Three-dimensional metamaterials exhibiting extreme isotropy and negative Poisson's ratio

•A discrete topology optimization method for designing mechanical metamaterials.•Novel 3D metamaterials with ideal isotropy and Poisson's ratios approaching −1.•A series of strut-based microstructures with tunable auxeticity and tunable stiffness.•Numerical simulations for demonstrating effective properties of optimized designs. This research will develop a new set of mechanical metamaterials with simultaneous ideal elastic isotropy and extreme negative Poisson's ratio through a topology optimization approach. Firstly, we propose a new method to generate a three-dimensional ground structure with geometric cubic symmetry. Secondly, a more efficient topology optimization method is developed by utilizing a small number of both integer and continuous design variables for topological description of the design based on the discrete ground structure. Thirdly, two new geometric constraints are incorporated into the mathematical formulation to ensure creation of designs that are practical. Finally, the genetic algorithm is used to solve this type of mixed-integer programming problems. Several numerical examples have been used to demonstrate the generation of a series of three-dimensional microarchitectures from a single integrated design. The microlattices can also be customized to obtain a broad range of negative Poisson's ratios across different relative densities. The effective material properties and macroscopic deformation behaviors of the designs are numerically validated using commercial software. [Display omitted]