The Solution of Continuum Mechanics Plane Problem in the Polar Coordinates Using the Argument Functions of Complex Variable

Abstract

The general approaches to the solution of the plane problem of continuum mechanics, which have been successfully tested in the theory of plasticity, elasticity, dynamic problems of the theory of elasticity, are considered. 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. As a result of this, dependencies of existence of solutions in a form of the invariant Cauchy–Riemann conditions and Laplace’s equations are determined. The result obtained is conveniently applied for simplification, allowing linearization of boundary conditions. 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. The solutions that determine not the functions themselves, but the conditions of their existence using Cauchy–Riemann differential conditions are obtained. 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. 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.