PSI - Issue 33
Domenico Ammendolea et al. / Procedia Structural Integrity 33 (2021) 858–870 Domenico Ammendolea et al./ Structural Integrity Procedia 00 (2019) 000 – 000
863
u
T
T
A
A
j
(9)
with ,
1, 2
J
1 , i i W q dA
qdA
dA
i j
ii
ii
=
−
+
+
=
ij
x
x
x
1
1
1
A
0
where, with reference to Fig. 2, A and A 0 are the area confined by the closed path S ( S=S 0 +S 1 +S + +S - ) and arbitrary path S 0 , respectively. Also, q(x 1 ,x 2 ) is a scalar function that assumes the value of the unity on the inner contour of A ( i.e. , S 0 ) and zero on the outer one ( i.e., S 1 ). In the first integral at the right-hand side of Eq.(9), W is the elastic strain energy, which is defined as follows: ( ) 1 with (planestress) or (planestrain) 2 1 1 2 ij ij ii E E W T = − = = − − (10) In Eq.(10), E , , and are the Young’s Modulus, Poisson’s ratio, and the coefficient of thermal expansion of the material, respectively. By shrinking the area A 0, and applying the superimposed state to the Eq. (9), one achieves the M -integral expression:
act
aux
u
u
act
T
A
A
j
j
aux
act
act aux
aux
with ,
1, 2
M
q dA
qdA
i j
(11)
=
+
−
+
=
ij
ii
1 , i
ij
ij
ij
i
x
x
x
1
1
1
By using the relationship between the J -integral and SIFs for two-dimensional problems, one arrives to: ( ) 2 2 act aux act aux I I II II K K K K M E + =
(12)
( ) 2 1 E E = − for plane strain.
where, E E = for plane stress and The SIFs of the actual state (i.e.,
, act act I II K K ) are computed using two interaction integrals, derived by using a pure
mode-I and a pure mode-II auxiliary fields, thus achieving the following expressions of the actual SIFs:
E M
E M
, act aux I −
, act aux II −
(
)
(
)
(13)
act
aux I −
act
aux II −
0 and
0
K
K
K
K
=
=
=
=
I
II
II
I
aux I −
aux II −
2
2
K
K
I
II
Fig. 2. J -integral and a schematic of the arbitrary function q(x 1 , x 2 )
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