Nonlinear boundary integral equations method for contact problems of the elasticity theory

Authors

  • Александр Иванович Александров Zaporizhzhya National University Zhukovsky str., 66, Zaporizhzhya, Ukraine, 69600, Ukraine https://orcid.org/0000-0001-9265-1991
  • Юрий Михайлович Стреляев Zaporizhzhya National University Zhukovsky str., 66, Zaporizhzhya, Ukraine, 69600, Ukraine

DOI:

https://doi.org/10.15587/1729-4061.2014.24853

Keywords:

elastic body, contact problem, Coulomb friction, integral equation, iterative method

Abstract

When implementing variational methods for solving complex contact problems, there are difficulties, associated with non-convexity of minimized energy function of the system of interacting bodies and non-differentiability of this function at the desired point of its minimum. These difficulties do not allow to use gradient methods and convex analysis methods to minimize the energy function, therefore, numerical procedures for finding minimum points of such functions are cumbersome in program implementation and sometimes make it impossible to obtain the contact problem solution with sufficient accuracy. Non-variational method, based on using nonlinear operator equations with no difficulties during its implementation is proposed in the paper. Applying these equations allows to use modern achievements of nonlinear functional analysis, fixed-point theory of continuous mappings, theory of iterative methods for solving operator equations for both proving theorems of existence of solutions to contact problems, and developing effective iterative procedures for approximate solutions. Nonlinear boundary integral equations, used in this paper to simulate the contact interaction of elastic bodies, allow (unlike other similar equations) to take into account both the linkage and partial slip on the contact surface of bodies, and loading history of these bodies. Based on these equations, simple and efficient iterative procedures for approximate solutions to the contact problems are developed. A numerical solution of the contact problem on indenting the elastic sphere in the elastic half-space is obtained, and comparison of results with the known problem solution is made. 

Author Biographies

Александр Иванович Александров, Zaporizhzhya National University Zhukovsky str., 66, Zaporizhzhya, Ukraine, 69600

Docent

Department of calculus

Юрий Михайлович Стреляев, Zaporizhzhya National University Zhukovsky str., 66, Zaporizhzhya, Ukraine, 69600

Senior lecturer

Department of calculus

References

  1. Кравчук, А. С. Вариационный метод в контактных задачах. Состояние проблемы, направления развития [Текст] / А. С. Кравчук // Прикл. математ. и механика. – 2009. – Вып. 73, № 3. – С. 492–502.
  2. Reina, S. A quadratic programming formulation for the solution of layered elastic contact problems: Example applications and experimental validation [Text] / S. Reina, D. Dini, D. A. Hills, Y. Lida // European Journal of Mechanics – A/Solids. – 2011. – Vol. 30, Issue 3. – P. 236–247.
  3. Галанов, Б. А. О приближенном решении некоторых задач упругого контакта двух тел [Текст] / Б. А. Галанов // Изв. АН СССР. Механика твёрдого тела. – 1981. – № 5. – С. 61–67.
  4. Галанов, Б. А. Метод граничных уравнений типа Гаммерштейна для контактных задач теории упругости в случае неизвестных областей контакта [Текст] / Б. А. Галанов // Прикл. математ. и механика. – 1985. – Т. 49, Вып. 5. – С. 827–835.
  5. Александров, В. М. Пространственная контактная задача для двухслойного упругого основания с заранее неизвестной областью контакта [Текст] / В. М. Александров, J. J. Kalker, Д. А. Пожарский // Изв. РАН. Механика твёрдого тела. – 1999. – № 4. – С. 51–55.
  6. Александров, В. М. Трёхмерные контактные задачи при учёте трения и нелинейной шероховатости [Текст] / В. М. Александров, Д. А. Пожарский // Прикл. математ. и механика. – 2004. – Т. 68, Вып. 3. – С. 516–527.
  7. Чебаков, М. И. Трёхмерная контактная задача для слоя с учётом сил трения в области контакта [Текст] / М. И. Чебаков // Изв. РАН. Механика твёрдого тела. – 2002. – № 6. – С. 59–68.
  8. Sundaram, Naravan Mechanics of advancing pin-loaded contact with friction [Text] / Naravan Sundaram, T. N. Farris // Journal of the Mechanics and Physics of Solids. – 2010. – Vol. 58, Issue 11. – P. 1819–1833.
  9. Александров, А. И. Метод решения пространственной контактной задачи о взаимодействии двух упругих тел при наличии трения между ними [Текст] / А. И. Александров // Математичнi методи i фiзико-механiчнi поля. – 2013. – Т.56, № 3. – С. 29–42.
  10. Александров, А. И. Решение задач о контакте упругих тел с использованием нелинейных интегральных уравнений [Текст] / А. И. Александров // Доп. Національної академії наук України. – 2012. – № 11. –С. 47–52.
  11. Kalker, J. J. A survey of the mechanics of contact between solid bodies [Text] / J. J. Kalker // ZAMM. – 1977. – B.57, H.5. – P. T3–T17.
  12. Джонсон, К. Механика контактного взаимодействия [Текст] / Пер. с англ. В. Э. Наумова, А. А. Спектора под ред. Р. В. Гольдштейна. – М.: Мир, 1989. – 510 с.
  13. Kravchuk, A. S. (2009). Variational method in contact problems. State of problem, directions of development. Prikladnaia matematika i mekhanika, 73 (3), 492–502.
  14. Reina, S., Dini, D., Hills, D. A., Lida, Y. (2011). A quadratic programming formulation for the solution of layered elastic contact problems: Example applications and experimental validation. European Journal of Mechanics-A/Solids, 30 (3), 236–247.
  15. Galanov, B. A. (1981). On an approximate solution of some problems for two bodies elastic contact. Izvestiya AN USSR. Mekhanika Tverdogo Tela, (5), 61–67.
  16. Galanov, B. A. (1985). Method of boundary equations of Hammerstein kind for contact problems of linear elasticity in case unknown contact areas. Prikladnaia matematika i mekhanika, 49 (5), 827–835.
  17. Aleksandrov, V. M., Kalker, J. J., Pozharskii, D. A. (1999). Three-dimensional contact problem for a two-layered elastic base with an unknown contact area. Izvestiya RAN. Mekhanika Tverdogo Tela, (4), 51–55.
  18. Aleksandrov, V. M., Pozharskii, D. A. (2004). Three-dimensional contact problems with friction and non-linear roughness taken into account. Prikladnaia matematika i mekhanika, 68 (3), 516–527.
  19. Chebakov, M. I. (2002). Three-dimensional contact problem for a layer with allowance for friction in a contact area. Izvestiya RAN. Mekhanika Tverdogo Tela, (6), 59–68.
  20. Sundaram, N., Farris, T. N. (2010). Mechanics of advancing pin-loaded contacts with friction. Journal of the Mechanics and Physics of Solids, 58 (11), 1819–1833.
  21. Alexandrov, A. I. (2013). Method of the solution for three-dimensional contact problem of interaction two elastic bodies in presence of friction. Mathematical Methods and Physicomechanical Fields, 56 (3), 29–42.
  22. Alexandrov, A. I. (2012). Solution of problems for contact of elastic bodies by use non-linear integral equations. Reports of the National Academy of Sciences of Ukraine, 11, 47–52.
  23. Kalker, J. J. (1977) A survey of the mechanics of contact between solid bodies. ZAMM, 57 (5), 3–17.
  24. Johnson, K. (1989). Contact Mechanics. Moscow, USSR: Mir, 510.

Downloads

Published

2014-06-20

How to Cite

Александров, А. И., & Стреляев, Ю. М. (2014). Nonlinear boundary integral equations method for contact problems of the elasticity theory. Eastern-European Journal of Enterprise Technologies, 3(7(69), 36–40. https://doi.org/10.15587/1729-4061.2014.24853

Issue

Vol. 3 No. 7(69) (2014): Applied mechanics

Section

Applied mechanics

License

Copyright (c) 2014 Александр Иванович Александров, Юрий Михайлович Стреляев

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

The consolidation and conditions for the transfer of copyright (identification of authorship) is carried out in the License Agreement. In particular, the authors reserve the right to the authorship of their manuscript and transfer the first publication of this work to the journal under the terms of the Creative Commons CC BY license. At the same time, they have the right to conclude on their own additional agreements concerning the non-exclusive distribution of the work in the form in which it was published by this journal, but provided that the link to the first publication of the article in this journal is preserved.

A license agreement is a document in which the author warrants that he/she owns all copyright for the work (manuscript, article, etc.).
The authors, signing the License Agreement with TECHNOLOGY CENTER PC, have all rights to the further use of their work, provided that they link to our edition in which the work was published.
According to the terms of the License Agreement, the Publisher TECHNOLOGY CENTER PC does not take away your copyrights and receives permission from the authors to use and dissemination of the publication through the world's scientific resources (own electronic resources, scientometric databases, repositories, libraries, etc.).
In the absence of a signed License Agreement or in the absence of this agreement of identifiers allowing to identify the identity of the author, the editors have no right to work with the manuscript.
It is important to remember that there is another type of agreement between authors and publishers – when copyright is transferred from the authors to the publisher. In this case, the authors lose ownership of their work and may not use it in any way.