PSI - Issue 10

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

168

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

6



 3 2 2 ) 1] 1 , (  

C 

(22)

* * 2 [ (2 C H C

H y 

)





m

f

(actually, in Eq. (35) of the aforementioned paper - the present Eq. (21), brackets about the term 2 C * are missing while the power of D was inadvertently taken 4 instead of 2). Then according to the standard procedure, inserting the distance D between E f and C f , measured on caustics photos, in Eq. (21), yields ℓ . Eq. (22), though simpler, is not used due to the difficulty in specifying the line y ΄ =0 and elevation H of the points E f , C f on caustics photos (Fig.2b). 5. Two ways to specify y ΄ =0 line, the elevation H and the contact length 2 ℓ by the linear formula of Eq. (22) 5.1. 1 st way: from measuring the vertical distance between the points C f and C r That procedure is based on measuring on caustics photos the distance between the starting points C f and C r of the front and rear caustic. Namely, setting in Eqs. (17, 18) θ =0, yields the coordinates of these points as:  (23) whence it is seen that C f C r is for any λ m a vertical segment, bisected (under the consequence of w f = – w r ) by the y ΄ =0 line, equalling 2 H (Fig.3a). Thus, measuring C f C r on the photos and inserting C f C r /2= H in Eq. (22), ℓ is directly obtained. C r can be easier detected on photos by slightly turning the optical setup with respect to the specimen (in xz -plane) so that the rear caustic will shift relatively to the front one in x -direction, letting C r be seen more clearly. (a) (b) , f r 2 3 * )(2 ) C * ( )    (   ), ( )    (   2 f r f r f C C x x x W   m o C  C y y W m o r y r C                  

C f

y ΄

Front caustic

Rear caustic

Front caustic

Rear caustic

C f

H

Ο΄

( y ΄=0 )

H

x ΄

M r

( y ΄=0 )

r o (π /2)

x ΄

C r

Projection of initial curve

( M f M r )/ 2

M f

Tangent of the initial curve

λ m =1

λ m ≠1

Fig. 3. (a) Obtaining y ΄ =0 from ( C f C r )/2; (b) Obtaining y ΄ =0 from ( M f M r )/2.

5.2. 2 nd way: by the aid of points M f and M r In that procedure the mid-points M f and M r of the front and rear caustic are necessary, and as it is significantly simplified in case of parallel incident light, setting in Eqs. (17, 18) λ m =1, θ = π /2 their coordinates are obtained as:

r C 

1 2 

2 3









(24)

M M x x W   x

M M y y W    y

C

(

2) 0,

(

2)   

* (2 )

  

 





*









o

f

r

, f r

f

f

f

showing that M f M r is a vertical segment, bisected (under the consequence of w f = – w r ) by the initial curve. Thus, drawing on caustics photo a horizontal line bisecting M f M r , one actually has the tangent of the initial curve at its mid-point. Next, drawing from that tangent, vertically upwards, the segment r o ( θ = π /2), obtained from Eq. (16):

2 3

1  (with ℓ here the theoretical value, predicted by the third of Eqs. (2)) (25)

o r

* (2 ) C

(

2)    

one obtains the line y ΄ =0. Then, H is directly measured on the photo from C f downwards (or C r upwards) to the line