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Recent developments in numerical homogenization

Boso D. P., Lefik M., Schrefler B. A.

Computer Assisted Mechanics and Engineering Sciences

2009

Vol. 16, No. 3/4

161-183

This paper deals with homogenization of non linear fibre-reinforced composites in the coupled thermo-mechanical field. For this kind of structures, i.e. inclusions randomly dispersed in a matrix, the self consistent methods are particularly suitable to describe the problem. Usually, in the framework of the self consistent scheme the homogenized material behaviour is obtained with a symbolic approach. For the non linear case, that method may become tedious. This paper presents a different, fully numerical procedure. The effective properties are determined by minimizing a functional expressing the difference (in some chosen norm) between the solution of the heterogeneous problem and the equivalent homogenous one. The heterogeneous problem is solved with the Finite Element method, while the second one has its analytical solution. The two solutions are written as a function of the (unknown) effective parameters, so that the final global solution is found by iterating between the two single solutions. Further, it is shown that the considered homogenization scheme can be seen as an inverse problem and Artificial Neural Networks are used to solve it.

Effective yield strengths of random materials by an e-self-consistent method

Turgeman S., Guessab B., Doremus P.

Archives of Mechanics

2007

Vol. 59, nr 3

205-230

The problem of determining the effective yield strength domain of a material containing random distributed heterogeneities is dealt with. This material is represented by a set of microstructures, each occupying a volume of the order of the heterogeneities. A homogeneous comparison material is used, characterized by its own yield strength domain, in which these microstructures are placed. The equivalent homogeneous material is eivisaged as the solution of a system of self-consistent eąuations. The problems of non-existence or non-uniqueness of the solutions of this system lead to modifying it, using an equality to "within e". "Extremal" solutions are highlighted for each of the equations of the system transformed in this way, which bound the effective domain sought for. The proposed hornogenization method is applied to a defect material and the result is compared with a structure calculation.