Multiscale modeling of transient problems in periodic Cauchy materials: Asymptotic and spectro-hierarchical homogenization

This work presents a detailed and systematic analytical investigation of the transient elastic wave propagation in two-dimensional periodic heterogeneous composites. The study is conducted through two complementary methodologies: asymptotic homogenization and a recently proposed spectro-hierarchical approach, specifically designed to resolve higher-order microstructural effects. The asymptotic homogenization method derives effective macroscopic equations that accurately capture the averaged behavior of the periodic microstructure, providing a reduced-order but reliable representation of long-wave and low-frequency dynamics. In parallel, the spectro-hierarchical approach systematically reconstructs microstructural fluctuations using a combination of truncated Fourier expansions and a hierarchical sequence of differential problems, allowing the recovery of both first-order homogenized responses and higher-order corrections due to local heterogeneities. The analysis considers both zero and non-zero initial conditions, enabling the study of general transient excitations, including short-time dynamics and localized disturbances, rather than merely steady-state or frequency-limited responses.