An elliptical inhomogeneity under nonuniform heat flux

Abstract

We use complex variable techniques to study the decoupled two-dimensional steady-state heat conduction and thermoelastic problems associated with an elliptical elastic inhomogeneity perfectly bonded to an infinite matrix subjected to a nonuniform heat flux at infinity. Specifically, the nonuniform remote heat flux takes the form of a linear distribution. It is found that the internal temperature and thermal stresses inside the elliptical inhomogeneity are quadratic functions of the two in-plane coordinates. Explicit closed-form expressions of the analytic functions characterizing the temperature and thermoelastic field in the matrix are derived.

1. Introduction

The analysis of an elliptical elastic inhomogeneity under a heat flux at infinity has been carried out by a few authors [1−5]. In previous discussions, the heat flux at infinity is assumed to be uniform. A uniform heat flux at infinity will induce a linear stress distribution inside the elliptical inhomogeneity. Although the effects of a nonuniform mechanical loading have been thoroughly investigated [6−12], the influence of a nonuniform heat flux on the induced thermoelastic field in a composite has seldom been examined. As in the case of mechanical loading, highly variable thermal loadings can also develop in composites.

In this paper, we study the decoupled two-dimensional steady-state heat conduction and thermoelastic problems associated with an elliptical elastic inhomogeneity embedded in an infinite matrix under a nonuniform heat flux at infinity. The nonuniform remote heat flux has a linear distribution. By means of Muskhelishvili’s complex variable formulation [13,14], closed-form analytic solutions for both heat conduction and thermoelasticity are derived. It is found that the internal temperature and thermal stresses inside the elliptical inhomogeneity are quadratic functions of the two in-plane coordinates. With the aid of analytic continuation, explicit expressions of the analytic functions characterizing the temperature and thermoelastic field in the matrix are derived.

2. Complex variable formulations

A Cartesian coordinate system $\{x_i\}(i=1,2,3)$ is established. In the case of two-dimensional isotropic thermoelasticity, the in-plane heat flux $(q_1,q_2)$ , the temperature T, the three in-plane stresses $({\sigma }_{11},{\sigma }_{22},{\sigma }_{12})$ , the two in-plane displacements $(u_1,u_2)$ , and the two stress functions $({\varphi }_1,{\varphi }_2)$ can be expressed in terms of the complex temperature function $g(z)$ and the two Muskhelishvili’s complex stress functions ϕ(z) and $\psi (z)$ of the complex variable $z=x_1+\mathrm{i}{x}_2$ as [13,14]

$$q_1-\mathrm{i}{q}_2=-\mathrm{mg}\prime (z),T=\mathrm{Re}\{g(z)\},$$

(1)

$$\begin{array}{c}{\sigma }_{11}+{\sigma }_{22}=2[\phi \prime (z)+\overline{\phi \prime (z)}],\hfill \\ {\sigma }_{22}-{\sigma }_{11}+2\mathrm{i}{\sigma }_{12}=2[\overline{z}\phi \prime \prime (z)+\psi \prime (z)],\hfill \\ 2\mu (u_1+\mathrm{i}{u}_2)=\kappa \phi (z)-z\overline{\phi \prime (z)}-\overline{\psi (z)}+2\mu \alpha \prime \int g(z)\mathrm{d}z,\hfill \\ {\varphi }_1+\mathrm{i}{\varphi }_2=\mathrm{i}[\phi (z)+z\overline{\phi \prime (z)}+\overline{\psi (z)}],\hfill \end{array}$$

(2)

where m is the heat conductivity, µ is the shear modulus; $\kappa =3-4\nu$ , $\alpha \prime =(1+\nu )\alpha$ for plane strain and $\kappa =(3-\nu )/(1+\nu )$ , $\alpha \prime =\alpha$ for plane stress with $\alpha$ being the thermal expansion coefficient and ν the Poisson’s ratio.

In addition, the stresses are related to the stress functions through [15]

$$\begin{array}{c}{\sigma }_{11}=-{\varphi }_{1,2},{\sigma }_{12}={\varphi }_{1,1},\hfill \\ {\sigma }_{21}=-{\varphi }_{2,2},{\sigma }_{22}={\varphi }_{2,1}.\hfill \end{array}$$

(3)

3. Analysis

As shown in Figure 1, we consider a two-phase composite composed of an internal elliptical inhomogeneity surrounded by an infinite elastic matrix. Let $S_1:x_1^2/a^2+x_2^2/b^2\le 1$ (with a and b representing the semi-major and semi-minor axes of the ellipse, respectively) and $S_2:x_1^2/a^2+x_2^2/b^2\ge 1$ denote the inhomogeneity and the matrix, respectively, which are perfectly bonded through the elliptical interface $L:x_1^2/a^2+x_2^2/b^2=1$ . The matrix is subjected to only a linear distribution of heat flux at infinity, which will be specified below. In what follows, the subscripts 1 and 2 are used to identify the respective quantities in $S_1$ and $S_2$ . In addition, *′ and *′′ are the real and imaginary parts of the complex number *.

Figure 1.

An elliptical inhomogeneity under a nonuniform heat flux at infinity.

First, we introduce the following conformal mapping function:

$$z=\omega (\xi )=R(\xi +\lambda {\xi }^{-1}),$$

(4)

where

$$R=\frac{a+b}{2},\lambda =\frac{a-b}{a+b}(R>0,0\le \lambda <1).$$

(5)

Using the mapping function in equation (4), the domain occupied by the matrix is mapped onto $\left|\xi \right|\ge 1$ , and the elliptical interface L is mapped onto the unit circle $\left|\xi \right|=1$ .

The corresponding boundary value problems for decoupled heat conduction and thermoelasticity take the following form in the physical z-plane:

$$\begin{array}{c}{g}_2(z)+\overline{g_2(z)}=g_1(z)+\overline{g_1(z)},\hfill \\ g_2(z)-\overline{g_2(z)}=\mathrm{\Lambda }{g}_1(z)-\mathrm{\Lambda }\overline{g_1(z)},z\in L;\hfill \end{array}$$

(6a)

$$g_2(z)\cong \frac{Q}{R^3}{z}^2+O\left(\frac{1}{z}\right),\left|z\right|\to \infty ,$$

(6b)

and

$$\begin{array}{c}{\phi }_2(z)+z\overline{{\phi \prime }_2(z)}+\overline{{\psi }_2(z)}={\phi }_1(z)+z\overline{{\phi \prime }_1(z)}+\overline{{\psi }_1(z)},\hfill \\ {\kappa }_2{\phi }_2(z)-z\overline{{\phi \prime }_2(z)}-\overline{{\psi }_2(z)}=\mathrm{\Gamma }{\kappa }_1{\phi }_1(z)-\mathrm{\Gamma }z\overline{{\phi \prime }_1(z)}-\mathrm{\Gamma }\overline{{\psi }_1(z)}-2{\mu }_2[{\alpha \prime }_2\int g_2(z)\mathrm{d}z-{\alpha \prime }_1\int g_1(z)\mathrm{d}z],z\in L;\hfill \end{array}$$

(7a)

$${\phi }_2(z)\cong O(1),{\psi }_2(z)\cong O(1),\left|z\right|\to \infty ,$$

(7b)

where Q is a given complex constant, and the two ratios Λ and Γ are defined by

$$\mathrm{\Lambda }=\frac{m_1}{m_2},\mathrm{\Gamma }=\frac{{\mu }_2}{{\mu }_1}.$$

(8)

Equation (6a) describes the continuity of temperature and normal heat flux across the perfect elliptical interface L, equation (6b) gives the remote asymptotic behavior of $g_2(z)$ due to the linear distribution of heat flux at infinity which is specified as follows

$$q_1\cong \frac{2m_2}{R^3}(Q\prime \prime x_2-Q\prime x_1)+O\left(\frac{1}{{\left|z\right|}^2}\right),q_2\cong \frac{2m_2}{R^3}(Q\prime \prime x_1+Q\prime x_2)+O\left(\frac{1}{{\left|z\right|}^2}\right),\left|z\right|\to \infty .$$

(9)

Equation (7a) describes the continuity of tractions and displacements across the perfect elliptical interface L, and equation (7b) is the stress-free condition at infinity.

Using the mapping function in equation (4), the boundary value problems for heat conduction and thermoelasticity take the following form in the image ξ-plane:

$$\begin{array}{c}{g}_2(\xi )+\overline{g_2(\xi )}=g_1(\xi )+\overline{g_1(\xi )},\hfill \\ g_2(\xi )-\overline{g_2(\xi )}=\mathrm{\Lambda }{g}_1(\xi )-\mathrm{\Lambda }\overline{g_1(\xi )},\left|\xi \right|=1;\hfill \end{array}$$

(10a)

$$g_2(\xi )\cong \frac{Q}{R}({\xi }^2+2\lambda )+O\left(\frac{1}{\xi }\right),\left|\xi \right|\to \infty ,$$

(10b)

and

$$\begin{array}{c}{\phi }_2(\xi )+\frac{\omega (\xi )\overline{{\phi \prime }_2(\xi )}}{\overline{\omega \prime (\xi )}}+\overline{{\psi }_2(\xi )}={\phi }_1(\xi )+\frac{\omega (\xi )\overline{{\phi \prime }_1(\xi )}}{\overline{\omega \prime (\xi )}}+\overline{{\psi }_1(\xi )},\hfill \\ {\kappa }_2{\phi }_2(\xi )-\frac{\omega (\xi )\overline{{\phi \prime }_2(\xi )}}{\overline{\omega \prime (\xi )}}-\overline{{\psi }_2(\xi )}=\mathrm{\Gamma }{\kappa }_1{\phi }_1(\xi )-\mathrm{\Gamma }\frac{\omega (\xi )\overline{{\phi \prime }_1(\xi )}}{\overline{\omega \prime (\xi )}}-\mathrm{\Gamma }\overline{{\psi }_1(\xi )}\hfill \\ -2{\mu }_2[{\alpha \prime }_2\int g_2(\xi )\omega \prime (\xi )\mathrm{d}\xi -{\alpha \prime }_1\int g_1(\xi )\omega \prime (\xi )\mathrm{d}\xi ],\left|\xi \right|=1;\hfill \end{array}$$

(11a)

$${\phi }_2(\xi )\cong O(1),{\psi }_2(\xi )\cong O(1),\left|\xi \right|\to \infty ,$$

(11b)

where $g_i(\xi )=g_i(\omega (\xi )),{\phi }_i(\xi )={\phi }_i(\omega (\xi )),{\psi }_i(\xi )={\psi }_i(\omega (\xi ))(i=1,2)$

The analytic function $g_1(z)$ defined in the elliptical inhomogeneity is assumed to take the following form:

$$g_1(z)=\frac{C}{R^3}{z}^2+\frac{D}{R},z\in S_1,$$

(12)

where C and D are two complex constants to be determined.

By enforcing the continuity conditions in equation (10a) with the use of equation (12) and with the aid of analytic continuation, we arrive at

$$\begin{array}{c}\multicolumn{1}{c}{g_2(\xi )=\frac{1}{2R}[C(1+\mathrm{\Lambda })+\overline{C}{\lambda }^2(1-\mathrm{\Lambda })]{\xi }^2+\frac{1}{2R}[C{\lambda }^2(1+\mathrm{\Lambda })+\overline{C}(1-\mathrm{\Lambda })]{\xi }^{-2}}\\ \multicolumn{1}{c}{+\frac{\lambda }{R}[C(1+\mathrm{\Lambda })+\overline{C}(1-\mathrm{\Lambda })]+\frac{1}{2R}[D(1+\mathrm{\Lambda })+\overline{D}(1-\mathrm{\Lambda })],\left|\xi \right|\ge 1.}\end{array}$$

(13)

Using equation (13) to satisfy the remote asymptotic condition in equation (10b), the following set of linear algebraic equations can be obtained:

$$\begin{array}{c}C(1+\mathrm{\Lambda })+\overline{C}{\lambda }^2(1-\mathrm{\Lambda })=2Q,\hfill \\ D(1+\mathrm{\Lambda })+\overline{D}(1-\mathrm{\Lambda })+2\lambda [C(1+\mathrm{\Lambda })+\overline{C}(1-\mathrm{\Lambda })]=4Q\lambda .\hfill \end{array}$$

(14)

Consequently, the two complex constants C and D can be uniquely determined from equation (14) as follows:

$$\begin{array}{c}C=\frac{2[Q(1+\mathrm{\Lambda })-\overline{Q}{\lambda }^2(1-\mathrm{\Lambda })]}{{(1+\mathrm{\Lambda })}^2-{\lambda }^4{(1-\mathrm{\Lambda })}^2},\hfill \\ D=\frac{\lambda (1-{\lambda }^2)(1-\mathrm{\Lambda })\{Q(1-{\mathrm{\Lambda }}^2)(1+{\lambda }^2)-\overline{Q}[{(1+\mathrm{\Lambda })}^2+{\lambda }^2{(1-\mathrm{\Lambda })}^2]\}}{\mathrm{\Lambda }[{(1+\mathrm{\Lambda })}^2-{\lambda }^4{(1-\mathrm{\Lambda })}^2]}.\hfill \end{array}$$

(15)

Thus, the assumption of the form of solution for heat conduction inside the elliptical inhomogeneity in equation (12) is indeed valid. Furthermore, the temperature is distributed inside the elliptical inhomogeneity as follows:

$$T=\frac{1}{R^3}[C\prime (x_1^2-x_2^2)-2C\prime \prime x_1{x}_2]+\frac{D\prime }{R},z\in S_1.$$

(16)

It is seen from the above expression that the temperature inside the elliptical inhomogeneity is a quadratic function of the two in-plane coordinates $x_1$ and $x_2$ .

In addition, by using equations (12) and (13), the term $\alpha {\prime }_2\int g_2(\xi )\omega \prime (\xi )\mathrm{d}\xi -\alpha {\prime }_1\int g_1(\xi )\omega \prime (\xi )\mathrm{d}\xi$ appearing in equation (11a) is explicitly determined as

$$\alpha {\prime }_2\int g_2(\xi )\omega \prime (\xi )\mathrm{d}\xi -\alpha {\prime }_1\int g_1(\xi )\omega \prime (\xi )\mathrm{d}\xi ={\delta }_3{\xi }^3+{\delta }_1\xi +{\delta }_{-1}{\xi }^{-1}+{\delta }_{-3}{\xi }^{-3},$$

(17)

where the four complex constants ${\delta }_3,{\delta }_1,{\delta }_{-1}$ , and ${\delta }_{-3}$ are given by

$$\begin{array}{c}{\delta }_3=\frac{{\alpha \prime }_2}{6}[C(1+\mathrm{\Lambda })+\overline{C}{\lambda }^2(1-\mathrm{\Lambda })]-\frac{{\alpha \prime }_1}{3}C,\hfill \\ {\delta }_1=\frac{{\alpha \prime }_2}{2}[C\lambda (1+\mathrm{\Lambda })+\overline{C}\lambda (1-\mathrm{\Lambda })(2-{\lambda }^2)+D(1+\mathrm{\Lambda })+\overline{D}(1-\mathrm{\Lambda })]-{\alpha \prime }_1(C\lambda +D),\hfill \\ {\delta }_{-1}=\frac{{\alpha \prime }_2}{2}\{C{\lambda }^2(1+\mathrm{\Lambda })+\overline{C}(1-\mathrm{\Lambda })(2{\lambda }^2-1)+\lambda [D(1+\mathrm{\Lambda })+\overline{D}(1-\mathrm{\Lambda })]\}-{\alpha \prime }_1\lambda (C\lambda +D),\hfill \\ {\delta }_{-3}=\frac{{\alpha \prime }_2}{6}\lambda [C{\lambda }^2(1+\mathrm{\Lambda })+\overline{C}(1-\mathrm{\Lambda })]-\frac{{\alpha \prime }_1}{3}C{\lambda }^3.\hfill \end{array}$$

(18)

The pair of analytic functions ${\phi }_1(z)$ and ${\psi }_1(z)$ defined in the elliptical inhomogeneity is assumed to take the following form:

$${\phi }_1(z)=\frac{A_1}{R}z+\frac{A_3}{R^3}{z}^3,{\psi }_1(z)=\frac{B_1}{R}z+\frac{B_3}{R^3}{z}^3,z\in S_1,$$

(19)

where $A_1,A_3,B_1$ and $B_3$ are four complex constants to be determined.

By enforcing the continuity conditions in equation (11a) with the use of equations (17) and (19) and with aid of analytic continuation, we arrive at

$$\begin{array}{c}\multicolumn{1}{c}{{\phi }_2(\xi )=\frac{(\mathrm{\Gamma }{\kappa }_1+1)A_3+3{\lambda }^2(1-\mathrm{\Gamma }){\overline{A}}_3+{\lambda }^3(1-\mathrm{\Gamma }){\overline{B}}_3-2{\mu }_2{\delta }_3}{{\kappa }_2+1}{\xi }^3}\\ \multicolumn{1}{c}{+\frac{(\mathrm{\Gamma }{\kappa }_1+1)A_1+(1-\mathrm{\Gamma }){\overline{A}}_1+\lambda (1-\mathrm{\Gamma }){\overline{B}}_1+3\lambda (\mathrm{\Gamma }{\kappa }_1+1)A_3+3(1-\mathrm{\Gamma })(2\lambda +{\lambda }^3){\overline{A}}_3+3{\lambda }^2(1-\mathrm{\Gamma }){\overline{B}}_3-2{\mu }_2{\delta }_1}{{\kappa }_2+1}\xi }\\ \multicolumn{1}{c}{+\frac{\lambda (\mathrm{\Gamma }{\kappa }_1+1)A_1+\lambda (1-\mathrm{\Gamma }){\overline{A}}_1+(1-\mathrm{\Gamma }){\overline{B}}_1+3{\lambda }^2(\mathrm{\Gamma }{\kappa }_1+1)A_3+3(1-\mathrm{\Gamma })(1+2{\lambda }^2){\overline{A}}_3+3\lambda (1-\mathrm{\Gamma }){\overline{B}}_3-2{\mu }_2{\delta }_{-1}}{{\kappa }_2+1}{\xi }^{-1}}\\ \multicolumn{1}{c}{+\frac{{\lambda }^3(\mathrm{\Gamma }{\kappa }_1+1)A_3+3\lambda (1-\mathrm{\Gamma }){\overline{A}}_3+(1-\mathrm{\Gamma }){\overline{B}}_3-2{\mu }_2{\delta }_{-3}}{{\kappa }_2+1}{\xi }^{-3},}\end{array}$$

(20)

$$\begin{array}{c}{\psi }_2(\xi )+\frac{\overline{\omega }\left(\frac{1}{\xi }\right){\phi \prime }_2(\xi )}{\omega \prime (\xi )}=\frac{3\lambda (\mathrm{\Gamma }+{\kappa }_2)A_3+{\lambda }^3({\kappa }_2-\mathrm{\Gamma }{\kappa }_1){\overline{A}}_3+(\mathrm{\Gamma }+{\kappa }_2)B_3+2{\mu }_2{\overline{\delta }}_{-3}}{{\kappa }_2+1}{\xi }^3\hfill \\ +\frac{\left(\begin{array}{c}\lambda (\mathrm{\Gamma }+{\kappa }_2)A_1+\lambda ({\kappa }_2-\mathrm{\Gamma }{\kappa }_1){\overline{A}}_1+(\mathrm{\Gamma }+{\kappa }_2)B_1\hfill \\ \multicolumn{1}{c}{+3(\mathrm{\Gamma }+{\kappa }_2)(1+2{\lambda }^2)A_3+3{\lambda }^2({\kappa }_2-\mathrm{\Gamma }{\kappa }_1){\overline{A}}_3+3\lambda (\mathrm{\Gamma }+{\kappa }_2)B_3+2{\mu }_2{\overline{\delta }}_{-1}}\end{array}\right)}{{\kappa }_2+1}\xi \hfill \\ +\frac{\left(\begin{array}{c}(\mathrm{\Gamma }+{\kappa }_2)A_1+({\kappa }_2-\mathrm{\Gamma }{\kappa }_1){\overline{A}}_1+\lambda (\mathrm{\Gamma }+{\kappa }_2)B_1\hfill \\ \multicolumn{1}{c}{+3(\mathrm{\Gamma }+{\kappa }_2)(2\lambda +{\lambda }^3)A_3+3\lambda ({\kappa }_2-\mathrm{\Gamma }{\kappa }_1){\overline{A}}_3+3{\lambda }^2(\mathrm{\Gamma }+{\kappa }_2)B_3+2{\mu }_2{\overline{\delta }}_1}\end{array}\right)}{{\kappa }_2+1}{\xi }^{-1}\hfill \\ +\frac{3{\lambda }^2(\mathrm{\Gamma }+{\kappa }_2)A_3+({\kappa }_2-\mathrm{\Gamma }{\kappa }_1){\overline{A}}_3+{\lambda }^3(\mathrm{\Gamma }+{\kappa }_2)B_3+2{\mu }_2{\overline{\delta }}_3}{{\kappa }_2+1}{\xi }^{-3}.\hfill \end{array}$$

(21)

Using equations (20) and (21) to satisfy the remote asymptotic conditions in equation (11b), we obtain the following set of linear algebraic equations

$$\begin{array}{c}(\mathrm{\Gamma }{\kappa }_1+1)A_3+3{\lambda }^2(1-\mathrm{\Gamma }){\overline{A}}_3+{\lambda }^3(1-\mathrm{\Gamma }){\overline{B}}_3=2{\mu }_2{\delta }_3,\hfill \\ 3\lambda (\mathrm{\Gamma }+{\kappa }_2)A_3+{\lambda }^3({\kappa }_2-\mathrm{\Gamma }{\kappa }_1){\overline{A}}_3+(\mathrm{\Gamma }+{\kappa }_2)B_3=-2{\mu }_2{\overline{\delta }}_{-3},\hfill \\ (\mathrm{\Gamma }{\kappa }_1+1)A_1+(1-\mathrm{\Gamma }){\overline{A}}_1+\lambda (1-\mathrm{\Gamma }){\overline{B}}_1+3\lambda (\mathrm{\Gamma }{\kappa }_1+1)A_3+3(1-\mathrm{\Gamma })(2\lambda +{\lambda }^3){\overline{A}}_3+3{\lambda }^2(1-\mathrm{\Gamma }){\overline{B}}_3=2{\mu }_2{\delta }_1,\hfill \\ \lambda (\mathrm{\Gamma }+{\kappa }_2)A_1+\lambda ({\kappa }_2-\mathrm{\Gamma }{\kappa }_1){\overline{A}}_1+(\mathrm{\Gamma }+{\kappa }_2)B_1\hfill \\ +3(\mathrm{\Gamma }+{\kappa }_2)(1+2{\lambda }^2)A_3+3{\lambda }^2({\kappa }_2-\mathrm{\Gamma }{\kappa }_1){\overline{A}}_3+3\lambda (\mathrm{\Gamma }+{\kappa }_2)B_3=-2{\mu }_2{\overline{\delta }}_{-1}.\hfill \end{array}$$

(22)

The four complex constants $A_1,A_3,B_1$ , and $B_3$ can be uniquely determined from equation (21) as follows:

$$\begin{array}{c}{A\prime }_3=\frac{2{\mu }_2[{\delta \prime }_3(\mathrm{\Gamma }+{\kappa }_2)+{\delta \prime }_{-3}{\lambda }^3(1-\mathrm{\Gamma })]}{(\mathrm{\Gamma }+{\kappa }_2)[\mathrm{\Gamma }{\kappa }_1+1+3{\lambda }^2(1-\mathrm{\Gamma })]-{\lambda }^4(1-\mathrm{\Gamma })[3(\mathrm{\Gamma }+{\kappa }_2)+{\lambda }^2({\kappa }_2-\mathrm{\Gamma }{\kappa }_1)]},\hfill \\ {A″}_3=\frac{2{\mu }_2[{\delta ″}_3(\mathrm{\Gamma }+{\kappa }_2)-{\delta ″}_{-3}{\lambda }^3(\mathrm{\Gamma }-1)]}{(\mathrm{\Gamma }+{\kappa }_2)[\mathrm{\Gamma }{\kappa }_1+1+3{\lambda }^2(\mathrm{\Gamma }-1)]-{\lambda }^4(\mathrm{\Gamma }-1)[3(\mathrm{\Gamma }+{\kappa }_2)+{\lambda }^2(\mathrm{\Gamma }{\kappa }_1-{\kappa }_2)]},\hfill \\ {B\prime }_3=\frac{-2{\mu }_2\{{\delta \prime }_3\lambda [3(\mathrm{\Gamma }+{\kappa }_2)+{\lambda }^2({\kappa }_2-\mathrm{\Gamma }{\kappa }_1)]+{\delta \prime }_{-3}[\mathrm{\Gamma }{\kappa }_1+1+3{\lambda }^2(1-\mathrm{\Gamma })]\}}{(\mathrm{\Gamma }+{\kappa }_2)[\mathrm{\Gamma }{\kappa }_1+1+3{\lambda }^2(1-\mathrm{\Gamma })]-{\lambda }^4(1-\mathrm{\Gamma })[3(\mathrm{\Gamma }+{\kappa }_2)+{\lambda }^2({\kappa }_2-\mathrm{\Gamma }{\kappa }_1)]},\hfill \\ {B″}_3=\frac{2{\mu }_2\{-{\delta ″}_3[3\lambda (\mathrm{\Gamma }+{\kappa }_2)+{\lambda }^3(\mathrm{\Gamma }{\kappa }_1-{\kappa }_2)]+{\delta ″}_{-3}[\mathrm{\Gamma }{\kappa }_1+1+3{\lambda }^2(\mathrm{\Gamma }-1)]\}}{(\mathrm{\Gamma }+{\kappa }_2)[\mathrm{\Gamma }{\kappa }_1+1+3{\lambda }^2(\mathrm{\Gamma }-1)]-{\lambda }^4(\mathrm{\Gamma }-1)[3(\mathrm{\Gamma }+{\kappa }_2)+{\lambda }^2(\mathrm{\Gamma }{\kappa }_1-{\kappa }_2)]},\hfill \end{array}$$

(23)

and

$$\begin{array}{c}{A\prime }_1=\frac{2{\mu }_2{\delta \prime }_1(\mathrm{\Gamma }+{\kappa }_2)+2{\mu }_2{\delta \prime }_{-1}\lambda (1-\mathrm{\Gamma })+3{A\prime }_3\lambda \{{\lambda }^2(1-\mathrm{\Gamma })[\mathrm{\Gamma }(1-{\kappa }_1)+2{\kappa }_2]-(\mathrm{\Gamma }+{\kappa }_2)[\mathrm{\Gamma }({\kappa }_1-1)+2]\}}{(\mathrm{\Gamma }+{\kappa }_2)[\mathrm{\Gamma }({\kappa }_1-1)+2]-{\lambda }^2(1-\mathrm{\Gamma })[\mathrm{\Gamma }(1-{\kappa }_1)+2{\kappa }_2]},\hfill \\ {A″}_1=\frac{2{\mu }_2{\delta }_1(\mathrm{\Gamma }+{\kappa }_2)-2{\mu }_2{\delta ″}_{-1}\lambda (\mathrm{\Gamma }-1)+3{A″}_3\lambda \mathrm{\Gamma }({\kappa }_1+1)[{\lambda }^2(\mathrm{\Gamma }-1)-(\mathrm{\Gamma }+{\kappa }_2)]}{\mathrm{\Gamma }({\kappa }_1+1)[\mathrm{\Gamma }+{\kappa }_2-{\lambda }^2(\mathrm{\Gamma }-1)]},\hfill \\ {B\prime }_1=\frac{\left(\begin{array}{c}-2{\mu }_2{\delta \prime }_1\lambda [\mathrm{\Gamma }(1-{\kappa }_1)+2{\kappa }_2]-2{\mu }_2{\delta \prime }_{-1}[\mathrm{\Gamma }({\kappa }_1-1)+2]\hfill \\ \multicolumn{1}{c}{+3{A\prime }_3\{{\lambda }^4(1-\mathrm{\Gamma })[\mathrm{\Gamma }(1-{\kappa }_1)+2{\kappa }_2]-{\lambda }^2\mathrm{\Gamma }({\kappa }_1+1)({\kappa }_2+1)-(\mathrm{\Gamma }+{\kappa }_2)[\mathrm{\Gamma }({\kappa }_1-1)+2]\}}\\ \multicolumn{1}{c}{+3{B\prime }_3\lambda \{{\lambda }^2(1-\mathrm{\Gamma })[\mathrm{\Gamma }(1-{\kappa }_1)+2{\kappa }_2]-(\mathrm{\Gamma }+{\kappa }_2)[\mathrm{\Gamma }({\kappa }_1-1)+2]\}}\end{array}\right)}{(\mathrm{\Gamma }+{\kappa }_2)[\mathrm{\Gamma }({\kappa }_1-1)+2]-{\lambda }^2(1-\mathrm{\Gamma })[\mathrm{\Gamma }(1-{\kappa }_1)+2{\kappa }_2]},\hfill \\ {B″}_1=\frac{2{\mu }_2{\delta ″}_{-1}-2{\mu }_2{\delta }_1\lambda +3[{A″}_3({\lambda }^2+1)+{B\prime \prime }_3\lambda ][{\lambda }^2(\mathrm{\Gamma }-1)-(\mathrm{\Gamma }+{\kappa }_2)]}{\mathrm{\Gamma }+{\kappa }_2-{\lambda }^2(\mathrm{\Gamma }-1)}.\hfill \end{array}$$

(24)

Consequently, the assumption of the form of solution for thermoelasticity inside the elliptical inhomogeneity in equation (18) is indeed valid. In addition, the thermal stresses are nonuniformly distributed inside the elliptical inhomogeneity as follows:

$$\begin{array}{c}{\sigma }_{11}=\frac{6{A\prime }_3-3{B\prime }_3}{R^3}(x_1^2-x_2^2)-\frac{6{A\prime }_3}{R^3}(x_1^2+x_2^2)+\frac{6{B″}_3-12{A″}_3}{R^3}{x}_1{x}_2+\frac{2{A\prime }_1-{B\prime }_1}{R},\hfill \\ {\sigma }_{22}=\frac{6{A\prime }_3+3{B\prime }_3}{R^3}(x_1^2-x_2^2)+\frac{6{A\prime }_3}{R^3}(x_1^2+x_2^2)-\frac{12{A″}_3+6{B″}_3}{R^3}{x}_1{x}_2+\frac{2{A\prime }_1+{B\prime }_1}{R},\hfill \\ {\sigma }_{12}=\frac{6{A″}_3}{R^3}(x_1^2+x_2^2)+\frac{3{B″}_3}{R^3}(x_1^2-x_2^2)+\frac{6{B\prime }_3}{R^3}{x}_1{x}_2+\frac{{B″}_1}{R},z\in S_1.\hfill \end{array}$$

(25)

It is seen from the above expression that the internal thermal stresses inside the elliptical inhomogeneity are quadratic functions of the two in-plane coordinates $x_1$ and $x_2$ .

Now that the remote asymptotic behaviors of ${\phi }_2(\xi )$ and ${\psi }_2(\xi )$ in equation (11b) have been satisfied, the pair of analytic functions ${\phi }_2(\xi )$ and ${\psi }_2(\xi )$ is thus given by

$$\begin{array}{c}\multicolumn{1}{c}{{\phi }_2(\xi )=\frac{\left(\begin{array}{c}\lambda (\mathrm{\Gamma }{\kappa }_1+1)A_1+\lambda (1-\mathrm{\Gamma }){\overline{A}}_1+(1-\mathrm{\Gamma }){\overline{B}}_1\hfill \\ \multicolumn{1}{c}{+3{\lambda }^2(\mathrm{\Gamma }{\kappa }_1+1)A_3+3(1-\mathrm{\Gamma })(1+2{\lambda }^2){\overline{A}}_3+3\lambda (1-\mathrm{\Gamma }){\overline{B}}_3-2{\mu }_2{\delta }_{-1}}\end{array}\right)}{{\kappa }_2+1}{\xi }^{-1}}\\ \multicolumn{1}{c}{+\frac{{\lambda }^3(\mathrm{\Gamma }{\kappa }_1+1)A_3+3\lambda (1-\mathrm{\Gamma }){\overline{A}}_3+(1-\mathrm{\Gamma }){\overline{B}}_3-2{\mu }_2{\delta }_{-3}}{{\kappa }_2+1}{\xi }^{-3},}\\ \multicolumn{1}{c}{{\psi }_2(\xi )=\frac{\left(\begin{array}{c}(\mathrm{\Gamma }+{\kappa }_2)A_1+({\kappa }_2-\mathrm{\Gamma }{\kappa }_1){\overline{A}}_1+\lambda (\mathrm{\Gamma }+{\kappa }_2)B_1\hfill \\ \multicolumn{1}{c}{+3(\mathrm{\Gamma }+{\kappa }_2)(2\lambda +{\lambda }^3)A_3+3\lambda ({\kappa }_2-\mathrm{\Gamma }{\kappa }_1){\overline{A}}_3+3{\lambda }^2(\mathrm{\Gamma }+{\kappa }_2)B_3+2{\mu }_2{\overline{\delta }}_1}\end{array}\right)}{{\kappa }_2+1}{\xi }^{-1}}\\ +\frac{3{\lambda }^2(\mathrm{\Gamma }+{\kappa }_2)A_3+({\kappa }_2-\mathrm{\Gamma }{\kappa }_1){\overline{A}}_3+{\lambda }^3(\mathrm{\Gamma }+{\kappa }_2)B_3+2{\mu }_2{\overline{\delta }}_3}{{\kappa }_2+1}{\xi }^{-3}-\frac{\overline{\omega }\left(\frac{1}{\xi }\right){\phi \prime }_2(\xi )}{\omega \prime (\xi )},\hfill \\ \multicolumn{1}{c}{\left|\xi \right|\ge 1.}\end{array}$$

(26)

The induced thermal stresses in the matrix can be obtained by substituting equation (26) into equation (2).

4. Conclusion

We have solved the two-dimensional heat conduction and thermoelastic problems associated with an elliptical elastic inhomogeneity embedded in an infinite matrix subjected to nonuniform remote heat flux represented by a linear heat flux distribution. The temperature and the induced thermal stresses within the elliptical inhomogeneity given by equations (16) and (25) are quadratic functions of the two in-plane coordinates $x_1$ and $x_2$ . The analytic function $g_2(\xi )$ and the pair of analytic functions ${\phi }_2(\xi )$ and ${\psi }_2(\xi )$ characterizing the temperature and the thermoelastic field in the matrix are explicitly given by equations (13) and (26). The present solution method can be conveniently adapted to treat the case in which the nonuniform remote heat flux has an arbitrary order polynomial distribution.