Issue 50

Ch. F. Markides, Frattura ed Integrità Strutturale, 50 (2019) 451-470; DOI: 10.3221/IGF-ESIS.50.38 464 Obviously, obtaining ℓ H in the case of single caustics presents no difference from the case of double ones, since only the elevations H + f,r,t are used in both cases, that is, Eqs.(30,31) hold also true in the case of single caustics. In conclusion, having ℓ D from Eq.(27) (see previous paragraph) and ℓ H from Eq.(31), the final “experimental” ℓ exp can be quite accurately defined as:      exp ( ) 2 D H (32) There of course remains the problem of specifying the X f,t -axes positions on reflected and transmitted caustics’ photos in order that H + f,r,t can be measured. Specifying the position of X f,t -axes on reflected and transmitted caustics’ photos To measure H + f,r, (required in Eq.(30) to obtain ℓ H,f,r,t ), locating the positions of X f,t -axes on caustics’ photos is sine qua non. In this context, two ways are next proposed to locate X f,t -axes. Namely, in the case of double caustics, it has been seen that Eq.(29) provides the elevations H – f,r,t of the innermost end points E – f,r,t or L – f,r,t (Fig.7) of the double caustics independently of ℓ H,f,r,t ,; actually, H – f,r,t depend only on the experimental set-up. Thus, detecting on caustics’ photos the innermost end points E – f,r,t (or L – f,r,t ) of double caustics and then drawing from these points normally the respective ℓ H,f,r,t - independent elevations H – f,r,t , X f,t -axes are automatically obtained on the photos independently of ℓ H , and thus they can be safely considered as a basis in obtaining ℓ H,f,r,t ,. Indeed, having X f,t -axes on caustics’ photos, measuring H + f,r,t and in turn obtaining ℓ H,f,r,t , and ℓ H from Eqs. (30) and (31) is a trivial procedure. Obviously, the higher the number of detectable points E – f,r,t (or/and L – f,r,t ) on the photos is, the more accurate the location of X f,t -axes on caustics’ photos will be. Clearly, that simple approach does not apply to single caustics since in that case there are no any innermost ends E – f,r,t or L – f,r,t . However, there is an alternative approach to specify X f,t -axes on caustics’ photos graphically, at least in a first approximation, applicable to both cases (double and single caustics). Namely, it is seen that X f,t -axes pass approximately from the intersection of the lines E + t L + f or E + f L + t with the vertical symmetry axes of caustics, i.e., the Y f,t -axes either in the case of double (Fig.8a) or single (Fig.8b) caustics. Figure 8: Obtaining graphically the X ft -axes, (a) in the case of double and (b) in the case of single caustics. Of course, the points L + f and E + f are located on the front screen while the points E + t and L + t are located on the rear one; thus, in order that these points can be combined to form the above lines, the points E + t , L + t ( E + f , L + f ) from the rear (front) photos should be properly specified on the front (rear) photos containing the points E + f , L + f ( E + t , L + t ). For example, the equation of the line E + t L + f (in both cases of Fig.(8a,b), which is of the form Y f,t = αX f,t + β , with α the slope, and β the ordinate of the point the line E + t L + f intersects Y f,t -axis, is given as:                                                           , , , f t f t f t f t t t f f f t f t f t f t L E L E L E L E f t f t f t E E L L L E L E L E L E Y Y Y Y Y Y Y Y Y X Y X X Y X X X X X X X X X α α β β (33) where