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

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Category: Content №1 2020

Last Updated on 09 April 2020

Published on 10 March 2020

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Authors:

V.V.Pabyrivskyi, Cand. Sc. (Phys.-Math.), Assoc. Prof., orcid.org/0000-0002-6071-3817, Lviv Polytechnic National University, Lviv, Ukraine, e-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.; This email address is being protected from spambots. You need JavaScript enabled to view it.; This email address is being protected from spambots. You need JavaScript enabled to view it.; This email address is being protected from spambots. You need JavaScript enabled to view it.

I.V.Kuzio, Dr. Sc. (Tech.), Prof., orcid.org/0000-0001-9271-6505, Lviv Polytechnic National University, Lviv, Ukraine, e-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.; This email address is being protected from spambots. You need JavaScript enabled to view it.; This email address is being protected from spambots. You need JavaScript enabled to view it.; This email address is being protected from spambots. You need JavaScript enabled to view it.

N.V.Pabyrivska, Cand. Sc. (Phys.-Math.), Assoc. Prof., orcid.org/0000-0003-4631-0189, Lviv Polytechnic National University, Lviv, Ukraine, e-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.; This email address is being protected from spambots. You need JavaScript enabled to view it.; This email address is being protected from spambots. You need JavaScript enabled to view it.; This email address is being protected from spambots. You need JavaScript enabled to view it.

P.Ya.Pukach, Dr. Sc. (Tech.), Prof., orcid.org/0000-0002-0359-5025, Lviv Polytechnic National University, Lviv, Ukraine, e-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.; This email address is being protected from spambots. You need JavaScript enabled to view it.; This email address is being protected from spambots. You need JavaScript enabled to view it.; This email address is being protected from spambots. You need JavaScript enabled to view it.

Naukovyi Visnyk Natsionalnoho Hirnychoho Universytetu. 2020, (1):46-51
https://doi.org/10.33271/nvngu/2020-1/046

 повний текст / full article

Abstract:

Purpose. To find the stressed state and describe the operating regimes of an important element of mining industrial equipment – 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 three-dimensional boundary value problems of the theory of elasticity are based on the representation of the fundamental solution of the Lame equations in the Papkovich-Neuber 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 complex-conjugate boundary value problems of spatial theory of elasticity in terms of holomorphic 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 cross-section, 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 two-dimensional boundary value problem that describes the distribution of stresses and corresponding external loads on the lateral surface of a cylindrical drilling bit of a canonical cross-section. Such mathematical models and analysis of the external load structure can be effectively used to describe the safe operating modes of mining boring equipment.

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