In this work, a multi-scaling homogenization process using boundary element formulation (BEM) for modeling a two-dimensional multi-phase microstructure containing irregular’s inclusions is presented. The BEM is very attractive for multiscale modeling tools for heterogeneous materials. In this approach, the iterative inhomogeneity discretization of the external boundary is disregarded, leading to a computational low cost. This approach was used for solving the elastic problem of a representative volume element (RVE) and the field theory medium. The main goal relies on finding the effective properties of micro-heterogeneous materials within a homogeneous and orthotropic matrix. Expressions for evaluating the effective properties under Plane Stress (PT) for orthotropic materials were also presented. Generally, the numerical models consider the graphite nodules as voids for GGG-40 and the roundness is close circular geometry. In this sense, a nodular cast iron GGG-40 microgram was obtained by X-ray computed tomography and Laser Confocal Microscope System, allowing the modeling of the true nodule shape. The numerical results showed good agreement with the experimental tests. The inclusions of graphite were considered as voids in the material matrix. Experimental stress–strain tests and micrographic analysis were used to determine the Young’s modulus, spatial distributions, as well as, nodule shape. The numerical in this work was compared with the obtained experimental results in this work. The comparison between the obtained experimental data with those available in the literature also showed good agreement.
Simple Solutions of Lattice Sums for Electric Fields Due to Infinitely Many Parallel Line Charges
