## 2 Tailoring Stiffness via Cellular Materials
In general, 2D cellular structures used in airfoil morphing applications showed lower levels of in-plane stiffness than their parent material, and higher levels of out-of-plane stiffness than fully dense elements of comparable mass, as well as mass reduction percentages above 90% and higher strain capabilities [120]. In airfoil cores, auxetic cellular structures are preferred due to their higher in-plane flexibility which causes lower stress in cell walls than other types of cellular cores [119]. On the other hand, airfoil-skin morphing commonly uses sandwiched skins with cellular structure cores and thin, high-strain-capable face sheets that provide a smooth aerodynamic surfaces. However, this sandwich configurations also involve dealing with the boundary effects and Poisson's contractions or expansions. Therefore, zero-Poisson cellular structures are preferred to deal with this restriction in unidirectional morphing applications such as airfoil-skin morphing. Tables 1 and 2 present the principal works of airfoil-core and airfoil-skin morphing applications using 2D Cellular Materials. Works in these tables are categorized by author and cellular structure used. Stiffness reductions are presented by in-plane Young's modulus ratios, \(E_{cx}/E\) and \(E_{cy}/E\), for airfoil-core morphing (see Table 1), and by main-direction Young's modulus ratio, \(E_{cx}/E\), and shear-stiffness ratio, \(G_{c}/E\), for airfoil-skin morphing (see Table 2), where \(E_{c}\) and \(G_{c}\) are the Young and shear moduli of the Cellular Material, while \(E\) is the Young's modulus of the parent material. Additionally, the percentage of weight reduction is reported in Table 1. Tables 1 and 2 are presented to show the potential to perform high reductions of the stiffness levels in flexure elements through the substitution of fully dense solids with cellular patterns. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Author**} & \multicolumn{2}{c}{**Cellular**} & \multicolumn{2}{c}{**Young’s Modulus Ratios**} & \multicolumn{2}{c}{**Weight Reduction**} \\ \cline{2-5}  & **Structure** & \(\left(E_{cx}/E\right)\times 10^{5}\) & \(\left(E_{cy}/E\right)\times 10^{5}\) & **(\%)** \\ \hline [PERSON] et al. [116] & A—Hex. Chiral & 1.18–0.60 & 1.18–0.60 & 90–95 \\ [PERSON] et al. [117] & A—Re-entrant & 2.51 & 1.35 & – \\ [PERSON] et al. [118] & P—Cross-shaped & 0.25 & 0.13 & – \\ [PERSON] et al. [119] & A—Hex. Chiral & 11.58 & 11.58 & 90 \\  & P—Hexagonal & 69.32 & 69.32 & 94 \\  & A—Re-entrant & 427.19 & 427.19 & 81 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Stiffness reduction via Cellular Materials: Airfoil-core morphing. Principal works found in literature by author and cellular structure used (P: positive Poisson, Z: zero-Poisson, A: auxetic). [PERSON]’s modulus ratios \(E_{c}/E\) between the obtained by the application of Cellular Material, \(E_{c}\), and that of the parent material \(E\) for in-plane \(E_{cx}/E\) and \(E_{cy}/E\) stiffness, and percentage of weight reduction. “\(-\)” indicates that there is no sufficient information reported to obtain that parameter. The percentage of weight reduction was obtained based on the reported relative density \(\bar{\rho}\) of the cellular structure. Figure 5: (**a**) Complete schematic overview of airfoil morphing and main categories of 2D cellular structures, and (**b**) unit-cell of the zero-Poisson’s ratio cellular material termed as”Fish cell” (figure adapted from in [125]). Other important application of the compliance behavior of Cellular Materials is in energy absorption. Cellular Materials are suitable for energy absorption applications as their particular structures permit high levels of elastic deflections without permanent deformations or high values of stress, allowing the material to spring back to its original configuration after the load is released.