Mohr Circles Non-Linear Cover Curve for results obtained from Triaxial test on Intact Rocks
#miningengineering #rockmechanics #soilmechanics #geotechnicalengineering
Three years ago, I developed a method and made a tool to calculate the Non-Linear Failure Cover Curve of Mohr Circles for results obtained from triaxial tests. Today, I decided to share it with others; it could be helpful for someone! The related research paper, open source code, and .exe file are found via the following links:
www.researchgate.net/publicat...
www.researchgate.net/publicat...
github.com/ahmadmehri/Mohr-Co...
Failure criterion is the heart of each geomechanical design. Although, thriaxial experiments typically show non-linear behavior for rocks under different confining stress, still the Mohr-Coulomb linear criterion is a most applicable theoretical criterion in rock mechanics. But it is clear that when high accuracy is required in stability analysis, non-linear criterion is needed. Therefore, the different non-linear empirical equations were developed during the past decades. 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 goal of this research. In this research, the general differential equation of Mohr circles set was solved and it was demonstrated that the unusual answer of this differential equation is the equation of cover curve of Mohr circles. Summary Here the system of equations parametrically were solved and the obtained non-linear equation experimentally was validated. Results obtained from the theoretical solution have a good accordance with the actual observations. In addition, parameters used in this criterion easily could be measured through regression analysis of triaxial test data. This non-linear failure criterion could be used in stability analysis simulations to accurately predict the states of failure in rocks. The Hoek-Brown (1980) empirical criterion is a most well-known failure equation in rock mechanics. This criterion using several empirical parameters accurately predicts the failure states of σ3 and σ1 in rocks. But sometimes in stability analysis it is necessary to predict the failure states of τn and σn on failure plane. In order to overcome this problem, Hoek and Brown used the Balmer (1952) equations and they modified and rewritten the equations based on their own empirical factors. Since the applicability of these empirical equations was difficult, they developed a computer program (named RockData) to found the data points (τn, σn) of failure states. In this research, differentially it is tried to develop a non-linear theoretical failure criterion to accurately predict the equation of failure states of τn and σn on failure plane. 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 circles as a general differential equation, the unusual answer of this differential equation is the equation of cover curve of Mohr circles which is known as failure criterion.