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

2662

4

2

2

2

F

F

F

 

 



(8)

,

,

 

 

  

x

y

xy

2

2

y

x

x y

 

Eq.(5) provides the generalized biharmonic equation in F:

4

4

4

F

F

F

 



 





(9)

2

0

   

  

22

12

66

11

4

2 2

4

x

x y

y

 

with a characteristic equation:   4 2 11 12 66 22 2 0          

(10)

Eq.(10) has four purely imaginary roots, the so-called complex parameters, as follows:

i , i                   ;

(11)

1 i ,

2 i ,

1

2

3

1

1

4

2

2

1 2     



  







2

;

(12)

2     

2

4

2

,

     

    



12

66

12

66

11 22

11

1

2

over-bar denotes the complex conjugate. Then, following Lekhnitskii (1981) it can be written:

  11 1 12 2 F x, y 2 F z F z           

(13)

with z 1 =x+μ 1 y and z 2 =x+μ 2 y the so-called complicated complex variables, and:

  z F z ,    11 1

  z F z  12 2  

(14)

1 1  

2 2  

the Lekhnitskii’s complex potentials (where prime denotes the first derivative). Φ 1 and Φ 2 are obtained from the given values of stresses on the disc’s boundary. Namely, stresses (Eqs.(8)) and their values on L (Eqs.(6)), expressed in terms of Φ 1 and Φ 2 as (Lekhnitskii 1981):

 

z , 

 

 

 

 

  z 1 1 1         2  2 2 x

 











(15)

2

,

2

              



2 2 2

y

1

2

xy

1 1

2 2

S  0

S  0

  z

  z

  z 1 1 1          2 2 2 2   z

(16)

2

Y dS,

X dS

       









1 1

2 2

n

n

with S the arc length on L. According to Lekhnitskii (1968), Φ 1 and Φ 2 are sought in series form as:



  z A A z       1 1 0 1 1 n 2

  A P z , n 1n 1

(17)



  z B B z           1n,2n 1,2 1,2 P z R 1 i      2 2 0 1 2 n 2

  B P z ; n 2n 2

n         

  

n

n









     

  

(18)

2 z R 1          2 2 2 z R 1 2 z

2

z



1,2

1,2

1,2

1,2

1,2

1,2