# Photoelasticity as a Tool for Stress Analysis of Re-Entrant Auxetic Structures
## 4 Experimental Measurement  
For experimental measurements, transmission photoelasticity was chosen. The basic concept of photoelasticity is that, when a photoelastic model is loaded, the resulting principal stresses create an optical effect that appears as an isochromatic fringe order when observed through a polariscope. Applying the load in increments, fringes will appear at the most highly stressed points. As the load gradually increases, new fringes appear, and the earlier fringes are pushed towards areas with lower stress. The fringes are numbered sequentially (first, second, etc.) according to the order in which they appeared, and this  
Figure 9: Stress values for Variant 2 at measured locations for: (**a**) symmetrical and (**b**) asymmetrical loading. order (fringe order) remains throughout the entire loading sequence. The fringes always follow one another, they never cross or merge with one another, and they maintain their position in the ordered sequence. The observed order of the fringes is proportional to the difference in principal stresses. Experimental full-field analysis using photoelasticity is also useful for anisotropic structures, but for anisotropic materials, the stress-optic law requires three independent stress-optic coefficients [46]. For this reason, the authors decided to validate the numerical modeling results only in selected locations where uniaxial stress is present. This linear relationship can be expressed by the following Equation (4) [46,47]. \[\sigma_{1}-\sigma_{3}=\frac{N\cdot C}{t}, \tag{4}\]  
where \(N\) is the fringe order, \(C\) is the stress-optic constant of the model material, t is the thickness of the model, and \(\sigma_{1}\) and \(\sigma_{3}\) are principal stresses, respectively. The key point is that the value of the difference in principal stresses or the value of the maximum shear stress in the analyzed model can be easily obtained by multiplying the observed fringe order N by the constant \(C\cdot t^{-1}\). In practice, we often encounter situations where the stress state is uniaxial with zero stress \(\sigma_{1}\) or \(\sigma_{3}\). In this case, there is only one non-zero principal stress in the plane of the surface of the analyzed model, and it can be determined directly from Equation (4). For instance, if \(\sigma_{3}\) = 0, then the principal stress \(\sigma_{1}\) can be expressed by Equation (5) [46,47]. \[\sigma_{1}=\frac{N\cdot C}{t} \tag{5}\]  
One of the stresses is zero in the case of straight, prismatic profiles loaded in tension or compression, except at points where external loads are applied. From the perspective of practical stress analysis, the points where the highest stresses are most likely to occur are at the edges and free boundaries of the model. At every point along the free edge of the model, the principal axes are perpendicular and tangential to the edge. Since the principal stress perpendicular to the edge is necessarily zero, the stress at the point on the unloaded contour is uniaxial, and the non-zero principal stress is always in the direction of the tangent to the edge. For the experimental measurement, a transmission circular polariscope model M060 (Vishay, Malvern, PA, USA) with a null-balance compensator model 067 (Vishay, Malvern, PA, USA) was used. The models were made from an optically sensitive sheet, type PSM-1. To produce the models, the authors chose the waterjet cutting method, which, from their experience, does not introduce residual stresses into the model. The magnitude of the applied force was recorded by an RSCC force sensor (HBM, Darmstadt, Germany), connected to the P3 measuring apparatus (Micro-Measurements, Wendell, NC, USA), as shown in Figure 10a. For each of the analyzed variants, two pieces were produced, as seen in Figure 10b. Figure 11 shows a view of the unloaded models placed in the working space of the circular polariscope, which indicates that no residual stresses were generated during the production of the models.