Issue 41

Ch. F. Markides et alii, Frattura ed Integrità Strutturale, 41 (2017) 396-411; DOI: 10.3221/IGF-ESIS.41.51 401 T HE ROLE OF CRITICAL PARAMETERS ON THE DEFORMATION OF “ TRANSTROPIC ” DISCS aking now advantage of the closed-form expressions for the displacements presented in previous section it is possible to explore crucial features of the deformation induced on a “transtropic” disc under parabolic pressure. As it was mentioned the parameters that will be considered include the anisotropy ratio δ , the inclination of the loading with respect to the planes of transverse isotropy and the length of the two loaded arcs. Before analytic data are presented attention is drawn to the following issues:  The anisotropy ratio δ is defined as δ = Ε / Ε ΄ and it is assumed that δ >1 since E corresponds to the strong anisotropy direction.  The term “material” represents, in fact, a specific combination of the values assigned to the material properties char- acterizing a transversely isotropic material, namely E , ν , E ΄, ν ΄ and G ΄. Various combinations of these values (mainly of E and E ΄) are considered, in order for the analysis to cover the broadest possible range of rocks and rock-like materials. These materials, when do not correspond to a specific natural material, are denoted as “fictitious” materials. It is underlined that when considering various combinations of the values assigned to E , ν , E ΄ and ν ΄, attention must be paid to fulfill the necessary condition: xy E ΄ E΄    (18) characterizing the magnitude of dilatation in y -direction due to compression in x -direction, dictated by the symmetry of the strain tensor  Under the term “actual material” a serpentinous schist with E =58 GPa, ν =0.34, E ΄=27 GPa, ν ΄=0.12 [11] is implied.  In all applications use is made of an approximating formula between the material properties which was proposed by Lekhnitskii [18] and is valid for a relatively wide class of rocks:   1 2 EE΄ G΄ E ΄ E΄     (19)  Allthough Eqs.(5, 6) refer to the points z 1 and z 2 , all results obtained on the disc cross-section and on L , refer to the point ( x , y ) or ( r , θ ), through Eqs.(7). The role of the anisotropy ratio, δ and the load inclination angle ϕ o In order for a global overview of the deformation imposed on the “transtropic” disc under parabolic radial pressure to be obtained, it was decided as a first step to pay attention at the immediate vicinity of the disc center, given that this is the point where fracture is expected (or where fracture “must” start), for the results of the Brazilian-disc test to be reliable. In this direction, the components of the strain tensor are plotted in Figs.2 and 3 versus the anisotropy ratio δ , for a series of “fictitious” “transtropic” materials. The values assigned to δ vary from δ =1 (which corresponds to a disc made of an iso- tropic material) to δ =5, covering well the range of anisotropy ratios that can be met in real rocks and rock-like materials. The mechanical properties of the materials considered are recapitulated in Tab. 1. The radius of the disc was set equal to 50 mm and its thickness (length) equal to 10 mm. A constant overall load equal to P frame =20 kN was exerted on the discs. Angle ϕ o , i.e., the inclination of the axis of symmetry of the parabolic distribution of radial stresses with respect to x -axis, was set equal to ϕ o =0 o , 15 o , 30 o , 45 o , 60 o , 75 o and 90 o . The dependence of the normal radial strain component ε rr (i.e., that along the loading axis) and the normal transverse strain component ε θθ (i.e., that normal to the loading axis) on the anisotropy ratio δ is shown in Figs.2(a, b) for all ϕ o - values. It is observed from Fig.2a that the normal radial strain ε rr is negative (contractive) for all δ -values, as it is obviously expected. For relatively low δ -values ( δ <1.3) ε rr decreases rapidly, reaching a minimum value for δ ≈1.3 for all ϕ o -values without any exception. For higher δ -values ε rr increases smoothly. The dependence of ε rr on the inclination ϕ o of the anisotropy planes with respect to the loading line is smooth and monotonous. The respective behaviour of the transverse strain component ε θθ is somehow “symmetric” to that of ε rr (Fig.2b): It is positive (dilatative) for all δ - and ϕ o -values, it exhibits a clear maximum (again for δ ≈1.3) and its dependence on angle ϕ o is also smooth and monotonous. Of higher importance is the variation of the shear strain γ rθ against the anisotropy ratio δ which is shown in Fig.3. Contrary to what happens at the center of an isotropic ( δ =1) disc (where γ rθ is constantly zero for obvious symmetry reasons), T