The celebrated solution of the Eshelby ellipsoidal inclusion has laid the cornerstone for many fundamental aspects of micromechanics. A well-known difficulty of this classical solution is to determine the elastic field outside the ellipsoidal inclusion. In this paper, we first analytically present the full displacement field of an ellipsoidal inclusion subjected to uniform eigenstrain. It is demonstrated that the displacements inside inclusion are linearly related to the coordinates and continuous across the interface of inclusion and matrix. The exterior displacement, which is less detailed in existing literatures, may be expressed in a more compact, explicit, and simpler form through utilizing the outward unit normal vector of an auxiliary confocal ellipsoid. Other than many practical applications in geological engineering, the displacement solution can be a convenient starting point to derive the deformation gradient, and subsequently in a straightforward manner to accomplish the full-field solutions of the strain and stress. Following Eshelby's definition, a complete set of the Eshelby tensors corresponding to the displacement, deformation gradient, strain, and stress are expressed in explicit analytical form. Furthermore, the jump conditions to quantify the discontinuities across the interface are discussed and a benchmark problem is provided to validate the present formulation.

References

References
1.
Mura
,
T.
,
1982
,
Micromechanics of Defects in Solids
,
Springer
,
Dordrecht, The Netherlands
.
2.
Mura
,
T.
,
1988
, “
Inclusion Problems
,”
ASME Appl. Mech. Rev.
,
41
(
1
), pp.
15
20
.
3.
Eshelby
,
J. D.
,
1957
, “
The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems
,”
Proc. R. Soc. London, Ser. A
, 241(1226), pp.
376
396
.
4.
Eshelby
,
J.
,
1959
, “
The Elastic Field Outside an Ellipsoidal Inclusion
,”
Proc. R. Soc. London, Ser. A
,
252
(
1271
), pp.
561
569
.
5.
Rudnicki
,
J. W.
,
2007
, “
Models for Compaction Band Propagation
,”
Rock Physics and Geomechanics in the Study of Reservoirs and Repositories
, Vol.
284
,
Geological Society, Special Publications
,
London
, pp.
107
125
.
6.
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.
,
66
(
2
), pp.
570
574
.
7.
Li
,
S.
, and
Wang
,
G.
,
2008
,
Introduction to Micromechanics and Nanomechanics
,
World Scientific
,
Singapore
.
8.
Ferrers
,
N.
,
1877
, “
On the Potentials of Ellipsoids, Ellipsoidal Shells, Elliptic Laminae and Elliptic Rings of Variable Densities
,”
Q. J. Pure Appl. Math.
,
14
(
1
), pp.
1
22
.
9.
Dyson
,
F. W.
,
1891
, “
The Potentials of Ellipsoids of Variable Densities
,”
Q. J. Pure Appl. Math.
,
25
, pp.
259
288
.
10.
Cai
,
W.
,
2007
, “
Lecture 18: Potential Field of a Uniformly Charged Ellipsoid
,” Class Notes of Theory and Applications of Elasticity, Stanford University, Stanford, CA, accessed July 5, 2016, http://micro.stanford.edu/∼caiwei/me340/
11.
Jin
,
X. Q.
,
Keer
,
L. M.
, and
Wang
,
Q.
,
2011
, “
A Closed-Form Solution for the Eshelby Tensor and the Elastic Field Outside an Elliptic Cylindrical Inclusion
,”
ASME J. Appl. Mech.
,
78
(
3
), p.
031009
.
12.
Zhou
,
Q.
,
Jin
,
X.
,
Wang
,
Z.
,
Wang
,
J.
,
Keer
,
L. M.
, and
Wang
,
Q.
,
2016
, “
Numerical EIM With 3D FFT for the Contact With a Smooth or Rough Surface Involving Complicated and Distributed Inhomogeneities
,”
Tribol. Int.
,
93
(A), pp.
91
103
.
13.
Jin
,
X.
,
Wang
,
Z.
,
Zhou
,
Q.
,
Keer
,
L. M.
, and
Wang
,
Q.
,
2014
, “
On the Solution of an Elliptical Inhomogeneity in Plane Elasticity by the Equivalent Inclusion Method
,”
J. Elasticity
,
114
(
1
), pp.
1
18
.
14.
Healy
,
D.
,
2009
, “
Elastic Field in 3D Due to a Spheroidal Inclusion—MATLAB™ Code for Eshelby's Solution
,”
Comput. Geosci.
,
35
(
10
), pp.
2170
2173
.
15.
Meng
,
C.
,
Heltsley
,
W.
, and
Pollard
,
D. D.
,
2012
, “
Evaluation of the Eshelby Solution for the Ellipsoidal Inclusion and Heterogeneity
,”
Comput. Geosci.
,
40
, pp.
40
48
.
16.
Gradsteyn
,
I.
, and
Rhyzik
,
I.
,
1965
,
Table of Integrals, Series and Products
,
Academic Press
,
New York
.
17.
Tanaka
,
K.
, and
Mori
,
T.
,
1972
, “
Note on Volume Integrals of the Elastic Field Around an Ellipsoidal Inclusion
,”
J. Elasticity
,
2
(
3
), pp.
199
200
.
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