The family of elastically isotropic stretching-dominated cubic truss lattices

Abstract

We present a family of elastically isotropic stretching-dominated cubic truss lattices. Their Young’s, shear, bulk moduli approach nearly 1/3 of the theoretical Hashin-Shtrikman upper bounds at low density limit, which are the maximum isotropic stiffness that truss lattices can achieve. The proposed lattices are designed by combination of two elementary anisotropic lattices based on theoretical criteria for elastically isotropic stretching-dominated truss lattices. The elementary truss lattices are identified by exhaustive search of the Euclidean Patterns in Non-Euclidean Tilings database. Matrix analysis formulas for periodic pin-jointed assemblies are developed to prove that all combined lattices exhibit stretching-dominated behaviors under arbitrary macroscopic loads, i.e., their corresponding periodic pin-jointed assemblies have no macroscopic strain-producing mechanisms. The stiffness of the combined lattices is further evaluated through C3D10 solid element based finite element models and numerical homogenization. Numerical results show good agreement with theoretical calculations at low densities, thus validating the stretching-dominated behavior and elastic isotropy. The lattices exhibit different elastic moduli and anisotropy at moderate relative densities due to their different joint connections, among which the sqc1 + sqc5432 lattice possesses both nearly isotropic stiffness and relatively high elastic moduli at 30% relative density. This work presents a systematic method to design elastically isotropic cubic truss lattices with optimal isotropic stiffness, and reveals that the optimal designs are infinite.