# Negative Poisson's Ratio Lattice Structure with Chiral and Re-Entrant Properties
## 2 Structural Design and Methods
### Finite Element Analysis  
The static simulation was performed using the commercially available software ANSYS 14.0. First, a model was created with identical dimensional and material parameters, as described in Section 2.3 regarding the specimens. In the simulation of the static tensile test, the fixed constraints were applied to the bottom of the ancient window grill system, while the upper end of the test piece was subjected to the same loading conditions as outlined in Section 2.4. The schematic diagram is depicted in Figure 7a.  
Secondly, it was important to maintain consistent boundary and loading conditions while employing finite element techniques to analyze the negative Poisson's ratios of 90 distinct structures, as depicted in Figure 3. The objective of this study was to examine the impact of geometric parameters on the negative Poisson's ratio values and understand how changes in these parameters influenced the mechanical behavior of the structures.  
Figure 6: **(a)** Schematic diagram of the measurement point and **(b)** the experiment process.  
The numerical simulation of the window grill structure fell under the category of thin plate bending problems, primarily due to its dot matrix structure which exhibited a high level of regularity. Thus, the element utilized for meshing was plane-182, which fell under the category of linear elements.  
To exclusively examine the impact of load on the NPR, we opted to analyze the central region of the specimen as the designated calculation area (see Figure 7b). The blue arrow symbolizes the direction of active stretching, while the green arrow represents the direction of passive expansion. The red box denotes the calculation area for the NPR. To minimize potential errors, we performed calculations for two Poisson's ratios and subsequently obtained their average value. The calculation process can be outlined as follows:  
\[\mathrm{v}_{1}=-\frac{\varepsilon_{1}^{\prime}}{\varepsilon_{1}}=-\frac{ \frac{\Delta\Delta C}{\Delta C}}{\frac{\Delta AB}{AB}} \tag{3}\]  
\[\mathrm{v}_{2}=-\frac{\varepsilon_{2}^{\prime}}{\varepsilon_{2}}=-\frac{ \frac{\Delta BD}{BD}}{\frac{\Delta CD}{CD}} \tag{4}\]  
\[\mathrm{v}=\frac{\mathrm{v}_{1}+\mathrm{v}_{2}}{2} \tag{5}\]  
when utilizing ANSYS software to analyze stress and strain in the stretching process, it was observed that a finer mesh division led to more precise calculation results. However, once the grid had been divided to a certain extent, additional division had a minimal effect on the outcomes.  
In order to mitigate the occurrence of redundant and erroneous calculation durations, we consulted the research conducted by [PERSON] et al. [33] and imposed a constraint on the maximum length of the segment, denoted as dmax. The dependence of the result on the mesh was plotted, as depicted in Figure 8b. The findings suggested that the influence on the negative Poisson's ratio was not substantial when the grid length ranged from 0.5  
Figure 7: Schematic of the simulation (**a**) and calculation of Poisson’s ratio (**b**).  
to 3. In order to ensure the precision of the outcomes and minimize the utilization of computational resources, the maximum length of the element edge (dmax) was established at 1.5.