Homogenization of high-frequency wave propagation in linearly elastic layered media using a continuum Irving-Kirkwood theory

This article presents an application of a recently developed continuum homogenization theory, inspired by the classical work of Irving and Kirkwood, to the homogenization of plane waves in layered linearly elastic media. The theory explicitly accounts for the effects of microscale dynamics on the macroscopic definition of stress. It is shown that for problems involving high-frequency wave propagation, the macroscopic stress predicted by the theory differs significantly from classical homogenized stress definitions. The homogenization of plane waves is studied to illustrate key aspects and implications of the theory, including the characteristics of the homogenized macroscopic stress and the influence of frequency on the determination of an intermediate asymptotic length scale. In addition, a method is proposed for predicting the homogenized stress field in a one-dimensional bar subjected to a frequency-dependent forced vibration using only knowledge of the boundary conditions and the material’s dispersion solution. Furthermore, it is shown that due to the linearity of the material, the proposed method accurately predicts the homogenized stress for any time-varying displacement or stress boundary condition that can be expressed as a sum of time-periodic signals.