# Negative Poisson's Ratio Lattice Structure with Chiral and Re-Entrant Properties
## 1 Introduction  
With the advancement of contemporary engineering technology, there is a growing need for materials that possess specialized functionalities. However, specialized materials are not readily available in nature. Therefore, the challenge lies in the logical design of artificial microstructure metamaterials that possess advanced physical and mechanical properties. In recent years, there has been a significant increase in the design of mechanical metamaterials with both two- and three-dimensional structures. These designs have been inspired by biological characteristics and macrostructures [1; 2; 3; 4; 5; 6]. As a result, these metamaterials exhibit a wide range of functional characteristics [7; 8; 9; 10; 11; 12; 13; 14; 15] and have the potential to be applied in various fields. For instance, structures can be designed to obtain materials with a negative Poisson's ratio (NPR). NPR materials have been specifically engineered to possess several notable mechanical properties, including improved resistance to in-plane indentation, increased fracture toughness, an elevated transverse shear modulus, and enhanced dynamic performance. Hence, these materials possess significant potential for applications in medical devices, sports engineering, shock absorption, and the aerospace industry. In 1985, [PERSON] [16] made an attempt to establish structures with an NPR theoretically by utilizing the ideal structures of rods, hinges, and springs. In the same year, [PERSON] [17] introduced a novel method that could be utilized for calculating the NPR of a structure. In 1991, [PERSON] [18] called materials with an NPR by the name \"auxetics\". Then, the classification of auxetic was introduced by [PERSON] [19] and [PERSON] [20]. NPR materials have been found to exhibit distinct mechanical properties that deviate from those of conventional materials, leading to significant interest in them among researchers [21; 22]. For instance, [PERSON] [23] conducted an analysis on the occurrence of coupled NPR and thenegative coefficient of thermal expansion. [PERSON] [24] conducted an analysis on the energy absorption properties of NPR materials. The properties of NPR materials primarily rely on the structural system designs. However, there are also alternative mechanisms that result in a negative Poisson's ratio [25; 26; 27; 28; 29; 30; 31; 32; 33; 34]. Numerous studies have been undertaken to investigate the effects of various NPR topologies [35; 36; 37; 38; 39; 40; 41; 42]. The two primary structures observed in NPR materials are the re-entrant topology [43; 44; 45] and the chiral topology [46; 47; 48]. In recent years, there has been a surge in the proposal of different types of two-dimensional and three-dimensional chiral auxetic materials. These designs capitalize on the displacement and ligament bending deformation characteristics exhibited by chiral components. [PERSON] [25] demonstrated the existence of a negative Poisson's ratio within a chiral and elastically isotropic phase of a thermodynamic model at the molecular level. And, he successfully solved a specific category of chiral and elastically isotropic molecular models at absolute zero temperature, thereby demonstrating the existence of a negative Poisson's ratio [49]. The author also later discussed the role of the center of symmetry in planar models of auxetics [50]. [PERSON] et al. [51] introduced the concept of six-fold chiral negative Poisson's ratio structures and conducted an investigation into their mechanical properties using a linear elastic model. [PERSON] et al. [52] proposed designs for four-fold chiral, anti-four-fold chiral, three-fold chiral, and anti-three-fold chiral structures with a negative Poisson's ratio. In addition to the conventional unit designs, various researchers have put forth innovative metamaterials that exhibit local chiral deformation characteristics. For instance, a two-phase anti-four-chiral structure with an adjustable Poisson's ratio and optimized mechanical performance was designed using a bandgap penalization method [53].