Generalized Eshelby's inclusion and inhomogeneity problems for transient heat transfer

TL;DR

This paper generalizes Eshelby's inclusion problem to arbitrary shapes in transient heat transfer, deriving analytical tensors and formulas applicable to complex inhomogeneities in 2D and 3D, enabling advanced thermal modeling.

Contribution

It introduces a unified analytical framework for Eshelby's tensors applicable to arbitrary-shaped inclusions in transient heat transfer, extending classic solutions.

Findings

Derived Eshelby's tensors for arbitrary shapes in transient heat transfer.

Verified polyhedral inclusion formulas against spherical solutions.

Analyzed temporal effects and discontinuities in domain integrals.

Abstract

Eshelby's inclusion problems have been generalized to arbitrary shape of polygonal, polyhedral, and ellipsoidal inclusions embedded in an infinite isotropic domain under transient heat transfer, and Eshelby's tensors have been analytically derived to evaluate disturbed thermal fields caused by inclusions with a polynomial-form eigen-field. Transformed coordinates are applied to arbitrarily shaped inclusions for domain integrals of transient fundamental solutions. This formulation is for general transient heat transfer, and it can recover classic Eshelby's tensor for the ellipsoidal subdomain with explicit expression for the spherical domain in the steady state, Michelitsch's solution in the harmonic state, and recent solution in the transient state. The formulae for a polyhedral inclusion is verified by comparison to closed-form solutions of a spherical inclusion when the sphere is divided into many polyhedrons. The discontinuity of domain integrals for Eshelby's tensor are investigated the temporal effects are elaborated. The generalized formulation for Eshelby's problems enables the simulation and modeling of particulate composites containing inhomogeneities of various shapes for steady-state, harmonic and transient heat transfer in both two- and three-dimensional space through the equivalent inclusion method.