## 1 Introduction
In that work, in addition to the importance of auxetic materials, a potential shift from ordered microstructures to disordered ones using the homogenization method was pointed out as future work. More recently, [PERSON] et al. (2023) [11] examined the principles of auxeticity, mechanical properties, and applications in medicine, automotive manufacturing, and protective gear, highlighting future development directions for auxetic metamaterials. These authors categorized auxetic honeycomb structures into re-entrant, chiral, and rotational rigid types, comparing their mechanisms and properties. Different authors addressed the problem of characterizing honeycomb and auxetic materials' properties, adopting experimental and/or numerical/analytical approaches. [PERSON] and [PERSON] (1996) [12] developed a theoretical model to predict the elastic constants of honeycombs by considering deformation mechanisms, hence obtaining expressions for the tensile moduli, shear moduli, and Poisson's ratios. [PERSON] et al. (2015) [13] studied the effective properties and dynamic response of a sandwich panel consisting of two face sheets and an auxetic core with a filler material, having found that auxetic sandwich panels could be created with two solid materials having positive Poisson ratios, showing significant structural and dynamic benefits. Numerical simulations were performed by [PERSON] and [PERSON] (2000) [14] to determine the transverse shear modulus of honeycombs with in-plane negative Poisson's ratios. The results revealed that the modulus lies between the theoretical upper and lower bounds and varies with the relative density and the gauge-thickness ratio. [PERSON] et al. (2022) [15] proposed an analytical framework to predict the in-plane mechanical properties of auxetic structures. To reduce stress shielding at the bone-implant interface by adjusting the elastic modulus to better match the bone, [PERSON] et al. (2024) [16] recently used experimental and numerical methods to obtain the elastic modulus of gradient structures with auxetic unit cells. [PERSON] et al. (2024) [17] used the Euler-Bernoulli beam theory to derive the equivalent elastic properties of auxetic cellular structures with irregular configurations. The impact of cell height and inclination angle on the in-plane linear elastic properties was examined. Under quasi-static compressive loading, the results showed that the gradient cell size minimally affects elastic behavior and energy absorption. Bidirectional gradient structures exhibited higher platform stress, while homogeneous small-size structures exhibited the highest load-bearing and energy-absorption capacities. In a complementary field, it is also possible to find studies mainly focused on the design and development of auxetic unit cells. This is the case of the work of [PERSON] et al. (2021) [18], who proposed an asymmetric 3D auxetic structure, designed using Timoshenko beam theory and based on a 2D cell configuration. The new design, tested through quasi-static compression tests of ABS samples obtained by fused filament fabrication, was shown to be aligned with the theoretical predictions. [PERSON] et al. (2022) [19] developed and characterized a stainless-steel stent with a re-entrant hexagonal cell geometry, micromachined using a high-precision femtosecond laser. The mechanical behaviors under various loadings were analyzed, and the stent demonstrated superior bending, twisting, and buckling resistance, ensuring maintained blood flow. More recently, an architected lattice system inspired by ancient Chinese window grills, combining chiral and re-entrant properties was investigated by [PERSON] et al. (2023) [20]. Experiments and simulations demonstrated a significant negative Poisson's ratio effect, influenced by the structure's geometric parameters. Another relevant research direction regards mechanical behavior prediction, wherein several published studies on auxetic structures are found.