Multi-scale first-order and second-order computational homogenization of microstructures towards continua

This paper addresses a first-order and a second-order framework for the multiscale modelling of heterogeneous and multiphase materials. The macroscopically required (first-order or second-order) constitutive behavior is retrieved directly from the numerical solution of a boundary value problem at the level of the underlying microstructure. The most important features of computational homogenization schemes are: no constitutive assumptions on the macro level; large deformations and rotations on the micro and macro level; arbitrary physically nonlinear and time-dependent material behavior on the micro level; independent of the solution technique used on the micro level; applicable to evolving and transforming microstructures. In particular, a second-order computational homogenization scheme deals with localization and size effects in heterogeneous or multiphase materials. Higher-order continua are naturally retrieved in the presented computational multiscale model, through which the analysis of size and localization effects can be incorporated. The paper sketches a brief introductory overview of the various classes of multiscale models. Higher-order multiscale methods, as typically required in the presence of localization, constitute the main topic. Details on the second-order approach are given, whereas several higher-order issues are addressed at both scales, with a particular emphasis on localization phenomena. Finally, the applicability and limitations of the considered first-order and second-order computational multiscale schemes for heterogeneous materials are high-lighted.