Issue 38

N. Vaysfeld et alii, Frattura ed Integrità Strutturale, 38 (2016) 1-11; DOI: 10.3221/IGF-ESIS.38.01









y v x y u x y ( , ) ,  are the displacements which satisfy the Lame’s equations. The Lame’s

x u x y u x y ( , ) ,  ,

here

equations are written in the following form [24]

2

2

2

 





u x y

u x y

v x y x y u x y x y    

( , )

( , )

( , ) 0







* 

0 

     

2

2





x

y

(3)

2

2

2







v x y

v x y

( , )

( , )

( , ) 0

* 

0 







2

2





x

y

1

0 * ,   through the Muskchelishvili constant

0 



* 

0   

where

. After the expression of the constants

,

1





1 2 



3 4   



, one obtains the system (3) in the another form

2

2

2

 





   

   

u x y

u x y

v x y x y u x y x y

( , )

( , )

( , )

1 1 1 1

2







0

     

2

2

1   







x

y

(4)

2

2

2







v x y

v x y

( , )

( , )

2

( , )







0

2

2

1   







x

y

The boundary conditions on the semi-strip’s edge are reformulated with the terms of the displacements

   , 0



   v x , 0  

 

  

u x

 

1  

 

 

0   

G

p x

x a

(5)

2

, 0





x

y

  , 0

  , 0





u x

v x



0, 0   

x a

(6)





y

x

One needs to solve the boundary value problem (2), (4)-(6) to estimate the stress state of the semi-strip.

T HE GENERAL SOLVING SCHEME OF THE PROBLEMS ON THE SEMI - STRIP STRESS STATE ESTIMATION

T

he Fourier’s transformation is applied to the system of Lame’s equation and to the boundary conditions by the scheme     u x y u x y dy v x y v x y 0 ( ) cos , ( ) sin ,                          (7)

with the inverse formula

 u x y v x y , , 



0  

u x v x  

( ) cos ( ) sin    



y

  

  

  

  

2





d

(8)





y







The initial problem has the form after this

3