High-Order Two-Scale Asymptotic Paradigm for the Elastodynamic Homogenization of Periodic Composites

The classical two-scale asymptotic paradigm provides macroscopic and microscopic analyses for the elastodynamic homogenization of periodic composites based on the spatial or/and temporal variable, which offers an approximate framework for the asymptotic homogenization analysis of the motion equation. However, in this framework, the growing complexity of the homogenization formulation gradually becomes an obstacle as the asymptotic order increases. In such a context, a compact, fast, and accurate asymptotic paradigm is developed. This work reviews the high-order spatial two-scale asymptotic paradigm with the effective displacement field representation and optimizes the implementation by symmetrizing the tensor to be determined. Remarkably, the modified implementation gets rid of the excessive memory consumption required for computing the high-order tensor, which is demonstrated by representative one- and two-dimensional cases. The numerical results show that (1) the contrast of the material parameters between media in composites directly affects the convergence rate of the asymptotic results for the homogenization of periodic composites, (2) the convergence error of the asymptotic results mainly comes from the truncation error of the modified asymptotic homogenized motion equation, and (3) the excessive norm of the normalized wavenumber vector in the two-dimensional inclusion case may lead to a non-convergence of the asymptotic results.