Structural Engineering and Mechanics

Techno-Press (테크노프레스)
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Analytical solution of a contact problem and comparison with the results from FEM

  • Oner, Erdal (Department of Civil Engineering, Bayburt University) ;
  • Yaylaci, Murat (Department of Civil Engineering, Recep Tayyip Erdogan University) ;
  • Birinci, Ahmet (Department of Civil Engineering, Karadeniz Technical University)
  • Received : 2014.06.10
  • Accepted : 2014.10.29
  • Published : 2015.05.25
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Abstract

This paper presents a comparative study of analytical method and finite element method (FEM) for analysis of a continuous contact problem. The problem consists of two elastic layers loaded by means of a rigid circular punch and resting on semi-infinite plane. It is assumed that all surfaces are frictionless and only compressive normal tractions can be transmitted through the contact areas. Firstly, analytical solution of the problem is obtained by using theory of elasticity and integral transform techniques. Then, finite element model of the problem is constituted using ANSYS software and the two dimensional analysis of the problem is carried out. The contact stresses under rigid circular punch, the contact areas, normal stresses along the axis of symmetry are obtained for both solutions. The results show that contact stresses and the normal stresses obtained from finite element method (FEM) provide boundary conditions of the problem as well as analytical results. Also, the contact areas obtained from finite element method are very close to results obtained from analytical method; disagree by 0.03-1.61%. Finally, it can be said that there is a good agreement between two methods.

Keywords

References

  1. Adams, G.G. (1978), "An elastic strip pressed against an elastic half plane by a steadily moving force", J. Appl. Mech., 45(1), 89-94. https://doi.org/10.1115/1.3424279
  2. Adibelli, H., Comez, I. and Erdol, R. (2013), "Receding contact problem for a coated layer and a half-plane loaded by a rigid cylindrical stamp", Arch. Mech., 65(3), 219-236.
  3. Alexandrov, V.M. (1970), "On plane contact problems of the theory of elasticity in the presence of adhesion or friction", J. Appl. Math. Mech., 34(2), 232-243. https://doi.org/10.1016/0021-8928(70)90137-1
  4. ANSYS (2008), Swanson Analysis System, USA.
  5. Argatov, I. (2013), "Contact problem for a thin elastic layer with variable thickness: application to sensitivity analysis of articular contact mechanics", Appl. Math. Model., 37(18-19), 8383-8393. https://doi.org/10.1016/j.apm.2013.03.042
  6. Birinci, A. and Erdol, R. (2001), "Continuous and discontinuous contact problem for a layered composite resting on simple supports", Struct. Eng. Mech., 12(1), 17-34. https://doi.org/10.12989/sem.2001.12.1.017
  7. Birinci, A. and Erdol, R. (2003), "A frictionless contact problem for two elastic layers supported by a winkler foundation", Struct. Eng. Mech., 15(3), 331-344. https://doi.org/10.12989/sem.2003.15.3.331
  8. Brizmer, V., Kligerman, Y. and Etsion, I. (2006), "The effect of contact conditions and material properties on the elasticity terminus of a spherical contact", Int. J. Solid. Struct., 43, 5736-5749. https://doi.org/10.1016/j.ijsolstr.2005.07.034
  9. Cakiroglu, F., Cakiroglu, M. and Erdol, R. (2001), "Contact problems for two elastic layers resting on elastic half-plane", J. Eng. Mech., ASCE, 127(2), 113-118. https://doi.org/10.1061/(ASCE)0733-9399(2001)127:2(113)
  10. Chan, S.K. and Tuba, I.S. (1971), "A finite element method for contact problems of solid bodies-1: theory and validation", Int. J. Mech. Sci., 13(7), 615-625. https://doi.org/10.1016/0020-7403(71)90032-4
  11. Comez, I. and Erdol, R. (2013), "Frictional contact problem of a rigid stamp and an elastic layer bonded to a homogeneous substrate", Arch. Appl. Mech., 83(1), 15-24. https://doi.org/10.1007/s00419-012-0626-4
  12. Dag, S., Guler, M.A., Yildirim, B. and Ozatag, A.C. (2009), "Sliding frictional contact between a rigid punch and a laterally graded elastic medium", Int. J. Solid. Struct., 46(22-23), 4038-4053. https://doi.org/10.1016/j.ijsolstr.2009.07.023
  13. Dempsey, J.P., Zhao, Z.G., Minnetyan, L. and Li, H. (1990), "Plane contact of an elastic layer supported by a winkler foundation", J. Appl. Mech., 57(4), 974-980. https://doi.org/10.1115/1.2897670
  14. Dini, D. and Nowell, D. (2004), "Flat and rounded fretting contact problems incorporating elastic layers", Int. J. Mech. Sci., 46(11), 1635-1657. https://doi.org/10.1016/j.ijmecsci.2004.10.003
  15. El-Borgi, S., Abdelmoula, R. and Keer, L. (2006), "A receding contact plane problem between a functionally graded layer and a homogeneous substrate", Int. J. Solid. Struct., 43(3-4), 658-674. https://doi.org/10.1016/j.ijsolstr.2005.04.017
  16. Erdogan, F. and Gupta, G. (1972), "On the numerical solutions of singular integral equations", Q. Appl. Math., 29, 525-534. https://doi.org/10.1090/qam/408277
  17. Etsion, I., Kligerman, Y. and Kadin, Y. (2005), "Unloading of an elastic-plastic loaded spherical contact", Int. J. Solid. Struct., 42, 3716-3729. https://doi.org/10.1016/j.ijsolstr.2004.12.006
  18. Francavilla, A. and Zienkiewicz, O.C. (1975), "A note on numerical computation of elastic contact problems", Int. J. Numer. Meth. Eng., 9(4), 913-924. https://doi.org/10.1002/nme.1620090410
  19. Galin, L.A. (2008), Contact Problems, Springer.
  20. Garrido, J.A., Foces, A. and Paris, F. (1991), "BEM applied to receding contact problems with friction", Math. Comput. Model., 15(3-5), 143-153. https://doi.org/10.1016/0895-7177(91)90060-K
  21. Garrido, J.A. and Lorenzana, A. (1998), "Receding contact problem involving large displacements using the BEM", Eng. Anal. Bound. Elem., 21(4), 295-303. https://doi.org/10.1016/S0955-7997(98)00018-6
  22. Gecit, M.R. (1986), "Axisymmetric contact problem for a semi-infinite cylinder and a half space", Int. J. Eng. Sci., 24(8), 1245-1256. https://doi.org/10.1016/0020-7225(86)90054-6
  23. Gun, H. and Gao, X.W. (2014), "Analysis of frictional contact problems for functionally graded materials using BEM", Eng. Anal. Bound. Elem., 38, 1-7. https://doi.org/10.1016/j.enganabound.2013.10.004
  24. Jing, H.S. and Liao, M.L. (1990), "An improved finite element scheme for elastic contact problems with friction", Comput. Struct., 35(5), 571-578. https://doi.org/10.1016/0045-7949(90)90385-F
  25. Kahya, V., Ozsahin, T.S., Birinci, A. and Erdol, R. (2007), "A receding contact problem for an anisotropic elastic medium consisting of a layer and a half plane", Int. J. Solid. Struct., 44(17), 5695-5710. https://doi.org/10.1016/j.ijsolstr.2007.01.020
  26. Kumar, N. and Dasgupta, A. (2013), "On the contact problem of an inflated spherical hyperelastic membrane", Int. J. Nonlin. Mech., 57, 130-139. https://doi.org/10.1016/j.ijnonlinmec.2013.06.015
  27. Li, X.Y., Zheng, R.F. and Chen, W.Q. (2014), "Fundamental solutions to contact problems of a magnetoelectro- elastic half-space indented by a semi-infinite punch", Int. J. Solid. Struct., 51(1), 164-178. https://doi.org/10.1016/j.ijsolstr.2013.09.020
  28. Liao, X. and Wang, G.G. (2007), "Non-linear dimensional variation analysis for sheet metal assemblies by contact modeling", Finite Elem. Anal. Des., 44(1-2), 34-44. https://doi.org/10.1016/j.finel.2007.08.009
  29. Long, J.M. and Wang, G.F. (2013), "Effects of surface tension on axisymmetric hertzian contact problem", Mech. Mater., 56, 65-70. https://doi.org/10.1016/j.mechmat.2012.09.003
  30. Ma, L.F. and Korsunsky, A.M. (2004), "Fundamental formulation for frictional contact problems of coated systems", Int. J. Solid. Struct., 41(11-12), 2837-2854. https://doi.org/10.1016/j.ijsolstr.2003.12.022
  31. Nowell, D. and Hills, D.A. (1988), "Contact problems incorporating elastic layers", Int. J. Solid. Struct., 24(1), 105-115. https://doi.org/10.1016/0020-7683(88)90102-3
  32. Oysu, C. (2007), "Finite element and boundary element contact stress analysis with remeshing technique", Appl. Math. Model., 31(12), 2744-2753. https://doi.org/10.1016/j.apm.2006.11.001
  33. Porter, M.I. and Hills, D.A. (2002), "Note on the complete contact between a flat rigid punch and an elastic layer attached to a dissimilar substrate", Int. J. Mech. Sci., 44(3), 509-520. https://doi.org/10.1016/S0020-7403(01)00106-0
  34. Ratwani, M. and Erdogan, F. (1973), "On the plane contact problem for a frictionless elastic layer", Int. J. Solid. Struct., 9(8), 921-936. https://doi.org/10.1016/0020-7683(73)90021-8
  35. Rhimi, M., El-Borgi, S. and Lajnef, N. (2011), "A double receding contact axisymmetric problem between a functionally graded layer and a homogeneous substrate", Mech. Mater., 43(12), 787-798. https://doi.org/10.1016/j.mechmat.2011.08.013
  36. Roncevic, B. and Siminiati, D. (2010), "Two dimensional receding contact analysis with nx nastran", Adv. Eng., 4, 69-74.
  37. Satish Kumar, K., Dattaguru, B., Ramamurthy, T.S. and Raju, K.N. (1996), "Elastoplastic contact stress analysis of joints subjected to cyclic loading", Comput. Struct., 60(6), 1067-1077. https://doi.org/10.1016/0045-7949(95)00413-0
  38. Soldatenkov, I.A. (2013), "The periodic contact problem of the plane theory of elasticity. Taking friction, wear and adhesion into account", Pmm. J. Appl. Math. Mech., 77(2), 245-255. https://doi.org/10.1016/j.jappmathmech.2013.07.017
  39. Weitsman, Y. (1972), "A tensionless contact between a beam and an elastic half space", Int. J. Eng. Sci., 10(1), 73-81. https://doi.org/10.1016/0020-7225(72)90075-4
  40. Yaylaci, M. and Birinci, A. (2013), "The receding contact problem of two elastic layers supported by two elastic quarter planes", Struct. Eng. Mech., 48(2), 241-255. https://doi.org/10.12989/sem.2013.48.2.241
  41. Zhang, W.M. and Meng, G. (2006), "Numerical simulation of sliding wear between the rotor bushing and ground plane in micromotors", Sens. Actuat. A Phys., 126(1), 15-24. https://doi.org/10.1016/j.sna.2005.08.004

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