Skip to main content
Log in

Strain gradient solutions of half-space and half-plane contact problems

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

General solutions for the problems of an elastic half-space and an elastic half-plane, respectively, subjected to a symmetrically distributed normal force of arbitrary profile are analytically derived using a simplified strain gradient elasticity theory (SSGET) that contains one material length scale parameter. Mindlin’s potential function method and Fourier transforms are employed in the formulation, and the half-space and half-plane contact problems are solved in a unified manner. The specific solutions for the problems of a half-space/plane subjected to a concentrated normal force or a uniformly distributed normal force are obtained by directly applying the general solutions, which recover the existing classical elasticity-based solutions of the Flamant and Boussinesq problems as special cases. In addition, the indentation problems of an elastic half-space indented by a flat-ended cylindrical punch, a spherical punch, and a conical punch, respectively, are solved using the general solutions, leading to hardness formulas that are indentation size- and material microstructure-dependent. Numerical results reveal that the displacement and stress fields in a half-space/plane given by the current SSGET-based solutions are smoother than those predicted by the classical elasticity-based solutions and do not exhibit the discontinuity and/or singularity displayed by the latter. Also, the indentation hardness values based on the newly obtained half-space solution are found to increase with decreasing indentation radius and increasing material length scale parameter, thereby explaining the microstructure-dependent indentation size effect.

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

Access this article

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. Altan B.S., Aifantis E.C.: On some aspects in the special theory of gradient elasticity. J. Mech. Behav. Mater. 8(3), 231–282 (1997)

    Article  Google Scholar 

  2. Arinstein A., Burman M., Gendelman O., Zussman E.: Effect of supramolecular structure on polymer nanofibre elasticity. Nat. Nanotechnol. 2, 59–62 (2007)

    Article  Google Scholar 

  3. Barber J.R.: Elasticity, 2nd edn. Kluwer, Dordrecht (2002)

    MATH  Google Scholar 

  4. Barbot S., Fialko Y.: Fourier-domain Green’s function for an elastic semi-infinite solid under gravity, with applications to earthquake and volcano deformation. Geophys. J. Int. 182, 568–582 (2010)

    Article  Google Scholar 

  5. Boussinesq J.: Application des potentiels à l’étude de l’équilibre et du mouvement des solides élastiques. Gauthiers-Villars, Paris (1885)

    MATH  Google Scholar 

  6. Chen W.Q., Pan E.N., Wang H.M., Zhang C.Z.: Theory of indentation on multiferroic composite materials. J. Mech. Phys. Solids 58, 1524–1551 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cho J., Joshi M.S., Sun C.T.: Effect of inclusion size on mechanical properties of polymeric composites with micro and nano particles. Compos. Sci. Tech. 66, 1941–1952 (2006)

    Article  Google Scholar 

  8. Cosserat E., Cosserat F.: Théorie des Corps Déformables. A. Hermann et Fils, Paris (1909)

    Google Scholar 

  9. Dhaliwal R.S.: The axisymmetric Boussinesq problem for a semi-space in couple-stress theory. Int. J. Eng. Sci. 11, 1161–1174 (1973)

    Article  MATH  Google Scholar 

  10. Gao X.-L., Liu M.Q.: Strain gradient solution for the Eshelby-type polyhedral inclusion problem. J. Mech. Phys. Solids 60, 261–276 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gao X.-L., Ma H.M.: Green’s function and Eshelby’s tensor based on a simplified strain gradient elasticity theory. Acta Mech. 207, 163–181 (2009)

    Article  MATH  Google Scholar 

  12. Gao X.-L., Ma H.M.: Solution of Eshelby’s inclusion problem with a bounded domain and Eshelby’s tensor for a spherical inclusion in a finite spherical matrix based on a simplified strain gradient elasticity theory. J. Mech. Phys. Solids 58, 779–797 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gao X.-L., Ma H.M.: Strain gradient solution for Eshelby’s ellipsoidal inclusion problem. Proc. R. Soc. A 466, 2425–2446 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gao X.-L., Ma H.M.: Strain gradient solution for the Eshelby-type anti-plane strain inclusion problem. Acta Mech. 223, 1067–1080 (2012)

    Article  MathSciNet  Google Scholar 

  15. Gao X.-L., Park S.K.: Variational formulation of a simplified strain gradient elasticity theory and its application to a pressurized thick-walled cylinder problem. Int. J. Solids Struct. 44, 7486–7499 (2007)

    Article  MATH  Google Scholar 

  16. Gao X.-L., Park S.K., Ma H.M.: Analytical solution for a pressurized thick-walled spherical shell based on a simplified strain gradient elasticity theory. Math. Mech. Solids 14, 747–758 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Georgiadis H.G.: The mode-III crack problem in microstructured solids governed by dipolar gradient elasticity: static and dynamic analysis. ASME J. Appl. Mech. 70, 517–530 (2003)

    Article  MATH  Google Scholar 

  18. Georgiadis H.G., Anagnostou D.S.: Problems of the Flamant–Boussinesq and Kelvin type in dipolar gradient elasticity. J. Elast. 90, 71–98 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Harding J.W., Sneddon I.N.: The elastic stresses produced by the indentation of the plane surface of a semi-infinite elastic solid by a rigid punch. Proc. Cambridge Philos. Soc. 41, 16–26 (1945)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hertz H.: Über die Berührung fester elastischer Körper. J. Reine Angew. Math. 92, 156–171 (1882)

    MATH  Google Scholar 

  21. Hutchinson J.W.: Plasticity at the micron scale. Int. J. Solids Struct. 37, 225–238 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  22. Karlis G.F., Charalambopoulos A., Polyzos D.: An advanced boundary element method for solving 2D and 3D static problems in Mindlin’s strain-gradient theory of elasticity. Int J. Numer. Method Eng. 83, 1407–1427 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Koiter W.T.: Couple-stresses in the theory of elasticity, I & II. Proc. K. Ned. Akad. Wet. (B) 67, 17–44 (1964)

    MATH  Google Scholar 

  24. Lakes, R.: Experimental methods for study of Cosserat elastic solids and other generalized elastic continua. In: Mühlhaus, H. (ed.) Continuum models for materials with micro-structure, pp. 1–22. Wiley, New York (1995)

  25. Lazar, M.: The fundamentals of non-singular dislocations in the theory of gradient elasticity: dislocation loops and straight dislocations. Int. J. Solids Struct. (2012). doi:10.1016/j.ijsolstr.2012.09.017

  26. Lazar M., Maugin G.A.: Nonsingular stress and strain fields of dislocations and disclinations in first strain gradient elasticity. Int. J. Eng. Sci. 43, 1157–1184 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Li S., Miskioglu I., Altan B.S.: Solution to line loading of a semi-infinite solid in gradient elasticity. Int. J. Solids Struct. 41, 3395–3410 (2004)

    Article  MATH  Google Scholar 

  28. Little R.W.: Elasticity. Prentice-Hall, Englewood Cliffs (1973)

    MATH  Google Scholar 

  29. Liu M.Q., Gao X.-L.: Strain gradient solution for the Eshelby-type polygonal inclusion problem. Int. J. Solids Struct. (2012). doi:10.1016/j.ijsolstr.2012.09.010

  30. Ma H.M., Gao X.-L.: Eshelby’s tensors for plane strain and cylindrical inclusions based on a simplified strain gradient elasticity theory. Acta Mech. 211, 115–129 (2010)

    Article  MATH  Google Scholar 

  31. Ma H.M., Gao X.-L.: Strain gradient solution for a finite-domain Eshelby-type plane strain inclusion problem and Eshelby’s tensor for a cylindrical inclusion in a finite elastic matrix. Int. J. Solids Struct. 48, 44–55 (2011)

    Article  MATH  Google Scholar 

  32. Mindlin R.D.: Influence of couple-stresses on stress concentrations. Exp. Mech. 3, 1–7 (1963)

    Article  Google Scholar 

  33. Mindlin R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51–78 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  34. Mindlin R.D.: Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct. 1, 417–438 (1965)

    Article  Google Scholar 

  35. Mindlin R.D., Eshel N.N.: On first strain-gradient theories in linear elasticity. Int. J. Solids Struct. 4, 109–124 (1968)

    Article  MATH  Google Scholar 

  36. Mindlin R.D., Tiersten H.F.: Effects of couple-stresses in linear elasticity. Arch. Ration. Mech. Anal. 11, 415–448 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  37. Nix W.D., Gao H.: Indentation size effects in crystalline materials: a law for strain gradient plasticity. J. Mech. Phys. Solids 46, 411–425 (1998)

    Article  MATH  Google Scholar 

  38. Park S.K., Gao X.-L.: Variational formulation of a modified couple stress theory and its application to a simple shear problem. Z. angew. Math. Phys. 59, 904–917 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  39. Polyzos D., Tsepoura K.G., Tsinopoulos S.V., Beskos D.E.: A boundary element method for solving 2-D and 3-D static gradient elastic problems. Part I. Integral formulation. Comput. Methods Appl. Mech. Eng. 192, 2845–2873 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  40. Sadd M.H.: Elasticity: theory, applications, and numerics, 2nd edn. Academic Press, Burlington (2009)

    Google Scholar 

  41. Selvadurai A.P.S.: Partial differential equations in mechanics 2—The Biharmonic equation, Poisson’s equation. Springer, Berlin (2000)

    Book  MATH  Google Scholar 

  42. Shi M.X., Huang Y., Hwang K.C.: Fracture in a higher-order elastic continuum. J. Mech. Phys. Solids 48, 2513–2538 (2000)

    Article  MATH  Google Scholar 

  43. Shodja H.M., Tehranchi A.: A formulation for the characteristic lengths of fcc materials in first strain gradient elasticity via the Sutton–Chen potential. Phil. Mag. 90, 1893–1913 (2010)

    Article  Google Scholar 

  44. Sneddon I.N.: The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile. Int. J. Eng. Sci. 3, 47–57 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  45. Sun, L., Han, R.P.S., Wang, J., Lim, C.T.: Modeling the size-dependent elastic properties of polymeric nanofibers. Nanotechnology 19, 455706-1~8 (2008)

    Google Scholar 

  46. Toupin R.A.: Elastic materials with couple-stresses. Arch. Ration. Mech. Anal. 11, 385–414 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  47. Toupin R.A.: Theory of elasticity with couple-stress. Arch. Rat. Mech. Anal. 17, 85–112 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  48. Vardoulakis I., Exadaktylos G., Aifantis E.: Gradient elasticity with surface energy: mode-III crack problem. Int. J. Solids Struct. 33, 4531–4559 (1996)

    Article  MATH  Google Scholar 

  49. Wang G.F., Feng X.Q.: Effects of surface stresses on contact problems at nanoscale. J. Appl. Phys. 101, 013510–1~6 (2007)

    Google Scholar 

  50. Yang F., Chong A.C.M., Lam D.C.C., Tong P.: Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39, 2731–2743 (2002)

    Article  MATH  Google Scholar 

  51. Yang F.Q.: Analysis of the axisymmetric indentation of a semi-infinite piezoelectric material: the evaluation of the contact stiffness and the effective piezoelectric constant. J. Appl. Phys. 103, 074115–1~8 (2008)

    Google Scholar 

  52. Zhou D., Jin B.: Boussinesq–Flamant problem in gradient elasticity with surface energy. Mech. Res. Commun. 30, 463–468 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  53. Zhou, S.-S., Gao, X.-L.: Solutions of half-space and half-plane contact problems based on surface elasticity. Z. Angew. Math. Phys. (published on-line) (2012). doi: 10.1007/s00033-012-0205-0

  54. Zhou S.-S., Gao X.-L., He Q.-C.: A unified treatment of axisymmetric adhesive contact problems using the harmonic potential function method. J. Mech. Phys. Solids 59, 145–159 (2011)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Xin-Lin Gao.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gao, XL., Zhou, SS. Strain gradient solutions of half-space and half-plane contact problems. Z. Angew. Math. Phys. 64, 1363–1386 (2013). https://doi.org/10.1007/s00033-012-0273-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00033-012-0273-1

Mathematics Subject Classification

Keywords

Navigation