Multiscale numerical homogenization to predict the effective electrical conductivity of matrix-inclusion composites

Electrical conductivity is a critical physical property due to its significant role in various industrial applications. Precise determination of this property enables optimized material selection and supports the advancement of innovative technologies. In this study, the effective electrical conductivity of three-dimensional (3D) matrix–inclusion composites containing randomly distributed, non-overlapping spherical inclusions is investigated. A multiscale numerical homogenization scheme based on the finite element method (FEM) is employed to quantify the effects of conductivity contrast and inclusion volume fraction (P = 0.05, 0.10, 0.15, 0.20). Two configurations are considered: composites with a more conductive matrix and those with more conductive inclusions. To ensure the representativeness of the predicted effective properties, two types of boundary conditions are applied, namely the Uniform Electric Potential Gradient (UGE) and the Uniform Current Flux (UCE). The simulation protocol includes a mesh convergence analysis and the Representative Volume Element (RVE) size determination. Validation against experimental data from the literature confirms the reliability of the numerical procedure. It is observed that the inclusion shape, modified via mesh refinement, has negligible influence on the effective conductivity. Moreover, periodic microstructures comprising a single inclusion yield similar results to those of random configurations, suggesting that, in terms of electrical response, the RVE size is on the order of a single inclusion. The contrast C, defined as the ratio of the electrical conductivities of the inclusion and matrix phases, governs the system’s behavior. Numerical results show that the effective conductivity becomes insensitive to further contrast variation for C ≥200 (conductive inclusions) and C ≤1/200 (conductive matrices), indicating a saturation regime where the response depends primarily on the volume fraction of the dominant phase. Furthermore, the effective conductivity values are found to closely follow the Hashin–Shtrikman bounds: the upper bound in the case of a conductive matrix (C <1) and the lower bound when the inclusions are more conductive (C >1). In both regimes, the results are consistent with an analytical model, confirming its accuracy in predicting the effective behavior of 3D composites with inclusion volume fractions below the percolation threshold (approximately 23%).