On the accurate analysis of linear elastic meta-material properties for use in design optimization problems

In the field of meta-materials engineering, asymptotic homogenization has proven to be a valuable tool to estimate material properties for many different design needs. Homogenization is used in the design of an object to tie the global material properties to the local material microstructure of the meta-material. The governing equations of the theory are derived from a small-parameter expansion that assumes the scaling length of the unit cell, the smallest repeatable structure found in the material, is much smaller than the entire material length scale. However, for those design problems that do not meet this scaling requirement, the meta-material properties given by homogenization may be inaccurate, leading to error in the material design process. To demonstrate some of these limitations, the more general set of macroscopic analyses derived from the average stress and average strain theorems are applied to several designs in which the assumptions of homogenization theory are no longer valid, those chiefly being materials made of a single or a few layers of unit cells. Then, by applying both materials analyses to simple design problems, the necessity of using a meta-material design method to link global and local properties in a manner consistent with global design methods is demonstrated. In particular, the use of homogenization in designing single layer materials is shown to incur a very large error in the design process, whereas the averaging method is much more accurate.