An analytical model for star-shaped re-entrant lattice structures with the orthotropic symmetry and negative Poisson's ratios

A new analytical model is developed for three types of 2-D periodic star-shaped re-entrant lattice structures that possess the orthotropic symmetry and exhibit negative Poisson's ratios. Contributions from both the re-entrant and connection struts are considered using an energy method based on Castigliano's second theorem. Each re-entrant strut is treated as a Timoshenko beam, and stretching, transverse shearing and bending deformations are all incorporated in the formulation. Unlike existing studies, the overlapping of struts at joints is included in determining the relative density, which is analytically expressed for each lattice type. Closed-form formulas are derived for the effective Young's moduli and Poisson's ratios of each type of lattice structure, which contain three non-dimensional length ratios, two re-entrant angles, one shear correction factor, and Young's modulus and Poisson's ratio of the strut material. The new analytical model is validated against finite element simulations conducted in the current study and two existing analytical models for simpler square lattice structures without re-entrant struts. To illustrate the newly developed analytical model, a parametric study is conducted to quantitatively show the effects of the five geometrical parameters on the effective properties of each type of lattice structure. It is found that for the effective Poisson's ratios the key controlling parameters are the two re-entrant angles, while for the effective Young's moduli all of the geometrical parameters can have significant effects except for the external connection length ratio. It is demonstrated that through a proper selection of the geometrical parameters, vertex connections and strut material, it is possible to tailor the effective Poisson's ratios and Young's moduli of each type of lattice structure over wide ranges to satisfy different needs in various applications.