The mechanics and optimal design of micro-architectured stepped hexagonal lattices

Abstract

This paper conceptualises and analyses a new class of stepped hexagonal lattice achieved through modifying the cross-section of the constituent beams in a controlled manner. The main idea lies in the redistribution of the mass of the constituent beam to obtain a range of equivalent elastic properties, which is not possible within the scope of regular hexagonal lattices. The mass of the stepped lattices is kept the same as the regular hexagonal lattices with equivalent geometry. The in-plane mechanics of such mass-conserved hexagonal lattices are investigated, considering stepped beams as constituent members. The mechanics of a unit cell is exploited to derive the closed-form analytical expressions of equivalent elastic properties of the lattice. The derivation utilises the stiffness elements of the constituent stepped beam members. The stiffness matrices are obtained using two different approaches. They include a semi-analytical condensed stiffness matrix approach based on static substructuring and an analytical method based Castigliano’s energy formulation. Both of these approaches are general and can handle arbitrary geometry for the constituent beams. New closed-form analytical expressions of equivalent elastic properties of the lattice are derived. The optimum geometric parameters are obtained by formulating an analytical optimisation problem. It is shown that a unique solution is possible by solving two simultaneous nonlinear equations. The general closed-form expressions of the equivalent elastic properties can be considered benchmark solutions. In the particular case of the regular lattice, they reduce to the well-known classical expressions. Numerical results show that the values of equivalent elastic properties of the lattice can change significantly by redistributing the mass of the constituent beams. It is demonstrated that up to a 37% increase in the equivalent elastic modulus can be achieved compared to the regular lattice by optimally choosing a stepped profile of the constituent elements.