Based on the analytical solution of Crouch to the problem of a constant discontinuity in displacement over a finite line segment in an infinite elastic solid, in the present paper, the crack-tip displacement discontinuity elements, which can be classified as the left and the right crack-tip elements, are presented to model the singularity of stress near a crack tip. Furthermore, the crack-tip elements together with the constant displacement discontinuity elements presented by Crouch and Starfied are used to develop a numerical approach for calculating the stress intensity factors (SIFs) of general plane cracks. In the boundary element implementation, the left or the right crack-tip element is placed locally at the corresponding left or right crack tip on top of the constant displacement discontinuity elements that cover the entire crack surface and the other boundaries. The method is called the hybrid displacement discontinuity method (HDDM). Numerical examples are given and compared with the available solutions. It can be found that the numerical approach is simple, yet very accurate for calculating the SIFs of branched cracks. As a new example, cracks emanating from a rhombus hole in an infinite plate under biaxial loads are taken into consideration. The numerical results indicate the efficiency of the present numerical approach and can reveal the effect of the biaxial load on the SIFs. In addition, the hybrid displacement discontinuity method together with the maximum circumferential stress criterion (Erdogan and Sih) becomes a very effective numerical approach for simulating the fatigue crack propagation process in plane elastic bodies under mixed-mode conditions. In the numerical simulation, for each increment of crack extension, remeshing of existing boundaries is not required because of an intrinsic feature of the HDDM. Crack propagation is simulated by adding new boundary elements on the incremental crack extension to the previous crack boundaries. At the same time, the element characters of some related elements are adjusted according to the manner in which the boundary element method is implemented.

1.
Cotterel
,
B.
, and
Rice
,
J. R.
,
1980
, “
Slightly Curved or Kinked Cracks
,”
Int. J. Fract.
,
16
, pp.
155
169
.
2.
Anderdon
,
H.
, and
Erratum
,
W. G.
,
1969
, “
Stress Intensity Factors at the Tips of a Star-Shape Contour in an Infinite Tensile Sheet
,”
J. Mech. Phys. Solids
,
17
, pp.
405
417
.
3.
Palaniswamy, K., and Knauss, W. G., “On the Problem of Crack Extension in Brittle Solids under General Loading,” Report No. SM74-8, Graduate Aeronautical Lab., Caltech.
4.
Billy
,
B. A.
, and
Cardew
,
G. E.
,
1975
, “
The Crack with a Kinked Tip
,”
Int. J. Fract.
,
11
, pp.
708
711
.
5.
Billy, B. A., Cardew, G. E., and Howard, I. C., 1977, “Stress Intensity Factors at the Tips of Kinked and Forked Cracks,” in Fracture 1977, Taplin, D. M. R., ed., University of Waterloo Press, Canada, Vol. 3, pp. 197–200.
6.
Kitagawa, H., and Yuuki, R., 1977, “Analysis of Branched Cracks Under Biaxial Stresses,” in Fracture 1977, Taplin D. M. R., eds., University of Waterloo Press, Canada, Vol. 3, pp. 201–211.
7.
Kitagawa
,
H.
,
Yuuki
,
R.
, and
Ohira
,
T. C.
,
1975
, “
Crack-Morphological Aspects in Fracture Mechanics
,”
Eng. Fract. Mech.
,
7
, pp.
515
529
.
8.
Chatterjee
,
S. N.
,
1975
, “
The Stress Field in the Neighborhood of a Branched Crack in an Infinite Sheet
,”
Int. J. Solids Struct.
,
11
, pp.
521
538
.
9.
Lo
,
K. K.
,
1978
, “
Analysis of Branched Cracks
,”
ASME J. Appl. Mech.
,
45
, pp.
797
802
.
10.
Theocaris
,
P. S.
, and
Loakimidis
,
N.
,
1976
, “
The Symmetrically Branched Cracks in an Infinite Elastic Medium
,”
ZAMP
,
27
, pp.
801
814
.
11.
Wilson
,
W. K.
, and
Cherepko
,
J.
,
1983
, “
Analysis of Cracks With Multiple Branches
,”
Int. J. Fract.
,
22
, pp.
302
315
.
12.
Vitek
,
V.
,
1977
, “
Plane Strain Stress Intensity Factors for Branched Cracks
,”
Int. J. Fract.
,
13-4
, pp.
481
510
.
13.
Blandford
,
G. E.
,
Ingraffea
,
A. R.
, and
Liggett
,
J. A.
,
1981
, “
Two-Dimensional Stress Intensity Factor Computations Using the Boundary Element Method
,”
Int. J. Numer. Methods Eng.
,
17
, pp.
387
404
.
14.
Balas, J., Sladek, J., and Sladek, V., 1989, Stress Analysis by Boundary Element Methods, Elsevier, Amsterdam.
15.
Hong
,
H.
, and
Chen
,
J.
,
1988
, “
Derivatives of Integral Equations of Elasticity
,”
J. Eng. Mech.
,
114
, pp.
1028
1044
.
16.
Portela
,
A.
, and
,
M. H.
,
1992
, “
The Dual Boundary Element Method: Effective Implementation for Crack Problems
,”
Int. J. Numer. Methods Eng.
,
33
, pp.
1269
1287
.
17.
Tanaka
,
M.
, and
Itoh
,
H.
,
1987
, “
New Crack Elements for Boundary Element Analysis of Elastostatics Considering Arbitrary Stress Singularities
,”
Appl. Math. Model.
,
11
, pp.
357
363
.
18.
Cruse, T. A., 1989, Boundary Element Analysis in Computational Fracture Mechanics, Kluwer, Dordrecht.
19.
Aliabadi, M. H., and Rooke, D. P., 1991, Numerical Fracture Mechanics, Computational Mechanics Publications, Southampton and Kluwer, Dordrecht.
20.
Crouch, S. L., and Starfield, A. M., 1983, Boundary Element Method in Solid Mechanics with Application in Rock Mechanics and Geological Mechanics, Allen and Unwin, London.
21.
Crouch
,
S. L.
,
1976
, “
Solution of Plane Elasticity Problems by Displacement Discontinuity Method
,”
Int. J. Numer. Methods Eng.
,
10
, pp.
301
343
.
22.
Ingraffea, A. R., Blandford, G., and Liggett, J. A., 1987, “Automatic Modeling of Mixed-Mode Fatigue and Quasi-Static Crack Propagation Using the Boundary Element Method,” 14th Natl. Symp. on Fracture, ASTM STP 791, pp. 1407–1426.
23.
,
M. H.
,
1997
, “
Boundary Element Formulation in Fracture Mechanics
,”
Appl. Mech. Rev.
,
50
, pp.
83
96
.
24.
Erdogan
,
F.
, and
Sih
,
G. C.
,
1963
, “
On the Crack Extension in Plates Under Plane Loading and Transverse Shear
,”
J. Basic Eng.
,
85
, pp.
519
527
.
25.
Charambides
,
P. G.
, and
McMeeking
,
R. M.
,
1987
, “
Finite Element Method Simulation of a Crack Propagation in a Brittle Microcracked Solid
,”
Mech. Mater.
,
6
, pp.
71
87
.
26.
Huang
,
X.
, and
Karihaloo
,
B. L.
,
1993
, “
Interaction of Penny Shaped Cracks With a Half Plane Crack
,”
Int. J. Solids Struct.
,
25
, pp.
591
607
.
27.
Scavia
,
C.
,
1992
, “
A Numerical Technique for the Analysis of Cracks Subjected to Normal Compressive Stresses
,”
Int. J. Numer. Methods Eng.
,
33
, pp.
929
942
.
28.
Pan
,
E.
,
1997
, “
A General Boundary Element Analysis of 2-D Linear Elastic Fracture Mechanics
,”
Int. J. Fract.
,
88
, pp.
41
59
.
29.
Murakami, Y., 1987, Stress Intensity Factors Handbook, Pergamon, New York.
30.
Chen
,
Y. Z.
,
1999
, “
Stress Intensity Factors for Curved and Kinked Cracks in Plane Extension
,”
Theor. Appl. Fract. Mech.
,
31
, pp.
223
232
.
31.
Liu
,
N.
,
Altiero
,
N. J.
, and
Sur
,
U.
,
1990
, “
An Alternative Integral Equation Approach Applied to Kinked Cracks in Finite Plane Bodies
,”
Comput. Methods Appl. Mech. Eng.
,
84
, pp.
211
226
.
32.
Denda
,
M.
, and
Dong
,
Y. F.
,
1999
, “
Analytical Formulas for a 2-D Crack-Tip Singular Boundary Element for Rectilinear Cracks and Crack Growth Analysis
,”
Eng. Anal. Boundary Elem.
,
23
, pp.
35
49
.
33.
Sih
,
G. C.
, and
Barthelemy
,
B. M.
,
1980
, “
Mixed Mode Fatigue Crack Growth Prediction
,”
Eng. Fract. Mech.
,
13
, pp.
439
451
.
You do not currently have access to this content.