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Vibrations in engineering and technology

space SCIENTIFIC JOURNALS OF VINNITSA NATIONAL AGRARIAN UNIVERSITY

Issue №: 4 (103)

Published: 2021.12.24
DOI: 10.37128/2306-8744-2021-4


Description:
The journal deals with the problems of vibration technologies and machines, mathematical methods of vibration process studies, information on design and technological development, presents teaching and methodological aspects of teaching in the Higher School of Applied Sciences, where vibration machines and technologies are studied.

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STUDY OF THE KINEMATICS OF PLASTIC FLOW OF A METAL USING THE EULER AND LAGRANGE VARIABLES

DOI: 10.37128/2306-8744-2021-4-8
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Sivak Roman – Doctor of Technical Sciences, Associate Professor of the Department of General Technical Disciplines of Vinnytsia national agrarian university (st. Soniachna, 3, Vinnytsia, Ukraine, 21008, e-mail: sivak_r_i@ukr.net)

Zalizniak Roman – graduate student (st. Soniachna, 3, Vinnytsia, Ukraine, 21008, e-mail: pacifistroma@gmail.com)

 

Annotation

In continuum mechanics, two equivalent points of view on the flow of a metal in the process of deformation - Euler and Lagrange - have found application. When attention is focused on a given point in space, where different particles of a continuous medium come in the process of deformation, this is the essence of Euler's point of view on the studied motion of a continuous medium. From the point of view of Lagrange, the motion is considered known if the speed, acceleration, temperature and other quantities are given as functions of the coordinates of a point in space and time. The essence of Lagrange's view of the studied motion lies in the fact that they focus on a specific particle of a continuous medium and investigate the history of its deformation and motion. The coordinates that individualize the points of the continuous medium are called Lagrange variables. Lagrange's view on the study of the motion of a continuous medium underlies physical laws, since it is associated with the motion of individual particles. At the initial moment, the coordinates of a certain point according to Euler and Lagrange coincide.

The article discusses the possibility of using mixed Euler and Lagrange coordinates to determine the component of the strain rate tensor in plastic deformation processes. The expediency of the approach of Lagrange and Euler is clarified in specific problems. It should be borne in mind that in the Euler coordinates, the equilibrium equations have a simpler form (they are linear), and the boundary conditions are more complex due to the fact that it is impossible, without solving the problem, to determine the displacement components that are usually included in the boundary conditions. In Lagrange coordinates, the boundary conditions, as a rule, are written easier, and the equilibrium equation is more complicated.

When processing metals by pressure, the metal undergoes significant deformation. However, it is convenient to consider each process at each fixed moment of its course, therefore, it is necessary to know the basic provisions and dependencies related to small deformations.

 

Keywords: Lagrange and Euler variables, deformations, strain rate tensor, plastic flow, deformed state, Cartesian coordinate system, partial derivative.

List of references

1. Аlіevа L. І., Аlіeв І. S., Grudkіnа N. S., Маlіy H. V. Моdеliuvаnnia prоtsеsu kоmbіnоvаnоgо rаdіаlnо-zvоrоtnоgо vydаvliuvаnnia dеtаlеy z flаntsеm. Оbrаbоtkа маtеriаlоv dаvlеniеm. Krаmаtоrsk. 2019. № 1 (48). S. 23-34.

2. Маtvіychuk V. А., Bubnоvskа І. А. Оtsіnkа dеfоrmоvаnоstі mаtеrіаlu kryvоlіnіynyh zаgоtоvоk pry hоlоdnоmu vаltsuvаnnі. Теhnіkа, еnеrgеtykа, trаnspоrt АPК. Vіnnytsia. 2017. Vypusk 4. S. 92-97.

3. Мyhаlеvych V. М., Dоbrаniuk Iu. V., Кrаevskyi V. О. Pоrіvnialnе dоslіdzhеnnia mоdеlеy grаnychnyh plаstychnyh dеfоrmаtsіy. Vіsnyk mаshynоbuduvаnnia tа trаnspоrtu. Vіnnytsia. 2018. № 2. S. 56-64.

4. Grushkо О. V., Оgоrоdnіkоv V. А., Slоbоdianiuk Iu. О. Dеfоrmоvnіst malоvuglеtsеvоgо drоtu v prоtsеsі yоgо bаgаtоstupіnchаstоgо hоlоdnоgо vоlоchіnnia. Vіsnyk Vіnnytskоgо pоlіtеhnіchnоgо іnstytutu. Vіnnytsia. 2019. №3. S. 103-110.

5. Sivak R. Evaluation of metal plasticity and research of the mechanics of pressure treatment processes under complex loading. Eastern-European Journal of Enterprise Technologies. Kharkiv. 2017. №6/7 (90). P. 34-41.

6. Rekechynskyi V. І. Zastosuvannia metody funktsiy toku v statsionarnyh protsessah plastychnogo deformuvannia. Vibratsii v tehnitsi і tehnologiah. Vinnytsia. 2020. №2 (97). S. 157-163.

7. Syvаk R. І., Gunkо І. V., Zаlіzniak R. О. Zаstоsuvаnnia lіnіy tоku pry vyznаchеnnі kіnеmаtychnyh hаrаktеrystyk v stаtsіоnаrnyh prоtsеsаh plаstychnоy tеchiy mеtаlu. Vibratsii v tehnitsi і tehnologiah. Vinnytsia. 2021. №2 (97). S. 157-163.

 

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About journal

Topics of the journal:

The journal "Vibrations   in engineering  and technology" presents materials on the following issues

• Theory of processes and machines
• Mechanical Engineering and materialprocessing
• Processing and food production

Key information:
ISSN (print): 2306-8744
DOI: 10.37128/2306-8744

The certificateof massmediaState registration:kv no 16643-5115 from 30.04.2010 .
Founder of the journal: Vinnytsia National Agrarian University

Kind of publication: journal
Type of publication : Scientific
Publication status: Domestic
Year of founding:
Periodicity: 4 times a year
Extent: 18.75 nominal printed pages
ISSN: 2306-8744 (printed version), (online)
Language of edition  : (mixed languages) Ukrainian, English
The scope of the distribution and the category of readers: national, foreign, teaching staff, scientists, businessmen.
Periodical is included in the list of scientific professional editions of Ukraine approved by the Order of Ministry of Education and Sciences of Ukraine from 21.12.2015 No. 1328.
The journal "Vibrations in engineering and technology" is included in the "Catalogue of periodicals of Ukraine".
Journal subscription can be executed in each post office department.
 Subscription Index is  99720.

 

Old version of site: http://vibrojournal.vsau.edu.ua/

History of journal:

In June 1994 the 2nd International Scientific and Technical Conference "Application of vibrations for technological purposes" was organized on the basis of Vinnytsia State Agricultural Institute. Leading experts in this field, noting the significant contribution to the school of Vibration Engineering under the leadership of P. S. Bernyk, proposed to create a professional all-Ukrainian scientific and technical journal "Vibration in engineering and technology..The journal was foundedat Vinnytsia State Agricultural Institute and P.S. Bernyk was elected to be the chief editor .
For all these years (since 1994) theJournal "vibration in engineering and technology" published  94 issues wherestudy of vibration effects, the creation of progressive energy saving technologies and equipment for their implementation were highlighted.
Currently Kaletnik H.M  PhD , professor, academician NAAS is the chief editor of the "Vibrations in engineering  and Technology"
The journal "Vibration in Engineering and technology", which has no analogues on the territory of Ukraine, is well known abroad.