We study the internal stress field of a three-phase two-dimensional inclusion of arbitrary shape bonded to an unbounded matrix through an intermediate interphase layer when the matrix is subjected to remote uniform in-plane stresses. The elastic materials occupying all three phases belong to a particular class of compressible hyperelastic harmonic materials. Our analysis indicates that the internal stress field can be uniform and hydrostatic for some nonelliptical shapes of the inclusion, and all of the possible shapes of the inclusion permitting internal uniform hydrostatic stresses are identified. Three conditions are derived that ensure an internal uniform hydrostatic stress state. Our rigorous analysis indicates that for the given material and geometrical parameters of the three-phase inclusion of a nonelliptical shape, at most, eight different sets of remote uniform Piola stresses can be found, leading to internal uniform hydrostatic stresses. Finally, the analytical results are illustrated through an example.

References

References
1.
Hardiman
,
N. J.
, 1954, “
Elliptic Elastic Inclusion in an Infinite Elastic Plate
,”
Q. J. Mech. Appl. Math.
,
7
(
2
), pp.
226
230
.
2.
Eshelby
,
J. D.
, 1957, “
The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems
,”
Proc. R. Soc. London
,
A241
, pp.
376
396
.
3.
Mura
,
T.
, 1987,
Micromechanics of Defects in Solids
,
Martinus Nijhoff
,
Dordrecht
.
4.
Ru
,
C. Q.
,
Schiavone
,
P.
, and
Mioduchowski
,
A.
, 1999, “
Uniformity of the Stresses Within a Three-Phase Elliptical Inclusion in Anti-Plane Shear
,”
J. Elast.
,
52
, pp.
121
128
.
5.
Ru
,
C. Q.
, 1999, “
Three-Phase Elliptical Inclusions With Internal Uniform Hydrostatic Stresses
,”
J. Mech. Phys. Solids
,
47
, pp.
259
273
.
6.
Antipov
,
Y.
A.
,
and
Schiavone
,
P.
, 2003, “
On the Uniformity of Stresses Inside an Inhomogeneity of Arbitrary Shape
,”
IMA J. Appl. Math.
,
68
, pp.
299
311
.
7.
Wang
,
X.
, and
Gao
,
X. L.
, 2011, “
On the Uniform Stress State Inside an Inclusion of Arbitrary Shape in a Three-Phase Composite
,”
Z. Angew. Math. Phys.
(in press).
8.
Ru
,
C. Q.
, 2002, “
On Complex-Variable Formulation for Finite Plane Elastostatics of Harmonic Materials
,”
Acta Mech.
,
156
, pp.
219
234
.
9.
Li
,
X.
, and
Steigmann
,
D. J.
, 1993, “
Finite Plane Twist of an Annular Membrane
,”
Q. J. Mech. Appl. Math.
,
46
, pp.
601
625
.
10.
Ru
,
C. Q.
,
Schiavone
,
P.
,
Sudak
,
L. J.
, and
Mioduchowski
,
A.
, 2005, “
Uniformity of Stresses Inside an Elliptical Inclusion in Finite Elastostatics
,”
Int. J. Nonlinear Mech.
,
40
, pp.
281
287
.
11.
Wang
,
X.
, 2011, “
Three-Phase Elliptical Inclusions With Internal Uniform Hydrostatic Stresses in Finite Plane Elastostatics
,”
Acta Mech.
,
219
, pp.
77
90
.
12.
Varley
,
E.
, and
Cumberbatch
,
E.
, 1980, “
Finite Deformation of Elastic Materials Surrounding Cylindrical Holes
,”
J. Elast.
,
10
, pp.
341
405
.
13.
Abeyaratne
,
R.
, 1983, “
Some Finite Elasticity Problems Involving Crack Tips
,”
Modelling Problems in Crack Tip Mechanics
,
J. T.
Pindera
, ed.,
University of Waterloo
,
Waterloo
, pp.
3
24
.
14.
England
,
A. H.
, 1971,
Complex Variable Methods in Elasticity
,
Wiley
,
London
.
15.
Knowles
,
J. K.
, and
Sternberg
,
E.
, 1975, “
On the Singularity Induced by Certain Mixed Boundary Conditions in Linearized and Nonlinear Elastostatics
,”
Int. J. Solids Struct.
,
11
, pp.
1173
1201
.
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