Dynamical homogenization of microstructured media towards micromorphic effective continua

Abstract

The virtual power principle for materials with microstructure modeled in the framework of micromorphic effective media is written from a micromechanical perspective, considering recent contributions of (Alavi et al., 2021, 2023) however restricting to statics. In the present work, we extend the micromorphic framework to dynamics by deriving the macroscopic kinetic energy from the upscaling of microscopic classical principles. Our methodology focuses on identifying a homogeneous velocity that captures the microscale kinematics of the micromorphic effective medium, defined by the macroscopic velocity and the rates of the displacement gradient, distortion tensor, and its first gradient. Unit cell boundary value problems are formulated for the velocity fluctuation that leads to the computation of the effective dynamical micromorphic properties. The predictions of kinetic energy, incorporating higher-order contributions, are exemplified by the square and tetrachiral unit cells, emphasizing the crucial role of structural design in micromorphic media. Internal length scales, comparable to the unit cell size, further validate the micromorphic model's capability to capture scale-dependent effects and demonstrate its effectiveness in predicting the dynamic behavior of architected materials.