Two-dimensional elastic theory methods for describing the stress state and the modes of elastic boring

Abstract

Purpose. To find the stressed state and describe the operating regimes of an important element of mining industrial equip ment -an elastic drilling bit -based on the method of finding the solutions of problems of solid mechanics using holomorphic functions of two complex variables.Methodology. Methods for constructing the basic solutions of threedimensional boundary value problems of the theory of elasticity are based on the representation of the fundamental solution of the Lame equations in the PapkovichNeuber form using a scalar spatial harmonic function and a vectoral spatial harmonic function. This made it possible to formulate the boundary value problem in terms of holomorphic functions of complex variables. Based on the representation of the above holomorphic functions in the form polynomial of the n th order in accordance with degrees of the complex variables, corresponding boundary conditions are formulated for basic solutions and integral conditions for the principle moment of the stress vector on the lateral surface being equal to zero are additionally specified.findings. The paper formulates complexconjugate boundary value problems of spatial theory of elasticity in terms of holomor phic functions of two complex variables. We have considered the case when there is one of the spatial coordinates on which the stress tensor does not depend on, for a body whose surface is described by standard curves. For the given finite elastic cylindrical solid (a drilling bit) of canonical crosssection, a set of the basic complex solutions of the n th order and of corresponding vectors of external loads has been constructed. originality. For the first time, a scheme and a method for constructing the basic solutions of static boundary value problems of spatial theory of elasticity have been suggested and the corresponding boundary conditions have been imposed; the real and imaginary parts of the solutions of the basic boundary value problems for a cylindrical drilling bit have been constructed and an analysis of these solutions has been carried out.Practical value. An example of using the developed methods for constructing solutions of boundary value problems of spatial theory of elasticity has been considered for finding an exact analytical solution of a twodimensional boundary value problem that describes the distribution of stresses and corresponding external loads on the lateral surface of a cylindrical drilling bit of a ca nonical crosssection. Such mathematical models and analysis of the external load structure can be effectively used to describe the safe operating modes of mining boring equipment.

The study of mathematical models of the linear theory of elasticity by presenting the fundamental solution in harmonic potentials

“…In [18], an example of using the developed methods for constructing a set of exact analytical solutions of boundary-value problems of spatial theory of elasticity is considered. The example is based on the representation of the general solution of the equation of the linear theory of elasticity in the form of Papkovich-Neuber through scalar and vector harmonic functions [2,3].…”