattice-type materials represent a promising solution, providing both lightweight properties and significant possibilities of tailoring mechanical and dynamic properties. Additionally, lattice structures also enable the generation of hierarchical architectures, in which basic constit...

Materials that are lightweight yet exhibit superior mechanical properties are of compelling importance for several technological applications that range from aircrafts to household appliances. Lightweight materials allow energy saving and reduce the amount of resources required for manufacturing. Researchers have expended significant efforts in the quest for such materials, which require new concepts in both tailoring material microstructure as well as structural design. Arch...

Metamaterials exhibit materials response deviation from conventional elasticity. This phenomenon is captured by the generalized elasticity as a result of extending the theory at the expense of introducing additional parameters. These parameters are linked to internal length scales. Describing on a macroscopic level a material possessing a substructure at a microscopic length scale calls for introducing additional constitutive parameters. Therefore, in principle, an asymptotic...

Framework materials and their deformations provide a compelling relation between materials science and algebraic geometry. Physical distance constraints within the material transform into polynomial constraints, making algebraic geometry and associated numerical strategies feasible for finding equilibrium configurations and deformation pathways. In this paper, we build the necessary geometric formulations and numerical strategies to explore the mechanics of two examples of 3-...

A collection of thin structures buckle, bend, and bump into each-other when confined. This contact can lead to the formation of patterns: hair will self-organize in curls; DNA strands will layer into cell nuclei; paper, when crumpled, will fold in on itself, forming a maze of interleaved sheets. This pattern formation changes how densely the structures can pack, as well as the mechanical properties of the system. How and when these patterns form, as well as the force required...

Reinforced elastic sheets surround us in daily life, from concrete shell buildings to biological structures such as the arthropod exoskeleton or the venation network of dicotyledonous plant leaves. Natural structures are often highly optimized through evolution and natural selection, leading to the biologically and practically relevant problem of understanding and applying the principles of their design. Inspired by the hierarchically organized scaffolding networks found in p...