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International Journal for Multiscale Computational Engineering
HOMOGENIZATION OF MATERIALS HAVING INCLUSIONS SURROUNDED BY LAYERS MODELED BY THE EXTENDED FINITE ELEMENT METHOD
Department of Structural Engineering, Ben-Gurion University, Beer-Sheva, 84105, Israel
Department of Structural Engineering, Ben-Gurion University, Beer-Sheva 84105, Israel
Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, New York 10027, USA
We study the homogenization of materials having three phases: matrix, inclusions, and inclusion coatings. To model the inclusions and their coatings we investigate two weakly discontinuous enrichment functions within the extended finite element method (XFEM) framework described by a single level set function. In both formulations the inclusion and coating shapes are independent of each other. The first approach, denoted as a V-type enrichment, combines several weak discontinuities by stacking the corresponding inclusion and coating enriched degrees of freedom in a single node (one on top of the other) while the second, denoted as a zigzag-type enrichment, only adds one additional degree of freedom per each direction. The XFEM approach is extremely efficient and avoids excessive remeshing compared to standard FEM, in particular when the coatings are very thin as in the case of aggregates surrounded by the interface transition zone (ITZ) in concrete materials. Comprehensive verification studies are presented including two-dimensional continuum problems and homogenization of concrete. Herein we mainly focus on the microscopic material response via homogenization of the unit cell. Nonetheless, once the homogenized material properties are obtained, the application to a full multiscale analysis is straightforward. While both enrichment types are different possible extensions to XFEM applied to such three-phase materials, both methods seem to work well and provide significant reduction in degrees of freedom and computation time as compared to standard FEM.
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