PSI - Issue 52

T. Profant et al. / Procedia Structural Integrity 52 (2024) 455–471

462

8

T. Profant et al/ Structural Integrity Procedia 00 (2023) 000 – 000

and

3 3 2 2

1 2

3 2

5 2

 

 

, in IIE r

sin2

sin

sin

sin ,  

u

R =  

R B 

B

B

  +





−

+

3

1

3

2

3 3 2 2

1 2

3 2

5 2

 

  

, in IIE

cos2

cos

cos

cos

,

u

R =  

R B 

B l

B

  +







+

+

(24)

3

4

3 12

2



1 1

( f C R B l a   = +  0 2 2 3 2

  

1 3 3 5sin sin   + +

  

)

, in IIE

,





 

12

2

2



where

( (

) )

2

2

1 3 8 1 2 l l l l + − 2 2 2 5 8 1 2

 

.

l

=−

12

1 + − The amplitude factors 1 , 2 , 3 and 0 are associated with the so-called lower-order terms of the asymptotic solution and the Cauchy type stresses and electric potential at the crack tip ( R = ⁄ = 0 ), respectively. The amplitude factors 1,…,4 and 1,…,4 are the amplitude factors associated with the dominant terms of the asymptotic solution for the case of mode I and mode II loadings. The outer solution ,⋅ and the inner solution ,⋅ (the dot in the upper index means the symbol or in the following) represent the solution of the same problem in two regions of different dimensions. In addition, neither the outer nor inner solution offers its intermediate form, which would describe the transition between them. This transition is somewhere in the region whose length is of order ( ) behind the tip of the crack, if ≪ . Let 0 be the distance ′ = 0 from the tip of the crack of the outer solution ,⋅ as the lowest limit value of the radius ′ , at which the error is still of the order ( 34 ) ≼ ( 4 ) , where 3 ≡ 1 ≈ 2 . From this follows that the value of the shift of the coordinate system ( ′ , ′ ) is 0 ≈− 0 ,see Fig. 3. The matching procedure described in details in (Profant et al, 2023) leads to the overdetermined system of algebraic equations with unknows 1,…,4 or 1,…,4 for loading modes I or II, respectively. the expressions assessing the amplitude factors 1,…,4 and 1,…,4 can be written as ( ) ( ) ( ) ( ) ( ) 2 2 12 2 1 2 1/2 1/2 12 4 5 1 4 7 3 12 7 5 , , 8 1 8 9 6 2 6 2 I I l K K A A l El El           + + + + +  + − − − − + 2

0

0

(25)

2

2

K

K

2

1

(

)

 

+ −

2

I

I

,

12

11

A

A

−



 

+ −

(

)

3

4

1/2

1/2

( ) 2 

( ) 

8 1 8 9   − − +

l

6 2 El

El

12

0

0

(

) )

2

2

4 5 1 4 7 3     + + + + +

l

K

K

(

)

12

2

II

II

12 7 5 ,   + −

,

B

B

−

−

(

1

2

1/2

1/2

( ) 

( ) 

8 1 8 9   − − +

l

6 2 El

2 2 El

12

0

0

(26)

2

6

K

K

2

1

(

)

 

+ −

2

II

II

,

12

11 ,

B

B

−



 

+ −

(

)

3

4

1/2

1/2

( ) 2 

( ) 

8 1 8 9   − − +

l

6 2 El

El

12

0

0

where is Young’s modulus and is the Poisson ratio.