PSI - Issue 3

Christos F. Markides et al. / Procedia Structural Integrity 3 (2017) 334–345 Christos F. Markides, Stavros K. Kourkoulis / Structural Integrity Procedia 00 (2017) 000–000

341

8

Eventually, combining Eqs.(27-29) and Eqs.(41) standing for displacements due to the terms  1 1 1 2 1 2 2 q A z q B z   , provides the complete displacement field of the orthotropic disc as:         0 0 66 0 u x, y x y    



1 1 1 2 1 2 2 p A z p B z ,  

11 x              12 y xy 2

(42)

    

 

  

  

  

  





  1 3 13 1 2 p A P z   

  A P z p B P z    m 1m 1 2 3 23 2

 

m 2m 2 B P z





m 5,7,9,... 

m 5,7,9,... 



 

  0   xy              66 0 12 x 22 y y x 2    0

v x, y

(43)

    

 

  

  

  

  





  1 3 13 1 2 q A P z   

  A P z q B P z    m 1m 1 2 3 23 2

 

m 2m 2 B P z





m 5,7,9,... 

m 5,7,9,... 

Finally, notice the following two choices for values of angle ω ο (the half-loaded rim on L) and P c (the amplitude of parabolic pressure), entering previous formulae via Eqs.(24-26): (a) ω ο can be arbitrarily predefined; in that case P c will read as (Markides & Kourkoulis, 2012):

 Rd sin 2 2 cos 2       frame 2P sin o 2

(44)

P



c

o

o

o

(b) ω ο and P c are obtained by resorting to the intact disc-ISRM’s jaw related contact problem; in that case it will, approximately, hold that (Markides & Kourkoulis 2012):

  6K P 

  o   

1

3 P 

1

 

     o o

 

  o

(45)

o frame

Arcsin

, P

, K

frame

J

 

 



  o 32K Rd 

c o

Rd

4G 4G



xy

J

It is mentioned here that the respective equations obtained by Markides & Kourkoulis (2012), which constitute the basis for deriving Eqs.(45), concerned isotropic materials for both the disc and the jaw. Therefore ω ο and P c were there considered independent of the inclination angle ϕ o of P frame . In the present case, however, orthotropy dictates different elastic response of the disc for various ϕ o -values so that ω ο and P c in Eqs.(45) must be expressed in terms of ϕ o . In the same formulae, κ(ϕ o )=3–4ν(ϕ o ) and κ J =3–4ν J stand for Muskhelishvili’s (1963) constants (in plane strain) for the disc and jaw respectively, as if both were made of isotropic materials for each particular ϕ o -value; moreover, ν stands for Poisson’s ratio while G xy and G J are the disc’s and jaw’s shear moduli. An example of

determining ω ο (ϕ o ) and P c (ϕ o ) according to Eqs.(45) is given later in the applications section. 3. Particularization of the solution in the case of the Transversely Isotropic disc

Due to its practical interest, the case of the transversely isotropic disc, called for brevity “transtropic”, is studied in this section, as a particular case of the orthotropic disc. In this context, introducing in Eqs.(6) the substitutions:

(46)

E E E, E E΄, G G΄,   

,

΄

       

x

z

y

xy

zx

yx

yz

one obtains:

2

1

1

1

1

 

 

  

(47)

2 ΄ , 

΄

,

1

,

11  

  

   

66  

 

12

22

E΄

E΄

G΄

E

E΄



for the reduced elastic constants of a transtropic disc such that: (a) all of its planes parallel to xz-plane are planes of isotropy constituting the so-called strong direction with Young’s modulus E and Poisson’s ratio ν, (b) all of its planes normal to the planes of isotropy constitute the so-called weak direction with Young’s modulus E΄ and shear modulus G΄ and (c) tension/compression in y-direction causes contraction/dilatation of xz-planes of isotropy