# Photoelasticity as a Tool for Stress Analysis of Re-Entrant Auxetic Structures
## 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 [PERSON] et al. in [4]. Subsequently, [PERSON] proposed and manufactured thefirst 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. [PERSON] 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 [PERSON] 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 [PERSON] 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.