In order to create materials for the basis of new products, engineers must learn to model and design the macroscale by taking into account the relevant micro structural features. The goal of multiscale modelling is to capture the microscopic phenomena, while still preserving the macroscopic conservation principles.

One popular multiscale method is the mathematical theory of asymptotic homogenization [1], which uses asymptotic expansions of field variables about macroscopic values. However, the homogenization method suffers from a major limitation stemming from the fact that it breaks down in critical regions of high gradients such as cracks.

To overcome the drawbacks of the existing methods we propose an enrichment method based on the principles of partition of unity [2], which uses an enrichment technique for the modelling of heterogeneous media in the presence of singularities in order to compute the microscopic fields near the crack edge within the macroscale computations. A structural enrichment-based homogenization method is introduced in which the approximation space at the macroscopic scale is enriched by functions generated at the microscopic scale using the asymptotic homogenization technique. To confine the enrichment to a region around the crack we implement a geometry independent enrichment technique developed in [3]. The method is examined for both one and two-dimensional problems.

[1]

A. Bensoussan, J.L. Lions, G. Papanicolaou, "Asymptotic analysis for periodic structures", Amsterdam: Noth-Holland, 1978.

[2]

K. Yosida, "Functional analysis", Springer-Verlag: Berlin Heidelberg, 5th Edn., 1978.

[3]

M. Macri, S. De, "Enrichment of the method of finite spheres using geometry independent localized scalable bubbles", Int. J. Num. Meth. Eng., 69, 1-32, 2006.

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