PSI- Issue 9

Ch.F. Markides et al. / Procedia Structural Integrity 9 (2018) 108–115 Author name / Structural Integrity Procedia 00 (2018) 000–000

110

3

a a 0       

A 0,

(2)

1

1

with

2 E [(1 )(1 2 )], plane strain E (1 ), plane stress        

: shear mod ulus



3 ,

 

  

(3)

,

 

  

E : Young΄s mod ulus,

: Poisson΄s ratio



  

Muskhelishvili showed that by admitting multi-valued displacements, their components, after a circuit along any contour around the inner ring’s hole, undergo an increase given as: u u y , v v x               (4) where (+), (-) indicate the two sides of a cut (joining its perimeters), introduced to convert the ring into a simply connected region. He also showed that:

A(1 )   

i



1 ( a a )    1

,

i

 

     

(5)





According to Muskhelishvili, ε, α and β (ascribed certain physical interpretation by A. Timpe (1905) in the case of the ring and later by V. Volterra (1907) in the general case of multiply connected regions), are the so-called “ characteristics of the dislocation ” (as denoted after A. E. H. Love (1927)). Moreover, Muskhelishvili showed that by admitting multi-valued displacements (so that A and 1 1 a a      are given through Eqs.(5), for fixed values of ε, α and β, instead of Eqs.(2)), the complex potentials and the corresponding stresses assume definite non-zero values in the ring even in the case of absence of external forces. For example, for {ε≠0, α=β=0} and {ε=α=0, β≠0}, namely the two cases of particular interest for the problem solved here, it holds, respectively, that (Muskhelishvili (1963)):

2 2 2 R ogR R ogR    2

(         2 ( 2 ) 2 ) 1 

  

(z)  

ogz

(6a)

1 2

1







 



2 2 R R 

1

2 2 1 2 )R R R 1 2

(    

(6b)

(z)   

og





2 ( 2 )(R R )     

2 2

2

R

z

1

1

2 2 1 2 R R R R ogR R ogR    2 2 2 2 2

) ogr 1    

  

(        

( )   

og

(6c)

1 2

1





r

2 2 r R R 

2

2 2 R R 

( 2 )

R



2

1

1

1

2 2 1 2 R R R R ogR R ogR    2 2 2 2 2

) ogr 1    

  

(        

( )   

og

1

(6d)

1 2

1









2 2 r R R 

2

2 2 R R 

( 2 )

R



2

1

1

1

(6e)

r ( )    

0

and

) 1 2z 

  

(    

(z)  

(7a)

 



2 2 ( 2 ) z R R     

2 2



1

2 2 1 2 1 R R 1

 

  



(z)  

(7b)

 



2 2 ( 2 ) z R R z      2 3

1

2