On solving the problem of contact problems with deformable stamp

Authors

  • Evdokimova O.V. Kuban State University, Krasnodar, Российская Федерация ORCID 0000-0003-1283-3870
  • Mukhin A.S. Kuban State University, Krasnodar, Российская Федерация ORCID 0000-0001-8935-0151
  • Uafa S.B. Kuban State University, Krasnodar, Российская Федерация
  • Bushueva O.A. Kuban State University, Krasnodar, Российская Федерация

UDC

539.3

DOI:

https://doi.org/10.31429/vestnik-20-3-42-49

Abstract

In this paper, for the first time, an approach is proposed that allows not only to construct exact solutions to contact problems with a deformable stamp for some types of non-classical domains, but also to explicitly obtain important relations describing new phenomena arising in these problems. By applying the block element method to the study of contact problems for a multilayer medium on the effect of a classical, semi-infinite, stamp and non-classical stamps in the form of a strip and a quarter plane, it was possible to obtain important properties not previously described. These include the following features and results. 1. The contact problem for a deformable stamp is available for solution only after the contact problem for an absolutely rigid stamp has been solved. 2. For the first time, exact solutions have been constructed for the cases of absolutely rigid stamps and deformable stamps of these shapes, the material of which is described by the Helmholtz equation. 3. In cases of dynamic tasks on harmonic effects on stamps, isolated resonances do not occur in contact problems with absolutely rigid stamps. Isolated resonances, first predicted by academician I.I. Vorovich, are present in contact problems with a deformable stamp. 4. The use of the block element method for solving contact problems allows, depending on the shape of the stamp, to obtain an explicit or integral dispersion equation describing resonant frequencies. 5. By the block element method using solutions of a system of Wiener-Hopf integral equations and a universal (fractal) modeling method can be used to solve the considered contact problems with deformable stamps consisting of materials of complex rheologies.

Keywords:

contact problem, block element, deformable stamp, fractals, rheology, Lame equations, Wiener-Hopf equations

Acknowledgement

The work was supported by the Russian Science Foundation (project no. 22-21-00129).

Author Infos

Olga V. Evdokimova

д-р физ.-мат. наук, главный научный сотрудник научно-исследовательской части Кубанского государственного университета

e-mail: evdokimova.olga@mail.ru

Aleksey S. Mukhin

канд. физ.-мат. наук, старший научный сотрудник Научно-исследовательской части Кубанского государственного университета

e-mail: muhin@mail.kubsu.ru

Samir B. Uafa

младший научный сотрудник Кубанского государственного университета

e-mail: uafa70@mail.ru

Olga A. Bushueva

аспирантка факультета компьютерных технологий и математики Кубанского государственного университета

e-mail: olyabushuyeva@gmail.com

References

  1. Papangelo, A., Ciavarella, M., Barber, J.R., Fracture mechanics implications for apparent static friction coefficient in contact problems involving slip-weakening laws. Proc. R. Soc. A., 2015, vol. 471. DOI: 10.1098/rspa.2015.0271
  2. Zhou, S., Gao, X.L., Solutions of half-space and half-plane contact problems based on surface elasticity. Zeitschrift fr angewandte Mathematik und Physik, 2013, vol. 64, pp. 145–166. DOI: 10.1007/s00033-012-0205-0
  3. Almqvist, A., An LCP solution of the linear elastic contact mechanics problem, 2013. URL: https://www.mathworks.com/matlabcentral/fileexchange/43216-an-lcp-solution-of-the-linear-elastic-contact-mechanics-problem
  4. Cocou, M., A class of dynamic contact problems with Coulomb friction in viscoelasticity. Nonlinear Analysis: Real World Applications, 2015, vol. 22, pp. 508–519. DOI: 10.1016/j.nonrwa.2014.08.012
  5. Ciavarella, M., The generalized Cattaneo partial slip plane contact problem. I–Theory. Int. J. Solids Struct., 1998, vol. 35, pp. 2349–2362. DOI: 10.1016/S0020-7683(97)00154-6
  6. Ciavarella, M., The generalized Cattaneo partial slip plane contact problem. II–Examples. Int. J. Solids Struct., 1998, vol. 35, pp. 2363–2378. DOI: 10.1016/S0020-7683(97)00155-8
  7. Guler, M.A., Erdogan, F., The frictional sliding contact problems of rigid parabolic and cylindrical stamps on graded coatings. Int. J. Mech. Sci., 2007, vol. 49, pp. 161–182. DOI: 10.1016/j.ijmecsci.2006.08.006
  8. Ke, L.-L., Wang, Y.-S., Two-dimensional sliding frictional contact of functionally graded materials. Eur. J. Mech. A/Solids, 2007, vol. 26, pp. 171–188. DOI: 10.1016/j.euromechsol.2006.05.007
  9. Almqvist, A., Sahlin, F., Larsson, R., Glavatskih, S., On the dry elasto-plastic contact of nominally flat surfaces. Tribology International, 2007, vol. 40, iss. 4, pp. 574–579. DOI: 10.1016/j.triboint.2005.11.008
  10. Andersson, L.E., Existence results for quasistatic contact problems with Coulomb friction. Appl. Math. Optim., 2000, vol. 42, pp. 169–202. DOI: 10.1007/s002450010009
  11. Cocou, M., Rocca, R., Existence results for unilateral quasistatic contact problems with friction and adhesion. Math. Modelling and Num. Analysis, 2000, vol. 34, pp. 981–1001.
  12. Kikuchi, N., Oden, J., Contact problems in elasticity: a study of variational inequalities and finite element methods. SIAM Studies in Applied Mathematics. SIAM, Philadelphia, 1988.
  13. Raous, M., Cangémi, L., Cocu, M., A consistent model coupling adhesion, friction, and unilateral contact. Comput. Meth. Appl. Mech. Engrg., 1999, vol. 177, pp. 383–399. DOI: 10.1016/S0045- 7825(98)00389-2
  14. Shillor, M., Sofonea, M., Telega, J.J., Models and analysis of quasistatic contact. Lect. Notes Phys., vol. 655. Springer, Berlin, Heidelberg, 2004.
  15. Бабешко, В.А., Евдокимова, О.В., Бабешко, О.М., О контактных задачах с деформируемым штампом. Проблемы прочности и пластичности, 2022, т. 84, №1, с. 25–34. [Babeshko, V.A., Evdokimova, O.V., Babeshko, O.M., On contact problems with a deformable stamp. Problemy prochnosti i plastichnosti = Problems of strength and plasticity, 2022, vol. 84, no. 1, pp. 25–34. (in Russian)] DOI: 10.32326/1814-9146-2022-84-1-25-34
  16. Ворович, И.И., Бабешко, В.А., Динамические смешанные задачи теории упругости для неклассических областей. Москва, Наука, 1979. [Vorovich, I.I., Babeshko, V.A., Dinamicheskie smeshannye zadachi teorii uprugosti dlya neklassicheskikh oblastey = Dynamic mixed problems of elasticity theory for nonclassical domains. Nauka, Moscow, 1979. (in Russian)]
  17. Бабешко, В.А., Евдокимова, О.В., Бабешко, О.М., Фрактальные свойства блочных элементов и новый универсальный метод моделирования. Доклады Академии наук, 2021, т. 499, с. 21–26. [Babeshko, V.A., Evdokimova, O.V., Babeshko, O.M., Fractal properties of block elements and a new universal modeling method. Doklady Akademii nauk = Rep. of the Academy of Sciences, 2021, vol. 499, pp. 21–26. (in Russian)] DOI: 9.31857/S2686740021040039
  18. Бабешко, В.А., Евдокимова, О.В., Бабешко, О.М., Об одном методе решения граничных задач динамической теории упругости в четвертьплоскости. ПММ, 2021, т. 85, № 3, с. 275–282. [Babeshko, V.A., Evdokimova, O.V., Babeshko, O.M., On a method for solving boundary value problems in the dynamic theory of elasticity in a quarter-plane. Prikladnaya matematika i mekhanika = Applied Mathematics and Mechanics, 2021, vol. 85, no. 3, pp. 275–282.] DOI: 10.31857/S0032823521030024
  19. Бабешко В.А., Евдокимова О.В., Бабешко О.М. Об одной факторизационной задаче Гильберта-Винера и методе блочного элемента. Доклады Академии наук, 2014, т. 459, № 5, с. 557–561. [Babeshko, V.A., Evdokimova, O.V., Babeshko, O.M., On one Hilbert-Wiener factorization problem and the block element method. Doklady Akademii nauk = Reports of the Academy of Sciences, 2014, vol. 459, no. 5, pp. 557–561. (in Russian)]
  20. Babeshko, V.A., Evdokimova, O.V., Babeshko, O.M., On the problem of evaluating the behavior of multicomponent materials in mixed boundary conditions in contact problems. Materials Physics and Mechanics, 2022, vol. 48, iss. 2, pp. 379–385. DOI: 10.18720/MPM.48(3)2022_8
  21. Ворович, И.И., Спектральные свойства краевой задачи теории упругости для неоднородной полосы.Доклады АН СССР, 1979, т. 245, № 4, с. 817–820. [Vorovich, I.I., Spectral properties of the boundary value problem of elasticity theory for an inhomogeneous strip.Doklady AN SSSR = Rep. of the Academy of Sciences of the USSR, 1979, vol. 245, no. 4, pp. 817–820. (in Russian)]
  22. Ворович, И.И., Резонансные свойства упругой неоднородной полосы. Доклады АН СССР, 1979, т. 245, № 5, с. 1076–1079. [Vorovich, II, Resonance properties of an elastic inhomogeneous strip. Doklady AN SSSR = Rep. of the Academy of Sciences of the USSR, 1979, vol. 245, no. 5, pp. 1076–1079. (in Russian)]

Downloads

Issue

2023. Vol. 20. No. 3

Section

Mechanics

Pages

42-49

Submitted

2023-08-16

Published

2023-09-29

How to Cite

Evdokimova O.V., Mukhin A.S., Uafa S.B., Bushueva O.A. On solving the problem of contact problems with deformable stamp. Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation, 2023, vol. 20, no. 3, pp. 42-49. DOI: https://doi.org/10.31429/vestnik-20-3-42-49 (In Russian)

Copyright (c) 2023 Evdokimova O.V., Mukhin A.S., Uafa S.B., Bushueva O.A.

Creative Commons License

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