Issue 50

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

from the span

Figure 7: The general case for double reflected and transmitted caustics.

of a double or single caustic while two different ways are proposed to locate X f,t -axis on the experimental caustics’ photos, necessary for the use of the existing linear formula [26] for obtaining ℓ from the elevation of caustic’s extreme points. Obtaining the contact length from the span of the general contact caustic curves Let us first consider the more general case of double caustics. In this context, the spans of the caustics, i.e., the distances D + f,r,t between the outermost ends E + f,r,t and L + f,r,t of the caustic curves and the distances D – f,r,t between the innermost ends E – f,r,t and L – f,r,t (Fig.7) are defined by the aid of the first of Eqs. (13), as:

 2 3

   

C

 ρ

1

, , f r t

  2 ( f r t D X θ  , , f r t , ,

  

(15)

 

 1 2

λ

0) 2



, , , m f r t

λ

KR

, , , m f r t

2 3

   

   

1 3

   

   

2 3

λ

C





 ρ

 ρ

1

1

  

  

, , , m f r t

, , f r t

  2 ( f r t D X θ  , , f r t , ,

 



 1 2

λ

C

0) 2

2



(16)

, , , m f r t

, , f r t

KR

C

λ

KR

, , f r t

, , , m f r t

The superscripts + and – in the above formulae, are in accordance with the previous discussion and refer to the value respectively, describing the radius of the corresponding double initial curve. In this context, the abscissas, r  and , , , o f r t r  , , , o f r t

, , ( 0) f r t X θ   and

, , f r t X θ    , of E + f,r,t ( π)

r 

i 0

, , , e o f r t

and L + f,r,t

, correspond respectively to the outermost points

and



, , ( 0) f r t X θ   and



, , , e o f r t 

-i

π)   , of E – f,r,t 

r

X

(

and L – f,r,t

of the relevant initial curve, while the abscissas,

, correspond

, , f r t



r 

, , , e o f r t 

i 0

-i

r

, , , e o f r t

respectively to the innermost points

and

of the same initial curve. What is more, there will be in for ℓ upon using Eqs. (15) and (16), need to be combined

and ℓ – D

general two (slightly) different experimental values ℓ + D

, i.e., the half contact length ℓ obtained from the span D . In this direction, solving Eq.

to yield the final experimental ℓ D (15) for ℓ , one can write the expression:

2 3

2

   

   



   

   

C

D



ρ

1

, , f r t

, , f r t





4 3

2

(17)







(

)

2

(

)

0





, , , D f r t

, , , D f r t

λ

KR

λ

2

, , , m f r t

, , , m f r t

460