Eshelby’s inclusion and inhomogeneity problems under harmonic heat transfer

Eshelby’s equivalent inclusion method (EIM) has been formulated to solve harmonic heat transfer problems of an infinite or semi-infinite domain containing an inclusion or inhomogeneity. For the inclusion problem, the heat equation is reduced to a modified Helmholtz’s equation in the frequency domain through the Fourier transform, and the harmonic Eshelby’s tensor is derived from the domain integrals of the corresponding Green’s function in the form of Helmholtz’s potential. Using the convolution property of the Fourier space, Helmholtz’s potential with polynomial-form source densities is integrated over an ellipsoidal inclusion, which is reduced to a one-dimensional integral for spheroids and an explicit, exact expression for spheres. The material mismatch in the inhomogeneity problem is simulated by continuously distributed eigen-fields, namely, the eigen-temperature-gradient (ETG) and eigen-heat-source (EHS) for thermal conductivity and specific heat, respectively. The proposed EIM formulation is verified by the conventional boundary integral method with the harmonic Green’s function and multi-domain interfacial continuity, and the accuracy and efficacy of the solution are discussed under different material and load settings.