Issue 41
V. Rizov, Frattura ed Integrità Strutturale, 41 (2017) 491-503; DOI: 10.3221/IGF-ESIS.41.61
y varies in the interval
/ 2, / 2 b b .
where 1
The J -integral solution in segment, 2
, of the integration contour (Fig. 1) can be determined by (30). For this purpose, r ,
1 1 n z , 1
z
, 2
and 1
have to be replaced with 0 ,
, 1 u
, 2 u
and 2
, respectively. Besides, the sign of (30) must be
2 2 n
set to „minus” because the integration contour is directed upwards in segment, 2 . The J -integral final solution is found by substituting of 1 J and 2 J in (25)
2 y B
2
m J m 1 m
1
B
2 B r
4
m
m
1
1
0
1 1
u
u
1
2
u
h m 2
2
1 m b m
1
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
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
2
q
q
u
1
2
(31)
f
q
u u
m
1
B
y B
4
m
m
m
1
1
0
1 1
2
u
u
1
2
u
u
u
2
m
1 m b m
1
1
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
2
q
q
q
q
u
2
u
u
u
u
1
2
1
2
u
u
u
u
2
f
q
f
q
3
2
h
u
u
u
u
f q
f q
u u
u u
2 2 2 n
z
q
q
u
1
2
u
u
f
q
u u
Formula (31) describes the distribution of the J -integral value along the crack front. The average value of the J -integral along the crack front is written as
2 b
1
b
J
J dy
(32)
AV
1
2 b
It should be noted that the J -integral solution derived by substituting of (31) in (32) is exact match of the strain energy release rate (22). This fact verifies the fracture analysis developed in the present paper.
R ESULTS
he distribution of J -integral value along the crack front is analyzed. For this purpose, calculations are performed by using formula (31). It is assumed that b =0.020 m, h =0.004 m, m =0.7, f =7, q =10, 1.7 u m , 17 u f , 10 u q and 20 y M Nm. The J -integral value is presented in non-dimensional form by using the formula, 0 / N J J B b . The material gradient along the beam width is characterized by 1 0 / B B ratio. Fig. 5 shows the distribution of the J -integral value in non T
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