Size of a representative volume element in a second-order computational homogenization framework

In this paper the intrinsic role of the size of the microstructural representative volume element (RVE) in a second-order computational homogenization is investigated. The presented second-order computational homogenization is an extension of the classical first-order computational homogenization scheme and is based on a proper incorporation of the macroscopic gradient of the deformation tensor and the associated higher-order stress measure into the multiscale framework. The macroscopic homogenized continuum obtained through this scheme is the full second gradient continuum. It is demonstrated with several examples that the size of the microstructural RVE used in a second-order computational homogenization scheme may be related to the length scale of the associated macroscopic homogenized higher-order continuum. It is shown that the analytical second-order homogenization of a microstructurally homogeneous linearly elastic material leads to the second gradient elastic Mindlin's continuum on the macroscale, where the resulting macroscopic length scale parameter is proportional to the RVE size. Several numerical microstructural and multiscale analyses reveal the significance of the contribution of the physical and geometrical nonlinearities in the relation between the RVE size and the calculated macroscopic response. Based on the obtained results, some conclusions are drawn with respect to the choice of the microstructural RVE in the second-order computational homogenization analysis.