# Photoelasticity as a Tool for Stress Analysis of Re-Entrant Auxetic Structures
## 1 Introduction
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 [PERSON] et al. [37] and [PERSON] et al.