{"id":29060,"date":"2016-08-29T16:55:02","date_gmt":"2016-08-29T16:55:02","guid":{"rendered":"https:\/\/www.padtinc.com\/2016\/08\/29\/classification-of-cellular-solids-and-why-it-matters\/"},"modified":"2022-03-24T22:50:49","modified_gmt":"2022-03-24T22:50:49","slug":"classification-of-cellular-solids-and-why-it-matters","status":"publish","type":"post","link":"https:\/\/www.padtinc.com\/2016\/08\/29\/classification-of-cellular-solids-and-why-it-matters\/","title":{"rendered":"Classification of Cellular Solids (and why it matters)"},"content":{"rendered":"

Updated (8\/30\/2016)<\/strong>: Two corrections made following suggestions by Gilbert Peters: the first corrects the use of honeycomb structures in radiator grille applications as being for flow conditioning, the second corrects the use of the Maxwell stability criterion, replacing the space frame example with an octet truss.<\/p>\n

~<\/p>\n

This is my first detailed post in a series on cellular structures for additive manufacturing, following an introductory post I wrote where I classified the research landscape in this area into four elements: design, analysis, manufacturing and implementation.<\/a><\/p>\n

Within the design element, the first step in implementing cellular structures in Additive Manufacturing (AM) is selecting the appropriate unit cell(s). The unit cell is selected based on the performance desired of it\u00a0as well as the\u00a0manufacturability of the cells. In this post, I wish to delve deeper into the different types of cellular structures and why the classification is important. This will set the stage for defining criteria for why certain unit cell designs are preferable over others, which I will attempt in future posts. This post\u00a0will also explain in greater detail what a “lattice”\u00a0structure, a term that is often erroneously used to describe all cellular solids,\u00a0truly is.<\/p>\n

1. Honeycomb<\/h1>\n

1.1 Definition <\/strong>
\nHoneycombs are prismatic, 2-dimensional<\/strong><\/em> cellular designs extruded in the 3rd dimension, like the well-known hexagonal honeycomb shown in Figure 1. All cross-sections through the 3rd dimension are thus identical, making honeycombs somewhat easy to model. Though the hexagonal honeycomb is most well known, the term applies to all designs that have this prismatic property, including square and triangular honeycombs.\u00a0Honeycombs\u00a0have a strong\u00a0anisotropy in the 3rd dimension – in fact, the modulus of regular hexagonal and triangular honeycombs is transversely isotropic – equal in all directions in the plane but very different out-of-plane.<\/p>\n

\"\" <\/picture><\/a>
Figure 1<\/strong>. Honeycomb structure showing two-dimensional, prismatic nature (Attr: modified from work done by George William Herbert, Wikipedia)<\/figcaption><\/figure>\n
\"honeycomb_bmwi3\" <\/picture><\/a>
Figure 2<\/strong>. Honeycomb design in use as part of a BMW i3 crash structure (Attr: adapted from youkeys, Wikipedia)<\/figcaption><\/figure>\n

1.2\u00a0Design Implications
\n<\/strong>The 2D nature of honeycomb structures means that their use is beneficial when the environmental conditions are predictable and the honeycomb design can be oriented in such a way to extract maximum benefit<\/em><\/strong>.\u00a0One such example is the crash structure in Figure 2 as well as a range of\u00a0sandwich panels. Several\u00a0automotive radiator grilles are also of a honeycomb design to condition the flow of air. In both cases, the direction of the environmental stimulus is known – in the former, the impact load, in the latter, airflow.<\/p>\n

2. Open-Cell Foam<\/h1>\n
\"openfoam\" <\/picture>
Figure 3<\/strong>. Open cell foam unit cell, following Gibson & Ashby (1997)<\/figcaption><\/figure>\n

2.1 Definition<\/strong>
\nFreeing up the prismatic requirement on the honeycomb brings us to a fully 3-dimensional<\/em> <\/strong>open-cell foam design as shown in one representation of a unit cell in Figure 3.\u00a0Typically, open-cell foams are bending-dominated<\/strong><\/em>, distinguishing them from stretch-dominated lattices, which are discussed in more detail in\u00a0a following section on lattices.<\/p>\n

2.2 Design Implications
\n<\/strong>Unlike the honeycomb, open cell foam designs are more useful\u00a0when\u00a0the\u00a0environmental stimulus (stress, flow, heat) is\u00a0not as predictable and\u00a0unidirectional.\u00a0<\/strong><\/em>The bending dominated mechanism of deformation make open-cell foams ideal for energy absorption – stretch dominated structures tend to be stiffer.\u00a0As a result of this, applications that require energy absorption such as mattresses\u00a0and\u00a0crumple zones in complex structures. The interconnectivity of open-cell foams also makes them a candidate for applications requiring fluid flow through the structure.<\/p>\n

\"Metal_Foam\" <\/picture>
Figure 4<\/strong>. SEM image of a metallic open-cell foam (Attr: SecretDisc, Wikipedia)<\/figcaption><\/figure>\n
\"openfoam-deform\" <\/picture><\/a>
Figure 5<\/strong>. FEA simulation of open cell foam unit cell under compression, showing predominant mode of deformation is on account of bending<\/figcaption><\/figure>\n

3. Closed-Cell Foam<\/h1>\n

\"closedfoam\" <\/picture>
Figure 6<\/strong>. Open cell foam unit cell representation [following Gibson and Ashby, 1997]<\/figcaption><\/figure>3.1 Definition
\n<\/strong>As the name suggests, closed cell foams are open-cell foams with enclosed cells, such as the representation shown in Figure 6. This typically involves a membrane like structure that may be of varying thickness from the strut-like structures, though this is not necessary. Closed-cell foams arise from a lot of natural processes\u00a0and are commonly found in nature. In man-made entities, they are commonly found in the food industry (bread, chocolate) and in engineering applications where the enclosed cell is filled with some fluid (like air in bubble wrap, foam for bicycle helmets and\u00a0fragile packaging).<\/p>\n

3.2 Design Implications<\/strong>
\nThe primary benefit of closed cell foams is the ability to encapsulate a fluid of different properties for compressive resilience. From a structural standpoint, while the membrane is a load-bearing part of the structure under certain loads, the additional material\u00a0and manufacturing burden can be\u00a0hard to justify. Within the AM context, this is a key area of interest for those exploring 3D printing food products in particular but may also have value for biomimetic applications.<\/p>\n

\"Closed_cell_metal_foam_with_large_cell_size\" <\/picture>
Figure 8<\/strong>. Closed cell Aluminum foam with very large cells [Shinko Wire Company, Attr: Curran2, Wikimedia Commons]<\/figcaption><\/figure>\n

\u00a04. Lattice<\/h1>\n

4.1 Definition<\/strong>
\nLattices are in appearance very similar to\u00a0open cell foams but differ\u00a0in that\u00a0lattice member deformation is\u00a0stretch<\/strong>-dominated<\/strong><\/em>, as opposed to bending*. This is important since for the same material allocation, structures\u00a0tend to be stiffer<\/strong><\/em> in tension and\/or compression compared to bending – by contrast, bending dominated structures typically absorb more energy and are more compliant.<\/p>\n

So the question is – when does an open cell foam become stretch dominated and therefore, a lattice? Fortunately, there is an app<\/del> equation for that.<\/p>\n

Maxwell’s Stability Criterion<\/strong>
\nMaxwell’s stability criterion involves the computation of a metric M<\/em>\u00a0for a lattice-like structure with b <\/em>struts and\u00a0j<\/em> joints\u00a0as follows:<\/p>\n

In 2D structures: <\/em>M = b – 2j + 3
\nIn 3D structures: <\/em>M = b – 3j + 6<\/p><\/blockquote>\n

Per Maxwell’s criterion, for our purposes here where the joints are locked (and not pinned), if M < 0, we get a structure that is bending dominated. If M >= 0, the structure is stretch dominated. The former constitutes an open-cell foam, the latter a lattice.<\/p>\n

There are several approaches to establishing the appropriateness of a lattice design for a structural applications (connectivity, static and kinematic determinism etc.) and how they are applied to periodic structures and space frames. It is easy for one\u00a0(including for this author) to confuse these ideas and their applicability. For the purposes of AM, Maxwell’s Stability Criterion for 3D structures is a sufficient condition for static determinancy. Further, for a periodic structure to be truly space-filling (as we need for AM applications),\u00a0there is no simple rigid polyhedron that fits the bill – we need a combination of polyhedra (such as an octahedron and tetrahedron in the octet truss shown in the video below) to generate true space filling, and rigid structures. The 2001 papers by Deshpande, Ashby and Fleck illustrate these ideas in greater detail and are referenced at the end of this post.<\/p>\n