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

Determining the patterns of asymmetric interaction of plastic medium with counter-directional metal flow

“…The closed problem of plasticity theory was proposed in [15]. The method of argument of functions of a complex variable is used, which is generalized to various problems of continuum mechanics.…”

“…In dynamic problems of the theory of elasticity, a wave equation of the hyperbolic type is often used. However, the function argument method proposed in [15] was used to solve another hyperbolic type equation in a dynamic problem. Generalized approaches to solving these equations were obtained using the Cauchy-Riemann differential relations, which made it possible to specify d'Alembert's result [28].…”

“…In [15], a closed plane problem of plasticity theory was stated and solved. The statement of the closed problem is a system of equations of the plane theory of plasticity, including the heat equation:…”

“…A distinctive feature of fundamental substitution (9) from [34] is that this function is not specified but determined. Expression (9) has worked well for solving symmetric problems [15], however, with different boundary conditions at the ends, it is not possible to link the solution into a single mathematical model under asymmetric loading. In this case, it is advisable to use in the solution not the constant value С σ in the product (9) but to switch to a variable, i.e.…”

“…Using expressions ( 14), (15) in equation ( 8), taking into account the second derivative of the y coordinate, we have:…”