## 3 Results
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Unit Cell** & \multicolumn{2}{c}{**Relative Elastic Modulus, \(EEs\)**} \\ \cline{2-3}  & **Euler-Bernoulli Theory** & **Timoshenko Theory** \\ \hline BCC & \(\left(\frac{\Delta t^{3}}{\beta}+\frac{1}{2\pi}\right)\) & \(\frac{\Delta t^{3}}{\beta}\) \\  & \(\frac{1}{2\beta}\) & \(\frac{1}{2\beta}\) \\ \hline Diamond & \(\frac{1}{\beta}(1)^{\frac{3}{3}}\) & \(\frac{1}{2\beta}\) \\  & \(\frac{1}{2\beta}(1)^{\frac{3}{3}}\) & \(\frac{1}{2\beta}\) \\ \hline Heagonal packing & \(\frac{1}{2}(1)^{\frac{3}{3}}\Big{(}1+\frac{1}{2\beta}(1)^{\frac{3}{3}}\Big{)}\) & \(\frac{1}{2\beta}\) \\  & \(\frac{1}{2\beta}(1)^{\frac{3}{3}}\Big{(}1+\frac{1}{2\beta}(1)^{\frac{3}{3}} \Big{)}\) & \(\frac{1}{2\beta}\) \\ \hline Rhombicuboctahedron & \(\frac{4\pi(1)^{2}}{3(1+\sqrt{2})}\) & \(\frac{4\pi(1)^{2}}{3(1+\sqrt{2})}\) \\  & \(\frac{1}{2\beta}(1)^{\frac{3}{3}}\Big{(}1+\sqrt{2})\) & \(\frac{4\pi(1)^{2}}{3(1+\sqrt{2})}\) \\  & \(\frac{1}{2\beta}(1)^{\frac{3}{3}}\Big{(}1+\sqrt{2})\) & \(\frac{1}{2\beta}\) \\ \hline Truncated cube & \(\frac{2\pi}{\sqrt{2}(1)^{\frac{3}{3}}\beta(1)^{\frac{3}{3}}}\) & \(\frac{1}{2\beta(1)^{\frac{3}{3}}}\) \\  & \(\frac{1}{2\beta}\) \\ \hline Truncated octahedron & \(\frac{\Delta t^{3}}{\beta(1+\frac{1}{2\beta})}\) & \(\frac{1}{\sqrt{2}\beta(1)^{\frac{3}{3}}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Normalized elastic modulus relationships based on Euler–Bernoulli and Timoshenko beam theories for different unit cell types (S = \(\frac{4\pi}{T}\), \(\mathbf{T}=\frac{1}{4\pi^{2}},\mathbf{T}=\frac{1}{4\pi^{2}+4\pi^{2}},\mathbf{V }=\frac{1}{4\pi^{2}+4\pi^{2}}\), and \(\mathbf{W}=\frac{5\pi^{2}}{T}\)). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Unit Cell** & \multicolumn{2}{c}{**Poisson’s Ratio,**} \\ \cline{2-3}  & **Euler-Bernoulli Theory** & **Timoshenko Theory** \\ \hline BCC & \(\frac{-\frac{3}{2}+\frac{\mu^{2}}{2}}{\frac{3}{2}+\frac{\mu^{2}}{2}}\) & \(\frac{-\frac{1}{2}+\frac{1}{2}}{1+\frac{1}{2}}\) \\ \hline Diamond & \(\frac{1-3(1)^{2}}{2-3(1)^{2}}\) & \(\frac{-1+\frac{1}{2}}{1+\frac{1}{2}}\) \\ \hline Hexagonal packing & \(\frac{-\frac{1}{2}+\frac{\mu}{2}(1)^{2}}{\frac{3}{2}+\frac{\mu}{2}(1)^{2}}\) & \(\frac{\frac{1}{2}-\frac{1}{2}}{5+\frac{1}{2}}\) \\ \hline Rhombicuboctahedron & \(\frac{1}{3}\left[\frac{8-12(1)^{2}-36(1)^{2}+\frac{\mu}{2}(1)^{2}+\frac{\mu} {2}(1)^{2}+(1)^{2}+(1)^{2}+(1)^{2}+(1)^{2}+(1)^{2}+(1)^{2}+(1)^{2})^{2}}\right]\) & \(\frac{5(5-7(4)(85+185+147)}{18795+3897+52087+23897+23897+3897+3897+3897+3897+3897+ 3897+3897+3897+3897+3897+3897+3897+3897+3897+3897+3897+3897}\) \\ \hline Truncated cube & \(\frac{1-3(1)^{2}}{5+\frac{1}{2}}\) & \(\frac{1-\frac{1}{2}}{5+\frac{1}{2}}\) \\ \hline Truncated octahedron & \(0.5\frac{4\frac{C}{2-182}}{\frac{3}{2}+\frac{1}{2}}\) & \(\frac{1}{2}\frac{5-\frac{1}{2}}{5+\frac{1}{2}}\) \\ \hline \end{tabular}
\end{table} 
Table 3: [PERSON]’s ratio relationships based on Euler-Bernoulli and Timoshenko beam theories for different unit cell types (\(S=\frac{A_{P}}{P}\), \(T=\frac{1}{\frac{1}{\frac{1}{\frac{1}{\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1} {1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1} {1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+{1}}{1}{1}{1}{1}{1}{1}{1}{1}{1}{1}{1}{1}}{1}}{1}}{1}{1}}{1}{}{1}}{1}{}{1}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{ }}}}}}\), and \(W=\frac{G_{P}}{P}\)). \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Unit Cell**} & \multicolumn{2}{c}{**Relative Yield Stress, \(\frac{\sigma_{k}}{\sigma_{m}}\)**} \\ \cline{2-3}  & **Euler-Bernoulli Theory** & **Timoshenko Theory** \\ \hline BCC & \(\frac{1}{\frac{1}{\mu\alpha^{2}}(1)^{\frac{1}{2}+\frac{2}{2}}(1)^{\  
Figure 6: Comparison of analytical (Euler–Bernoulli and [PERSON]) and numerical values of normalized elastic modulus for different unit cell types: (**a**) BCC, (**b**) diamond, (**c**) hexagonal packing, (**d**) rhombicuboctahedron, (**e**) truncated cube, and (**f**) truncated octahedron.