In 1957, Eshelby proved that the strain field within a homogeneous ellipsoidal inclusion embedded in an infinite isotropic media is uniform, when the eigenstrain prescribed in the inclusion is uniform. This property is usually referred to as the Eshelby property. Although the Eshelby property does not hold for the non-ellipsoidal inclusions, in recent studies we have successfully proved that the arithmetic mean of Eshelby tensors at N rotational symmetrical points inside an N-fold rotational symmetrical inclusion is constant and equals the Eshelby tensor for a circular inclusion, when N3 and N4. The property is named the quasi-Eshelby property or the arithmetic mean theorem of Eshelby tensors for interior points. In this paper, we investigate the elastic field outside the inclusion. By the Green formula and the knowledge of complex variable functions, we prove that the arithmetic mean of Eshelby tensors at N rotational symmetrical points outside an N-fold rotational symmetrical inclusion is equal to zero, when N3 and N4. The property is referred to as the arithmetic mean theorem of Eshelby tensors for exterior points. Due to the quality of the Green function for plane strain problems, the fourfold rotational symmetrical inclusions are excluded from possessing the arithmetic mean theorem. At the same time, by the method proposed in this paper, we verify the quasi-Eshelby property which has been obtained in our previous work. As corollaries, two more special properties of Eshelby tensor for N-fold rotational symmetrical inclusions are presented which may be beneficial to the evaluation of effective material properties of composites. Finally, the circular inclusion is used to test the validity of the arithmetic mean theorem for exterior points by using the known solutions.

1.
Eshelby
,
J. D.
, 1957, “
The Determination of the Elastic Field of an Ellipsoidal Inclusion and Related Problems
,”
Proc. R. Soc. London, Ser. A
1364-5021,
241
, pp.
376
396
.
2.
Mura
,
T.
, 1987,
Micromechanics of Defects in Solids
,
2nd ed.
,
Nijhoff
, Dordrecht.
3.
Willis
,
J. R.
, 1981, “
Variational and Related Methods for the Overall Properties of Composites
,”
Adv. Appl. Mech.
0065-2156,
21
, pp.
1
78
.
4.
Reid
,
A. C. E.
, and
Gooding
,
R. J.
, 1992, “
Inclusion Problem in a Two-Dimensional Nonlocal Elastic Solid
,”
Phys. Rev. B
0163-1829,
46
(
10
), pp.
6045
6049
.
5.
Sun
,
Y. Q.
,
Gu
,
X. M.
, and
Hazzledine
,
P. M.
, 2002, “
Displacement Field Inside a Spherical Dislocation Cage and the Eshelby Tensor
,”
Phys. Rev. B
0163-1829,
65
(
22
), pp.
220103
–1–220103-
4
.
6.
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.
0021-8936,
66
(
2
), pp.
315
322
.
7.
Owen
,
D. R. J.
, 1972, “
Analysis of Fibre-Reinforced Materials by an Initial Strain Method
,”
Fibre Sci. Technol.
0015-0568,
5
, pp.
37
-
59
.
8.
Chiu
,
Y. P.
, 1977, “
On the Stress Field due to Initial Strains in a Cuboid Surrounded by an Infinite Elastic Space
,”
ASME J. Appl. Mech.
0021-8936,
44
, pp.
587
590
.
9.
Rodin
,
G.
, 1996, “
Eshelby’s Inclusion Problem for Polygons and Polyhedra
,”
J. Mech. Phys. Solids
0022-5096,
44
, pp.
1977
1995
.
10.
Mura
,
T.
,
Shodja
,
H. M.
,
Lin
,
T. Y.
,
Safadi
,
A.
, and
Makkawy
,
A.
, 1994, “
The Determination of the Elastic Field of a Pentagonal Star Shaped Inclusion
,”
Bull. Technical University of Istanbul
,
47
, pp.
267
280
.
11.
Lubarda
,
V. A.
, and
Markenscoff
,
X.
, 1998, “
On the Absence of Eshelby Property for Non-ellipsoidal Inclusions
,”
Int. J. Solids Struct.
0020-7683,
35
, pp.
3405
3411
.
12.
Downes
,
J. R.
, and
Faux
,
D. A.
, 1995, “
Calculation of Strain Distributions in Multiple-Quantum-Well Strained-Layer Structures
,”
J. Appl. Phys.
0021-8979,
77
(
6
), pp.
2444
2447
.
13.
Faux
,
D. A.
,
Downes
,
J. R.
, and
O’Reilly
,
E. P.
, 1996, “
A Simple Method for Calculating Strain Distributions in Quantum-Wire Structures
,”
J. Appl. Phys.
0021-8979,
80
(
4
), pp.
2515
2517
.
14.
Faux
,
D. A.
,
Downes
,
J. R.
, and
O’Reilly
,
E. P.
, 1997, “
Analytic Solutions for Strain Distributions in Quantum-Wire structures
,”
J. Appl. Phys.
0021-8979,
82
(
8
), pp.
3754
3762
.
15.
Faux
,
D. A.
, and
Pearson
,
G. S.
, 2000, “
Green’s Tensors for Anisotropic Elasticity: Application to Quantum Dots
,”
Phys. Rev. B
0163-1829,
62
(
8
), pp.
R4798
R4801
.
16.
Andreev
,
A. D.
,
Downes
,
J. R.
,
Faux
,
D. A.
, and
O’Reilly
,
E. P.
, 1999, “
Strain Distributions in Quantum Dots of Arbitrary Shape
,”
J. Appl. Phys.
0021-8979,
86
(
1
), pp.
297
305
.
17.
Bergman
,
D. J.
,
Strelniker
,
Y. M.
, and
Sarychev
,
A. K.
, 1997, “
Recent Advances in Strong Field Magneto-Transport in a Composite Medium
,”
Physica A
0378-4371,
241
, pp.
278
283
.
18.
Coffey
,
M. W.
, 2002, “
Logarithmically Perturbed Two Dimensional Oscillator Model of a Quantum-Dot Nanostructure
,”
Appl. Phys. Lett.
0003-6951,
80
(
7
), pp.
1219
1221
.
19.
Nozaki
,
H.
, and
Taya
,
M.
, 1997, “
Elastic Fields in a Polygon-Shaped Inclusion with Uniform Eigenstrains
,”
ASME J. Appl. Mech.
0021-8936,
64
, pp.
495
502
.
20.
Kawashita
,
M.
, and
Nozaki
,
H.
, 2001, “
Eshelby Tensor of a Polygonal Inclusion and Its Special Properties
,”
J. Elast.
0374-3535,
64
, pp.
71
84
.
21.
Wang
,
M. Z.
, and
Xu
,
B. X.
, 2004, “
The Arithmetic Mean Theorem of Eshelby Tensor for a Rotational Symmetrical Inclusion
,”
J. Elast.
0374-3535,
77
, pp.
13
23
.
22.
Xu
,
B. X.
, and
Wang
,
M. Z.
, 2005, “
The Quasi Eshelby Property for Rotational Symmetrical Inclusions of Uniform Eigencurvatures within an Infinite Plate
,”
Proc. R. Soc. London, Ser. A
1364-5021,
461
, pp.
2899
2910
.
23.
Xu
,
B. X.
, and
Wang
,
M. Z.
, 2005, “
Special Properties of Eshelby Tensor for a Regular Polygonal Inclusion
,”
Acta Mech. Sin.
0459-1879 (in English),
21
, pp.
267
271
.
24.
Ferrers
,
N. M.
, 1877, “
On the Potentials of Ellipsoids, Ellipsoidal Shells, Elliptic Laminae and Elliptic Rings of Variable Densities
,”
Q. J. Pure Appl. Math.
,
14
, pp.
1
22
.
25.
Dyson
,
F. W.
, 1891, “
The Potentials of Ellipsoids of Variable Densities
,”
Q. J. Pure Appl. Math.
,
25
, pp.
259
288
.
26.
Holland
,
A. S. B.
, 1980,
Complex Function Theory
,
Elsevier
, New York.
27.
Aboudi
,
J.
, 1991,
Mechanics of Composite Materials: A Unified Micromechanical Approach
,
Elsevier
, Amsterdam.
28.
Nemat-Nasser
,
S.
, and
Hori
,
M.
, 1999,
Micromechanics: Overall Properties of Heterogeneous Elastic Solids
,
2nd ed.
,
North-Holland
, Amsterdam.
29.
Milton
,
G. W.
, 2002,
The Theory of Composites
,
Cambridge University Press
, Cambridge, UK.
30.
Torquato
,
S.
, 2002,
Random Heterogeneous Materials: Microstructure and Macroscopic Properties
,
Springer
, New York.
You do not currently have access to this content.