Numerical methods for modeling heterogeneous materials

Abstract

The present work used the Finite Element Method (FEM) and its variations to study heterogeneous materials. The conventional FEM, the Voronoi cell finite element method (VCFEM) and the Extended Finite Element Method (XFEM) were explored. Initially, the conventional FEM was applied to study some functionally graded material (FGM) plates. The plates were modeled with the aid of the commercial software ABAQUS. The conventional FEM was flexible due to the existing tools. However, there were certain restrictions in modeling the FGM plates. So it was necessary to look for an alternative method. The VCFEM was applied in various examples of heterogeneous materials. The reciprocal stress correction was added to the stress approximation formulation. Therefore, the effect of the inclusionsb shape on the stresses was added in order to improve the solution. The VCFEM was applied in studying several unit cells of composite materials with embedded inclusions. The inclusions had either circular or elliptical shape. An FGM specimen and a FGM cantilever beam with more inclusions was also studied. The VCFEM gave good prediction of displacements at low computational cost. However, the stresses were not captured properly. In some applications, the Voronoi tessellation was used for the mesh. In some others, the quadratic elements was used as Voronoi cell finite elements. The quadratic mesh proved to be more flexible than the Voronoi tessellation since it could be generated easier. The XFEM was also applied to studied the heterogeneous materials. Though the XFEM was more expensive than the VCFEM, the XFEM gave better stresses results. By applying the penalty method, the XFEM could model the porous structures as well, which the VCFEM could not do. An example of trabecular bone was conducted to show the advantage of the XFEM.