引用本文: 付自国, 李化, 王佳信等. 岩石强度准则的主应力显式表达式几何推导及应用. 力学与实践, 2023, 45(3): 526-534.
Fu Ziguo, Li Hua, Wang Jiaxin, et al. A geometric derivation of the explicit expressions with respect to the principal stresses for rock strength criteria and its applications. Mechanics in Engineering, 2023, 45(3): 526-534.
 Citation: Fu Ziguo, Li Hua, Wang Jiaxin, et al. A geometric derivation of the explicit expressions with respect to the principal stresses for rock strength criteria and its applications. Mechanics in Engineering, 2023, 45(3): 526-534.

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

• 摘要: 主应力是岩石强度准则的基本量，然而，给定应力张量求解主应力的三次特征方程通常难以求解。本文通过几何关系将主应力的三次代数方程转化为三元一次方程组的求解，最终建立了三个主应力关于应力张量不变量的显式表达式，并着重推演了其在静水压力效应的刻画、三维强度准则的推导和空间屈服面的绘制中的应用。可为岩石真三轴试验数据可视化提供参考，同时丰富了弹塑性力学教学内容。

Abstract: 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.

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