Issue 44

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

 

 

 

 







0  



0  





 

, x B

, x B

, x B

c

x c

0,

xy

xy

xy

0

1

(7)







0  



0  





 

, x B

, x B

, x B

c

x c

0,

y

y

y

0

1

One needs to solve the corresponding boundary value problems to estimate the stress state of the semi-strip and the concentration of the stresses at the crack’s tips.

G ENERAL SOLVING SCHEME FOR THE SEMI - STRIP STRESS STATE ESTIMATION

A

ccording to the approach [29], the Fourier’s transformation was applied to the system of Lame’s equilibrium Eqs. (2) and to the boundary conditions (1), (3)-(4), (1), (5) by the generalized scheme [30]. The initial problem was reduced to a vector boundary problem [31]         2 0 0, 0 L y x f x y y a           (8)









  L y x Iy x    "

 

 

2









is the differential operator of the second order, I is an identity

here

' Qy x 

Py x 

2





2

matrix,

   

u x v x  

  

   



 

 

y x 

 

    

1 0 1

     



P

 

 

1 1

  

0

1

 



0





1

 

Q

   

1





0

1  







 

 

 

 

3

1 1

3 1

    

     

  x

  x

















'( ) x

b

b

sin

cos

'



1

2





1

 

 

f x

 

 

 

 

1 1

1 1

 

 





( ) sin ' x    





 

b x

b x

cos

1

2

,

, and the second boundary



u

   x v x y   ,



is an unknown function. So  ' , v x y



  x   ' 

 



' x 

y





y

y

0

0



y

0

  y x  

condition in (3) is satisfied automatically. The components of the vector

are the Fourier transformation of the

displacements

 , u x y v x y , 



   

u x v x  

( ) ( )



y

cos sin

  

  

 

  

  

0 

dy

(9)





y

 

The solution of the vector boundary problem was obtained in the form [27, 28]

85