PSI - Issue 68

M. Dudyk et al. / Procedia Structural Integrity 68 (2025) 53–58 M. Dudyk et al. / Structural Integrity Procedia 00 (2025) 000–000

55 3

! " ############

!

;

(3) (4) (5)

! =

" = #

=

!

!

!

!" # ! = <

$ ### %&' ( )" # ! " = " # $ = $ # '*+ ( ) !

!" #  = >

$ ###

!

;

;

! "

! "

# # 

= =

!

!

!

!

!

!

!"# ! + " ! !" !"# ! !

!"# ! + " ! !" !"# ! !

;

!"&&

' &&&& $% &

!

"&

!

"

$ #"

$ #"

% # "

% # "

# =

+

'

+

"

#

#

#" $ # #" $ # ! !

$ $

"

" " ! " &

! ! + ! " + ! + ! + + ! !

"# + #

! " "" #$%C # !#%% !% !" # = = ! = + " ! #$$ #$$ !

"

$ =

;

;

% &&

'( % &&

!

% =

# =

!

) #" $ &' "#

!

"

!

!

+ ! + !

!

"

% &&

' ( = # !

;

"

=

"

!#"$

!#"$

!

!

"

!

! " ! !

!

!

is the jump of ;

is the phase angle of stress in the process zone, which will be considered constant and

! !

! !

!

equal to its average value in what follows ; the dash above

and

is the complex conjugate;

is the

!

complex SIF at the tip of the interface crack. Using the Mellin integral transformation, the formulated boundary value problem (2-5) is reduced to the Wiener Hopf matrix equation (6) in the strip ( are the small positive numbers) which contains the imaginary axis: ! " #$ ! !" < < " ! " # $ ! !

! " ! " #$% ! " ! " ! " ! ! " ! # ! ! ! ! + " + = " ! " # #

,

(6)

+

$

&   +

' * ,

%   +

& * ,

! -

! - %

# $%& # $%& # #

$ % $ % ! !

# & # & ! !

$ ( % &  * + , " $ !

" ! " # $ # !

( (

' '

& +

' * ,

%

!

!

, ! # # $

, ! " " &

!

* = 

* =

" ! # $ (

&$! % #

" +

" $

" " # " =

!"#

!"# % %

& ( ( ( ( ( *

' ) ) ) ) ) +

!

!

%

+

!

!

&'( " #

$ " #

$ " #

!

%

%

#! $#

! $ ! !

$

!"# + +

!"#

! ! ! + + + + + + ! "# $#! $ ! ! ! ! !

$

$

!

$

!

!

!

+

+

%

"

,

,

"

=

!

) * $

=

!"#

!"# % %

!

!

%

+

!

!

(+, $ #

% " #

% " #

!

"

!

%

%

$

!"# + +

!"#

$

$

!

$

!

!

!

+

+

% "

!

#$

%$" %&'( !

!

+ +

! !

"

,

"

!

$ %

# "

=

!

!

$

% $" % !$ ! ! ! + + + + + +

%$" %&'(! !

!

"

" !

! !

"

!

"

!

! " " !

! ! " ! # % & ' ! !

! %

" $ &

!

!

= #

= $

! " ! " ! = + ! " # ! "

, ! " #$% ! " # " ! " =

,

,

.

The exact analytical solution of equation (6) is obtained taking into account the features of the behavior of stresses and displacements near the tip of the interface discontinuity line and using the principle of analytical continuation, theorems of the Abelian type and Liouville and the method of Khrapkov factorization of the matrix coefficient (Khrapkov, 1971) similarly to the solution of problems in works by Kaminskii et al. (1995, 1999, 2023). Having applied to the found solution the requirement of the stress finiteness at the end of the process zone, an equation was derived for calculating its relative length and the phase angle ψ of the stress in the zone: ! " ! ! ! ! " # =

!

& ( ) ( +

' ( * ( ,

"#$ % &'

(

! # ! " # $ + + + !

!

!

,

(7)

$)' % &'

( ! ! " # $ + + +

! " =

%

+

!"#

!"#$ %#

& " # $ % + + + "

,

(8)

! = &

!

where

!

! ! " # " " # # = + + ! ! $ #% & ! "#$% $ $$$$ $$$$ C$$$$ = $ % & = + & ! ! ' (( ' (( $ % $ # " " # + # # #

! (

" )

%' (

)*+ ' $

' =