PSI - Issue 10

Ch.F. Markides / Procedia Structural Integrity 10 (2018) 163–170 Ch.F. Markides / Structural Integrity Procedia 00 (2018) 000 – 000

167

5

4 ( ), m W x C Φ W y C Φ              4 ( ) x m y

(13)

, f r

, f r

, y W W   , considering also the in plane deformation of the specimen (Kourkoulis et al. x

Resent formulae for

f

f

(2013)) are easily seen not to influence the extent of the contact length considered here. Under the above assumptions, the Jacobians of the transformations of Eqs. (13), coincide:   2 2 , 4 ( ) f r m J J C Φ      

(14)

and it is known that for J =0 optical mappings between P f and P f ΄ and between P r and P r ΄ cease to be one-to-one so that light received at the points P f ΄ and P r ΄ on the screen will emanate from reflections from more than one point P f and P r on the specimen, leading to the creation of the caustics. In this context, setting J =0 in Eqs. (14), yields:

4 ( ) m C Φ    

(15)

which solved for r ( r =| ζ |=( x 2 + y 2 ) 1/2 ) provides the radius r o of the so-called initial curves, i.e., the loci of points P f , P r on the disc providing the caustics on the screen; clearly, under the present assumption of no refraction, initial curves on the front and rear face are similar. Inserting r o in Eqs. (13), yields the parametric equations of the front and rear caustics (colored red and green in Fig. 2b). Substituting in Eqs. (15) and (13), from Eqs. (2), for Φ ( ζ ), the radius of the initial curves and parametric equations of caustics are obtained as (assuming λ m ≥1) :

2

4 3

(16)

o r

* cos 2 cos 2 1 (2 ) C     

( ) 



     

  

  

  

1 2   

2 3

(17)

W

r

r C

cos

* (2 ) sin











, f r x 

m o

o

2

  

  

  

1 2   

2 3

(18)

W

r

* C r C 

sin 2 

* (2 ) cos

  





, f r y 

m o

o

2

1,2 - i  defined in Fig. 1b. The above

where C *= C /(3 λ m KR ) and use has been made of the auxiliary variables ζ 1,2 = r 1,2 e formulae are, for the subscript f , according to Theocaris and Stassinakis (1978).

4. The standard procedure for estimating the contact length 2 ℓ by measuring the distance D Setting θ = π and 0 in Eqs. (16-18), for the front face, Theocaris and Stassinakis (1978) obtained the coordinates of the end points A , B of the initial curve and the corresponding points E f , C f of the respective caustic (Fig.2b), as:

2 3

(19)

x

* 1 (2 ) , C

y

0

 



, A B

, A B

2 3

[





*] 2 , ( C r x 

)

(20)

x

W

, m o E C r y  ,

W

r C

(

)    

(

)    

* (2 )

















, E C x

y

m o

, o A B

f

f

f

f

f

f

whence two formulas were derived for the contact length as follows:

2

2 3 4 3

2

2

2 ) f C 

(21)

* (2 ) C

D

D x 

(4 ) 0, (  





m