Transient computational homogenization for heterogeneous materials under dynamic excitation

This paper presents a novel transient computational homogenization procedure that is suitable for the modelling of the evolution in space and in time of materials with non-steady state microstructure, such as metamaterials. This transient scheme is an extension of the classical (first-order) computational homogenization framework. It is based on an enriched description of the micro–macro kinematics by allowing large spatial fluctuations of the microscopic displacement field in contrast to the macroscopic displacement field, as a result of possible transient phenomena. From the microstructural analysis, the macroscopic stress and the macroscopic linear momentum are obtained from an extended Hill–Mandel macrohomogeneity condition. In particular, the full balance of linear momentum is solved at both scales. Consistent time discretization towards the numerical implementation of the framework as well as a condensed formulation for linear elasticity are provided to ensure a more efficient calculation of the effective response. As an example, the transient computational homogenization approach is applied to the multi-scale analysis of a metamaterial subjected to dynamic uniaxial loading. Consistency of the framework with respect to Direct Numerical Simulations is shown for a large range of loading frequencies and microstructural sizes. Attenuation of the macroscopic waves at specific loading frequencies as well as size effects are highlighted.