Skip to main content
SpringerLink
Log in

Mechanics of sliding frictional contact for a graded orthotropic half-plane

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

The main interest in this study is the crack initiation in graded orthotropic materials under sliding contact conditions. We consider the two-dimensional sliding contact problem between a graded orthotropic half-plane and a rigid punch with an arbitrary profile. The orthotropic graded half-plane is modeled as a linearly elastic and locally inhomogeneous orthotropic material with an exponentially varying Young’s modulus in the depth direction. The principal axes of orthotropy are assumed to be parallel and perpendicular to the contact surface. The problem is formulated under plane strain or generalized plane stress conditions. Using the standard Fourier transform, the problem is reduced to a singular integral equation, which is solved numerically using Jacobi polynomials. Extensive parametric study is done to determine the effect of the inhomogeneity parameter, β, the friction coefficient between the half-plane and the stamp, η, as well as the material orthotropic elastic parameters: the stiffness ratio, δ, the effective Poisson’s ratio, ν, and the shear parameter, κ, on the contact stress distribution and stress intensity factors at the sharp edges of the stamps that may have a bearing on the fatigue and fracture of the graded orthotropic half-plane.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Mindlin R.D.: Compliance of elastic bodies in contact. J. Appl. Mech. 16, 259–268 (1949)

    MathSciNet  MATH  Google Scholar 

  2. Sneddon I.N.: Fourier Transforms. McGraw-Hill, New York (1951)

    Google Scholar 

  3. Muskhelishvili N.I.: Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff, Groningen (1953)

    MATH  Google Scholar 

  4. Spence D.A.: The Hertz problem with finite friction. J. Elast. 5, 297–319 (1975)

    Article  MATH  Google Scholar 

  5. Keer L.M., Mowry D.B.: The stress field created by a circular sliding contact on transversely isotropic spheres. Int. J. Solids Struct. 15, 33–39 (1979)

    Article  MATH  Google Scholar 

  6. Gladwell G.M.L.: Contact Problems in the Classical Theory of Elasticity. Sijtho and Noordho, Alphen aan den Rijn (1980)

    Book  MATH  Google Scholar 

  7. Johnson K.L.: Contact Mechanics. Cambridge University Press, London (1985)

    Book  MATH  Google Scholar 

  8. Hills D.A., Nowell D., Sackfield A.: Mechanics of Elastic Contact. Butterworth-Heinemann, Oxford (1993)

    Google Scholar 

  9. Barber J.R.: Elasticity, 3rd edn. Springer, Berlin (2009)

    Google Scholar 

  10. Barber J.R., Ciavarella M.: Contact mechanics. Int. J. Solids Struct. 37, 29–43 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Giannakopoulos A.E., Pallot P.: Two-dimensional contact analysis of elastic graded materials. J. Mech. Phys. Solids 48, 1597–1631 (2000)

    Article  MATH  Google Scholar 

  12. Guler M.A., Erdogan F.: Contact mechanics of graded coatings. Int. J. Solids Struct. 41, 3865–3889 (2004)

    Article  MATH  Google Scholar 

  13. 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, 5779–5798 (2006)

    Article  MATH  Google Scholar 

  14. 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, 4038–4053 (2009)

    Article  MATH  Google Scholar 

  15. Guler M.A.: Contact Mechanics of FGM Coatings: An Integral Equation Approach. VDM Verlag, Saarbrücken (2009)

    Google Scholar 

  16. 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, 1673–1692 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ke L.L., Wang Y.S., Yang J.: Sliding frictional contact analysis of functionally graded piezoelectric layered half-plane. Acta Mech. 209, 249–268 (2010)

    Article  MATH  Google Scholar 

  18. Chen P., Chen S.: Contact behaviors of a rigid punch and a homogeneous half-space coated with a graded layer. Acta Mech. 223, 563–577 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Chen P.J., Chen S.H.: Thermo-mechanical contact behavior of a finite graded layer under a sliding punch with heat generation. Int. J. Solids Struct. 50, 1108–1119 (2013)

    Article  Google Scholar 

  20. Chen P.J., Chen S.H.: Partial slip contact between a rigid punch with an arbitrary tip-shape and an elastic graded solid with finite thickness. Mech. Mater. 59, 24–35 (2013)

    Article  Google Scholar 

  21. Giannakopoulos A.E., Pallot P.: Two-dimensional contact analysis of elastic graded materials. J. Mech. Phys. Solids 48, 1597–1631 (2000)

    Article  MATH  Google Scholar 

  22. Chen S.H., Yan C., Soh A.: Adhesive behavior of two-dimensional power-law graded materials. Int. J. Solids Struct. 46, 3398–3404 (2009)

    Article  MATH  Google Scholar 

  23. Chen, S.H., Yan, C., Zhang, P., Gao, H.J.: Mechanics of adhesive contact on a power-law graded elastic half-space. J. Mech. Phys. Solids 57, 1437–1448 (2009)

  24. Green A.E., Zerna W.: Theoretical Elasticity. Oxford University Press, Oxford (1954)

    MATH  Google Scholar 

  25. Leknitskii S.G.: Theory of Elasticity of an Anisotropic Elastic Body. Holden-Day, San Francisco (1963)

    Google Scholar 

  26. Sveklo V.A.: Boussinesq type problems for the anisotropic half-space. PMM 28, 908–913 (1964)

    MathSciNet  Google Scholar 

  27. Turner J.R.: Contact on a transversely isotropic half-space, or between two transversely isotropic bodies. Int. J. Solids Struct. 16, 409–419 (1966)

    Article  Google Scholar 

  28. Willis J.R.: Hertzian contact of anisotropic bodies. J. Mech. Phys. Solids 14, 163–176 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  29. Dahan M., Zarka J.: Elastic contact between a sphere and a semi infinite transversely isotropic body. Int. J. Solids Struct. 13, 229–238 (1977)

    Article  MATH  Google Scholar 

  30. Hwu C., Fan C.W.: Solving the punch problems by analogy with the interface crack problems. Int. J. Solids Struct. 35, 3945–3960 (1998)

    Article  MATH  Google Scholar 

  31. Pan E., Yuan F.: Three-dimensional Green’s functions in anisotropic bimaterials. Int. J. Solids Struct. 37, 5329–5351 (2000)

    Article  MATH  Google Scholar 

  32. Pan E.: Three-dimensional Green’s functions in anisotropic magneto-electro-elastic bimaterials. Z. Angew. Math. Phys. 53, 815–838 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  33. Pan E., Yang B.: Three-dimensional interfacial Green’s functions in anisotropic bimaterials. Appl. Math. Model. 27, 307–326 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  34. Borodich F.: Some contact problems of anisotropic elastodynamics: integral characteristics and exact solutions. Int. J. Solids Struct. 37, 3345–3373 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  35. Ciavarella M., Demelio G., Schino M., Vlassak J.J.: The general 3D Hertzian contact problem for anisotropic materials. Key Eng. Mater. 221, 281–292 (2001)

    Google Scholar 

  36. Barnett D., Lothe J.: Line force loadings on anisotropic half-spaces and wedges. Phys. Nor. 8, 13–22 (1975)

    Google Scholar 

  37. Shi D., Lin Y., Ovaert T.C.: Indentation of an orthotropic half-space by a rigid ellipsoidal indenter. J. Tribol. Trans. ASME 125, 223–231 (2003)

    Article  Google Scholar 

  38. Swanson S.R.: Hertzian contact of orthotropic materials. Int. J. Solids Struct. 41, 1945–1959 (2004)

    Article  MATH  Google Scholar 

  39. Willis J.R.: Hertzian contact of anisotropic bodies. J. Mech. Phys. Solids 14, 163–176 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  40. Pagano N.J.: Exact solutions for rectangular bidirectional composites and sandwich plates. J. Compos. Mater. 4, 20–34 (1970)

    Google Scholar 

  41. Srinivas S., Rao A.K.: Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates. Int. J. Solids Struct. 6, 1463–1481 (1970)

    Article  MATH  Google Scholar 

  42. Lin Y., Ovaert T.: A rough surface contact model for general anisotropic materials. J. Tribol. 126, 41–49 (2004)

    Article  Google Scholar 

  43. Rand O., Rovenskii V.: Analytical Methods in Anisotropic Elasticity: With Symbolic Computational Tools. Birkhauser, Basel (2005)

    Google Scholar 

  44. Ning X., Lovell M.R., Slaughter W.S.: Asymptotic solutions for axisymmetric contact of a thin, transversely isotropic elastic layer. Wear 260, 693–698 (2006)

    Article  Google Scholar 

  45. Li X., Wang M.: Hertzian contact of anisotropic piezoelectric bodies. J. Elast. 84, 153–166 (2006)

    Article  MATH  Google Scholar 

  46. Ramirez G.: Frictionless contact in a layered piezoelectric medium characterized by complex eigenvalues. Smart Mater. Struct. 15, 1287–1295 (2006)

    Article  MathSciNet  Google Scholar 

  47. Galin L.: Contact Problems: The Legacy of LA Galin. Springer, Berlin (2008)

    Google Scholar 

  48. He L., Ovaert T.: Three-dimensional rough surface contact model for anisotropic materials. J. Tribol. 130, 021402-1–021402-6 (2008)

    Article  Google Scholar 

  49. Batra R.C., Jiang W.: Analytical solution of the contact problem of a rigid indenter and an anisotropic linear elastic layer. Int. J. Solids Struct. 45, 5814–5830 (2008)

    Article  MATH  Google Scholar 

  50. Jiang W., Batra R.C.: Indentation of a laminated composite plate with an interlayer rectangular void. Compos. Sci. Technol. 70, 1023–1030 (2010)

    Article  Google Scholar 

  51. Erbas B., Yusufoglu E., Kaplunov J.: A plane contact problem for an elastic orthotropic strip. J. Eng. Math. 70, 399–409 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  52. 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. A 92, 1937–1965 (2012)

    Article  Google Scholar 

  53. Bagault C., Nélias D., Baietto M.C.: Contact analyses for anisotropic half space: effect of the anisotropy on the pressure distribution and contact area. J. Tribol. 134, 031401–10314018 (2012)

    Article  Google Scholar 

  54. Bagault C., Nélias D., Baietto M.C., Ovaert T.C.: Contact analyses for anisotropic half-space coated with an anisotropic layer: effect of the anisotropy on the pressure distribution and contact area. Int. J. Solids Struct. 50, 743–754 (2013)

    Article  Google Scholar 

  55. Blazquez A., Mantic V., Pars F.: Application of BEM to generalized plane problems for anisotropic elastic materials in presence of contact. Eng. Anal. Bound. Elem. 30, 489–502 (2006)

    Article  MATH  Google Scholar 

  56. Rodríguez-Tembleque L., Buroni F.C., Abascal R., Sáez A.: 3D frictional contact of anisotropic solids using BEM. Eur. J. Mech. A Solids 30, 95–104 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  57. Guler M.A.: Contact stresses in an orthotropic medium: a closed-form solution. Int. J. Mech. Sci. 87, 72–88 (2014)

    Article  Google Scholar 

  58. Kucuksucu A., Guler M.A., Avci A.: Closed-form solution of a frictional sliding contact problem for an orthotropic elastic half-plane indented by a wedge-shaped punch. Key Eng. Mater. 618, 203–225 (2014)

    Article  Google Scholar 

  59. Bakırtaş I.: The contact problem of an orthotropic non-homogeneous elastic half-space. Int. J. Eng. Sci. 22, 347–359 (1984)

    Article  MATH  Google Scholar 

  60. Krenk S.: On the elastic constants of plane orthotropic elasticity. J. Compos. Mater. 13, 108–116 (1979)

    Article  Google Scholar 

  61. Cinar A., Erdogan F.: The crack and wedging problem for an orthotropic strip. Int. J. Fract. 19, 83–102 (1983)

    Google Scholar 

  62. Kucuksucu, A.: Contact Mechanics Analysis of Orthotropically Graded Materials. Ph.D.Dissertation. Mechanical Engineering Department, Selçuk University, Konya, Turkey (2011)

  63. Erdogan, F.: Mixed boundary value problems in mechanics. In: Nemat-Nasser, S. (ed.) Mechanics Today, vol. 4, pp 1–86. Pergamon Press, Oxford (1978)

  64. Erdogan F., Gupta G.D.: On the numerical solution of singular integral equations. Q. Appl. Math. 29, 525–534 (1972)

    MathSciNet  MATH  Google Scholar 

  65. Ozturk M., Erdogan F.: Mode I crack problem in an inhomogeneous orthotropic medium. Int. J. Eng. Sci. 35, 869–883 (1997)

    Article  MATH  Google Scholar 

  66. Ozturk M., Erdogan F.: The mixed mode crack problem in an inhomogeneous orthotropic medium. Int. J. Fract. 98, 243–261 (1999)

    Article  Google Scholar 

  67. Guler, M.A.: The Problem of a Rigid punch with Friction on a Graded Elastic Medium. MS Thesis, Mechanical Engineering Department, Lehigh University, Bethlehem (1996)

  68. Elloumi R., Guler M.A., Kallel-Kamoun I., El-Borgi S.: Closed-form solutions of the frictional sliding contact problem for a magneto-electro-elastic half-plane indented by a rigid conducting punch. Int. J. Solids Struct. 50, 3778–3792 (2013)

    Article  Google Scholar 

  69. Elloumi R., Kallel-Kamoun I., El-Borgi S., Guler M.A.: On the frictional sliding contact problem between a rigid circular conducting punch and a magneto-electro-elastic half-plane. Int. J. Mech. Sci. 87, 1–17 (2014)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Bursa Orhangazi University, Bursa, 16310, Turkey

    Aysegul Kucuksucu

  2. Department of Mechanical Engineering, TOBB University of Economics and Technology, Ankara, 06560, Turkey

    Aysegul Kucuksucu & Mehmet A. Guler

  3. Department of Mechanical Engineering, Selçuk University, Konya, 42003, Turkey

    Ahmet Avci

Authors
  1. Aysegul Kucuksucu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mehmet A. Guler
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ahmet Avci
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mehmet A. Guler.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kucuksucu, A., Guler, M.A. & Avci, A. Mechanics of sliding frictional contact for a graded orthotropic half-plane. Acta Mech 226, 3333–3374 (2015). https://doi.org/10.1007/s00707-015-1374-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-015-1374-7

Keywords

Search

Navigation