An approximate analytical solution to the nearly circular inclusion problems of arbitrary shape in plane thermoelasticity is provided. The shape of the inclusion boundary considered in the present study is assumed to have the form $r=a0[1+Aθ],$ where $a0$ is the radius of the unperturbed circle and $Aθ$ is the radius perturbation magnitude that is represented by a Fourier series expansion. The proposed method in this study is based on the complex variable theory, analytical continuation theorem, and the boundary perturbation technique. Originating from the principle of superposition, the solution of the present problem is composed of the reference and the perturbation terms that the reference term is the known exact solution pertaining to the case with circular inclusion. First-order perturbation solutions of both temperature and stress fields are obtained explicitly for elastic inclusions of arbitrary shape. To demonstrate the derived general solutions, two typical examples including elliptical and smooth polygonal inclusions are discussed in detail. Compared to other existing approaches for elastic inclusion problems, our methodology presented here is remarked by its efficiency and applicability to inclusions of arbitrary shape in a plane under thermal load.

1.
Florence
,
A. L.
, and
Goodier
,
J. N.
,
1959
, “
Thermal Stress at Spherical Cavities and Circular Holes in Uniform Heat Flow
,”
ASME J. Appl. Mech.
,
26
, pp.
293
294
.
2.
Florence
,
A. L.
, and
Goodier
,
J. N.
,
1960
, “
Thermal Stress due to Disturbance of Uniform Heat Flow by an Insulated Ovaloid Hole
,”
ASME J. Appl. Mech.
,
27
, pp.
635
639
.
3.
Chen
,
W. T.
,
1967
, “
Plane Thermal Stress at an Insulated Hole Under Uniform Heat Flow in an Orthotropic Medium
,”
ASME J. Appl. Mech.
,
34
, pp.
133
136
.
4.
Green, A. E., and Zerna, W., 1954, Theoretical Elasticity, Oxford University Press, London.
5.
Hwu
,
C.
,
1990
, “
Thermal Stresses in an Anisotropic Plate Disturbed by an Insulated Elliptic Hole or Crack
,”
ASME J. Appl. Mech.
,
57
, pp.
916
922
.
6.
Stroh
,
A. N.
,
1958
, “
Dislocations and Cracks in Anisotropic Elasticity
,”
Philos. Mag.
,
7
, pp.
625
646
.
7.
Kattis
,
M. A.
, and
Mequid
,
S. A.
,
1995
, “
Two-Phase Potentials for the Treatment of an Elastic Inclusion in Plane Thermoelasticity
,”
ASME J. Appl. Mech.
,
62
, pp.
7
12
.
8.
Chao
,
C. K.
, and
Shen
,
M. H.
,
1998
, “
Thermal Stresses in a Generally Anisotropic Body With an Elliptic Inclusion Subject to Uniform Heat Flow
,”
ASME J. Appl. Mech.
,
65
, pp.
51
58
.
9.
Hwu
,
C.
, and
Yen
,
W. J.
,
1993
, “
On the Anisotropic Elastic Inclusions in Plane Elastostatics
,”
ASME J. Appl. Mech.
,
60
, pp.
626
632
.
10.
Jaswon
,
M. A.
, and
Bhargave
,
R. D.
,
1961
, “
Two-Dimensional Elastic Inclusion Problem
,”
Proc. Cambridge Philos. Soc.
,
57
, pp.
669
680
.
11.
Sendeckyi
,
G. P.
,
1970
, “
Elastic Inclusion Problems in Plane Elastostatics
,”
Int. J. Solids Struct.
,
6
, No.
6
, pp.
1535
1543
.
12.
Ru
,
C. Q.
,
1999
, “
Analytic Solution for Eshelby’s Problem of an Inclusion of Arbitrary Shape in a Plane or Half-Plane
,”
ASME J. Appl. Mech.
,
66
, pp.
315
322
.
13.
Ru
,
C. Q.
,
1998
, “
Interface Design of Neutral Elastic Inclusions
,”
Int. J. Solids Struct.
,
35
, pp.
557
572
.
14.
Benveniste
,
Y.
, and
Miloh
,
T.
,
1999
, “
Neutral Inhomogeneities in Conduction Phenomena
,”
J. Mech. Phys. Solids
,
47
, pp.
1873
1892
.
15.
Gao
,
H.
,
1991
, “
A Boundary Perturbation Analysis for Elastic Inclusions and Interfaces
,”
Int. J. Solids Struct.
,
28
, No.
6
, pp.
703
725
.
16.
Kattis
,
M. A.
,
1991
, “
Thermoelastic Plane Problems With Curvilinear Boundaries
,”
Acta Mech.
,
87
, pp.
93
103
.
17.
Bogdanoff
,
J. L.
,
1954
, “
Note on Thermal Stress
,”
ASME J. Appl. Mech.
,
21
, p.
88
88
.
18.
Chao
,
C. K.
, and
Shen
,
M. H.
,
1997
, “
On Bonded Circular Inclusions in Plane Thermoelasticity
,”
ASME J. Appl. Mech.
,
64
, pp.
1000
1004
.