Abstract
We obtained an analytical solution for the effective thermal conductivity of composites composed of orthotropic matrices and spherical inhomogeneities with interfacial thermal resistance using a micromechanics-based homogenization. We derived the closed form of a modified Eshelby tensor as a function of the interfacial thermal resistance. We then predicted the heat flux of a single inhomogeneity in the infinite media based on the modified Eshelby tensor, which was validated against the numerical results obtained from the finite element analysis. Based on the modified Eshelby tensor and the localization tensor accounting for the interfacial resistance, we derived an analytical expression for the effective thermal conductivity tensor for the composites by a mean-field approach called the Mori-Tanaka method. Our analytical prediction matched very well with the effective thermal conductivity obtained from finite element analysis with up to 10% inhomogeneity volume fraction.
Highlights
We obtained an analytical solution for the effective thermal conductivity of composites composed of orthotropic matrices and spherical inhomogeneities with interfacial thermal resistance using a micromechanics-based homogenization
When the thermal conductivity of a spherical inhomogeneity is twenty time larger than that of a matrix, an finite element analysis (FEA) study based on a representative volume element (RVE) containing ten inhomogeneities showed that the standard deviation of thermal conductivity became as large as 10% of the predicted thermal conductivity[16]
The Eshelby tensor is constant for the ellipsoidal inclusion in heat conduction in the absence of an interfacial resistance, as it is constant for analogous elasticity problems
Summary
Effective Thermal Conductivity in the Absence of Interfacial Resistance. where g is a heat source, T is temperature field, and K0 is a symmetric second order thermal conductivity tensor. Effective Thermal Conductivity in the Absence of Interfacial Resistance. Where g is a heat source, T is temperature field, and K0 is a symmetric second order thermal conductivity tensor. The Green’s function G(x − y) in the steady state heat conduction equation is defined as the temperature field at a position x in the presence of a unit heat source at another position y in an infinite medium, K0ij. Thermal resistance, the Green’s function is introduced to derive the Eshelby tensor Sik of an eigen-intensity problem that relates the intensity field e = −∇T and the eigen-intensity field e* within the inclusion as[30,47],. Where V is the volume of an inclusion whose thermal conductivity is identical to the matrix (see Fig. 1(a)). By adopting the classical potential theory and the mathematical analogy with electrostatics[48] (See Appendix B in the reference), one can simplify Eq (4) for an ellipsoidal inclusion as
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