A GEOMETRIC DERIVATION OF THE EXPLICIT EXPRESSIONS WITH RESPECT TO THE PRINCIPAL STRESSES FOR ROCK STRENGTH CRITERIA AND ITS APPLICATIONS

Principal stresses are essential variables to establish strength criteria for rocks. However, it is usually difficult to solve the cubic characteristic equation to obtain the principal stresses for a given stress tensor. In this paper, explicit expressions of the principal stresses with respect to stress invariants are derived by transforming the cubic algebraic equation of principal stresses into the three-dimensional first-order equations through a geometric relation. Then, we emphatically introduce the applications of the derived formula in the domain of rock strength criteria, including quantitative characterization of hydrostatic pressure effect for rocks, derivation of three-dimensional failure criteria and drawing yield surface of the rock in the principal stress space. The research not only presents a visualization method for triaxial test data of rock samples but also enhances the teaching content of elastic-plastic mechanics.