This work is concerned with the precise characterization of the elastic fields due to a spherical inclusion embedded within a spherical representative volume element (RVE). The RVE is considered having finite size, with either a prescribed uniform displacement or a prescribed uniform traction boundary condition. Based on symmetry and group theoretic arguments, we identify that the Eshelby tensor for a spherical inclusion admits a unique decomposition, which we coin the “radial transversely isotropic tensor.” Based on this notion, a novel solution procedure is presented to solve the resulting Fredholm type integral equations. By using this technique, exact and closed form solutions have been obtained for the elastic disturbance fields. In the solution two new tensors appear, which are termed the Dirichlet–Eshelby tensor and the Neumann–Eshelby tensor. In contrast to the classical Eshelby tensor they both are position dependent and contain information about the boundary condition of the RVE as well as the volume fraction of the inclusion. The new finite Eshelby tensors have far-reaching consequences in applications such as nanotechnology, homogenization theory of composite materials, and defects mechanics.

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.
Eshelby
,
J. D.
, 1959, “
The Elastic Field Outside an Ellipsoidal Inclusion
,”
Proc. R. Soc. London, Ser. A
1364-5021,
252
, pp.
561
569
.
3.
Eshelby
,
J. D.
, 1961, “
Elastic Inclusions and Inhomogeneities
,”
Progress in Solid Mechanics
,
N. I.
Sneddon
and
R.
Hill
, eds., North Holland, Amsterdam, The Netherlands, Vol.
2
, pp.
89
104
.
4.
Thostenson
,
E. T.
,
Ren
,
Z.
, and
Chou
,
T. W.
, 2001, “
Advances in Science and Technology of Carbon Nanotubes and their Composites: A Review
,”
Compos. Sci. Technol.
0266-3538,
61
, pp.
1899
1912
.
5.
Shi
,
D.-L.
,
Feng
,
X.-Q.
,
Huang
,
Y.-Y.
,
Hwang
,
K.-C.
, and
Gao
,
H.
, 2004, “
The Effect of Nanotube Waviness and Agglomeration on the Elastic Property of Carbon Nanotube-Reinforced Composites
,”
J. Eng. Mater. Technol.
0094-4289,
126
, pp.
250
257
.
6.
Kinoshita
,
N.
, and
Mura
,
T.
, 1984, “
Eigenstrain Problems in a Finite Elastic Body
,”
SIAM J. Appl. Math.
0036-1399,
44
, pp.
524
535
.
7.
Kröner
,
E.
, 1986, “
Statistical Modeling
,”
Modeling Small Deformation of Polycrystals
,
J.
Gittus
and
J.
Zarka
, eds.,
Elsevier Applied Science
,
New York
, pp.
229
291
.
8.
Kröner
,
E.
, 1990, “
Modified Green’s Function in the Theory of Heterogeneous and/or Anisotropic Linearly Elastic Media
,”
Micromechanics and Inhomogeneity, The Toshio Mura 65th Anniversary Volume
,
G. J.
Weng
,
M.
Taya
, and
H.
Abe
, eds.,
Springer
,
New York
, pp.
599
622
.
9.
Mazilu
,
P.
, 1972, “
On the Theory of Linear Elasticity in Statically Homogeneous Media
,”
Rev. Roum. Math. Pures Appl.
0035-3965,
17
, pp.
261
273
.
10.
Luo
,
H. A.
, and
Weng
,
G. J.
, 1987, “
On Eshelby’s Inclusion Problem in a Three-Phase Spherically Concentric Solid and a Modification of Mori-Tanaka’s Method
,”
Mech. Mater.
0167-6636,
6
, pp.
347
361
.
11.
Li
,
S.
,
Wang
,
G.
, and
Sauer
,
R. A.
, 2007, “
The Eshelby Tensors in a Finite Spherical Domain—Part II: Applications to Homogenization
,”
ASME J. Appl. Mech.
0021-8936,
74
, pp.
784
797
.
12.
Love
,
A. E. H.
, 1927,
A Treatise on the Mathematical Theory of Elasticity
,
4th ed.
,
Cambridge University Press
,
Cambridge, UK
.
13.
Mura
,
T.
, 1987,
Micromechanics of Defects in Solids
,
2nd ed.
,
Martinus Nijhoff
,
Boston
.
14.
Ju
,
J. W.
, and
Sun
,
L. Z.
, 1999, “
A Novel Formulation for the Exterior Point Eshelby’s Tensor of an Ellipsoidal Inclusion
,”
ASME J. Appl. Mech.
0021-8936,
66
, pp.
570
574
.
15.
Walpole
,
L. J.
, 1981, “
Elastic Behavior of Composite Materials: Theoretical Foundations
,”
Advances in Applied Mechanics
,
C. S.
Yih
, ed.,
Academic
,
New York
, Vol.
21
, pp.
169
242
.
16.
Sauer
,
R. A.
,
Wang
,
G.
, and
Li
,
S.
, 2007, “
The Composite Eshelby Tensors and Their Application to Homogenization
,” submitted.
17.
Li
,
S.
,
Sauer
,
R. A.
, and
Wang
,
G.
, 2005, “
Circular Inclusion in a Finite Domain, I: Direchlet-Eshelby Problem
,”
Acta Mech.
0001-5970,
179
, pp.
67
90
.
18.
Wang
,
G.
,
Li
,
S.
, and
Sauer
,
R. A.
, 2005, “
Circular Inclusion in a Finite Domain, II: Neumann-Eshelby Problem
,”
Acta Mech.
0001-5970,
179
, pp.
91
110
.
You do not currently have access to this content.