Key Takeaway: The number of cells in a test specimen composed of cellular structures greatly influences its mechanical response, and the nature of this relationship depends on the unit cell design and the imposed loading condition

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Fig 1. A 3D-printed Voronoi honeycomb specimen under compression

Problem Statement:

The implementation of cellular (or lattice) materials in functional parts depends on our ability to predict their behavior under different loading and environmental conditions. This is typically studied by constructing test specimens filled with the unit cell under consideration and then subjecting that specimen to controlled tests (as shown in Figure 1) to determine an effective property that represents the behavior of that specific unit cell design, in combination with the specific Additive Manufacturing (AM) process and material used to construct it. A quick evaluation of testing the identical unit cell design with specimens of varying size quickly reveals a troubling size dependence – for example, under compression, test specimens with more cells produce a higher effective stiffness, all other variables being held constant. This represents a challenge to anyone who wants to characterize these materials – in this post, I attempt to answer two questions: Why is there a size effect? and What can we do about it? This is a short summary of a much more detailed peer-reviewed paper we published in the proceedings of the 2017 SFF Symposium.

Empirical Observations

First, let us see some evidence of this size effect. In figure 2, a strong increasing trend is seen for the effective modulus as a function of the number of cells in both axial and transverse directions for a hexagonal honeycomb structure. The effective modulus (calculated from measured load-displacement response normalized by the cross-sectional area and gauge length of the cellular specimen) is a homogenized property that is intended to represent the behavior of structures composed of these unit cells. However, the plot below clearly shows such a homogenized estimation is highly dependent on the number of cells needed to construct it. Similar effects have also been demonstrated in numerical and experimental studies conducted by Onck et al. (2001), Andrews et al. (2001)Tekoglu et al. (2011), and Yang (2016).

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Fig 2. Size dependence in a hexagonal honeycomb specimen – increasing cells in both axial and transverse directions has the effect of increase the calculated effective modulus

Origins

Now let us move on to the first question posed at the top of this post: Why is there a size effect with regard to characterizing cellular materials? The answer to this lies in the gap between a theoretical model that assumes that a given cell is completely surrounded by other cells in all directions ad infinitum, and the reality that experiments involve finite sized specimens that introduce two edge effects as shown in Figure 3: free surfaces that have cells that are “partially constrained,” and the interface with the testing equipment that generates a frictional interaction. Neither of these is explicitly accounted for in the calculations of a homogeneous effective property.

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Fig 3. The two primary sources of the size effect – free surfaces and frictional constraints

Generating Valid Data

Given this challenge, the question now is this: What can you do about this size effect? One option is to not use a homogenization approach at all – in other words, do not seek to use such testing procedures to extract any meaningful data. Alternatives to this approach would be to use bulk properties (if it can be shown that they are representative) or model the cellular material as an assemblage of beams and junctions – which are then characterized independent of the structure they constitute. However, the reality is that homogenization approaches have a powerful advantage in that they are far less expensive computationally to implement in analysis – as a result it is important that a method be developed to extract meaningful data that is ideally size-independent or at a minimum mitigates the effect of this dependence.

Brezny and Green (1990) proposed a method to compensate for the size effect by assuming that the material in the middle of the specimen equals the bulk value, but that the outer edges have a reduced modulus. It is more common however, to study this size effect to identify the size at which the property of interest (such as modulus or strength) stabilizes – and use this size for subsequent material modeling. Towards this, Onck et al. (2001), Andrews et al. (2001) and Tekoglu et al. (2011) have all plotted measured properties as a function of an L/d or H/d ratio, where L and H represent the length (or width) and height of the specimens, and d represents an average unit cell size.

An alternative way of representing the size effect, as we did in Le et al. (2017), is to count the number of constrained and unconstrained cells in the specimen, each term defined in Figure 4. This allows for the definition of a “percentage of constrained cells” in the specimen, which has the effect of combining the L/d and H/d terms into one value.

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Fig 4. Defining cells as being either partially or fully constrained

In compression tests conducted on over two dozen different sizes of hexagonal, square and triangular honeycombs each (such as shown in Figure 5 for the square honeycomb), we extracted a material modulus by resolving the globally applied load and measured displacement at the level of each individual beam and then using beam theory to develop a relationship between this applied load and measured displacement.

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Fig 5. Typical load-displacement response for a square honeycomb under compression

For the square honeycomb, the relationship between this material modulus and the percentage of constrained cells is shown in Figure 6. Using a percentage of constrained cells as a criterion, it appears that the material modulus approaches saturation beyond 70%. The letters A-E correspond to groupings of specimens with the number of cells increasing in either the axial or transverse cells: Group A contained all the specimens that had only 4 axial cells (but an increasing number of transverse cells), while group B contained all the specimens that had 4  transverse cells (but an increasing number of axial cells). Group C contained all of the specimens that had 7 axial cells, and group D contained all the specimens that had 7 transverse cells. Note how the two groups bifurcate – addition of more cells in different directions do not follow the same trends (hence the need to treat L/d and H/d separately in the previously reported studies). This bifurcation of modulii converges on a smaller range of values beyond the 70% constrained condition. These trends are also observed for hexagonal and triangular honeycombs with slight differences.

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Fig 6. Predicted material modulus as a function of the percentage of constrained cells (material modulus is NOT the same as an effective modulus, though the above trend holds for either)

Whether one characterizes these modulii (or any other property) on the basis of the L/d or H/d ratio, or as we propose, using the percentage of constrained cells as a metric, the fact remains that such a study is necessary to ensure the measured quantity is an accurate representation of the fundamental behavior of the unit cell by doing a size effect study specific to the unit cell of interest.

Practical Implications

So what does this all mean in practice? Quite simply, if you are a designer or an analyst relying on experimental data for modeling cellular materials – be they honeycombs or foams, ensure that the data you are using is coming from specimens that are large enough to where edge effects can be ignored. Keep in mind that the size at which this effect can be ignored varies both by the design of the unit cell and the nature of the loading conditions (compression, shear or bending). This is similar to doing a mesh refinement study in Finite Element Analysis to show that you have arrived at a solution that is, for all practical purposes, mesh-insensitive – there is no single element size that works for all geometries and conditions.

If you are studying these materials experimentally, consider alternative ways of building up these models that do not involve homogenization – in addition to introducing the size-dependence discussed here – which itself is unit cell design and loading condition dependent, homogenization models are themselves only good enough for the specific unit cell they represent. In a future post I will discuss this point in more detail and review some alternative methods to homogenization techniques.

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Thank you for reading! For comments on this post, or if you’d like to collaborate on similar ideas, please contact me.

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