This paper deals with numerical solutions of singular integral equations in interaction problems of elliptical inclusions under general loading conditions. The stress and displacement fields due to a point force in infinite plates are used as fundamental solutions. Then, the problems are formulated as a system of singular integral equations with Cauchy-type or logarithmic-type singularities, where the unknowns are the body force densities distributed in infinite plates having the same elastic constants as those of the matrix and inclusions. To determine the unknown body force densities to satisfy the boundary conditions, four auxiliary unknown functions are derived from each body force density. It is found that determining these four auxiliary functions in the range 0φkπ/2 is equivalent to determining an original unknown density in the range 0φk2π. Then, these auxiliary unknowns are approximated by using fundamental densities and polynomials. Initially, the convergence of the results such as unknown densities and interface stresses are confirmed with increasing collocation points. Also, the accuracy is verified by examining the boundary conditions and relations between interface stresses and displacements. Randomly or regularly distributed elliptical inclusions can be treated by combining both solutions for remote tension and shear shown in this study.

1.
Barrett
R. F.
,
Sheth
P. R.
, and
Patel
G. C.
,
1971
, “
Effect of Two Circular Holes in a Plate Subjected to Pure Shear Stress
,”
ASME JOURNAL OF APPLIED MECHANICS
, Vol.
93
, pp.
528
530
.
2.
Donnel, L. H., 1974, Stress Concentration Factors, R. E. Peterson, ed., John Wiley and Sons, New York, p. 223.
3.
Erdogan
F.
, and
Gupta
G. D.
,
1972
, “
On the Numerical Solution of Singular Integral Equations
,”
Quarterly Applied Mathematics
, Vol.
30
, pp.
525
534
.
4.
Eshelby
J. D.
,
1957
, “
The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems
,”
Proceedings Royal Society of London
., Vol.
A241
, pp.
376
396
.
5.
Eshelby
J. D.
,
1959
, “
The Elastic Field Outside an Ellipsoidal Inclusion
,”
Proceedings Royal Society of London
., Vol.
A252
, pp.
561
569
.
6.
Mura, T., 1987, Micromechanics of Defects in Solids, 2nd Ed., Martinus Nijhoff, Boston, MA, pp. 129–176.
7.
Mura
T.
,
1988
, “
Inclusion Problems
,”
ASME Applied Mechanics Reviews
, Vol.
41
, pp.
129
176
.
8.
Nisitani
H.
,
1963
, “
On the Tension of an Infinite Plate Containing an Infinite Row of Elliptical Holes
,”
Bull Japan Soc. Mech. Eng.
, Vol.
6
, pp.
635
638
.
1.
Nisitani
H.
,
1967
, “
The Two-Dimensional Stress Problem solved using an Electrical Digital Computer
,”
Jour. Japan Soc. Mech. Eng.
, Vol.
70
, pp.
627
632
. (
2.
Bull. Japan Soc. Mech. Eng.
, Vol.
11
,
1968
, pp.
14
23
).
1.
Nisitani
H.
,
1968
, “
Method of Approximate Calculation for Interference Effects and Its Application
,”
J. Japan Soc. Mech. Eng.
, Vol.
11
, pp.
725
738
.
2.
Nisitani, H., 1974, “Solution of Notch Problems by Body Force Method,” Stress Analysis of Notch Problem G. C. Sih, ed., Leyden, pp. 1–68.
3.
Nisitani, H., and Chert, D. H., 1987, Tatisekiryokuhou, Baifukan, p. 89 (in Japanese).
4.
Noda, N.-A., and Matsuo, T., 1995a, “Singular Integral Equation Method in the Analysis of Interaction between Crack and Defects,” Fracture Mechanics ASTM STP 1220, F. Erdogan, ed., American Society for Testing and Materials, Philadelphia, pp. 591–605.
5.
Noda
N.-A.
, and
Matsuo
T.
,
1995
b, “
Singular Integral Equation Method in Optimization of Stress-Relieving Hole: A New Approach Based on the Body Force Method
,”
Int. Jour. Fract.
, Vol.
70
, pp.
147
165
.
6.
Noda
N.-A.
, and
Matsuo
T.
,
1997
, “
Numerical Solution of Singular Integral Equations in Stress Concentration Problems
,”
International Journal Solids Structures
, Vol.
34
, No.
19
, pp.
2429
2444
.
7.
Shioya
S.
,
1971
, “
On the Tension of an Infinite Thin Plate Containing a Pair of Circular Inclusions
,”
Bulletin of the JSME
, Vol.
14
, No.
68
. pp.
117
126
.
This content is only available via PDF.
You do not currently have access to this content.