## Abstract

Eshelby’s inclusion problem is solved for non-elliptical inclusions in the context of two-dimensional thermal conduction and for cylindrical inclusions of non-elliptical cross section within the framework of generalized plane elasticity. First, we consider a two-dimensional infinite isotropic or anisotropic homogeneous medium with a non-elliptical inclusion subjected to a prescribed uniform heat flux-free temperature gradient. Eshelby’s conduction tensor field and its area average are first expressed compactly in terms of two boundary integrals avoiding the usual singularity and then specified analytically for arbitrary polygonal inclusions and for inclusions characterized by the finite Laurent series. Next, we are interested in a three-dimensional infinite isotropic or transversely isotropic homogeneous medium with a cylindrical inclusion of a non-elliptical cross section that undergoes uniform generalized plane eigenstrains. The solution to this problem is obtained by decomposing a generalized plane eigenstrain tensor into a plane strain part and an anti-plane strain part, exploiting the mathematical similarity between two-dimensional thermal conduction and anti-plane elasticity, and combining the relevant results of Zou *et al.* (Zou *et al.* 2010 *J. Mech. Phys. Solids* **58**, 346–372. (doi:10.1016/j.jmps.2009.11.008)) with those derived in the present work for Eshelby’s conduction tensor field and its area average.

## 1. Introduction

The thermal counterpart of the well-known Eshelby elastic inclusion problem (Eshelby 1957, 1959) corresponds to an infinite homogeneous medium *Ω* containing a subdomain *ω*, called a thermal inclusion, over which a uniform heat flux-free temperature gradient (or an eigen thermal gradient) is prescribed (e.g. Hatta & Taya 1985; Le Quang *et al.* 2008). Eshelby’s elastic inclusion problem is of prominent importance to a large variety of mechanical and physical phenomena and plays an important role in particular in micromechanics (e.g. Mura 1987; Nemat-Nasser & Hori 1993). Eshelby’s thermal inclusion problem is mathematically simpler than its elastic counterpart but physically far reaching since numerous transport phenomena (e.g. Beleggia *et al.* 2009; De Graef & Beleggia 2009), such as electric conduction, dielectrics, magnetism, diffusion and flow in porous media, are analogous to thermal conduction (e.g. Beleggia *et al.* 2009; De Graef & Beleggia 2009). In addition, anti-plane elasticity is mathematically identical to two-dimensional thermal conduction. It is useful to note that boundary layer techniques and the notions of polarization and moment tensors, which are related to the solutions to Eshelby’s elastic and thermal conduction problems, have been developed and applied by Ammari *et al.* (2005) and Ammari & Kang (2007) to determine the effective elastic and transport properties of composite materials.

The analytical or closed-form solutions reported for Eshelby’s thermal inclusion problem have been up to now limited to the situation where inclusions are elliptical in the two-dimensional case or ellipsoidal in the three-dimensional case. Under this limitation, Eshelby’s conduction tensor field, which is by definition the second-order tensor *Σ*^{ω} relating the uniform heat flux-free temperature gradient prescribed over *ω* to the resulting temperature gradient over *Ω*, is uniform inside *ω*. As in elasticity, this uniformality makes it possible to use Eshelby’s equivalent inclusion idea. More precisely, the problem of determining the temperature gradient field over an infinite homogeneous medium containing an elliptical or ellipsoidal inhomogeneity and undergoing a remote uniform temperature gradient can be reduced to Eshelby’s thermal inclusion problem by imposing a suitable uniform heat flux-free temperature gradient. A number of studies dealing with estimation of the effective transport properties of inhomogeneous media have considered only elliptical or ellipsoidal inhomogeneities and resorted to Eshelby’s equivalent inclusion idea (Hatta & Taya 1985, 1986; McLachlan 1987; Miloh & Benveniste 1988; Whitehouse *et al.* 1991; Dunn *et al.* 1993; Lu 1998, 1999; Shafiro & Kachanova 2000; Berryman 2005; Giordano & Palla 2008).

However, as conjectured early by Eshelby (1961) and proved recently by Kang & Milton (2008) and Liu (2008) (see also Liu 2009; Ammari *et al.* 2010), Eshelby’s tensor field inside an inclusion is uniform if and only if this inclusion is elliptical or ellipsoidal. This important result has the consequence that Eshelby’s equivalent inclusion idea is no longer applicable when a non-elliptical or non-ellipsoidal inhomogeneity is concerned. At the same time, the inclusions and inhomogeneities encountered in problems of practical interest are mostly often non-elliptical or non-ellipsoidal. In view of this rather unsatisfactory situation, we have recently carried out a thorough study of Eshelby’s problem of non-elliptical inclusions and answered some fundamental questions in the context of two-dimensional elasticity (Zou *et al.* 2010). The present work is a continuation of our previous study and has the following three objectives:

— First, it aims at elaborating an approach similar to the one of Zou

*et al.*(2010), to obtain analytically and explicitly the expressions of Eshelby’s conduction tensor field for a wide class of non-elliptical inclusions. Owing to the fact that Eshelby’s thermal inclusion problem is mathematically simpler than its elastic counterpart, the limitative assumption that the infinite medium is isotropic is not made in the present work.— Second, it consists in exploiting the mathematical similarity between two-dimensional thermal conduction and anti-plane elasticity to solve Eshelby’s elastic inclusion problem in the case of anti-plane strains by directly using the relevant thermal conduction results.

— Third, it has the purpose of determining Eshelby’s tensor field and its area average for cylindrical elastic inclusions of non-elliptical cross section by combining the relevant results of Zou

*et al.*(2010) with those of the present work concerning Eshelby’s thermal inclusion problem.

To reach the foregoing threefold objective, the paper is organized as follows. In the next section, the definition of Eshelby’s conduction tensor field *Σ*^{ω} and an irreducible decomposition of *Σ*^{ω} are given, as in Le Quang *et al.* (2008). In §3, two simple and compact boundary integral expressions are derived for *Σ*^{ω} and the area average of *Σ*^{ω}, avoiding the singularity involved in the usual expressions of *Σ*^{ω} and . The case of an isotropic medium is investigated directly while that of an anisotropic medium is treated indirectly with the help of a linear mapping transforming the anisotropic medium into an isotropic medium. In §4, the explicit analytical expressions of *Σ*^{ω} and are derived for inclusions of different shapes characterized by the finite Laurent series and for arbitrary polygonal inclusions in the case of an isotropic medium. Section 5 is concerned with the case of an anisotropic medium. However, to avoid cumbersomeness, the expressions of *Σ*^{ω} and are not specified and only an example is examined in detail to illustrate the effects of material anisotropy. In §6, Eshelby’s inclusion problem is solved first in the case of anti-plane elasticity and then in the case of generalized plane elasticity by using the mathematical similarity between two-dimensional thermal conduction and anti-plane elasticity, decomposing a tensor of generalized plane strains into a plane strain part and an anti-plane strain part, and combining the relevant results derived in the present work with those provided in our previous study (Zou *et al.* 2010). In §7, A few concluding remarks are given.

## 2. Eshelby’s conduction tensor

Let *Ω* be a two-dimensional infinite homogeneous medium whose thermal conduction is characterized by Fourier’s Law. We are now interested in the thermal counterpart of the well-known Eshelby inclusion problem in elasticity. Thus, by analogy, a (negative) heat flux-free or eigen temperature gradient **e*** is prescribed over a subdomain *ω* of *Ω*, called inclusion. More precisely, **e*** is required to be uniform inside *ω*, so that **e*** can be conveniently expressed as
2.1
where **e**^{0} is a constant two-dimensional vector and *χ*^{ω} is the characteristic function of *ω* such that *χ*^{ω}(**x**)=1 for **x**∈*ω* and *χ*^{ω}(**x**)=0 for **x**∉*ω*.

In Le Quang *et al.* (2008), it is shown that the temperature field *T*(**x**) over *Ω* owing to **e**^{0} can be calculated by
2.2
In this formula, *K*_{ij} are the matrix components of the thermal conductivity tensor **K** of the material forming *Ω*, which is symmetric and definite positive; *G*(**y**−**x**) is the Green function giving the temperature at point **x** generated by a unit point heat source at point **y** of *Ω*; the subscript *i* following a comma stands for the derivative with respect to *y*_{i}. It is immediate from equation (2.2) that
2.3
In this expression, the second-order tensor *Σ*^{ω} is called *Eshelby’s conduction tensor* and is given by
2.4

It is proved in Le Quang *et al.* (2008) that *Σ*^{ω} admits the decomposition
2.5
in which the first term in the right-hand side is the isotropic part of *Σ*^{ω} with *Σ*^{0} being identical to Eshelby’s conduction tensor field inside a circular inclusion, namely
2.6
and the second term in the right-hand side is the traceless part of *Σ*^{ω}. The formulae (2.5) and (2.6) imply that, independently of the geometry of *ω* and of the thermal anisotropy of the material forming *Ω*, the isotropic part of *Σ*^{ω} corresponds to Eshelby’s conduction tensor field in the case of a circular inclusion, which is uniform inside *ω* and vanishes outside *ω*. Consequently, the problem to be solved is to determine the traceless part **d**^{ω} of *Σ*^{ω} for non-elliptical inclusions.

## 3. Boundary integral expression for Eshelby’s conduction tensor

Inspired by the recent work of Zou *et al.* (2010), we proceed now to derive a boundary integral expression for Eshelby’s conduction tensor *Σ*^{ω}. As in the elastic case, this boundary integral expression allows the singularity met in the area integral expression (2.4) of *Σ*^{ω} to be avoided and rends it much easier to obtain explicit analytical expressions of *Σ*^{ω} for non-elliptical inclusions.

### (a) Isotropic media

When the material forming *Ω* is isotropic, the thermal conductivity tensor takes the simple form *K*_{ij}=*kδ*_{ij} with *k* being a positive constant and the Green function *G*(**y**−**x**) is given by
3.1
Introducing this expression and *K*_{ij}=*kδ*_{ij} into equation (2.4) yields
3.2
where **z**=**y**−**x**. It follows from equation (3.2) that Eshelby’s conduction tensor *Σ*^{ω} in the isotropic case is symmetric and independent of the material parameter *k*. This immediately implies that the traceless part **d**^{ω} involved in formula (2.5) is also symmetric; precisely,
3.3
where ‘Ir’ denotes the operation of taking the traceless-symmetric part.

Among the components of **d**^{ω}, there are the following relationships:
3.4
Consequently, **d**^{ω} has at maximum two independent components. Setting
3.5
formulae (2.5) and (2.6) can be recast into the convenient matrix form
3.6

Letting **i**_{1} and **i**_{2} be the two orthonormal vectors associated with a Cartesian coordinate system and setting **w**=**i**_{1}+*ι***i**_{2} and *γ*_{2}=*p*_{2}+*ιq*_{2} with as the unit imaginary number, we have the following expression (cf. Zheng 1993; Zheng & Zou 2000):
3.7
where the overbar is used to denote the complex conjugation and **P**_{2} and **Q**_{2} are the real and imaginary parts, respectively, of **w**^{⊗2}. With the representation (3.7), we can write the symmetric-traceless part of any second-order tensor **U** as
3.8
Since *w*_{i}*w*_{i}=0, it follows from equation (3.8)_{2} that the symmetric-traceless part of *aδ*_{ij} is equal to zero. With the aid of this simple property as well as *z*_{i,j}=*δ*_{ij} and |**z**|_{,i}=*z*_{i}/|**z**|, we can rewrite equation (3.3) in the alternative form
3.9

Using the Green theorem, we obtain from the first relation of equation (3.9) that
where ds^{y}=**n** d*s*^{y} with the unit outward vector **n** normal to ∂*ω* and the arc length element *ds*^{y} at the boundary point *y*. Setting
3.10
and resorting to equation (3.8), we finally have the expression for the complex coefficient *γ*_{2},
3.11
where *x*=*x*_{1}+*ιx*_{2}, *y*=*y*_{1}+*ιy*_{2} and *z*=*y*−*x* are the complex representations of the position vectors **x**, **y** and the relative position vector **z**=**y**−**x**. Remark that the increasing direction of *dy* is defined so as to keep *ω* on the left-hand side as the Cartesian coordinate system is anticlockwise oriented.

For the average of *γ*_{2} over the inclusion *ω*, denoted by , we start from the second relations of equation (3.9), note that ()_{,i}=−∂()/∂*x*_{i}, and apply the Green theorem. Thus,
where |*ω*| denotes the inclusion area. Using equations (3.8) and (3.10), we obtain
3.12

The average of *Σ*^{ω} over *ω*, referred to as the average Eshelby conduction tensor and designated by , admits the decomposition similar to formula (2.5); namely,
3.13
where **1** is the two-dimensional second-order identity tensor and the average of **d**^{ω} over *ω* is determined once is calculated by equation (3.12).

### (b) Anisotropic media

In the general case, when the material forming *Ω* is anisotropic, we first make the spectral decomposition of the conductivity tensor **K**,
3.14
where the eigenvalues *k*_{1} and *k*_{2} of **K** are ordered so that *k*_{2}>*k*_{1}>0 and and are two perpendicular unit vectors defining the principal directions of **K**. It is easy to see that **K** can be rewritten in the form
3.15
where and
3.16
with *c*=(*k*_{1}/*k*_{2})^{1/4}.

Following Le Quang *et al.* (2008), Eshelby’s problem for an inclusion *ω* in an infinite anisotropic medium characterized by **K** is now transformed into Eshelby’s problem for the inclusion
3.17
in the infinite isotropic medium with the conductivity tensor . Consequently, Eshelby’s conduction tensor field *Σ*^{ω} for the anisotropic medium is related to the one for the corresponding isotropic medium by Le Quang *et al.* (2008)
3.18
where . Note that, in general, while .

Formula (3.18) provides a direct and simple way to determine *Σ*^{ω} via which can be calculated by using the method presented in the last subsection. Since the transformation tensor **B** is constant and preserves area, the average of *Σ*^{ω} over *ω* is simply connected to the average of over by
3.19

## 4. Analytical solutions for inclusions of various shapes in an infinite isotropic medium

With the help of the boundary integral expressions (3.11) and (3.12), we proceed now to analytically and explicitly find Eshelby’s conduction tensor and its average for inclusions of various shapes. Since the procedure and technical details involved are similar to those presented in our recent work on Eshelby’s problem of elastic non-elliptical inclusions (Zou *et al.* 2010), they are omitted here to alleviate the presentation.

### (a) Inclusions characterized by the Laurent series

It is known from the Riemann mapping theorem (e.g. Henrici 1986) that, for any simple-closed curve *Γ*, there exists a unique Laurent series
4.1
which maps the unit circle *U* centred at the origin onto *Γ* and the outer domain of *U* onto the one of *Γ*. In formula (4.1), *f*_{0} is a unique inner point of the domain *ω* bounded by *Γ*, *R* is a positive real number, and every complex coefficient *b*_{k} satisfies |*b*_{k}|<1/*m*. For Eshelby’s inclusion problem, we can, without loss of generality, shift and zoom *ω* so that the parameters *R* and *f*_{0} defining the size and centre of *ω* become *f*_{0}=0 and *R*=1. Thus, the Laurent series expression for the boundary ∂*ω*=*Γ* of *ω* takes the simple form
4.2
The expression (4.2) has very good convergence in approximating various two-dimensional shapes (e.g. Zou *et al.* 2010). In practice, we can use the finite Laurent series
4.3
to approximate an arbitrarily shaped inclusion.

Based on the boundary integral formulae (3.11), the explicit expressions of *γ*_{2} for various inclusions characterized by the finite Laurent series can be obtained with the aid of the residue theorem from the complex analysis. As examples, the expressions of *γ*_{2} evaluated inside elliptical, hypocycloidal quasi-parallelogram inclusions are specified below:

— when

*ω*is an ellipse (=2-gonal hypocycloid) with*y*(*θ*)=*e*^{ιθ}+*b*_{1}*e*^{−ιθ}, 4.4— when

*ω*is a 3-gonal hypocycloid with*y*(*θ*)=*e*^{ιθ}+*b*_{2}*e*^{−ι2θ}, 4.5— when

*ω*is an (*n*+1)-gonal hypocycloid with*y*(*θ*)=*e*^{ιθ}+*b*_{n}*e*^{−ιnθ}and*n*≥3, 4.6— when

*ω*is a quasi-parallelogram with , 4.7

The constant solution (4.4) corresponds to Eshelby’s one for an elliptical inclusion. Interestingly, the expressions (4.5–4.7) of *γ*_{2} inside hypocycloidal and quasi-parallelogram inclusions are polynomial functions of coordinates and thus bounded.

For more complicated analysis of inclusions described by
4.8
with non-zero coefficients *b*_{1} and *b*_{n}, the expression of *γ*_{2} evaluated inside *ω* is provided by
4.9
where are the binomial expansion coefficients and [*n*/2] denotes the integer part of *n*/2 such that, for example, [3/2]=1 and [1/2]=0.

With the help of expression (3.12) and the residue theorem, the average Eshelby conduction tensor can also be specified for various inclusions characterized by the finite Laurent series. For instance, symbolizing the area of *ω* by |*ω*|, the average values of for inclusions characterized by equation (4.9) have the following explicit expressions:

— when

*n*is an even number ≥2, 4.10— when

*n*is an odd number such that*n*=2*m*+1≥3, 4.11

It is particularly interesting to note the solution for a quasi-parallelogram inclusion corresponding to *n*=3,
4.12
For hypocycloidal inclusions with *b*_{1}=0 in equations (4.10 and 4.11), it is inferred that for any integer *n*≥2. This result is in agreement with the general one presented by Le Quang *et al.* (2008) on the average Eshelby conduction tensor.

### (b) Polygonal inclusions

#### (i) General solutions

Consider a generic polygonal inclusion *ω* whose boundary consists of *N* rectilinear sides ∂*ω*_{(k)} with *k*=1,2,…,*N*. As in Zou *et al.* (2010), the *k*th side ∂*ω*_{(k)} of *ω* is parametrized through
4.13
where *y*_{(k)} and *y*_{(k+1)} are the end points of ∂*ω*_{(k)} (figure 1). Defining and with (Zill & Shanahan 2003), we carry out the boundary integral (3.11) and obtain the compact formula
4.14
Because of the logarithmic terms involved in this formula, the prescription of the arguments *θ*_{(k)} of *z*_{(k)} with *k*=1,2,…,*N* has to comply with the following rule. The range of *θ*_{(1)} is chosen to be (−*π*,*π*]. If *z*_{(2)} is anticlockwise/clockwise rotated from *z*_{(1)} through an angle smaller than *π*, then *θ*_{(2)} is assigned to be larger/smaller than *θ*_{(1)}. In the same way, *θ*_{(k+1)} is considered as being larger/smaller than *θ*_{(k)} when the anticlockwise/clockwise rotation from *z*_{(k)} to *z*_{(k+1)} is less than *π*. For a simply connected polygonal inclusion with *N* sides, the complex point *z*_{(N+1)} can be superposed to *z*_{(1)} but must possess argument if *x* is an interior point. The ranges of *ϕ*_{(k)} are defined in the same way when calculating the average Eshelby tensor.

Expressing *z*_{(k)} and *s*_{(k)} as
where *R*_{(k)} and *θ*_{(k)} are the norm and argument of *z*_{(k)} and *L*_{(k)} and *ϕ*_{(k)} are those of *s*_{(k)}, it follows from formula (4.14) that
4.15
which is sometimes more convenient.

To carry out the boundary integral (3.12) related to the average Eshelby conduction tensor, let *x* be a point on the *j*th side and *y* a point on the *k*th side defined by
4.16
Thus, we have
4.17
with
4.18
Next, the boundary integral (3.12) can be calculated exactly as
4.19
In this formula, *C*_{(jk)} is symmetric in *j* and *k* and specified as follows:
4.20
when *k*=*j*+1, and
4.21
when *k*>*j*+1. Remark that, in the foregoing formulae, when *j* (or *k*) is equal to *N*, we have to set *j*+1=1 (or *k*+1=1).

#### (ii) Special polygonal inclusions

Formulae (4.14) and (4.19) allow us to explicitly calculate Eshelby’s conduction tensor and its average for an arbitrary convex or non-convex polygonal inclusion. In this subsection, for simplicity, we apply them only to a few convex inclusions as illustrative examples.

*Parallelogram, rectangle and square inclusions.* Let *ω* be a parallelogram inclusion whose side lengths are equal to 2*a* and 2*b* (*a*≥*b*) and whose phase angle is *ϕ*≤*π*. Since the Eshelby tensor field is size-independent, *a* and *b* can without loss of generality be chosen to be *a*=1 and *b*=*η*≤1. The larger sides are set to be parallel to the orthonormal basis vector **i**_{1}. As illustrated in figure 2, the four vertices can be defined by
4.22
Substituting
into equation (4.15), and noting *R*_{(5)}=*R*_{(1)} and *θ*_{(5)}=*θ*_{(1)} for an exterior point and *θ*_{(5)}=2*π*+*θ*_{(1)} for an interior point, we obtain
4.23
In this expression, *χ*^{ω}(*x*) is the characteristic function of *ω*. By formula (4.19), we can deduce
4.24

In the case where *ω* reduces to a rectangular inclusion, it suffices to set *ϕ*=*π*/2. Correspondingly, formulae (4.23) and (4.24) are simplified into
4.25
and
4.26
If *ω* reduces further to a square-shaped inclusion, we pose *η*=1 in formulae (4.25) and (4.26). Interestingly, we obtain , which means that the traceless part of the average Eshelby conduction tensor is null for a square-shaped inclusion. This is in agreement with the results of Le Quang *et al.* (2008).

*Regular polygonal inclusions.* Let *ω* be an *N*-fold (*N*≥3) regular polygonal inclusion inscribed into a unit circle. Thus, the vertices of *ω* are specified by
4.27
with *k*=1,2,…,*N* and *θ*=2*π*/*N*. Inserting equation (4.27) and *s*_{(k)}=*e*^{ι(k+1)θ}−*e*^{ιkθ} into equation (4.15), we derive
4.28
which is valid for any point inside and outside *γ*_{2}. In particular, the interior solution for a regular triangle inclusion takes the simple form
4.29
It follows from equation (4.28) that the value of *γ*_{2} evaluated at the centre of the inclusion, namely *x*=0, vanishes for all *N*-fold (*N*≥3) regular polygonal inclusions.

Directly from expression (3.12) and the geometrical symmetry of an *N*-fold (*N*≥3) regular polygonal inclusion, we can infer that the traceless part of its average Eshelby conduction tensor is equal to zero.

## 5. Analytical solutions for inclusions of various shapes in an infinite anisotropic medium

In the last section, the explicit analytical expressions of Eshelby’s conduction tensor are derived for any inclusion characterized by finite Laurent series and any polygonal inclusion in an infinite isotropic medium. With the help of these expressions and by means of formulae (3.18) and (3.19), we can directly derive the explicit analytical expressions of Eshelby’s conduction tensor for the relevant inclusions in an infinite anisotropic medium. To avoid rendering the present paper lengthy, we do not specify these results in the anisotropic case. However, to illustrate the effect of anisotropy, we examine a simple example in detail.

Consider a unit square-shaped inclusion *ω* whose sides are parallel to the orthonormal basis vectors **i**_{1} and **i**_{2} and which is in an infinite medium characterized by the conductivity tensor **K** having the spectral decomposition (3.14). By hypothesis, we have *k*_{2}>*k*_{1} so that *c*=(*k*_{1}/*k*_{2})^{1/4}<1. With no loss of generality, it is assumed that |*θ*|≤*π*/4. The transformed inclusion given by equation (3.17) is a parallelogram of the same area as *ω* (figure 3).

The vertices of the unit square-shaped inclusion *ω* being defined by {*e*^{ι(π/4)},*e*^{ι(3π/4)},*e*^{−ι(3π/4)},*e*^{−ι(π/4)}}, those of the transformed inclusion are specified by
with
According to equation (3.17), the angle parameters *α* and *β* of as shown in figure 3 can be determined by
In addition, the side ratio *η* is given by
In summary, the transformed inclusion in the isotropic medium characterized by is a parallelogram with the phase angle *β*, side ratio *η* and rotation angle *α* determined before.

Applying formulae (3.18) and (3.19), the complex coefficient and its average associated with the square-shaped inclusion *ω* can be determined from the counterparts and of the transformed inclusion by
Note that formulae (4.23) and (4.24) can be directly used to obtain the expressions of and . Then, it follows that
5.1
and
5.2

To see the effects of the material anisotropy and the inclusion orientation with respect to the material symmetry axes, let us discuss four cases according to the values of *c* and *θ*.

—

*c*<1 and*θ*=0. In this case, we have*α*=0,*β*=*π*/2,*η*=*c*^{2}<1, and the transformed inclusion is rectangular. Correspondingly, and—

*c*<1 and*θ*=±*π*/4. In this case, ,*η*=1 and is a diamond. Thus, and— When , the material tends to being isotropic. Consequently, , , and and

— When , the material exhibits high anisotropy. In such a case, we have and

The foregoing four cases show that the degree of the material anisotropy and the inclusion orientation relative to the material symmetry axes affect Eshelby’s conduction tensor and its average as expected.

## 6. Elastic cylindrical inclusions of non-elliptical cross section

Now, let *Φ* be a cylindrical domain of finite cross section but infinite length in an infinite body *Ω*. With no loss of generality, the generators of *Φ* are taken to be along the unit vector **i**_{3} of the orthonormal basis {**i**_{1},**i**_{2},**i**_{3}} of a Cartesian coordinate system. A uniform anti-plane eigenstrain field of the form
is prescribed over *Φ*. The linearly elastic and homogeneous material forming the infinite body *Ω* is assumed to be such that the strain field **e** over *Ω* owing to **e**^{0} remains anti-plane and takes the form
This is possible when the constituent material of *Ω* is isotropic, cubic, transversely isotropic, tetragonal, orthotropic or monoclinic (e.g. Le Quang *et al.* 2008). Then, the corresponding Eshelby tensor field **S**^{Φ} is, by definition, the second-order tensor relating **e** to **e**^{0},
6.1

Let *ω* denote the cross section of *Φ*, which is in general non-elliptical. It is known that the problem of determining Eshelby’s tensor field **S**^{Φ} in the case of anti-plane elasticity is mathematically equivalent to the one of finding Eshelby’s tensor field **S**^{ω} in the case of two-dimensional thermal conduction (e.g. Le Quang *et al.* 2008). This equivalence allows us to use the above results derived for **S**^{ω} directly in determining **S**^{Φ} not only in the isotropic situation but also in the aforementioned anisotropic situations. The expressions thus obtained for **S**^{Φ} are new in the context of elasticity.

Next, consider the general case where a uniform generalized plane eigenstrain tensor
is imposed over the cylindrical domain *Φ*. This eigenstrain tensor can be decomposed into a plane strain part and an anti-plane strain part ,
6.2
where and are two complementary orthogonal projection operators such that
6.3
The strain tensor field ** ε** over

*Ω*owing to

*ε*^{0}is formally given by Eshelby’s three-dimensional fourth-order tensor as follows: 6.4

If the material forming *Ω* is isotropic or transversely isotropic about the axis along **i**_{3}, we can write
and
so that
6.5
with
6.6
The expressions of Eshelby’s tensor field given by Zou *et al.* (2010) in the context of two-dimensional isotropic elasticity can be directly applied to determine while those deduced above for **S**^{Φ} can be immediately used to specify . Thus, we have completely solved Eshelby’s problem of an infinite cylindrical inclusion of non-elliptical cross section in an isotropic elastic medium or a transversely isotropic elastic medium whose symmetry axis is perpendicular to the cylindrical cross section.

To avoid cumbersomeness, here we do not specify the general result (6.5) for all possible cylindrical elastic inclusions of non-elliptical section. However, as an example of illustration, we consider a cylindrical elastic inclusion *Φ* whose cross section *ω* is rectangular and has sides parallel to the orthonormal basis vectors **i**_{1} and **i**_{2}. In this case, formula (4.25) allows us to specify equation (6.1) as
6.7
with
6.8
In addition, from Zou *et al.* (2010), we know that
6.9
with
6.10
Using equation (6.5), we finally obtain Eshelby’s tensor for a cylindrical elastic inclusion of rectangular cross section
in which
6.11
and
6.12
with
These results can be applied, for example, directly to determine the strain and stress fields induced by uniform thermal eigenstrains in a cylindrical inclusion of rectangular cross section in an infinite isotropic medium.

## 7. Conclusion

Eshelby’s inclusion problem has received much less attention in studying transport phenomena, even though it is far reaching as in elasticity. Spurred by our recent work (Zheng *et al.* 2006; Zou *et al.* 2010) on Eshelby’s problem of non-elliptical elastic inclusions and observing that most of the existing analytical results about estimation of the effective transport properties of heterogeneous media are based on the solution to an elliptical or ellipsoidal inclusion in an infinite isotropic medium, we have in the present work solved Eshelby’s problems of non-elliptical thermal inclusions. The approach used is similar to that of Zou *et al.* (2010) but the results obtained are more general in the sense that no limitation is imposed on the material symmetry of the infinite homogeneous medium in which an inclusion is embedded. New results have been also obtained for Eshelby’s inclusion problem in the context of anti-plane elasticity by exploiting the mathematical equivalence between two-dimensional thermal conduction and anti-plane elasticity. Further, by splitting generalized plane eigenstrains into a plane eigenstrain part and an anti-plane eigenstrain part and by combining the relevant results of Zou *et al.* (2010) with those of the present work, we have solved Eshelby’s problem of cylindrical elastic inclusions of non-elliptical cross section.

In light of the analytical results obtained in the present work for Eshelby’s problem of non-elliptical thermal inclusions, we have also numerically examined the validity of the elliptical or ellipsoidal approximation widely used in estimating the effective transport properties of heterogeneous media. The corresponding numerical results, which have not been given here to avoid rendering the paper lengthy, support, in the context of thermal conduction, the same conclusions as those reached by Zou *et al.* (2010) within the framework of elasticity.

## Acknowledgements

Z.W.N. acknowledges the financial support from NSFC (grant no. 10872086).

- Received May 27, 2010.
- Accepted July 15, 2010.

- This journal is © 2010 The Royal Society