Issue 47

S. K. Kourkoulis et alii, Frattura ed Integrità Strutturale, 47 (2019) 247-265; DOI: 10.3221/IGF-ESIS.47.19 263 about P f =1.79 kN. Substituting this value of P f into Eqn. (22), the maximum tensile stress at point A (in other words the tensile strength of the CSR’s material) was calculated equal to about 60.2 MPa. A preliminary protocol of direct tension tests with standardized dog-bone specimens (of the same batch from which the CSR-specimens were cut), provided a value for the tensile strength of the material equal to about 58.8 MPa, i.e., only 2.4% lower than that determined with the aid of the CSR-test. On the other hand, a series of tests with compact discs (again made of the same batch of material and having diameter 2 R equal to 100 mm and thickness t equal to 10 mm) subjected to Brazilian disc test [1, 2], in accordance with the ISRM suggestion, provided a fracture force equal to P Br =87.65 kN. Adopting Hondros’ familiar formula [3], the respective tensile strength is calculated equal to σ Br =P/(πRt) =55.8 MPa, i.e., 5.1% lower than the value provided by the direct tension tests. The above mentioned inconsistency, between the data of the direct tension test and the CSR-test, even though it is relatively small, could be justified by taking into account the role of another parameter (which was ignored up to this point), namely the thickness of the CSR-specimens. In fact, the specimens tested in the experimental protocol had a thickness of 10 mm. They were neither very thin (plane stress) nor very thick (plane strain) for obvious practical reasons. The role of the specific parameter cannot be analytically explored (the analysis described is not actually three dimensional and the configurations considered must be characterized either as plane stress or plane strain). Inevitably testing specimens of moderate thickness introduces some discrepancies. In this context, the role of thickness is here explored by taking advantage of the numerical model (previously validated), and more specifically of the respective reference configuration. In Fig. 17a the variation of the horizontal displacement u x ≡ u along the axis of symmetry of the CSR-specimen (locus AB) is plotted for both the front face (identical to the rear one) and the central section of the three dimensional numerical model. The difference between the two plots are by no means negligible, however, it is almost constant all along the locus considered. In the same figure the results of the analytical solution for the variation of u x ≡ u along locus AB are plotted, assuming either 2D-plane stress or 2D-plane strain conditions. The difference between the two cases is, now, not constant along the AB locus. It is zeroed for r =25 mm (inner contour of the CSR) and increases gradually until r ≈35 mm. Then it starts decreasing until r =50 mm (outer contour of the CSR). The respective plots for the transverse stress developed along the AB locus are exhibited in Fig. 17b. As it is, perhaps, ex- pected, the difference for the stresses between the central section and the two surface faces of the three dimensional nu- merical model are negligible. On the contrary, the difference between the plane strain and the plane stress assumptions is quite considerable, approaching 13% at points A and B (the points at which the maximum difference is observed). More specifically, the maximum tensile stress (i.e., that at point A) is equal to about 29.7 MPa for the 2D-plane stress scheme, while for the respective 2D-plane strain scheme it is equal to about 33.6 MPa. 1.000 1.025 1.050 1.075 1.100 25 30 35 40 45 50 u x [mm] r [mm] 3D_central 3D_front 3D_back 2D_plane stress 2D_plane strain A B B -80 -40 0 40 25 30 35 40 45 50 σ θ [MPa] r [mm] 3D_central 3D_front 3D_back 2D_plane stress 2D_plane strain A B A B (a) (b) Figure 17 : The variation of (a) the horizontal displacement u x ≡ u and (b) the normal transverse stress along the AB locus, according to the reference configuration of the numerical model, assuming either three dimensional conditions (the plots are realized for both the front and rear surfaces and also the central section) or two dimensional conditions (plane stress and plane strain).