Issue 48

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

    3 4 . The unknown constant matrices



0 1 , , i i C C i



0,1

 ( ) z Y are shown in Appendix 3.

and

The basic matrices are expressed by the formulas after transformations



  

  (11)

(12)

1

i

i

 

( ) z

(28)

Ψ

i

(21)

(22)

D

  

N i

i

 2 2



where N D sh Nh Nh . The components of the matrix (28) are shown in Appendix 4. The expression for Green’s matrix function is proposed in [39]              0 0 1 1 ( , ) ( , ) ( ) ( , ) ( ) ( , ) z z z U z z U z G Φ Ψ Φ Ψ Φ where  ( , ) z Φ is a fundamental matrix of the corresponding homogeneous equation (26). For the construction of the fundamental matrix   4

(29)

 ( , ) z Φ , let’s continue the function ( ) y z on all the real axes and apply

the Fourier integral transform



    ( ) i z y y z e dz 

(30)

The same transform (30) should be applied to the equation in (24)

2

    2 I y

 i y N y Q P f   



0

Hence, one derives



y

   M f 1 ( ) i





After the inverse integral Fourier transform the correspondence for ( ) y z is obtained

 

    1 2

y z

( )

   i M f 1 ( )

   ( i z

)

 

e

d d

(31)



 

According to the fundamental matrix definition, this matrix is

 



   (

) i z

1

)

   )

 



z

i e

d

(

(

M

1

Φ



2



Or after changing the variable    s i , the matrix has the form

     

z z

0 0

*

( ) s

 s N M 2





( s z

)

   )

z

e

(

,

(32)

1

Φ



i

2

2 2

(

)

C



here  C are the closed contours that cover the poles   s N or    iN . The residuals at   iN and    iN are correspondingly equal

 3 1 (4 ) (1 ), N Ny



 3 1 (4 ) (1 ),

Ny

Ny

 

  

 N Ny y z

  

e

e

(33)





777