A nonstationary contact problem for a stamp and a membrane in an axisymmetric formulation


Аuthors

Kireenkov A. A.1*, Okonechnikov A. S.2**, Feoktistova E. S.***

1. Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, 101-1, prospekt Vernadskogo, Moscow, 119526, Russia
2. Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: greghome@mail.ru
**e-mail: leon_lionheart@mail.ru
***e-mail: feoktistova0206@mail.ru

Abstract

This work is devoted to the development of an algorithm for solving contact problems of interaction between a rigid indenter and a membrane in an axisymmetric formulation. The problem assumes the presence of two stages of interaction at different velocity - supersonic and subsonic. The subsonic stage of contact interaction has a number of features, for the solution of which two hypotheses were introduced. According to these hypotheses, the shape of the membrane under the indenter will coincide with the shape of the indenter itself, and the membrane area outside the contact will take the form of a linear function. Note that these hypotheses are formulated in such a way that they do not apply to the supersonic stage of contact interaction. The introduction of the above hypotheses makes it possible to determine the position of the boundaries of the contact area of the indenter and the membrane at any time, if the displacement of the frontal point of the indenter is known. Further, the developed algorithm makes it possible to determine the deviations of the membrane points from the initial state, as well as the contact pressure under the indenter. It is worth noting that the expression for the contact pressure obtained in the course of solving the problem makes it possible to speak about the presence of both distributed and concentrated loads. In this case, concentrated loads will appear only if the expansion of the contact area coincides with the speed of sound. Based on the data obtained when solving the subsonic stage of interaction, we proceed to the solution at the supersonic stage. The problem statement will be similar to the subsonic stage. However, taking into account the features of the supersonic stage of contact interaction, we will assume that the deflection of the membrane at supersonic speeds of the indenter is completely determined by the shape of the indenter and the depth of its penetration. Further, taking into account the presence of the same carrier in the structure of the contact pressure and the shape of the indenter, the pressure under the indenter is determined. It is worth noting that the obtained structure of distributed loads under the indenter at the supersonic stage of contact interaction will coincide with the results obtained for the subsonic stage. Thus, we can conclude that the structure of the distributed pressure at different speeds of contact interaction remains unchanged, however, at velocity exceeding sonic, concentrated forces will arise at the contact boundaries.

Keywords:

non-stationary contact, rigid stamp, membrane, membrane deflection, contact area, influence function

References

  1. Krasnyuk P.P. Trenie i iznos, 2005, vol. 26, no. 2, pp. 117-127.
  2. Nemirovich-Danchenko M.M., Khudorozhko I.N. Mezhdunarodnaya konferentsiya «Fizicheskaya mezomekhanika materialov. Fizicheskie printsipy formirovaniya mnogourovnevoi struktury i mekhanizmy nelineinogo povedeniya»: tezisy dokladov. Novosibirsk, Novosibirskii natsional'nyi issledovatel'skii gosudarstvennyi universitet, 2022, pp. 328-329. DOI: 10.25205/978-5-4437-1353-3-197
  3. Pozharskii D.A. Doklady akademii nauk, 2014, vol. 455, no. 2, pp. 158. DOI: 10.7868/S0869565214080118
  4. Tarlakovskii D.V., Fedotenkov G.V. Uchenye zapiski Kazanskogo universiteta. Seriya: fiziko-matematicheskie nauki, 2016, vol. 158, no. 1, pp. 141-151.
  5. Vestyak A.V., Igumnov L.A., Tarlakovskii D.V., Fedotenkov G.V. Vychislitel'naya mekhanika sploshnykh sred, 2016, vol. 9, vol. 4, pp. 443-452.
  6. Arutyunyan A.M., Tarlakovskii D.V., Fedotenkov G.V. Materialy X Vserossiiskoi konferentsii po mekhanike deformiruemogo tverdogo tela: sbornik trudov. Samara, Samarskii gosudarstvennyi tekhnicheskii universitet, 2017, vol. 1, pp. 50-53.
  7. Gorshkov A.G., Medvedskii A.L., Rabinskii L.N., Tarlakovskii D.V. Volny v sploshnykh sredakh (Waves in continuous media), Moscow, Fizmatlit, 2004, 472 p.
  8. Okonechnikov A.S., Tarlakovsky D.V., Fedotenkov G.V. Transient Interaction of Rigid Indenter with Elastic Half-plane with Adhesive Force, Lobachevskii Journal of Mathematics, 2019, vol. 40, no. 4, 489-498. DOI: 10.1134/S1995080219030132
  9. Mikhailova E.Y., Fedotenkov G.V., Tarlakovskii D.V. Transient contact problem for spherical shell and elastic half-space, Shell Structures. Proceedings of the 11th International Conference on Shell Structures: Theory and Applications, 2017, pp. 301-304. DOI: 10.1201/9781315166605-67
  10. Lai Tkhan' Tuan, Tarlakovskii D.V. Trudy MAI, 2012, no. 53. URL: http://trudymai.ru/eng/published.php?ID=29267
  11. Davydov S.A., Akhmetova E.R., Zemskov A.V. Thermoelastic diffusion multicomponent half-space under the effect of surface and bulk unsteady perturbations, Mathematical and Computational Applications, 2019, vol. 24 (1), pp. 26. DOI: 10.3390/mca24010026
  12. Yulong L.I, Arutiunian A.M., Kuznetsova El.L., Fedotenkov G.V. Method for solving plane unsteady contact problems for rigid stamp and elastic half-space with a cavity of arbitrary geometry and location, INCAS BULLETIN, 2020, vol. 12, special issue, pp. 99-113. DOI: 10.13111/2066-8201.2020.12.S.9
  13. Gol'dshtein R.V. Prikladnaya matematika i mekhanika, 1965, vol. 29, no. 3, pp. 516–525.
  14. Tarlakovskii D.V., Fedotenkov G.V. Two-Dimensional Nonstationary Contact of Elastic Cylindrical or Spherical Shells, Journal of Machinery Manufacture and Reliability, 2014, vol. 43, no. 2, pp. 145–152. DOI: 10.3103/S1052618814010178
  15. Soldatenkov I.A. Prikladnaya matematika i mekhanika, 2017, vol. 81, no. 3, pp. 257-277.
  16. Aver'yanov I.O. Zinin A.V. Trudy MAI, 2022, no. 124. URL: https://trudymai.ru/eng/published.php?ID=167067. DOI: 10.34759/trd-2022-124-12
  17. Lokteva N.A., Ivanov S.I. Trudy MAI, 2021, no. 117. URL: https://trudymai.ru/eng/published.php?ID=122234. DOI: 10.34759/trd-2021-117-05
  18. Martirosov M.I., Khomchenko A.V. Trudy MAI, 2022, no. 126. URL: https://trudymai.ru/eng/published.php?ID=168990. DOI: 10.34759/trd-2022-126-04
  19. Firsanov V.V., Fam V.T., Chan N.D. Trudy MAI, 2021, no. 114. URL: https://trudymai.ru/eng/published.php?ID=118893. DOI: 10.34759/trd-2020-114-07
  20. Lokteva N.A., Serdyuk D.O., Skopintsev P.D., Fedotenkov G.V. Trudy MAI, 2021, no. 120. URL: https://trudymai.ru/eng/published.php?ID=161423. DOI: 10.34759/trd-2021-120-09


Download

mai.ru — informational site MAI

Copyright © 2000-2024 by MAI

 Вход