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

Cellular Design Strategies in Nature: A Classification

What types of cellular designs do we find in nature?

Cellular structures are an important area of research in Additive Manufacturing (AM), including work we are doing here at PADT. As I described in a previous blog post, the research landscape can be broadly classified into four categories: application, design, modeling and manufacturing. In the context of design, most of the work today is primarily driven by software that represent complex cellular structures efficiently as well as analysis tools that enable optimization of these structures in response to environmental conditions and some desired objective. In most of these software, the designer is given a choice of selecting a specific unit cell to construct the entity being designed. However, it is not always apparent what the best unit cell choice is, and this is where I think a biomimetic approach can add much value. As with most biomimetic approaches, the first step is to frame a question and observe nature as a student. And the first question I asked is the one described at the start of this post: what types of cellular designs do we find in the natural world around us? In this post, I summarize my findings.

Design Strategies

In a previous post, I classified cellular structures into 4 categories. However, this only addressed “volumetric” structures where the objective of the cellular structure is to fill three-dimensional space. Since then, I have decided to frame things a bit differently based on my studies of cellular structures in nature and the mechanics around these structures. First is the need to allow for the discretization of surfaces as well: nature does this often (animal armor or the wings of a dragonfly, for example). Secondly, a simple but important distinction from a modeling standpoint is whether the cellular structure in question uses beam- or shell-type elements in its construction (or a combination of the two). This has led me to expand my 4 categories into 6, which I now present in Figure 1 below.

Figure 1. Classification of cellular structures in nature: Volumetric – Beam: Honeycomb in bee construction (Richard Bartz, Munich Makro Freak & Beemaster Hubert Seibring), Lattice structure in the Venus flower basket sea sponge (Neon); Volumetric – Shell: Foam structure in douglas fir wood (U.S. National Archives and Records Administration), Periodic Surface similar to what is seen in sea urchin skeletal plates (Anders Sandberg); Surface: Tessellation on glypotodon shell (Author’s image), Scales on a pangolin (Red Rocket Photography for The Children’s Museum of Indianapolis)

Setting aside the “why” of these structures for a future post, here I wish to only present these 6 strategies from a structural design standpoint.

  1. Volumetric – Beam: These are cellular structures that fill space predominantly with beam-like elements. Two sub-categories may be further defined:
    • Honeycomb: Honeycombs are prismatic, 2-dimensional cellular designs extruded in the 3rd dimension, like the well-known hexagonal honeycomb shown in Fig 1. All cross-sections through the 3rd dimension are thus identical. Though the hexagonal honeycomb is most well known, the term applies to all designs that have this prismatic property, including square and triangular honeycombs.
    • Lattice and Open Cell Foam: Freeing up the prismatic requirement on the honeycomb brings us to a fully 3-dimensional lattice or open-cell foam. Lattice designs tend to embody higher stiffness levels while open cell foams enable energy absorption, which is why these may be further separated, as I have argued before. Nature tends to employ both strategies at different levels. One example of a predominantly lattice based strategy is the Venus flower basket sea sponge shown in Fig 1, trabecular bone is another example.
  2. Volumetric – Shell:
    • Closed Cell Foam: Closed cell foams are open-cell foams with enclosed cells. This typically involves a membrane like structure that may be of varying thickness from the strut-like structures. Plant sections often reveal a closed cell foam, such as the douglas fir wood structure shown in Fig 1.
    • Periodic Surface: Periodic surfaces are fascinating mathematical structures that often have multiple orders of symmetry similar to crystalline groups (but on a macro-scale) that make them strong candidates for design of stiff engineering structures and for packing high surface areas in a given volume while promoting flow or exchange. In nature, these are less commonly observed, but seen for example in sea urchin skeletal plates.
  3. Surface:
    • Tessellation: Tessellation describes covering a surface with non-overlapping cells (as we do with tiles on a floor). Examples of tessellation in nature include the armored shells of several animals including the extinct glyptodon shown in Fig 1 and the pineapple and turtle shell shown in Fig 2 below.
    • Overlapping Surface: Overlapping surfaces are a variation on tessellation where the cells are allowed to overlap (as we do with tiles on a roof). The most obvious example of this in nature is scales – including those of the pangolin shown in Fig 1.
Figure 2. Tessellation design strategies on a pineapple and the map Turtle shell [Scans conducted at PADT by Ademola Falade]

What about Function then?

This separation into 6 categories is driven from a designer’s and an analyst’s perspective – designers tend to think in volumes and surfaces and the analyst investigates how these are modeled (beam- and shell-elements are at the first level of classification used here). However, this is not sufficient since it ignores the function of the cellular design, which both designer and analyst need to also consider. In the case of tessellation on the skin of an alligator for example as shown in Fig 3, was it selected for protection, easy of motion or for controlling temperature and fluid loss?

Figure 3. Varied tessellation on an alligator conceals a range of possible functions (CCO public domain)

In a future post, I will attempt to develop an approach to classifying cellular structures that derives not from its structure or mechanics as I have here, but from its function, with the ultimate goal of attempting to reconcile the two approaches. This is not a trivial undertaking since it involves de-confounding multiple functional requirements, accounting for growth (nature’s “design for manufacturing”) and unwrapping what is often termed as “evolutionary baggage,” where the optimum solution may have been sidestepped by natural selection in favor of other, more pressing needs. Despite these challenges, I believe some first-order themes can be discerned that can in turn be of use to the designer in selecting a particular design strategy for a specific application.

References

This is by no means the first attempt at a classification of cellular structures in nature and while the specific 6 part separation proposed in this post was developed by me, it combines ideas from a lot of previous work, and three of the best that I strongly recommend as further reading on this subject are listed below.

  1. Gibson, Ashby, Harley (2010), Cellular Materials in Nature and Medicine, Cambridge University Press; 1st edition
  2. Naleway, Porter, McKittrick, Meyers (2015), Structural Design Elements in Biological Materials: Application to Bioinspiration. Advanced Materials, 27(37), 5455-5476
  3. Pearce (1980), Structure in Nature is a Strategy for Design, The MIT Press; Reprint edition

As always, I welcome all inputs and comments – if you have an example that does not fit into any of the 6 categories mentioned above, please let me know by messaging me on LinkedIn and I shall include it in the discussion with due credit. Thanks!

Modeling 3D Printed Cellular Structures: Approaches

How can the mechanical behavior of cellular structures (honeycombs, foams and lattices) be modeled?

This is the second in a two-part post on the modeling aspects of 3D printed cellular structures. If you haven’t already, please read the first part here, where I detail the challenges associated with modeling 3D printed cellular structures.

The literature on the 3D printing of cellular structures is vast, and growing. While the majority of the focus in this field is on the design and process aspects, there is a significant body of work on characterizing behavior for the purposes of developing analytical material models. I have found that these approaches fall into 3 different categories depending on the level of discretization at which the property is modeled: at the level of each material point, or at the level of the connecting member or finally, at the level of the cell. At the end of this article I have compiled some of the best references I could find for each of the 3 broad approaches.

1. Continuum Modeling

The most straightforward approach is to use bulk material properties to represent what is happening to the material at the cellular level [1-4]. This approach does away with the need for any cellular level characterization and in so doing, we do not have to worry about size or contact effects described in the previous post that are artifacts of having to characterize behavior at the cellular level. However, the assumption that the connecting struts/walls in a cellular structure behave the same way the bulk material does can particularly be erroneous for AM processes that can introduce significant size specific behavior and large anisotropy. It is important to keep in mind that factors that may not be significant at a bulk level (such as surface roughness, local microstructure or dimensional tolerances) can be very significant when the connecting member is under 1 mm thick, as is often the case.

The level of error introduced by a continuum assumption is likely to vary by process: processes like Fused Deposition Modeling (FDM) are already strongly anisotropic with highly geometry-specific meso-structures and an assumption like this will generate large errors as shown in Figure 1. On the other hand, it is possible that better results may be had for powder based fusion processes used for metal alloys, especially when the connecting members are large enough and the key property being solved for is mechanical stiffness (as opposed to fracture toughness or fatigue life).

Fig 1. Load-displacement curves for ULTEM-9085 Honeycomb structures made with different FDM toolpath strategies

2. Cell Level Homogenization

The most common approach in the literature is the use of homogenization – representing the effective property of the cellular structure without regard to the cellular geometry itself. This approach has significantly lower computational expense associated with its implementation. Additionally, it is relatively straightforward to develop a model by fitting a power law to experimental data [5-8] as shown in the equation below, relating the effective modulus E* to the bulk material property Es and their respective densities (ρ and ρs), by solving for the constants C and n.

homogenizationeqn

While a homogenization approach is useful in generating comparative, qualitative data, it has some difficulties in being used as a reliable material model in analysis & simulation. This is first and foremost since the majority of the experiments do not consider size and contact effects. Secondly, even if these were considered, the homogenization of the cells only works for the specific cell in question (e.g. octet truss or hexagonal honeycomb) – so every new cell type needs to be re-characterized. Finally, the homogenization of these cells can lose insight into how structures behave in the transition region between different volume fractions, even if each cell type is calibrated at a range of volume fractions – this is likely to be exacerbated for failure modeling.

3. Member Modeling

The third approach involves describing behavior not at each material point or at the level of the cell, but at a level in-between: the connecting member (also referred to as strut or beam). This approach has been used by researchers [9-11] including us at PADT [12] by invoking beam theory to first describe what is happening at the level of the member and then use that information to build up to the level of the cells.

membermodeling
Fig 2. Member modeling approach: represent cellular structure as a collection of members, use beam theory for example, to describe the member’s behavior through analytical equations. Note: the homogenization equations essentially derive from this approach.

This approach, while promising, is beset with some challenges as well: it requires experimental characterization at the cellular level, which brings in the previously mentioned challenges. Additionally, from a computational standpoint, the validation of these models typically requires a modeling of the full cellular geometry, which can be prohibitively expensive. Finally, the theory involved in representing member level detail is more complex, makes assumptions of its own (e.g. modeling the “fixed” ends) and it is not proven adequately at this point if this is justified by a significant improvement in the model’s predictability compared to the above two approaches. This approach does have one significant promise: if we are able to accurately describe behavior at the level of a member, it is a first step towards a truly shape and size independent model that can bridge with ease between say, an octet truss and an auxetic structure, or different sizes of cells, as well as the transitions between them – thus enabling true freedom to the designer and analyst. It is for this reason that we are focusing on this approach.

Conclusion

Continuum models are easy to implement and for relatively isotropic processes and materials such as metal fusion, may be a good approximation of stiffness and deformation behavior. We know through our own experience that these models perform very poorly when the process is anisotropic (such as FDM), even when the bulk constitutive model incorporates the anisotropy.

Homogenization at the level of the cell is an intuitive improvement and the experimental insights gained are invaluable – comparison between cell type performances, or dependencies on member thickness & cell size etc. are worthy data points. However, caution needs to be exercised when developing models from them for use in analysis (simulation), though the relative ease of their computational implementation is a very powerful argument for pursuing this line of work.

Finally, the member level approach, while beset with challenges of its own, is a promising direction forward since it attempts to address behavior at a level that incorporates process and geometric detail. The approach we have taken at PADT is in line with this approach, but specifically seeks to bridge the continuum and cell level models by using cellular structure response to extract a point-wise material property. Our preliminary work has shown promise for cells of similar sizes and ongoing work, funded by America Makes, is looking to expand this into a larger, non-empirical model that can span cell types. If this is an area of interest to you, please connect with me on LinkedIn for updates. If you have questions or comments, please email us at info@padtinc.com or drop me a message on LinkedIn.

References (by Approach)

Bulk Property Models

[1] C. Neff, N. Hopkinson, N.B. Crane, “Selective Laser Sintering of Diamond Lattice Structures: Experimental Results and FEA Model Comparison,” 2015 Solid Freeform Fabrication Symposium

[2] M. Jamshidinia, L. Wang, W. Tong, and R. Kovacevic. “The bio-compatible dental implant designed by using non-stochastic porosity produced by Electron Beam Melting®(EBM),” Journal of Materials Processing Technology214, no. 8 (2014): 1728-1739

[3] S. Park, D.W. Rosen, C.E. Duty, “Comparing Mechanical and Geometrical Properties of Lattice Structure Fabricated using Electron Beam Melting“, 2014 Solid Freeform Fabrication Symposium

[4] D.M. Correa, T. Klatt, S. Cortes, M. Haberman, D. Kovar, C. Seepersad, “Negative stiffness honeycombs for recoverable shock isolation,” Rapid Prototyping Journal, 2015, 21(2), pp.193-200.

Cell Homogenization Models

[5] C. Yan, L. Hao, A. Hussein, P. Young, and D. Raymont. “Advanced lightweight 316L stainless steel cellular lattice structures fabricated via selective laser melting,” Materials & Design 55 (2014): 533-541.

[6] S. Didam, B. Eidel, A. Ohrndorf, H.‐J. Christ. “Mechanical Analysis of Metallic SLM‐Lattices on Small Scales: Finite Element Simulations versus Experiments,” PAMM 15.1 (2015): 189-190.

[7] P. Zhang, J. Toman, Y. Yu, E. Biyikli, M. Kirca, M. Chmielus, and A.C. To. “Efficient design-optimization of variable-density hexagonal cellular structure by additive manufacturing: theory and validation,” Journal of Manufacturing Science and Engineering 137, no. 2 (2015): 021004.

[8] M. Mazur, M. Leary, S. Sun, M. Vcelka, D. Shidid, M. Brandt. “Deformation and failure behaviour of Ti-6Al-4V lattice structures manufactured by selective laser melting (SLM),” The International Journal of Advanced Manufacturing Technology 84.5 (2016): 1391-1411.

Beam Theory Models

[9] R. Gümrük, R.A.W. Mines, “Compressive behaviour of stainless steel micro-lattice structures,” International Journal of Mechanical Sciences 68 (2013): 125-139

[10] S. Ahmadi, G. Campoli, S. Amin Yavari, B. Sajadi, R. Wauthle, J. Schrooten, H. Weinans, A. Zadpoor, A. (2014), “Mechanical behavior of regular open-cell porous biomaterials made of diamond lattice unit cells,” Journal of the Mechanical Behavior of Biomedical Materials, 34, 106-115.

[11] S. Zhang, S. Dilip, L. Yang, H. Miyanji, B. Stucker, “Property Evaluation of Metal Cellular Strut Structures via Powder Bed Fusion AM,” 2015 Solid Freeform Fabrication Symposium

[12] D. Bhate, J. Van Soest, J. Reeher, D. Patel, D. Gibson, J. Gerbasi, and M. Finfrock, “A Validated Methodology for Predicting the Mechanical Behavior of ULTEM-9085 Honeycomb Structures Manufactured by Fused Deposition Modeling,” Proceedings of the 26th Annual International Solid Freeform Fabrication, 2016, pp. 2095-2106

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.

Modeling 3D Printed Cellular Structures: Challenges

In this post, I discuss six challenges that make the modeling of 3D printed cellular structures (such as honeycombs and lattices) a non-trivial matter. In a following post, I will present how some of these problems have been addressed with different approaches.

At the outset, I need to clarify that by modeling I mean the analytical representation of material behavior, primarily for use in predictive analysis (simulation). Here are some reasons why this is a challenging endeavor for 3D printed cellular solids – some of these reasons are unique to 3D printing, others are a result of aspects that are specific to cellular solids, independent of how they are manufactured. I show examples with honeycombs since that is the majority of the work we have data for, but I expect that these ideas apply to foams and lattices as well, just with varying degrees of sensitivity.

1. Complex Geometry with Non-Uniform Local Conditions

I state the most well-appreciated challenge with cellular structures first: they are NOT fully-dense solid materials that have relatively predictable responses governed by straightforward analytical expressions. Consider a dogbone-shaped specimen of solid material under tension: it’s stress-strain response can be described fairly well using continuum expressions that do not account for geometrical features beyond the size of the dogbone (area and length for stress and strain computations respectively). However, as shown in Figure 1, such is not the case for cellular structures, where local stress and strain distributions are non-uniform. Further, they may have variable distributions of bending, stretching and shear in the connecting members that constitute the structure. So the first question becomes: how does one represent such complex geometry – both analytically and numerically?

non-uniform-strain
Fig 1. Honeycomb structure under compression showing non-uniform local elastic strains [Le & Bhate, under preparation]

2. Size Effects

A size effect is said to be significant when an observed behavior varies as a function of the size of the sample whose response is being characterized even after normalization (dividing force by area to get stress, for example). Here I limit myself to size effects that are purely a mathematical artifact of the cellular geometry itself, independent of the manufacturing process used to make them – in other words this effect would persist even if the material in the cellular structure was a mathematically precise, homogeneous and isotropic material.

It is common in the field of cellular structure modeling to extract an “effective” property – a property that represents a homogenized behavior without explicitly modeling the cellular detail. This is an elegant concept but introduces some practical challenges in implementation – inherent in the assumption is that this property, modulus for example, is equivalent to a continuum property valid at every material point. The reality is the extraction of this property is strongly dependent on the number of cells involved in the experimental characterization process. Consider experimental work done by us at PADT, and shown in Figure 2 below, where we varied both the number of axial and longitudinal cells (see inset for definition) when testing hexagonal honeycomb samples made of ULTEM-9085 with FDM. The predicted effective modulus increases with increasing number of cells in the axial direction, but reduces (at a lower rate) for increasing number of cells in the longitudinal direction.

This is a significant challenge and deserves a full form post to do justice (and is forthcoming), but the key to remember is that testing a particular cellular structure does not suffice in the extraction of effective properties. So the second question here becomes: what is the correct specimen design for characterizing cellular properties?

sizeeffect
Fig 2. Effective modulus under compression showing a strong dependence on the number of cells in the structure [Le & Bhate, under preparation]

3. Contact Effects

In the compression test shown in the inset in Figure 2, there is physical contact between the platen and the specimen that creates a local effect at the top and bottom that is different from the experience of the cells closer the center. This is tied to the size effect discussed above – if you have large enough cells in the axial direction, the contribution of this effect should reduce – but I have called it out as a separate effect here for two reasons: Firstly, it raises the question of how best to design the interface for the specimen: should the top and bottom cells terminate in a flat plate, or should the cells extend to the surface of contact (the latter is the case in the above image). Secondly, it raises the question of how best to model the interface, especially if one is seeking to match simulation results to experimentally observed behavior. Both these ideas are shown in Figure 3 below. This also has implications for product design – how do we characterize and model the lattice-skin interface? As such, independent of addressing size effects, there is a need to account for contact behavior in characterization, modeling and analysis.

contact
Fig 3. Two (of many possible) contact conditions for cellular structure compression – both in terms of specimen design as well as in terms of the nature of contact specified in the simulation (frictionless vs frictional, for example)

4. Macrostructure Effects

Another consideration related to specimen design is demonstrated in an exaggerated manner in the slowed down video below, showing a specimen flying off the platens under compression – the point being that for certain dimensions of the specimen being characterized (typically very tall aspect ratios), deformation in the macrostructure can influence what is perceived as cellular behavior. In the video below, there is some induced bending on a macro-level.

5. Dimensional Errors

While all manufacturing processes introduce some error in dimensional tolerances, the error can have a very significant effect for cellular structures – a typical industrial 3D printing process has tolerances within 75 microns (0.003″) – cellular structures (micro-lattices in particular) very often are 250-750 microns in thickness, meaning the tolerances on dimensional error can be in the 10% and higher error range for thickness of these members. This was our finding when working with Fused Deposition Modeling (FDM), where on a 0.006″ thick wall we saw about a 10% larger true measurement when we scanned the samples optically, as shown in Figure 4. Such large errors in thickness can yield a significant error in measured behavior such as elastic modulus, which often goes by some power to the thickness, amplifying the error. This drives the need for some independent measurement of the manufactured cellular structure – made challenging itself by the need to penetrate the structure for internal measurements. X-ray scanning is a popular, if expensive approach. But the modeler than has the challenge of devising an average thickness for analytical calculations and furthermore, the challenge of representation of geometry in simulation software for efficient analysis.

Fig 4. (Clockwise from top left): FDM ULTEM 9085 honeycomb sample, optical scan image, 12-sample data showing a mean of 0.064″ against a designed value of 0.060″ – a 7% error in thickness

6. Mesostructural Effects

The layerwise nature of Additive Manufacturing introduces a set of challenges that are somewhat unique to 3D Printed parts. Chief among these is the resulting sensitivity to orientation, as shown for the laser-based powder bed fusion process in Figure 5 with standard materials and parameter sets. Overhang surfaces (unsupported) tend to have down-facing surfaces with different morphology compared to up-facing ones. In the context of cellular structures, this is likely to result in different thickness effects depending on direction measured.

Fig 5. 3D Printed Stainless Steel Honeycomb structures showing orientation dependent morphology [PADT, 2016]
For the FDM process, in addition to orientation, the toolpaths that effectively determine the internal meso-structure of the part (discussed in a previous blog post in greater detail) have a very strong influence on observed stiffness behavior, as shown in Figure 6. Thus orientation and process parameters are variables that need to be comprehended in the modeling of cellular structures – or set as constants for the range of applicability of the model parameters that are derived from a certain set of process conditions.

Figure
Fig 6. Effects of different toolpath selections in Fused Deposition Modeling (FDM) for honeycomb structure tensile testing  [Bhate et al., RAPID 2016]

Summary

Modeling cellular structures has the above mentioned challenges – most have practical implications in determining what is the correct specimen design – it is our mission over the next 18 months to address some of these challenges to a satisfactory level through an America Makes grant we have been awarded. While these ideas have been explored in other manufacturing contexts,  much remains to be done for the AM community, where cellular structures have a singular potential in application.

In future posts, I will discuss some of these challenges in detail and also discuss different approaches to modeling 3D printed cellular structures – they do not always address all the challenges here satisfactorily but each has its pros and cons. Until then, feel free to send us an email at info@padtinc.com citing this blog post, or connect with me on LinkedIn so you get notified whenever I write a post on this, or similar subjects in Additive Manufacturing (1-2 times/month).

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.

The Additive Manufacturing Cellular Solids Research Landscape

I am writing this post after visiting the 27th SFF Symposium, a 3-day Additive Manufacturing (AM) conference held annually at the University of Texas at Austin. The SFF Symposium stands apart from other 3D printing conferences held in the US (such as AMUG, RAPID and Inside3D) in the fact that about 90% of the attendees and presenters are from academia. This year had 339 talks in 8 concurrent tracks and 54 posters, with an estimated 470 attendees from 20 countries – an overall 50% increase over the past year.

As one would expect from a predominantly academic conference, the talks were deeper in their content and tracks were more specialized. The track I presented in (Lattice Structures) had a total of 15 talks – 300 minutes of lattice talk, which pretty much made the conference for me!

In this post, I wish to summarize the research landscape in AM cellular solids at a high level: this classification dawned on me as I was listening to the talks over two days and taking in all the different work going on across several universities. My attempt in this post is to wrap my arms around the big picture and show how all these elements are needed to make cellular solids a routine design feature in production AM parts.

Classification of Cellular Solids

First, I feel the need to clarify a technicality that bothered me a wee bit at the conference: I prefer the term “cellular solids” to “lattices” since it is more inclusive of honeycomb and all foam-like structures, following Gibson and Ashby’s 1997 seminal text of the same name. Lattices are generally associated with “open-cell foam” type structures only – but there is a lot of room for honeycomb structures and close-cell foams, each having different advantages and behaviors, which get excluded when we use the term “lattice”.

CellularSolids
Figure 1. Classification of Cellular Solids [Gibson & Ashby, 1997]

The AM Cellular Solids Research Landscape

The 15 papers at the symposium, and indeed all my prior literature reviews and conference visits, suggested to me that all of the work in this space falls into one or more of four categories shown in Figure 2. For each of the four categories (design, analysis, manufacturing & implementation), I have listed below the current list of capabilities (not comprehensive), many of which were discussed in the talks at SFF. Further down I list the current challenges from my point of view, based on what I have learned studying this area over the past year.

AMcellular
Figure 2. AM Cellular Solid Research Landscape

Over the coming weeks I plan to publish a post with more detail on each of the four areas above, summarizing the commercial and academic research that is ongoing (to the best of my knowledge) in each area. For now, I provide below a brief elaboration of each area and highlight some important research questions.

1. Representation (Design)

This deals with how we incorporate cellular structures into our designs for all downstream activities. This involves two aspects: the selection of the specific cellular design (honeycomb or octet truss, for example) and its implementation in the CAD framework. For the former, a key question is: what is the optimum unit cell to select relative to performance requirements, manufacturability and other constraints? The second set of challenges arises from the CAD implementation: how does one allow for rapid iteration with minimal computational expense, how do cellular structures cover the space and merge with the external skin geometry seamlessly?

2. Optimization (Analysis)

Having tools to incorporate cellular designs is not enough – the next question is how to arrange these structures for optimum performance relative to specified requirements? The two most significant challenges in this area are performing the analysis at reasonable computational expense and the development of material models that accurately represent behavior at the cellular structure level, which may be significantly different from the bulk.

3. Realization (Manufacturing)

Manufacturing cellular structures is non-trivial, primarily due to the small size of the connecting members (struts, walls). The dimensions required are often in the order of a few hundred microns and lower, which tends to push the capabilities of the AM equipment under consideration. Additionally, in most cases, the cellular structure needs to be self-supporting and specifically for powder bed fusion, must allow for removal of trapped powder after completion of the build. One way to address this is to develop a map that identifies acceptable sizes of both the connecting members and the pores they enclose. For this, we need robust ways of monitoring quality of AM cellular solids by using in-situ and Non-Destructive techniques to guard against voids and other defects.

4. Application (Implementation)

Cellular solids have a range of potential applications. The well established ones include increasing stiffness-to-weight ratios, energy absorption and thermal performance. More recent applications include improving bone integration for implants and modulating stiffness to match biological distributions of material (biomimicry), as well as a host of ideas involving meta-materials. The key questions here include how do we ensure long term reliability of cellular structures in their use condition? How do we accurately identify and validate these conditions? How do we monitor quality in the field? And how do we ensure the entire life cycle of the product is cost-effective?

So What?

I wrote this post for two reasons: I love to classify information and couldn’t help myself after 5 hours of hearing and thinking about this area. But secondly, I hope it helps give all of us working in this space context to engage and communicate more seamlessly and see how our own work fits in the bigger picture.

A lot of us have a singular passion for the overlapping zone of AM and cellular solids and I can imagine in a few years we may well have a conference, an online journal or a forum of some sort just dedicated to this field – in fact, I’d love to assess interest in such an effort or an equivalent collaborative exercise. If this idea resonates with you, please connect with me on LinkedIn and drop me a note, or send us an email (info@padtinc.com) and cite this blog post so it finds its way to me.

Additive Manufacturing – Back to the Future!

Paul Nigh's 'TeamTimeCar.com' Back to the Future DeLorean Time Machine
Production of Back to the Future began in 1984 – it was recently announced that the DeLorean is to go back in production with new cars rolling out in 2017

Most histories of Additive Manufacturing (3D printing) trace the origins of the technology back to Charles Hull’s 1984 patent, the same year production began on the first of the Back to the Future movies. Which is something of a shock when you see 3D printing dotting the Gartner Hype Cycle like it was invented in the post-Seinfeld era. But that is not what this post is about.

When I started working on Additive Manufacturing (AM), I was amazed at the number of times I was returning to text books and class notes I had used in graduate school a decade ago. This led me to reflect on how AM is helping bring back to the forefront disciplines that had somehow lost their cool factor – either by becoming part of the old normal, or because they contained ideas that were ahead of their time. I present three such areas of research that I state, with only some exaggeration, were waiting for AM to come along.

  • Topology Optimization: I remember many a design class where we would discuss topology optimization, look at fancy designs and end with a conversation that involved one of the more cynical students asking “All that’s fine, but how are you going to make that?”. Cue the elegant idea of building up a structure layer-by layer. AM is making it possible to manufacture parts with geometries that look like they came right out of a stress contour plot. And firms such as ANSYS, Autodesk and Altair, as well as universities and labs are all working to improve their capabilities at the intersection of topology optimization and additive manufacturing.
Topology optimization applied to the design of an automobile upper control-arm done with GENESIS Topology for ANSYS Mechanical (GTAM) from Vanderplaats Research & Development and ANSYS SpaceClaim
Topology optimization applied to the design of an automobile upper control-arm done with GENESIS Topology for ANSYS Mechanical (GTAM) from Vanderplaats Research & Development and ANSYS SpaceClaim
And we printed that!
And we printed that!
  • Lattice Structures: One of the first books I came across when I joined PADT was a copy of Cellular Solids by Lorna Gibson and M.F. Ashby. Prof. Gibson’s examples of these structures as they occur in nature demonstrate how they provide an economy of material usage for the task at hand. Traditionally, in engineering structures, cellular designs are limited to foams or consistent shapes like sandwich panels where the variation in cell geometry is limited – this is because manufacturing techniques do not normally lend themselves well to building complex, three dimensional structures like those found in nature. With AM technologies however, cell sizes and structures can be varied and densities modified depending on the design of the structure and the imposed loading conditions, making this an exciting area of research.

    Lattice specimens made with the Fused Deposition Modeling (FDM) process
    Lattice specimens made with the Fused Deposition Modeling (FDM) process
  • Metallurgy: As I read the preface to my “Metallurgy for the Non-Metallurgist” text book, I was surprised to note the author openly bemoan the decline of interest in metallurgy, and subsequently, fewer metallurgists in the field. And I guess it makes sense: materials science is today mostly concerned with much smaller scales than the classical metallurgist trained in. Well, lovers of columnar grain growth and precipitation hardening can now rejoice – metallurgy is at the very heart of AM technology today – most of the projected growth in AM is in metals. The science of powder metallurgy and the microstructure-property-process relationships of the metal AM technologies are vital building blocks to our understanding of metal 3D printing. Luckily for me, I happen to possess a book on powder metallurgy. And it too, is from 1984.
This book was printed in 1984, and is very relevant today
Published 1984