PSI - Issue 26

Christos F. Markides et al. / Procedia Structural Integrity 26 (2020) 53–62 Ch. F. Markides et al. / Structural Integrity Procedia 00 (2019) 000 – 000

57

5

and the problem is reduced to obtaining the functions Φ o (ζ), Ψ o (ζ) in the ζ -plane. The demand that, for large |z|, the functions Φ o (z), Ψ o (z) must be of the form given by Eqs.(9), is now translated (in conjunction with Eq.(5)) to the conditions (Muskhelishvili 1963):

( )     =    =            ( ) o o 2 2 1   1 ,

,

(11)

( ) o 3 1     =       which are here somehow relaxed in order to obtain the present approximate solution. Substituting from Eqs.(10) and (5) in the boundary conditions for the stresses on BCD (Muskhelishvili 1963):

( ) ( )

( ) ( )



( )     +  =   +   +   +         ( ) ( ) ( ) i    

(12)

and postulating that the parabo lic notch BCD is to be free from external stresses, i.e., for σ ηη +iτ ξη =0, one arrives at the following conjugate expressions for the boundary values of Φ o (ζ), Ψ o (ζ) on BCD , or, what is the same thing in the [ξ Β ,ξ D ] interval of ξ -axis in the ζ -plane:

)

2

ia

 −

(

)

( ) ( ) ( 3 A ia  +   = −  +  +  −   +  −   + ia A c a ia ) ( ) ( 2 ) ( ) o ( o o

( )

o 

(13)

 

2

)

2

ia

 +

(

)

( ) ( ) ( 3 A ia  −   = −  +  +  +   +  +   + ia A c a ia ) ( ) ( 2 ) ( ) o ( o o

( )

o 

(14)

 

2

) ( ) o ia  −   ( holomorphic

In this context, considering that (

) ( ) o ia  −   is the boundary value of the function (

) ( ) o ia  +   and (

) ( ) ia   +   2

in the upper ζ half -plane and vanishing at infinity via the third of Eqs.(11)), whereas (

o

) ( ) ia   +   holomorphic in the lower ζ 2 o

) ( ) o ia  +   and (

are, respectively, the boundary values of the functions (

half- plane and vanishing at infinity via the first and second of Eqs.(11), Eq.(4) yields (for ξ Β = – ξ D ):







   

D

D

3   d

A

d  

) (   

)

( )   = o





2 a c −

(15)

−

(

2 i

ia

  +

 −   − 

−

−

D

D



D    −   1 d 2 i −

throughout Eq.(13), yields:

Similarly, taking the Cauchy integrals

D







   

)

2

ia

 −

D

D

3    − d

A

d  

ia ia

) (   

)

( )   = o

( ) (





) ( ) o   

2

(16)

c a −

+

+   +

(

(

o

2 i

ia

2 ia  +

  +

 −   −   +

−

−

D

D

whence performing the integrations:

3

2

3 

A a c 2 log

) ( 

)

( )   = o

(

)

(

)

2

2

D −  +   −  −  − −  −   −     D D D log 2

D

(

2 i

ia

  +

(17)

) 

(

)

(

3

3

log −  −  +  − −  log

D

D