Email this article 
On the Contact Problem with Deformable Stamp in the Quarter Plain
- Authors: Babeshko V.A.1,2, Evdokimova O.V.1, Babeshko O.M.2, Zaretskaya M.V.2, Evdokimov V.S.2
-
Affiliations:
- Southern Scientific Center of the Russian Academy of Sciences
- Kuban State University
- Issue: Vol 87, No 2 (2023)
- Pages: 303-313
- Section: Articles
- URL: https://jdigitaldiagnostics.com/0032-8235/article/view/675164
- DOI: https://doi.org/10.31857/S0032823523020030
- EDN: https://elibrary.ru/TYZHJI
- ID: 675164
Cite item
Open Access
Access granted
Subscription Access
Abstract
In this paper, for the first time, a two-dimensional dynamic contact problem on the action of a deformable stamp on a quarter of the plane of a multilayer medium is strictly mathematically investigated. In contrast to the case of an absolutely solid stamp, a deformable stamp introduces additional features, consisting in the possibility of the occurrence of discrete resonances predicted by academician I.I. Vorovich. The paper shows that the use of a method based on the use of block elements makes it possible to obtain an equation describing resonant frequencies. To study contact problems with a deformable stamp made of materials of complex rheology, including smart materials, it is proposed in the paper to first conduct a study for the case of a deformable stamp made of a material of simple rheology described by Helmholtz equations. Solutions of boundary value problems for stamps of complex rheology, after that, are represented by a combination of solutions of boundary value problems for stamps of simple rheology.
About the authors
V. A. Babeshko
Southern Scientific Center of the Russian Academy of Sciences; Kuban State University
Author for correspondence.
Email: babeshko41@mail.ru
Russia, Rostov-on-Don; Russia, Krasnodar
O. V. Evdokimova
Southern Scientific Center of the Russian Academy of Sciences
Author for correspondence.
Email: evdokimova.olga@mail.ru
Russia, Rostov-on-Don
O. M. Babeshko
Kuban State University
Author for correspondence.
Email: babeshko49@mail.ru
Russia, Krasnodar
M. V. Zaretskaya
Kuban State University
Author for correspondence.
Email: zarmv@mail.ru
Russia, Krasnodar
V. S. Evdokimov
Kuban State University
Author for correspondence.
Email: evdok_vova@mail.ru
Russia, Krasnodar
References
- Vorovich I.I. Spectral properties of the boundary value problem of elasticity theory for an inhomogeneous band // Dokl. akad. nauk SSSR, 1979, vol. 245, no. 4, pp. 817–820. (in Russian)
- Vorovich I.I. Resonant properties of an elastic inhomogeneous band // Dokl. akad. nauk SSSR, 1979, vol. 245, no. 5, pp. 1076–1079. (in Russian)
- Vorovich I.I., Babeshko V.A., Prakhina O.D. Dynamics of Massive Bodies and Resonant Phenomena in Deformable Media. Moscow: Nauka, 1999. 246 p. (in Russian).
- Babeshko V.A., Evdokimova O.V., Babeshko O.M. Fractal properties of block elements and a new universal modeling method // Dokl. Phys., 2021, vol. 66, iss. 8, pp. 218–222.
- Babeshko V.A., Evdokimova O.V., Babeshko O.M. On contact problems with a deformable stamp // Problems of Strength&Plasticity, 2022, vol. 84, no. 1, pp. 25–34. doi: 10.32326/1814-9146-2022-84-1-25-34 (in Russian)
- Goracheva I.G., Dobichin M.N. Contact Problems of Tribology. Moscow: Mashinostroenie, 1988. 256 p. (in Russian)
- Papangelo A., Ciavarella M., Barber J.R. Fracture Mechanics implications for apparent static friction coefficient in contact problems involving slip-weakening laws // Proc. Roy. Soc., 2015, A 471, iss. 2180, Art. No. 20150271.
- Ciavarella M. The generalized Cattaneo partial slip plane contact problem. I-Theory, II-Examples // Int. J. Solids Struct., 1998, vol. 35, pp. 2349–2378.
- 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.
- 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.
- 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.
- Almqvist A., Sahlin F., Larsson R., Glavatskih S. On the dry elasto-plastic contact of nominally flat surfaces // Tribol. Int., 2007, vol. 40 (4), pp. 574–579. doi: 10.31857/S0032823522050046
- Almqvist A. An lcp solution of the linear elastic contact mechanics problem. // http://www.mathworks.com/matlabcentral/fileexchange/43216.
- Andersson L.E. Existence results for quasistatic contact problems with Coulomb friction // Appl. Math. Optim., 2000, vol. 42, pp. 169–202.
- Cocou M. A class of dynamic contact problems with Coulomb friction in viscoelasticity // Nonlin. Anal.: Real World Appl., 2015, vol. 22, pp. 508–519.
- Babeshko V.A., Evdokimova O.V., Babeshko O.M. Exact Solution to the Contact Problem in a Quarter-Plane of a Multilayer Medium by the Universal Simulation Method // Mech. Solids, 2022, vol. 57, no. 8, pp. 2058–2065. doi: 10.3103/S0025654422080039
Supplementary files
