Development of an analytical method for the calculation of the Mohr-Coulomb failure envelope

Summary
It is demonstrated that the mechanical behavior of rocks is a function of the 3D stress state. But the limitations in the true-triaxial compression test have resulted in more use of the two-dimensional failure criteria in rock mechanics. Mohr-Coulomb criterion is the most applicable theoretical criterion in rock mechanics but the calculation of the equation of the failure envelope is still the main matter in this regard.
Introduction
The failure criterion of intact rocks is a matter of fundamental importance in rock engineering design and a substantial amount of research on the failure criteria of intact rock has been developed during the past years. Among them all, one of the most significant suggestions was made by Mohr and Coulomb. The major limitation of the Coulomb criterion is that it is a linear criterion and expresses the strength of the rock as a linear function of confining pressure or normal stress. On the other hand, a large amount of experimental observation suggests that Mohr failure envelopes of most of the intact rocks and soils are not linear, particularly under relatively low or extremely high amounts of confining stresses. In addition, difficulties involved in developing a theoretical model which satisfactorily predicts non-linear behavior of intact rocks under different stress conditions led engineers to propose some empirical relationships between principal stresses or between shear and normal stresses at failure. But for practical applications, it is more important, how easily the parameters of a strength criterion can be obtained in the field and whether corresponds to the applied field situation by the specific field conditions which empirical criterion was developed. Then, this article aims to present a simple and accurate analytical procedure for calculating Mohr failure envelop based on at least three triaxial experiment data obtained from core samples.
 
Methodology and Approaches
Finding the equation of the tangent of the general equation of a curve set is a solved problem in mathematic. Then, in rock mechanics, if we consider the system equations of Mohr’s circles as a general differential equation, the unusual answer of this differential equation is the equation of cover curve (or failure envelop) of Mohr circles which is known as failure criterion. In mathematics, the abnormal (or unusual) answer (or solution) of the first-order differential equation is a curve that is tangent to all curves generated from the general equation. Then, by finding a correlation between centers and radius of circles, substituting the parameters and few mathematical calculations the new non-linear Mohr’s failure envelope can be expressed by a parabolic equation.
Results and Conclusions
The proposed theoretical failure criterion in this paper follows Mohr’s hypothesis and is expressed in functional form τ = f (σ). in order to compare the new proposed model by Hoek-Brown and Coulomb linear model a computer code was developed to plot all of these criteria in a same τ-σ coordinate system. The results obtained from the new parabolic Mohr failure envelope have good accordance with the data points presented by the Hoek-Brown failure criterion. It should be noted that the new technic directly results from the failure equation while the Hoek-Brown model only represents the locus of data points of the normal and shear stresses on the failure plane.