Application of catastrophe theory for mathematical modeling of landslide process on concave slopes of mountain territories

Abstract

Introduction. Analysis of existing theoretical and experimental studies has shown that the model approach is the main method of landslide research. The existing mathematical model of landslides does not meet the requirements of the necessary adequacy. Materials and methods. This study uses a mathematical modeling methodology based on catastrophe theory. Results and discussion. To solve this actual problem, in this article authors developed a mathematical model of the landslide process on the concave slopes of mountainous territories. The developed model contains two components. The first of them is a mathematical model of the stress field in the volume of rocks located inside the slope section. This model uses the framework of fractal and multifractal modeling methods developed by the authors. The results of this model research are final expressions for calculating the stress field used rock pressure and bending stress as the external stress field. The superposition of the field induced by these external stresses gives the stress field in the volume of rocks located inside the slope section. Analysis of program implementation of this model showed that there are two areas in the slope section: compressive and tensile stresses adjacent to each other. At the boundary between these areas, there is a discontinuity of the stress field. A displacement surface passes along this boundary, forming a potentially landslide body. Moreover, it was found that a potentially landslide body on a slope is in a state of local and global instabilities. A potentially landslide body tends to occupy the position of the minimum potential energy. Local instability is expressed as the tendency of movement to a stable equilibrium without changing its location in the rock mass. The tendency of landslide body to move down the slope is a demonstration of global instability. The second mathematical model describes the realization of local instability that leads to the formation of a landslide body. Conclusion. According to the model analysis, it was found that the implementation of instability leads to the formation of a landslide body. At the same time, according to this analysis a landslide body can take up three stable equilibrium positions, allowing it to stay on the slope without global instability. Suggestions for practical application and direction of future research. The research results can be used to predict landslides on concave slopes of mountainous territories and to develop new mathematical models allowing to make the analysis of concave slopes of mountainous territories taking into account fracturing.