PSI - Issue 28

Christos F. Markides et al. / Procedia Structural Integrity 28 (2020) 710–719 Christos F. Markides and Stavros K. Kourkoulis / Structural Integrity Procedia 00 (2019) 000–000

714

5

y

P=const.

η

ζ 2

ζ 1

z 2

z 1

k z

k 

z= ω (z)=R ζ

z=re i θ

θ Θ

θ Θ

ζ = ρ e i θ

ρ

r

1

R

x

ξ

Ο

Ο

γ

L

k z

k 

ζ 3

ζ 4

z 3

z 4

Fig. 4. The conformal mapping. What is more, and considering again the chord z 1 z 2 , n is chosen to be an odd number, so that the same number of points z k to exist on both sides of z 1 z 2 on the right and left of y -axis, with the mid z k -point (i.e., the z k for k =( n +1)/2) being located at the intersection of z 1 z 2 with y -axis. To solve the problem, the method of conformal mapping is adopted. In fact, the actual disc of radius R is mapped onto the unit disc in the ζ = ξ + i η = ρ e i θ plane, as it is shown in the above Fig.4. The mapping is described by the relation:   z R      (2) The mapping is implemented in conjunction with supplementary conditions that should be satisfied by the analytic functions sought. These conditions are deduced from well known properties of Cauchy integrals (Muskhelishvili 1963). In this context, Muskhelishvili’s complex potentials for the solution of the present problem read, in terms of ζ , as:



 

 

   

  

s s

s   

 

iP

iP

k      

n



 





 

1

2

log

log

1 2 3 4 s s s s      





 

 



 2 1



o

(3)

2

3 s due to forces at z z z z     4 1 2 3 4 , , , 2

  



1

k



k









, due to forces at z z

k k



 

 

   



due to forces at

s

s

 

 

 

 



  

iP

s

s

s

s

 

1

2

log

 











1

2

3

4

    



1 2 3 4 , , , z z z z

2

s

s

s

s

s

s



















(4)

3

4

1

2

3

4

   

   

  

  

1

i P 

  





n



  



log

, due to forces at z z

 









k

k

k

 2 1



o

k k

  

         

1

k



k

k

k

where φ ο ( ζ ) and ψ ο ( ζ ) are subjected to determination. In the above formulae, s 1 = e i Θ , s 2 = – e – i Θ , s 3 = – e i Θ , s 4 = e –iΘ , are the points on the unit circle γ ( ρ =1), corresponding through Eq.(2) to the points z 1 , z 2 , z 3 , z 4 on the actual disc’s periphery L. Analogously, points , k k   in the interior of γ correspond to the internal points , k k z z of the actual disc, i.e., the points of application of the point forces (0, ) P  . In addition, κ is Muskhelishvili’s constant, equal to (3–4 ν ) for plane strain conditions, and (3–ν)/(1+ν), for plane stress ones, with ν denoting the Poisson’s ratio of the material. The functions φ ο ( ζ ) and ψ ο ( ζ ), holomorphic everywehere in the unit disc, are obtained from the second problem, i.e., that of the elastic equilibrium of the disc under vertical point forces at the internal points , k k z z . It can be seen, after some lengthy algebra, that these functions are eventually expressed as:

   

   

  

  

1 1

1

iP

 

       





n







 

(5)

log









   

 





k

k

k

 2 1



o

k

k

1

1

2

  

k  

k  

k  





1

k



   

  

         

  

1 1

1

iP

 

k     k

k 

n



 

log







    



 

k

k

k

 2 1



o

1

1      

1

1      

  





1

k



k

k

k

k

(6)





   





1

  

    

1

  

    

1



  





k

k

k

k

k

k

















k

k

2

2



1



k  

1



k  