# Photoelasticity as a Tool for Stress Analysis of Re-Entrant Auxetic Structures
## 2 Materials and Methods  
In this paper, two configurations of auxetic models with re-entrant cell types were utilized, as they represent the most commonly employed structure demonstrating a negative Poisson's ratio. This structure is designed to be easily manufactured at minimal cost. Regarding re-entrant cells, the main determinant of their auxetic behavior is the angle \(\theta\) (Figure 1), as described by [39]. They constrained the angle within the range from \(-5^{\circ}\) to \(-70^{\circ}\) to prevent the early contact of individual rods under different loading conditions. The re-entrant cells are additionally defined by the vertical wall length (\(l\)), the length of the inclined wall (\(l\)), and the cell-wall thickness (\(t\)) [40]. The analyzed samples are characterized by an angle \(\theta=-30^{\circ}\), a vertical length \(h=30\) mm, a thickness \(t=5\) mm, and an inclined wall length \(l=20\) mm. Figure 1: The shape and dimensions of the re-entrant unit cell to be analyzed. Based on Equation (1)  
\[\rho*=\frac{(2l+h)t}{2l\cos\theta(h-l\sin\theta)}, \tag{1}\]  
described by [41], we can further determine the relative density \(\rho*\) of the auxetic sample, which in our case will be 25.26%. The sample was chosen to be stiffer, thereby ensuring that stresses become visible on the photoelastic sheet while also preventing premature damage to the sample. The Poisson ratio of the re-entrant cells is given by equations  
\[\
u_{12}=\frac{\sin\theta\Big{(}\frac{h}{7}+\sin\theta\Big{)}}{\cos^{2}\theta}, \tag{2}\]  
\[\
u_{21}=\frac{\cos^{2}\theta}{\sin\theta\Big{(}\frac{h}{7}+\sin\theta\Big{)}}, \tag{3}\]  
where the resulting values are calculated as \(\
u_{12}=-0.666\) and \(\
u_{21}=-1.5\). The first configuration is shown in Figure 2a. In this configuration, the number of bottom holes B1-B5 and upper holes T1-T5 (for boundary conditions) was set at 5 to symmetrically align with the number of cells in the end parts. These holes were utilized for fixing and applying external force (tensile stress) during the experiment. The total number of re-entrant cells was 32 whole cells and 6 half cells. The second configuration with sample dimensions is shown in Figure 2b. The total number of re-entry cells was the same as in the previous case, i.e., 32 whole cells and 6 half cells, but they were rotated at 90\({}^{\circ}\). The number and location of holes for the realization of experimental measurements were adjusted so that they corresponded with the number of cells in the end parts (see Figure 2). The lower holes are labeled as Bottom-B and the upper holes as Top-T. Both samples were made from a photoelastic sheet type PSM-1 (Micro-Measurements, Raleigh, NC, USA). PSM-1 is a specially isotropic polycarbonate material [42]. This material is easy to process, with no creep and no edge effect as well as excellent transparency and optical sensitivity. It is supplied in three basic thicknesses (3.2 mm, 6.25 mm, and 9.5 mm) depending on the range of the measured stress levels. The basic information is given in Table 1.  
Figure 2: Configuration of auxetic re-entrant cell sample (**a**) Variant 1; (**b**) Variant 2. Figure 2 illustrates the configurations of the models that were the subject of the research. This article presents the result for the chosen symmetric and asymmetric loading of the model. The value of the maximum loading force was selected based on an assessment of the sensitivity of the optically sensitive material used (depending on the thickness of the sheet). In this case, the force value was selected equal to \(F\) = 150 N. One advantage of numerical modeling is the ability to determine the value at any point, which to some extent depends on the mesh parameters. On the other hand, it should be noted that optimal model dimensions are used in modeling, contacts do not account for manufacturing inaccuracies, and the results depend on correctly defined boundary conditions, among other factors. Therefore, appropriate experimental methods are used in practice to verify the accuracy of numerical simulations.