Indentation of an Anisotropic Elastic Half-Space

Chapter
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 91)

Abstract

This chapter is devoted to the effect of material anisotropy on the incremental indentaion stiffness in the frictionless indentation of elastic materials. In particular, the cases of cylindrical flat-ended (circular and elliptical), paraboloidal, and conical indenters are considered. We present simple approximations for the introduced indentation moduli.

References

  1. 1.
    Argatov, I.: Frictionless and adhesive nanoindentation: Asymptotic modeling of size effects. Mech. Mater. 42, 807–815 (2010)CrossRefGoogle Scholar
  2. 2.
    Argatov, I.I.: Asymptotic models of contact interaction among elliptic punches on a semiclassical foundation. Int. Appl. Mech. 42, 67–83 (2006)CrossRefGoogle Scholar
  3. 3.
    Argatov, I.I., Dmitriev, N.N.: Fundamentals of the Theory of Elastic Discrete Contact. Polytechnics, St. Petersburg (2003). [in Russian]Google Scholar
  4. 4.
    Argatov, I., Mishuris, G.: Contact Mechanics of Articular Cartilage Layers. Springer, Cham (2015)CrossRefGoogle Scholar
  5. 5.
    Barber, J.R.: Determining the contact area in elastic indentation problems. J. Strain Anal. 9, 230–232 (1974)CrossRefGoogle Scholar
  6. 6.
    Barnett, D.M., Lothe, J.: Line force loadings on anisotropic half-spaces and wedges. Physica Norvegica 8, 13–22 (1975)Google Scholar
  7. 7.
    Borodich, F.M.: Solution of contact problems of elasticity theory for an anisotropic body by the method of similarity. Sov. Appl. Mech. 26, 631–636 (1990)CrossRefGoogle Scholar
  8. 8.
    Borodich, F.M.: The Hertz frictional contact between nonlinear elastic anisotropic bodies (the similarity approach). Int. J. Solids Struct. 30, 1513–1526 (1993)CrossRefGoogle Scholar
  9. 9.
    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
  10. 10.
    Davtyan, D.B., Pozharskii, D.A.: The action of a strip punch on a transversely isotropic half-space. J. Appl. Math. Mech. 76, 558–566 (2012)CrossRefGoogle Scholar
  11. 11.
    Delafargue, A., Ulm, F.-J.: Explicit approximations of the indentation modulus of elastically orthotropic solids for conical indenters. Int. J. Solids Struct. 41, 7351–7360 (2004)CrossRefGoogle Scholar
  12. 12.
    Dovnorovich, V.I.: Spatial Contact Problems of the Theory of Elasticity. Belorus. University, Minsk (1959). [in Russian]Google Scholar
  13. 13.
    Fabrikant, V.I.: Non-traditional contact problem for transversely isotropic half-space. Quart. J. Mech. Appl. Mat. 64, 151–170 (2011)CrossRefGoogle Scholar
  14. 14.
    Fabrikant, V.I.: Contact problem for a transversely isotropic half-space limited by a plane perpendicular to its planes of isotropy. IMA J. Appl. Math. 81, 199–227 (2015)CrossRefGoogle Scholar
  15. 15.
    Galin, L.A.: Contact problems in the theory of elasticity. In: Sneddon, I.N. (ed.) Dept. of Math., North Carolina State College, Raleigh (1961)Google Scholar
  16. 16.
    Gao, Y.F.: Passing stiffness anisotropy in multilayers and its effects on nanoscale surface self-organization. Int. J. Solids Struct. 40, 6429–6444 (2003)CrossRefGoogle Scholar
  17. 17.
    Gao, Y.F., Pharr, G.M.: Multidimensional contact moduli of elastically anisotropic solids. Scr. Mater. 57, 13–16 (2007)CrossRefGoogle Scholar
  18. 18.
    Garcia, J.J., Altiero, N.J., Haut, R.C.: An approach for the stress analysis of transversely isotropic biphasic cartilage under impact load. J. Biomech. Eng. 120, 608–613 (1998)CrossRefGoogle Scholar
  19. 19.
    Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Academic Press, New York (1980)Google Scholar
  20. 20.
    Itskov, M., Aksel, N.: Elastic constants and their admissible values for incompressible and slightly compressible anisotropic materials. Acta Mech. 157, 81–96 (2002)CrossRefGoogle Scholar
  21. 21.
    Jahnke, E., Emde, F., Lösch, F.: Special Functions: Formulae, Graphs, Tables [Russian translation]. Nauka, Moscow (1977)Google Scholar
  22. 22.
    Krein, S.G. (ed.): Functional Analysis. Nauka, Moscow (1972). [in Russian]Google Scholar
  23. 23.
    Lekhnitskii, S.G.: Theory of Elasticity of an Anisotropic Body. Mir publishing, Moscow (1981)Google Scholar
  24. 24.
    Lothe, J., Barnett, D.M.: On the existence of surface-wave solutions for anisotropic elastic half-spaces with free surface. J. Appl. Phys. 47, 428–433 (1976)CrossRefGoogle Scholar
  25. 25.
    Lurie, A.I.: Three Dimensional Problems of the Theory of Elasticity. Interscience, New York (1964)Google Scholar
  26. 26.
    Mossakovskii, V.I.: Estimating displacements in spatial contact problems [in Russian]. J. Appl. Math. Mech. (PMM) 15, 635–636 (1951)Google Scholar
  27. 27.
    Shi, D., Lin, Y., Ovaert, T.C.: Indentation of an orthotropic half-space by a rigid ellipsoidal indenter. ASME J. Trib. 125, 223–231 (2003)CrossRefGoogle Scholar
  28. 28.
    Suo, Z.: Singularities, interfaces and cracks in dissimilar anisotropic media. Proc. Roy. Soc. Lond. A 427, 331–358 (1990)CrossRefGoogle Scholar
  29. 29.
    Sveklo, V.A.: Boussinsq type problems for the anisotropic half-space. J. Appl. Math. Mech. 28, 1099–1105 (1964)CrossRefGoogle Scholar
  30. 30.
    Sveklo, V.A.: The action of a stamp on an elastic anisotropic half-space. J. Appl. Math. Mech. 34, 165–171 (1970)CrossRefGoogle Scholar
  31. 31.
    Sveklo, V.A.: Hertz problem on compression of anisotropic bodies. J. Appl. Math. Mech. 38, 1023–1027 (1974)CrossRefGoogle Scholar
  32. 32.
    Swadener, J.G., Pharr, G.M.: Indentation of elastically anisotropic half-spaces by cones and parabolae of revolution. Philos. Mag. A 81, 447–466 (2001)CrossRefGoogle Scholar
  33. 33.
    Swanson, S.R.: Hertzian contact of orthotropic materials. Int. J. Solids Struct. 41, 1945–1959 (2004)CrossRefGoogle Scholar
  34. 34.
    Ting, T.C.T.: Anisotropic Elasticity. Oxford Univ. Press, Oxford, UK (1996)Google Scholar
  35. 35.
    Vlassak, W.D., Nix, W.D.: Indentation modulus of elastically anisotropic half spaces. Phil. Mag. A 67, 1045–1056 (1993)CrossRefGoogle Scholar
  36. 36.
    Vlassak, W.D., Nix, W.D.: Measuring the elastic properties of anisotropic materials by means of indentation experiments. J. Mech. Phys. Solids 42, 1223–1245 (1994)CrossRefGoogle Scholar
  37. 37.
    Vlassak, J.J., Ciavarella, M., Barber, J.R., Wang, X.: The indentation modulus of elastically anisotropic materials for indenters of arbitrary shape. J. Mech. Phys. Solids 51, 1701–1721 (2003)CrossRefGoogle Scholar
  38. 38.
    Willis, J.R.: Hertzian contact of anisotropic bodies. J. Mech. Phys. Solids 14, 163–176 (1966)CrossRefGoogle Scholar
  39. 39.
    Willis, J.R.: Boussinesq problems for an anisotropic half-space. J. Mech. Phys. Solids 15, 331–339 (1967)CrossRefGoogle Scholar
  40. 40.
    Zhupanska, O.I.: Indentation of a rigid sphere into an elastic half-space in the direction orthogonal to the axis of material symmetry. J. Elast. 99, 147–161 (2010)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of MechanicsTechnical University of BerlinBerlinGermany
  2. 2.Department of Mathematics, IMPACSAberystwyth UniversityAberystwythUK

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