Issue 41

Ch. F. Markides et alii, Frattura ed Integrità Strutturale, 41 (2017) 396-411; DOI: 10.3221/IGF-ESIS.41.51 405 stant biaxiality ratio k , equal to k = σ compressive / σ tensile =3, independently of the actual boundary conditions which prevail along the loaded arcs of the disc. This was suggested, perhaps for the first time, by Peltier [21], who arrived at the conclusion that, as long as the loaded arc is not “too large” (indicatively, he mentioned as an upper limit for the length of the loaded arcs a value less than one fifth of the disc diameter), the tensile stresses remain uniform almost all along the length of the loaded diameter. Barenbaum and Brodie [22] as well as Rudnick et al. [24] drew similar conclusions. However, the fact that the stress field at the disc center is more or less insensitive to the actual conditions along the loaded arcs is by no means a necessary and sufficient condition for the validity of the results provided by the Brazilian-disc test: It has been in- dicated by many researchers that depending on the actual stress field in the immediate vicinity of the loaded arcs it is quite possible that fracture could start far from the disc center, unless certain precautions are taken, rendering the results of the test erroneous [3-7, 24-26]. In the direction of minimizing the as above risk, Mellor and Hawkes [5] suggested, for the first time, the use of curved jaws (instead of plane loading platens) of radius exceeding that of the specimens, suggestions which were the seed for the respective ISRM standard [1]. It is therefore evident that, in spite of the insensitivity of the stress field at the disc center to the actual boundary conditions, the reliability of the final outcomes of the Brazilian-disc test strongly depends on the actual conditions prevailing along the disc loaded arcs and on their length. As a result, one should consider, at least qualitatively, the role of the specific parameter (length of the contact arc), independently of whether the disc is made of isotropic or “transtropic” material. Given that the problem for isotropic materials was recently considered thoroughly [17], a first attempt is described here to explore the role of the loaded arcs’ length on the displacement field in case of “transtropic” discs. This attempt is possible because the displacement field in a “transtropic” disc under parabolic pressure is, for the first time, provided here in terms of analytic, closed-form expressions. In this direction, a disc made of a serpentinous schist (with E =58 GPa, ν =0.34, E ΄=27 GPa, ν ΄=0.12) [11] was considered, compressed between the ISRM jaws, by an overall load equal to P frame =20 kN, which is kept constant for all ω ο -values assigned to the contact semi-arc. Given now that the length of the two contact arcs (2 ω ο ) is arbitrarily prescribed (rather than being determined from the respective contact problem) the amplitude of the parabolic distribution must be determined through Eqs.(4) (it is here recalled that the two formulae of Eq.(4) are almost equivalent, especially for relatively small ω o -values). As a first step, it was decided to plot the components of the displacement field along the disc loaded diameter, which is, perhaps, the locus of utmost importance for practical applications. A relatively wide interval of ω o -values was studied, ranging from a very small one equal to ω o =2 o (which in fact approaches the diametral compression by a pair of “point” forces, or equivalently to compression of very stiff discs which are not significantly deformed at the disc-jaw interface) to a rather very high one equal to ω o =30 o (which corresponds to diametral compression of discs with extremely low stiffness). Concerning angle ϕ o , three characteristic values were studied, namely ϕ o =0 o (loading parallel to the planes of isotropy, or equivalently x -axis parallel to the loading line), ϕ o =90 o (loading normal to the planes of isotropy, or equivalently x -axis normal to the loading line) and ϕ o =30 o (corresponding to the geometry for which the shear strain at the disc center is maximized independently of the anisotropy ratio, δ (Fig.3)). The results of the analysis are shown in Figs.5(a, b, c), in which the polar components u r and u θ of the displacement field are plotted along the loaded diameter from r =0 (the center of the disc) to r = R (or in other words to the point where the disc is in contact to the loading jaw). As it is expected, both for ϕ o =0 o and ϕ o =90 o the transverse (tangential) component of the displacement field is zero for obvious symmetry reasons. In addition, as it was also expected, the magnitude of the non-zero component of the displacement, i.e., u r , increases with increasing ϕ o (for the same point and the same ω ο -value) since for ϕ o =0 o loading is exerted along the strong anisotropy axis (characterized by high value of the elastic modulus or equivalently by increased stiffness) while for ϕ o =90 o loading is exerted along the weak anisotropy axis (characterized by low value of the elastic modulus or equivalently by decreased stiffness). In accordance to similar conclusions for isotropic discs [15, 16], it is also here concluded that the influence of angle ω o on the displacement (and therefore on the strain- and stress-fields) is definitely negligible at the disc center. The role of ω o becomes significant only for r → R . The lower limit of r for which ω o plays some role on the displacement field appears very sensitive to the value of angle ϕ o . From a quantitative point of view, it can be seen from Fig.5a that for ϕ o =0 o and r = R it holds that u r =6.5x10 -5 for ω ο =2 o while for ω ο =30 o the respective value is u r =3.5x10 -5 . In other words, increasing ω ο from 2 o to 30 o results to a decrease of the radial displacement in the vicinity of the loaded arc by more than 45%. Similar conclusions are drawn also for ϕ o =90 o and ϕ o =30 o . In general, increasing the length of the loading arc weakens the intensity of the displacement field. Concerning the influence of angle ω ο on the transverse (tangential) component of the displacement field, it can be said that it is almost negligible all along the loaded radius. Indeed, as it can be seen from Fig.5b only for r >0.45 R some differ- ences can be detected, which are much lower compared to the respective ones of u r .