PSI - Issue 81

Ivan Zvizlo et al. / Procedia Structural Integrity 81 (2026) 109–115

113

dS

1



3

  

ξ

( )

,

, S m k n     , 2, .

B

B

ξ X

x

u

N



B

0

3 01 j

0

1

1

j

x

ξ

G



1

j



10

B

S

0

Using the representation (7) for unknown densities and the method of analytical calculation of two-dimensional integrals over infinite domains (Stankevich (1996)), we reduce the BIE (9) to the final form

dS

dS

 

2   



ξ

ξ

( ) ξ

( ) ξ

B

B

u

u

  



(10)

31

2

31

1 x ξ 

1 m x ξ 

2

m



S

S

1

1

1 11 1      d x 2

1

1

e



  

  

 



0 

B

( ) ξ

1 x ξ ( , , )        ( , , ) m d dS ξ 1 x ξ

,

,

x

u

N

S







В

31

0

1

1

2

2(

)(1 )

b b

G

 



2

m



3 4

B

В

S

1

where

2

i



2



1 x ξ , , )      x ξ ( , , ) i (

4

( | 

|) ,      ()8 (1 )(12 )

,

x ξ

J

b

0

0

4

B

B

B

1       1 m

1       1 m

2

i

x



0

i



31





2 () 2 (1 )2 (12 )(14 ) x            

2

 

(1 2 )

,

b

b

  



1

301

4

3

B

B

B

B





()2(1 )4 (12 ) b x b        (1 4 ) (1 2 )(1 ) , x

2

301

4

301 3

B

B

B

B

B

kd  

2

1 x ξ

2 21 2     | ( ) x

2

2 21         2 2 ( ) x nd |

2

|

, |

.

x ξ

x

x

31

31

1       1 k m

A characteristic feature of equation (10) is the absence of integration over an infinite interface 0 S , which is important for the application of numerical methods. 4. Numerical solution of BIE By performing the differentiation operation in the first term of equation (10), we obtain the integrand kernel, which contains a hypersingularity of the form 3 1   x ξ . Further methods for its regularization are described in (Zvizlo and Stankevych (2024). The unknown densities 31 B u  were chosen in the form

2 B u a x x      х х 2 2 31 1 11 21 ( )

( ) ,

(11)

1

where 1 ( )  х is the unknown twice non-differentiable function in the domain 1 S . Representation (11) satisfies the conditions for the closure of the crack surfaces on its contour. During the numerical solution of the BIE, the circular defect regions 1 S were divided into quadrilateral boundary elements with division steps 0.1 а in polar r and 12  angular  coordinates. Within each boundary element, the discrete values of the unknown function 1 ( )  х were considered constant. The problem was reduced to solving a system of linear algebraic equations with respect to the nodal values of the function 1 ( )  х . The stress intensity factor (SIF) for mode I in the vicinity of the crack contour was determined

2

G a  

( )

( cos , sin ). a a   

K

 

I

1



where  is the angular coordinate of the crack contour point. The semi-infinite integral in (10) was calculated using Laguerre ’s quadrature.