Issue 50

Ch. F. Markides, Frattura ed Integrità Strutturale, 50 (2019) 451-470; DOI: 10.3221/IGF-ESIS.50.38

1     27        

   

4

2

           

   

   

   

   

C

, , f r t C λ

, , f r t C λ

λ

C

ρ  

1

1 6

, , f r t

, , , m f r t

, , , m f r t

, , , m f r t

, , f r t



1 2 

1 2 

 



2

1 2

λ

KR

λ

C

λ

C

, , f r t C λ

, , , m f r t

, , , m f r t

, , f r t

, , , m f r t

, , f r t

, , , m f r t

3 2 1 3 1 2 2 2





  

     



  

  

KRD

1 1 2  

, , f r t

 

             

(25)

4(1 ) 

ρ C

, , f r t



and

          , , , ( D D f D r D t 

) 3

(26)

Eventually, supposing that all D + f,r,t obtained as the average value of ℓ + D

and D – f,r,t and ℓ – D :

are measurable on caustics’ photos, the respective “experimental” ℓ D

will be

       ( ) 2 D D D

(27)

Clearly, in case of simple caustics where only D + f,r,t

, exists, the “experimental” ℓ D

will coincide with ℓ + D

of Eq.(21). It

should be mentioned however, that usually in practice it is not feasible to measure all of the D ± f,r,t to the partially overlapped reflected caustics (front and rear) and thus not all of the experimental ℓ ± D,f,,r,t calculated. Obtaining the contact length from the elevations of the extreme points of the general contact caustic curves In the more general case of double caustics, the elevations H ± f,r,t of the end points E ± f,r,t , L ± f,r,t from Eqs. (20, 25); in that case the final “experimental” ℓ D will be the average value of those ℓ ± D,f,,r,t

on caustic’s photos due

can be calculated that could be

of caustics (Fig.7), will be

defined by the aid of the second of Eqs. (13), as:

    

    

2 3

1 3

   

   

2 3

λ

C







ρ

 ρ

 ρ

1

1

1

  

  

, , , m f r t

, , f r t





(28)

 H Y θ , , f r t , , f r t (

  0)



 1 2

C

2

2



, , f r t

KR KR C

λ

KR

, , f r t

, , , m f r t



ρ

  1 ( 0) 2





 H Y θ , , f r t , , f r t

C

(29)

, , f r t

KR

Eq.(29) does not include ℓ and it cannot be used directly to obtain ℓ , however, it can be proven very useful in specifying the location of X f,t -axes (see below) which is the basic requirement for measuring the elevations H + f,r,t in experimental caustic’s photos. Thus, in the case of double caustics, only Eq.(28) can be directly used to obtain ℓ (from only the elevations H + f,r,t ). In this context, solving Eq.(28) for ℓ yields:

3 2

   

    

2

 KRH ρ

  

   

C

 

1

, , f r t

, , f r t



 

2

1 1



(30)

 2 1

 ρ C  

, , , H f r t

λ

KR

, , , m f r t

, , f r t

which with few inessential modifications (for the sake of generality) is the second formula obtained in [26] for determining ℓ . As previously, writing ℓ H,f,r,t , instead of just ℓ (superscript + has now been omitted for obvious reasons), is to indicate that now ℓ is obtained from the elevations H + f,r,t and to stress the distinction that should be made among the experimental values ℓ H,f , ℓ H,r and ℓ H,t with respect to each other and with respect to the theoretical ℓ of Eqs. (1), due to reasonable slight differences expected between theoretical and experimental H + f , H + r and H + t -values Thus, supposing that H + f , H + r and H + t have been measured on caustics’ photos then introducing these values in Eq.(30), ℓ H,f , ℓ H,r and ℓ H,t are calculated; the final “experimental” ℓ H will be their average value:        , , , ( ) 3 H H f H r H t (31)

463