Issue 50

Ch. F. Markides, Frattura ed Integrità Strutturale, 50 (2019) 451-470; DOI: 10.3221/IGF-ESIS.50.38 466 through standardized uniaxial tests in rectangular specimens of dimensions (25x2x0.2) cm, of the same batch of the material, resulting to a stress-optical constant c f =1.19 x 10 –7 m 2 /kN (first of Eqs. (5)). In addition, for the particular batch of the material, k was estimated equal to 1.50, resulting to a c r =1.79 x 10 –7 m 2 /kN (second of Eqs. (5)). The experimental protocol In this first series of experiments the parameters of the experimental/optical set-up were properly chosen so that double transmitted and single reflected caustics will be obtained. The choice of receiving the transmitted double caustics instead of the reflected ones is to be attributed to the fact that the revealing for example of the double front reflected caustics would require a shorter distance Z i between the focus point of the second lens and the specimen or a high level of the externally applied load. In the first case, as the focus of the impinging divergent light is too close to the specimen, so that the area illuminated is very confined, quality of caustics is not very good with secondary optical phenomena, difficult to describe, shadowing in most of the experimental attempts undertaken the results. In the second case, the load would be required to obtain the double reflected front curves would be, unfortunately, for the particular material, too high to suffice the material would remain in its linearly elastic state, a crucial assumption in the deduction of the analytic formulae (see also the discussion in the concluding section). In this context, and for a k =1.5 for the particular batch of the material (second of Eqs. (5)), the following settings were made: Z o,f =1.0 m, Z o,t =0.2 and Z i =0.2 m. In addition, the load level was quasi-statically increased from zero up to P frame =3 kN. During loading, a series of photos of caustics taken from both the front and rear screens were obtained (Fig.9). Namely, in Figs. 9(a1, a2), 9(b1, b2) and 9(c1, c2), the photos of reflected and transmitted caustics are shown for three specific loading levels, P frame =1 kN, 2.5 kN and 3 kN, respectively. Then, by the aid of these experimental results and the previous analytic formulae, the contact length was obtained from the expressions involving the spans D ± f,r,t of the caustics that were available/measurable on the photos of experimental caustics (the formulae involving the elevations H ± f,r,t were not used here). Actually, the experimental results, viz., the distances D ± f,r,t used for estimating the contact length, were exclusively pumped out of Figs. 9(b1) and 9(b2). In the other figures, apart from presenting the gradual evolution of caustics upon load increment, relevant information is given for clarity, as for example the dimensions of the 2- and 1-cent of euro coins attached to the front and rear screens, respectively, for the necessary measurements to be feasible. Namely, from Fig.9(b1) it is found that D + f =7 cm and D + r =8.5 cm, while from Fig.9(b2) it is seen that D + t =2.4 cm and D – t =0.3 cm. Then introducing the values measured, D + f =7 cm, D + r =8.5 cm and D + t =2.4 cm in Eq.(20), the experimental values for ℓ + D,f , ℓ + D,r and ℓ + D,t are calculated as ℓ + D,f =0.41 cm, ℓ + D,r =0.39 cm and ℓ + D,t =0.46 cm, respectively, whence by Eq.(21) one takes the experimental value ℓ + D =0.42 cm. On the other hand, introducing the value D – t =0.3 cm in Eq.(25), the experimental value for ℓ – D,t is calculated as ℓ – D,t =0.43, whence Eq.(26) obviously yields again ℓ – D =0.43 cm (since the transmitted caustic is here the only double one). Finally, combining ℓ + D =0.42 cm and ℓ – D =0.43 cm, Eq.(27) provides here the experimental value ℓ D =0.425 cm for the half contact length, which is in a quite good agreement with the respective theoretical value ℓ =0.388 cm, provided by the second expression of Eqs. (1), for the particular data. C ONCLUSIONS n the present paper, an effort was undertaken to describe the nature of double initial and caustic curves, generated in the contact region realized between two cylindrical bodies when compressed against each other, and then based on the above description to extend/complete existing formulae [26] for obtaining the contact length. As it is shown, single curves are actually a particular case of double ones with the latter corresponding to the whole range of possible solutions of Eq.(7) that provides the radius of the initial curve. Namely, Eq.(7) admits in general three solutions, a real, a complex and a purely imaginary one (Fig.4b), where of course only the real one can be justified to stand as the radius of the initial curve. It is seen that in that case the initial curve splits into two parts, the left and the right, each one of which further consists of two branches the outermost and the innermost (Fig.4c). In the limiting case where there is only a real solution to Eq.(7), the initial curve consists of one part with one branch, which is the usual case of single initial and in turn caustic curve. In the same direction, it was shown that under certain assumptions six independent parameters can be distinguished influencing the occurrence of double or single curves, viz., the load level, the material’s compliance and various features of the optical set-up. In this context, it was shown that is not always an extended contact length responsible for the appearance of double curves; for example, a small distance Z i between the focus point of the last (2 nd ) lens and the specimen or a small distance Z o,t between the specimen’s cross-section and the rear reference screen may lead to double curves even in the case of low load levels and small contact lengths. Namely, as it was seen from the experimental protocol, similar single and double curves exist simultaneously and it is up to the particular choice of the I