Development of Calculation Method for Tensile Strength of Rock Samples Under Compression

Abstract

The paper describes a procedure for improving the approach to determining the ultimate strength of rock samples, the basic method is L. Prandtl’s method. In earlier works of the authors of this paper, the L. Prandtl’s method was improved in relation to rocks, using a general algebraic equation for calculating horizontal normal stresses depending on the vertical component and mechanical characteristics of the rock. Here, the authors reject the assumption that the tangent contact stresses resulting from L. Prandtl's solution do not depend on the abscissa x and on the variable vertical stresses, in contrast to the well-known Coulomb–Amonton law, according to which the tangent stresses are directly proportional to the vertical pressure. The regularities of contact normal and tangential stresses distribution at failure of prismatic samples are clarified. As result, the method was developed for calculating the tensile strength of samples with three characteristic values of mechanical properties: kп, yield strength of material in shear; ρ, angle of internal friction; fс, coefficient of contact friction. These characteristics can be established experimentally by simple experimental studies. The proposed approach is effective for constructing ultimate curves and diagrams of “stress — longitudinal strain” of rocks in the case of contact friction. A comparative evaluation is given for the proposed method for calculating the tensile strength of samples during their destruction with experimental data. Comparison of the calculated tensile strength with experimental data shows that the relative error of calculations according to the proposed method is significantly lower than the relative error of calculations performed using the E. Unksov’s method. The proposed method can be recommended for practical application.

Comparison of the reliability of strength limit calculation methods for prismatic samples with different spreading of normal contact stresses in their wedge failure shape

“…Let's clarify the assumptions outlined in [1]. There, the side walls of the specimen are assumed to be straight.…”

“…In other cases, they employed the linear distribution according to L. Prandtl [1], following the formula…”

“…Both of these expressions have been derived by the authors from the joint solution of one ordinary differential equation and one algebraic equation. In solid mechanics, there is a recommendation that, when solving the stress-strain state of a body under plane deformation, the conditions for the joint solution of two partial differential equations should be satisfied at any point of the body [1]…”

“…where σy, σх, τхуvertical and horizontal normal and tangential stresses in the specimen body, Pa; and one algebraic equation of equilibrium [1,2,3].…”

“…To meet the requirements of solid mechanics, the authors of this article [2] have developed a theoretically more accurate law for the distribution of contact stresses based on the solution of the mentioned equations ( 3) -( 5) for rock materials.…”