Theoretical stress analyses for rock spheres and cylinders under the point load strength test.

摘要

The Point Load Strength Test (PLST) is an extremely convenient and useful method for rock classification and strength estimation. Although the PLST has been extensively studied by experimental approach, there are relatively few theoretical studies for the PLST. Analytic studies for the PLST include analyses of isotropic spheres under the diametral PLST by Hiramatsu and Oka (1966), finite cylinders under the axial PLST by Wijk (1978) and by Peng (1976), and finite cylinders under the diametral PLST by Wijk (1980) and Chau (1998a).; However, the exact stress field within a cylinder under the axial or diametral PLST has not been solved analytically. The only analytic results are the approximate solutions by Wijk (1978, 1980), in which the interaction between the indentors and the surfaces of the cylinder was idealized by two point forces. The finite element method has been applied to the axial PLST (e.g. Peng, 1976), but the contact problem between the indentors and the end surfaces was not considered either. The analytic solution by Chau (1998a) is for finite isotropic cylinders with zero shear displacement on the two end surfaces under the diametral PLST. Moreover, all of these analyses are restricted to considering rock as isotropic solids, there is no analytic solution for anisotropic rocks under the PLST.; Therefore, this dissertation presents a series of exact analytic solutions for anisotropic spheres and finite isotropic cylinders under the PLST. More specifically, the dissertation presents: (I) an exact analytic solution for spherically isotropic spheres under the diametral PLST; (II) an exact analytic solution for finite isotropic cylinders under the axial PLST; (III) an exact analytic solution for finite isotropic cylinders under the diametral PLST; and (IV) a general analytic solution for finite isotropic cylinders under arbitrary surface load. In addition, a series of the PLST experiments have also been done on plaster, a kind of artificial rock-like material, to verify the theoretical solutions.; The method of solution for spheres uses the displacement potential approach together with the Fourier-Legendre expansion for the boundary loads. The solution reduces to the classical solution by Hiramatsu and Oka (1966) in isotropic case. Numerical results show that the maximum tensile stress along the axis of loading is very sensitive to the anisotropy in Young's modulus, Poisson's ratio and shear modulus, while the pattern of the stress distribution is relatively insensitive to anisotropy of rocks. The method for finite isotropic cylinders under the axial or diametral PLST expresses displacement functions in terms of series of the Bessel and modified Bessel functions; and the contact problem between the surfaces of the cylinder and the indentors, through which the point loads are applied, is considered. Numerical results show that the tensile stress distribution along the axis of loading within isotropic cylinders, either under the axial or diametral PLST, similar to that within isotropic spheres under PLST, is not uniform, tensile stress concentrations are developed near the point loads, the maximum tensile stress increases with the decrease of Poisson's ratio and the size of loading area. The theoretical prediction for the size and shape effects of the specimen on the PLST agrees well with experimental results. More importantly, if the sizes of the specimens are comparable (ISRM, 1985), the tensile stress distributions along the axis of loading in a sphere, in a cylinder under the axial or diametral PLST, have the similar pattern. This conclusion indicates that the PLST is insensitive to the exact shape of the specimen, thus we provide the first theoretical basis for irregular lumps under the PLST.; In addition, by generalizing the method of solution for axisymmetric; problem of the axial PLST and for non-axisymmetric problem of the diametral PLST, a new analytic framework for analyzing stresses for finite isotropic sol