Advancing a generalized method for solving problems of continuum mechanics as applied to the Cartesian coordinate system

Solving the problem of continuum mechanics has revealed the defining generalizations using the function argument method. The aim of this study was to devise new approaches to solving problems of continuum mechanics using defining generalizations in the Cartesian coordinate system. Additional funct...

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Autores principales: Valeriy Chigirinsky, Olena Naumenko
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Publicado: PC Technology Center 2021
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spelling oai:doaj.org-article:db6ae7bb0fdb467488050f9d027cd24c2021-11-04T14:08:25ZAdvancing a generalized method for solving problems of continuum mechanics as applied to the Cartesian coordinate system1729-37741729-406110.15587/1729-4061.2021.241287https://doaj.org/article/db6ae7bb0fdb467488050f9d027cd24c2021-10-01T00:00:00Zhttp://journals.uran.ua/eejet/article/view/241287https://doaj.org/toc/1729-3774https://doaj.org/toc/1729-4061Solving the problem of continuum mechanics has revealed the defining generalizations using the function argument method. The aim of this study was to devise new approaches to solving problems of continuum mechanics using defining generalizations in the Cartesian coordinate system. Additional functions, or the argument of the coordinates function of the deformation site, are introduced into consideration. The carriers of the proposed function arguments should be basic dependences that satisfy the boundary or edge conditions, as well as functions that simplify solving the problem in a general form. However, there are unresolved issues related to how not the solutions themselves should be determined but the conditions for their existence. Such generalized approaches make it possible to predict the result for new applied problems, expand the possibilities of solving them in order to meet a variety of boundary and edge conditions. The proposed approach makes it possible to define a series of function arguments, each of which can be a condition of uniqueness for a specific applied problem. Such generalizations concern determining not the specific functions but the conditions of their existence. From these positions, the flat problem was solved in the most detailed way, was tested, and compared with the studies reported by other authors. Based on the result obtained, a mathematical model of the flat applied problem of the theory of elasticity with complex boundary conditions was built. Expressions that are presented in coordinateless form are convenient for analysis while providing a computationally convenient context. The influence of the beam shape factor on the distribution of stresses in transition zones with different intensity of their attenuation has been shown. By bringing the solution to a particular result, the classical solutions have been obtained, which confirms its reliability. The mathematical substantiation of Saint-Venant's principle has been constructed in relation to the bending of a beam under variable asymmetric loadingValeriy ChigirinskyOlena NaumenkoPC Technology Centerarticlegeneralized approachesfunction argumentcartesian coordinateslaplace equationscauchy-riemann relationsTechnology (General)T1-995IndustryHD2321-4730.9ENRUUKEastern-European Journal of Enterprise Technologies, Vol 5, Iss 7 (113), Pp 14-24 (2021)
institution DOAJ
collection DOAJ
language EN
RU
UK
topic generalized approaches
function argument
cartesian coordinates
laplace equations
cauchy-riemann relations
Technology (General)
T1-995
Industry
HD2321-4730.9
spellingShingle generalized approaches
function argument
cartesian coordinates
laplace equations
cauchy-riemann relations
Technology (General)
T1-995
Industry
HD2321-4730.9
Valeriy Chigirinsky
Olena Naumenko
Advancing a generalized method for solving problems of continuum mechanics as applied to the Cartesian coordinate system
description Solving the problem of continuum mechanics has revealed the defining generalizations using the function argument method. The aim of this study was to devise new approaches to solving problems of continuum mechanics using defining generalizations in the Cartesian coordinate system. Additional functions, or the argument of the coordinates function of the deformation site, are introduced into consideration. The carriers of the proposed function arguments should be basic dependences that satisfy the boundary or edge conditions, as well as functions that simplify solving the problem in a general form. However, there are unresolved issues related to how not the solutions themselves should be determined but the conditions for their existence. Such generalized approaches make it possible to predict the result for new applied problems, expand the possibilities of solving them in order to meet a variety of boundary and edge conditions. The proposed approach makes it possible to define a series of function arguments, each of which can be a condition of uniqueness for a specific applied problem. Such generalizations concern determining not the specific functions but the conditions of their existence. From these positions, the flat problem was solved in the most detailed way, was tested, and compared with the studies reported by other authors. Based on the result obtained, a mathematical model of the flat applied problem of the theory of elasticity with complex boundary conditions was built. Expressions that are presented in coordinateless form are convenient for analysis while providing a computationally convenient context. The influence of the beam shape factor on the distribution of stresses in transition zones with different intensity of their attenuation has been shown. By bringing the solution to a particular result, the classical solutions have been obtained, which confirms its reliability. The mathematical substantiation of Saint-Venant's principle has been constructed in relation to the bending of a beam under variable asymmetric loading
format article
author Valeriy Chigirinsky
Olena Naumenko
author_facet Valeriy Chigirinsky
Olena Naumenko
author_sort Valeriy Chigirinsky
title Advancing a generalized method for solving problems of continuum mechanics as applied to the Cartesian coordinate system
title_short Advancing a generalized method for solving problems of continuum mechanics as applied to the Cartesian coordinate system
title_full Advancing a generalized method for solving problems of continuum mechanics as applied to the Cartesian coordinate system
title_fullStr Advancing a generalized method for solving problems of continuum mechanics as applied to the Cartesian coordinate system
title_full_unstemmed Advancing a generalized method for solving problems of continuum mechanics as applied to the Cartesian coordinate system
title_sort advancing a generalized method for solving problems of continuum mechanics as applied to the cartesian coordinate system
publisher PC Technology Center
publishDate 2021
url https://doaj.org/article/db6ae7bb0fdb467488050f9d027cd24c
work_keys_str_mv AT valeriychigirinsky advancingageneralizedmethodforsolvingproblemsofcontinuummechanicsasappliedtothecartesiancoordinatesystem
AT olenanaumenko advancingageneralizedmethodforsolvingproblemsofcontinuummechanicsasappliedtothecartesiancoordinatesystem
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