**Adrián Betancur 1, Carla Anflor 1,\*, André Pereira 2,\* and Ricardo Leiderman <sup>3</sup>**


Received: 21 June 2018; Accepted: 21 July 2018; Published: 15 August 2018

**Abstract:** In this work, a multiscale homogenization procedure using the boundary element method (BEM) for modeling a two-dimensional (2D) and three-dimensional (3D) multiphase microstructure is presented. A numerical routine is specially written for modeling nodular cast iron (NCI) considering the graphite nodules as cylindrical and real geometries. The BEM is used as a numerical approach for solving the elastic problem of a representative volume element from a mean field model. Numerical models for NCI have generally been developed considering the graphite nodules as voids due to their soft feature. In this sense, three numerical models are developed, and the homogenization procedure is carried out considering the graphite nodules as non-voids. Experimental tensile, hardness, and microhardness tests are performed to determine the mechanical properties of the overall material, matrix, and inclusion nodules, respectively. The nodule sizes, distributions, and chemical compositions are determined by laser scanning microscopy, an X-ray computerized microtomography system (micro-CT), and energy-dispersive X-ray (EDX) spectroscopy, respectively. For the numerical model with real inclusions, the boundary mesh is obtained from micro-CT data. The effective properties obtained by considering the real and synthetic nodules' geometries are compared with those obtained from the experimental work and the existing literature. The final results considering both approaches demonstrate a good agreement.

**Keywords:** boundary element method (BEM); periodic boundary conditions; representative volume elements (RVEs); effective elastic properties; homogenization
