PSI - Issue 2_A

Christos F. Markides et al. / Procedia Structural Integrity 2 (2016) 2659–2666 Christos F. Markides, Stavros K. Kourkoulis / Structural Integrity Procedia 00 (2016) 000–000

2663

5

When z=re iθ is on L (r=R), then z –1 ]/ 2 with s=e iθ . Substituting in Eqs.(16) from Eqs. (17), (18), expanding the right-hand sides of Eqs.(16) in Fourier series form and comparing coefficients of the same order of s=e iθ , the following systems of equations are obtained for determining the real (  ) and imaginary (  ) parts of coefficients of Φ 1 and Φ 2 of Eqs.(17):                                   n n 2 2 2 n n n n 1 1 1 n n n n n n 2 1 2 1 1 2 n n 2 2 2 n n n n 1 1 1 n n n n n n 2 1 2 1 1 2 1 t 1 t a b 1 t 1 t A B 1 t 1 t 1 t 1 t 1 t 1 t a b 1 t 1 t A , n 3,5, ... B 1 t 1 t 1 t 1 t                                                                      (19) 1,2 =R[(1– iμ 1,2 )+(1+ iμ 1,2 )s

with t 1,2 =(1+ iμ 1,2 ) / (1– iμ 1,2 ) and:

a b

1 P R

    

e   

sin 2 cos 2

sin 4

 

1

 



 

  

3

i2  

sin 2

sin 2

e

c

o

o

o



 

   



o

o

o

o

i 6  

2 2sin sin 4 2cos 2 sin 4 sin 2 cos 4         o 2

4

  

3

(20)

 

i2  

 

 

o





o e ,   i2

n 3 

o

o

o

o

o





2

2sin

2

3

 

 

o









  

a b

c 1 P R i 2 n

sin n 1

sin n 1

   

 

 

 

1

n

o

o







   

2 2sin n 1 cos 2 sin n 1 2sin 2 cos n 1          o n 1

n 1 

n















        

 i n 1       

 i n 1    

 

o

o

o

o

e

e





    

o

o





2

4 n 1  

(21)









    

c 1 P R i 2 n

sin n 1

sin n 1

        

 

  

 

1





o

o

n 5, 7,... 



2

n 1 

2sin

n 1 



o

 











 

o 2 n 1 cos 2 sin n 1 2sin 2 cos n 1  o o

        

 i n 1       

 i n 1    

 

o

e

e



    

o

o





4 n 1  

the Fourier series coefficients of the parabolic pressure induced on the disc (a n =b n =0, n=0,2,4,…). For n=1, one obtains the following relations (Lekhnitskii 1968):             1 1 1 1 1 1 1 1 1 2 1 1 1 2 1 1 1 1 2 2 2 2 1 1 1 2 1 1 1 2 1 1 A B A B a a R A B A B a a Ri b b R A B A B b b Ri                          (22)

with the respective Fourier series coefficients of parabolic pressure given as:

a b

1 P R

    

  

  

2 sin 2

sin 2 cos 2

i 2  

e

 

  

 

 

  

o

(23)

1

o   i 2  

2

sin 2 e

sin 2

c

o

o

o

o



 

   



o

o

o

o

i 2  

2

2

2sin

2sin

2



 

  

1

o

o