In this paper, an analytical solution is developed of the stress and displacement components in a three-dimensional solid disc subjected to Brazilian and point load test boundary conditions. The methodology is initially based on introducing the load distribution in both tests using the double Fourier series technique. The displacement vector in cylindrical polar coordinates (r, θ, z) is then presented by means of harmonic potentials according to the Papkovich-Neuber scheme. Stresses are then recovered by searching for acceptable potentials that satisfy both the domain Laplacian equation and the prescribed Fourier boundary expression. Several solutions are developed in terms of design graphs, and the effect of influencing parameters on the calculated results, e.g. material elastic properties, loading configurations and specimen geometries, are discussed.
The Brazilian test (Hondros 1959, ISRM 1978,ASTM 008a) and Point load test (Wijk 1978, ISRM 1985, ASTM 008b) have a rich history in the field of engineering applications. Both tests provide a measure of the indirect tensile strength of a given material for which testing under direct tension is difficult. Investigation for the critical value of stresses induced in a target brittle material subjected to these tests, has long been studied experimentally, theoretically and numerically. A review of the state of art, especially for the testing of rock specimen, can be found in Chau (1998) and Diyuan and Louis (2012). However, a generic three-dimensional (3D), elastic, closed-form solution has not yet been proposed for the Brazilian test. Moreover, reported results obtained by analytical methods for the point load test generally employ excessively complex 3D mathematical derivations (Wei and Chau 2000). Therefore, in the study presented herein, the mathematical hurdles for the calculation of elastic field variables in a solid disc subjected to Brazilian and point load boundary tractions are investigated with a relatively simple 3D elasticity procedure. The proposed formulation can also be conveniently used to solve a variety of other problems in which the curved boundary of a solid disc is under arbitrary loading with different frequencies applied in the axial and circumferential directions., e.g. strip-loaded (Barton 1941) or fully-loaded cylinders (Serati et al. 2012).