New approaches to solving plane problems of continuum mechanics in polar coordinates using the argument functions method

Abstract

Purpose. Development of an algorithm of simplification of solving of continuum mechanics plane problem in the polar coordinates using the argument functions of complex variable. Methodology of research. Based on the argument function method and the method of a complex variable, new approaches to the determination of components of the stress tensor in polar coordinates have been developed. The equilibrium equation systems were used to solve the flat problem. A fundamental substitution is suggested. Use of a trigonometric substitution that connects integral characteristics of a stressed state with components of a stress tensor is demonstrated. Argument functions of basic variables are introduced. When substituting into differential equations, operators are formed, which are characterized by these argument functions and that perform a role of special search regulators. Findings. Analytical dependencies of existence of solutions in a form of the invariant Cauchy-Riemann conditions and Laplace’s equations are determined. The solution uses generalized relations in the differential form for specific functions - functions of harmonic type. The trigonometric shape of the shearing stress distribution diagram is actually confirmed by theoretical and experimental data. Scientific novelty is in establishment of the solutions that determine not the functions themselves, but the conditions of their existence using Cauchy-Riemann differential conditions. The solution is a more general case with the feature that is represented not by the product of functions, each of which is determined by one coordinate, but by the product of different functions simultaneously dependent on two coordinates. Practical significance. Obtained resultis conveniently applied for simplification, allowing linearization of boundary conditions. Comparison of the obtained results with the solutions of other authors shows that the presented solution after simple transformations can be simplified and consider the obtained solution as more generalized.