Kama ile Destekli Elastik Tabakanın Temas Analizi

Öz

Bu çalışmada düzlemde kama ile destekli elastik tabakanın temas analizi yapılmıştır. Bu problem kapalı formda integral denklemleri ile formüle edilmiştir. Temas bölgesinin uzunluğu, kama ve tabakalar arasındaki basınç ile tabaka ve panç arasındaki basınç problemin bilinmeyenleridir. Bu problemde hem malzeme hem de tabaka elastik kabul edilmiştir. Bu problem dik açılı iki kama tarafından destekli bir tabaka olarak tanımlanabilir. Tabakanın kalınlığı üst yüzeyin çok büyük yarıçaplı bir çember olduğu varsayımı ile sonlu olarak kabul edilmiştir. Problemin çözümü için tekil integral denklemleri kullanılmıştır. Bu formülasyonun yararları şöyle açıklanabilir ; problem basit bir şekilde kuvvetlerin bir çok panç tarafından etki ettirildiği durum için genelleştirilebilir. Çözüm temas gerilmesini direkt olarak verir ve tekil integral denklemlerinin çözümü sayısal çözüm tekniği bakımından uygun bir yoldur. Bu problemin uygulaması rayların birleşim bölgeleri ile temas eden tren tekerleği örneğidir. Denge koşullarını göz önüne alarak çekirdeklerdeki ıraksak terimlerin birbirini sadeleştirdiği gösterilmiştir. Sayısal örnek olarak teker ve ray arasındaki temas problemi rayların birbiri arasında aralıklı hali için incelenmiştir.

Anahtar Kelimeler

• temas problemi
• kama
• elastik tabaka
• basınç
• uygulamalı mekanik

Contact Analysis of Elastic Layer Supported by a Wedge

Abstract

In this paper the contact analysis of elastic layer supported by a wedge is considered in plane. The problem is formulated with closed formed integral equations. The length of the contact region, the pressure between the wedge and the layers, and the pressure between the layer and the punch are unknowns. Both the material and layer are elastic in this problem. This problem can be defined as a layer supported over two wedges with perpendicular angle. The upper surface of the layer is assumed as circle with a large radius. The thickness of the layer is finite. Singular integral equations are used in the formulation of the problem. The benefits of this formulation are the following: the problem can easily be generalized for the case of forces acting through many rigid punches. The solution gives the contact stress directly and the solution of the singular integral equations is an appropriate way in terms of numerical solution technique. The application of this problem is the train wheel that is contacted to the connection part of the rails. It is shown that the divergent terms at the kernels cancel each other by considering the equilibrium conditions. As a numerical example, the contact problem between the wheel and rails is investigated.

Keywords

• contact problem
• wedge
• elastic layer
• pressure
• applied mechanics

Kaynakça

1. Boussinesq, J., Application des Potentials a L’Etude de L’Equilibre et due Mouvement des Solides Elastiques. Gauthier-Villars, Paris, France 1885.
2. J Hertz, H., Ueber die beruehrung elastischer koerper (On Contact Between Elastic Bodies), in Gesammelte Werke (Collected Works), 1, Leipzig-Germany, 1895, 179-195.
3. Dinnik, A.N., Hertz’s formula and its experimental verification. Zhural russk. Fiz Russk. Fiz.-Khim. ob-va, fiz. otd. 38(1), 1906, 242-249.
4. Belyaev, N.M., Calculation of the maximum design stresses resulting from the pressure between bodies in contact, Sbornik Leningradskago Instituta Inzhenerov Putei Soobscheniya, 1929, n102.
5. Belyaev, N.M., Application of the theory of Hertz to the calculation of the local stresses at the point of contact of a wheel and a rail, Vestnik Inzhenerov i Tekhnikov, 1917, n2.
6. Belyaev, N.M., On the question of local stresses in connection with the resistance of rails to crushing, Sbornik Leningradskago Instituta Inzhenerov Putei Soobscheniya, 1929, n90.
7. Shtaerman, I.J., The Contact Problem of the Theory of Elasticity. GOSTEKHIZ-DAT, 1949, Moscow-Leningrad.
8. Muskhelishvili, N.I., Some Basic Problems of the Mathematical Theory of Elasticity. 1953, Noordhoff-Groningen, Holland.

9. Barber, J.R., Davies, M., Hills, D.A., Frictional elastic contact with periodic loading. International Journal of Solids and Structures, 2011, 48(13), 2041-2047.
10. Comez, I., Frictional contact problem for rigid cylindrical stamp and an elastic layer resting on a half plane. International Journal of Solids and Structures, 2010, 47(7-8), 1090-1097.
11. Comez, I. ,Birinci, A., Erdol,R., Double receding contact problem for a rigid stamp and two elastic layers. European Journal of Mechanics - A/Solids, 2004, 23, 301-309.
12. Kahya, V, Ozsahin, T.S., Birinci, A., Erdol R., Receding contact problem for an anizotropic elastic medium consisting of a layer and a half plane, International Journal of Solids and Structures,2007, 44, 5695-5710.
13. Dag, S., Guler, M.A., Yildirim, B., Ozatag, A.C., Sliding frictional contact between a rigid punch and a laterally graded elastic medium. International Journal of Solids and Structures, 2009, 46(22-23), 4038-4053.
14. Jang, Y.H., Cho, H., Barber, J.R.. The thermoelastic Hertzian contact problem. International Journal of Solids and Structures, 2009, 46(22-23), 4073-4078.
15. Malanchuk, N., Martynyak, R., Monastyrskyy B., Thermally induced local slip of contacting solids in vicinity of surface groove. International Journal of Solids and Structures, 2011, 48(11-12), 1791-1797.
16. 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, 45(22-23), 5814-5830.
17. Yanik, A., Bakioglu, M., Contact Problem for Layered Medium Supported by a Wedge. In IOP Conference Series: Materials Science and Engineering (Vol. 603, No. 2, p. 022002). IOP Publishing, 4th World Multidisciplinary Civil Engineering-Architecture-Urban Planning Symposium (WMCAUS 2019), 17-21 June 2019, Prague-Czech Republic.
18. Erdogan, F, Gupta, G.D., Contact and Crack Problems for an Elastic Wedge. International Journal of Engineering Science , 1976, 14(2), 154-164.
19. Ratwani, M., Erdogan F., On the plane contact problem for a frictionless elastic layer, International Journal of Solids and Structures, 1973, 9(8), 921-936.
20. Erdogan, F. Ratwani,M. The contact problem for an elastic layer supported by two elastic quarter planes. Journal of Applied Mechanics, 1974, 41(3), 673-678.