Journal of Elasticity

, Volume 136, Issue 1, pp 103–121 | Cite as

An Adhesive Contact Problem for a Semi-plane with a Surface Elasticity in the Steigmann-Ogden Form

  • Anna Y. ZemlyanovaEmail author


The main goal of the present paper is to study a two-dimensional problem of adhesive contact of a rigid stamp with an elastic upper semi-plane. Surface elasticity in the Steigmann-Ogden form acts on the free boundary of the semi-plane. The mechanical problem is solved using integral representations of the stresses and the displacements. The resulting system of singular integro-differential equations is regularized using Fourier transform method. The numerical procedure of the solution of the system of singular integro-differential equations is presented, and the numerical results are obtained for different values of the mechanical parameters. The size-dependency of the solutions of the problem is studied. It can be seen that taking into account curvature-dependency of the surface energy is increasingly important at the very small scales (characteristic length of the contact interval is under \(20~\mbox{nm}\)).


Surface elasticity Surface energy Curvature-dependence Integral equations Contact problems 

Mathematics Subject Classification

74B05 45J05 



  1. 1.
    Benveniste, Y., Miloh, T.: Imperfect soft and stiff interfaces in two-dimensional elasticity. Mech. Mater. 33(6), 309–323 (2001) CrossRefGoogle Scholar
  2. 2.
    Chahapadia, P., Mohammadi, P., Sharma, P.: Erratum to: curvature-dependent surface energy and implications for nanostructures. J. Mech. Phys. Solids 60, 1241–1242 (2012) ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Cheng, Y.T., Cheng, C.M.: Scaling, dimensional analysis, and indentation measurements. Mater. Sci. Eng. 44, 91–149 (2004) CrossRefGoogle Scholar
  4. 4.
    Chhapadia, P., Mohammadi, P., Sharma, P.: Curvature-dependent surface energy and implications for nanostructures. J. Mech. Phys. Solids 59, 2103–2115 (2011) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dai, M., Gharani, A., Schiavone, P.: Analytic solution for a circular nano-inhomogeneity with interface stretching and bending resistance in plane strain deformations. Appl. Math. Model. 55, 160–170 (2018) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Duan, H., Wang, J., Huang, Z., Karihaloo, B.: Eshelby formalism for nano-inhomogeneities. Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 461, 3335–3353 (2005) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Duan, H., Wang, J., Huang, Z., Luo, Z.: Stress concentration tensors of inhomogeneities with interface effects. Mech. Mater. 37, 723–736 (2005) CrossRefGoogle Scholar
  8. 8.
    Fleck, N., Hutchinson, J.: A phenomenological theory for strain gradient effects in plasticity. J. Mech. Phys. Solids 41, 1825–1857 (1993) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gakhov, F.: Boundary Value Problems. Dover, New York (1990) zbMATHGoogle Scholar
  10. 10.
    Gao, X.L.: An expanding cavity model incorporating strain-hardening and indentation size effects. Int. J. Solids Struct. 43, 6615–6629 (2006) CrossRefzbMATHGoogle Scholar
  11. 11.
    Gao, X.L.: A new expanding cavity model for indentation hardness including strain-hardening and indentation size effects. J. Mater. Res. 21, 1317–1326 (2006) ADSCrossRefGoogle Scholar
  12. 12.
    Gao, X., Huang, Z., Fang, D.: Curvature-dependent interfacial energy and its effects on the elastic properties of nanomaterials. Int. J. Solids Struct. 113–114, 100–107 (2017) CrossRefGoogle Scholar
  13. 13.
    Gurtin, M., Murdoch, A.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57(4), 291–323 (1975) MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gurtin, M., Murdoch, A.: Surface stress in solids. Int. J. Solids Struct. 14(6), 431–440 (1978) CrossRefzbMATHGoogle Scholar
  15. 15.
    He, L., Lim, C.: Surface Green function for a soft elastic half-space: influence of surface stress. Int. J. Solids Struct. 43, 217–235 (2006) zbMATHGoogle Scholar
  16. 16.
    He, L., Lim, C., Wu, B.: A continuum model for size-dependent deformation of elastic films of nano-scale thickness. Int. J. Solids Struct. 41, 847–857 (2004) CrossRefzbMATHGoogle Scholar
  17. 17.
    Hong, S., Weil, R.: Low cycle fatigue of thin copper foils. Thin Solid Films 283, 175–181 (1996) ADSCrossRefGoogle Scholar
  18. 18.
    Horstemeyer, M., Baskes, M.: Atomistic finite deformation simulations: a discussion on length scale effects in relation to mechanical stresses. Trans. Am. Soc. Mech. Eng. 121, 114–119 (1999) Google Scholar
  19. 19.
    Huang, D.: Size-dependent response of ultra-thin films with surface effects. Int. J. Solids Struct. 45, 568–579 (2008) CrossRefzbMATHGoogle Scholar
  20. 20.
    Judelewicz, M., Künzi, H., Merk, N., Ilschner, B.: Tensile and fatigue strength of ultrathin copper films. Mater. Sci. Eng. A 186, 135–142 (1994) CrossRefGoogle Scholar
  21. 21.
    Kim, C., Schiavone, P., Ru, C.Q.: Analysis of a mode III crack in the presence of surface elasticity and a prescribed non-uniform surface traction. Z. Angew. Math. Phys. 61, 555–564 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kim, C., Schiavone, P., Ru, C.Q.: The effects of surface elasticity on an elastic solid with mode III crack: complete solution. J. Appl. Mech. 77, 021011 (2010) (7 pages) ADSCrossRefGoogle Scholar
  23. 23.
    Kim, C., Schiavone, P., Ru, C.Q.: Analysis of plane-strain crack problems (mode I and mode II) in the presence of surface elasticity. J. Elast. 104, 397–420 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kim, C., Schiavone, P., Ru, C.Q.: Effect of surface elasticity on an interface crack in plane deformations. Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 467, 3530–3549 (2011) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Kim, C., Schiavone, P., Ru, C.Q.: The effects of surface elasticity on a mode III interface crack. Arch. Mech. 63, 267–286 (2011) MathSciNetzbMATHGoogle Scholar
  26. 26.
    Kim, C., Ru, C.Q., Schiavone, P.: A clarification of the role of crack-tip conditions in linear elasticity with surface effects. Math. Mech. Solids 18(1), 59–66 (2012) MathSciNetCrossRefGoogle Scholar
  27. 27.
    King, F.: Hilbert Transforms, vol. 1. Encyclopedia of Mathematics and Its Applications, vol. 124. Cambridge University Press, Cambridge (2009) CrossRefzbMATHGoogle Scholar
  28. 28.
    Krasnov, M.: Integral Equations: Introduction Into the Theory, 1st edn. Nauka, Moscow (1975) (in Russian) Google Scholar
  29. 29.
    Kress, R.: Linear Integral Equations, 3rd edn. Springer, New York (2014) CrossRefzbMATHGoogle Scholar
  30. 30.
    Ma, Q., Clarke, D.: Size dependent hardness of silver single crystals. J. Mater. Res. 10, 853–863 (1995) ADSCrossRefGoogle Scholar
  31. 31.
    Mikhlin, S., Prössdorf, S.: Singular Integral Operators. Springer, Berlin (1986) CrossRefzbMATHGoogle Scholar
  32. 32.
    Miller, R., Shenoy, V.: Size dependent elastic properties of nanosized structural elements. Nanotechnology 11, 139–147 (2000) ADSCrossRefGoogle Scholar
  33. 33.
    Muskhelishvili, N.: Some Basic Problems of the Mathematical Theory of Elasticity; Fundamental Equations, Plane Theory of Elasticity, Torsion, and Bending. Noordhoff, Groningen (1963) zbMATHGoogle Scholar
  34. 34.
    Nix, W., Gao, H.: Indentation size effects in crystalline materials: a law for strain gradient plasticity. J. Mech. Phys. Solids 46(3), 411–425 (1998) ADSCrossRefzbMATHGoogle Scholar
  35. 35.
    Oliver, W., Pharr, G.: An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. J. Mater. Res. 7(6), 1564–1583 (1992) ADSCrossRefGoogle Scholar
  36. 36.
    Orav-Puurand, K., Pedas, A., Vainikko, G.: Nyström type methods for Fredholm integral equations with weak singularities. J. Comput. Appl. Math. 234, 2848–2858 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Pinyochotiwong, Y., Rungamornrat, J., Senjuntichai, T.: Analysis of rigid frictionless indentation on half-space with surface elasticity. Proc. Eng. 14, 2403–2410 (2011) CrossRefzbMATHGoogle Scholar
  38. 38.
    Pinyochotiwong, Y., Rungamornrat, J., Senjuntichai, T.: Rigid frictionless indentation on elastic half space with influence of surface stresses. Int. J. Eng. Sci. 71, 15–35 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Qu, S., Huang, Y., Nix, W., Jiang, H., Zhang, F., Hwang, K.: Indenter tip radius effect on the Nix-Gao relation in micro- and nanoindentation hardness experiments. J. Mater. Res. 19, 3423–3434 (2004) ADSCrossRefGoogle Scholar
  40. 40.
    Read, D.: Tension-tension fatigue of copper films. Int. J. Fatigue 20, 203–209 (1998) CrossRefGoogle Scholar
  41. 41.
    Savruk, M.: Two-Dimensional Problems of Elasticity for Cracked Solids. Naukova dumka, Kiev (1981) (in Russian) Google Scholar
  42. 42.
    Sendova, T., Walton, J.R.: A new approach to the modeling and analysis of fracture through extension of continuum mechanics to the nanoscale. Math. Mech. Solids 15, 368–413 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Sharma, P., Ganti, S.: Size-dependent Eshelby’s tensor for embedded nano-inclusions incorporating surface/interface energies. J. Appl. Mech. 71, 663–671 (2004) ADSCrossRefzbMATHGoogle Scholar
  44. 44.
    Sharma, P., Ganti, S., Bhate, N.: Effect of surfaces on the size-dependent elastic state of nano-inhomogeneities. Appl. Phys. Lett. 82, 535–537 (2003) ADSCrossRefGoogle Scholar
  45. 45.
    Shenoy, V.: Size-dependent rigidities of nanosized torsional elements. Int. J. Solids Struct. 39, 4039–4052 (2002) CrossRefzbMATHGoogle Scholar
  46. 46.
    Steigmann, D., Ogden, R.: Plain deformations of elastic solids with intrinsic boundary elasticity. Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 453, 853–877 (1997) CrossRefzbMATHGoogle Scholar
  47. 47.
    Steigmann, D., Ogden, R.: Elastic surface substrate interactions. Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 455, 437–474 (1999) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Stelmashenko, N., Walls, M., Brown, L., Milman, Y.: Microindentation on W and Mo oriented single crystals: an STM study. Acta Metall. Mater. 41, 2855–2865 (1993) CrossRefGoogle Scholar
  49. 49.
    Walton, J.: A note on fracture models incorporating surface elasticity. J. Elast. 109(1), 95–102 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Walton, J.: Plane-strain fracture with curvature-dependent surface tension: mixed-mode loading. J. Elast. 114(1), 127–142 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Walton, J., Zemlyanova, A.: A rigid stamp indentation into a semi-plane with a curvature-dependent surface tension on the boundary. SIAM J. Appl. Math. 76(2), 618–640 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Wang, C., Feng, X.: Effects of surface stress on contact problems at nanoscale. J. Appl. Phys. 101, 013510 (2007) (6 pages) ADSCrossRefGoogle Scholar
  53. 53.
    Zemlyanova, A.: The effect of a curvature-dependent surface tension on the singularities at the tips of a straight interface crack. Q. J. Mech. Appl. Math. 66(2), 199–219 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Zemlyanova, A.: Curvilinear mode-i/mode-ii interface fracture with a curvature-dependent surface tension on the boundary. IMA J. Appl. Math. 81(6), 1112–1136 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Zemlyanova, A.: A straight mixed mode fracture with the Steigmann-Ogden boundary condition. Q. J. Mech. Appl. Math. 70(1), 65–86 (2017) MathSciNetCrossRefGoogle Scholar
  56. 56.
    Zemlyanova, A.: Frictionless contact of a rigid stamp with a semi-plane in the presence of surface elasticity in the Steigmann-Ogden form. Math. Mech. Solids 23(8), 1140–1155 (2018) MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Zemlyanova, A., Mogilevskaya, S.: Circular inhomogeneity with Steigmann-Ogden interface: local fields and Maxwell’s type approximation formula. Int. J. Solids Struct. 135, 85–98 (2018) CrossRefGoogle Scholar
  58. 58.
    Zemlyanova, A., Walton, J.: Modeling of a curvilinear planar crack with a curvature-dependent surface tension. SIAM J. Appl. Math. 72, 1474–1492 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Zhao, X., Rajapakse, R.: Analytical solutions for a surface-loaded isotropic elastic layer with surface energy effects. Int. J. Eng. Sci. 47, 1433–1444 (2009) MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Zhou, S., Gao, X.L.: Solutions of half-space and half-plane contact problems based on surface elasticity. Z. Angew. Math. Phys. 64, 145–166 (2013) MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of MathematicsKansas State UniversityManhattanUSA

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