Issue 44

V. Reut et alii, Frattura ed Integrità Strutturale, 44 (2018) 82-93; DOI: 10.3221/IGF-ESIS.44.07

was constructed with the help of the matrix differential calculation, , 1, 2 i c i   is the Green’s matrix function which was constructed by the use of the matrix semi-infinite Fourier transformation. The components of the Green’s matrix function have the following form are known constants,   , G x  

 x   



  

   



x   









e

e

1

1

 x   



 x   



x   

x   









 G x  11





 x e 







 

 



x e 

e

e

,





 



2 1 



2





1

 x   

 



x   



 G x  12





 x e  

 x e 





 

,





2 1  





1

 x   

 



x   



 G x  21





 x e  

 x e 





 

,





2 1  

 x   



  

   



x   









e

e

1

1

 x   



 x   





x   





x   



 G x  22





 x e 

 





 

 



x e 

e

e

,





 



2 1 



2

After inverting the expression (18), and the summation of the weak-convergent integrals, the formulae for the displacements in the quarter plane have the following form

2

2

   

   

0  









   

















x

x

1   



2

1

  

2

2





    '









2

2



ln x y       x y  





, u x y

ln

 





2

2









4 1  

1    



2

2

 





x y

x y





  

    

   

2 2

  



 

x y

2



   x 





1

x







d





  



2

2







2

1

x y    

2





2

 



x y





   

    

0  









   

  

  









y

x

y

x

  

  

y

y

2

1

1 2





    '





 





sign x arctg  



, v x y

arctg

2

2











2 2  









2

2

x

x





 

 

x y

x y

 

 

  

    

   

 

















xy

x

xy

4

1

  

  



y

y

2

2

    x sign y 











 

arctg

d

2





  



2

2

2















2

2

x

1





 

 

x y

x y

2



2



 

x y



These expressions will describe the displacements in the quarter plane if the function   '   is found. To get it, the formulae for the displacements were put in the boundary condition   ( , 0) , 0 y x p x x a     . After changing the variables, the singular integral equation was derived

   

   

1

3 h x 

h

2 h x

1

 

 

 

1  

 

d q x  



(19)

  2

3







x   

x









x

x

0

 2 a 

 1   

2





3

2

4

here     

,    ,   q x is the known function. h

'   

 

h

h

,

,

 

1

2

3







2



91