Issue 41

Ch. F. Markides et alii, Frattura ed Integrità Strutturale, 41 (2017) 396-411; DOI: 10.3221/IGF-ESIS.41.51 409 solutions. The first one is related to the kind of the loading scheme considered (i.e., the parabolic distribution of radial stresses), which approaches closely the actual distribution developed along the disc-jaw interface when the Brazilian-disc test is implemented according to the ISRM standard [1]. In fact the specific distribution is of “cyclic” nature [15, 27], how- ever, it is quite accurately simulated by a parabolic one, like the one adopted here. In any case the parabolic distribution is much closer to experimental reality compared to either a “uniform” one or to a loading scheme consisting of “point” diametral forces. The second advantage of the present solution is the fact that the loaded arc is not arbitrarily prescribed but rather it is de- termined from the solution of the respective disc-jaw contact problem. As a result, the present solution takes into account the relative stiffness of the disc and jaw materials, providing, in a first approximation, realistic values for the length of the contact arc. To make the importance of the specific issue clear, it is mentioned characteristically that, depending on modulus of elasticity of the disc material, the length of the contact arc varies within extremely broad limits, ranging from 2 o to 3 o (for very stiff discs made, for example, of Dionysos marble) [16, 28] to 20 o or even 30 o (for discs of low stiffness made, for example, of shell-stones) [16, 29]. Another issue that should be thoroughly discussed (also in future studies) is the “suitability” of the anisotropy ratio, δ , to act as a proper parameter that could properly describe the dependence of the strain-field, developed in a “transtropic” disc, on the anisotropy degree of the disc material. To make this point clear, the variation of the shear strain γ rθ , developed at the disc center, θ = ϕ o =30ο, is plotted in Fig.7 against an “average stiffness” of the “transtropic” disc material. This “average stiffness” is here defined as the mean value E av =( E + E ')/2 of the two elastic moduli characterizing a “transtropic” material. To draw Fig.7 a series of fictitious “transtropic” materials were considered with constant anisotropy δ equal to δ =2. The modulus of elasticity along the “strong” anisotropy axis ranges from E =4 GPa to E =80 GPa, while that along the “weak” anisotropy axis ranges from E '=2 GPa to E '=40 GPa in such a way that δ is kept constant equal to δ =2. For the Poisson ratios it is considered (for all combination of E and E ΄) that ν =0.30, and ν '=0.15. An average external load equal Figure 7 : The dependence of the shear strain γ rθ at the disc center on the “average stiffness” ( E + E ')/2, i.e., on the mean value of the two elastic moduli characterizing “transtropic” materials. to P frame =20 kN is imposed on the discs for all materials’ combinations. The non-linear nature of the dependence shown in Fig.7 is quite striking, indicating that, even for materials with the same δ -value, the shear strain does not decrease linearly with linearly increasing “average stiffness” of the disc material (the term “average stiffness” is a “neologism” and is introduced in an attempt to somehow quantify the stiffness of materials which are characterized by more than one elastic moduli). Before concluding some limitations, restricting the general applicability of the present solution, should be mentioned. The assumption of zero shear stresses along the loaded arc (in other words the assumption of perfectly smooth interface) is, perhaps, the most critical one. It is widely accepted that these stresses do not influence the stress field at the disc center, however, under specific conditions they could be responsible for premature fracture in the vicinity of the loaded arc, jeo- pardizing, thus, the validity of the results of the Brazilian-disc test [30-32]. Although it is an extremely complicated -5.0E-03 -4.0E-03 -3.0E-03 -2.0E-03 -1.0E-03 0.0E+00 0 15 30 45 60 γ rθ - strain at the disc's centre ( Ε + Ε ΄)/2 ϕ o =30 o