Application of the Christensen Failure Criterion to Intact Rock

The Christensen criterion, originally introduced in materials science, has a simple mathematical form and uniaxial tensile and compressive strength as the only parameters, making it an attractive candidate for rock engineering purposes. In this study, the applicability of the criterion to rock materials is examined. Explicit equations for application of the criterion under biaxial, triaxial compression, triaxial extension, and polyaxial states of stresses are derived. A comprehensive strength data set including the results of tests on synthetic rock, chert dyke, Carrara marble and Westerly granite is utilized to examine the accuracy of the Christensen criterion to the failure of rock material. The two surprising findings about the Christensen criterion are the zero values of tensile strength and the very low slopes of the failure envelope obtained from fitting analyses for chert dyke and Westerly granite. It is shown that the two problems are interrelated and the values of tensile strength tend to zero to produce higher slopes. It is then mathematically proven that the maximum initial slope of the Christensen failure envelope is limited to 4 in triaxial compression and 2.5 in triaxial extension which is considerably lower than the slope of experimental data. The accuracy of the Christensen criterion was found to be significantly lower than the well-established Hoek–Brown criterion. The circular π-plane representations and brittle-to-ductile transition limits from the Christensen criterion are also inconsistent with the observed behavior of rocks.     

Continuous Monitoring of Pin Tip Wear and Penetration into Rock Surface Using a New Cerchar Abrasivity Testing Device

Evaluation of rock abrasivity is important when utilizing mechanized excavation in various mining and civil projects in hard rock. This is due to the need for proper selection of the rock cutting tools, estimation of the tool wear, machine downtime for cutter change, and costs. The Cerchar Abrasion Index (CAI) test is one of the simplest and most widely used methods for evaluating rock abrasivity. In this study, a new device for the determination of frictional forces and depth of pin penetration into the rock surface during a Cerchar test is discussed. The measured parameters were used to develop an analytical model for calculation of the size of the wear flat (and hence a continuous measure of CAI as the pin moves over the sample) and pin tip penetration into the rock during the test. Based on this model, continuous curves of CAI changes and pin tip penetration into the rock were plotted. Results of the model were used for introduction of a new parameter describing rock–pin interaction and classification of rock abrasion.     

Comparison of linear and nonlinear forms of isotherm models for Zn(II) and Cu(II) sorption on a kaolinite

The most appropriate method in designing the adsorption systems and assessing the performance of the adsorption systems is to have an idea on adsorption isotherms. Comparison analysis of linear least square method and nonlinear method for estimating the isotherm parameters was made using the experimental equilibrium data of Zn(II) and Cu(II) onto kaolinite. Equilibrium data were fitted to Freundlich, Langmuir, and Redlich–Peterson isotherm equations. In order to confirm the best-fit isotherms for the adsorption system, the data set using the chi-square (χ ²), combined with the values of the determined coefficient (r ²) was analyzed. Nonlinear method was found to be a more appropriate method for estimating the isotherm parameters. The best fitting isotherm was the Langmuir and Redlich–Peterson isotherm. The Redlich–Peterson is a special case of Langmuir when the Redlich–Peterson isotherm constant g was unity. The sorption capacity of kaolinite to uptake metal ions in the increasing order was given by Cu (4.2721 mg/g)?<?Zn (4.6710 mg/g).