, Volume 54, Issue 14, pp 2183–2206 | Cite as

Double frictional receding contact problem for an orthotropic layer loaded by normal and tangential forces

  • B. YildirimEmail author
  • K. B. Yilmaz
  • I. Comez
  • M. A. Guler


With the increasing research in the field of contact mechanics, different types of contact models have been investigated by many researchers by employing various complex material models. To ascertain the orthotropy effect and modeling parameters on a receding contact model, the double frictional receding contact problem for an orthotropic bilayer loaded by a cylindrical punch is taken into account in this study. Assuming plane strain sliding conditions, the governing equations are found analytically using Fourier integral transformation technique. Then, the resulting singular integral equations are solved numerically using an iterative method. The weight function describing the asymptotic behavior of the stresses are investigated in detail and powers of the stress singularities are provided. To control the trustworthiness and correctness of the analytical formulation and to compare the resulting stress distributions and contact boundaries, a numerically efficient finite element method was employed using augmented Lagrange contact algorithm. The aim of this paper is to investigate the orthotropy effect, modeling parameters and coefficients of friction on the surface and interface stresses, surface and interface contact boundaries, powers of stress singularities, weight function and to provide highly parametric benchmark results for tribological community in designing wear resistant systems.


Receding contact Metal-matrix material Polymer-matrix material Singular integral equation (SIE) Finite element method 


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Hwu C, Fan CW (1998) Solving the punch problems by analogy with the interface crack problems. Int J Solids Struct 35(30):3945–3960CrossRefzbMATHGoogle Scholar
  2. 2.
    Shi D, Lin Y, Ovaert TC (2003) Indentation of an orthotropic half-space by a rigid ellipsoidal indenter. J Tribol 125(2):223CrossRefGoogle Scholar
  3. 3.
    Swanson SR (2004) Hertzian contact of orthotropic materials. Int J Solids Struct 41(7):1945–1959CrossRefzbMATHGoogle Scholar
  4. 4.
    Willis JR (1966) Hertzian contact of anisotropic bodies. J Mech Phys Solids 14(3):163–176CrossRefADSMathSciNetzbMATHGoogle Scholar
  5. 5.
    Srinivas S, Rao AK (1970) Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates. Int J Solids Struct 6(11):1463–1481CrossRefzbMATHGoogle Scholar
  6. 6.
    Batra RC, Jiang W (2008) Analytical solution of the contact problem of a rigid indenter and an anisotropic linear elastic layer. Int J Solids Struct 45(22):5814–5830CrossRefzbMATHGoogle Scholar
  7. 7.
    Guler MA (2014) Closed-form solution of the two-dimensional sliding frictional contact problem for an orthotropic medium. Int J Mech Sci 87:72–88CrossRefGoogle Scholar
  8. 8.
    Zhou YT, Kim TW (2014) Closed-form solutions for the contact problem of anisotropic materials indented by two collinear punches. Int J Mech Sci 89:332–343CrossRefGoogle Scholar
  9. 9.
    Alinia Y, Hosseini-nasab M, Güler MA (2018) The sliding contact problem for an orthotropic coating bonded to an isotropic substrate. Eur J Mech A Solids 70:156–171CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Yilmaz KB, Çömez Güler MA, Yildirim B (2019) Analytical and finite element solution of the sliding frictional contact problem for a homogeneous orthotropic coating-isotropic substrate system. ZAMM 99(3):e201800117CrossRefMathSciNetGoogle Scholar
  11. 11.
    Keer LM, Mowry DB (1979) The stress field created by a circular sliding contact on transversely isotropic spheres. Int J Solids Struct 15(1):33–39CrossRefzbMATHGoogle Scholar
  12. 12.
    Kuo CH, Keer LM (1992) Contact stress analysis of a layered transversely isotropic half-space. J Tribol 114(2):253CrossRefGoogle Scholar
  13. 13.
    Hanson MT (1992) The elastic field for spherical Hertzian contact including sliding friction for transverse isotropy. J Tribol 114(3):606CrossRefGoogle Scholar
  14. 14.
    Ning X, Lovell M, Slaughter WS (2006) Asymptotic solutions for axisymmetric contact of a thin, transversely isotropic elastic layer. Wear 260(7):693–698CrossRefGoogle Scholar
  15. 15.
    Liu H, Pan E (2018) Indentation of a flat-ended cylinder over a transversely isotropic and layered half-space with imperfect interfaces. Mech Mater 118:62–73CrossRefADSGoogle Scholar
  16. 16.
    Binienda WK, Pindera MJ (1994) Frictionless contact of layered metal–matrix and polymer–matrix composite half planes. Compos Sci Technol 50(1):119–128CrossRefGoogle Scholar
  17. 17.
    Comez I, Yilmaz KB (2019) Mechanics of frictional contact for an arbitrary oriented orthotropic material. ZAMM 99(3):e201800084CrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Dundurs J (1975) The mechanics of the contact between deformable bodies, 1st edn. Springer, AmsterdamzbMATHGoogle Scholar
  19. 19.
    Dundurs J, Stippes M (1970) Role of elastic constants in certain contact problems. J Appl Mech 37:965–970CrossRefGoogle Scholar
  20. 20.
    Keer LM, Dundurs J, Tsai KC (1972) Problems involving a receding contact between a layer and a half space. J Appl Mech 39(4):301–309Google Scholar
  21. 21.
    Gladwell GML (1976) On some unbonded contact problems in plane elasticity theory. J Appl Mech 43(2):263–267CrossRefzbMATHGoogle Scholar
  22. 22.
    Comez I, Birinci A, Erdol R (2004) Double receding contact problem for a rigid stamp and two elastic layers. Eur J Mech A Solids 23(2):301–309CrossRefzbMATHGoogle Scholar
  23. 23.
    El-Borgi S, Abdelmoula R, Keer L (2006) A receding contact plane problem between a functionally graded layer and a homogeneous substrate. Int J Solids Struct 43(3):658–674CrossRefzbMATHGoogle Scholar
  24. 24.
    Kahya V, Ozsahin TS, Birinci A, Erdol R (2007) A receding contact problem for an anisotropic elastic medium consisting of a layer and a half plane. Int J Solids Struct 44(17):5695–5710CrossRefzbMATHGoogle Scholar
  25. 25.
    Rhimi M, El-Borgi S, Saïd WB, Jemaa FB (2009) A receding contact axisymmetric problem between a functionally graded layer and a homogeneous substrate. Int J Solids Struct 46(20):3633–3642CrossRefzbMATHGoogle Scholar
  26. 26.
    Comez I (2010) Frictional contact problem for a rigid cylindrical stamp and an elastic layer resting on a half plane. Int J Solids Struct 47(7):1090–1097CrossRefzbMATHGoogle Scholar
  27. 27.
    El-Borgi S, Usman S, Guler MA (2014) A frictional receding contact plane problem between a functionally graded layer and a homogeneous substrate. Int J Solids Struct 51(25):4462–4476CrossRefGoogle Scholar
  28. 28.
    Yan J, Li X (2015) Double receding contact plane problem between a functionally graded layer and an elastic layer. Eur J Mech A Solids 53:143–150CrossRefMathSciNetzbMATHGoogle Scholar
  29. 29.
    Parel K, Hills D (2016) Frictional receding contact analysis of a layer on a half-plane subjected to semi-infinite surface pressure. Int J Mech Sci 108–109:137–143CrossRefGoogle Scholar
  30. 30.
    Adibelli H, Comez I, Erdol R (2013) Receding contact problem for a coated layer and a half-plane loaded by a rigid cylindrical stamp. Arch Mech 65(3):219–236MathSciNetzbMATHGoogle Scholar
  31. 31.
    Comez I, El-Borgi S, Kahya V, Erdol R (2016) Receding contact problem for two-layer functionally graded media indented by a rigid punch. Acta Mech 227(9):2493–2504CrossRefMathSciNetzbMATHGoogle Scholar
  32. 32.
    Yan J, Mi C (2017) On the receding contact between an inhomogeneously coated elastic layer and a homogeneous half-plane. Mech Mater 112(Supplement C):18–27CrossRefGoogle Scholar
  33. 33.
    El-Borgi S, Comez I (2017) A receding frictional contact problem between a graded layer and a homogeneous substrate pressed by a rigid punch. Mech Mater 114(Supplement C):201–214CrossRefGoogle Scholar
  34. 34.
    McDevitt TW, Laursen TA (2000) A mortar-finite element formulation for frictional contact problems. Int J Numer Methods Eng 48(1):1525–1547CrossRefMathSciNetzbMATHGoogle Scholar
  35. 35.
    Li C, Zou Z, Duan Z (2000) Multiple isoparametric finite element method for nonhomogeneous media. Mech Res Commun 27(2):137–142CrossRefzbMATHGoogle Scholar
  36. 36.
    Kim JH, Paulino GH (2002) Isoparametric graded finite elements for nonhomogeneous isotropic and orthotropic materials. J Appl Mech 69(1):502–514CrossRefzbMATHGoogle Scholar
  37. 37.
    Dag S, Guler MA, Yildirim B, Ozatag AC (2009) Sliding frictional contact between a rigid punch and a laterally graded elastic medium. Int J Solids Struct 46:4038–4053CrossRefzbMATHGoogle Scholar
  38. 38.
    Guler MA, Gulver YF, Nart E (2012) Contact analysis of thin films bonded to graded coatings. Int J Mech Sci 55:50–64CrossRefGoogle Scholar
  39. 39.
    Brezeanu LC (2014) Contact stresses: analysis by finite element method (FEM). In: Procedia technology, the 7th international conference interdisciplinarity in engineering (INTER-ENG 2013), vol 12, pp 401–410Google Scholar
  40. 40.
    Guler MA, Kucuksucu A, Yilmaz KB, Yildirim B (2017) 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 120:12–29CrossRefGoogle Scholar
  41. 41.
    Yilmaz KB, Comez I, Yildirim B, Güler MA, El-Borgi S (2018) Frictional receding contact problem for a graded bilayer system indented by a rigid punch. Int J Mech Sci 141:127–142CrossRefGoogle Scholar
  42. 42.
    Yang Z, Deng X, Li Z (2019) Numerical modeling of dynamic frictional rolling contact with an explicit finite element method. Tribol Int 129:214–231CrossRefGoogle Scholar
  43. 43.
    Comez I, Erdol R (2013) Frictional contact problem of a rigid stamp and an elastic layer bonded to a homogeneous substrate. Arch Appl Mech 83(1):15–24CrossRefzbMATHGoogle Scholar
  44. 44.
    Erdogan F, Gupta G (1972) On the numerical solution of singular integral equations. Q Appl Math 29:525–539CrossRefMathSciNetzbMATHGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Mechanical Engineering DepartmentHacettepe UniversityAnkaraTurkey
  2. 2.Civil Engineering DepartmentKaradeniz Technical UniversityTrabzonTurkey
  3. 3.College of Engineering and TechnologyAmerican University of the Middle EastKuwait CityKuwait

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