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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