Remarks on Linear and Nonlinear Failure Criteria

ABSTRACT: To explore the nature of stress states at failure, published data are analyzed for three rock types – Dunnville sandstone, Red Wildmoor sandstone, and Taiwan siltstone - within the framework of three stress invariants. Data include a series of axisymmetric (conventional triaxial) compression, axisymmetric (conventional triaxial) extension, and multiaxial strength tests. The data are fitted to linear and nonlinear failure criteria: Paul-Mohr-Coulomb (PMC), a generalized linear criterion containing all three principal stresses; Hoek-Brown, a popular nonlinear criterion containing major and minor principal stresses; and a simplified version of PMC, where the friction angle in extension is assumed to be equal to the friction angle in compression. The results show that both versions of PMC provide a better approximation of the test data and successfully capture, in a piecewise-linear manner, the well-known nonlinear nature of the failure surface in the meridian and equipressure planes. 1. INTRODUCTION Mohr-Coulomb (MC) failure criterion considers principal stresses σI and σIII with a linear form: (Equation) where ϕ = friction angle, Co = uniaxial compressive strength, and σI, σIII are major, minor principal stresses. The sign convention is compression positive. Note that the friction angle is independent of axisymmetric compression (σII = σIII) or axisymmetric extension (σII = σI) stress paths, where σII is the intermediate principal stress; i.e. in the framework of MC, the friction angles in compression and extension are equal. Conventional triaxial compression tests show that the failure envelope is nonlinear when considering a large range of mean stress (Mogi 1971; Labuz et al. 2018; Hoek and Brown 2019). Many investigators have proposed nonlinear failure criteria, one being Hoek and Brown (1980): (Equation) where m and Co are material constants determined through fitting. Because the axisymmetric tests cannot simulate general stress states, multiaxial or true-triaxial devices have been developed for investigating failure. Results of true-triaxial tests suggest that failure is affected by three principal stresses or another three stress invariants (Mogi 1971; Wawersik et al. 1997; Zeng et al. 2019).