On the Functions of Cellular Structures in Nature

WHY did nature evolve cellular structures?

In a previous post, I laid out a structural classification of cellular structures in nature, proposing that they fall into 6 categories. I argued that it is not always apparent to a designer what the best unit cell choice for a given application is. While most mechanical engineers have a feel for what structure to use for high stiffness or energy absorption, we cannot easily address multi-objective problems or apply these to complex geometries with spatially varying requirements (and therefore locally optimum cellular designs). However, nature is full of examples where cellular structures possess multi-objective functionality: bone is one such well-known example. To be able to assign structure to a specific function requires us to connect the two, and to do that, we must identify all the functions in play. In this post, I attempt to do just that and develop a classification of the functions of cellular structures.

Any discussion of structure in nature has to contend with a range of drivers and constraints that are typically not part of an engineer’s concern. In my discussions with biologists (including my biochemist wife), I quickly run into justified skepticism about whether generalized models associating structure and function can address the diversity and nuance in nature – and I (tend to) agree. However, my attempt here is not to be biologically accurate – it is merely to construct something that is useful and relevant enough for an engineer to use in design. But we must begin with a few caveats to ensure our assessments consider the correct biological context.

1. Uniquely Biological Considerations

Before I attempt to propose a structure-function model, there are some legitimate concerns many have made in the literature that I wish to recap in the context of cellular structures. Three of these in particular are relevant to this discussion and I list them below.

1.1 Design for Growth

Engineers are familiar with “design for manufacturing” where design considers not just the final product but also aspects of its manufacturing, which often place constraints on said design. Nature’s “manufacturing” method involves (at the global level of structure), highly complex growth – these natural growth mechanisms have no parallel in most manufacturing processes. Take for example the flower stalk in Fig 1, which is from a Yucca tree that I found in a parking lot in Arizona.

Figure 1. The flower stalk (before bloom) of a Yucca plant in Arizona with overlapping surface cellular structure (Author’s image)

At first glance, this looks like a good example of overlapping surfaces, one of the 6 categories of cellular structures I covered before. But when you pause for a moment and query the function of this packing of cells (WHY this shape, size, packing?), you realize there is a powerful growth motive for this design. A few weeks later when I returned to the parking lot, I found many of the Yucca stems simultaneously in various stages of bloom – and captured them in a collage shown in Fig 2. This is a staggering level of structural complexity, including integration with the environment (sunlight, temperature, pollinators) that is both wondrous and for an engineer, very humbling.

Figure 2. From flower stalk to seed pods, with some help from pollinators. Form in nature is often driven by demands of growth. (Author’s images)

The lesson here is to recognize growth as a strong driver in every natural structure – the tricky part is determining when the design is constrained by growth as the primary force and when can growth be treated as incidental to achieving an optimum functional objective.

1.2 Multi-functionality

Even setting aside the growth driver mentioned previously, structure in nature is often serving multiple functions at once – and this is true of cellular structures as well. Consider the tessellation of “scutes” on the alligator. If you were tasked with designing armor for a structure, you may be tempted to mimic the alligator skin as shown in Fig. 3.

Figure 3. The cellular scutes on the alligator serve more than just one function: thermal regulation, bio-protection, mobility, fluid loss mitigation are some of the multiple underlying objectives that have been proposed (CCO public domain, Attr. Republica)

As you begin to study the skin, you see it is comprised of multiple scutes that have varying shape, size and cross-sections – see Fig 4 for a close-up.

Figure 4. Close-up of alligator scutes (Attr: Hans Hillewaert, Flickr, Creative Commons)

The pattern varies spatially, but you notice some trends: there exists a pattern on the top but it is different from the sides and the bottom (not pictured here). The only way to make sense of this variation is to ask what functions do these scutes serve? Luckily for us, biologists have given this a great deal of thought and it turns out there are several: bio-protection, thermoregulation, fluid loss mitigation and unrestricted mobility are some of the functions discussed in the literature [1, 2]. So whereas you were initially concerned only with protection (armor), the alligator seeks to accomplish much more – this means the designer either needs to de-confound the various functional aspects spatially and/or expand the search to other examples of natural armor to develop a common principle that emerges independent of multi-functionality specific to each species.

1.3 Sub-Optimal Design

This is an aspect for which I have not found an example in the field of cellular structures (yet), so I will borrow a well-known (and somewhat controversial) example [3] to make this point, and that has to do with the giraffe’s Recurrent Laryngeal Nerve (RLN), which connects the Vagus Nerve to the larynx as shown in Figure 5, which it is argued, takes an unnecessarily long circuitous route to connect these two points.

Figure 5. Observe how the RLN in the giraffe emerges from the Vagus Nerve far away from the thorax: a sub-optimal design that was likely carried along through the generations in aid of prioritizing neck growth (Attr: Vladimir V. Medeyko)

We know that from a design standpoint, this is sub-optimal because we have an axiom that states the shortest distance between two points is a straight line. And therefore, the long detour the RLN makes in the giraffe’s neck must have some other evolutionary and/or developmental basis (fish do not have this detour) [3]. However, in the case of other entities such as the cellular structures we are focusing on, the complexity of the underlying design principles makes it hard to identify cases where nature has found a sub-optimal design space for the function of interest to us, in favor of other pressing needs determined by selection. What is sufficient for the present moment is to appreciate that such cases may exist and to bear them in mind when studying structure in nature.

2. Classifying Functions

Given the above challenges, the engineer may well ask: why even consider natural form in making determinations involving the design of engineering structures? The biomimic responds by reminding us that nature has had 3.8 billion years to develop a “design guide” and we would be wise to learn from it. Importantly, natural and engineering structures both exist in the same environment and are subject to identical physics and further, are both often tasked with performing similar functions. In the context of cellular structures, we may thus ask: what are the functions of interest to engineers and designers that nature has addressed through cellular design? Through my reading [1-4], I have compiled the classification of functions in Figure 6, though this is likely to grow over time.

Figure 6. A proposed classification of functions of cellular structures in nature (subject to constant change!)

This broad classification into structural and transport may seem a little contrived, but it emerges from an analyst’s view of the world. There are two reasons why I propose this separation:

  1. Structural functions involve the spatial allocation of materials in the construction of the cellular structures, while transport functions involve the structure AND some other entity and their interactions (fluid or light for example) – thus additional physics needs to be comprehended for transport functions
  2. Secondly, structural performance needs to be comprehended independent of any transport function: a cellular structure must retain its integrity over the intended lifetime in addition to performing any additional function

Each of these functions is a fascinating case study in its own right and I highly recommend the site AskNature.org [1] as a way to learn more on a specific application, but this is beyond the scope of the current post. More relevant to our high-level discussion is that having listed the various reasons WHY cellular structures are found in nature, the next question is can we connect the structures described in the previous post to the functions tabulated above? This will be the attempt of my next post. Until then, as always, I welcome all inputs and comments, which you can send by messaging me on LinkedIn.

Thank you for reading!

References

  1. AskNature.org
  2. Foy (1983), The grand design: Form and colour in animals, Prentice-Hall, 1st edition
  3. Dawkins (2010), The greatest show on earth: the evidence for evolution, Free Press, Reprint of 1st edition
  4. Gibson, Ashby, Harley (2010), Cellular Materials in Nature and Medicine, Cambridge University Press; 1st edition
  5. Ashby, Evans, Fleck, Gibson, Hutchinson, Wadley (2000), Metal Foams: A Design Guide, Butterworth-Heinemann, 1st edition

SFF Symposium 2016 Paper: Predicting the Mechanical Behavior of ULTEM-9085 Honeycomb Structures

Our work on  3D printed honeycomb modeling that started as a Capstone project with students from ASU in September 2015 (described in a previous blog post), was published in a peer-reviewed paper released last week in the proceedings of the SFF Symposium 2016. The full title of the paper is “A Validated Methodology for Predicting the Mechanical Behavior of ULTEM-9085 Honeycomb Structures Manufactured by Fused Deposition Modeling“. This was the precursor work that led to a us winning an 18-month award to pursue this work further with America Makes.

Download the whole paper at the link below:
http://sffsymposium.engr.utexas.edu/sites/default/files/2016/168-Bhate.pdf

Abstract
ULTEM-9085 has established itself as the Additive Manufacturing (AM) polymer of choice for end-use applications such as ducts, housings, brackets and shrouds. The design freedom enabled by AM processes has allowed us to build structures with complex internal lattice structures to enhance part performance. While solutions exist for designing and manufacturing cellular structures, there are no reliable ways to predict their behavior that account for both the geometric and process complexity of these structures. In this work, we first show how the use of published values of elastic modulus for ULTEM-9085 honeycomb structures in FE simulation results in 40- 60% error in the predicted elastic response. We then develop a methodology that combines experimental, analytical and numerical techniques to predict elastic response within a 5% error. We believe our methodology is extendable to other processes, materials and geometries and discuss future work in this regard.

Figure
Fig 1. Honeycomb tensile test behavior varying as a function of manufacturing parameters
The ASU Capstone team (left to right): Drew Gibson, Jacob Gerbasi, John Reeher, Matthew Finfrock, Deep Patel and Joseph Van Soest.
Fig 2. The ASU Capstone team (left to right): Drew Gibson, Jacob Gerbasi, John Reeher, Matthew Finfrock, Deep Patel and Joseph Van Soest.

Classification of Cellular Solids (and why it matters)

Updated (8/30/2016): 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.

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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.

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 as well as the manufacturability 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 will also explain in greater detail what a “lattice” structure, a term that is often erroneously used to describe all cellular solids, truly is.

1. Honeycomb

1.1 Definition
Honeycombs are prismatic, 2-dimensional 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. Honeycombs have a strong anisotropy 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.

Figure 1. Honeycomb structure showing two-dimensional, prismatic nature (Attr: modified from work done by George William Herbert, Wikipedia)
honeycomb_bmwi3
Figure 2. Honeycomb design in use as part of a BMW i3 crash structure (Attr: adapted from youkeys, Wikipedia)

1.2 Design Implications
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. One such example is the crash structure in Figure 2 as well as a range of sandwich panels. Several automotive 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.

2. Open-Cell Foam

openfoam
Figure 3. Open cell foam unit cell, following Gibson & Ashby (1997)

2.1 Definition
Freeing up the prismatic requirement on the honeycomb brings us to a fully 3-dimensional open-cell foam design as shown in one representation of a unit cell in Figure 3. Typically, open-cell foams are bending-dominated, distinguishing them from stretch-dominated lattices, which are discussed in more detail in a following section on lattices.

2.2 Design Implications
Unlike the honeycomb, open cell foam designs are more useful when the environmental stimulus (stress, flow, heat) is not as predictable and unidirectional. The bending dominated mechanism of deformation make open-cell foams ideal for energy absorption – stretch dominated structures tend to be stiffer. As a result of this, applications that require energy absorption such as mattresses and crumple zones in complex structures. The interconnectivity of open-cell foams also makes them a candidate for applications requiring fluid flow through the structure.

Metal_Foam
Figure 4. SEM image of a metallic open-cell foam (Attr: SecretDisc, Wikipedia)
openfoam-deform
Figure 5. FEA simulation of open cell foam unit cell under compression, showing predominant mode of deformation is on account of bending

3. Closed-Cell Foam

closedfoam
Figure 6. Open cell foam unit cell representation [following Gibson and Ashby, 1997]
3.1 Definition
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 and 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 fragile packaging).

3.2 Design Implications
The 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 and manufacturing burden can be hard 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.

Closed_cell_metal_foam_with_large_cell_size
Figure 8. Closed cell Aluminum foam with very large cells [Shinko Wire Company, Attr: Curran2, Wikimedia Commons]

 4. Lattice

4.1 Definition
Lattices are in appearance very similar to open cell foams but differ in that lattice member deformation is stretch-dominated, as opposed to bending*. This is important since for the same material allocation, structures tend to be stiffer in tension and/or compression compared to bending – by contrast, bending dominated structures typically absorb more energy and are more compliant.

So the question is – when does an open cell foam become stretch dominated and therefore, a lattice? Fortunately, there is an app equation for that.

Maxwell’s Stability Criterion
Maxwell’s stability criterion involves the computation of a metric M for a lattice-like structure with b struts and j joints as follows:

In 2D structures: M = b – 2j + 3
In 3D structures:
M = b – 3j + 6

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.

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 (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), there 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.

Video: The octet truss is a classic stretch-dominated structure, with b = 36 struts, j = 14 joints and M = 0 [Attr. Lawrence Livermore National Labs]

4.2 Design Implications
Lattices are the most common cellular solid studied in AM – this is primarily on account of their strong structural performance in applications where high stiffness-to-weight ratio is desired (such as aerospace), or where stiffness modulation is important (such as in medical implants). However, it is important to realize that there are other cellular representations that have a range of other benefits that lattice designs cannot provide.

Conclusion: Why this matters

It is a fair question to ask why this matters – is this all just semantics? I would like to argue that the above classification is vital since it represents the first stage of selecting a unit cell for a particular function. Generally speaking, the following guidelines apply:

  • Honeycomb structures for predictable, unidirectional loading or flow
  • Open cell foams where energy absorption and compliance is important
  • Closed cell foams for fluid-filled and hydrostatic applications
  • Lattice structures where stiffness and resistance to bending is critical

Finally, another reason it is important to retain the bigger picture on all cellular solids is it ensures that the discussion of what we can do with AM and cellular solids includes all the possibilities and is not limited to only stiffness driven lattice designs.

Note: This blog post is part of a series on “Additive Manufacturing of Cellular Solids” that I am writing over the coming year, diving deep into the fundamentals of this exciting and fast evolving topic. To ensure you get each post (~2 a month) or to give me feedback for improvement, please connect with me on LinkedIn.

References

[1] Ashby, “Materials Selection in Mechanical Design,” Fourth Edition, 2011
[2] Gibson & Ashby, “Cellular Solids: Structure & Properties,” Second Edition, 1997
[3] Gibson, Ashby & Harley, “Cellular Materials in Nature & Medicine,” First Edition, 2010
[4] Ashby, Evans, Fleck, Gibson, Hutchinson, Wadley, “Metal Foams: A Design Guide,” First Edition, 2000
[5] Deshpande, Ashby, Fleck, “Foam Topology Bending versus Stretching Dominated Architectures,” Acta Materialia 49, 2001
[6] Deshpande, Fleck, Ashby, “Effective properties of the octet-truss lattice material,”  Journal of the Mechanics and Physics of Solids, 49, 2001

Notes

* We defer to reference [1] in distinguishing lattice structures as separate from foams – this is NOT the approach used in [2] and [3] where lattices are treated implicitly as a subset of open-cell foams. The distinction is useful from a structural perspective and as such is retained here.