Summary: A synthesis of different taxonomies of cellular materials. Part 1 of this post discusses 2D prismatic structures.
In a previous post, I discussed four central questions in the design of cellular materials for Additive Manufacturing (AM). The first of these questions involved identifying an appropriate unit cell for the task at hand. In this blog post, I contribute my thoughts on the subject of building a formal taxonomy of cellular materials, something I have attempted before but I now feel needs a significant revision. But first you may well (rightly) ask…
Terminology and taxonomy are probably not high on the list of things most engineers deem critical to getting their jobs done: after all, what we call something is not as important as what we do with that something. The value proposition for a taxonomy for cellular materials is hinted at when you are faced with the challenge of selecting a specific unit cell for one or more functions in design. This is particularly true now with Additive Manufacturing (AM) opening up the design opportunities: there are infinite designs and variations that one can select from, or create – and many of these are manufacturable with AM. This is why I think the time is ripe for revisiting our classification of cellular materials, which may make the selection process more tractable.
Several approaches have been developed, either explicitly or implicitly, to classify cellular materials:
- The most common approach follows the work done by Gibson and Ashby and presented in their seminal book on Cellular Solids, who proposed classifying cellular materials as either honeycombs or foams.
- A more mathematical approach is developed in the field of tessellation, which deals with the partitioning of space into smaller units, or cells.
- Crystallography, the science of studying the structure of crystalline solids, has also developed nomenclature schemes that are very amenable to the study of ordered, lattice-like cellular materials.
- There has been at least one effort in the literature to develop design guidance for selection of prismatic cellular materials (Fazelpour et al., 2016)
- Finally, some software companies have developed a rationale for guiding a designer in selecting a unit cell design (nTopology, to cite one example).
A New Synthesis
In this post, I’ve attempted to synthesize ideas from all the above mentioned sources in developing a taxonomy for cellular materials that is relevant to AM design – this is presented in Figure 2.
The first level of separation in the taxonomy is a trivial one and concerns whether the cellular material is formulated in two dimensions – 2D (prismatic), or three – 3D (volumetric). Some have suggested a 2.5D option as well, which is essentially a prismatic structure that is extruded along a curve instead of a line (Yang et al., 2017). For the sake of simplicity, we shall consider this as a special case of a prismatic cellular material, with a nonlinear path of extrusion. Cellular materials on a surface (tiling on a wall, or the top of a turtle’s shell) can be thought of special instances of prismatic cellular materials with vanishingly small out-of-plane thicknesses.
Prismatic cellular materials can be completely described by the strategy used in tessellation – nothing further needs to be said about connectivity, since it is assumed that the connectivity coincides with the edges of the polygon – if it was otherwise, the tessellation would naturally change to accommodate it. However, for volumetric cellular materials, the case is a bit more complex since tessellation is only the first part of the consideration. We then need to specify the elements we will use to form connections (beams or surfaces) and further, the vertices and their connectivity that will yield the structure. As a result, Figure 2 shows two choices for prismatic cellular structures at the first level, but volumetric cellular materials show three separate decisions that all need to be made, with choices made within each decision. In this post, we examine the two categories listed in Figure 2 for 2D cellular materials – in a following post we extend this study to 3D materials.
Designing with cellular materials ultimately involves filling space. From a mathematical perspective, this is a question that falls within the purview of geometry, and more specifically in the domain of tiling, or tessellation. Broadly speaking, 2D space can either be tessellated in periodic, non-stochastic shapes (where the shape of each cell and its connectivity to others is prescribed), or stochastic shapes (where the shape emerges as a result of an underlying stochastic function). We deal with each of these two approaches in turn.
The key underlying concepts that describe different tessellation schemes are demonstrated in Figure 3. Most commonly studied (and tractable) tessellations are Edge-to-Edge – i.e. when two polygons intersect at more than one point, they share an edge. This is true of the first three tessellations in Figure 3, but not of the fourth. Further, within Edge-to-Edge tessellations, the polygons can be regular (all identical – of which there are only three permissible shapes: hexagon, square and triangle) or semi-regular (when there is more than one polygon). Focusing on the symmetry around vertices allows the introduction of a k-uniform classification, where k represents broadly the different possible surroundings around a vertex. Thus, regular tessellations are 1-uniform, and the third tessellation in Figure 2 is a 2-uniform tiling since its vertices fall into one of two different symmetry conditions with regard to the polygons around them.
In stochastic tessellation, we do not rely on the pre-supposition of one or more polygons. The most common way of representing stochastic tiling is using the Voronoi diagram, attributed to Ukranian mathematician Georgy Voronoy, who defined and generalized the n-dimensional case in 1908. In its simplest manifestation, a Voronoi diagram emerges from a distribution of a finite number of points in space, around which a cell is drawn such that every point is equidistant from the lines formed at the intersections of the cells, as shown in Figure 4.
Prismatic cellular materials, be they stochastic or periodic, have very different behavior in the out-of-plane direction compared to the behavior in-plane. From an application standpoint, the 2D nature of these structures means that their use is beneficial when the directionality of the environmental conditions is predictable and the cellular design can be oriented in such a way as to extract maximum benefit. Examples of this include crash panels in the automotive industry, sandwich panels in construction and automotive radiator grilles. In all these cases, the direction of the environmental stimulus is unidirectional and predictable – whether it be the mechanical load or fluid flow. However, there are several applications where this is not the case – and the reason we need 3D cellular materials – and it is to this that we will turn to in part 2, where I will attempt to justify the classification scheme proposed in Figure 2 for these materials.
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