Issue 49

A. Baryakh et alii, Frattura ed Integrità Strutturale, 49 (2019) 257-266; DOI: 10.3221/IGF-ESIS.49.25

where . A general solution of this linear differential equation with constant coefficients takes the following form:   1 2 3 4 ( ) ( ) ( ) ( ) f y C ch y C sh y C ych y C ysh y         (13) m l   

Then a stress function is determined according to the following expression:   1 2 3 4 sin ( ) ( ) ( ) ( ) x C ch y C sh y C ych y C ysh y          

(14)

and the relevant stress components are calculated according to the following formulas:

2













 

 

2

2



 

( ) x C ch y C sh y C sh y          ( ) 2 ( )



( ) ych y C ch y      2 ( )





( ) 

ysh y

sin

x

1

2

3

4

2



y

2     





2





( ) x C ch y C sh y C ych y C ysh y         ( ) ( ) ( )

sin

(15)

y

1

2

3

4

2



x

2















 

 



( ) x C sh y C ch y C ch y          ( ) ( )



( ) ysh y C sh y      ( )





( ) 

 

ych y

cos



xy

1

2

3

4

 

x y

where constants 1 , , C C C C are determined by relevant stress boundary conditions. For the test problem (Fig. 2), the boundary conditions can be represented in the form of symmetric expansion into a Fourier series: 2 3 4 ,



m x 

0     m q A A 1

(16)

cos m

l

where the expansion coefficients are defined by the following expressions:

m a 

2 sin q

1 cos a

m x 

, qa A l

l



A q 





dx

(17)

m

0



l

l

m



a

, C C C C in expressions for stresses (15):

With account of the boundary conditions, the constants 1 2 3 4 , ,

2 q sh h hch h     ( ) ( )

 



C

1

2

(2 ) 2  



sh h

h





a

cos

( ) (2 ) 2 sh h    

q

2





C

(18)

4

2

c 

sh h





a

cos

 

C C

0

2

3

where q- is determined by expression (16). The results of an analytical solution for middle line (y=0) on which only a normal stress takes place:

262