# Four Questions in Cellular Material Design

## Abstract

**:**

## 1. Introduction

- What is (are) the optimum unit cell(s)?
- How should the size of the cells vary spatially?
- What are the optimal cell parameters?
- How best should the cells be integrated with the larger form?

## 2. Question 1: What is the Optimum Unit Cell?

## 3. Question 2: How Should the Size of the Cells Vary Spatially?

## 4. Question 3: What Are the Optimal Cell Parameters?

_{s}, where ρ* is the density of the cellular material, and ρ

_{s}the density of the material of which the cellular structures are made. The relative density can then be calculated from the geometry of the shape, and for beam-based structures, it is typically some function of the ratio of the thickness of the member (edge or wall) to its length (t/l). For honeycombs and foams, for example, these relationships typically take the following forms, where C

_{1}, C

_{2}, and C

_{3}are constants [1]:

_{s}are the effective modulus of the cellular material and of the bulk solid, respectively [1]:

## 5. Question 4: How Best Should the Cells Be Integrated with the Larger Form?

## 6. Discussion: Research Opportunities

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Classification of cellular materials unit cell design based on a three level decision-making process first proposed in reference [4].

**Figure 2.**Four examples of how the three design decisions in Figure 1 translate into the realization of cellular shape designs.

**Figure 3.**Cells can be distributed in a prescriptive manner, (

**a**) shown for a periodic lattice, or (

**b**) by using a function, shown for a stochastic lattice.

**Figure 4.**Gradients occur commonly in natural cellular materials, as shown in (

**a**) sections of bamboo, (

**b**) a pine leaf, and (

**c**) implemented in design software.

**Figure 5.**The hexagonal honeycomb can be described in terms of the lengths of its walls (l), their thicknesses (t), and further, the radius at the corner (r).

**Figure 6.**Different cross-sections for the beams that constitute the lattice: (

**a**) circular section, (

**b**) square section, and (

**c**) teardrop shape to aid in self-supporting overhangs.

**Figure 7.**Three methods of prescribing cell thickness: (

**a**) global prescription of a thickness value, (

**b**) thickness specified in terms of a function, and (

**c**) thickness optimized by the solver in response to local stresses (screen captures using nTopology software [9]).

**Figure 8.**(

**a**) Polistes wasp’s nest (made of paper), (

**b**) bee’s nest (made of beeswax) and (

**c**) termite nest (made of mud) (Attr: P. Asman and J. Lenoble, creative commons 2.0 Generic (CC BY 2.0)).

**Figure 9.**High magnification images of cells comprising a radiolaria (

**left**, attr: Hannes Grobe, Wikimedia Commons) and honeycomb (

**right**). Clearly visible are nuances such as the corner radius and variations in the beam thicknesses.

**Figure 10.**Uniform lattice infill: (

**a**) basic concept, (

**b**) uniform infill with every unit cell completely represented inside the boundary, (

**c**) aligning of the centroid of the cells with the boundary, and (

**d**) ensuring the entire structure is completely filled with cells.

**Figure 11.**(

**a**) Basic conformal lattice infill, and (

**b**) wrapping cells such that they conform to the boundary.

**Figure 12.**Multi-functional wing concept made out of a cellular material that is both locally and globally optimized for structural and thermal requirements, adapted from reference [6].

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**MDPI and ACS Style**

Bhate, D.
Four Questions in Cellular Material Design. *Materials* **2019**, *12*, 1060.
https://doi.org/10.3390/ma12071060

**AMA Style**

Bhate D.
Four Questions in Cellular Material Design. *Materials*. 2019; 12(7):1060.
https://doi.org/10.3390/ma12071060

**Chicago/Turabian Style**

Bhate, Dhruv.
2019. "Four Questions in Cellular Material Design" *Materials* 12, no. 7: 1060.
https://doi.org/10.3390/ma12071060