Investigation of the possibility of a discrete spectrum in the block structure of the base and stamp and the nature of the wave field emitted outside the deformable stamp

Authors

  • Evdokimova O.V. Kuban State University, Krasnodar, Российская Федерация ORCID 0000-0003-1283-3870
  • Babeshko O.M. Kuban State University, Krasnodar, Российская Федерация ORCID 0000-0003-1869-5413
  • Babeshko V.A. Kuban State University, Krasnodar, Российская Федерация ORCID 0000-0002-6663-6357
  • Khripkov D.A. Kuban State University, Krasnodar, Российская Федерация ORCID 0000-0002-2161-121X
  • Mukhin A.S. Kuban State University, Krasnodar, Российская Федерация ORCID 0000-0001-8935-0151
  • Evdokimov V.S. Kuban State University, Krasnodar, Российская Федерация
  • Uafa S.B. Kuban State University, Krasnodar, Российская Федерация

UDC

539.3

DOI:

https://doi.org/10.31429/vestnik-19-4-57-67

Abstract

In the previously published work of the authors, the formulation and solution of the contact problem with a deformable stamp is investigated. Correctness is shown the formulation of the problem, which consists in the possibility of determining all the parameters of the solution. In particular, the functionals arising in these contact problems from the desired contact voltages. In this paper, the study of the features of solutions of contact problems with a deformable stamp continues. In contrast to the case of contact problems with an absolutely solid stamp, in the case of deformable stamps, discrete spectra may appear in the operator of a mixed problem. The paper identifies transcendental type relations that may contain this spectrum. In the case of a semi-infinite deformable stamp, this relation does not contain real points of the spectrum. The question of the behavior of wave fields excited on the surface by a semi-infinite deformable stamp is studied. The study was based on a newly developed universal modeling method that allows application both in boundary value problems for differential equations and in some types of integral equations. The construction of packed block elements in the quadrants of the Cartesian coordinate system is demonstrated.

Keywords:

contact problem, deformable stamp, integral equation, multilayer medium, block elements

Acknowledgement

The work was supported by the Russian Science Foundation (project 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

Olga M. Babeshko

д-р физ.-мат. наук, главный научный сотрудник научно-исследовательского центра прогнозирования и предупреждения геоэкологических и техногенных катастроф Кубанского государственного университета

e-mail: babeshko49@mail.ru

Vladimir A. Babeshko

академик РАН, д-р физ.-мат. наук, заведующий кафедрой математического моделирования Кубанского государственного университета

e-mail: babeshko41@mail.ru

Dmitry A. Khripkov

научный сотрудник Кубанского государственного университета

e-mail: vestnik@fpm.kubsu.ru

Aleksey S. Mukhin

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

e-mail: muhin@mail.kubsu.ru

Vladimir S. Evdokimov

магистрант Кубанского государственного университета

e-mail: evdok_vova@mail.ru

Samir B. Uafa

младший научный сотрудник Кубанского государственного университета

e-mail: uafa70@mail.ru

References

  1. 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 akademii nauk SSSR = Proc. of the Academy of Sciences of the USSR, 1979, vol. 245, no. 4, pp. 817–820. (in Russian)]
  2. Ворович, И.И., Резонансные свойства упругой неоднородной полосы. Доклады АН СССР, 1979, т. 245, №5, с. 1076–1079. [Vorovich, I.I., Resonance properties of an elastic inhomogeneous band. Doklady akademii nauk SSSR = Proc. of the Academy of Sciences of the USSR, 1979, vol. 245, no. 5, pp. 1076–1079. (in Russian)]
  3. 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
  4. 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)]
  5. 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
  6. Zhou, S., Gao, X.L., Solutions of half-space and half-plane contact problems based on surface elasticity. Zeitschrift für angewandte Mathematik und Physik, 2013, vol. 64, pp. 145–166. DOI 10.1007/s00033-012-0205-0
  7. 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
  8. 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
  9. 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
  10. 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
  11. 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
  12. 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
  13. 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
  14. 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
  15. 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.
  16. 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.
  17. 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.
  18. Shillor, M., Sofonea, M., Telega, J.J., Models and analysis of quasistatic contact. Lect. Notes Phys., vol. 655. Springer, Berlin, Heidelberg, 2004.
  19. 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
  20. 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
  21. 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

Downloads

Issue

2022. Vol. 19. No. 4

Section

Mechanics

Pages

57-67

Submitted

2022-11-15

Published

2022-11-30

How to Cite

Evdokimova O.V., Babeshko O.M., Babeshko V.A., Khripkov D.A., Mukhin A.S., Evdokimov V.S., Uafa S.B. Investigation of the possibility of a discrete spectrum in the block structure of the base and stamp and the nature of the wave field emitted outside the deformable stamp. Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation, 2022, vol. 19, no. 4, pp. 57-67. DOI: https://doi.org/10.31429/vestnik-19-4-57-67 (In Russian)

Copyright (c) 2022 Evdokimova O.V., Babeshko O.M., Babeshko V.A., Khripkov D.A., Mukhin A.S., Evdokimov V.S., Uafa S.B.

Creative Commons License

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