Model-reduction in multiscale problems for composite and polycrystalline materials

Abstract

International audienceA common practice in multiscale problems for heterogeneous materials with well separated scales, is to look for homogenized, or effective, constitutive relations. In linear elasticity the structure of the homogenized constitutive relations is strictly preserved in the change of scales. The linear effective properties can be computed once for all by solving a finite number of unit-cell problems. Unfortunately there is no exact scale-decoupling in multiscale nonlinear problems which would allow one to solve only a few unit-cell problems and then use them subsequently at a larger scale. Computational approaches developed to investigate the response of representative volume elements along specific loading paths, do not provide constitutive relations. Most of the huge body of information generated in the course of these costly computations is often lost. Model reduction techniques, such as the Non Uniform Transformation Field Analysis ([1]), may be used to exploit the information generated along such computations and, at the same time, to account for the commonly observed patterning of the local plastic strain field. A new version of the model [2] will be proposed in this talk, with the aim of preserving the underlying variational structure of the constitutive relations (similar objective in [3]), while using approximations which are common in nonlinear homogenization.[1] J.C. Michel, P. Suquet, Int. J. Solids Structures 40, 6937-6955 (2003)[2] J.C. Michel, P. Suquet, J. Mech. Phys. Solids, In press (2016)[3] F. Fritzen, M. Leuschner, Comput. Meth. Appl. Mech. Eng. 260, 143–154 (2013