PSI - Issue 16

Zinoviy Nazarchuk et al. / Procedia Structural Integrity 16 (2019) 11–18 Zinoviy Nazarchuk, Leonid Muravsky, Dozyslav Kuryliak/ Structural Integrity Procedia 00 (2019) 000 – 000

13 3

( i y u u y x L       ) 0, 0;

( ,0)

,

(3)

0, t i u u u y x L        0;

( , ) (0, )  .

(4)

Here j  is the exciting wave propagating in the negative direction of the Ox axis, 2 2 1/ 2 i k k k     is the wave number ( , 0 k k    ). We find a solu tion of the boundary-value problem (1) – (4) in the class of functions realizing the limit absorption as x  and satisfying the conditions at the crack tips: 1/ 2 u  , 1/ 2 y u    , if 2 2 1/ 2 [ ] 0 x y     , 2 2 1/ 2 [( ) ] 0 x L y      . Applying the Fourier transform to Eqs. (1) – (4) we derive the Wiener-Hopf equation of the problem as [( (2 1) / 2 ) j ] j d k      , Re 0 j   , 1, 2,... j  ; ( , ) u x y e   i sin( ) j x y

( )           , i α L e [ ( α) (α)] ( ) ( ) 0 M J

(5)

1

( ) ( α)   and

 

( α)

0 :{ }       , 0 Im k   ; 0

1 ( ) J  are unknown

Re i Im ( i )       ,

,

where

( ) ( α)   is

( α)   is regular in the complex plane

0    ,

functions with known regularity properties, namely,

i j    ,

0 Re j    , where it has a simple pole. Function 1 ( ) J  ( ) M  is a known function regular in the

0    , except the point

regular in the half plane

is an entire function expressed via an integral within certain finite limits, strip  and admitting simple zeros and poles outside the strip  ;

( ) ch( ) /[ γsh( )] M d d     ,

(α,0) iβ /[ 2π(α iγ )] j L j j e    

( ) ( α)      U

( α)      U

(α,0) iβ /[ 2π(α iγ )] j j  

,

,

( α, 0) U   ,

( α, 0) U   are unknown regular functions in overlapping half-planes

2 ( ) k     , Re 0   ; 2 1/ 2

0 Im    respectively:

0 Im     and

L





1/ 2        ( , 0) (2 )  

1/ 2       ( , 0) (2 )  

i ( , 0) x u x e dx 

i ( u x e dx   ) ( , 0) x L

U

U

,

.

y

y

0

Nazarchuk et al. (2011) have shown that the solution of the functional Wiener-Hopf equation (5) is reduced to the following infinite system of linear algebraic equations (ISLAE): [ ] I A X F   . (6)

0 { } r r F f   

, if n  ; I is the identity matrix,

( ) ( ) ( ), i

1 ( )

0 { } n n X x   

ns      

n x O n  

,

,

Here

x M i

r f 

n

ns

, 0 :{ } rn r n A a   ,

i s k    ;

2 [ (2 1) / (4 ) n 2

2 d k 

2 1/ 2

2 ns n d k     [( / )

2 1/ 2

M i    

)]     ,

,

, 0

( ) / [ 2 (

]

]

   

j

j

rs

j

nc

 

L

L

,

(7)





2 i d e 

1

2

2 {[ (i )] ( M

1







a

[ (i )] M 

e

)(         ms rs ms ms ms

)}

 





ns

ms

rn

n

ns

ns

m

ns





m

0



0 n  and

0 n  ;

1/ 2

1 n   for

n  

for

where









i /( ) d n  

i /( ) 1 d n   

M

kd

[1 /(i )] e   

{i sin [1 /(i )] [1 /(i )] k kd e      

( ) cos  

}

.

nc

s

ns

0



n

n

1

1



