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

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.

Figures (13)

• Figure 1: Schematic illustration of three-dimensional transformation coordinate (3DTC) on an arbitrary polyhedron, (a) a tetrahedral on the $I^{th}$ surface with $J^{th}$ edge; (b) dimensions in 3DTC of a tetrahedral based on field point $\textbf{x}$, its projection point $\textbf{x}_p$ and the $J^{th}$ edge of the $I^{th}$ surface
• Figure 2: Verification of domain integrals of Helmholtz's potential over an approximated sphere by $N_I = 229$, $683$, $3,664$, and $18,549$-surface polyhedrons, when $x_3 \in [-2.5, 2.5]$ a. (a), (b) are real and imaginary parts of $\Phi$ by direct volume integral; (c), (d) are real and imaginary parts of $\Phi_{,3}$ by Stokes' theorem
• Figure 3: Reproduction and verification of the analytical spherical integral with radius $a = 0.1$ m, $t = 2 s$ , $\alpha = 0.05 m^2 / s$ along the vertical center line $x_3 \in [-2.5, 2.5] a$ on (a) different $N_I$-surface polyhedron ($229$, $683$, $3,664$, and $18,549$) using direct volume integral scheme as Eq. (\ref{['eq:direct_vol']}); (b) different $N_I$-surface polyhedron using Stokes' theorem integral scheme as Eq. (\ref{['eq:stokes_spatial']}). ($n_{max} = 10$)
• Figure 4: Reproduction and verification of the analytical spherical integral with radius $a = 0.1$ m, $\alpha = 0.05 m^2 / s$ containing a uniform heat source (existing within [0, 1] s), along the vertical center line $x_3 \in [-2.5, 2.5] a$ on (a) different $N_I$-surface polyhedron ($229$, $683$, $3,664$, and $18,549$) with $t = 2 s$; (b) different $N_I$-surface polyhedron with $t = 5 s$. ($n_{max} = 10$)
• Figure 5: Comparison of domain integrals of transient Green's functions over a cuboid inclusion (a) $L$, (b) $L_{,3}$, (c) $L_{,33}$ by the method of Fourier transform powered with fast-Fourier-transform (FFT), and (d) $L$, (e) $L_{,3}$, (f) $L_{,33}$ by series-form solution. Time $t = 2$ s, $\alpha = 0.05 m^2 / s$, $n = 10$, and $x_2, x_3 \in [-0.5, 0.5]$ m, while $x_1 = 0$.