The content of this review consists of recent developments covering an advanced treatment of multiple crack problems in plane elasticity. Several elementary solutions are highlighted, which are the fundamentals for the formulation of the integral equations. The elementary solutions include those initiated by point sources or by a distributed traction along the crack face. Two kinds of singular integral equations, three kinds of Fredholm integral equations, and one kind of hypersingular integral equation are suggested for the multiple crack problems in plane elasticity. Regularization procedures are also investigated. For the solution of the integral equations, the relevant quadrature rules are addressed. A variety of methods for solving the multiple crack problems is introduced. Applications for the solution of the multiple crack problems are also addressed. The concept of the modified complex potential (MCP) is emphasized, which will extend the solution range, for example, from the multiple crack problem in an infinite plate to that in a circular plate. Many multiple crack problems are addressed. Those problems include: (i) multiple semi-infinite crack problem, (ii) multiple crack problem with a general loading, (iii) multiple crack problem for the bonded half-planes, (iv) multiple crack problem for a finite region, (v) multiple crack problem for a circular region, (vi) multiple crack problem in antiplane elasticity, (vii) T-stress in the multiple crack problem, and (viii) periodic crack problem and many others. This review article cites 187 references.

1.
Griffith
,
A. A.
, 1921, “
The Phenomena of Rupture and Flow in Solids
,”
Philos. Trans. R. Soc. London
0962-8428,
A221
, pp.
163
197
.
2.
Muskhelishvili
,
N. I.
, 1953,
Some Basic Problems of the Mathematical Theory of Elasticity
,
Noordhoff
,
Groningen
.
3.
Liebowitz
,
H.
, ed., 1968,
Fracture An Advanced Treatise, Vol. 2, Mathematical Fundamentals
,
,
New York
.
4.
Sneddon
,
I. N.
, and
Lowengrub
,
M.
, 1969,
Crack Problems in Classical Theory of Elasticity
,
Wiley
,
New York
.
5.
England
,
A. H.
, 1971,
Complex Variable Methods in Elasticity
,
Wiley
,
London
.
6.
Sih
,
G. C.
, ed., 1973,
Mechanics of Fracture, Vol. 1, Method of Analysis and Solutions of Crack Problems
,
Noordhoff
,
Leyden
.
7.
Savruk
,
M. P.
, 1981,
Two-Dimensional Problems of Elasticity for Body With Crack
,
Naukava Dumka
,
Kiev
(in Russian).
8.
Parton
,
V. Z.
, and
Perlin
,
P. I.
, 1982,
Integral Equations in Elasticity
,
Mir Publishers
,
Moscow
.
9.
Kanninen
,
M. F.
, and
Popelar
,
C. H.
, 1985,
,
Oxford University Press
,
London
.
10.
Parton
,
V. Z.
, and
Morozov
,
E. M.
, 1989,
Mechanics of Elastic-Plastic Fracture
,
Hemisphere
,
Washington
.
11.
,
M. H.
, and
Rooke
,
D. P.
, 1991,
Numerical Fracture Mechanics
,
Computational Mechanics Publications
,
Southampton
.
12.
Chen
,
Y. Z.
, 1995, “
A Survey of New Integral Equations in Plane Elasticity Crack Problem
,”
Eng. Fract. Mech.
0013-7944,
51
, pp.
97
134
.
13.
,
A. M.
, 2002,
Boundary Integral Equations in Elasticity Theory
,
Kluwer
,
Dordrehet
.
14.
Chen
,
Y. Z.
,
Hasebe
,
N.
, and
Lee
,
K. Y.
, 2003,
Multiple Crack Problems in Elasticity
,
WIT Press
,
Southampton
.
15.
Tang
,
R. J.
, 1996,
Theory of Cracked Torsion Bars
,
Jiao Tong University Press
,
Shanghai
(in Chinese).
16.
Li
,
Z. L.
,
Wang
,
Y. H.
, and
Li
,
T. J.
, 1996,
Numerical Methods in Fracture Mechanics
,
Di Zheng Press
,
Beijing
(in Chinese).
17.
Fan
,
T. Y.
, 2003,
Foundation of Fracture Theory
,
Science Press
,
Beijing
(in Chinese).
18.
Murakami
,
Y.
, ed., 1987,
Stress Intensity Factors Handbook
,
Pergamon
,
Oxford
, Vol.
1
.
19.
Murakami
,
Y.
, ed., 1987,
Stress Intensity Factors Handbook
,
Pergamon
,
Oxford
, Vol.
2
.
20.
Murakami
,
Y.
, ed., 1992,
Stress Intensity Factors Handbook
,
Pergamon
,
Oxford
, Vol.
3
.
21.
Murakami
,
Y.
, ed., 2001,
Stress Intensity Factors Handbook
,
Pergamon
,
Oxford
, Vol.
4
.
22.
Murakami
,
Y.
, ed., 2001,
Stress Intensity Factors Handbook
,
Pergamon
,
Oxford
, Vol.
5
.
23.
Erdogan
,
F.
, 1975,
Complex Function Technique
,
,
New York
.
24.
Woods
,
L. C.
, 1975,
Analytic Function Theory
,
,
New York
.
25.
Muskhelishvili
,
N. I.
, 1953,
Singular Integral Equations
,
Noordhoff
,
Groningen
.
26.
Chen
,
Y. Z.
, 1993, “
Derivation of Two-Dimensional Green’s Functions for Bimaterials by Means of Complex Variable Function Technique
,”
Int. J. Fract.
0376-9429,
60
, pp.
R9
R13
.
27.
Chen
,
Y. Z.
, 1998, “
Complex Potentials in Plane Elasticity by Distribution of Dislocation or Force Doublet Along a Curve
,”
Int. J. Eng. Sci.
0020-7225,
36
, pp.
23
31
.
28.
Nisitani
,
H.
, 1978,
Solution of Notch Problems by Body Force Method
, Vol.
5
, Mechanics of Fracture,
Sih
,
G. C.
, ed.,
Noordhoff
,
Leyden
, pp.
1
68
.
29.
Nisitani
,
H.
, and
Chen
,
D. H.
, 1990,
Body Force Method
,
Zhejiang University Press
,
Hangzhou
(in Chinese, translation from Japanese).
30.
Carvalho
,
J. L.
, and
Curran
,
J. H.
, 1992, “
Two Dimensional Green’s Functions for Elastic Bi-Materials
,”
ASME J. Appl. Mech.
0021-8936,
59
, pp.
321
327
.
31.
Erdogan
,
F.
, 1983, “
Stress Intensity Factors
,”
ASME J. Appl. Mech.
0021-8936,
50
, pp.
992
1002
.
32.
Sih
,
G. C.
,
Paris
,
P. C.
, and
Erdogan
,
F.
, 1962, “
Crack-Tip, Stress Intensity Factors for Plane Extension and Plate Bending Problem
,”
ASME J. Appl. Mech.
0021-8936,
29
, pp.
306
311
.
33.
Dempsey
,
J. P.
, 1981, “
The Wedge Subjected to Tractions: A Paradox Resolved
,”
J. Elast.
0374-3535,
11
, pp.
1
10
.
34.
Ting
,
T. C. T.
, 1984, “
The Wedge Subjected to Tractions: A Paradox Re-Examined
,”
J. Elast.
0374-3535,
14
, pp.
235
247
.
35.
Panasyuk
,
V. V.
,
Savruk
,
M. P.
, and
Datsyshyn
,
A. P.
, 1977, “
A General Method of Solution of Two-Dimensional Problems in the Theory of Cracks
,”
Eng. Fract. Mech.
0013-7944,
9
, pp.
481
497
.
36.
Chen
,
Y. Z.
, 1993, “
Numerical Solution for a Cruciform Crack Problem
,”
Int. J. Fract.
0376-9429,
63
, pp.
R31
R34
.
37.
Chen
,
Y. Z.
, and
Hasebe
,
N.
, 1995, “
New Integration Scheme for Branch Crack Problem
,”
Eng. Fract. Mech.
0013-7944,
52
, pp.
791
801
.
38.
Chen
,
Y. Z.
, and
Hasebe
,
N.
, 1992, “
An Alternative Fredholm Integral Equation Approach for Multiple Crack Problem and Multiple Rigid Line Problem in Plane Elasticity
,”
Eng. Fract. Mech.
0013-7944,
43
, pp.
257
268
.
39.
Erdogan
,
F.
,
Gupta
,
G. D.
, and
Cook
,
T. S.
, 1973,
Numerical Solution of Singular Integral Equation
,
Sih
,
G. C.
, ed.,
Noordhoff
,
Leyden
, Mechanics of Fracture Vol.
1
, pp.
368
425
.
40.
Rooke
,
D. P.
, and
Tweed
,
J.
, 1973, “
The Stress Intensity Factors of a Radial Crack in a Point Loaded Disc
,”
Int. J. Eng. Sci.
0020-7225,
11
, pp.
285
290
.
41.
Chen
,
Y. Z.
, 1984, “
A Fredholm Integral Equation Approach for Multiple Crack Problem in an Infinite Plate
,”
Eng. Fract. Mech.
0013-7944,
20
, pp.
767
776
.
42.
Yarema
,
S. Y.
, 1979, “
Analysis of Cracked Disk Specimens
,”
Eng. Fract. Mech.
0013-7944,
12
, pp.
365
375
.
43.
Gross
,
D.
, 1982, “
Stress Intensity Factors of Systems of Cracks
,”
Ing.-Arch.
0020-1154,
51
, pp.
301
310
.
44.
Horii
,
H.
, and
Nemat-Nasser
,
S.
, 1985, “
Elastic Fields of Interacting Inhomogeneities
,”
Int. J. Solids Struct.
0020-7683,
21
, pp.
731
745
.
45.
Benveniste
,
Y.
,
Dvorak
,
G. J.
,
Zarzour
,
J.
, and
Wung
,
C. J.
, 1989, “
On Interacting Cracks and Complex Crack Configurations in Linear Elastic Media
,”
Int. J. Solids Struct.
0020-7683,
25
, pp.
1279
1293
.
46.
Kachanov
,
M.
, 1987, “
Elastic Analysis With Many Cracks: A Simple Method of Analysis
,”
Int. J. Solids Struct.
0020-7683,
23
, pp.
23
43
.
47.
Kachanov
,
M.
, and
Montagut
,
E.
, 1986, “
Interaction of a Crack With Certain Microcrack Arrays
,”
Eng. Fract. Mech.
0013-7944,
25
, pp.
625
636
.
48.
Kachanov
,
M.
, 2003, “
On the Problems of Interactions and Crack Coalescence
,”
Int. J. Fract.
0376-9429,
120
, pp.
537
543
.
49.
Gorbatikh
,
L.
, and
Kachanov
,
M.
, 2000, “
A Simple Technique for Constructing the Full Stress and Displacement Fields in Elastic Plates With Multiple Cracks
,”
Eng. Fract. Mech.
0013-7944,
66
, pp.
51
63
.
50.
Li
,
Y. P.
,
Tham
,
L. G.
,
Wang
,
Y. H.
, and
Tsui
,
Y.
, 2003, “
A Modified Kachnov Method for Analysis of Solids With Many Cracks
,”
Eng. Fract. Mech.
0013-7944,
70
, pp.
1115
1129
.
51.
Tamate
,
O.
, 1976, “
Two Arbitrarily Situated Cracks in an Elastic Plate Under Flexure
,”
Int. J. Solids Struct.
0020-7683,
12
, pp.
287
298
.
52.
Tweed
,
J.
, and
Rooke
,
D. P.
, 1979, “
The Stress Intensity Factor for a Crack at the Edge of a Loaded Hole
,”
Int. J. Solids Struct.
0020-7683,
15
, pp.
899
906
.
53.
Boduroglu
,
H.
, and
Erdogan
,
F.
, 1983, “
Internal and Edge Cracks in a Plate of Finite Width Under Bending
,”
ASME J. Appl. Mech.
0021-8936,
50
, pp.
621
629
.
54.
Cinar
,
A.
, and
Erdogan
,
F.
, 1983, “
The Crack and Wedging Problem for an Orthotropic Strip
,”
Int. J. Fract.
0376-9429,
23
, pp.
83
102
.
55.
Civelek
,
M. B.
, and
Erdogan
,
F.
, 1982, “
Crack Problem for a Rectangular Plate and an Infinite Strip
,”
Int. J. Fract.
0376-9429,
19
, pp.
139
159
.
56.
Cook
,
T. S.
, and
Erdogan
,
F.
, 1972, “
Stresses in Bonded Materials With a Crack Perpendicular to the Interface
,”
Int. J. Eng. Sci.
0020-7225,
10
, pp.
677
697
.
57.
Delale
,
F.
, and
Erdogan
,
F.
, 1979, “
Bonded Orthotropic Strips With Cracks
,”
Int. J. Fract.
0376-9429,
15
, pp.
343
364
, 1979.
58.
Delale
,
F.
, and
Erdogan
,
F.
, 1982, “
Stress Intensity Factors in a Hollow Cylinder Containing a Radial Crack
,”
Int. J. Fract.
0376-9429,
20
, pp.
251
265
.
59.
Erdogan
,
F.
, and
Aksogan
,
O.
, 1974, “
Bonded Half-planes Containing an Arbitrarily Oriented Crack
,”
Int. J. Solids Struct.
0020-7683,
10
, pp.
569
585
.
60.
Erdogan
,
F.
, and
Bircikoglu
,
V.
, 1973, “
Two Bonded Half-planes With a Crack Going Through the Interface
,”
Int. J. Eng. Sci.
0020-7225,
11
, pp.
745
766
.
61.
Erdogan
,
F.
,
Gupta
,
G. D.
, and
Ratwani
,
M.
, 1974, “
Interaction Between a Circular Inclusion and an Arbitrarily Oriented Crack
,”
ASME J. Appl. Mech.
0021-8936,
41
, pp.
1007
1013
.
62.
Keer
,
L. M.
,
Lee
,
J. C.
, and
Mura
,
T.
, 1983, “
Stress Distributions for a Quarter Plane Containing an Arbitrarily Oriented Crack
,”
ASME J. Appl. Mech.
0021-8936,
50
, pp.
43
49
.
63.
Kishida
,
M.
, and
Asano
,
M.
, 1984, “
A Study of Interface of Three Parallel Cracks
,”
Eng. Fract. Mech.
0013-7944,
19
, pp.
531
538
.
64.
Krenk
,
S.
, 1975, “
On the Elastic Strip With an Internal Crack
,”
Int. J. Solids Struct.
0020-7683,
11
, pp.
693
708
.
65.
Rubinstein
,
A. A.
, and
,
A. M.
, 1986, “
Analysis of a Crack Emanating From a Circular Hole in a Loaded Plane
,”
Int. J. Fract.
0376-9429,
32
, pp.
47
57
.
66.
Sekine
,
H.
, 1977, “
Crack Problem for a Semi-infinite Solid With Heated Bounding Surface
,”
ASME J. Appl. Mech.
0021-8936,
44
, pp.
637
642
.
67.
Erdogan
,
F.
, 1978,
Mixed Boundary-Value Problems in Mechanics
,
Nemat-Nasser
,
S.
, ed.,
Pergamon
,
New York
, Mechanics Today Vol.
4
, pp.
1
84
.
68.
Erdogan
,
F.
, 1969, “
Approximate Solutions of Systems of Singular Integral Equations
,”
SIAM J. Appl. Math.
0036-1399,
17
, pp.
1041
1059
.
69.
Erdogan
,
F.
, and
Gupta
,
G. D.
, 1972, “
On the Numerical Solution of Singular Integral Equations
,”
Q. Appl. Math.
0033-569X,
29
, pp.
525
534
.
70.
Boiko
,
A. V.
, and
Karpenko
,
L. N.
, 1981, “
On Some Numerical Methods for the Solution of the Plane Elasticity Problem for Bodies With Cracks by Means of Singular Integral Equations
,”
Int. J. Fract.
0376-9429,
17
, pp.
381
388
.
71.
Chen
,
Y. Z.
, 2004, “
Collinear Crack Problem in Antiplane Elasticity for a Strip of Functionally Graded Materials
,”
J. Mech.
1727-7191,
20
, pp.
167
175
.
72.
Sneddon
,
I. N.
, 1973,
Integral Transform Methods
,
Sih
,
G. G.
, ed.,
Noordhoff
,
Leyden
, Mechanics of Fracture Vol.
1
, pp.
315
367
.
73.
Sneddon
,
I. N.
, 1972,
The Use of Integral Transforms
,
Wiley
,
New York
.
74.
Sneddon
,
I. N.
, 1975,
Application of Integral Transforms in the Theory of Elasticity
,
Spinger
,
New York
.
75.
Chen
,
Y. Z.
, 1989, “
Crack Problem in Plane Elasticity Under Antisymmetric Loading
,”
Int. J. Fract.
0376-9429,
41
, pp.
R29
R34
.
76.
Chudnovsky
,
A.
,
Dolgopolsky
,
A.
, and
Kachanov
,
M.
, 1987, “
Elastic Interaction of a Crack With a Microcrack Array—I. Formulation of the Problem and General Form of the Solution
,”
Int. J. Solids Struct.
0020-7683,
23
, pp.
1
10
.
77.
Chudnovsky
,
A.
,
Dolgopolsky
,
A.
, and
Kachanov
,
M.
, 1987, “
Elastic Interaction of a Crack With a Microcrack Array—II. Elastic Solution for Two Crack Configurations (Piecewise Constant and Linear Approximations)
,”
Int. J. Solids Struct.
0020-7683,
23
, pp.
11
21
.
78.
Wang
,
X. M.
,
Gao
,
S.
, and
Chen
,
Y. H.
, 1996, “
Further Investigation for the Macro-Microcrack Interaction in the Infinite Isotropic Body
,”
Int. J. Solids Struct.
0020-7683,
33
, pp.
4051
4063
.
79.
Rubinstein
,
A. A.
, 1985, “
Macrocrack Interaction With Semi-Infinite Microcrack Array
,”
Int. J. Fract.
0376-9429,
27
, pp.
113
119
.
80.
Rubinstein
,
A. A.
, 1986, “
Macrocrack-Microdefect Interaction
,”
ASME J. Appl. Mech.
0021-8936,
53
, pp.
505
510
.
81.
Kachanov
,
M.
, 1993, “
Elastic Solids With Many Cracks and Related Problem
,”
Advance in Applied Mechanics
,
Hutchinson
,
J. W.
, and
Wu
,
T.
, ed.,
,
New York
, Vol.
30
, pp.
259
445
.
82.
Feng
,
X. Q.
,
Li
,
J. Y.
, and
Yu
,
S. W.
, 2003, “
A Simple Method for Calculating Interaction of Numerous Microcracks and Its Applications
,”
Int. J. Solids Struct.
0020-7683,
40
, pp.
447
464
.
83.
Zhao
,
L. G.
, and
Chen
,
Y. H.
, 1997, “
On the Contribution of Subinterface Microcracks Near the Tip of an Interface Macrocrack to the J-Integral in Bimaterial Solids
,”
Int. J. Eng. Sci.
0020-7225,
35
, pp.
387
407
.
84.
Chen
,
Y. H.
, and
Hasebe
,
N.
, 1998, “
A Consistency Check for Strongly Interacting Multiple Crack Problem in Isotropic, Bimaterial and Orthotropic Bodies
,”
Int. J. Fract.
0376-9429,
89
, pp.
333
353
.
85.
Chen
,
Y. H.
, 2001, “
M-Integral Analysis for Two-Dimensional Solids With Strongly Interacting Microcracks. Part I: In an Infinite Brittle Solid
,”
Int. J. Solids Struct.
0020-7683,
38
, pp.
3193
3212
.
86.
Chen
,
Y. H.
, and
Lu
,
T. J.
, 2003, “
Recent Developments and Applications of Invariant Integrals
,”
Appl. Mech. Rev.
0003-6900,
56
, pp.
515
552
.
87.
Chen
,
Y. Z.
, 2004, “
Analysis of the M-Integral in Plane Elasticity
,”
ASME J. Appl. Mech.
0021-8936,
71
, pp.
572
574
.
88.
Budiansky
,
B.
, and
O’Coonnell
,
R. J.
, 1976, “
Elastic Moduli of a Cracked Body
,”
Int. J. Solids Struct.
0020-7683,
12
, pp.
81
97
.
89.
Wang
,
J.
,
Fang
,
J.
, and
Karihaloo
,
B. L.
, 2000, “
Asymptotics of Multiple Crack Interaction and Prediction of Effective Modulus
,”
Int. J. Solids Struct.
0020-7683,
37
, pp.
4261
4273
.
90.
Wang
,
J.
,
Fang
,
J.
, and
Karihaloo
,
B. L.
, 2000, “
Asymptotic Bounds on Overall Moduli of Cracked Bodies
,”
Int. J. Solids Struct.
0020-7683,
37
, pp.
6221
6237
.
91.
Chen
,
Y. Z.
, 1993, “
New Fredholm Integral Equation for Multiple Crack Problem in Plane and Antiplane Elasticity
,”
Int. J. Fract.
0376-9429,
64
, pp.
63
77
.
92.
Chen
,
Y. Z.
, 1994, “
Various Integral Equations for a Single Crack Problem of Elastic Half-Plane
,”
Eng. Fract. Mech.
0013-7944,
49
, pp.
849
858
.
93.
Denda
,
M.
, and
Dong
,
Y. F.
, 1997, “
Complex Variable Approach to the BEM for Multiple Crack Problems
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
141
, pp.
247
264
.
94.
Denda
,
M.
, and
Kosaka
,
I.
, 1997, “
Dislocation and Point-Force-Based Approach to the Special Green’s Function BEM for Elliptic Hole and Crack Problem in Two Dimensions
,”
Int. J. Numer. Methods Eng.
0029-5981,
40
, pp.
2857
2889
.
95.
Chen
,
Y. Z.
, 1984, “
Elastic Analysis of an Infinite Plate Containing Hole With Cusps and Applied by Concentrated Forces
,”
Eng. Fract. Mech.
0013-7944,
20
, pp.
573
582
.
96.
Yan
,
X. Q.
, 2004, “
A Numerical Analysis of Perpendicular Cracks Under General In-Plane Loading With a Hybrid Displacement Discontinuity Method
,”
Mech. Res. Commun.
0093-6413,
31
, pp.
175
183
.
97.
Helsing
,
J.
, 1999, “
Fast and Accurate Numerical Solution to an Elastostatic Problem Involving Ten Thousand Randomly Oriented Cracks
,”
Int. J. Fract.
0376-9429,
100
, pp.
321
327
.
98.
Helsing
,
J.
, and
Peters
,
G.
, 1999, “
Integral Equation Methods and Numerical Solutions of Crack and Inclusion Problems in Plane Elastostatics
,”
SIAM J. Appl. Math.
0036-1399,
59
, pp.
965
982
.
99.
,
J.
, 1923,
Lectures on Cauchy’s Problem in Linear Partial Differential Equation
,
Yale University Press
,
New Haven, CT
.
100.
Kaya
,
A. C.
, and
Erdogan
,
F.
, 1987, “On the Solutions of Integral Equations With Strongly Singular Kernels,” Quant. Appl. Mech., 45, pp. 105–122.
101.
Mayrhofer
,
K.
, and
Fischer
,
F. D.
, 1992, “
Derivation of a New Analytical Solution for a General Two-Dimensional Finite-Part Integral Applicable in Fracture Mechanics
,”
Int. J. Numer. Methods Eng.
0029-5981,
33
, pp.
1027
1047
.
102.
Chen
,
Y. Z.
, 1995, “
Hypersingular Integral Equation for Multiple Crack Problem in an Infinite Plate
,”
Acta Mech.
0001-5970,
108
, pp.
121
131
.
103.
Chen
,
Y. Z.
, 1997, “
Numerical Solution of Multiple Crack Problem by Using Hypersingular Integral Equation
,”
Int. J. Fract.
0376-9429,
88
, pp.
L9
L4
.
104.
Chen
,
Y. Z.
, 1998, “
Numerical Solution of Circular Cracked Plate Problem by Using Hypersingular Integral Equation Approach
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
14
, pp.
451
461
.
105.
Ioakimidis
,
N. I.
, 1988, “
Mangler-Type Principal Value Integrals in Hypersingular Integral Equations for Crack in Plane Elasticity
,”
Eng. Fract. Mech.
0013-7944,
31
, pp.
895
898
.
106.
,
A. M.
, and
Mogilevskaya
,
S. G.
, 1986, “
Finite-Part Integrals in Problems of Three-Dimensional Cracks
,”
Prikl. Mat. Mekh.
0032-8235,
50
, pp.
652
658
.
107.
,
A. M.
, and
Mogilevskaya
,
S. G.
, 1994, “
Complex Hypersingular Integrals and Integral Equations in Plane Elasticity
,”
Acta Mech.
0001-5970,
105
, pp.
189
205
.
108.
Krenk
,
S.
, 1975, “
On the Use of the Interpolation Polynomial for Solutions of Singular Integral Equations
,”
Q. Appl. Math.
0033-569X,
32
, pp.
479
484
.
109.
Martin
,
P. A.
, 2000, “
Perturbed Cracks in Two Dimensions: An Integral-Equation Approach
,”
Int. J. Fract.
0376-9429,
100
, pp.
317
327
.
110.
Chan
,
Y. S.
,
Finnjiang
,
A. C.
, and
Paulino
,
G. H.
, 2003, “
Integral Equations With Hypersingular Kernels—Theory and Application to Fracture Mechanics
,”
Int. J. Eng. Sci.
0020-7225,
41
, pp.
683
720
.
111.
Nied
,
H. F.
, 1987, “
Periodic Array of Cracks in a Half-Plane Subjected to Arbitrary Loading
,”
ASME J. Appl. Mech.
0021-8936,
54
, pp.
642
648
.
112.
Chen
,
Y. Z.
, 1992, “
Hypersingular Integral Equation for a Curved Crack Problem in Half-Plane
,”
Int. J. Fract.
0376-9429,
57
, pp.
R41
R45
.
113.
Chen
,
Y. Z.
, 1993, “
Numerical Solution of a Curved Crack Problem by Using Hypersingular Integral Equation Approach
,”
Eng. Fract. Mech.
0013-7944,
46
, pp.
275
283
.
114.
Chen
,
Y. Z.
, 1994, “
Hypersingular Integral Equation for Curved Crack Problem in Antiplane Elasticity
,”
Int. J. Fract.
0376-9429,
66
, pp.
R19
R21
.
115.
Chen
,
Y. Z.
, and
Hasebe
,
N.
, 1996, “
Hypersingular Integral Equation for a Curved Crack Problem of Circular Region in Antiplane Elasticity
,”
ASME J. Appl. Mech.
0021-8936,
63
, pp.
645
649
.
116.
Chen
,
Y. Z.
, 2003, “
A Numerical Solution Technique of Hypersingular Integral Equation for Curve Cracks
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
19
, pp.
645
655
.
117.
Chen
,
Y. Z.
,
Lin
,
X. Y.
, and
Wang
,
Z. X.
, 2004, “
Numerical Solutions of Hypersingular Integral Equation for Antiplane Elasticity Curved Crack Problems of Circular Regions
,”
Acta Mech.
0001-5970,
173
, pp.
1
11
.
118.
Chen
,
Y. Z.
, 2005, “
Numerical Solutions of Hypersingular Integral Equation for Curved Cracks in Circular Regions
,”
Int. J. Fract.
0376-9429,
132
, pp.
205
222
.
119.
Mogilevskaya
,
S. G.
,
Rothenburg
,
L.
, and
Dusseault
,
M. B.
, 2000, “
Interaction Between a Circular Opening and Fractures Originating From its Boundary in a Piecewise Homogenous Plane
,”
Int. J. Numer. Analyt. Meth. Geomech.
0363-9061,
24
, pp.
947
970
.
120.
Mogilevskaya
,
S. G.
, 2000, “
Complex Hypersingular Integral Equation for the Piece-Wise Homogeneous Half-Plane With Cracks
,”
Int. J. Fract.
0376-9429,
102
, pp.
177
204
.
121.
Chen
,
W. H.
, and
Chen
,
T. C.
, 1995, “
An Efficient Dual Boundary Element Technique for a Two-Dimensional Fracture Problem With Multiple Cracks
,”
Int. J. Numer. Methods Eng.
0029-5981,
38
, pp.
1739
1756
.
122.
Ammons
,
B. A.
, and
Vable
,
M.
, 1996, “
Boundary Element Analysis of Cracks
,”
Int. J. Solids Struct.
0020-7683,
33
, pp.
1853
1865
.
123.
Wang
,
Y. B.
, and
Chau
,
K. T.
, 1997, “
A New Boundary Element for Plane Elastic Problem Involving Cracks and Holes
,”
Int. J. Fract.
0376-9429,
87
, pp.
1
20
.
124.
Chau
,
K. T.
, and
Wang
,
Y. B.
, 1999, “
A New Boundary Integral Formulation Element for Plane Elastic Bodies Containing Cracks and Holes
,”
Int. J. Solids Struct.
0020-7683,
36
, pp.
2041
2074
.
125.
Cheeseman
,
B. A.
, and
Santare
,
M. H.
, 2000, “
The Interaction of a Curve Crack With a Circular Elastic Inclusion
,”
Int. J. Fract.
0376-9429,
103
, pp.
259
277
.
126.
Pan
,
E.
, 1997, “
A General Boundary Element Analysis of 2-D Linear Elastic Fracture Mechanics
,”
Int. J. Fract.
0376-9429,
88
, pp.
41
59
.
127.
Wang
,
J. L.
,
Crough
,
S. L.
, and
Mogilevsakya
,
S. G.
, 2003, “
A Complex Boundary Integral Method for Multiple Circular Holes in an Infinite Plate
,”
Eng. Anal. Boundary Elem.
0955-7997,
27
, pp.
789
802
.
128.
Aparicio
,
N. D.
, 2000, “
Elastic Complex Analysis and Its Applications in Fracture Mechanics
,”
Int. J. Theor. Phys.
0020-7748,
37
, pp.
3873
3895
.
129.
Chen
,
Y. Z.
, 1985, “
Multiple Semi-Infinite Crack Problems in Elastic Plane or Half-Plane
,”
Int. J. Fract.
0376-9429,
27
, pp.
R55
R58
.
130.
Chen
,
Y. Z.
, 1984, “
General Case of Multiple Crack Problems in an Infinite Plate
,”
Eng. Fract. Mech.
0013-7944,
20
, pp.
591
598
.
131.
Chen
,
Y. Z.
, 1986, “
Multiple Crack Problems for Two Bonded Half-Planes in Plane and Antiplane Elasticity
,”
Eng. Fract. Mech.
0013-7944,
25
, pp.
1
9
.
132.
Chen
,
Y. Z.
, and
Hasebe
,
N.
, 1996, “
Properties of Eigenfunction Expansion Form for the Rigid Line Problem in Dissimilar Media
,”
Int. J. Solids Struct.
0020-7683,
33
, pp.
611
628
.
133.
Chen
,
Y. Z.
, and
Hasebe
,
N.
, 1992, “
Stress-Intensity Factors for Curved Circular Crack in Bonded Dissimilar Materials
,”
Theor. Appl. Fract. Mech.
0167-8442,
17
, pp.
189
196
.
134.
Han
,
X. L.
,
Ellyin
,
F.
, and
Xia
,
Z. H.
, 2002, “
Interaction Among Interface, Multiple Crack and Dislocation
,”
Int. J. Solids Struct.
0020-7683,
39
, pp.
1575
1590
.
135.
Isida
,
M.
, and
Noguhi
,
H.
, 1993, “
Arbitrary Array of Cracks in Bonded Half Planes Subjected to Various Loading
,”
Eng. Fract. Mech.
0013-7944,
46
, pp.
365
380
.
136.
Wang
,
W. C.
, and
Chen
,
J. T.
, 1989, “
Stress Analysis of Finite Interfacially Bimaterial Plates by Using the Variational Method
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
73
, pp.
153
171
.
137.
Chen
,
Y. Z.
, and
Hasebe
,
N.
, 1993, “
Singularity Eigenvalue Analysis of Crack Along a Wedge-Shaped Interface
,”
ASME J. Appl. Mech.
0021-8936,
60
, pp.
781
783
.
138.
Chen
,
Y. Z.
, and
Hasebe
,
N.
, 1994, “
Eigenfunction Expansion and Higher Order Weight Functions of Interface Cracks
,”
ASME J. Appl. Mech.
0021-8936,
61
, pp.
843
849
.
139.
Chen
,
Y. Z.
, 1985, “
Solutions of Multiple Crack Problems of Elastic Half-Plane
,”
ASME J. Appl. Mech.
0021-8936,
52
, pp.
979
981
.
140.
Chen
,
Y. Z.
, and
Hasebe
,
N.
, 1995, “
Solution of Multiple Edge Crack Problem of Elastic Half-Plane by Using Singular Integral Equation Approach
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
11
, pp.
607
617
.
141.
Ioakimidis
,
N. I.
, and
Theocaris
,
P. S.
, 1979, “
A System of Curvilinear Cracks in an Isotropic Elastic Half-Plane
,”
Int. J. Fract.
0376-9429,
15
, pp.
299
309
.
142.
Chen
,
Y. Z.
, 1995, “
Weaker Singular Integral Equation Approach for an Oblique Edge Crack Problem of Half-Plane
,”
Int. J. Fract.
0376-9429,
72
, pp.
R77
R80
.
143.
Chen
,
Y. Z.
, and
Cheung
,
Y. K.
, 1990, “
New Integral Equation Approach for the Crack Problem in Elastic Half-Plane
,”
Int. J. Fract.
0376-9429,
46
, pp.
57
69
.
144.
Chen
,
Y. Z.
, 1991, “
Multiple Crack Problem for an Infinite Strip
,”
Eng. Fract. Mech.
0013-7944,
40
, pp.
9
16
.
145.
Chen
,
Y. Z.
, 1988, “
Multiple Crack Problems for Finite Plate With Arbitrary Contour Configuration
,”
Eng. Fract. Mech.
0013-7944,
31
, pp.
289
295
.
146.
Shen
,
D. W.
, and
Fan
,
T. Y.
, 2003, “
Exact Solutions of Two Semi-Infinite Collinear Cracks in a Strip
,”
Eng. Fract. Mech.
0013-7944,
70
, pp.
813
822
.
147.
Chen
,
Y. Z.
, 1983, “
Reducing Crack Problem of a Circular Plate or an Infinite Plate Containing a Circular Hole Into Fredholm Integral Equation
,”
Int. J. Fract.
0376-9429,
23
, pp.
R101
R104
.
148.
Chen
,
Y. Z.
, 1984, “
Solutions of Multiple Crack Problems of a Circular Plate or an Infinite Plate Containing a Circular Hole by Using Fredholm Integral Equation
,”
Int. J. Fract.
0376-9429,
25
, pp.
155
168
.
149.
Chen
,
Y. Z.
, and
Lin
,
W. X.
, 1985, “
Elementary Solutions of Multiple Crack Problems for Circular Region With Fixed Boundary
,”
Int. J. Fract.
0376-9429,
28
, pp.
R79
R82
.
150.
Lin
,
W. X.
, and
Chen
,
Y. Z.
, 1989, “
Multiple Crack Problems Inside and Outside Circular Region
,”
Theor. Appl. Fract. Mech.
0167-8442,
11
, pp.
199
208
.
151.
Chen
,
Y. Z.
, and
Liu
,
H. Y.
, 1988, “
Multiple Cracks in Pressurized Hollow Cylinder
,”
Theor. Appl. Fract. Mech.
0167-8442,
10
, pp.
213
218
.
152.
Chen
,
Y. Z.
, 1985, “
Multiple Crack Problem for a Rotating Disc
,”
Acta Mech. Sin.
0459-1879,
17
, pp.
577
580
(in Chinese).
153.
Gross
,
D.
, and
Chen
,
Y. Z.
, 1991, “
A New Integral Equation Approach for the Curved Crack Problem in Circular Plate
,”
Advance in Continuum Mechanics
,
O.
Bruller
, ed.
Springer
,
Heideberg
, pp.
267
273
.
154.
Theotokoglou
,
E. N.
, 1992, “
Integral Equation Solution of the Eccentrically Rotating Cracked Finite Disc
,”
Eng. Fract. Mech.
0013-7944,
41
, pp.
299
308
.
155.
Xu
,
Y. L.
, and
Dalale
,
F.
, 1992, “
Stress Intensity Factors for an Internal or Edge Crack in a Circular Disk Subjected to Concentrated or Distributed Loads
,”
Eng. Fract. Mech.
0013-7944,
42
, pp.
757
787
.
156.
Tracy
,
P.
, 1980, “
Stress Intensity Factors for Multiple Edge Cracks in Rotating Disc
,”
Int. J. Fract.
0376-9429,
16
, pp.
85
93
.
157.
Sekine
,
H.
, and
Koizumi
,
R.
, 1982, “
Stress Intensity Factors for an Embedded Crack in a Thick Walled Cylinder Subjected to Internal Pressure
,”
Int. J. Fract.
0376-9429,
18
, pp.
R3
R8
.
158.
Fett
,
T.
, 1997, “
A Semi-Analytical Study of the Edge-Cracked Circular Disc by Use the Boundary Collocation Method
,”
Eng. Fract. Mech.
0013-7944,
56
, pp.
331
346
.
159.
Dong
,
S. M.
,
Wang
,
Y.
, and
Xia
,
Y. M.
, 2004, “
Stress Intensity Factors for Central Cracked Circular Disk Subjected to Compression
,”
Eng. Fract. Mech.
0013-7944,
71
, pp.
1155
1168
.
160.
Williams
,
M. L.
, 1957, “
On the Stress Distribution at the Base of a Stationary Crack
,”
ASME J. Appl. Mech.
0021-8936,
24
, pp.
109
114
.
161.
Rice
,
J. R.
, 1974, “
Limitations to the Small Scale Yielding Approximation for Crack Tip Plasticity
,”
J. Mech. Phys. Solids
0022-5096,
22
, pp.
17
26
.
162.
Betegon
,
C.
, and
Hancock
,
J. W.
, 1991, “
Two-Parameter Characterization of Elastic-Plastic Crack-Tip Fields
,”
ASME J. Appl. Mech.
0021-8936,
58
, pp.
104
110
.
163.
Chen
,
Y. Z.
, 1994, “
T Stress in Multiple Crack Problem for an Infinite Plate
,”
Eng. Fract. Mech.
0013-7944,
48
, pp.
641
647
.
164.
Chen
,
Y. Z.
, 2000, “
Closed Form Solutions of T-Stress in Plane Elasticity Crack Problems
,”
Int. J. Solids Struct.
0020-7683,
37
, pp.
1629
1637
.
165.
Chen
,
Y. Z.
, and,
Lin
,
X. Y.
, 1997, “
Novel Weight Function Approach for Evaluating T-Stress in Plane Elasticity Crack Problem
,”
Int. J. Fract.
0376-9429,
85
, pp.
L35
L40
.
166.
Zhao
,
L. G.
, and
Chen
,
Y. H.
, 1998, “
T-Stress of an Interface Macrocrack Induced by Near Tip Subinterface Microcracks
,”
Int. J. Fract.
0376-9429,
90
, pp.
275
285
.
167.
Han
,
J. J.
, and
Chen
,
Y. H.
, 1998, “
T-Effect for the Interaction Problem of an Interface Macrocrack With a Near Microvoid
,”
Int. J. Fract.
0376-9429,
102
, pp.
205
222
.
168.
Ma
,
H.
,
Zhao
,
L. G.
, and
Chen
,
Y. H.
, 1997, “
Non-Singular Terms for Multiple Crack in Anisotropic Elastic Solids
,”
Theor. Appl. Fract. Mech.
0167-8442,
27
, pp.
129
134
.
169.
Tan
,
C. L.
, and
Wang
,
X.
, 2003, “
The Use of Quarter-Point Crack-Tip Elements for T-Stress Determination in Boundary Element Method Analysis
,”
Eng. Fract. Mech.
0013-7944,
70
, pp.
2247
2252
.
170.
Chen
,
Y. Z.
, 1985, “
Multiple Crack Problems of Antiplane Elasticity in an Infinite Body by Using Fredholm Integral Equation Approach
,”
Eng. Fract. Mech.
0013-7944,
21
, pp.
473
478
.
171.
Chen
,
Y. Z.
, and
Wang
,
Z. X.
, 1986, “
Solutions of Multiple Crack Problems of Circular Region With Free or Fixed Boundary Condition in Antiplane Elasticity
,”
Int. J. Fract.
0376-9429,
30
, pp.
287
293
.
172.
Chen
,
Y. Z.
, and
Wang
,
Z. X.
, 1990, “
Anti-Plane Shear of Rectangular Region With Two Cracks
,”
Theor. Appl. Fract. Mech.
0167-8442,
12
, pp.
225
229
.
173.
Chen
,
Y. Z.
, 1984, “
Solutions of Multiple Crack Problems of a Circular Region for Antiplane Elastic Problem or Torsion Problem by Using Fredholm Integral Equation
,”
Int. J. Fract.
0376-9429,
27
, pp.
R15
R19
.
174.
Chen
,
Y. Z.
, 1980, “
Solutions of Torsion Crack Problems of a Rectangular Bar by Harmonic Function Continuation Technique
,”
Eng. Fract. Mech.
0013-7944,
13
, pp.
193
212
.
175.
Chen
,
Y. Z.
, 1999, “
Multiple Crack Problem for Circular Torsion Cylinder
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
15
, pp.
557
563
.
176.
Sih
,
G. C.
, 1963, “
Strength of Stress Singularities at Crack Tips for Flexural and Torsional Problems
,”
ASME J. Appl. Mech.
0021-8936,
30
, pp.
419
425
.
177.
Yue
,
J. C.
, and
Tang
,
R. J.
, 1996, “
Integral Equation Method for the Torsion of a Composite Cylinder With Crack and Inclusion
,”
Eng. Fract. Mech.
0013-7944,
55
, pp.
763
775
.
178.
Li
,
Y. L.
,
Hu
,
S. Y.
, and
Tang
,
R. L.
, 1995, “
The Stress Intensity of Crack-Tip and Notch-Tip in Cylinder Under Torsion
,”
Int. J. Eng. Sci.
0020-7225,
33
, pp.
447
455
.
179.
Chen
,
Y. Z.
, 1987, “
Plane Problem of Two Rows of Periodic Crack
,”
Theor. Appl. Fract. Mech.
0167-8442,
7
, pp.
185
188
.
180.
Chen
,
Y. Z.
,
Lin
,
X. Y.
, and
Wang
,
Z. X.
, 2005, “
Solution of Periodic Group Crack Problems by Using the Fredholm Integral Equation
,”
Acta Mech.
0001-5970,
178
, pp.
41
51
.
181.
Chen
,
Y. Z.
, and
Lin
,
X. Y.
, 2005, “
Periodic Group Crack Problems in an Infinite Plate
,”
Int. J. Solids Struct.
0020-7683,
42
, pp.
2837
2850
.
182.
Delameter
,
W. R.
,
Herrmann
,
G.
, and
Barnett
,
D. M.
, 1975, “
Weakening of Elastic Solid by a Rectangular Array of Cracks
,”
ASME J. Appl. Mech.
0021-8936,
42
, pp.
74
80
.
183.
Isida
,
M.
,
Usijima
,
N.
, and
Kishine
,
N.
, 1981, “
Rectangular Plate, Strips and Wide Plates Containing Internal Cracks Under Various Boundary Conditions
,”
Trans. Jpn. Soc. Mech. Eng., Ser. A
0387-5008,
47
, pp.
27
35
.
184.
Chen
,
Y. Z.
, 1983, “
An Investigation of the Stress Intensity Factor for a Finite Internally Cracked Plate by Using Variational Method
,”
Eng. Fract. Mech.
0013-7944,
17
, pp.
387
394
185.
Chen
,
Y. Z.
, and
Lee
,
K. Y.
, 2002, “
An Infinite Plate Weakened by Periodic Cracks
,”
ASME J. Appl. Mech.
0021-8936,
69
, pp.
552
555
.
186.
Karihaloo
,
B. L.
, and
Wang
,
J.
, 1997, “
On the Solution of Doubly Periodic Array of Cracks
,”
Mech. Mater.
0167-6636,
26
, pp.
209
212
.
187.
Wang
,
G. S.
, 2004, “
The Interaction of Doubly Periodic Arrays
,”
Theor. Appl. Fract. Mech.
0167-8442,
42
, pp.
249
294
.
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