Contact problem for functionally graded orthotropic strip. (Russian. English summary) Zbl 1505.74166
Summary: 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.
Keywords:
contact problem; functionally graded strip; orthotropic material; asymptotic analysis; boundary element methodReferences:
| [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, 50:12 (2008), 2205-2236 · doi: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, 119:1 (2015), 325-335 · doi: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, 23:3 (1959), 637-650 · Zbl 0091.40004 · doi:10.1016/0021-8928(59)90158-3 |
| [4] | Vorovich I. I., Alexandrov V. M., Babeshko V. A., Non-classical Mixed Problems in Elasticity Theory, Nauka, M., 1974, 456 pp. (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, 14 (1969), 529-531 · Zbl 0188.17702 |
| [6] | Alexandrov V. M., Babeshko V. A., “Contact problems for an elastic strip of small thickness”, Izv. USSR Academy of Sciences. Mechanics, 1965, no. 2, 95-107 (in Russian) |
| [7] | Aizikovich S. M., Aleksandrov V. M., Belokon A. V., Krenev L. I., Trubchik I. S., Contact Problems of Theory of Elasticity for Inhomogeneous Media, Fizmatlit, M., 2006, 240 pp. (in Russian) |
| [8] | Alexandrov V. M., Mhitaryan S. M., Contact Problems for Bodies with Thin Coatings and Layers, Nauka, M., 1983, 488 pp. (in Russian) |
| [9] | Argatov I. I., Asymptotic Models of Elastic Contact, Nauka, St. Petersburg, 2005, 447 pp. (in Russian) · Zbl 1107.74001 |
| [10] | Vatulyan A. O., Plotnikov D. K., “A model of indentation for a functionally graded strip”, Doklady Physics, 64:4 (2019), 173-175 · doi: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, 18:4 (2018), 421-432 (in Russian) · Zbl 1454.74007 · doi: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, 4:4 (1966), 343-359 · doi: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, 20-34 (in Russian) · doi: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, 35:1 (2018), 175-180 · doi: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, 56:7 (2021), 1379-1387 · Zbl 1482.74128 · doi:10.3103/S0025654421070268 |
| [16] | Vatulyan A. O., “On the action of a rigid stamp on an anisotropic half-space”, Static and Dynamic Mixed Problems of Elasticity Theory, ed. I. I.Vorovich, Rostov University Publ., Rostov-on-Don, 1983, 112-115 (in Russian) |
| [17] | Pozharskii D. A., “Contact problem for an orthotropic half-space”, Mechanics of Solids, 52 (2017), 315-322 · doi: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, 45:22-23 (2008), 5814-5830 · Zbl 1381.74162 · doi:10.1016/j.ijsolstr.2008.06.016 |
| [19] | Erbaş B., Yusufoǧlu E., Kaplunov J., “A plane contact problem for an elastic orthotropic strip”, Journal of Engineering Mathematics, 70 (2011), 399-409 · Zbl 1254.74088 · doi: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, 49:21 (2012), 2962-2977 · doi: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, 49 (2015), 299-307 · Zbl 1406.74484 · doi: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, no. 4(29), 74-82 (in Russian) |
| [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, no. 2(35), 51-56 (in Russian) |
| [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, 89 (2019), 1403-1419 · doi: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, 54:4 (2019), 254-275 · doi:10.1177/0309324719859110 |
| [26] | Babeshko V. A., Glushkov E. V., Zinchenko Zh. F., Dynamics of Inhomogeneous Linearly Elastic Media, Nauka, M., 1989, 344 pp. (in Russian) · Zbl 0698.73016 |
| [27] | Bakhvalov N. S., Numerical Methods (Analysis, Algebra, Ordinary Differential Equations), Nauka, M., 1975, 632 pp. (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, 12:5(77) (1957), 3-122 (in Russian) · Zbl 0087.29602 |
| [29] | Benerdzhi P., Batterfild R., Boundary Element Methods in Applied Sciences, Mir, M., 1984, 244 pp. (in Russian) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
