A general analytical framework for the mechanics of heterogeneous hexagonal lattices

Abstract

The in-plane mechanics of two-dimensional heterogeneous hexagonal lattices are investigated. The heterogeneity originates from two physically realistic considerations: different constituent materials and different wall thicknesses. Through the combination of multi-material and multi-thickness elements, the most general form of 2D heterogeneous hexagonal lattices is proposed in this paper. By exploiting the mechanics of a unit cell with multi-material and multi-thickness characteristics, exact closed-form analytical expressions of equivalent elastic properties of the general heterogeneous lattice have been derived. The equivalent elastic properties of the 2D heterogeneous lattice are Young’s modulli and Poisson’s ratios in both directions and the shear modulus. Two distinct cases, namely lattices with thin and thick constituent members, are considered. Euler–Bernoulli beam theory is employed for the thin-wall case, and Timoshenko beam theory is employed for the thick-wall case. The closed-form expressions are validated by independent finite element simulation results. The generalized expressions can be considered as benchmark solutions for validating future numerical and experimental investigations. The conventional single-material and single-thickness homogeneous lattice appears as a special case of the heterogeneous considered here. By introducing the Material Disparity Ratio (MDR) and Geometric Disparity Ratio (GDR), variability in the equivalent elastic properties has been graphically demonstrated. As opposed to classical homogeneous lattices, heterogeneous lattices significantly expand the design space for 2D lattices. Orders-of-magnitude of variability in the equivalent elastic properties is possible by suitably selecting material and geometric disparities within the lattices. The general closed-form expressions proposed in this paper open up the opportunity to design next-generation heterogeneous lattices with highly tailored effective elastic properties.