## 3 Poisson's Ratio
### Poisson's Ratio of 3D AH  
The Poisson's ratio of 3D AH, as presented in Figure 6, is confirmed numerically considering a geometric topology of \(L\)/\(R\) = 0.85 and \(L\) = 50 mm. The same type of beam elements (B31) used for modelling 2D AH is adopted herein to simulate the dynamic  
Figure 4: Compression force\(-\)displacement curves and stress–strain curves of chiral structures measured from FE simulation and experimental test for (**a**) 2D and (**b**) 3D auxetic structures. Reprinted with permission from ref. [30], Copyright 2020 Elsevier.  
Figure 5: Poisson’s ratio \(v_{xy}\) and \(v_{yx}\) of 2D AH for loading along X (horizontal) and Y (vertical) directions, respectively.  
crushing process of 3D AH, which has a circular cross-section of radius equal to 0.5 mm. The elastic-ideally plastic material model is adopted with a Young's modulus (E) of 70 GPa, yield stress (\(\sigma_{\text{Y}}\)) of 130 MPa, Poisson's ratio (\(\
u\)) of 0.3 and density (\(\rho\)) of 2.7g/cm\({}^{3}\) to represent the material properties of aluminum. There are 5 \(\times\) 5 \(\times\) 8 3D unit cells involved in the 3D AH model with a dimension of 536.26 \(\times\) 536.26 \(\times\) 505.57 mm\({}^{3}\). The 3D AH is sandwiched between two rigid plates, namely the top rigid (plate) and the bottom rigid (plate), which are constrained to travel along the unique direction (-Y) and fixed at the reference node, respectively. In addition, a set of central nodes at the bottom of 3D AH are fixed as well to eliminate the rigid displacements. The \"General Contact\" is used to consider all possible contact interactions during the crushing processes, and no friction is considered in the present study. The accuracy of the numerical model is verified through a mesh convergence study. It turns out that discretizing the whole model with 5 mm length B31 is reasonable to evaluate the dynamic performance of 3D AH, as shown in Figure 7. In comparison with the crushing force and plastic energy dissipation, the B31 model agrees very well with the B32 (3-node quadratic Timoshenko beam elements) model, but the B31 model is more time-efficient with the same mesh control as that of the B32 model.  
Figure 8a shows the Poisson's ratio (PR) behavior of 3D AHs subjected to various impact velocities. Almost the same magnitude of Poisson's ratios (\(v_{\text{vz}}\) and \(v_{\text{zx}}\)) is observed when the 3D AH is loaded along the Y direction up to a crushing strain of 63%. Following the increase in crushing strain, the original orthogonally placed 2D AH presents large deflections along the out-of-plane directions because of the contractility, which hauls the cross-section of the 3D AH to be diamond under relatively low impact velocities. However, this phenomenon is not apparent when the impact velocity is higher than 10 m/s. As plotted in Figure 8b, the PR-strain curves exhibit initial small absolute values of NPR, which increase with strain and then go down with the densification of materials. The 3D AH presents a maximum NPR approximately equal to \(-0.4\), corresponding to impact velocities ranging from 3 to 20 m/s.  
Figure 6: FE model of 3D AH in (**a**) front view, and (**b**) 3D view.  
Figure 7: Force\(-\)deformation curves and plastic energy dissipation\(-\)deformation curves of 3D AH corresponding to (**a**) different sizes and (**b**) different types of beam elements.