PSI - Issue 3

Christos F. Markides et al. / Procedia Structural Integrity 3 (2017) 334–345 Christos F. Markides, Stavros K. Kourkoulis / Structural Integrity Procedia 00 (2017) 000–000

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isotropic disc, i.e. δ=1 (Fig.5a), and the disc made of the fictitious material with δ=4 (Fig.5b) were here considered for brevity. The same as previously numerical data were used and the only difference is that the angle ϕ o is now set equal to 45 o . In addition, for clarity reasons, an unrealistically high value, equal to 2500 kN, was assigned to P frame . However, given that the solution introduced is linearly elastic, this choice is not expected to influence at all the results of a comparative consideration of the two cases (isotropic versus anisotropic). Besides the deformed and unde formed boundaries of the discs, the loaded diameter, the diameter normal to it and, also, the diameters corresponding to the end points of the loaded rims, are also drawn in Fig.5, for both the unstressed and the stressed states, in order to clearly show the deformation trend of the discs. As it is expected, in the case of isotropy (Fig.5a), the deformed shape of the disc is completely symmetric. On the contrary, symmetry is lost in the case of the transtropic disc. As it was concluded, by considering also the deformed shape of the serpentinous schist, i.e. the material with intermediate anisotropy ratio δ=2.14 (which is not plotted here), the asymmetry imposed increases dramatically with increasing δ. 5. Discussion and concluding remarks An analytic full-field solution for the displacement field developed in an orthotropic disc, compressed between the jaws of the device suggested by ISRM for the standardized Brazilian-disc test, was obtained. Besides its origin ality, regarding the determination of the complete displacement field in an orthotropic disc under parabolic pressure, the solution introduced is also characterized by two additional innovations, concerning the simulation of the boun dary conditions prevailing along the Brazilian disc - jaw interface: (i) The load exerted to the disc, is described by a parabolic distribution of radial stresses rather than by uniform pressure or by a pair of point forces and (ii) The length of the loaded arc is not arbitrarily pre-defined, but rather it is a function of the relative stiffness of the discs’ and jaws’ materials and also of the inclination angle of the load with respect to the anisotropy planes. These innovations cure some critical deficiencies of previous analytic approaches that consider rather unrealistic and relatively arbitrary conditions on the boundary of the disc. Moreover and in spite of the fact that these innovations made the mathematical problem far more complicated, it was finally managed to arrive at easily programmable ex pressions for the displacement components, permitting relatively easy and exhaustive parametric studies, which in turn could reveal critical aspects of the mechanical response of anisotropic discs anti-diametrically compressed. From a quantitative point of view it was proven that the dependence of the displacement field components on the anisotropy ratio is strongly non-linear. For a ratio δ=2.15 the difference of the radial displacement at r=R from that of the isotropic disc is equal to about 10% while for an anisotropy ratio equal to δ=4 the difference is almost 70%. The solution introduced is quite attractive because while it is valid for the general case of orthotropic materials it is quite easily transformed to a more convenient form, describing, with much simpler expressions, the elastic response of discs made of transtropic materials, which correspond to materials widely encountered in practical applications. Along the same line, it is emphasized that, although the solution is expressed in series form, it con vergences rapidly and, in addition, its limiting case, i.e. for anisotropy ratio equal to one or equivalently for isotropic materials, its results are identical to the respective ones for isotropic discs. Before concluding it is, also, worth mentioning that the accurate simulation of the boundary conditions renders the present solution a unique tool for validation/calibration of numerical models used to study the mechanical response of the disc and the jaws complex as an integrated elastic system. References Amadei, B., Savage, W.Z., Swolfs, H.S., 1987. Gravitational stress in anisotropic rock masses. International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts 24(1), 5-14. ASTM D3967-08, 2014. Standard test method for splitting tensile strength of intact rock core specimens. ASTM 04.08 Soil & Rock (I): D420 D5876. Barla, G., Innaurato, N., 1973. Indirect Tensile Testing of Anisotropic Rocks, Rock Mechanics 5, 215-230. Exadaktylos, G.E., Kaklis, K.N., 2001. Applications of an explicit solution for the transversely isotropic circular disc compressed diametrically. International Journal of Rock Mechanics & Mining Sciences 38, 227-243. Hondros, G., 1959. The evaluation of Poisson’s ratio and the modulus of materials of a low tensile resistance by the Brazilian (indirect tensile) test with particular reference to concrete. Australian Journal of Applied Sciences 10, 243-268.