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.