With the use of contact stress theory and complex variable methods in two dimensions, the transmission of a compressive stress through a circular cross section of a small material particle is calculated in the infinite plane of material below the circular cross section. The circular cross section of the particle is embedded in and completely bonded to an infinite plane of matrix material. It is shown that part of the stress is transmitted with a dependence of 1r, where r is a radial coordinate. Additionally, the stress is calculated in two dimensions for the interior of an ellipse that could model a cross section of a grain or particle. The boundary of the ellipse is loaded with the stress holding an elliptic kernel in place in an elastic matrix material after the kernel has undergone a small rotation under an applied tensile load. The resulting stresses are shown in contour plots for elliptic cross sections of varying shapes and orientations.

1.
Ohji
,
T.
,
Kusunose
,
T.
, and
Niihara
,
K.
, 1998, “
Threshold Stress in Creep of Alumina-Silicon Carbide Nanocomposites
,”
J. Am. Ceram. Soc.
0002-7820,
81
, pp.
2713
2716
.
2.
Oh
,
S.-T.
,
Sando
,
M.
, and
Niihara
,
K.
, 1998, “
Preparation and Properties of Alumina/Nickel-Cobalt Alloy Nanocomposites
,”
J. Am. Ceram. Soc.
0002-7820,
81
, pp.
3013
3015
.
3.
Davis
,
L. C.
, and
Allison
,
J. E.
, 1993, “
Residual Stresses and Their Effects on Deformation in Particle-Reinforced Metal-Matrix Composites
,”
Metall. Trans. A
0360-2133,
24
, pp.
2487
2496
.
4.
Kovalev
,
S.
,
Ohji
,
T.
,
Yamauchi
,
Y.
, and
Sakai
,
M.
, 2000, “
Grain Boundary Strength in Non-Cubic Polycrystals with Misfitting Intragranular Inclusions
,”
J. Mater. Sci.
0022-2461,
35
, pp.
1405
1412
.
5.
Mizushima
,
I.
,
Hamada
,
M.
, and
Shakudo
,
T.
, 1978, “
Tensile and Compressive Stress Problems for a Rigid Circular Disk in an Infinite Plate
,”
Bull. JSME
0021-3764,
21
, pp.
1325
1333
.
6.
Wang
,
J.
,
Andreasen
,
J. H.
, and
Karihaloo
,
B. L.
, 2000, “
The Solution of an Inhomogeneity in a Finite Plane Region and its Application to Composite Materials
,”
Compos. Sci. Technol.
0266-3538,
60
, pp.
75
82
.
7.
Mizushima
,
I.
,
Hamada
,
M.
, and
Kusano
,
N.
, 1979, “
Tensile and Compressive Stress Problems for a Circular Disk in an Infinite Plate
,”
Bull. JSME
0021-3764,
22
, pp.
1175
1181
.
8.
Gladwell
,
G. M. L.
, and
Iyer
,
K. R. P.
, 1974, “
Unbonded Contact Between a Circular Plate and an Elastic Half-Space
,”
J. Elast.
0374-3535,
4
, pp.
115
130
.
9.
Miller
,
G. R.
, and
Keer
,
L. M.
, 1983, “
Interaction Between a Rigid Indenter and a Near-Surface Void or Inclusion
,”
J. Appl. Mech.
0021-8936,
50
, pp.
615
620
.
10.
Wang
,
J.
,
Andreasen
,
J. H.
, and
Karihaloo
,
B. L.
, 2000, “
The Solution of an Inhomogeneity in a Finite Plane Region and its Application to Composite Materials
,”
Compos. Sci. Technol.
0266-3538,
60
, pp.
75
82
.
11.
Onaka
,
S.
, and
Kato
,
M.
, 1999, “
Effects of Elastic Modulus, Shape and Volume Fraction of an Elastically Inhomogeneous Second Phase on Stress States in a Loaded Composite
,”
Mater. Trans., JIM
0916-1821,
40
, pp.
1102
1107
.
12.
Batista
,
M.
, 1999, “
Stresses in a Confocal Elliptic Ring Subject to Uniform Pressure
,”
J. Strain Anal. Eng. Des.
0309-3247,
34
, pp.
217
221
.
13.
Chen
,
D.-H.
, 1996, “
Green’s Functions for a Point Force and Dislocation Outside an Elliptic Inclusion in Plane Elasticity
,”
Z. Angew. Math. Mech.
0044-2267,
47
, pp.
894
905
.
14.
Gross
,
R. S.
,
Goree
,
J. G.
, 1991, “
Torsion of a Rigid Smooth Elliptic Insert in an Infinite Elastic Plane
,”
J. Appl. Mech.
0021-8936,
58
, pp.
370
375
.
15.
Sendeckyj
,
G. P.
, 1970, “
Elastic Inclusion Problems in Plane Elastostatics
,”
Int. J. Solids Struct.
0020-7683,
6
, pp.
1535
1543
.
16.
Karihaloo
,
B. L.
, and
Viswanathan
,
K.
, 1988, “
A Partially Debonded Ellipsoidal Inclusion in an Elastic Medium. Part I: Stress and Displacement Fields
,”
Mech. Mater.
0167-6636,
7
, pp.
191
197
.
17.
Noda
,
N.-A.
,
Tomari
,
K.
, and
Matsuo
,
T.
, 1999, “
Interaction Effect Between Ellipsoidal Inclusions in an Infinite Body Under Asymmetric Uniaxial Tension
,”
JSME Int. J., Ser. A
1340-8046,
42
, pp.
372
380
.
18.
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
.
19.
Eshelby
,
J. D.
, 1959, “
The Elastic Field Outside an Ellipsoidal Inclusion
,”
Proc. R. Soc. London, Ser. A
1364-5021,
252
, pp.
561
569
.
20.
Edwardes
,
D.
, 1893, “
Steady Motion of a Viscous Liquid in Which an Ellipsoid is Constrained to Rotate About a Principal Axis
,”
Q. Appl. Math.
0033-569X,
26
, pp.
70
78
.
21.
Huang
,
Y.
,
Hu
,
K. X.
, and
Chandra
,
A.
, 1995, “
Stiffness Evaluation for Solids Containing Dilute Distributions of Inclusions and Microcracks
,”
ASME J. Appl. Mech.
0021-8936,
62
, pp.
71
77
.
22.
Erdogan
,
F.
,
Gupta
,
G. D.
, and
Ratwani
,
M.
, 1974, “
Interaction Between an Inclusion and an Arbitrarily Oriented Crack
,”
ASME J. Appl. Mech.
0021-8936,
41
, pp.
1007
1013
.
23.
Dundurs
,
J.
and
Mura
,
T.
, 1964, “
Interaction Between an Edge Dislocation and a Circular Inclusion
,”
J. Mech. Phys. Solids
0022-5096,
12
, pp.
177
189
.
24.
Boniface
,
V.
, and
Hasebe
,
N.
, 1998, “
Solution of the Displacement Boundary Value Problem of an Interface Between Two Dissimilar Half-Planes and a Rigid Elliptic Inclusion at the Interface
,”
ASME J. Appl. Mech.
0021-8936,
65
, pp.
880
888
.
25.
Ballarini
,
R.
, 1990, “
A Rigid Line Inclusion at a Bimaterial Interface
,”
Eng. Fract. Mech.
0013-7944,
37
, pp.
1
5
.
26.
Muskhelishvili
,
N. I.
, 1953,
Some Basic Problems of the Mathematical Theory of Elasticity
,
P. Noordhoff
, Ltd.
27.
Frocht
,
M. M.
, 1948,
Photoelasticity
,
Wiley
, New York, Vol.
II
.
28.
Wylie
,
C. R.
, and
Barrett
,
L. C.
, 1982,
Advanced Engineering Mathematics
,
McGraw–Hill
, New York.
29.
Churchhill
,
R. V.
,
Brown
,
J. W.
, and
Verhey
,
R. F.
, 1976,
Complex Variables and Applications
,
McGraw–Hill
, New York.
You do not currently have access to this content.