PSI - Issue 2_A

Christos F. Markides et al. / Procedia Structural Integrity 2 (2016) 2659–2666 Christos F. Markides, Stavros K. Kourkoulis / Structural Integrity Procedia 00 (2016) 000–000

2661

3





 





  P P , P 0      

(2)

2 P 1 sin            P

2 sin ,  





r

c

o

o

c

max

The elastic equilibrium of that transtropic disc is to be determined. In this context, the disc’s cross-section is con sidered lying in the z=x+iy=re iθ complex plane (Fig. 2). Assuming that w is comparable to R, plane strain conditions are adopted. Taking now into account Lekhnitskii’s approach for anisotropic cylindrical bodies, with their faces being planes of elastic symmetry (Lekhnitskii 1981), the equilibrium equations, Hooke’s generalized law, the equations of compatibility and the respectice boundary conditions, for zero body forces, are reduced respectively to:









(3)

xy  

xy

0,

0

x  

x

x y

x y

 

 

x                             11 x 12 y y 12 x 22 y xy u v y x , ,   u x v y

(4)

66 xy

2

2

2

                       11 x 12 y 12 x 22 y 66 xy 2 2 x y y x  

(5)

0

 

  cos n, y X , 

 

  cos n, y Y 

(6)

cos n, x

cos n, x



 



 

x

xy

n

xy

y

n

where u, v are the Cartesian components of displacement, X n , Y n are the components of σ r on L and β ij =α ij –α i3 α j3 / α 33 are the so-called reduced elastic constants, which from Eqs.(1) equal:

2

1

1

1

1

 

 

  

(7)

2 ΄ , 

,

΄

,

1

11  

   

   

66  

 

12

22

E΄

E΄

G΄

E

E΄



According to the theory of elasticity, Eqs.(3) imply the existence of the Airy’s function F(x,y) so that:

P frame

σ r = – P(θ)

Plane of elastic symmetry

Planes of isotropy

x

y

2ω ο

ϕ ο

θ

Ε΄ ν΄ G΄

z=re iθ

L

R

r

w

y

O

x

z

Ε, ν, G

2ω ο

Ε, ν, G

z

– P frame

Fig. 2. The isolated transtropic disc under parabolic pressure: Configuration of the problem and definition of symbols.