Issue 50

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

which agrees with the respective formula derived in [26] (after some minor revisions in [26], see [49], and few modifications introduced here for the sake of generality of the procedure proposed). In Eq.(17), the use of the notation ℓ + D,f,r,t , instead of simply writing ℓ , has been made to stress the distinction among ℓ + D,f , ℓ + D,r and ℓ + D,t (corresponding to D + f , D + r and D + t ) when obtained experimentally. Actually, while in theory all three ℓ + D,f , ℓ + D,r and ℓ + D,t equal ℓ given by the second of Eqs. (1) thus fulfilling Eq.(17) by identity, in practice reasonable differences between the theoretically predicted and experimentally measured D + f , D + r and D + t lead inevitably (via Eq.(20), see below) to “experimental” values for ℓ + D,f , ℓ + D,r and ℓ + D,t , slightly different to each other and from the theoretical ℓ . Thus, in applying the experimental method of caustics to obtain ℓ D there will be in general three different values for ℓ D , viz., ℓ + D,f , ℓ + D,r and ℓ + D,t which should be properly combined to provide the experimental ℓ D . Now setting:





3 2



m

(

)



(18)

, , , D f r t

, , f r t

Eq.(17) becomes:

2 3

2

   

   



   

   

C

D



ρ

1

, , f r t

, , f r t





3

2

(19)







m

m

(

)

2

(

)

0

, , f r t

, , f r t

λ

KR

λ

2

, , , m f r t

, , , m f r t

Eq.(19) admits two complex and one real solution, the latter of which, upon inserted into Eq.(18), yields ℓ + D,f,r,t as:

    

 

2 3

2

2

   

   

   

   



 

   

C

D

C





ρ

ρ

1 3

1

1

1

1

, , f r t

, , f r t

, , f r t



 







2

2



   

, , , D f r t

λ

KR

λ

λ

KR

2 2

27

, , , m f r t

, , , m f r t

, , , m f r t



1 3 1 2 2 2

 

    



4

   

   

2

   

   

   

   



   

   

C

D

C





ρ

ρ

1

1

1

1

1

       

, , f r t

, , f r t

, , f r t









2

2

λ

KR

λ

λ

K

R

729

2 2

27

, , , m f r t

, , , m f r t

, , , m f r t

(20)

2

 

2

   

   



 

   

D

C



ρ

1

1

1 2

, , f r t

, , f r t





   

λ

λ

KR

2 2

27

, , , m f r t

, , , m f r t



3 2 1 3 1 2 2 2

 

    



   

   

4

2

   

   

   

   



   

   

C

D

C





ρ

ρ

1

1

1

1

1

            

, , f r t

, , f r t

, , f r t







2

2

λ

KR

λ

λ

KR

729

2 2

27

, , , m f r t

, , , m f r t

, , , m f r t

According to the method of caustics, measuring the distance D + f,r,t that value in Eq.(20), will yield the half contact length ℓ + D,f,r,t

on the experimental caustics’ photos and introducing in question. Of course, as it has already been mentioned,

reasonable differences between the theoretical and the experimentally measured D + f,r,t approximations and experimental errors, will lead (via Eq.(20)) to slightly different values ℓ + D,f , ℓ + D,r and ℓ + D,t , in which case ℓ + D can be taken as the average value of the three experimental values ℓ + D,f , ℓ + D,r and ℓ + D,t : , attributed to both theoretical

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

) 3

(21)

In complete analogy with the previous case, solving Eq.(16) for ℓ , one takes:

461