## 3 Results and Discussion
### Discussions
#### 3.3.2 Blastworthiness Analysis  
Figure 18 illustrates the deformation of four models with varying \(\rho_{r}\), from the lowest (left) to the highest (right), at 5 ms. The ideal model with maximum SEA shows significant compression in the central area, whereas the ideal model for minimum \(\delta_{p}\) exhibits minimal compression. As seen in Figure 18 and Table 8, the ASP with higher values of \(\rho_{r}\) tends to exhibit smaller values of \(\delta_{p}\), whereas those with lower \(\rho_{r}\) tend to have larger values of \(\delta_{p}\). This correlation between \(\rho_{r}\) and \(\delta_{p}\) is further supported by the SHAP values depicted in Figure 13, where higher values of design variables correspond to smaller \(\delta_{p}\). These visual representations illustrate that increasing the design variables or \(\rho_{r}\) results in a stiffer structure, thereby enhancing its resistance to blast impulses. However, higher values of \(\rho_{r}\) also lead to lower SEA values due to the increased mass of the ASP. As seen in Figure 18, global negative Poisson's ratio (NPR) behavior is not consistently observed in the ASP. During high-velocity blast impulses impacting the ASP center, the upper part of the auxetic core collapses rapidly, limiting full shrinkage. NPR behavior is more likely to occur in models with lower \(\rho_{r}\), where unit cells are less constrained by neighboring cells, allowing freer deformation. Another influencing factor in NPR behavior of the ASP is the corner angle or node radius in CH models. Increased corner angles tend to promote NPR behavior, while smaller node radii increase the likelihood of NPR behavior in the CH because slender ligament struts make it easier for nodes to rotate. Figure 19 shows the displacement vector of the ASP when subjected to air-blast loading. From Figures 18 and 19, REH and CH show the most dominant NPR behavior, followed by DAH, while SH very rarely exhibits NPR behavior. In general, auxetic structure configurations enhance blastworthiness as the material flows toward the impact zone (central area), as seen in Figure 19.  
Figure 17: Comparison of the Pareto front for all auxetic models. Based on the global sensitivity analysis results, as indicated by the SHAP values (Figures 12 and 13), the most influential design variable in ASP under air-blast loading is the thickness (\(t\)). This finding aligns well with the analytical solutions for dynamic crushing stress or plateau stress of auxetic structures, as discussed in studies by [66; 67]. According to the formula for dynamic crushing stress in structures such as REH and SH, the variable \(t\) exhibits a quadratic relationship, reflecting its role in the plastic bending moment: \(M_{p}=\sigma_{y}bt^{2}/4\), where \(\sigma_{y}\) is the yield strength and \(b\) is the strut length. Therefore, increasing \(t\) results in a quadratic increase in crushing stress. Additionally, the variable \(N_{y}\) significantly influences blastworthiness performance. Figure 14 illustrates that design points with higher \(N_{y}\) values tend to cluster near the Pareto front. This is attributed to the increased number of cells stacked at the central blast point, enhancing ASP resistance. In contrast, variables \(N_{z}\) and \(\theta/r\) show opposite effects compared to \(N_{y}\), with higher values typically located in the bottom-right region, away from the Pareto front. Although increasing \(N_{y}\) and \(\theta/r\) reduces \(\delta_{p}\), this reduction is relatively minor and primarily affects SEA reduction, as depicted in Figure 13.  
Figure 19: Displacement vectors of ASP: (**a**) REH, (**b**) DAH, (**c**) SH, and (**d**) CH. The red lines indicate the areas that commonly exhibit negative Poisson’s ratio (NPR) behavior except for the SH model. Figure 18: Deformed shapes of four model types for each ASP, (**a**) REH, (**b**) DAH, (**c**) SH, and (**d**) CH, at 5 ms.