# Photoelasticity as a Tool for Stress Analysis of Re-Entrant Auxetic Structures
## 5 Discussion and Results  
The presented article describes transmission photoelasticity as a suitable tool for validating the results of numerical modeling using the finite element method. The presented results compare stress analysis results on planar models with auxetic structures, known for their effective mechanical properties with a negative Poisson's ratio. The negative Poisson's ratio significantly influences the stress distribution within the material, which can be effectively analyzed using photostress methods. When an auxetic structure is loaded, the resulting stress fields are complex due to the material's expansion in both longitudinal and transverse directions. This complexity is captured in the form of isochromatic fringes, which represent the paths of constant difference in principal stresses. The results obtained from numerical modeling were for models that were loaded and fixed at precisely defined locations. The experimental measurements were carried out on dimensionally identical models made from an optically sensitive material type PSM-1. The advantage of experimental measurement is that it takes into account not only real constraints but also the manufacturing inaccuracies associated with the production of the models. The principal stress values at the investigated points, obtained from numerical modeling and experimental measurements are processed in Tables 5-8 for the individual Variants 1 and 2 under symmetrical and asymmetrical loading. The deviations of the values are also included in the tables mentioned. A relatively high agreement between experimental measurements and numerical analysis was obtained (approximately within 5%). Higher deviations occur in areas with lower stress levels, which is directly related to the accuracy of fringe order readings. Also, the authors of the paper [48] indicate that when the fringe orders are \(\leq\)2, the errors in some points are \(>\)5%. However, when the fringe orders are \(\geq\)2.5, the errors are \(<\)5%. By this error analysis, the accuracy and effectiveness of the fringe orders in multiple-holes model were verified. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \multirow{2}{*}{**Point**} & \multicolumn{2}{c}{**Fringe Order N (-)**} & \multicolumn{2}{c}{**Standard Error (\%)**} & \multirow{2}{*}{**Point**} & \multicolumn{2}{c}{**Fringe Order N (-)**} & \multicolumn{2}{c}{**Standard Error (\%)**} \\ \cline{2-3} \cline{5-10}  & **SYM** & **ASYM** & **SYM** & & **ASYM** & & **SYM** & **ASYM** & **ASYM** \\ \hline
1 & - & - & - & - & 16 & 1.42 & 3.48 & 1.7 & 0.4 \\
2 & 4.27 & - & 0.3 & - & 17 & 1.33 & 2.63 & 1.7 & 0.9 \\
3 & - & 4.40 & - & 0.3 & 18 & 1.33 & 1.23 & 1.7 & 1.8 \\
4 & - & 2.38 & - & 0.9 & 19 & 1.33 & 0.82 & 1.7 & 2.2 \\
5 & 4.38 & 0.95 & 0.3 & 2.0 & 20 & 1.40 & 0.02 & 1.7 & 4.0 \\
6 & - & 0.45 & - & 2.9 & 21 & 1.38 & 0.59 & 1.7 & 2.9 \\
7 & 4.22 & - & 0.3 & - & 22 & 1.65 & 1.19 & 1.6 & 1.8 \\
8 & 4.49 & 3.02 & 0.3 & 0.6 & 23 & 1.53 & - & 1.6 & - \\
9 & - & - & - & - & 24 & 1.46 & 3.87 & 1.7 & 0.4 \\
10 & - & - & - & - & 25 & 1.33 & 2.44 & 1.7 & 0.9 \\
11 & - & 4.40 & - & 0.3 & 26 & 1.25 & 1.38 & 1.8 & 1.7 \\
12 & - & 2.44 & - & 0.9 & 27 & 1.30 & 0.33 & 1.7 & 3.1 \\
13 & - & 0.15 & - & 3.5 & 28 & 1.35 & 0.19 & 1.7 & 3.5 \\
14 & - & 1.96 & - & 1.1 & 29 & 1.47 & 0.87 & 1.7 & 2.1 \\
15 & 1.66 & - & 1.6 & - & 30 & 1.53 & 0.80 & 1.6 & 2.2 \\ \hline \end{tabular} Color scheme corresponds to Figure 7.  
\end{table} 
Table 4: Comparison of experimentally determined fringe orders N for symmetrically and asymmetrically loaded models—Variant 2. In Tables 5-8, high-stress values that were not validated by experimental measurement are marked in italics. Nevertheless, it can be concluded that transmission photoelasticity provides quick and valuable information in the deformation-stress analysis of geometrically complex structures, taking into account not only real constraints but also manufacturing inaccuracies. Based on the presented results, the authors conclude that it can be con