Issue 50

Ch. F. Markides, Frattura ed Integrità Strutturale, 50 (2019) 451-470; DOI: 10.3221/IGF-ESIS.50.38 465       2 3 , , 2 3 1 3 2 3 , , Eq. 13(a) Eq. 13(b) 1 ( π) 1 2 1 1 1 ( π) 2 2 1 2 t t t t m t E m t m t t t t E t m t C ρ X X θ λ λ KR λ C ρ ρ ρ Y Y θ C KR KR C λ KR                                                           (34)       2 3 , , 2 3 1 3 2 3 , , Eq. 13(a ) Eq. 13( b) 1 ( 0) 1 2 1 1 1 ( 0) 2 2 1 2 f f f f m f L m f m f f f f L f m f C ρ X X θ λ λ KR λ C ρ ρ ρ Y Y θ C KR KR C λ KR                                                        (34 cont) Usually, β is too small so that in a first approximation, it can be considered zero, in which case the line E + t L + f passes from the origin thus defining the X f,t -axes. So having X f,t -axes and any of the outermost end points E + f,r,t (or L + f,r,t ) on caustics’ photos, H + f,r,t can be directly measured and in turn ℓ H is obtained by the aid of Eqs. (30) and (31). E XPERIMENTAL PROCEDURE series of experiments were implemented to assess the efficiency of the previously provided general formulae and approach for obtaining the contact length in the case of double caustics. In this context, taking advantage of the previous analysis with regard to the conditions for the development of double curves, the imposed loading and features of the optical set-up were suitably chosen so that double caustics can occur, trying at the same time not to exceed the linearity limit of the material of the specimen. It is to be mentioned that the experimental protocol described below is only a first step of an ongoing research the complete results of which will be presented in a future work. The experimental arrangement The experimental set-up is shown in the sketch of Fig.2. It was consisted of a He-Ne laser tube emitting a narrow beam of red coherent light (of a wave length about 600 nm), refined and spread out as passing through a pin hole of 50 μm mounted before the output coupler of the laser. The outspread light is received by a first lens transforming it into a parallel wide coherent light beam that converges again as passing through the second lens. Moving the second lens back and forth its focus point may be located at the desired distance Z i in front of or behind the cylindrical specimen, resulting to a divergent or convergent incident light beam respectively and defining at one’s convenience the magnitude and sigh of the magnification factors λ m,f,r,t (Eqs. (3)). A semi-reflector, placed between the second lens and the specimen at an angle 45 o with respect the specimen’s cross-section, facilitates receiving the reflected light on the front reference screen; in that case Z o,f is the total distance: specimen’s middle cross-section – semi-reflector – front screen. Light passing through the transparent specimen is received on the rear reference screen, placed directly behind and parallel to the specimen’s middle cross-section at a distance Z o,r . The specimen was fixed within the jaws of the ISRM apparatus for the implementation of the Brazilian-disc test which was mounted at an electromechanical INSTRON 1125 loading frame of 50 kN capacity. The movable traverse of the loading frame moved downwards, at a rate of 0.1 mm/min, compressing the upper jaw of the ISRM apparatus, and in turn the cylindrical specimen. Normality of the loading axis was ensured by a semi-spherical head interposed between the traverse of the loading frame and the jaw bearing on its upper face a suitably perforated hollow cavity fitting the semi- spherical head. The lower jaw was placed on a 50 kN compression load cell calibrated with a verified Wykeham Farrance compression ring of 10.62 N sensitivity. The cell exhibited a linear behavior for the whole loading range with a deviation less than 0.2% while the displacement rate was also calibrated by a High Mag micrometric calibrator, exhibiting also a linear behavior with a deviation less than 0.4%. Regarding the specimen, it was a cylinder of radius R =5 cm and thickness t =1 cm, made of PMMA with Young’s modulus E =3.20 GPa and a Poisson’s ratio ν =0.38. The specific values of the elastic constants of the material were determined A