PSI - Issue 10

Ch.F. Markides / Procedia Structural Integrity 10 (2018) 163–170

166 4

Ch.F. Markides / Structural Integrity Procedia 00 (2018) 000 – 000

 

 

(3)

,     W O P w W O P w

f

f

r

r

where, given the equations of the faces at the points P f , P r as F f ( x , y , z ) = z – g ( x , y ) = 0, F r ( x , y , z ) = z + t + g ( x , y ) = 0, it is readily seen that ( i , j are the unit vectors along x-, y-axis, respectively):

(4)

w

n w

( , )] tan 2 }  n

Z g x y

Z t g x y

{[  

( , )] tan 2 } , 

{[   

f

o

r

o

w

w

f

r

  

  

2 ( , ) g x y 

g x y g x y

( , ) ( , )

 

x y     i

(5)

n

    n

j

, tan 2

,



 





w

w

2

f

r

1 ( , ) g x y  

with w f n , w r n being the unit vectors along w f , w r . Combining Eqs. (3-5) and considering that O ΄ P ΄ = OP , yield:

g x y

g x y

( , ) g x y

( , ) g x y





(6)

W OP

W OP

Z g x y

Z t g x y

2[   

( , )]

,

2[    

( , )]

f

o

r

o

2

2

1 ( , )  

1 ( , )  

in agreement, for W f , with Rosakis and Zehnder (1985). Moreover, neglecting t and g , compared to Z o , Eqs. (6), written in compact form, reduce to (implying now that w f = – w r , as a consequence of zero refraction in the disc):

(7)

 W OP

2 ( , ) o Z g x y 

, f r

agreeing, in W f , to the formula of Theocaris (1981). In turn, considering plain stress conditions:

(8)

( , ) g x y t

2             2 ( ) (2 t t

z

x

y

where Δ t is the thickness change and ε z is the strain component in the z -direction, Eqs. (7) become:



(9)

W OP

o Z t

(

)

  

 



, f r

x

y



Moreover, using Muskhelishvili’s stress function Φ of the complex variable ζ = x +i y = r e i θ (  is the real part):

(10)

4 ( ) y Φ       x

and expressing OP in complex form as ζ = x +i y , y <0 (Fig. 2b), Eqs. (9) are written in complex form (over-bar denotes the complex conjugate and prime the first derivative):   , 4 ( ), f r o W C Φ C Z t         (11)

or, when λ m ≠1 (in which case in place of C in Eqs. (11) it must be understood the value C ΄ = Z o t ν /( λ m Ε )):

4 ( ) C Φ  

(12)

W

m    

, f r

Separating in Eq. (12) real from imaginary (  ) part, the parametric equations of the optical mapping between P f ( x , y ), P r ( x , y ) and P f ΄ ( f x W  , f y W  ), P r ΄ ( r x W  , r y W  ) (with , f r x W  , , f r y W  the components of W f,r ), read as: