On the plane frictional contact problem of a homogeneous orthotropic layer loaded by a rigid cylindrical stamp

  • I. ComezEmail author
  • K. B. Yilmaz
  • M. A. Güler
  • B. Yildirim


In this study, contact problem for a homogeneous orthotropic layer loaded by a rigid cylindrical stamp is considered. The rigid cylindrical stamp slides over the contacting medium whose bottom surface is fixed to the ground in all directions. Using the integral transformation technique, the contact problem is formulated analytically into a singular integral equation. The resulting integral equation is converted to algebraic equations by using Gauss–Jacobi integration formulas and solved numerically. In addition to the analytical formulation, a finite element method (FEM) study is also conducted. The results that are obtained using FEM are compared with the results found using analytical formulation. It is found that the results obtained from analytical formulation and FEM study are in good agreement with each other. The primary intention of this paper is to demonstrate the effects of orthotropic material properties, geometrical properties and the coefficient of friction on the stresses generated due to the sliding motion of the rigid cylindrical stamp. The results of this study may provide benchmark results for engineers to be used in tribology applications involving friction and wear mechanisms.


Sliding contact Orthotropic layer Singular integral equation Finite element method Friction 



  1. 1.
    ANSYS: ANSYS Mechanical APDL Modeling and Meshing Guide, p. 15317. ANSYS Inc, Canonsburg, PA (2013)Google Scholar
  2. 2.
    ANSYS: ANSYS Mechanical APDL Contact Technology Guide, 17th edn, p. 15317. ANSYS Inc., Canonsburg, PA (2016)Google Scholar
  3. 3.
    ANSYS: ANSYS Mechanical APDL 18.1 Documentation, p. 15317. ANSYS Inc., Canonsburg, PA (2017)Google Scholar
  4. 4.
    Abhilash, M.N., Murthy, H.: Finite element analysis of 2-d elastic contacts involving fgms. Int. J. Comput. Methods Eng. Sci. Mech. 15(3), 253–257 (2014)CrossRefGoogle Scholar
  5. 5.
    Alinia, Y., Beheshti, A., Guler, M.A., El-Borgi, S., Polycarpou, A .A.: Sliding contact analysis of functionally graded coating/substrate system. Mech. Mater. 94(Supplement C), 142–155 (2016)CrossRefGoogle Scholar
  6. 6.
    Bagault, C., Nélias, D., Baietto, M.: Contact analyses for anisotropic half space: effect of the anisotropy on the pressure distribution and contact area. Int. J. Solids Struct. 134(3), 031401–031401–8 (2012)Google Scholar
  7. 7.
    Balci, M.N., Dag, S., Yildirim, B.: Subsurface stresses in graded coatings subjected to frictional contact with heat generation. J. Thermal Stresses 40(4), 517–534 (2017)CrossRefGoogle Scholar
  8. 8.
    Barber, J., Ciavarella, M.: Contact mechanics. Int. J. Solids Struct. 37(1), 29–43 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Binienda, W., Pindera, M.: Frictionless contact of layered metal-matrix and polymer-matrix composite half planes. Compos. Sci. Technol. 50(1), 119–128 (1994)CrossRefGoogle Scholar
  10. 10.
    Blazquez, A., Mantič, V., París, F.: Application of bem to generalized plane problems for anisotropic elastic materials in presence of contact. Eng. Anal. Bound. Elements 30(6), 489–502 (2006)CrossRefzbMATHGoogle Scholar
  11. 11.
    Chidlow, S., Teodorescu, M.: Two-dimensional contact mechanics problems involving inhomogeneously elastic solids split into three distinct layers. Int. J. Eng. Sci. 70(Supplement C), 102–123 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chidlow, S., Teodorescu, M.: Sliding contact problems involving inhomogeneous materials comprising a coating-transition layer-substrate and a rigid punch. Int. J. Solids Struct. 51(10), 1931–1945 (2014)CrossRefGoogle Scholar
  13. 13.
    Chidlow, S., Chong, W., Teodorescu, M.: On the two-dimensional solution of both adhesive and non-adhesive contact problems involving functionally graded materials. Eur. J. Mech. A/Solids 39(Supplement C), 86–103 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Choi, H.J.: On the plane contact problem of a functionally graded elastic layer loaded by a frictional sliding flat punch. J.Mech. Sci. Technol. 23(10), 2703–2713 (2009)CrossRefGoogle Scholar
  15. 15.
    Choi, H.J., Paulino, G.H.: Thermoelastic contact mechanics for a flat punch sliding over a graded coating/substrate system with frictional heat generation. J. Mech. Phys. Solids 56(4), 1673–1692 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Choi, H.J., Thangjitham, S.: Stress analysis of multilayered anisotropic elastic media. J. Appl. Mech. 58, 382–387 (1991)CrossRefzbMATHGoogle Scholar
  17. 17.
    Chong, W., Chidlow, S.: Analysing the effects of sliding, adhesive contact on the deformation and stresses induced within a multi-layered elastic solid. Mech. Mater. 101(Supplement C), 1–13 (2016)CrossRefGoogle Scholar
  18. 18.
    Dag, S., Guler, M.A., Yildirim, B., Ozatag, A.C.: Sliding frictional contact between a rigid punch and a laterally graded elastic medium. Int. J. Solids Struct. 46(22), 4038–4053 (2009)CrossRefzbMATHGoogle Scholar
  19. 19.
    Demirhan, N., Kanber, B.: Finite element analysis of frictional contacts of fgm coated elastic members#. Mech. Based Des. Struct. Mach. 41(4), 383–398 (2013)CrossRefGoogle Scholar
  20. 20.
    England, A .H.: Complex and Variable Methods in Elasticity. Wiley, Hoboken (1971)zbMATHGoogle Scholar
  21. 21.
    Erbas, B., Yusufoğlu, E., Kaplunov, J.: A plane contact problem for an elastic orthotropic strip. J. Eng. Math. 70(4), 399–409 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Erdogan, F., Delale, F.: The problem of internal and edge cracks in an orthotropic strip. J. Appl. Mech. 44(2), 237–242 (1978)zbMATHGoogle Scholar
  23. 23.
    Guler, M.A.: Closed-form solution of the two-dimensional sliding frictional contact problem for an orthotropic medium. Int. J. Mech. Sci. 87(Supplement C), 72–88 (2014)CrossRefGoogle Scholar
  24. 24.
    Guler, M.A., Gulver, Y.F., Nart, E.: Contact analysis of thin films bonded to graded coatings. Int. J. Mech. Sci. 55(1), 50–64 (2012)CrossRefGoogle Scholar
  25. 25.
    Guler, M.A., Kucuksucu, A., Yilmaz, K., Yildirim, B.: On the analytical and finite element solution of plane contact problem of a rigid cylindrical punch sliding over a functionally graded orthotropic medium. Int. J. Mech. Sci. 120(Supplement C), 12–29 (2017)CrossRefGoogle Scholar
  26. 26.
    He, L., Ovaert, T.C.: Three-dimensional rough surface contact model for anisotropic materials. J. Tribol. 130(2), 021402 (2008)CrossRefGoogle Scholar
  27. 27.
    Hertz, H. R.: Über die berührung fester elastischer körper und über die härte. Verhandlungen des Vereins zur Beförderung des Gewerbfleis̈es, Berlin : Verein zur Beförderung des Gewerbefleisses 1882, 449–463 (1896)Google Scholar
  28. 28.
    Johnson, K.L.: Contact Mechanics. Cambridge University Press, Cambridge (2012)Google Scholar
  29. 29.
    Ke, L.-L., Wang, Y.-S.: Two-dimensional contact mechanics of functionally graded materials with arbitrary spatial variations of material properties. Int. J. Solids Struct. 43(18), 5779–5798 (2006)CrossRefzbMATHGoogle Scholar
  30. 30.
    Ke, L.-L., Wang, Y.-S.: Two-dimensional sliding frictional contact of functionally graded materials. Eur. J. Mech. A/Solids 26(1), 171–188 (2007)CrossRefzbMATHGoogle Scholar
  31. 31.
    Keer, L., Mowry, D.: The stress field created by a circular sliding contact on transversely isotropic spheres. Int. J. Solids Struct. 15(1), 33–39 (1979)CrossRefzbMATHGoogle Scholar
  32. 32.
    Krenk, S.: On the elastic constants of plane orthotropic elasticity. J. Compos. Mater. 13(2), 108–116 (1979)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Kucuksucu, A., Guler, M., Avci, A.: Closed-form solution of the frictional sliding contact problem for an orthotropic elastic half-plane indented by a wedge-shaped punch. Key Eng. Mater. 618, 203–225 (2014)CrossRefGoogle Scholar
  34. 34.
    Lekhnitskii, S.G.: Theory of Elasticity of an Anisotropic Elastic Body (Holden-Day Series in Mathematical Physics). Tbh/Yes Dee, Chennai (1963). Please check and confirm the inserted publisher location is correct for the reference [34] and amend if necessaryGoogle Scholar
  35. 35.
    Mindlin, R.: Compliance of elastic bodies in contact. J. Appl. Mech. 16, 259–268 (1949)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Muskhelishvili, N.I.: Singular Integral Equations: Boundary Problems of Function Theory and Their Application to Mathematical Physics (Dover Books on Mathematics). Dover Publications, Mineola (2011)Google Scholar
  37. 37.
    Muskhelishvili, N.I.: Some Basic Problems of the Mathematical Theory of Elasticity. Springer, Netherlands (2013)Google Scholar
  38. 38.
    Pagano, N.: Exact solutions for rectangular bidirectional composites and sandwich plates. J. Compos. Mater. 4(1), 20–34 (1970)CrossRefGoogle Scholar
  39. 39.
    Popov, V.: Contact Mechanics and Friction: Physical Principles and Applications, 1st edn. Springer, Berlin (2010)CrossRefzbMATHGoogle Scholar
  40. 40.
    Rodriguez, N., Masen, M., Schipper, D.: A contact model for orthotropic-viscoelastic materials. Int. J. Mech. Sci. 74(Supplement C), 91–98 (2013)CrossRefGoogle Scholar
  41. 41.
    Rodríguez-Tembleque, L., Buroni, F., Abascal, R., Sáez, A.: 3d frictional contact of anisotropic solids using bem. Eur. J. Mech. A/Solids 30(2), 95–104 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Shi, D., Lin, Y., Ovaert, T.C.: Indentation of an orthotropic half-space by a rigid ellipsoidal indenter. J. Tribol. 125(2), 223–231 (2003)CrossRefGoogle Scholar
  43. 43.
    Sneddon, I .N.: Fourier Transforms. Dover Publications Inc, Mineola (1951)zbMATHGoogle Scholar
  44. 44.
    Spence, D.A.: The hertz contact problem with finite friction. J. Elast. 5(3), 297–319 (1975)CrossRefzbMATHGoogle Scholar
  45. 45.
    Srinivas, S., Rao, A.: Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates. Int. J. Solids Struct. 6, 1463–1481 (1970)CrossRefzbMATHGoogle Scholar
  46. 46.
    Swanson, S.R.: Hertzian contact of orthotropic materials. Int. J. Solids Struct. 41, 1945–1959 (2004)CrossRefzbMATHGoogle Scholar
  47. 47.
    Turner, J.: Contact on a transversely isotropic half-space, or between two transversely isotropic bodies. Int. J. Solids Struct. 16, 409–419 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Willis, J.: Hertzian contact of anisotropic bodies. J. Mech. Phys. Solids 14(3), 163–176 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Yang, B., Pan, E.: Three-dimensional green’s functions in anisotropic trimaterials. Int. J. Solids Struct. 39(8), 2235–2255 (2002)CrossRefzbMATHGoogle Scholar
  50. 50.
    Zhou, Y.T., Lee, K.Y.: Exact solutions of a new, 2d frictionless contact model for orthotropic piezoelectric materials indented by a rigid sliding punch. Philos. Mag. 92(15), 1937–1965 (2012)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • I. Comez
    • 1
    Email author
  • K. B. Yilmaz
    • 2
  • M. A. Güler
    • 3
  • B. Yildirim
    • 2
  1. 1.Department of Civil EngineeringKaradeniz Technical UniversityTrabzonTurkey
  2. 2.Department of Mechanical EngineeringHacettepe UniversityAnkaraTurkey
  3. 3.College of Engineering and TechnologyAmerican University of the Middle EastEqailaKuwait

Personalised recommendations