The paper describes a procedure for improving the approach to determining the ultimate strength of rock samples, the basic method is L. Prandtl’s method. In earlier works of the authors of this paper, the L. Prandtl’s method was improved in relation to rocks, using a general algebraic equation for calculating horizontal normal stresses depending on the vertical component and mechanical characteristics of the rock. Here, the authors reject the assumption that the tangent contact stresses resulting from L. Prandtl's solution do not depend on the abscissa x and on the variable vertical stresses, in contrast to the well-known Coulomb–Amonton law, according to which the tangent stresses are directly proportional to the vertical pressure. The regularities of contact normal and tangential stresses distribution at failure of prismatic samples are clarified. As result, the method was developed for calculating the tensile strength of samples with three characteristic values of mechanical properties: kп, yield strength of material in shear; ρ, angle of internal friction; fс, coefficient of contact friction. These characteristics can be established experimentally by simple experimental studies. The proposed approach is effective for constructing ultimate curves and diagrams of “stress — longitudinal strain” of rocks in the case of contact friction. A comparative evaluation is given for the proposed method for calculating the tensile strength of samples during their destruction with experimental data. Comparison of the calculated tensile strength with experimental data shows that the relative error of calculations according to the proposed method is significantly lower than the relative error of calculations performed using the E. Unksov’s method. The proposed method can be recommended for practical application.
Subject of study. the processes of cracking and destruction of rocks under the action of the working bodies of machines for disintegration. Methodology. A complex method of generalizing the laws of the theory of elasticity and plasticity was used; theoretical and experimental confirmation of the regularities of the distribution of contact normal and tangential stresses, equations of the limiting state of materials based on the Coulomb strength criterion; slip line theory; comparison of theoretical results with experimental diagrams "normal stresslongitudinal deformation" of samples; facts and phenomena of destruction of rocks; generalization of the theoretical regularities arising from the power contact of the tool with the rock in crushers. Purpose. Reducing energy consumption and increasing the efficiency of rock disintegration by controlling its stress-strain state in crushers on the basis of mathematical modeling and using the established regularities of stresses and deformations in rocks when interacting with a working tool. Output. In the contact area, with an increase in the tangential load, the zone of uniform compression of the material decreases, the depth of the most stressed point approaches the contact surface. There is a significant zone of shear deformations, which are the decisive factor in crack initiation. The development of the crack in depth and complete destruction occurs along the shear lines. Such conditions of rock loading are observed in jaw crushers with complex jaw movement, in cone crushers, in roller crushers with different roll rotation speeds and correspond to the model of the most effective sliding compression. The creation of asymmetric loading conditions using the forces of contact friction, frictional and strength characteristics of the destroyed material can reduce the energy consumption of disintegration.
Purpose. Development of an analytical method for calculating the parameters of complete diagrams longitudinal tension deformation for the truncated-wedge shape of destruction of cylindrical rock samples to control the stress-strain state of the rock mass and effective destruction of these materials during disintegration. Methodology. Analytically, by developing a mathematical model of the fracture process of cylindrical rock samples with their truncated-wedge form of crack development, an algorithm is created for calculating the full deformation diagram from the acting tension using the experimental values of four indicators of material properties the shear resistance limit, internal and external friction coefficients and elastic modulus. The method is based on the improved Coulomb strength criterion, supplemented by the parameters of contact friction and allowing one, using the theory of slip lines, to calculate the limiting state of the material at the tip of cracks developing from the edges of the sample of the correct geometry, taking into account the release of part of the material from the load and compliance with Hookes law deformation of the sample bearing area and the specific force on it. Findings. The method of mathematical modeling makes it possible to determine the ultimate strength and residual strength of cylindrical rock samples using four property indicators that can be experimentally established by simple methods. Originality. For the first time, analytical modeling of the process of destruction of cylindrical rock samples with their truncated-wedge form of destruction was carried out, taking into account the internal contact friction depending on the properties of the rock material and external contact friction. Practical value. The proposed method of mathematical modeling and the calculation algorithm make it possible to determine the limit and residual strength of rock samples using four property indicators, which can be established experimentally by simple methods under laboratory conditions of enterprises of the mining and metallurgical complex. The calculation results can be quickly used to control the state of the rock mass and effective destruction upon disintegration.
Приведены исследования распределения напряжений, действующих при разрушении крепких материалов в зоне контакта с бронеплитами в щековых дробилках со сложным движением щеки и с конусами в конусных дробилках. Проанализировано влияние направления сил контактного трения при сжатии на эффективность разрушения горной массы. Разработана математическая модель для определения напряженного состояния породы. Показана возможность управления режимами нагружения для повышения эффективности процесса дробления.
МОДЕЛИРОВАНИЕ ПРОЦЕССА РАЗРУШЕНИЯ ЦИЛИНДРИЧЕСКИХ ОБРАЗЦОВ ГОРНЫХ ПОРОД Аннотация. Разработана математическая модель разрушения цилиндрических образцов горных пород при наличии контактного трения. Разработан метод расчета предельного вертикального напряжения в вершине трещины и расчета предела прочности цилиндрических образцов горных пород с использованием трех показателей свойств (предел сопротивления материала сдвигу, угол внутреннего трения, коэффициент контактного трения), которые простыми способами могут быть установлены экспериментально в условиях горнорудных предприятий, где результаты расчета могут быть оперативно использованы для управления процессами дезинтеграции. Ключевые слова: математическая модель разрушения, цилиндрические образцы горных пород, трещина, внутреннее трение, контактное трение. . где τе -эффективные касательные напряжения на линиях скольжения, Пa; kппредел прочности материала на сдвиг; µ -коэффициент внутреннего трения; τ α -сопротивляемость породы сдвигу, Пa; σ α -нормальное напряжение на наклонной площадке, Пa. ��������������������������������� ��Васильев Л.М., Васильев Д.Л., Малич Н.Г., Назаров А.Е, 2020� «Системні технології» 5 (130) 2020 «System technologies» ISSN 1562-9945 (Print) ISSN 2707-7977 (Online) 48Из первого равенства следует, что разрушение не произойдет, если левая часть меньше правой. Для определения предела прочности пород σ (Па) на основании критерия при условии прямолинейности развития трещины -линии скольжения (ЛС) получена известная аналитическая формула [1,2,4 -6]
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