Issue 41

Ch. F. Markides et alii, Frattura ed Integrità Strutturale, 41 (2017) 396-411; DOI: 10.3221/IGF-ESIS.41.51 397 At this point, it is emphatically underlined that the analytic solutions, which have been used for the derivation of the as above classical formulae, are valid exclusively for linearly-elastic isotropic materials. In spite of this strict restriction, the as above formulae are quite often used, also, for the determination of the tensile strength of transversely isotropic materials, although it is well known that for non-isotropic materials the tensile strength is not a unique constant but rather it is a function of the orientation of the loading axis with respect to the anisotropy axes [11]. The main reason for this “compro- mise” is the fact that generally accepted analytic solutions for the stress and displacement fields developed in a finite circular disc made of a material characterized by even the simplest kind of anisotropy, i.e., transverse isotropy, are not as yet avail- able. Indeed, the respective mathematical problem, i.e., that of a finite disc made of transversely isotropic material that is subjected to diametral pressure is extremely complicated and the analytic solutions available for the stress and displace- ment field are based on quite a few simplifying assumptions which unavoidably restrict seriously their applicability [12, 13]. Among these assumptions, the one perhaps most distanced from experimentally reality, is the simulation of the load- ing scheme by either a pair of diametral point forces or by a distribution of uniform radial pressure acting along two very “small” arcs of the disc periphery, symmetric with respect to the disc center, of arbitrarily predefined length. In the direction of relieving some of these restrictions, an analytic, closed-form solution was recently introduced [14] con- sidering an orthotropic disc loaded by a parabolic distribution of radial stresses acting along two finite arcs of the disc periphery. The specific loading scheme is very close to the actual distribution of radial stresses [15, 16] developed along the disc-jaw contact arcs, when an intact isotropic disc is compressed between the curved jaws of the device suggested by the International Society for Rock Mechanics (ISRM) [1], in which case the length of these arcs is not constant but rather it is a function of the load level imposed and the relative stiffness of the disc and jaws materials, as it is expressed by the ratio of the respective elastic moduli [17]. The problem of the orthotropic disc [14] was solved with the aid of Lekhnitskii [18] complex potentials technique whereas regarding a first approximation of the loading scheme and contact length pre- viously mentioned formulae [16], based on Muskhelishvili formalism for isotropic bodies [19], were properly modified. In the work described here, advantage is taken of the above mentioned solution [14], in an effort to enlighten and quantify the influence of some crucial material and geometric parameters on the displacement field developed in a disc made of a transversely isotropic material. Among the parameters considered here is the degree of anisotropy, or in other words the ratio δ of the two elastic moduli characterizing a transversely isotropic material (recall that δ =1 corresponds to an isotropic material while increasing δ -values indicate stronger anisotropy). The role of the specific parameter is studied in-depth in order to quantify the limit of δ for which the use of analytic solutions developed for isotropic materials could be, perhaps, considered satisfactory approximations, also for transversely isotropic materials. The second parameter, the role of which is explored, is the inclination of the loading axis (in fact the axis of symmetry of the parabolic distribution) with respect to the axis of anisotropy. The role of the specific parameter is decisive since it governs the magnitude of shear deformation developed at the disc center in case the material layers are neither normal nor parallel to the loading direction. In other words, it dictates the shear stresses developed at the center of the disc, which in turn modify the fracture mechanism (with respect to that of the isotropic disc) rendering the validity of the Brazilian-disc test questionable in case of anisotropy. The last parameter studied here is the length of the loaded arcs. Although it is generally accepted that the specific quantity plays a rather minor role concerning the stress and strain fields at the center of the disc, it is known that it strongly influences the disc deformation in the immediate vicinity of the disc-jaw contact area [15, 16]. From this point of view it is definitely concluded that ignoring the length of the contact arc could yield erroneous results, since it is quite possible that fracture starts from the disc-jaw interface rather than from the center of the disc. The results of the study indicate that the dependence of the strain and displacement fields on the as above parameters (and especially on the degree of anisotropy) is strongly non-linear and the room for maneuver (i.e., using solutions for iso- tropic discs to approximately describe the tensile strength of transversely isotropic materials) is rather limited. It is there- fore concluded that there is a demanding need for novel standards for the determination of the tensile strength of the specific type of materials which should lie on analytical solutions capable to properly describe the stress and displacement fields in a transversely isotropic finite circular disc under parabolic diametral compression. T HE DISPLACEMENT FIELD IN A TRANSVERSELY ISOTROPIC DISC UNDER DIAMETRAL COMPRESSION circular disc of radius R and thickness d , made of a transversely isotropic material (or for brevity “transtropic” material, a term quoted in [18]) is in equilibrium, in the absence of any kind of friction, between two curved jaws, compressed against each other by a force P frame (Fig.1). The radius of curvature of the jaws ranges from 1.5 R (case corresponding to the ISRM standardization for implementing the Brazilian-disc test [1]) to infinity (case for the relevant ASTM standardized experimental procedure [2], i.e., plane loading platens). P frame acts in the disc cross-section, which, A