## 4 Results and Discussion  
### Auxetic Design of Laminated Rectangular Plate with Membrane Lamination Parameters  
In this section, feasible Poisson's ratios are discussed in the design space of the four membrane lamination parameters. All the layers are made from T300/5208 carbon-epoxy composites [9]; the material properties of the lamina are given in Table 1.  
The Poisson's ratio contours are plotted in Figure 3 on the following four planes in the design space of the four membrane lamination parameters (the Poisson's ratios are determined with an interval of 0.5 for lamination parameters):  
\begin{table}
\begin{tabular}{c c c c c} \hline \(\mathbf{E_{11}}\) (GPa) & \(\mathbf{E_{22}}\) (GPa) & \(\mathbf{v_{12}}\) & \(\mathbf{G_{12}}\) (GPa) & \(\mathbf{\rho}\) (kg/m\({}^{3}\)) & \(\mathbf{t_{\text{ply}}}\) (mm) \\ \hline
181 & 10.3 & 0.28 & 4.47 & 1580 & 0.1 \\ \hline \end{tabular}
\end{table}
Table 1: Material properties of the lamina.  
Figure 3: _Cont._  
In each plane, the contours are shown in the feasible region of the membrane lamination parameters given by Equation (9). According to Figure 3a, the Poisson's ratios of the orthotropic laminate (\(V_{3}^{A}=V_{4}^{A}\)\(=0\)) vary from 0.015 to 1.450, that is, the orthotropic laminate with \(V_{3}^{A}=V_{4}^{A}\)\(=0\) do not have the opportunity to obtain NPR no matter the ply angle or orientation change. According to Equation (6), for a laminate composed of \(0^{\circ}\), \(90^{\circ}\), \(45^{\circ}\), and \(-45^{\circ}\) fiber angles, the membrane lamination parameters can be formulated as follows:  
\[V_{1}^{A}=\frac{n_{0}-n_{90}}{n} \tag{25}\]  
\[V_{2}^{A}=\frac{n_{0}+n_{90}-n_{45}-n_{-45}}{n} \tag{26}\]  
\[V_{3}^{A}=\frac{n_{45}-n_{-45}}{n} \tag{27}\]  
where \(n\) denotes the number of total plies, and \(n_{0}\), \(n_{90}\), \(n_{45}\), and \(n_{-45}\) denote the number of plies for \(0^{\circ}\), \(90^{\circ}\), \(45^{\circ}\), and \(-45^{\circ}\), respectively. For the symmetric and balanced laminate composed of \(0^{\circ}\), \(90^{\circ}\), \(45^{\circ}\), and \(-45^{\circ}\) fiber angles, \(V_{3}^{A}=V_{4}^{A}=0\), the Poisson's ratio of the laminate can be determined according to Figure 3a and Equations (25) and (26). Furthermore, the boundary of the feasible region for the orthotropic laminate composed of \(0^{\circ},90^{\circ},45^{\circ}\) and \(-45^{\circ}\) fiber angles in the design space of \(V_{1}^{A},V_{2}^{A}\) is also shown in Figure 3a by a red dashed line.  
Comparing the Poisson's ratio contours on the four planes, it can be seen that the negative Poisson's ratio are more sensitive to \(V_{3}^{A}\). \(V_{3}^{A}\) represents the unbalanced degree of a laminate. The NPR regions for the two-lamination parameter plane are also given in Figure 2. The minimum NPR is \(-0.360\) for \([V_{1}^{A},V_{2}^{A},V_{3}^{A},V_{4}^{A}]=[-0.35,0.25,-0.6,0.25]\) or \([-0.35,0.25,0.6,-0.25]\). The maximum NPR is \(1.450\) for \([V_{1}^{A},V_{2}^{A},V_{3}^{A},V_{4}^{A}]=[-0.65,0.15,0,-0.05]\) or \([-0.65,0.15,0,0.05]\). Compared with the other three NPR regions, the NPR points for \(V_{2}^{A},V_{3}^{A}\) occupy the smallest proportion of whole \(V_{2}^{A},V_{3}^{A}\) region, which reveals that the \(V_{2}^{A}-V_{3}^{A}\) relationship takes the lead for NPR combinations of lamination parameters. NPR points for two-lamination parameter plane is shown in Figure 4.