Inspiration from Nature: Ultralight Fractal Designs for High Mechanical Efficiency

[From Left to Right] Daniel Rayneau-Kirkhope, Robert Farr, Yong Mao



Authors: Daniel Rayneau-Kirkhope¹, Robert Farr^2,3, Yong Mao⁴

Affiliation:
¹Open Innovation House, School of Science, Aalto University, Finland,
²Unilever R&D, Colworth House, Sharnbrook, Bedford, UK
³London Institute for Mathematical Sciences, Mayfair, London, UK
⁴School of Physics and Astronomy, University of Nottingham, UK

Hierarchical design is ubiquitous in nature [1]. Material properties can be tailored by having structural features on many length scales. The gecko, a lizard ranging from 2 to 60 cm in length, has a remarkable ability to walk on vertical walls and even upside-down on ceilings. This ability is brought about through the repeated splitting of the keratinous fibres on the bottom of the gecko’s foot, which increases the contact area so effectively that even the very weak van der Waals interactions can support the entire weight of the gecko [2].

A more specific form of hierarchical design is self-similar design, where one structural feature is found to be repeated on a number of different length scales. A natural example is the trabecular or spongy bone found around the joints in animals [3]. Here, a series of small beams are arranged in such a way that the stiffness and strength requirements are met while using minimal material. Regardless of the level of magnification, the same patterns are found in the structure. Interestingly, the exact configuration of the constituent beams in the trabecular bone is constantly changing: it is the result of a continuous opitimisation process that goes on throughout the lifetime of the bone and responds to change in stress levels [4]. It is found that when the animal’s bones support only small loads, many very slender pillars are present, and when the loading increases, fewer but stouter pillars are employed [5].

In our recently published work [6], we demonstrate that through the use of hierarchical, self-similar design principles, advantageous structural properties can be obtained. We show that the scaling of the amount of material required for stability against the loading can be altered in a systematic manner. A particular structure is fabricated through rapid prototyping, and we obtain the optimal generation number (for our specific structure) for any given value of loading.

Scaling

The volume of material required for stability can be related to the loading through a simple power law relationship. That is, the volume required is given by a dimensionless loading parameter raised to some power with a pre-factor (that is dependent only on material properties and specifics of the geometry). When the loading is small, it is the scaling (power) of the loading that dominates the relationship. Under tension, this power is one and a structure requires an amount of material that is proportional to the loading it must withstand; for a solid beam under compression, due to elastic buckling, the power is one-half. Given, for all realistic applications, the non-dimensional loading parameter is much less than 1, this means more material is required to support compressive than tensional loads. This one-half power law has direct consequences when one considers optimal structure: if a beam is bearing a compressive load, it is more efficient to use one beam rather than two, whereas in the case of tension, due to the linear relationship, splitting a tension member into more than one piece has no effect on the volume required for stability.

Fractal design

Our work centers on a very simple, iterative procedure that can be used to create designs of great complexity. The “generation” of a structure describes the number of iterations used to create the geometry. The simplest compression bearing structure is a solid slender beam. When loaded with a gradually increasing force, the beam will eventually buckle into a sinusoidal shape known as an “Euler buckling mode”. We can suppress this by using a hollow tube, but we introduce a second mode of a local failure of the tube wall – Koiter buckling. After optimizing for tube diameter and thickness, it is found that the scaling power increases to two-thirds, and the volume of material required for stability is reduced.

Figure 1: Showing the iterative process from low generation numbers to higher for structures bearing compression along their longest axis. At each step, all beams that are compressively loaded are replaced by a (scaled) generation-1 frame.

The next step is to replace the hollow beam with a space frame of hollow beams. The space frame used here is made up of n octahedra and two end tetrahedra. Optimising the number of octahedra, the radius and the wall thickness of the component beams (which are all assumed be identical) we find a new power law, and again, an improvement over the hollow beam design.

Continuing this procedure of replacing all beams under compressive load with (scaled) space frames constructed from hollow beams (figure 1), we find that the scaling law is always improved by the increased level of hierarchy. In general, the scaling is described by a (G+2)/(G+3) power-law relating non-dimensional volume to non-dimensional loading. Thus, as the generation number tends to infinity, the scaling relating material required for stability to loading approaches that of the tension member.

3D Printing

Working with Joel Segal, of the University of Nottingham, we fabricated an example of a generation-2 structure with solid beams, shown in figure 2. This was done through rapid prototyping technologies: micrometer-layer-by-micrometer-layer the structure was printed in a photosensitive polymer with each beam a fraction of a millimeter in radius. This structure shows the plausibility of the design and the extent to which modern manufacturing techniques allow for an increased creativity in design geometry. Through a process of 3-d printing and electro less deposition, it is believed that a metallic, hollow tubed structure could be created.

Figure 2: Showing a structure fabricated through rapid prototyping techniques. The inset shows the layering effect of the 3D printing technique. The structure shown in constructed in RC25 (Nanocure) material from envisionTEC on an envisionTEC perfactory machine.

Optimal generations

Although the scaling is always improved by increasing the generation number of the structure, the prefactor isn’t. The optimal structure is then obtained by balancing the scaling relationship with the prefactor in the expression. Generally, as the loading decreases (or the size of the structure increases), the scaling becomes more important and the optimal generation number increases. For large loads (or small structures) it can even be the case a simple, solid, beam is optimal.

Our work also formalises this relationship, for a long time engineers have created chair legs from hollow tubes or cranes out of space frames, Gustave Eiffel used three levels of structural hierarchy in designing the Eiffel tower. We show formally, that the optimal generation number has a set dependence on the loading conditions and allow future structures to be designed with this in mind. A further consequence of the alteration of the scaling law is that the higher the generations, the less difference it makes as to whether you have one structure holding a given load or two structures holding half the load each.

Reference:
[1] Robert Lakes, "Materials with structural hierarchy", Nature, 361, 511 (1993). Abstract.
[2] Haimin Yao, Huajian Gao, "Mechanics of robust and releasable adhesion in biology: Bottom–up designed hierarchical structures of gecko", Journal of the Mechanics and Physics of Solids, 54,1120 (2006). Abstract.
[3] Rachid Jennanea, Rachid Harbaa, Gérald Lemineura, Stéphanie Bretteila, Anne Estradeb, Claude Laurent Benhamouc, "Estimation of the 3D self-similarity parameter of trabecular bone from its 2D projection", Medical Image Analysis, 11, 91 (2007). Abstract.
[4] Rik Huiskes, Ronald Ruimerman, G. Harry van Lenthe, Jan D. Janssen, "Effects of mechanical forces on maintenance and adaptation of form in trabecular bone", Nature, 405, 704 (2000). Abstract.
[5] Michael Doube, Michał M. Kłosowski, Alexis M. Wiktorowicz-Conroy, John R. Hutchinson, Sandra J. Shefelbine, "Trabecular bone scales allometrically in mammals and birds", Proceedings of the Royal Society B, 278, 3067 (2011). Abstract.
[6] Daniel Rayneau-Kirkhope, Yong Mao, Robert Farr, "Ultralight Fractal Structures from Hollow Tubes", Physical Review Letters, 109, 204301 (2012). Abstract.