Bridging scales in non-standard thermo-viscoelastic periodic materials via dynamic non-local homogenization

This study aims to develop a new framework for analyzing the dynamic response of thermo-viscoelastic heterogeneous materials with periodic microstructure. To this end, a non-local asymptotic homogenization formulation designed to capture the influence of microstructural features on wave propagation is introduced. The approach is built upon a compact matrix-operator representation that facilitates the definition of the cell problems and the associated down-scaling relation, from which average field equations of infinite order are derived and structured according to increasing powers of the microstructural characteristic size. A central objective of the method is to obtain accurate dispersion relations in the frequency–wavenumber domain by evaluating these equations at different approximation orders. In doing so, the framework successfully reconstructs both the low-frequency acoustic branch and the onset of the first optical branch at higher frequencies, thus providing a precise description of the material’s complex spectrum. The validity of the formulation is demonstrated through direct comparison with the Floquet–Bloch spectrum of the heterogeneous medium, yielding excellent agreement. Furthermore, the method is conceived to be versatile and can be applied to a broad range of periodic microstructural configurations.