## 1 Introduction  
Honeycomb structures, nature-inspired by bee honeycombs, can be found in a diversity of fields, such as engineering, architecture, and biomedicine, and are a topic of interest both in terms of their structural analysis and their optimal design, as the research literature can show. Extending this structural configuration to auxeticity characteristics, namely by considering negative Poisson's ratio materials, it is possible to obtain materials and structures that are able to exhibit high specific strength, stiffness, superior energy absorption, and indentation resistance, making them highly valuable in industries, like aerospace, medicine, defense, and sports, among others. As far as it is possible to understand from the published works, and without aiming to establish an exhaustive historical line, it is important to refer to some key works in this context. The term auxetic was introduced into the scientific literature by [PERSON] (1991) [1], who emphasized the importance of the Poisson ratio to material performance improvement. However, some published works on structures with negative Poisson's ratio materials were already published. It is the case of the work of [PERSON] (1985) [2], where mechanical structures with a Poisson's ratio equal to minus one were proposed. It is also the case of the method first proposed by [PERSON] (1987) [3] to create the reentrant structure of an auxetic foam, which proved to be more resilient than conventional foams. [PERSON] (1989) [4] pioneered the computational study of isotropic two-dimensional chiral molecular models, considering the spontaneous formation of auxetic phases. In this sequence, a family of two-dimensional, biphasic composite materials with hexagonal symmetry was studied by [PERSON] (1992) [5], who showed that two- and three-dimensional composites with a Poisson's ratio approaching the value \(-1\) could be obtained through a selective design with specific layups involving separated length scales. Later, [PERSON] et al. (1998) demonstrated experimentally that negative Poisson ratios were common on cubic metals, a useful feature to reverse the compensation effect of positive Poisson ratios on the volume and area changes caused by uniaxial stress. In more recent years, it has been possible to find several published works where the design, development, properties characterization, and performance analysis of these material structures are often found, mainly in the context of sandwich configurations. However, only a few published studies consider that these material structures can be hybridized by graded mixtures of their material constituents [6]. Hence, the literature overview in this introductory section follows a categorization vector, starting with highlighting some of the review works. A review study on the design, fabrication, and optimization of honeycomb structures, developed by [PERSON] et al., 2015 [7], focused on the properties related to structure, scale, and materials and highlighted their potential in biomedical applications, like tissue engineering and regenerative medicine. Some years later, another review focused on sandwich structures, emphasizing their light weight, high performance, excellent strength-to-weight ratios, and energy absorption, was presented by [PERSON] et al. (2022) [8]. These authors considered various core types, face sheet materials, and joining methods. Other review studies in the context of auxetic structured materials were also published in parallel, such as the work of [PERSON] and [PERSON] (2000) [9], highlighting the significant advancements in auxetic materials, and noting their growing use in functional and structural applications. According to the authors, this field was rapidly evolving with new materials and processing methods, indicating substantial potential for future applications. [PERSON] (2012) [10] also did a review focused on auxetic materials, focusing on the computational and mechanical aspects.