Issue 41

Ch. F. Markides et alii, Frattura ed Integrità Strutturale, 41 (2017) 396-411; DOI: 10.3221/IGF-ESIS.41.51

2 E ΄

2

2

( ) 3 4 ( ) 3 4        









(3)

cos

sin





o

o

2

E΄

In the above formulae, κ is Muskhelishvili constant [19] while index J indicates the jaw. Alternatively, ignoring the contact problem, one may fix arbitrarily the length of the loaded arc, i.e., the value of ω ο , in which case (keeping P frame constant) [16] it would be written:

2 2 sin sin 2 2 cos 2 frame o P    

3 4 sin P Rd

frame





or

(4)

P

P

c

c





o 

o 

Rd

o

o

(the two expressions of Eq.(4) are equivalent, especially for small to moderate ω ο

-values, with the latter, however, being

the most adequate one in the complex form representation of the resultant force in [19]). Under the above considerations and assuming, also, that both the displacement and rigid body rotation of the disc center arte equal to zero, the displacement field on any point of the “transtropic” disc cross-section was recently obtained [14] by employing Lekhnitskii formalism [18, 20]. According to that solution [14], the Cartesian components of the displacement field were found to be described by the following equations (where  indicates the real part):



  0

  0

 

 

 





0 66









11   x

12  



, u x y

x

y

y

xy

2

(5)

    

    



   



   





  1 3 13 1 p A P z 

 

 

 









m m A P z

2 3 23 2 p B P z 

m m B P z

2

1 1

2 2

 

 





m

m

5,7,9,...

5,7,9,...



  0

  0

 

 

 





0 66









12   x

22  



, v x y

y

x

y

xy

2

(6)

    

    



   



   





  1 3 13 1 q A P z 

 

 

 









m m A P z

2 3 23 2 q B P z 

m m B P z

2

1 1

2 2

 

 





m

m

5,7,9,...

5,7,9,...

In the above formulae, z j

, j =1,2, are the so-called complicated complex variables (in contrast with the ordinary complex

variable z=x +i y = r e i θ , Fig.1), defined as:

or

(7)

, z x i y z x i    

1 

2 

1 z r 





1 i r 

sin , 

2 z r 





2 i r 

sin , 0 

 

y

r R

cos

cos

1

2

where:

2



  



2   





  

 

2

4

1  2 

  

12 66

12 66

11 22















(8)

,



1 2



2

11

and β ij

are the so-called reduced elastic constants defined as:

2









2   

1

1

1

1





΄ 



΄ 





 

1  



(9)

,

,

,

 

 

11

12

22

66

E

E΄

E΄

E΄

G΄

Moreover,

2

2

1 11 1   







 





1 12 1 22 1   q    



 







p

p

q

(10)

,

,

,

12

2 11 2 12

2 12 2 22 2

with μ j

, j =1,2 being the so-called complex parameters reading as:

(11)

1 



1 , i 

2 



2 

i

399