Based on the theory of autonomous dynamical systems, the developed mathematical model makes it possible to describe the external load-induced variation in crack density in a mineral as a textural component of half-rock. The theory enables describing the process of crack density variation in a mineral under external loading in terms of the parameter of state and an abstractfunction of time evolution of initial state. Interpretation of the phase portrait of the developed model yields a conclusion that, in conformity with the dynamics of change in crack density, there are two mechanisms of mineral disintegration in half-rocks. These mechanisms govern initiation and occurrence of rock falls on pitwall slopes. The first mechanism is failure of minerals as a result of unstable growth of cracks. The second mechanism is mineral failure due to formation of spatially distributed microcracks in its structure. It is found that in case that minerals in half-rocks composing pitwall slopes fail by the mechanism of unstable crack growth, it is advised to predict rock falls using the discrete element method. Otherwise, when half-rock minerals fail due to formation of spatially distributed microcracks (filled with water in real conditions), prediction of rock falls should use the finite element models.

For citation: Khalkechev R.K. Fuzzy mathematical model of crack density variation in mineral under external loading. MIAB. Mining Inf. Anal. Bull. 2019;(6):97-105. [In Russ]. DOI: 10.25018/0236-1493-



Rock falls, prediction, landslide processes, crack density, finite element method, discrete element method, dynamical system, phase portrait.

Issue number: 6
Year: 2019
ISBN: 0236-1493
UDK: 622.83
DOI: 10.25018/0236-1493-2019-06-0-97-105
Authors: Khalkechev R. K.

About authors: R.K. Khalkechev, Cand. Sci. (Phys. Mathem.), Assistant Professor, National University of Science and Technology «MISiS», 119049, Moscow, Russia, e-mail: syrus@list.ru.


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