PSI - Issue 72

Taras Dalyak et al. / Procedia Structural Integrity 72 (2025) 13–19

16

3

2

l d 2   ,

l    , t x l  is a non-dimensional variables; B Eh 2  ,

t     ,

Here

,

2 (3(1

))

 D Eh

 

(3 )(1 )      a , (1 ) (3 ) 0       , E and  are Young’s modulus and Poisson’s ratio for the plate material, C is an arbitrary constant. Substitute (4) into the boundary conditions (3) and, assuming that the closure of the cats occurs along their entire length, we obtain a system of integral equations of the problem: ( ) ( ) 0 3 1     f t hsf t  , 0 )( ( ))} ( ( )) )( { ( 1 1 1 2 22 1 21               d t hsf t hsf K K ,

 1  1

 1

1

1

 ( )) 

 2

 3 ) ( ) 

 4 ) ( )}

11 { ( K

)( t hsf

  (  K 12

)( t hsf

( ))}

K { ( 33

(

  d

t f

K

t f

d m 

















 

,

1

34





1

1





1

 3 ) ( ) 

 4 ) ( )}

( 1,1)   t .

K { ( 43

(

t f

K

t f

d C 











 

,

(5)

44



1



Here 3 sgn f s  is unknown constant, can be equal 1  or 1  . The system (5) should be solved under additional conditions of displacements and rotation angles unambiguousness: 3(1 ) (3 )       ,

1 

( 1) 0,  

( )

0,   i

1,4

f i

f

d  

.

(6)

4

1



3. Results and discussion 3.1 Asymptotic Analysis The asymptotic solution of problems (5), (6) at large distances between defects was constructed using the small parameter method (Savruk (1981)). After substituting the found solutions into the formulas:

2

2



lim 1

( )    t f t

lim ( ) 1 f t

,   i t

1, 4

,

K

l  

l 



i

i

i

1

1

t

t





we obtain the asymptotic expressions of the force and moment intensity factors near the lower crack tip:

   

2

   

   

    64(1 ) ( 3 1 2 ) 2 0   

1 5 1 4

128 9

0   

   

1 3 1 2

8 1

    (1 ) | | h l m    h l m  (1 ) | |

   0

   

  

4 

( ) 6 

2

 

O





,

1

K







1

   ,

  

16 3

128 3

    1 ( 3 1 2 ) 0   

64 15

  

  

   3 5

( ) 7 

  2

K

O





   

2

   

   

1 3 1 2

8 1

   0

   

    64(1 ) ( 3 1 2 ) 2 0   

1 5 1 4

m l

128 9

 

0   

   

  

2

4 

( ) 6 

 

,

1

K

O











3

1



  

   .

    1 ( 3 1 2 ) 0   

    m l 1 0

16 3

128 3

64 15

  

  

   3 5

( ) 7 

 

(7)

K

O



4