Issue 48

A. Fesenko et alii, Frattura ed Integrità Strutturale, 48 (2019) 768-792; DOI: 10.3221/IGF-ESIS.48.70

       

   

1 w ( ) z

1 F N z F N z

( , ) ( , )

  

  

β i b

  GD

a e

cos

βα 1



,

(38)

βα Z z

( )

2

2

N

  

2 w ( ) z

( ) T h F N z

( , ) ( , )

  

  

βα

1

βα 2

 

,

(39)

  

F N z

βα Z z

( )

D

2

2

N

( , ), ( , ) F N z F N z ,

N D are defined in (37).

functions

1

2

h

 









 0

  N z



          1 ( ) ( T

3 w ( ) z βα

2 z N T d ) ( )

 

  

e

N z

0

βα

*

βα

N

4

h

  ND 0





 0



      (11) 2 (12) ( ) ( ) , T N T d



βα

βα

2

N

(40)

h

 









 0

  N z

      ) ( )



3 Z z βα

1

 

       ( ) T d

e

( N z T

N z

( )

0

βα

*

βα

4

h

  ND 0





 0



      (21) 2 (22) ( ) ( ) . T N T d



βα

βα

2

N

In detail, the inversion of the Fourier transform for terms of the form (39) is described using the example of the term 2 βα w ( ) z . Proceeding from the boundary condition in (10) of the stationary heat conduction problem considered in part 3

   

   d d . Then the representation of the term 2 βα

w ( ) z will take the

 ( , ) cos i e

     βα f

βα T h f ( )

of the article, one gets

0

form

   

   



F N z

( , )

     ( , ) cos i e

 

i y

2 w ( , , ) x y z

 

     xd d d d

f

e

cos

1

2



D

2

N





0

0

Changing the order of integration, using the formula (401.06.,[41]), and application of the Euler formula allows to simplify the term 2 βα w ( ) z . The temperature is assumed to be constant, distributed over a finite interval        , 0 B B A , which leads to the expression                               2 ( ) ( ) ( ) 1 2 0 ( , ) w ( , , ) 8 B A i y i x i x N B F N z C x y z e e e d d d d D (41) Further, using formula (A3) in the Appendix 1, correspondence (41) can be written as

    B A

 

 t F t z

C

( , )

    2 2 t

2 w ( , , ) x y z

 * J t x y

t D sh t

 

   

d d dt

( , , , , )

,

1

0



D

4

t



B

0 0

Applying the procedure for calculating multiple integrals of the Bessel function to internal integral, one gets

     C

t F t z

( , )

2

  2 0 0

 ( )cos( cos )cos( sin ) A B t S tx ty    , d dt

2 w ( , , ) x y z

 

1

2



D

t

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