In this paper, the mechanics of a semi-infinite crack interacting with near crack-tip singularities (e.g., dislocations) in two-dimensional solids is investigated using the concept of potential energy release rate. The spontaneous relationship between the crack potential energy release rate and the well-known vector conservative integral Ji(i=1,2) is derived. It is demonstrated that J1 and J2 integrals are equally important in solving crack problems. This allows a more rational criterion to be proposed, based on the criterion of maximum energy release rate, to assess the so-called shielding/amplification effect on the crack tip due to the presence of the singularities. It is shown that the new criterion can not only assess the shielding/amplification effect under pure mode I or mode II remote loading, but also efficiently assess crack-singularity interaction under mixed mode remote loading. Simultaneously, it is found by re-examining the Ji integrals that there exists a simple but universal relation among the three values of the vector Ji integral corresponding separately to the contributions induced from the semi-infinite crack tip, the singularity, and the remote loading. Next, a multi-singularity-crack interaction model is addressed, and the closed-form solution is obtained. Finally, as an example, the problem of a single dislocation interacting with a main crack is solved to demonstrate the validity of the proposed model and the new criterion.

1.
Knowles
,
J. K.
, and
Stenberg
,
E.
, 1972, “
On a Class of Conservation Laws in Linearized and Finite Elastostatics
,”
Arch. Ration. Mech. Anal.
0003-9527,
44
, pp.
187
211
.
2.
Budiansky
,
B.
, and
Rice
,
J. R.
, 1973, “
Conservation Laws and Energy-Release Rates
,”
ASME J. Appl. Mech.
0021-8936,
40
, pp.
201
203
.
3.
Eshelby
,
J. D.
, 1951, “
The Force on an Elastic Singularity
,”
Proc. R. Soc. London, Ser. A
1364-5021,
244
, pp.
87
112
.
4.
Rice
,
J. R.
, 1968, “
A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks
,”
ASME J. Appl. Mech.
0021-8936,
35
, pp.
379
386
.
5.
Rice
,
J. R.
, 1968, “
Mathematical Analysis in the Mechanics of Fracture
,” in
Mathematical Fundamentals
, Fracture An Advanced Treatise Vol.
II
,
H.
Liebowitz
, ed.,
Pergamon
, New York, pp.
191
311
.
6.
Cherepanov
,
G. P.
, 1979,
Mechanics of Brittle Fracture
,
Nauka
, Moscow, English translation published by McGraw-Hill International, New York.
7.
Erdogan
,
F.
, and
Sih
,
G. C.
, 1963, “
On the Crack Extension in Plates Under Plane Loading and Transverse Shear
,”
ASME J. Basic Eng.
0021-9223,
91
, pp.
764
769
.
8.
Otsuka
,
A.
,
Mori
,
K.
, and
Miyata
,
T.
, 1975, “
The Condition of Fatigue Crack Growth in Mixed Mode Condition
,”
Eng. Fract. Mech.
0013-7944,
7
, pp.
429
432
.
9.
Sih
,
G. C.
, 1973, “
A Special Theory of Crack Propagation: Methods of Analysis and Solutions of Crack Problems
,” in
Mechanics of Fracture I
,
G. C.
Sih
, ed.,
Noordhoff
, Leyden, pp.
21
45
.
10.
Wu
,
C.-H.
, 1978, “
Fracture Under Combined Loads by Maximum-Energy-Release-Rate Criterion
,”
ASME J. Appl. Mech.
0021-8936,
45
, pp.
553
558
.
11.
Shen
,
M. S.
, and
Shen
,
M.-H. H.
, 1995, “
Direction of Crack Extension Under General Plane Loading
,”
Int. J. Fract.
0376-9429,
70
, pp.
51
58
.
12.
Ma
,
L. F.
, and
Korsunsky
,
A. M.
, 2005, “
On the Use of Vector J-Integral in Crack Growth Criteria for Brittle Solids
,” Int. J. Fracture, in press.
13.
Rice
,
J. R.
, and
Thomson
,
R.
, 1974, “
Ductile Versus Brittle Behavior of Crystals
,”
Philos. Mag.
0031-8086,
29
, pp.
73
97
.
14.
Hirth
,
J. P.
, and
Wagoner
,
R. H.
, 1976, “
Elastic Fields of Line Defects in a Cracked Body
,”
Int. J. Solids Struct.
0020-7683,
12
, pp.
117
123
.
15.
Lin
,
I.-H.
, and
Thomson
,
R.
, 1986, “
Cleavage, Dislocation Emission, and Shielding for Cracks Under General Loading
,”
Acta Metall.
0001-6160,
34
(
2
), pp.
187
206
.
16.
Lakshmanan
,
V.
, and
Li
,
J. C. M.
, 1988, “
Edge Dislocation Emitted Along Inclined Planes Form a Mode I Crack
,”
Mater. Sci. Eng., A
0921-5093,
104
, pp.
95
104
.
17.
Zhang
,
T.-Y.
,
Tong
,
P.
,
Ouyang
,
H.
, and
Lee
,
S.
, 1995, “
Interaction of an Edge Dislocation With a Wedge Crack
,”
J. Appl. Phys.
0021-8979,
78
, pp.
4873
4880
.
18.
Cleveringa
,
H. H. M.
,
Giessen
,
E. V. D.
, and
Needleman
,
A.
, 2000, “
A Dislocation Analysis of Mode I Crack Growth
,”
J. Mech. Phys. Solids
0022-5096,
48
, pp.
1133
1157
.
19.
Cai
,
H.
, and
Faber
,
K. T.
, 1992, “
On the Use of Approximation Methods for Micro-Crack Shielding Problems
,”
ASME J. Appl. Mech.
0021-8936,
59
, pp.
497
501
.
20.
Andersson
,
P.
, and
Ståhle
,
P.
, 1997, “
Shielding Effects and Residual Stresses at Cleavage due to Pre-Existing Dislocations
,”
Int. J. Fract.
0376-9429,
85
, pp.
365
380
.
21.
Chen
,
Y. H.
, 1996, “
On the Contribution of Discontinuities in a Near-Tip Stress Field to the J Integral
,”
Int. J. Eng. Sci.
0020-7225,
34
, pp.
819
829
.
22.
Irwin
,
G. R.
, 1957, “
Analysis of Stresses and Strains Near the End of a Crack Traversing a Plate
,”
ASME J. Appl. Mech.
0021-8936,
24
, pp.
361
364
.
23.
Suo
,
Z.
, 1989, “
Singularities Interacting with Interfaces and Cracks
,”
Int. J. Solids Struct.
0020-7683,
25
, pp.
1133
1142
.
You do not currently have access to this content.