Issue 41
V. Rizov, Frattura ed Integrità Strutturale, 41 (2017) 491-503; DOI: 10.3221/IGF-ESIS.41.61
where Γ is a contour of integration going from the lower crack face to the upper crack face in the counter clockwise direction, 0 u is the strain energy density, α is the angle between the outwards normal vector to the contour of integration and the crack direction, x p and y p are the components of stress vector, u and v are the components of displacement vector with respect to the crack tip coordinate system xy ( x is directed along the crack), ds is a differential element along the contour. The J -integral is solved by using an integration contour, Γ , that coincides with the beam contour (Fig. 1). It is obvious that the J -integral value is non-zero only in segments 1 and 2 of the integration contour. Therefore, the J -integral solution is written as
J J
J
(25)
1
2
J
J
are the J -integral values in segments 1
where
and
and
2 , respectively ( 1
and
2 coincide with the free
1
2
end of lower crack arm and the clamping, respectively). The components of J -integral in segment, 1
, of the integration contour are written as (Fig. 3)
m B
y p
0
,
(26)
p
x
1 ds dz , cos
(27)
1
where 1
z varies in the interval
.
h
h
[
/ 2,
/ 2]
1
1
/ u x , in (24) is found as
Partial derivative,
u x
z z
(28)
1 1 1 1 n
where 1 1 n z
and
1 are obtained from Eqs. (18) and (19). The strain energy density is calculated by substituting of (1) in
(10)
1 1 m B u m
(29)
0
By substituting of (1), (12), (16), (26), (27), (28) and (29) in (24) and integrating in boundaries from 1 / 2 h to 1 / 2 h , one derives
m J m 1 1 m 1
2 y B
2
B
2 B r
4
m
m
1
1
0
1 1
u
u
1
2
u
h m 2
2
1
1 m b m
1
1
u
u
z
B r
2
1
n
1
m
m
m
m
2
2
1
1
2 2
1
u
u
u
u
1
2
1
2
m
m
2
1
h
u
u
3 u u q
3 u u q
2 u u q
2 u u q
f
f
f
f
z
2
2 B q
1
n
1
q
q
q
q
u
1
u
u
u
u
1
2
1
2
(30)
2
f
q
f
q
3
2
h
u
u
u
u
f q
f q
u u
u u
2 1 1 n
z
q
q
u
1
2
f
q
u u
498
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