The boundary perturbation method is used to solve the problem of a nearly circular rigid inclusion in a two-dimensional elastic medium subjected to hydrostatic stress at infinity. The solution is taken to the fourth order in the small parameter epsilon that quantifies the magnitude of the variation of the radius of the inclusion. This result is then used to find the effective bulk modulus of a body that contains a dilute concentration of such inclusions. The corresponding results for a cavity are obtained by setting the Muskhelishvili coefficient κ equal to −1, as specified by the Dundurs correspondence principle. The results for nearly circular pores can be expressed in terms of the pore compressibility. The pore compressibilities given by the perturbation solution are tested against numerical values obtained using the boundary element method, and are shown to have good accuracy over a substantial range of roughness values.

1.
Mura
,
T.
, 1987,
Micromechanics of Defects in Solids
,
2nd ed.
,
Kluwer
, Amsterdam.
2.
Nemat-Nasser
,
S.
, and
Hori
,
M.
, 1999,
Micromechanics: Overall Properties of Heterogeneous Materials
,
2nd ed.
,
North-Holland
, Dordrecht, The Netherlands.
3.
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
.
4.
Savin
,
G. N.
, 1961,
Stress Concentration Around Holes
,
Pergamon
, Oxford, UK.
5.
Muskhelishvili
,
N. I.
, 1963,
Some Basic Problems of the Mathematical Theory of Elasticity
,
2nd ed.
,
Noordhoff
, Groningen, The Netherlands.
6.
Sisavath
,
S.
,
Jing
,
X. D.
, and
Zimmerman
,
R. W.
, 2001, “
Laminar Flow Through Irregularly-Shaped Pores in Sedimentary Rocks
,”
Transp. Porous Media
0169-3913,
45
, pp.
41
62
.
7.
Tsukrov
,
I.
, and
Novak
,
J.
, 2004, “
Effective Elastic Properties of Solids With Two-Dimensional Inclusions of Irregular Shapes
,”
Int. J. Solids Struct.
0020-7683,
41
, pp.
6905
6924
.
8.
Zimmerman
,
R. W.
, 1986, “
Compressibility of Two-Dimensional Cavities of Various Shapes
,”
ASME J. Appl. Mech.
0021-8936,
53
, pp.
500
504
.
9.
Ekneligoda
,
T. C.
, and
Zimmerman
,
R. W.
, 2006, “
Compressibility of Two-Dimensional Pores Having n-Fold Axes of Symmetry
,”
Proc. R. Soc. London, Ser. A
1364-5021,
462
, pp.
1933
1947
.
10.
Chang
,
C. S.
, and
Conway
,
H. D.
, 1968, “
A Parametric Study of the Complex Variable Method for Analyzing the Stresses in an Infinite Plate Containing a Rigid Rectangular Inclusion
,”
Int. J. Solids Struct.
0020-7683,
4
, pp.
1057
1066
.
11.
Jasiuk
,
I.
,
Chen
,
J.
, and
Thorpe
,
M. F.
, 1994, “
Elastic Moduli of Two Dimensional Materials With Polygonal and Elliptical Holes
,”
Appl. Mech. Rev.
0003-6900,
47
(
Supp 1
), pp.
18
21
.
12.
Kachanov
,
M.
,
Tsukrov
,
I.
, and
Shafiro
,
B.
, 1994, “
Effective Moduli of Solids With Cavities of Various Shapes
,”
Appl. Mech. Rev.
0003-6900,
47
(
Supp 1
), pp.
152
174
.
13.
van Dyke
,
M.
, 1975,
Perturbation Methods in Fluid Mechanics
,
Parabolic Press
, Stanford, CA.
14.
Low
,
E. F.
, and
Chang
,
F. W.
, 1967, “
Stress Concentrations Around Shaped Holes
,”
J. Eng. Mech.
0733-9399,
93
, pp.
33
44
.
15.
Wang
,
C. H.
, and
Chao
,
C. K.
, 2002, “
On Perturbation Solutions for Nearly Circular Inclusion Problems in Plane Thermoelasticity
,”
ASME J. Appl. Mech.
0021-8936,
69
, pp.
36
44
.
16.
Parnes
,
R.
, 1987, “
The Boundary Perturbation Method in Elastostatics—Investigation of Higher-Order Effects and Accuracy of Solutions
,”
J. Mec. Theor. Appl.
0750-7240,
6
, pp.
295
314
.
17.
Gao
,
H.
, 1990, “
A Boundary Perturbation Analysis for Elastic Inclusions and Interfaces
,”
Int. J. Solids Struct.
0020-7683,
28
, pp.
703
725
.
18.
Givoli
,
D.
, and
Elishakoff
,
I.
1992, “
Stress-Concentration at a Nearly Circular Hole With Uncertain Irregularities
,”
ASME J. Appl. Mech.
0021-8936,
59
, pp.
S65
-
S71
.
19.
Little
,
R. W.
, 1973,
Elasticity
,
Prentice Hall
, Englewood Cliffs, NJ.
20.
Barber
,
J. R.
, 1992,
Elasticity
,
Kluwer
, Dordrecht, The Netherlands.
21.
Jasiuk
,
I.
, 1995,
Cavities vis-a-vis Rigid Inclusions: Elastic Moduli of Materials With Polygonal Inclusions
,”
Int. J. Solids Struct.
0020-7683,
32
, pp.
407
422
.
22.
Dundurs
,
J.
, 1989, “
Cavities vis-à-vis Rigid Inclusions and Some Related General Results in Plane Elasticity
,”
ASME J. Appl. Mech.
0021-8936,
56
, pp.
786
790
.
23.
Goodier
,
J. N.
, 1933, “
Concentration of Stress around Spherical and Cylindrical Inclusions and Flaws
,”
Trans. ASME
0097-6822,
55
, pp.
39
44
.
24.
Zimmerman
,
R. W.
, 1991,
Compressibility of Sandstones
,
Elsevier
, Amsterdam, The Netherlands.
25.
Vigdergauz
,
S.
, 2006, “
Cross Relations Between the Planar Elastic Moduli of Perforated Structures
,”
ASME J. Appl. Mech.
0021-8936,
73
, pp.
163
166
.
26.
Martel
,
S. J.
, and
Muller
,
J. R.
, 2000, “
A Two-Dimensional Boundary Element Method for Calculating Elastic Gravitational Stresses in Slopes
,”
Pure Appl. Geophys.
0033-4553,
157
, pp.
989
1007
.
27.
Crouch
,
S. L.
, and
Starfield
,
A. M.
, 1983,
Boundary Element Method in Solid Mechanics
,
Allen and Unwin
, London, UK.
28.
Gol’dshtein
,
R. V.
, and
Entov
,
V. M.
1994,
Qualitative Methods in Continuum Mechanics
,
Wiley
, New York.
29.
Hashin
,
Z.
, 1983, “
Analysis of Composite Materials – a Survey
,”
ASME J. Appl. Mech.
0021-8936,
50
, pp.
481
505
.
30.
Warren
,
N.
, 1973, “
Theoretical Calculation of the Compressibility of Porous Media
,”
J. Geophys. Res.
0148-0227,
78
, pp.
352
362
.
31.
Rice
,
R. W.
, 1998,
Porosity of Ceramics
,
Marcel Dekker
, New York.
32.
Prokopiev
,
O.
, and
Sevostianov
,
I.
, 2006, “
On the Possibility of Approximation of Irregular Porous Microstructure by Isolated Spheroidal Pores
,”
Int. J. Fract.
0376-9429,
139
, pp.
129
136
.
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