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

337

4

2

2

2

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

(4)

0

xy dS dS dy dx

xy dS dS dx dy

or, with xy 0   on L, simply as

(5a)

X

, Y

   

   

n

x

n

y

n n r X cos , Y sin       (5b) In the above formulae u, v are the components of the displacement, β ij are the so-called reduced elastic constants, X n , Y n are the components of the stress vector acting at the point z(x,y)=z(R,θ) on L from the side of the outer normal unit vector n and S is the arc length on L; in addition, for the orthotropic disc considered here it holds that: r

2







2

1

1

1



(6)

yx

yz

yz

,

,

z E ,

  

    

  

66  

zx

11

12

zx

22

2 y

E E

E

E

E

G

E

x

z

y

y

y

xy

where E x , E y , E z are Young’s moduli along the respective directions, G xy is the shear modulus in xy-plane and ν ij , i,j=x, y, z, is Poisson’s ratio ruling contraction/dilatation in j-direction for tension/compression in i-direction. As usual, Eqs.(3) imply the existence of the Airy stress function F(x,y) so that:

2

2

2

F

F

F

 

 



(7)

,

,

 

 

  

x

y

xy

2

2

y

x

x y

 

Substitution in Eq.(4) from Eqs.(7), yields the “generalized” biharmonic differential equation in F:   4 4 4 22 12 66 11 4 2 2 4 F F F 2 0 x x y y                Its associated characteristic equation with its four imaginary roots, the so-called complex parameters, read as:   4 2 11 12 66 22 2 0         (9) (8)

(10)

i , i                  

1 i ,

2 i ,

1

2

3

1

1

4

2

2

where the overbar denotes the complex conjugate and for the parameters β 1 and β 2 it holds that:



  



2

12 2    

2

4

     

1      2





(11)

66

12

66

11 22

,     



1

2

2



11

Then, following Lekhnitskii (1981), F is here expressed with the aid of two functions of the so-called complicated complex variables z 1 =x+μ 1 y and z 2 =x+μ 2 y, as (  denotes the real part):       11 1 12 2 F x, y 2 F z F z        (12) with Lekhnitskii’s general complex potentials given as (prime denotes first derivative):         1 1 11 1 2 2 12 2 z F z , z F z       (13) The displacement components are then given as (Lekhnitskii 1981):         1 1 1 2 2 2 1 1 1 2 2 2 u 2 p z p z , v 2 q z q z                   (14) where 2 2 1 11 1 12 2 11 2 12 1 12 1 22 1 2 12 2 22 2 p , p , q , q                       (15) Concerning Φ 1 and Φ 2 , they are here sought in the form proposed by Lekhnitskii (1968) for circular regions, as: