The paper deals with a cohesive crack model in which the cohesive (crack-bridging) stress is a specified decreasing function of the crack-opening displacement. Under the assumption that no part of the crack undergoes unloading, the complementary energy and potential energy of an elastic structure which has a cohesive crack and is loaded by a flexible elastic frame is formulated using continuous influence functions representing compliances or stiffnesses relating various points along the crack. By variational analysis, in which the derivatives of the compliance or stiffness functions with respect to the crack length are related to the crack-tip stress intensity factors due to various unit loads, it is shown that the minimizing conditions reduce to the usual compatibility or equilibrium equations for the cohesive cracks. The variational equations obtained can be used as a basis for approximate solutions. Furthermore, the conditions of stability loss of a structure with a growing cohesive crack are obtained from the condition of vanishing of the second variation of the complementary energy or the potential energy. They have the form of a homogeneous Fredholm integral equation for the derivatives of the cohesive stresses or crack opening displacements with respect to the crack length. Loadings with displacement control, load control, or through a flexible loading frame are considered. Extension to the analysis of size effect on the maximum load or maximum displacement are left to a subsequent companion paper.

1.
Barenblatt
G. I.
,
1962
, “
The mathematical theory of equilibrium cracks in brittle fracture
,”
Advances in Applied Mechanics
, Vol.
7
, pp.
55
129
.
2.
Bazˇant
Z. P.
,
1976
, “
Instability, ductility, and size effect in strain softening concrete
,”
ASCE Journal of Engineering Mechanics
, Vol.
102
, No.
2
, pp.
331
344
.
3.
Bazˇant, Z. P., 1982, “Crack band model for fracture geomaterials,” Proceedings of the 4th International Conference on Numerical Method in Geomechanics, Z. Eisenstein, ed., Edmonton, Alberta, Vol. 3, pp. 1137–1152.
4.
Bazˇant
Z. P.
, and
Oh
B. H.
,
1983
, “
Crack band theory for fracture concrete
,”
Material and Structures (RILEM, Paris)
, Vol.
16
, pp.
155
177
.
5.
Bazˇant
Z. P.
,
1984
, “
Size effect in blunt fracture: Concrete, Rock, Metal
,”
ASCE Journal of Engineering Mechanics
, Vol.
110
, No.
4
, pp.
518
535
.
6.
Bazˇant, Z. P., and Cedolin, L., 1991, Stability of Structures: Elastic, Inelastic, Fracture, and Damage Theories, Oxford University Press, Oxford, U.K.
7.
Bilby
B. A.
,
Cottrell
A. H.
, and
Swinden
K. H.
,
1963
, “
The spread of plastic yield form a notch
,”
Proceedings of Royal Society London
, Vol.
A272
, pp.
304
314
.
8.
Dugdale
D. S.
,
1960
, “
Yielding of steel sheets containing slits
,”
Journal of the Mechanics and Physics of Solids
, Vol.
8
, pp.
100
104
.
9.
Foote
R. M. L.
,
Mai
Y.-W.
, and
Cottrell
B.
,
1986
, “
Crack growth resistance curves in strain-softening materials
,”
Journal of the Mechanics and Physics of Solids
, Vol.
34
, No.
6
, pp.
593
607
.
10.
Hillerborg
A.
,
Mode´er
M.
, and
Petersson
P. E.
,
1976
, “
Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements
,”
Cement and Concrete Research
, Vol.
6
, pp.
773
782
.
11.
Li
Y. N.
, and
Bazˇant
Z. P.
,
1994
, “
Eigenvalue analysis of size effect for cohesive crack model
,”
International Journal of Fracture
, Vol.
66
, pp.
213
226
.
12.
Li, Y. N., and Liang, R. Y., 1992, “Stability theory of the cohesive crack model,” ASCE Journal of Engineering Mechanics, Vol. 118, No. 3.
13.
Li
Y. N.
, and
Liang
R. Y.
,
1993
, “
The theory of boundary eigenvalue problem in the cohesive crack model and its application
,”
Journal of the Mechanics and Physics of Solids
, Vol.
41
, pp.
331
350
.
14.
Needleman
A.
,
1990
, “
An analysis of tensile decohesion along an interface
,”
Journal of the Mechanics and Physics of Solids
, Vol.
38
, No.
3
, pp.
289
324
.
15.
Petersson, P.-E., 1981, “Crack growth and development of fracture zones in plain concrete and similar materials,” Doctoral Dissertation, Lund Institute of Technology, Sweden.
16.
Rice
J. R.
,
1968
, “
Path Independent Integral and Approximate Analysis of Strain Concentration by Notches And Cracks
,”
ASME JOURNAL OF APPLIED MECHANICS
, Vol.
35
, pp.
379
386
.
17.
Rice
J. R.
,
1992
, “
Dislocation nucleation from a crack tip: An analysis based on the Peierls concept
,”
Journal of the Mechanics and Physics of Solids
, Vol.
40
, No.
2
, pp.
239
271
.
18.
Reinhardt
H. W.
,
1985
, “
Crack softening zone in plain concrete under static loading
,”
Cement and Concrete Research
, Vol.
15
, pp.
42
52
.
19.
Smith
E.
,
1974
, “
The structure in the vicinity of a crack tip: A general theory based on the cohesive zone model
,”
Engineering Fracture Mechanics
, Vol.
6
, pp.
213
222
.
20.
Suo
Z.
,
Bao
G.
, and
Fan
B.
,
1992
, “
Delamination R-curve phenomena due to damage
,”
Journal of the Mechanics and Physics of Solids
, Vol.
40
, No.
1
, pp.
1
16
.
21.
Tada, H., Paris, P. C., and Irwin, G. R., 1985, The stress analysis of cracks handbook, Del Research Corp., Hellertown, PA.
22.
Tvergaard
V.
, and
Hutchinson
J. W.
,
1992
, “
The relation between crack growth resistance and fracture process parameters in elastic-plastic solids
,”
Journal of the Mechanics and Physics of Solids
, Vol.
40
, No.
6
, pp.
1377
1397
.
23.
Willis
J. R.
,
1967
, “
A comparison of the fracture criteria of Griffith and Barenblatt
,”
Journal of the Mechanics and Physics of Solids
, Vol.
15
, pp.
151
162
.
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