Combined mesh superposition method and homogenization approach for a crack problem in periodic composites

The paper deals with numerical computation of a crack problem posed on microstructural heterogeneous materials that contain multiple phases in the microstructure. The mechanical failure of such materials is a natural multi–scale effect since cracks typically nucleate in regions of defects on the microscopic scale. The modeling strategy for solving the crack problem concerns simultaneously the macroscopic and microscopic models. Our approach is based on an efficient combination of the homogenization technique and the mesh superposition method (s-version of the finite element method). The homogenized model relies on a double–scale asymptotic expansion of the displacements field. The mesh superposition method uses two independent (global and local) finite element meshes and the concept of superposing the local mesh arbitrarily onto the global continuous mesh. The crack is treated by the local mesh and the homogenized material model is considered on the global mesh. Numerical experiments for problems on biomorphic microcellular ceramic templates with porous microstructures of multiple materials constituents are presented. MSC: 65M60; 74B05; 74Q05; 74Q15; 74R10