# Photoelasticity as a Tool for Stress Analysis of Re-Entrant Auxetic Structures
## 3 Numerical Analysis  
The finite element method is often used in the analysis of auxetic structures [43], examining effective mechanical properties [44], and a negative Poisson's ratio [45]. As mentioned above, the dimensions and shape of the investigated models take into account the experimental measurement possibilities (the working space of the polariscope and the size of the photoelastic surface). The FE model was created in the software SOLIDWORKS 2024 (Dassault Systemes, Paris, France). For the 3D stress analysis, the elements of the maximal dimension 1 mm with local concentrations on the edges up to 0.1 mm were applied. The size of the mesh was chosen after mesh sensitivity analysis. The final difference of the principal stresses was less than 1%. The Variant 1 finite element model was made from 388 179 nodes and 211 365 tetrahedral finite elements with quadratic approximation. The Variant 2 finite element model was made from 344 193 nodes and 199 863 tetrahedral finite elements with \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline Stress-optic constant \(C\) (MPa/fringe/mm) & 7 \\ Young’s modulus of Elasticity \(E\) (MPa) & 2500 \\ Poisson’s ratio \(\mu\) (-) & 0.38 \\ Elongation (\%) & 5 \\ Max usable temperature (\({}^{\circ}\)C) & 150 \\ Thickness \(t\) (mm) & 3.2 \\  & 254 \(\times\) 254 \\ Standardized plate sizes (mm \(\times\) mm) & 254 \(\times\) 508 \\  & 508 \(\times\) 508 \\  & 610 \(\times\) 762 \\ \hline \end{tabular}
\end{table} 
Table 1: Basic properties of a photoelastic sheet type PSM-1 (Micro-Measurements, USA). quadratic approximation. The validation of the results was carried out under static loading under two different load modes--symmetrical and asymmetrical (a description is given in Table 2). The top points of the model were loaded with a total force of 150 N and bottom points were fixed in all directions. The magnitude of the force was chosen so that the calculations were performed in the linear PSM-1 region of the material. For stress analysis, equivalent stresses according to the [PERSON] theory are most often used; the results of numerical modeling are presented in Figures 3 and 4. It should be mentioned that after producing the models by waterjet splitting, the authors found that some parts (places with acute angle) of the model were rounded by a radius \(r=0.5\) mm, so the 3D models were modified (rounded edges) before running the numerical simulations. In Figure 5, the field of principal stresses of symmetrical Variant 1 is expressed as a vector, which documents the change in the direction of the principal stresses around the concentrator. In regions where the stress is uniaxial, the vectors are parallel or perpendicular to the edge. The obtained results were used by the authors in the choice of the locations in which the validation of the results by experimental measurement was carried out. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multirow{2}{*}{**Model Variant**} & **Acting Force** & **Fixing Points** \\  & & **(Points According to** & **(Points According to** \\  & & **Figure 2)** & **Figure 2)** \\ \hline \multirow{2}{*}{Symmetric} & 1 & T1; T3; T5 & B1; B3; B5 \\  & 2 & T1; T4; T7 & B1; B4; B7 \\ \hline \multirow{2}{*}{Asymmetric} & 1 & T1 & B1; B3; B5 \\  & 2 & T1 & B1; B4; B7 \\ \hline \hline \end{tabular} Note: T—top side; B—bottom side.  
\end{table} 
Table 2: The basic properties of a photoelastic sheet type PSM-1. Figure 3: Variant 1: equivalent von Mises stress distribution (MPa), symmetrical and asymmetrical loading. For Variant 1, points in the center region of the model (Figure 6) were selected, specifically in the five vertical arms (points 1 to 10), in the stress concentrators (points 11 to 20), and in the regions near the cell tip (points 21 to 30). As it was not possible to measure stress concentrator ratios in the experiment, points 11-20 were chosen for Variant 1 based on the FEM values near the concentrators.