Using Non-Simply Connected Unit Cells in the Multiscale Analysis and Design of Meta-Materials

Abstract

A method to discretize materials into lattices using non-simply connected, rectangular unit cells such that the corners of the unit cells do not necessarily meet is proposed as an alternative to discretizing with non-square, parallelogram unit cells. When analyzing non-simply connected lattices, the methods commonly used to evaluate linear elastic properties, homogenization and volume averaging, are shown to require only minor changes in their nite element implementations. The linear elastic properties of honeycomb structures, decomposed into non-simply connected lattices, are shown to demonstrate the same convergence characteristics illustrated in the literature for periodic materials with simply connected unit cells. Additionally, these mechanical properties agree with those predicted by established analytical formulae. Finally, topology optimization examples are used to demonstrate the potential application of the proposed methods to meta-material design. Notably, an auxetic honeycomb structure is obtained as a local minimum to a simple optimization problem.