A multiscale approach to predict the effective conductivity of a suspension using the asymptotic homogenization method

Abstract

This work aims to implement the asymptotic homogenization method (AHM) to predict the effective thermal/electrical conductivity for suspensions with aligned inclusions. Exploiting the substantial separation of length scales between the macroscopic and microscopic structures, multiscale modeling using the AHM capitalizes on the perturbations of the potential field caused due to the presence of an inclusion under a macroscopic loading used to predict the effective property. The analytical formulation for the thermal/electrical conductivity problem is derived, and subsequently, the finite element formulation required to solve the unit cell problem is described. The results obtained for a cylindrical inclusion are validated against known analytical solutions for both the dilute [Mori–Tanaka (MT)] and concentrated volume fractions (ϕ) of the inclusion. This study revealed that MT estimate and AHM agree well at ϕ less than 0.4. However, in near-maximum packing fractions, the AHM results fared significantly better than MT when compared with known asymptotic forms [J. Keller, “Conductivity of a medium containing a dense array of perfectly conducting spheres or cylinders or nonconducting cylinders,” J. Appl. Phys. 34, 991 (1963)]. The proposed AHM method is then implemented in structures with aligned spheroidal inclusions of various aspect ratios and conductivity ratios, thus providing a more generalized approach to predict the effective thermal/electrical conductivity. The results obtained are systematically benchmarked and validated against known analytical expressions.