A dynamic homogenization framework predicting the spatiotemporal nonlocality and nonuniformity of field quantities in heterogeneous media

The homogenization of heterogeneous media has been attracting the attention of scientists for more than a century. The classical homogenization methods are focused on the prediction of the effective or overall properties of heterogeneous media such as the effective elastic tensors or potentials, and conductivity tensors. Recently, the topic becomes a focus of research in mathematics, mechanics, and materials science due to its importance in the modelling of advanced composite materials, in particular, metamaterials. However, the classical methodology produces local forms of governing equations and cannot describe complex dynamic responses of various heterogeneous media such as the dispersion and bandgaps of elastic waves. This lecture will present the recent work of the author's group on the dynamic homogenization of heterogeneous media. Starting from conventional local linear elastic constituents, we develop a dynamic homogenization framework and derive the macroscopic governing equations for heterogeneous media. The governing equations can reflect spatiotemporal nonlocality and nonuniformity of the field quantities and can correspond to conventional local models and the well-known nonlocal models including the Mindlin equation, Willis formalism, Eringen constitutive relation, and peridynamic formulation. For heat conduction, the governing equation of the average temperature can correspond to the Jeffreys-type equation, Nunziato equation, Gurtin and Pipkin equation, peridynamic formulation, and dual-phase-lag (DPL) equation. All the parameters in the governing equations can be determined from the geometrical and physical parameters of the constituents; thus the framework also sheds light on the physical mechanisms of the relevant formulations.