PSI - Issue 33

Anna Fesenko et al. / Procedia Structural Integrity 33 (2021) 509–527 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

526 18

n       .

Second case corresponds to a situation, when n belongs to the interval 1 1    

(1) ( ) m m m H z J z iN z   , where 3 i   was chosen. Real part ( ) ( )

( ) m J z - Bessel

In this case definition of Hankel function of first kind was used





( ) m N z - Neumann function. Instead of function 1  function

function,

0 Re , ;

    

was separated to provide reality of the solution while normal stress construction   2 2 1 3 ; i i n            2 1 2 2 2 1 ; n           

   4      ;



2 S M  

2

det

;

;

TS LM

n

n

     3 1 2 K                 2 * 2 3 0 * J     3 1 2 2 * 2 3 0 * N K   



  0     2   0 2

      2 1 3 ;

2    2 2    2 2 1 2 *  1 2 *

T

K J K N



2 1 3    

3

1





2 1  

2

;

L



3

1





2 1  

    2 1 3 1 2 J K                    K 2  2 2 2 * 2 3 0 3 1 2 * 2 4 2 J     2 1 3 1 2                    2 M N K  2 2 2 * 2 3 0 3 1 2 N K * 2 4 2 S

     2 0 3 0 2 J K  

;

     2 0 3 0 2 N K  

;

 

n



The last situation, when

.

1 1

 

 

Both function 1  and 2  were changed by minus imaginary unit by square roots that have positive expressions under them. Here real part of normal stress also has been separated.



 2





2

2 n          i i

;

;

i

i

n   

2 

    

1 2

 

 

1

3

4

1









2 A B  

2

det

;

;       4

;

CA DB

n

n





    J                    2 2 2 2 3 1 1 * * 0 4 1 3 2 2 1 J J                  2 * 3 4 0 J  3 1 4 0 3 1 4 N N

C



          4 1 3 N N 0

;







             N 3 1 4 0 3 1 4 J 

2 * 3 4 0 

D

N J 

  













              N 4 1 3 0 4 1 3 J

2  

2   

2

2 

;

N J





3

1 2

1





*

*

0

2 1  









    4 1 3 1 4 J J J N N                                2 2 2 * 0 4 1 3 0 4 1 3 2 ; J J N N                              2 2 4 1 3 1 4 1 3 1 4 * 3 4 0 3 1 4 0 3 1 4 2 4 B N J J N N J J N                                2 2 2 * 0 4 1 3 0 4 1 3 2 ; N J J N              1 3 1 4     * 3 4 0 3 1 4 J     2 2 0 3 1 4 2 4 A N N

Constructed formulas give opportunity to calculate the normal stress for large oscillation frequencies.