# Study of In-Plane Mechanical Properties of Novel Ellipse-Based Chiral Honeycomb Structure
## 1 Introduction
### Experimental Verification  
To further verify the correctness of the theoretical derivations and FEM simulations, a 3 \(\times\) 3 structured E-antitet sample with PLA as the base material was fabricated using a CREALITY 3D printer (Model: CR-6 max) with the fused deposition modeling (FDM) process. The parameters of the 3D printer setting are shown in Table 1 in Appendix A. The parameters of this sample are: \(l_{x}=30.00\) mm, \(l_{y}=30.00\) mm, \(a=10.00\) mm, \(b=5.00\) mm, \(t=3.00\) mm and \(d=150.00\) mm, which means \(\delta=2.00\), \(\alpha_{ax}=3.00\), \(\alpha_{ay}=3.00\), \(\alpha_{bx}=6.00\), \(\alpha_{by}=6.00\), \(\beta_{a}=0.30\), \(\beta_{b}=0.60\), \(\gamma_{a}=15.00\) and \(\gamma_{b}=30.00\). The \(3\times 3\) structured E-antitet samples were placed on the SANYU microcomputer electro-hydraulic servo pressure test machine (Model: HYE-300B) for in-plane [PERSON]'s ratio and [PERSON]'s modulus measurements (20 mm \(\times\) 20 mm \(\times\) 5 mm steel plates were placed on the top and bottom surfaces of the sample in contact with the pressure test machine), as shown in Figure 4. All in-plane compression experiments were performed at a constant displacement rate of 0.5 mm/min and no more than 3% strain. The horizontal displacements were measured by a SHISIWI digital multimeter (Model: CS-5311F) placed on one side. Finally, the theoretical derivations, FEM simulations and experimental results of [PERSON]'s ratio \(v_{yx}\) for the \(3\times 3\) structured E-antitet sample are \(-2.43\), \(-2.08\) and \(-2.48\pm 0.01\), respectively, which are shown in Table 2. The error between the experimental results and the theoretical derivations is only \(2.10\%\). The experimental verification of [PERSON]'s modulus differs from the theoretical derivations and FEM simulations due to the fact that the \(x\)-direction ligaments \(l_{x}\) on both sides of the edges of the \(3\times 3\) structured E-antitet sample are not constrained, which will lead to a decrease in [PERSON]'s modulus of the \(3\times 3\) structured E-antitet sample; therefore, we need to make modifications to Equations (17) and (18) in the analytical solutions obtained from the theoretical derivations of [PERSON]'s modulus of E-antitet:  
\[ \begin{array}{l}E_{x_{(m\times n)}}=n(m-1)\bigg{[}\frac{E_{x}\partial_{l}}{ 12l_{x}(a-t/2)^{2}l_{y}}+\frac{E_{x}\partial_{l_{x}}}{12l_{y}(a-t/2)^{2}l_{y} }\bigg{]}\\ \hskip 14.226378 pt+n\frac{E_{x}\partial_{l_{x}}}{12l_{y}(a-t/2)^{2}l_{y}}\end{array}  \tag{23}\] \begin{table}
\begin{tabular}{c c c c} \hline  & **Theory** & **FEM** & **Error** \\ \hline \(v_{xy}\) & \(-0.41\) & \(-0.35\) & \(14.29\%\) \\ \(v_{yx}\) & \(-2.43\) & \(-2.08\) & \(14.28\%\) \\ \(E_{x}\) (MPa) & \(13.12\) & \(11.65\) & \(11.20\%\) \\ \(E_{y}\) (MPa) & \(77.40\) & \(70.03\) & \(9.52\%\) \\ \hline \end{tabular}
\end{table} 
Table 1: Comparison of the results of theoretical derivations and FEM simulations. Figure 3: Stress and displacement cloud diagrams of E-antitet cell: (**a**) Stress and (**b**) displacement cloud of the cell under axial compression; (**c**) stress and (**d**) displacement cloud of the cell under transverse compression. \[E_{y_{(m\times x)}}=m(n-1)\bigg{[}\frac{E_{x}a^{3}l_{y}}{12l_{x}(b-l/2)^{2}l_{x}}+ \frac{E_{x}a^{3}l_{y}}{12l_{x}(b-l/2)^{2}l_{x}}\bigg{]} \tag{24}\] \[+m\frac{E_{x}a^{3}l_{y}}{12l_{x}(b-l/2)^{2}l_{x}}\]  
where \(m\) and \(n\), respectively, are the number of rows and columns of the E-antitet sample. We can see from Equations (23) and (24) that they are the same as Equations (17) and (18) when the number of rows and columns of E-antitet sample tends to infinity, which means:  
\[\lim_{\text{n,m}\rightarrow\infty}E_{x_{(m\times x)}}=E_{x} \tag{25}\]  
\[\lim_{\text{n,m}\rightarrow\infty}E_{y_{(m\times x)}}=E_{y} \tag{26}\]  
The theoretical derivations, FEM simulations and experimental results of [PERSON]'s modulus \(E_{y}\) for the 3 \(\times\) 3 structural E-antitet sample are also shown in Table 2. They are 68.68 MPa, 67.66 MPa and 58.16 \(\pm\) 0.1 MPa, respectively.