Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

For citation:

Vatulyan A. O., Plotnikov D. K. Contact problem for functionally graded orthotropic strip. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2022, vol. 22, iss. 4, pp. 479-493. DOI: 10.18500/1816-9791-2022-22-4-479-493, EDN: JDIVGD

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
30.11.2022
Full text:
download
(downloads: 953)
Language: 
Russian
Heading: 
Mechanics
Article type: 
Article
UDC: 
539.3
DOI: 
10.18500/1816-9791-2022-22-4-479-493
EDN: 
JDIVGD

Contact problem for functionally graded orthotropic strip

Autors: 
Vatulyan Alexander Ovanesovitsch, Southern Federal University
Plotnikov Dmitry K., Southern Mathematical Institute — the Affiliate of Vladikavkaz Scientific Centre of Russian Academy of Sciences
Abstract: 

Within the framework of plane elasticity, the equilibrium problem for an inhomogeneous orthotropic elastic strip under the action of a stamp with a smooth base is investigated. Based on the Fourier transform, a canonical system of differential equations with variable coefficients with respect to transformants of the displacement vector and stress tensor components is constructed. A connection between the vertical displacement and the normal boundary stress is constructed, with which an integral equation of the first kind with a difference kernel is formulated. Using the shooting method, the kernel symbol for the integral equation of the contact problem is constructed numerically. Based on the Vishik – Lyusternik method, an asymptotic analysis of the kernel symbol for large values of the transformation parameter is carried out. A computational scheme for solving an integral equation with an unknown contact area is~constructed. The solution of the contact problem for different laws of strip inhomogeneity is presented.

Key words: 
contact problem
functionally graded strip
orthotropic material
asymptotic analysis
boundary element method
Acknowledgments: 
This work was partially supported by the Russian Science Foundation (project No. 22-11-00265).
References: 
  1. Golovin Yu. I. Nanoindentation and mechanical properties of solids in submicrovolumes, thin near-surface layers, and films: a review. Physics of the Solid State, 2008, vol. 50, iss. 12, pp. 2205–2236. https://doi.org/10.1134/S1063783408120019
  2. Epshtein S. A., Borodich F. M., Bull S. J. Evaluation of elastic modulus and hardness of highly inhomogeneous materials by nanoindentation. Applied Physics A, 2015, vol. 119, iss. 1, pp. 325–335. https://doi.org/10.1007/s00339-014-8971-5
  3. Vorovich I. I., Ustinov Iu. A. Pressure of a die on an elastic layer of finite thickness. Journal of Applied Mathematics and Mechanics, 1959, vol. 23, iss. 3, pp. 637–650. https://doi.org/10.1016/0021-8928(59)90158-3
  4. Vorovich I. I., Alexandrov V. M., Babeshko V. A. Neklassicheskie smeshannye zadachi teorii uprugosti [Non-classical Mixed Problems in Elasticity Theory]. Moscow, Nauka, 1974. 456 p. (in Russian).
  5. Babeshko V. A. Asymptotic properties of the solutions of a class of integral equations occurring in elasticity theory and mathematical physics. Soviet Physics. Doklady, 1969, vol. 14, pp. 529–531.
  6. Alexandrov V. M., Babeshko V. A. Contact problems for an elastic strip of small thickness. Izv. USSR Academy of Sciences. Mechanics, 1965, iss. 2, pp. 95–107 (in Russian).
  7. Aizikovich S. M., Aleksandrov V. M., Belokon A. V., Krenev L. I., Trubchik I. S. Kontaktnie zadachi teorii uprugosti dlya neodnorodnyh sred [Contact Problems of Theory of Elasticity for Inhomogeneous Media]. Moscow, Fizmatlit, 2006. 240 p. (in Russian). EDN: OPWVHF
  8. Alexandrov V. M., Mhitaryan S. M. Kontaktnie zadachi dlya tel s tonkimi pokritiyami i podlozhkami [Contact Problems for Bodies with Thin Coatings and Layers]. Moscow, Nauka, 1983. 488 p. (in Russian).
  9. Argatov I. I. Asimptoticheskie modeli uprugogo kontakta [Asymptotic Models of Elastic Contact]. St. Petersburg, Nauka, 2005. 447 p. (in Russian). EDN: QJQFGH
  10. Vatulyan A. O., Plotnikov D. K. A model of indentation for a functionally graded strip. Doklady Physics, 2019, vol. 64, iss. 4, pp. 173–175. https://doi.org/10.1134/S1028335819040074
  11. Vatulyan A. O., Plotnikov D. K., Poddubny A. A. On some models of indentation for functionally-graded coatings. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2018, vol. 18, iss. 4, pp. 421–432 (in Russian). https://doi.org/10.18500/1816-9791-2018-18-4-421-432
  12. Conway H. D., Vogel S. M., Farnham K. A., So S. Normal and shearing contact stresses in indented strip and slabs. International Journal of Engineering Science, 1966, vol. 4, iss. 4, pp. 343–359. https://doi.org/10.1016/0020-7225(66)90036-X
  13. Volkov S. S., Vasilev A. S, Aizikovich S. M., Seleznev N. M., Leonteva A. V. Stress-strain state of an elastic soft functionally-graded coating subjected to indentation by a spherical punch. PNRPU Mechanics Bulletin, 2016, no. 4, pp. 20–34 (in Russian). https://doi.org/10.15593/perm.mech/2016.4.02
  14. Vasiliev A. S., Volkov S. S., Aizikovich S. M. Approximated analytical solution of contact problem on indentation of elastic half-space with coating reinforced with inhomogeneous interlayer. Materials Physics and Mechanics, 2018, vol. 35, iss. 1, pp. 175–180. https://doi.org/10.18720/MPM.3512018_20
  15. Vatulyan A. O., Plotnikov D. K. On a study of the contact problem for an inhomogeneous elastic strip. Mechanics Solids, 2021, vol. 56, no. 7, pp. 1379–1387. https://doi.org/10.3103/S0025654421070268
  16. Vatulyan A. O. On the action of a rigid stamp on an anisotropic half-space. In: I. I.Vorovich (ed.) Staticheskie i dinamicheskie smeshannye zadachi teorii uprugosti [Static and Dynamic Mixed Problems of Elasticity Theory]. Rostov-on-Don, Rostov University Publ., 1983, pp. 112–115 (in Russian). EDN: XQUWZM
  17. Pozharskii D. A. Contact problem for an orthotropic half-space. Mechanics of Solids, 2017, vol. 52, pp. 315–322. https://doi.org/10.3103/S0025654417030086
  18. Batra R. C., Jiang W. Analytical solution of the contact problem of a rigid indenter and an anisotropic linear elastic layer. International Journal of Solids and Structures, 2008, vol. 45, iss. 22–23, pp. 5814–5830. https://doi.org/10.1016/j.ijsolstr.2008.06.016
  19. Erbas B., Yusufoglu E., Kaplunov J. A plane contact problem for an elastic orthotropic strip. Journal of Engineering Mathematics, 2011, vol. 70, pp. 399–409. https://doi.org/10.1007/s10665-010-9422-8
  20. Greenwood J. A., Barber J. R. Indentation of an elastic layer by a rigid cylinder. International Journal of Solids and Structures, 2012, vol. 49, iss. 21, pp. 2962–2977. https://doi.org/10.1016/j.ijsolstr.2012.05.036
  21. Argatov I. I., Mishuris G. S., Paukshto M. V. Cylindrical lateral depth-sensing indentation testing of thin anisotropic elastic films. European Journal of Mechanics — A/Solids, 2015, vol. 49, pp. 299–307. https://doi.org/10.1016/j.euromechsol.2014.07.009
  22. Mozharovsky V. V., Kuzmenkov D. S. The technique for determining the parameters of a contact for indenter with the orthotropic coating on the elastic isotropic substrate. Problems of Physics, Mathematics and Technics, 2016, iss. 4 (29), pp. 74–82 (in Russian). EDN: XEEKKX
  23. Mozharovsky V. V., Maryina N. A., Kuzmenkov D. S. Realization of solution of the contact problem on indentation of rigid cylindrical indenter in isotropic viscoelastic strip on the orthotropic basis. Problems of Physics, Mathematics and Technics, 2018, iss. 2 (35), pp. 51–56 (in Russian). EDN: XUFGCL
  24. Comez I., Yilmaz K. B., Guler M. A., Yildirim B. On the plane frictional contact problem of a homogeneous orthotropic layer loaded by a rigid cylindrical stamp. Archive of Applied Mechanics, 2019, vol. 89, pp. 1403–1419. https://doi.org/10.1007/s00419-019-01511-6
  25. Yilmaz K. B., Comez I., Guler M. A., Yildirim B. The effect of orthotropic material gradation on the plane sliding frictional contact mechanics problem. The Journal of Strain Analysis for Engineering Design, 2019, vol. 54, iss. 4, pp. 254–275. https://doi.org/10. 1177/0309324719859110
  26. Babeshko V. A., Glushkov E. V., Zinchenko Zh. F. Dinamika neodnorodnykh lineino-uprugikh sred [Dynamics of Inhomogeneous Linearly Elastic Media]. Moscow, Nauka, 1989. 344 p. (in Russian).
  27. Bakhvalov N. S. Chislennye metody (analiz, algebra, obyknovennye differentsial’nye uravneniia) [Numerical Methods (Analysis, Algebra, Ordinary Differential Equations)]. Moscow, Nauka, 1975. 632 p. (in Russian).
  28. Vishik M. I., Ljusternik L. A. Regular degeneracy and boundary layer for linear differential equations with a small parameter. Uspekhi Matematicheskikh Nauk, 1957, vol. 12, iss. 5 (77), pp. 3–122 (in Russian).
  29. Benerdzhi P., Batterfild R. Metody granichnykh elementov v prikladnykh naukakh [Boundary Element Methods in Applied Sciences]. Moscow, Mir, 1984. 244 p. (in Russian).
Received: 
06.06.2022
Accepted: 
05.08.2022
Published: 
30.11.2022
  • 1268 reads