Issue 47
V. Rizov, Frattura ed Integrità Strutturale, (2047) 468-481; DOI: 10.3221/IGF-ESIS.47.37
1 , is written as
The J -integral in segment,
y
i n
u x
v
1 1 i
L
J
u
p
p
ds
cos
(34)
L
xi
yi
0
x
i
1
i
1
y
i
1
where 0 i L u is the strain energy density in the i -th layer of the left-hand crack arm, α is the angle between the outwards p are the components of stress vector in the i -th layer of the left-hand crack arm, u and v are the components of displacement vector with respect to the crack tip coordinate system xy ( x is directed along the delamination crack), ds is a differential element along the contour of integration. The strain energy density in the i -th layer of the left-hand crack arm is obtained by applying the following formula [17, 18]: normal vector to the contour of integration and the crack direction, xi p and yi
m
1
i
m
2
i
i
i
u
(35)
L
0
1
E
2
i
i
m
m H
1
i
i
i
By substituting of (7) in (35), one obtains
m
1
i
m
2
i
i
i
u
.
(36)
L
0
1
1 y y
i
E E
i
1
2
cos
m
m H
1
i
d
f
i
i
y
y
2
i
i
1 1 i
i
1
, are written as
The other components of the J -integral in segment, 1
i
p
(37)
xi
yi p
(38)
0
ds dy
(39)
1
(40)
cos
1
, in (37). The partial derivative,
/ u x , that is involved
It should be noted that formula (16) is used to obtain the stress, i
in (34) is expressed as
u x
1 1 z
.
(41)
2 , is written as
The J -integral is segment,
y
i n
u
v
2 1 i
J
u
p
p
ds
cos
(42)
R
R
xRi
yRi
R
0
x
x
i
2
i
1
R
R
y
i
2
u , in the i -th layer of the un-cracked beam portion is obtained by formula (36). For this
where the strain energy density, 0 i R
i , 1
R ,
y , 1 i
y and
y
2 y ,
2 i y and
y
J
are replaced, respectively, with
. The other components of
purpose,
1 1 i
2 1 i
i
2
are written as
476
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