Development of a mathematical model for the stability of a parabolic landslide body based on elementary catastrophe theory

Introduction. An analysis of scientific studies in the field of hazardous geological phenomena shows that the stability of formed landslide bodies is mainly assessed using the methods of limit equilibrium, finite element analysis, and catastrophe theory. Practice demonstrates that models developed within the framework of catastrophe theory provide the most accurate results. However, in these models, the stability of parabolic landslide bodies, which are common on mountain slopes, has not been considered. The aim of the study is to develop a new mathematical model of the stability of a landslide body with a parabolic translational shape using the elementary theory of catastrophes. Materials and methods. A landslide body with a shape obtained by translating a truncated parabola within a rock mass is considered. This is referred to as a parabolic landslide body. Two mathematical models were constructed, forming the basis for the theory of stability of a parabolic landslide body on a horizontal surface. These mathematical models are reduced to equations describing equilibrium surfaces or manifolds of catastrophe in the space of control parameters. They are expressed as incomplete cubic equations in terms of the state parameter with real coefficients. Results and discussion. According to René Thom’s topological concept of structural stability, a cusp catastrophe is observed. This indicates that the physical system, represented by the landslide situation, has been brought to a structurally stable form. The study of the obtained family of functions made it possible to determine the bifurcation set in the space of control parameters, which divides the system into areas with qualitatively different behaviors. Conclusions. The bifurcation set has a single cusp. When the gravity center locates inside the cusp of the parabolic landslide body, there are one unstable and two stable equilibrium positions. When the gravity center locates outside the cusp, there is only one stable equilibrium position. Thus, in the first case, the probability of landslide occurrence is lower than in the second. Resume. 1. A mathematical model of the stability of a parabolic landslide body has been developed within the framework of the mathematical theory of catastrophes. 2. The research results can be used to predict and prevent parabolic landslides on mountain territories and in open-pit mines. 3. Future studies should focus on improving methods for determining the gravity centers of landslide bodies

Catastrophe theory · Finite element method · Landslide · Mathematical model · Parabola · Parabolic partial differential equation · Earthquake Detection and Analysis · Geotechnical and Geomechanical Engineering · Landslides and related hazards