# Negative Poisson's Ratio Lattice Structure with Chiral and Re-Entrant Properties
## 3 Results and Discussions
### The Influence of Geometric Parameters on NPR  
The findings from Section 3.2 demonstrate a robust correlation between the numerical analysis results and the experimental findings. In order to conduct an analysis on the influence of geometric parameters on NPR, we chose the side length of the chiral cross (\(a\)), outer diameter of the re-entrant structure (\(b\)), and (\(\theta\)) as the variables, as depicted in Figure 1. However, due to the constant overall length of the individual cell, the value of b varied as \(a\) increased. Therefore, the \(a/b\) ratio, also known as the length ratio, and the  
parameters. The application of numerical analysis techniques facilitated the computation of negative values for Poisson's ratios. The results of the finite element numerical simulations for the NPRs of the window grill lattice structures are presented in Figure 12. The simulations were conducted for combinations of nine different length ratios (_a/b_) and ten deflection angles (\(\theta\)). Figure 12a depicts the variation in the NPR of a window grill lattice structure with an increasing deflection angle (\(\theta\)) and constant length ratio. As the deflection angle (\(\theta\)) increased, the influence of NPR tended to diminish. The observed decrease in strength could be ascribed to the constriction of the expansion area of the re-entrant structure when exposed to longitudinal static stretching at higher deflection angles. As a consequence, the application of compression mitigated the horizontal expansion phenomenon of the structure on a macroscopic scale. Consequently, as the deflection angle (\(\theta\)) increased, the presence of the re-entrant structure led to a weakening of the NPR effect. When the length ratio (_a/b_) fell within the range of 3.290 to 7.202 and the deflection angle (\(\theta\)) exceeded 30\({}^{\circ}\), the NPR effect was observed to be enhanced. This enhancement could be attributed to the presence of the largest geometric dimensions in the chiral structure and the smallest geometric dimensions in the re-entrant structure. As the deflection angle (\(\theta\)) increased, the re-entrant structure underwent further compression, resulting in a reduction in its geometric size. Consequently, the chiral structure played a crucial role in the non-planar re-entrant effect, thereby promoting the overall effect. It was observed that the NPR effect increased when the length ratio (_a/b_) was 0.343 and the deflection angles (\(\theta\)) ranged from 0\({}^{\circ}\) to 25\({}^{\circ}\). The observed outcome could be ascribed to the specific configuration of the chiral cross structure. To offer a comprehensive  
Figure 12: Structural design model of the evolution of the NPR at different deflection angles (**a**) and different length ratios (**b**), the 3D (a/b)-(\(\theta\))-(PR) plot (**c**), and a schematic diagram of the re-entrant chiral deflection angle (**d**). elucidation of this matter, this article delineates the concave chiral deflection angle, as illustrated in Figure 11d. When the length ratio was 0.343, the re-entrant chiral deflection angle approached 41\({}^{\circ}\), which was the value closest to 45\({}^{\circ}\). This phenomenon amplified the adverse [PERSON]'s effect on the structure. Additionally, the deflection angle did not impede the range of the concave chiral torsion angle, which spanned from 0 to 25\({}^{\circ}\). Figure 12b illustrates the variation of the NPR effect as the length ratio (_a/b_) increased while the deflection angle (\(\theta\)) was constant. As the length ratio (a/b) increased, the NPR effect exhibited a sharp increase followed by a sharp decrease and gradual increase. The influence of the change in the length ratio (_a/b_) on the NPR effect was found to be greater than that of the deflection angle. Through the finite element analysis results (see Figure 12), it was determined that the lattice structure of the ancient Chinese window grill demonstrated an NPR effect when exposed to static tension.