Photoelasticity as a Tool for Stress Analysis of Re-Entrant Auxetic Structures

Abstract

The presented study illustrates the use of photoelasticity as an effective tool for validating the results of finite element method (FEM) simulations of auxetic structures. This research focuses on comparing stress distributions in planar auxetic models under symmetrical and asymmetrical loading conditions. Experimental measurements, conducted using an optically sensitive material (PSM-1), were found to align closely with FEM predictions, with deviations within 5%. This agreement highlights the accuracy of both methods, though discrepancies were noted in areas with lower stress levels due to fringe order reading precision. The experimental process makes it possible to take into account real conditions and inaccuracies in production, while numerical modelling is based on ideal conditions. The findings affirm the value of photoelasticity for stress field analysis in complex geometries, particularly for auxetic structures, and underscore its role in verifying and refining computational models. The study concludes that photoelasticity can be a valuable tool for designers and engineers in verifying FEM simulations, even without the use of digital processing and the evaluation of measured data.

1. Introduction

A metamaterial (from the Greek “meta”, meaning “beyond” or “through”), refers to an artificial material characterized by extraordinary properties surpassing those of conventional natural materials. These exceptional characteristics stem from a unique geometrical structure rather than the inherent properties of its constituent materials. There has been a huge development in metamaterial field in recent years thanks to innovative manufacturing techniques such as 3D printing, laser sintering, and stereolithography, among others. Consequently, the study of metamaterials has emerged as a prominent and thriving field of research [1]. The physical properties of metamaterials are determined by the size, shape, and spatial arrangement of their building units. The application of optimization techniques during the design process of metamaterials can be used to develop materials with specific properties to suit applications. This versatile approach opens a wide array of potential uses for metamaterials across various engineering and technological domains [2].

Auxetic structures, known for their negative Poisson’s ratio, exhibit unique mechanical properties that make them particularly interesting for various engineering applications [3]. The relationship between auxetic structures and a negative Poisson’s ratio was first established by Gibson et al. in [4]. Subsequently, Lakes proposed and manufactured the first foam material exhibiting a negative Poisson’s ratio in [5]. Unlike conventional materials, which become thinner when stretched and thicker when compressed (exhibiting a positive Poisson’s ratio), auxetic materials expand laterally when stretched and contract when compressed. This counterintuitive behavior is due to their specific internal structure, often characterized by re-entrant or rotating geometries [6,7]. Therefore, it is necessary to investigate structures with a negative Poisson’s number and their behavior under loading conditions, as was conducted by the authors of [8], whose primary goal was to assess the effect of boundary conditions on the energy-absorbing properties of auxetic metamaterial, discovering that with optimized geometric parameters, auxetic metamaterials with a negative Poisson’s ratio are capable of absorbing over 90% of the impact energy. Wojciechowski introduced a model of a spontaneously forming structure—an isotropic, thermodynamically stable phase with a negative Poisson’s ratio/modulus—in [9], and later solved its elastic properties in the static (zero-temperature) limit in [10]. Finally, the term “auxeticity” was formally introduced by Evans in [11]. In contrast to isotropic structures, which can be either auxetic or nonauxetic, amongst the anisotropic structures the situation is not that simple, and one should distinguish between auxetic, partially auxetic, and nonauxetic structures [12].

Additional research [13] has centered on the comparison of three auxetic structures, re-entrant and arrowhead structures exhibiting a positive Poisson’s ratio and conventional honeycomb structures exhibiting a negative Poisson’s ratio, to investigate the effect of Poisson’s ratio on their mechanical behavior. The results showed that the ultimate forces and the amount of energy absorption both increase if the structure has a negative Poisson’s ratio. These exceptional characteristics result not only from the unique geometrical structure but also from the inherent properties of the constituent materials, as evidenced by Bilski et al. [14,15], who demonstrate that, contrary to common belief, both the structure and the materials significantly influence the Poisson’s ratio. Specifically, their studies highlight that elastically anisotropic or isotropic re-entrant structures made of certain materials can exhibit highly nonauxetic behavior.

The authors of [16] described auxetic structures with different configurations and found out that isotropic systems, like hexachiral or rotating places structures, have a limit on the maximum negative Poisson’s ratio achievable in any direction. On the other hand, anisotropic systems, like re-entrant structures, can achieve a more negative Poisson’s ratio in one direction and a lower negative Poisson’s ratio in the other in-plane direction.

A negative Poisson’s ratio significantly influences the stress distribution within the material, which can be effectively analyzed using photostress methods. Photostress analysis, a subset of photoelasticity, enables the visualization of stress patterns through polarized light, allowing the observation of isochromatic lines that reveal paths of constant stress differences [17]. When a structure is subjected to mechanical stress, 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 lines, which represent the paths of constant difference in principal stresses [18]. In auxetic materials, the stress concentration areas, often highlighted by these isochromatic lines, behave differently compared to conventional materials [19]. The negative Poisson’s ratio leads to a more uniform distribution of stress across the material, potentially reducing the likelihood of failure points [20], which is a critical insight provided by photostress analysis. Additionally, because auxetic structures often experience higher strains under the same load conditions compared to non-auxetic materials [21], the photoelastic response in these regions is more pronounced, allowing for a more detailed analysis of the stress–strain behavior [22]. In this article, we highlight the practical potential of using photoelasticity with standard measurement techniques that place lower demands on hardware and software equipment and personnel. However, advancements in digital photoelasticity have revolutionized the process, providing automated and more accurate stress analysis across the entire field of observation. Recent innovations in digital photoelasticity have significantly enhanced the clarity and resolution of fringe patterns, even in complex multicolored stress fields. For example, the Twelve Fringe Photoelasticity (TFP) method, developed to improve color accuracy in photoelastic images, enables the demodulation of highly intricate stress patterns. This method proves particularly effective in scenarios where multiple stress levels coexist, enhancing the interpretation of complex geometries and multicolored fringe patterns, and thereby improving the accuracy of stress quantification in materials with complex structures [23]. In recent years, the development of machine learning algorithms has added another layer of precision to digital photoelasticity. The introduction of deep learning models such as FringeNet, a U-Net model specifically designed for photoelastic applications, offers enhanced fringe demodulation. By applying advanced neural networks, FringeNet improves the resolution and continuity of fringe patterns, overcoming traditional limitations in digital photoelasticity such as fringe overlap or discontinuities. These advancements make it possible to analyze isochromatic fringe patterns with higher accuracy, particularly in complex stress fields common in auxetic materials [24]. Moreover, full-field automated techniques such as RGB photoelasticity have advanced the field further, allowing for the observation and analysis of stress distributions under white light conditions. Previously, such analyses were limited by the monochromatic light used in traditional photoelastic setups. RGB photoelasticity facilitates faster and more precise measurements of stress patterns, making it possible to automate the analysis of complex geometries [25], as is the case in auxetic structures. For auxetic materials, validating numerical models like those developed using the finite element method (FEM) is essential. While FEM simulations can predict stress distributions within auxetic structures, experimental verification is critical for ensuring the accuracy of these predictions, especially given real-world imperfections such as boundary conditions and manufacturing tolerances. By combining FEM simulations with experimental photoelasticity, it becomes possible to cross-validate stress distributions and identify any discrepancies caused by idealized conditions in simulations. This synergy between numerical and experimental methods is particularly valuable for analyzing re-entrant geometries and other auxetic structures, where stress distributions tend to be more complex than in conventional materials [26]. Several authors have chosen photoelasticity and the FEM numerical method as an experimental method, as we have in our article. The authors of [27] analyzed composite piles through FEM simulation and validated them through a photoelastic experiment. They discovered the influence of small pile diameter and spacing on the displacement and stress coefficient, which provides an important reference for studying the soil arching effect. In another experimental study [28], the researchers investigated the interaction between oblique incident blast stress waves and static cracks using dynamic photoelasticity and numerical simulations. The researchers designed an experimental setup to simulate the interaction while using dynamic photoelasticity techniques to visualize and analyze the stress wave propagation and its interaction with the static crack. The experimental part showed that the incident angle of the blast stress wave has an influence on the interaction between the stress wave and the crack. After that, they conducted a numerical analysis using FEM software, which provides good agreement and verifies the photoelastic experiment.

In mechanical engineering, auxetic metamaterials offer opportunities for developing lightweight structures of complex designs to enhance energy absorption by retaining materials in high-deformation regions and removing them from low-deformation areas, resulting in improved specific energy absorption without compromising mechanical properties [29]. Examples of these include auxetic foams with superior shear resistance calculated from compression tests and elasticity theory [30]; 4D-printed programmable auxetic metamaterials with tunable mechanical properties, where deformation, Poisson’s ratios, and elastic moduli can be controlled through topological parameters and temperature, offering potential applications in fields like flexible electronics and biomedical scaffolds [31]; and also metamaterials applied in biomedical engineering, particularly in bone tissue engineering and vascular stents, while also discussing future perspectives for the field [32]. With the growing interest in advanced materials, innovative manufacturing techniques are being explored to enhance the fabrication of auxetic structures. Additive manufacturing presents an opportunity to fabricate auxetic structures with improved properties and multifunctional capabilities. In contrast to conventional methods like braiding, molding, sintering, and spinning, additive manufacturing provides significant flexibility and versatility in producing these metamaterials. The integration of additive manufacturing has offered a sustainable balance between the cost-effectiveness and efficiency of the resulting structures [33].

With the development and availability of modern computational techniques (FEA) and 3D printing, research is advancing in the direction of the design of various shapes of auxetic structures (with a negative Poisson’s ratio), which find applications in areas such as sports equipment [34], shock absorbers [35], or auxetic fabrics [36].

While photoelasticimetry has been explored in the context of auxetic materials, as demonstrated in the works of Carta et al. [37] and Köllner et al. [38], its application to the analysis of such structures remains underexplored in the scientific literature. Therefore, we decided to present the potential of validating numerical modeling results of auxetic structures using photoelasticimetry. Two variants were selected for analysis, with “cells” oriented in the vertical and horizontal directions, respectively. The article compares results from measurements performed under symmetric and asymmetric loading. The experimental measurement results from five repeated tests show good agreement at selected points with the values obtained via FEM. In conclusion, the authors discuss the potential of using additional optical methods not only to validate numerical modeling results but also to play an essential role in refining boundary conditions in the simulation of complex 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 θ (Figure 1), as described by [39]. They constrained the angle within the range from −5° to −70° to prevent the early contact of individual rods under different loading conditions.

The re-entrant cells are additionally defined by the vertical wall length (h), the length of the inclined wall (l), and the cell-wall thickness (t) [40]. The analyzed samples are characterized by an angle θ = −30°, a vertical length h = 30 mm, a thickness t = 5 mm, and an inclined wall length l = 20 mm.

Based on Equation (1)

$$\rho *={\displaystyle \frac{\left(2l+h\right)t}{2l\cos \theta \left(h-l\sin \theta \right)}},$$

(1)

described by [41], we can further determine the relative density ρ* 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

$${\nu }_{12}={\displaystyle \frac{\sin \theta \left(\frac{h}{l}+\sin \theta \right)}{{\cos }^2\theta }},$$

(2)

$${\nu }_{21}={\displaystyle \frac{{\cos }^2\theta }{\sin \theta \left(\frac{h}{l}+\sin \theta \right)}},$$

(3)

where the resulting values are calculated as ν₁₂ = −0.666 and ν₂₁ = −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°. 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. Basic properties of a photoelastic sheet type PSM-1 (Micro-Measurements, USA).

Parameter	Value
Stress-optic constant C (MPa/fringe/mm)	7
Young’s modulus of Elasticity E (MPa)	2500
Poisson’s ratio μ (-)	0.38
Elongation (%)	5
Max usable temperature (°C)	150
Thickness t (mm)	3.2
Standardized plate sizes (mm × mm)	254 × 254 254 × 508 508 × 508 610 × 762

.

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. One advantage of experimental methods is primarily the consideration of real operating conditions, such as friction, manufacturing inaccuracies, and assembly inaccuracies. In this paper, the authors used an optical method—transmission photoelasticity—for validation. Its advantage is the full-field analysis of stress fields, including the fast identification of areas with high-stress gradients. The disadvantage of photoelasticity is the more demanding process of determining the stress value at the stress concentrator location. As mentioned in the introduction, many authors have developed digital techniques for processing and evaluating measured data [24,26]. This often requires higher requirements on device and hardware equipment. Despite certain limitations, the authors decided to highlight the possibilities of using manual measurement techniques. However, this required a careful selection of suitable measurement points. Their determination was made on the basis of the FEA results. Therefore, the authors chose such areas of the model for the comparison of numerical modeling results and experimental measurements to enable quantitative analysis (Figure 2 marked in red).

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 Systèmes, 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 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 basic properties of a photoelastic sheet type PSM-1.

Model	Model Variant	Acting Force (Points According to Figure 2)	Fixing Points (Points According to Figure 2)
Symmetric	1	T1; T3; T5	B1; B3; B5
	2	T1; T4; T7	B1; B4; B7
Asymmetric	1	T1	B1; B3; B5
	2	T1	B1; B4; B7

Note: T—top side; B—bottom side.

). 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 von Mises theory are most often used; the results of numerical modeling are presented in Figure 3 and Figure 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.

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. For Variant 2, points were also chosen in the center region of the model (Figure 7), specifically in the midline in the stress concentrators (points 1 to 14) and in the region of largest values near the inner and outer tips of the cell (points 15–30). At points 1 to 14, the readings were taken near the concentrators.

For each simulation, Figure 8 and Figure 9 graphically process the values of the principal stresses σ₁ and σ₃ and the equivalent stresses according to the von Mises theory. The graphs show the dependence between the principal stresses (tensile or compressive) and the equivalent stresses according to the von Mises theory, which is the most commonly used strength theory. If one of the principal stresses is equal to zero, the absolute value of the non-zero principal stress corresponds to the value of equivalent stress.

The values of the stresses at the selected locations were validated by experimental measurements using photoelasticity. The aim was to document the application of a well-known optical method in the strain–stress analysis of shape-complex auxetic structures, which is currently being addressed by many researchers.

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 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].

$${\sigma }_1-{\sigma }_3={\displaystyle \frac{N\cdot C}{t}},$$

(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 σ₁ and σ₃ 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·t⁻¹. In practice, we often encounter situations where the stress state is uniaxial with zero stress σ₁ or σ₃. 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 σ₃ = 0, then the principal stress σ₁ can be expressed by Equation (5) [46,47].

$${\sigma }_1={\displaystyle \frac{N\cdot C}{t}}$$

(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. It also documents the isochromatic at the maximum load induced by the force F = 150 N. The loading and fixing of the models were conducted according to Table 2. The advantage of transmission photoelasticity is the fast identification of the area with low and high-stress levels, which can serve as a tool for validating the distribution of stress fields. As can be seen in each of the investigated cases, the highest stress levels are in the area of concentrators. In such areas, it is difficult to accurately determine the line order; therefore, as already mentioned, the authors chose places near the concentrators for the experimental measurements based on the analysis of the FEM results.

At first glance, the difference in the distribution of isochromatics for each variant under symmetrical and asymmetrical loading is evident. A quick identification of stress levels can be realized by evaluating the corresponding color of the isochromatics. However, such a procedure requires extensive experience from the operator. A simpler and, of course, more accurate method is the use of a modern digital technique [26]. The authors used a manual technique with a null-balance compensator model 067 with a sensitivity of 48 counts per fringe. As already mentioned, at the edge of the model, one of the principal stresses is still equal to zero, and the other reaches its maximum value. It can be determined from the position of the compensator with respect to the edge of the model whether it is a tensile or compressive stress, as seen in Figure 12.

Table 3. Comparison of experimentally determined fringe orders N for symmetrically and asymmetrically loaded models—Variant 1.

Point	Fringe Order N (-)		Standard Error (%)		Point	Fringe Order N (-)		Standard Error (%)
	SYM	ASYM	SYM	ASYM		SYM	ASYM	SYM	ASYM
1	0.45	2.10	2.9	1.0	16	-	-	-	-
2	1.08	2.61	1.8	0.9	17	-	1.03	-	1.8
3	1.07	3.41	1.8	0.4	18	-	2.84	-	0.8
4	0.93	0.92	2.0	2.0	19	-	-	-	-
5	0.97	1.93	2.0	1.1	20	-	-	-	-
6	0.98	0.17	2.0	3.5	21	3.30	-	0.5	-
7	0.93	0.68	2.0	2.7	22	2.67	-	0.9	-
8	1.08	1.17	1.8	1.8	23	3.00	-	0.6	-
9	1.10	0.49	1.8	2.9	24	3.10	-	0.6	-
10	0.45	1.11	2.9	1.8	25	2.88	2.42	0.8	0.9
11	-	-	-	-	26	2.85	2.81	0.8	0.8
12	-	-	-	-	27	3.13	0.89	0.6	2.0
13	-	-	-	-	28	2.97	0.75	0.7	2.7
14	-	-	-	-	29	2.71	3.07	0.9	0.6
15	-	-	-	-	30	3.29	4.08	0.5	0.3

Color scheme corresponds to Figure 6.

and

Table 4. Comparison of experimentally determined fringe orders N for symmetrically and asymmetrically loaded models—Variant 2.

Point	Fringe Order N (-)		Standard Error (%)		Point	Fringe Order N (-)		Standard Error (%)
	SYM	ASYM	SYM	ASYM		SYM	ASYM	SYM	ASYM
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

Color scheme corresponds to Figure 7.

show the average edge order values taken at the analysis points from the five repeated measurements. The standard error associated with these measurements is given in these tables. The standard use of the null-balance compensator model 067 in combination with the camera is for the determination of fringe orders up to N = 4. For higher fringe orders, a microscope is needed. In this paper, the authors only used the null-balance compensator, which is why not all values are listed in the tables. The color coding of the points is consistent with Figure 6 and Figure 7.

It can be concluded that transmission photoelasticity provides valuable information about stress field distribution, especially in geometrically complex models, including the presented 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

Table 5. A comparison of the principal stresses from FEM and the experiment for Variant 1—symmetrical loading.

Point	FEM (MPa)	EXP (MPa)	Diff. (%)	Point	FEM (MPa)	EXP (MPa)	Diff. (%)	Point	FEM (MPa)	EXP (MPa)	Diff. (%)
1	0.91	0.99	8.5	11	11.11	-	-	21	−7.13	7.23	1.3
2	2.27	2.37	4.3	12	12.61	-	-	22	−5.79	5.85	1.0
3	2.24	2.35	4.5	13	12.35	-	-	23	−6.47	6.57	1.5
4	1.94	2.03	4.7	14	13.70	-	-	24	−6.68	6.78	1.5
5	2.01	2.12	5.1	15	11.45	-	-	25	−6.23	6.31	1.2
6	2.03	2.14	4.9	16	11.87	-	-	26	−6.19	6.25	1.0
7	1.95	2.04	4.6	17	12.28	-	-	27	−6.75	6.84	1.4
8	2.24	2.36	5.2	18	11.87	-	-	28	−6.43	6.50	1.0
9	2.27	2.40	5.4	19	11.70	-	-	29	−5.86	5.93	1.2
10	0.91	0.99	8.1	20	11.55	-	-	30	−7.12	7.20	1.1

Color scheme corresponds to Figure 6.

,

Table 6. A comparison of the principal stresses from FEM and the experiment for Variant 1—asymmetrical loading.

Point	FEM (MPa)	EXP (MPa)	Diff. (%)	Point	FEM (MPa)	EXP (MPa)	Diff. (%)	Point	FEM (MPa)	EXP (MPa)	Diff. (%)
1	4.49	4.59	2.2	11	38.47	-	-	21	−24.29	-	-
2	5.63	5.72	1.6	12	39.26	-	-	22	−19.51	-	-
3	7.36	7.46	1.4	13	29.38	-	-	23	−13.98	-	-
4	1.90	2.01	5.4	14	27.11	-	-	24	−15.40	-	-
5	4.09	4.22	3.0	15	13.72	-	-	25	−5.11	5.29	3.4
6	0.34	0.37	9.1	16	9.92	-	-	26	−6.05	6.15	1.6
7	1.45	1.50	3.1	17	−2.22	2.26	1.9	27	1.87	1.94	3.8
8	−2.47	2.56	3.3	18	−6.10	6.21	1.8	28	1.59	1.65	3.7
9	−1.04	1.08	3.4	19	−12.04	-	-	29	6.59	6.72	1.9
10	−2.30	2.43	5.2	20	−15.42	-	-	30	8.88	8.94	0.7

Color scheme corresponds to Figure 6.

,

Table 7. A comparison of the principal stresses from FEM and the experiment for Variant 2—symmetrical loading.

Point	FEM (MPa)	EXP (MPa)	Diff. (%)	Point	FEM (MPa)	EXP (MPa)	Diff. (%)	Point	FEM (MPa)	EXP (MPa)	Diff. (%)
1	11.05	-	-	11	9.96	-	-	21	−2.99	3.03	1.3
2	9.24	9.34	1.1	12	9.73	-	-	22	−3.55	3.61	1.8
3	10.19	-	-	13	10.01	-	-	23	−3.26	3.36	2.9
4	9.62	-	-	14	10.08	-	-	24	−3.12	3.20	2.6
5	9.53	9.59	0.6	15	−3.55	3.64	2.5	25	−2.79	2.90	3.9
6	11.17	-	-	16	−3.02	3.10	2.7	26	−2.65	2.75	3.5
7	9.17	9.23	0.7	17	−2.80	2.90	3.5	27	−2.74	2.84	3.4
8	9.69	9.83	1.4	18	−2.81	2.90	3.2	28	−2.85	2.95	3.3
9	11.35	-	-	19	−2.82	2.90	2.8	29	−3.12	3.21	2.9
10	10.44	-	-	20	−3.00	3.07	2.2	30	−3.26	3.35	2.7

Color scheme corresponds to Figure 7.

and

Table 8. A comparison of the principal stresses from FEM and the experiment for Variant 2—asymmetrical loading.

Point	FEM (MPa)	EXP (MPa)	Diff. (%)	Point	FEM (MPa)	EXP (MPa)	Diff. (%)	Point	FEM (MPa)	EXP (MPa)	Diff. (%)
1	28.34	-	-	11	9.52	9.65	1.3	21	−1.21	1.29	1.3
2	17.82	-	-	12	5.24	5.35	2.0	22	−2.53	2.61	2.0
3	9.49	9.64	1.5	13	0.29	0.32	9.2	23	−10.17	-	-
4	5.09	5.22	2.5	14	−4.23	4.29	1.4	24	−8.34	8.47	1.4
5	2.00	2.07	3.4	15	−11.40	-	-	25	−5.20	5.34	0.7
6	0.94	0.99	5.5	16	−7.52	7.62	1.3	26	−2.94	3.02	1.3
7	29.26	-	-	17	−5.64	5.75	1.9	27	−0.68	0.72	1.9
8	6.50	6.62	1.7	18	−2.63	2.70	2.6	28	−0.37	0.41	2.6
9	28.88	-	-	19	−1.72	1.79	3.8	29	1.81	1.90	3.8
10	20.76	-	-	20	−0.04	0.05	12.3	30	1.65	1.75	12.3

Color scheme corresponds to Figure 7.

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 ≤2, the errors in some points are >5%. However, when the fringe orders are ≥2.5, the errors are <5%. By this error analysis, the accuracy and effectiveness of the fringe orders in multiple-holes model were verified.

In Table 5, Table 6, Table 7 and Table 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 considered a valuable tool for designers and analysts in validating the results obtained by numerical modeling.

It is important to note that every experimental measurement requires thorough preparation of the measurement methodology, which in this case also considered the selection of a suitable optically sensitive material for the maximum load chosen by the authors, F = 150 N. In practice, there may be cases where the measurement setup is correctly designed but the sensitivity of the chosen optically sensitive material is outside the measured range.

The presented article documents that the experimental results obtained by photoelasticity can be used even without the use of state-of-the-art hardware and software. On the other hand, their use would allow a more detailed strain–stress analysis not only at the edge of the model but also at any point of the investigated object. Based on the findings, future research will focus on exploring the applicability of photoelasticity to a broader range of auxetic and geometrically complex materials, considering varying load conditions and material sensitivities. At the same time, there are possibilities for validation of the results on the same samples using modern optical methods (digital image correlation).