Issue 48

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

The formulas

 2 ( ) z N Z z ,

 2 ( ) z i N Z z

 u ( ) α βα

 v ( ) β βα

βα

βα

were used to find the originals of the displacements 2

2 u ( , , ), v ( , , ) x y z x y z

So,

 

 

βα T h

βα T h i D N ( ) β

( )



2 u ( ) z βα

2 ( , ), v ( ) z

 

 

F N z

F N z

.

( , )

2

βα

2

2

2

N D N

2

2

N

By analogy, it was found

 2



 

 

F t z

C

( , )

 2 u ( , , ) x y z 2

  0 0

, A B

 ( )cos( cos )cos( sin ) tx ty    d dt

S

,

2

                 2 2 x C y

t

 t D F t z t

 2



 2 v ( , , ) x y z 2

( , )

  0 0

, A B

 ( )cos( cos )cos( sin ) tx ty    d dt

S

.

2

t

 t D

t

Let’s consider the terms containing the integrals in (37). Using the solution of the stationary thermal conductivity problem (10), the temperature transformation and its derivative are written in the form    βα βα ( ) , chN T f chNh     βα βα ( ) . shN T Nf chNh After substitution of this transform into (40), the formula for 3 βα w ( ) z was constructed

h

 

 0

3 w ( ) z βα

 

 

f

( , , ) F z N d

0

βα

4

   

      (11) (12) shN N chN

1









  N z

          1

  F z N e ( , , )

 

N z

( shN z N chN )

.

where

 

*

chNh

D

N

In order to find final expression for displacement (40)

 

h

 

  

 

i y

3 w ( , , ) x y z

 

 ( , , ) F z N e

    xd d d

f

cos

(42)

0

βα 2



4



0 0

f should be substituted into (42)

the representation of the function 

   

 

h

 

  

   i y i

3 w ( , , ) x y z

 

  ( , )

 ( , , ) F z N e e

       d d d d d .

f

cos cos x

0

2



4





0

0 0

Using formula (A3) in the Appendix 1 and (401.06., [41]), it was derived

 B A h    

  

C

3 w ( , , ) x y z

 t F z t J t x y *

     d d dtd ,

 

  ( , , ) ( , , , , )

0

0



8



B

0 0 0

where the temperature is defined on a finite interval and supposed to be constant C . Further, after changing the order of integration, and calculating the double integral of the Bessel function, the integral is represented in the form

780