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International Journal for Multiscale Computational Engineering
IF: 1.016 5-Year IF: 1.194 SJR: 0.554 SNIP: 0.68 CiteScore™: 1.18

ISSN Print: 1543-1649
ISSN Online: 1940-4352

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.v1.i4.40
16 pages

MultiScale First-Order and Second-Order Computational Homogenization of Microstructures towards Continua

Marc Geers
Dept. of Mechanical Engineering Eindhoven University of Technology PO Box 513, 5600 MB Eindhoven The Netherlands
Varvara G. Kouznetsova
Department of Mechanical Engineering, Eindhoven University of Technology P.O. Box 513, 5600 MB Eindhoven; and Netherlands Institute for Metals Research, Rotterdamseweg 137 2628 AL Delft, The Netherlands
W. A. M. Brekelmans
Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands


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.