A generalized theory of elastodynamic homogenization for periodic media

For periodically inhomogeneous media, a generalized theory of elastodynamic homogenization is proposed so that even the long-wavelength and low-frequency asymptotic expansions of the resulting effective (or macroscopic) motion equation can, approximately but simultaneously, capture all the acoustic and some of the optical branches of the microscopic dispersion curve. The key to constructing the generalized theory resides in incorporating new kinematical degrees of freedom in conjunction with rapidly oscillating body forces as microscopic and macroscopic loadings while satisfying an energetical consistency constraint reminiscent of Hill–Mandel lemma. By this constraint, an effective displacement field is naturally defined as the projection of a microscopic one onto the dual to the space of body forces. To illustrate these results, a two-phase string is studied in detail.