## 3 Poisson's Ratio  
### Poisson's Ratio of 2D AH  
Before going ahead with the 3D model, the 2D AH, as shown in Figure 2, is simulated in ABAQUS 6.14 to identify the in-plane Poisson's ratio in two directions. According to the original research revealed in [37], the ultimate orthogonal strains in both X and Y directions are \(\varepsilon_{X}=\varepsilon_{Y}=\phi r/R\), where \(\phi\) is the angular deflection induced by the rotation of nodes. The shapes of nodes have no influence on the Poisson's ratio of chiral auxetics since \(\phi=\phi^{\prime}\), which is illustrated in Figure 2. Later on, [PERSON] and [PERSON] [38] refined the estimation of elasto-static behavior of chiral structures considering the axial and shear deformations of ligaments, and they found that the expression of [PERSON]'s ratio is \(v_{\rm m}=\frac{4(t/L)^{2}}{(t/L)^{4}\cos^{2}\beta+1-\cos^{2}\beta+3(t/L)^{2} }-1\), where \(t\) is the wall thickness of nodes and ligaments. Incorporating with Equation (1), the following equation (Equation (3)) is obtained. Then, it can be seen from above that the parameter L determines the different Poisson's ratio between AC and AH when replacing the circular nodes with hexagonal ones. \[v_{m}=\frac{4t^{2}}{L^{2}\left(-\frac{\left(L^{2}-4t^{2}+R^{2}\right)^{2}}{4L ^{2}R^{2}}+\frac{3t^{2}}{L^{2}}+\frac{t^{4}\left(L^{2}-4t^{2}+R^{2}\right)^{2 }}{4L^{6}R^{2}}+1\right)}-1 \tag{3}\]  
The FE model of 2D AH is set up with a geometric topology \(L/R=0.85\), and the value of L is initially set to 100 mm. The nodes and ligaments are modelled with 2-node linear Timoshenko beam elements (B31), and they share the same beam section profile of circular shape with a radius of 0.5 mm. The 2D AH is discretized with an element size of 5 mm. For the purpose of mimicking the ligament-bending-dominated deformation assumption considered by [PERSON] and [PERSON] [37], [PERSON]'s modulus of nodes is ten-times the one of the ligaments, as recommended by [PERSON] and [PERSON] [39]. The boundary conditions of 2D AH, as depicted in Figure 3a,b, are defined as below: only in-plane deformations are allowed for the whole model; displacement boundary conditions are applied at the center of each side node in either the X or Y direction, and nodes are constrained to be rigid. In order to eliminate rigid displacements, nodes on the loaded side are limited to travel along the loaded direction. The Poisson's ratio of 2D AH is obtained by measuring the shrinkage of the yellow rectangle, the sides of which pass through the half of ligaments embracing the central node of the model. In addition, force\(-\)displacement curves obtained from the explicit model developed using shell elements in ABAQUS were also assessed against the results of the experimental test, which was conducted in the laboratory. The specimens of chiral structure were manufactured via 3D printing, with a dimension of 140.8 \(\times\) 124.5 mm\({}^{2}\) in plane and 15 mm in depth. The geometric topology of test specimens has an \(L/R\) of 0.85, \(L\) of 30 mm and wall  
Figure 3: Measurement of Poisson’s ratio for 2D AH subject to displacement boundary conditions in (**a**) X direction, and (**b**) Y direction. thickness of 2 mm. The material used for 3D printing is DSM's Somos(r)14120, which has an elastic modulus of 2460 MPa, Poisson's ratio of 0.23 and tensile strength of 45 MPa. The MTS universal testing machine was used for the tests, with a compressive speed of 0.1 mm/s. The capability of FE models to capture the failure modes and stress-strain response of 3D auxetic structures is validated with the available experimental tests in the literature for 3D re-entrant lattices [30]. The specimen has a dimension of 65 \(\times\) 65 \(\times\) 56 mm\({}^{3}\) and a compressive velocity of 5 m/s. Figure 4 plots the compression forces-deformations curves for both the numerical simulation and experimental test. Good agreements are observed throughout, which shows that both the 2D and 3D explicit dynamic models used in the present study are effective and reasonable.