Failure criterion of intact rocks is a matter of fundamental importance in geotechnical and geomechanical design. Although, it has been widely reported that the behaviour for rocks under different confining stress follows a non-linear behaviour, still the Mohr-Coulomb linear criterion is a most applicable theoretical criterion in rock mechanics. But it is clear that when real assessment is required in stability analysis, non-linear criterion is needed. Therefore, the different non-linear empirical equations were developed during the past decades such as Hoek-Brown failure criterion.

The overall applicability of empirical equations is difficult because these equations usually have been developed for the specific conditions and also the empirical factors and constants used in these equations physically are difficult to understand. Therefore, the development of a non-linear theoretical failure criterion is the central goal of this research. Finding the equation of the tangent of the general equation of a curves 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 solution of this differential equation is the equation of cover curve of Mohr's circles which is known as failure envelope. In this paper the system of equations was solved and the method is introduced by developing a computer code open source program.

Results obtained from the theoretical solution, laboratory observations and the data points presented by Hoek-Brown failure criterion are plotted to comparison. The new proposed method found with the advantage of easier application than solving Ballmer's equations of Hoek-Brown criterion. Although in most of stress states, the results of both methods are consistent, but generally in failures under low stress states, the proposed new approach gives more conservative strength results. In addition, parameters used in this criterion easily could be measured through regression analysis of triaxial test data.

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