## 2 Materials and Methods  
### Geometry Description  
The configuration of the auxetic sandwich panel (ASP) is shown in Figure 1. The ASP consists of two face sheets, with a thickness of \(t_{f}\), and an auxetic core, which has the total dimensions of \(L\times W\times H\). The \(L\), \(W\), and \(H\) are the length, width, and height of the auxetic core in the \(x\), \(y\), and \(z\) directions, respectively. The auxetic core consists of repetitive 2D unit cells of auxetic geometry with the number of cells being \(N_{y}\) and \(N_{z}\) in the \(y\) and \(z\) directions, respectively. The auxetic core is oriented with its cells' longitudinal direction parallel to the two face sheets. As research by [PERSON] et al. [51]shows, this configuration yields better blastworthiness than when the cells' longitudinal direction is perpendicular to the face sheets. The size of the ASP is determined based on the floor size of the armored fighting vehicle (AFV) and the availability of additional space as a protective structure. Based on reference [52], the typical floor size of a small AFV is \(L\) = 1400 mm, \(W\) = 2500 mm, and \(H\) = 120 mm. The stand-off distance (SoD) from the blast charge position to the front face is 800 mm. However, to reduce the computational time, the half scale of ASP model was used, where \(L\) = 700 mm, \(W\) = 1250 mm, and \(H\) = 100 m with an SoD of 400 mm to compensate for the smaller size of ASP.  
In this study, four auxetic basic geometries were proposed as the core of the ASP, which are re-entrant honeycomb (REH), double-arrow honeycomb (DAH), star honeycomb (SH), and tetra-chiral honeycomb (CH). Figure 2 shows the unit cell configuration of each auxetic geometry. There are four independent geometric parameters which are defined in this study: the cell width \(w\), the cell height \(h\), the corner cell angle \(\theta\) and node radius \(r\) for CH, and the cell thickness \(t\). The variables \(w\) and \(h\) are proportional with \(N_{y}\) and \(N_{z}\), respectively, except for the variable \(h\) of DAH, which depends on \(N_{z}\) and \(\theta\). Other dependent geometric parameters include inner cell spacing \(d\) (\(d_{1}\) and \(d_{2}\) for SH), strut length \(l_{1}\) and \(l_{2}\), and inner corner angle \(\theta^{\prime}\) for DAH. Table 1 summarizes the geometric parameters and geometric constraints of each auxetic geometry. The relative density is defined as the ratio of the effective density of the structures to the density of the matrix or constitutive material. This parameter indicates the porosity of the cellular structure. Based on those geometric parameters, the relative density (\(\rho_{r}\)) of each auxetic geometry can be expressed as follows:  
\[\begin{split}&\rho_{r,REH}=\frac{t}{w(t+H)}\bigg{(}\frac{2h}{ \cos\theta}N_{z}+\frac{w+h\tan\theta}{2}(2N_{z}+1)\bigg{)}\\ &\rho_{r,DAH}=\frac{2 tN_{z}}{wH}\bigg{(}\frac{w}{2\cos\theta}+ \sqrt{h^{2}+\frac{w^{2}}{4}}\bigg{)}\\ &\rho_{r,SH}=\frac{3t}{wh}\frac{(w+h)(1+\cos\theta+\sin\theta)}{ 2\cos\theta(\cos\theta+1)}\\ &\rho_{r,CH}=\frac{t}{wh}\Big{(}2\pi r+\sqrt{w^{2}-4r^{2}}+\sqrt {h^{2}-4r^{2}}\Big{)}\end{split} \tag{1}\]  
Figure 1: (**a**) Configuration of the auxetic sandwich panel (ASP), which consists of two face sheets and an auxetic core. (**b**) Overall dimensions of the ASP: \(L\times W\times H\). (**c**) Zoomed-in view of the face sheets and auxetic core of the ASP, highlighting the region marked by the red dashed line in panel (**b**).