## Abstract

In the context of thermal conduction taken as a prototype of numerous transport phenomena, a general method is elaborated to study Eshelby's problem of inclusions inside a bounded homogeneous anisotropic medium. This method consists in: (i) recasting by a linear transformation the initial problem into Eshelby's problem of the transformed inclusion inside the transformed finite isotropic medium and (ii) decomposing Eshelby's problem of a thermal inclusion embedded in a finite isotropic medium into the sub-problem of the same inclusion inside the associated infinite medium and the sub-problem of the finite ambient isotropic medium including no inclusion but undergoing appropriate compensating boundary conditions. The general method is applied in the two-dimensional situation and the corresponding temperature field and Eshelby's conduction tensor are explicitly expressed in terms of some curvilinear complex integrals for the Dirichlet and Neumann boundary conditions. Thus, the difficulties owing to the unavailability or non-existence of Green's function are overcome. The general results in the two-dimensional case are finally specified and illustrated by considering a finite circular medium with circular or polygonal inclusions.

## 1. Introduction

The problem of inclusions occurs frequently in studying transport phenomena such as thermal conduction, electric conduction, diffusion, flow, dielectrics and magnetism [1–5]. Considering thermal conduction as a prototype of transport phenomena, the present work is concerned with the thermal counterpart of the elastic inclusion problem of Eshelby [6–8]. By analogy with elasticity, a thermal inclusion in a medium *Ω* refers to a subdomain *ω* of *Ω* subjected to a uniform eigen (or a heat flux-free) temperature gradient. Solving the so-called Eshelby's thermal inclusion problem amounts to finding the resulting temperature and heat flux fields inside and outside *ω*.

The widely used Eshelby's classical solution for the elastic inclusion problem presents two main limitations from the practical point of view. First, the inclusion is required to be elliptic in the two-dimensional case and ellipsoidal in the three-dimensional case. Second, the medium in which the inclusion is embedded has to be infinite. The issue of relaxing the limitation to elliptic and ellipsoidal inclusions has been studied in a variety of situations, and many quite general results have been reported recently. In this regard, we refer to the research paper of Zou *et al.* [9], the very recent review paper of Zhou *et al.* [10] and the relevant studies cited therein. By contrast, the issue of removing the restriction to an infinite medium has been much less investigated and the results obtained are quite limited [11–17], in spite of the fact that the works dedicated to it can be traced back to those of Kinoshita & Mura [18] and Kröner [19,20]. The second issue is probably much tougher than the first one. For relevant discussions and references, we refer to the introduction in the recent paper of Zou *et al*. [21] (see also [22]), §5 in the review of Zhou *et al.* [10] as well as the corresponding studies listed therein.

This study aims to extend to the thermal conduction context from our previous work [21] in which an efficient method has been proposed to solve the problem of inclusions inside a finite elastic medium. The basic idea underlying that work is to exploit the superposition principle so as to decompose the initial problem of an inclusion *ω* in a finite elastic medium *Ω* into a sub-problem of *ω* embedded in the associated infinite medium and a sub-problem of *Ω* containing no inclusion but subjected to the compensating boundary conditions provided by the solution to the former sub-problem. The method elaborated on the basis of this idea presents three advantages: (i) it allows directly using a number of results reported in the literature for the problem of an inclusion inside an infinite medium; (ii) it allows us to overcome the difficulties owing to the fact that Green's function is rarely available analytically for a finite medium *Ω* with Dirichlet boundary conditions and does not exist for a finite medium *Ω* with Neumann boundary conditions; and (iii) it explicitly identifies the boundary effects of a finite medium *Ω*.

From the mathematical standpoint, the problem of thermal inclusions is less complicated than that of elastic inclusions. However, from the physical point of view, the former is not at all less important than the latter. Indeed, transport phenomena similar to thermal conduction are numerous, and, in addition, a one-to-one correspondence exists between anti-plane elasticity and two-dimensional steady thermal conduction. Thus, the problem of thermal inclusions is far reaching while lending itself to more general solutions than its elastic counterpart. This fact, already used in our paper [23] dedicated to the problem of thermal inclusions in an infinite medium, will be once more exploited in the present work where the infinite medium is replaced by a finite one.

The rest of this paper is structured as follows. In §2, Eshelby's problem of a thermal inclusion is defined and formulated for a finite homogeneous anisotropic medium obeying Fourier's law. To solve this general problem, we elaborate a strategy consisting of two steps. First, by a linear transformation, the initial problem is recast into Eshelby's problem of the transformed inclusion inside the transformed finite isotropic medium, so that our attention can be, with no loss of generality, limited to isotropic media. Second, as explained above, the problem of an inclusion inside a finite isotropic medium is decomposed into the sub-problem of the same inclusion inside the associated infinite medium and the sub-problem of the finite isotropic medium containing no inclusion but undergoing appropriate compensating boundary conditions. In §3, the general method presented in §2 is applied to obtain the solution to Eshelby's two-dimensional problem of a thermal inclusion in a finite isotropic medium. Precisely, this solution is expressed, in a compact and unified way, in terms of some complex curvilinear integrals valid for the Dirichlet and Neumann boundary conditions. Section 4 is dedicated to illustrating the results obtained in §4 by considering circular and polygonal inclusions inside a finite circular medium. In §5, some concluding remarks are provided.

## 2. Eshelby's problem of thermal inclusions in a finite body

### (a) Formulation of the problem

Consider a finite body *Ω* consisting of a homogeneous medium whose thermal conduction behaviour is governed by Fourier's law. We are interested in the thermal counterpart of the well-known Eshelby's inclusion problem in elasticity. Thus, by analogy, a uniform eigen (or negative heat flux-free) temperature gradient **e*** is prescribed over a subdomain *ω*, called thermal inclusion, of the finite body *Ω*. The disturbance temperature field *T* in *Ω* due to **e*** can be determined by solving the following boundary value problem:
2.1where
2.2and
2.3In equation (2.3), **K** is a second-order positive definite symmetric tensor defining the thermal conductivity of the homogeneous medium; ∂*Ω* is the boundary of *Ω*; *χ*^{ω}(**x**) is the characteristic function of *ω* such that *χ*^{ω}(**x**)=1 for **x**∈*ω* and *χ*^{ω}(**x**)=0 for **x**∉*ω*; **n** is the outward unit normal to ∂*Ω*.

Equations (2.1)–(2.3) constitute a formulation of Eshelby's problem of thermal inclusion in a finite body with the isothermal (Dirichlet) or adiabatic (Neumann) boundary condition. Since this problem is linear, we can formally introduce a second-order tensor **S**^{(ω,Ω)}, called *finite Eshelby's conduction tensor*, which relates **e**(**x**) to **e***:
2.4

The material forming *Ω* is homogeneous but not necessarily isotropic. In the general anisotropic case, the thermal conductivity tensor **K**, as a second-order symmetric tensor, admits the spectral expansion
2.5where *k*_{1}, *k*_{2} and *k*_{3} are the ordered eigenvalues such that *k*_{3}≥*k*_{2}≥*k*_{1}>0 and **m**_{i} (*i*=1,2,3) are three orthonormal vectors defining three principal axes of **K**. Owing to the positive definiteness of **K**, we can express **K** in the following way:
2.6with and
2.7Following Le Quang *et al.* [3], Eshelby's problem for an inclusion *ω* in the finite anisotropic body *Ω* characterized by **K** is now transformed into Eshelby's problem for the inclusion
2.8in the finite isotropic body
2.9with the conductivity tensor
2.10

Correspondingly, Eshelby's conduction tensor field **S**^{(ω,Ω)} for the anisotropic body *Ω* is related to the one for the isotropic body by
2.11where .

Formula (2.11) provides a direct and simple way to determine **S**^{(ω,Ω)} via which may be easier to be found. Since the transformation tensor **B** is constant and preserves the volume, the relation connecting the volume average of **S**^{(ω,Ω)} over *ω* with the volume average of over follows immediately from (2.11):
2.12

### (b) Decomposition of the problem

The linearity of the problem formulated by equation (2.1) makes the superposition principle applicable. Thus, we can decompose the solution *T* of (2.1) as follows:
2.13where the temperature fields *T*^{a} and *T*^{b} fulfil the equations
2.14and
2.15respectively. In (2.14) and (2.15), the boundary function *g* on ∂*Ω* can be arbitrary under the condition that the boundary value problems defined by (2.14) and (2.15) are well posed.

The arbitrariness of *g* in the boundary conditions of (2.14) and (2.15) gives us the possibility of choosing *g* in such a way that the solutions *T*^{a} and *T*^{b} are available or can be obtained more easily. In particular, imagine that the inclusion *ω* is inside an infinite body made of the same homogeneous medium as the one constituting *Ω* and that *ω* is subjected to the uniform eigen temperature gradient **e***. The resulting temperature field over is denoted by . Then, we cut *Ω* containing *ω* out of and impose on the boundary ∂*Ω*, so that the temperature field in *Ω* remains unchanged and is equal to the restriction of to *Ω*. This amounts to posing
2.16Next, the sought temperature field *T* is obtained once the auxiliary boundary value problem formulated by
2.17is solved. Figure 1 illustrates how the solution to Eshelby's problem of an inclusion in a finite body can be decomposed into the one to the associated Eshelby's problem of an inclusion in an infinite body and the one to the corresponding auxiliary boundary value problem.

The foregoing strategy of using (2.16) and solving (2.17) to find *T* is efficient if is available and if the auxiliary problem (2.17) is simpler than the original one (2.1). In fact, has been analytically and explicitly obtained in the three-dimensional situation, where the inclusion *ω* is ellipsoidal and in the two-dimensional situations where *ω* is polygonal or characterized by the Laurent polynomial [23]. In addition, as will be shown in what follows, the problem (2.17) is indeed more straightforward than the original one (2.1) in a number of cases.

In §2*a*,*b*, the formulation of the problem and the problem solution strategy have been provided for a single inclusion inside a finite body. If the number of inclusions in a finite body is more than one, we can first use our method to treat each inclusion individually and then apply the superposition principle to obtain the final solution.

### (c) Structure of the solution

In §2*a*, it is seen that an anisotropic thermal conduction problem can be transformed into an equivalent isotropic one. For this reason, we shall confine our attention to the problem of an inclusion embedded in an isotropic material.

The thermal conductivity tensor for the isotropic material forming the finite body *Ω* is taken to be of the form *K*_{ij}=*kδ*_{ij} with *k* being a strictly positive scalar. Equation (2.1) governing the temperature field can then be written as
2.18It was shown [3,23] that, when *Ω* is an infinite body , the temperature field over due to **e*** is given by
2.19with Green's function *G*(**y**−**x**) representing the temperature at point **x** due to a unit point heat source at point **y** of . In formula (2.19), the derivative is made with respect to **y**. Owing to the fact that Green's function *G*(**y**−**x**) vanishes for boundary points and introducing the expression (2.2)_{2} of **q**^{p}, formula (2.19) yields
2.20where
2.21with *ds*_{i} being the components of an infinitesimal surface vector specified by
2.22In (2.19) and (2.21), use is made of Green's integral theorem. It is immediate from (2.20) that
2.23In this expression, the symmetric second-order tensor , called *Eshelby's conduction tensor*, is given by
2.24The vector field in (2.21) can be viewed as the potential of Eshelby's conduction tensor, because
2.25

The second-order symmetric tensor defined by (2.24) admits the irreducible decomposition (cf. [3])
2.26where is a scalar and is a second-order deviator. It was proved by Le Quang *et al*. [3] that the first part in (2.26) is independent of the geometry of *ω* and equal to Eshelby's conduction tensor in the case of a circular inclusion, namely
2.27with *d* (=2 or 3) being the dimension of *Ω*. Correspondingly, the second-order deviator in (2.26) occurs once the inclusion *ω* is neither circular nor spherical.

The solution to the auxiliary boundary value problem (2.17) has a similar structure as (2.20) and (2.23):
2.28However, Eshelby's conduction tensor **S**^{b} is in general not symmetric. In the two-dimensional case, **S**^{b} can be decomposed as follows [24]:
2.29where and are two scalars, ** ϵ** represents the two-dimensional permutation tensor and

**d**

^{b}is a two-dimensional second-order deviator. In the three-dimensional case, the irreducible decomposition of

**S**

^{b}takes the form: 2.30where

*α*

^{b}is a scalar,

*β*^{b}is a three-dimensional vector,

**stands for the three-dimensional permutation tensor and**

*ϵ***d**

^{b}corresponds to a three-dimensional second-order deviator.

## 3. General solution to Eshelby's problem of a two-dimensional thermal inclusion in a finite body

The method presented in §2 is now applied and detailed to give a general solution to Eshelby's two-dimensional thermal inclusion problem in a finite body. Apart from its own interest, this problem is meaningful for elasticity because of the mathematical analogy between two-dimensional thermal conduction and anti-plane elasticity.

### (a) Complex representation of the general solution

It is well known that Green's function for the two-dimensional thermal conduction in an infinite isotropic medium reads
3.1Substituting this expression into (2.21) and (2.24) yields
3.2and
3.3where **z**=**y**−**x**.

Denote by *i*_{1} and *i*_{2} two orthonormal base vectors associated to a Cartesian coordinate system and set **w**=*i*_{1}+*ι**i*_{2} with . Then, we have
3.4where *x*=*x*_{1}+*ιx*_{2}, *y*=*y*_{1}+*ιy*_{2} and *z*=*y*−*x*. To represent the fields and , let us introduce two complex functions:
3.5and
3.6In the same way, the complex representations
3.7
3.8
and
3.9can be constructed for the fields **U**^{b}, , and **d**^{b} involved in the auxiliary problem.

Now, setting in (2.13), the auxiliary boundary value problem (2.17) for *T*^{b} becomes
3.10This means that the field *T*^{b}(**x**) has to be a harmonic function satisfying the boundary condition imposed by the field .

Introduce the complex representations for the (negative) temperature gradient **e** and outward unit normal **n** of ∂*Ω*
3.11We can write
3.12and express and as
3.13

Letting the field *T*^{c}(**x**) be the harmonic conjugate of *T*^{b}, then an analytic function *H*(*x*) of *x* can be constructed such that
3.14We can then write the isothermal boundary condition on ∂*Ω* by
3.15For the adiabatic boundary condition on ∂*Ω*, we use (3.12) to obtain
3.16Note that
and can be understood to be function of for *x*∉*ω* since
Thus, the adiabatic boundary condition can be further expressed by
3.17so that
3.18a constant where the analytic function *H*(*x*) will be determined to be difference. This means that the two boundary conditions can be written in a unified way:
3.19with *η*=1 for the isothermal boundary and *η*=−1 for the adiabatic boundary.

From the above construction, it is seen that the solution to Eshelby's problem of a thermal inclusion in a finite body resides finally in determining an analytic function *H*(*x*) verifying the boundary condition (3.19) which involves the solution to the problem of a thermal inclusion in an infinite domain given by (3.5). The approach to finding the analytic function *H*(*x*) over *Ω* from (3.19) is to transform the involved problem into the Privalov or Riemann–Hilbert problem [25,26], as given in the following.

According to the Riemann mapping theorem [25], for any Jordan region *Ω*, there exists a unique conformal map
3.20which is analytic and one-to-one in a neighbourhood of the origin and normalized by *φ*(0)=0 and *φ*^{′}(0)=*R*>0. Remark also that the analytic function *H*(*x*) over *Ω* is still an analytic function *f*:=*H*°*φ* over the unit disc after the conformal mapping (3.20). Then by virtue of the symmetry extension, namely
3.21of the analytic function *f*(*w*) over the unit disc, which is analytic outside the unit disc, we introduce a piecewise analytic function
3.22to recast the boundary condition (3.19) into the jump relation
3.23The formulation of the Privalov or Riemann–Hilbert problem in (3.23) results in the integral formula of *F*(*w*) to within a constant:
3.24for |*w*|<1 and *x*=*φ*(*w*)∈*Ω*.

### (b) Explicit expression of the general solution

From the expressions (3.7)–(3.9) and the definition (3.11)_{1}, it is easy to deduce that
3.25Since *T*^{b}=Re *H* with *H* given by (3.24), a direct comparison with (3.25)_{1} results in
3.26

Using (3.26) and accounting for (3.8) and (3.9), we can derive 3.27and 3.28

Finally, the general solution to Eshelby's two-dimensional problem of a thermal inclusion *ω* in a finite body *Ω* can be summarized as
3.29where *U*_{i}(**x**) are determined through *x*=*φ*(*w*) by
3.30and
3.31with *η*=1 for the isothermal boundary and *η*=−1 for the adiabatic boundary. Then, Eshelby's conduction tensor **S** defined by can be derived as
3.32with
3.33and
3.34The general expressions of the temperature field given by (3.29)–(3.31) and Eshelby's conduction tensor indicated by (3.32)–(3.34) are remarkable owing to the fact that they are given in terms of three explicit integrals even Green's function is not available for a finite domain.

## 4. Examples: inclusions in a circular body

In §3, we have presented a general method for solving Eshelby's problem of a thermal inclusion in a two-dimensional finite body. In §2, the methodology is valid for both the two-dimensional and three-dimensional problems. After the formulation (2.1)–(2.4) of the problem, using a transformation procedure based on the decomposition (2.5)–(2.7) of the thermal conductivity tensor, the equivalence between the original anisotropic problem and an isotropic one of the transformed inclusion (2.8) embedded in the transformed body (2.9) with the conductivity tensor (2.10) is established in §2*a*. Then in §2*b*, exploiting the superposition principle, we decompose Eshelby's problem of an inclusion in a finite domain into two sub-problems (2.14) and (2.15) of which one is an infinite domain with the inclusion, and the other a finite domain without the inclusion but with the appropriate boundary compensation conditions derived from the first sub-problem. Since the solution of the former problem is available in many cases, we can focus our attention on the second problem. The tensor structure of its solution in addition to that of the former problem is given in §2*c*. In the two-dimensional case, the complex variable representation is introduced and the solution to the auxiliary boundary value problem can be ascribed to three complex variables (3.7)–(3.9). By introducing the analytic function (3.14), two kinds of boundary value problems can be condensed into (3.19), and the use of the symmetry extension (3.21)–(3.22) transforms the boundary value problem into the Privalov or Riemann–Hilbert problem (3.23), which admits the explicit integral solution (3.24). Finally, combining (3.29)–(3.34) with (2.27), (3.5) and (3.6), we obtain the explicit integral expression for Eshelby's two-dimensional thermal inclusion problem in §3.

In this section, the general two-dimensional results are illustrated by particularizing them in the situation where the finite body in question is circular. More precisely, we consider an arbitrarily shaped inclusion *ω* located inside a finite circular body *Ω* of radius *R*. With no loss of generality, assume that the centre of *Ω* coincides with the origin of the coordinate system. Then, using *φ*(*τ*)=*Rτ* and *w*=*x*/*R*, the field in (3.31) becomes
4.1and so
4.2Below, we study the cases that the inclusions are either circular or polygonal.

### (a) A circular inclusion inside a circular body

If the inclusion is circular and characterized by *y*=*h*+*r*e^{iϕ} with *h* defining its centre and *r* specifying its radius, we can obtain from (3.31), (4.1) and (4.2) that
4.3
4.4
and
4.5More specifically, if the inclusion has the same centre as *Ω*, namely *h*=0, the foregoing expressions reduce to
4.6and
4.7

### (b) Polygonal inclusions in a circular body

Polygonal inclusions are of particular importance for two reasons. First, they are involved in practical situations. Second, they can be used to approximate inclusions of complex shape.

Consider a generic polygonal inclusion *ω* with *N* rectilinear sides ∂*ω*_{(k)} (*k*=1,2,…,*N*). As in Zou *et al.* [9], the *k*th side ∂*ω*_{(k)} of *ω* is parametrized by
4.8where *y*_{(k)} and *y*_{(k+1)} are the end points of ∂*ω*_{(k)}. Defining with *z*_{(k)}=*y*_{(k)}−*x* and with [27], and performing the boundary integrals in (3.31) and (3.34) we obtain the compact formulae
4.9and
4.10Next, applying (4.1) and (4.2), it follows that
4.11and
4.12

As an example, we consider four square inclusions *ω*^{α} (*α*=1,2,3,4) embedded in the circular body *Ω*. The sides of these squares are parallel to the orthonormal basis vector *i*_{1} or *i*_{2}, and the length of each side is equal to 2*a*. The centres *c*_{(α)} (*α*=1,2,3,4) of the four squares are chosen to be the points (*b*,*b*), (−*b*,*b*), (−*b*,−*b*) and (*b*,−*b*) as illustrated in figure 2. The relations between *a*, *b* and *R* are taken to be
4.13

Denote by
4.14the complex coordinates of the four corners of a square centred to the origin. Then, the corner coordinates of the aforementioned square inclusion *α* can be conveniently calculated by . The solution to the resulting inclusion problem can be summarized as follows:
4.15
4.16
and
4.17The temperature and its gradient due to *e** are given by
4.18

The results (4.15)–(4.17) are plotted in figures 3 and 4 by taking *η*=1 for the isothermal boundary condition imposed on ∂*Ω*. As expected from
the component fields *U*_{1} and *U*_{2} of *γ*_{1} shown in figure 3 have the reflection symmetry with respect to the axes *i*_{1} and *i*_{2}, respectively. The real parts of *α* and *γ*_{2} illustrated in figure 4*a*,*c* have the reflection symmetry in regard to both the axes *i*_{1} and *i*_{2}, while their image parts shown in figure 4*b*,*d* exhibit the anti-reflection symmetry relative to both the axes *i*_{1} and *i*_{2}. Using (4.16), it is easy to prove that the isothermal boundary condition *T*=0 on ∂*Ω* is equivalent to , while the adiabatic boundary condition *n*_{i}∂_{i}*T*=0 on ∂*Ω* amounts to . In our illustrations, the former fact for the isothermal boundary is seen to be satisfied very well, which is also shown in figure 3 by the observation that *γ*_{1} vanishes when *x* approaches the boundary ∂*Ω*.

## 5. Concluding remarks

Eshelby's problem of inclusions inside a finite medium has been studied in the context of steady thermal conduction viewed as a prototype of a number of similar physical transport phenomena. The difficulties of this problem owing to the finiteness of the ambient medium are overcome by elaborating a method analogous to the one presented in our recent work [21] framed within the theory of linear elasticity. However, since steady thermal conduction is simpler than elasticity, the results obtained in the present work are more general than those provided in the context of elasticity; in particular, the ambient finite medium is not required to be isotropic. In addition, with the help of the fact that anti-plane elasticity is mathematically identical to the two-dimensional thermal conduction, the results obtained in §§3 and 4 can be directly transposed to those for anti-plane elasticity, which can be further combined with the results derived by Zou *et al.* [22] for plane elasticity to deliver the solution to the problem of a cylindrical inclusion in a cylindrical body of finite cross-section undergoing generalized plane strains.

The method elaborated in the present work to solve Eshelby's thermal inclusion problem in question is efficient in the cases where Green's function is unavailable analytically or does not exist. It is also consistent with the direct method in the rare cases where Green's function is available. To comment on these two points, let *G*(**x**,**y**) stand for the temperature variation evaluated at a point **x** of a finite medium *Ω* produced by a unit heat source located at a point **y** of *Ω*, and assume the material forming *Ω* to be isotropic and characterized by the scalar conductivity *k*. Then, the field equation to be satisfied by *G* is given by
with *δ* being the Dirac function. When the boundary conditions of *Ω* are adiabatic (or of Neumann-type), the corresponding boundary problem is ill-posed and Green's function *G* is non-existent. When the boundary conditions of *Ω* are isothermal (or of Dirichlet type), Green's function *G* exists but is seldom available analytically. Consider the interesting case where *Ω* is a disc of radius *R* subjected to isothermal boundary conditions. In this situation, Green's function is known to be given by [28]
5.1Using *G*(**x**,**y**)=0 on ∂*Ω*, formulae (3.4)_{3} and (2.21), we can obtain
5.2On the other hand, combining (3.5) and (4.1) yields
5.3Observing that
5.4it is easy to verify that (5.3) is identical to (5.2).

## Funding statement

The financial support from the National Natural Science Foundation of China (grant nos. 10872086 and 11072105) is acknowledged.

## Acknowledgements

Z.W.N. acknowledges valuable discussion with Prof. Ernian Pan of Akron University.

- Received April 7, 2013.
- Accepted June 7, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.