Solving applied problems of elasticity theory in geomechanics using the method of argument functions of a complex variable

Abstract

When solving many tasks related to mine workings, rock pressure management, development systems, support structures, the issues of strength and stability of rocks become relevant. Limitations and gaps are identified, emphasizing the need for further research and development of new methods for solving applied problems of elasticity theory. It is of theoretical and practical interest to determine the influence of half-space geometry on the stressed state of the medium and to assess whether it would suffice, in this case, to confine oneself to radial stress when characterizing the stressed state. To build a mathematical model of the stressed state of the array, a complex variable function argument method was used. Based on the developed complex variable function argument method, the applied problem of mechanics on loading the wedge with a concentrated force in polar coordinates was solved. A feature of the proposed approach is the introduction of tangential stresses with the need to meet boundary conditions along inclined faces. The introduction to the consideration of tangential stress shows that it cannot be neglected at a certain stage of the search for a solution. First of all, this is due to the half-space geometry, the angle at the apex, and the depth of the array. When changing the angle of the wedge, the interface surface changes fundamentally and can pass from a convex shape to a concave one. Simplification of the proposed expressions leads to a complete coincidence with the solutions by other authors obtained by the stress method, which indicates the reliability of the result reported here. This method may be advanced by complicating the half-space geometry, as well as loading, and by building a mathematical model for assessing the effect of tangent stresses on the strength and stability of soils