Indentation of a Poroelastic/Biphasic Half-Space

  • Ivan ArgatovEmail author
  • Gennady Mishuris
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 91)


This chapter is devoted to indentation testing of liquid saturated materials, which, for the sake of simplicity, are assumed to be isotropic and undergoing small deformations. In particular, analytical solutions are presented for two types of indenters (cylindrical and paraboloidal) and for two kinds of loading protocols (creep and load-relaxation).


Half-space Surface Drainage Boundary Conditions Paraboloidal Indentation Creep Cavities Pore Fluid Flow 
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  1. 1.
    Abramowitz, M., Stegun, I. (eds.): Handbook of Mathematical Functions. Dover, New York (1970)Google Scholar
  2. 2.
    Agbezuge, L.K., Deresiewicz, H.: On the indentation of a consolidating half-space. Isr. J. Technol. 12, 322–338 (1974)Google Scholar
  3. 3.
    Agbezuge, L.K., Deresiewicz, H.: Consolidation settlement of a circular footing. Isr. J. Technol. 13, 264–269 (1975)Google Scholar
  4. 4.
    Ateshian, G.A.: Mixture theory for modeling biological tissues: illustrations from articular cartilage. In: Holzapfel, G.A., Ogden, R.W. (eds.) Biomechanics: Trends in Modeling and Simulation. Studies in Mechanobiology, Tissue Engineering and Biomaterials, vol. 20. Springer, Cham, pp. 1–51 (2017)Google Scholar
  5. 5.
    Bargar, W.L., Nowinski, J.L.: The Hertz problem for rheological materials of a poroelastic class. Acta Mech. 20, 217–231 (1974)CrossRefGoogle Scholar
  6. 6.
    Biot, M.A.: General theory of three-dimensional consolidation. J. Appl. Phys. 12, 155–164 (1941)CrossRefGoogle Scholar
  7. 7.
    Biot, M.A.: Theory of elasticity and consolidation for a porous anisotropic solid. J. Appl. Phys. 26, 182–185 (1955)CrossRefGoogle Scholar
  8. 8.
    Biot, M.A.: Theory of finite deformations of porous solids. Indiana Univ. Math. J. 21, 597–620 (1972)CrossRefGoogle Scholar
  9. 9.
    Biot, M.A., Willis, D.G.: The elastic coefficients of the theory of consolidation. J. Appl. Mech. 24, 594–601 (1957)Google Scholar
  10. 10.
    Booker, J.R., Small, J.C.: The consolidation of a deep clay stratum subject to an impermeable axisymmetric surface loading. Comput. Geotech. 1(4), 245–261 (1985)CrossRefGoogle Scholar
  11. 11.
    Chen, S.L., Chen, L.Z., Zhang, L.M.: The axisymmetric consolidation of a semi-infinite transversely isotropic saturated soil. Int. J. Numer. Anal. Meth. Geomech. 29, 1249–1270 (2005)CrossRefGoogle Scholar
  12. 12.
    Cheng, A.H.-D.: Poroelasticity. Springer, Switzerland (2016)CrossRefGoogle Scholar
  13. 13.
    Chiarella, C., Booker, J.R.: The time-settlement behaviour of a rigid die resting on a deep clay layer. Int. J. Numer. Anal. Meth. Geomech. 8, 343–357 (1975)Google Scholar
  14. 14.
    De Josselin de Jong, G.: Application of stress functions to consolidation problems. In: Proceedings of the Fourth International Conference on Soil Mechanics and Foundation Engineering, London, vol. 1, pp. 320–323 (1957)Google Scholar
  15. 15.
    Deresiewicz, H.: On the indentation of a consolidationg half-space II. Effect of Poisson’s ratio. Isr. J. Technol. 15, 89–97 (1976)Google Scholar
  16. 16.
    Deresiewicz, H.: Effects of restricted flow at the surface of saturated clay. J. Numer. Anal. Meth. Geomech. 3, 1–11 (1979)CrossRefGoogle Scholar
  17. 17.
    Detournay, E., Cheng, H.-D.A.: Fundamentals of poroelasticity. In: Hudson, J.A. (ed.) Comprehensive Rock Engineering: Principles, Practice and Projects, pp. 113–171. Pergamon, Oxford (1993)Google Scholar
  18. 18.
    Doi, M.: Gel dynamics. J. Phys. Soc. Jpn. 78(5), 052001, 19 p (2009)Google Scholar
  19. 19.
    Galli, M., Oyen, M.L.: Fast identification of poroelastic parameters from indentation tests. Comput. Model. Eng. Sci. (CMES) 48, 241–268 (2009)Google Scholar
  20. 20.
    Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Academic Press, New York (1980)Google Scholar
  21. 21.
    Hahn, H.G.: Elastizitätstheorie. Teubner, Stuttgart (1985)CrossRefGoogle Scholar
  22. 22.
    Heinrich, G., Desoyer, K.: Theorie dreidimensionaler setznugsvorgänge in Tonschichten. Ing. Arch. 30(4), 225–253 (1961)CrossRefGoogle Scholar
  23. 23.
    Hu, Y., Zhao, X., Vlassak, J.J., Suo, Z.: Using indentation to characterize the poroelasticity of gels. Appl. Phys. Lett. 96, 121904, 3 p (2010)Google Scholar
  24. 24.
    Hui, C.-Y., Muralidharan, V.: Gel mechanics: a comparison of the theories of Biot and Tanaka, Hocker, and Benedek. J. Chem. Phys. 123, 154905, 7 p (2005)Google Scholar
  25. 25.
    Kim, J., Selvadurai, A.P.S.: A note on the consolidation settlement of a rigid circular foundation on a poroelastic halfspace. Int. J. Numer. Anal. Meth. Geomech. (2016).
  26. 26.
    Lai, Y., Hu, Y.: Unified solution of poroelastic oscillation indentation on gels for spherical, conical and cylindrical indenters. Soft Matter 13, 852–861 (2017)CrossRefGoogle Scholar
  27. 27.
    Lai, W.M., Mow, V.C.: Drug-induced compression of articular cartilage during a permeation experiment. Biorheology 17, 111–123 (1980)CrossRefGoogle Scholar
  28. 28.
    Lin, Y.-Y., Hu, B.-W.: Load Relaxation of a flat rigid circular indenter on a gel half space. J. Non. Cryst. Solids 352, 4034–4040 (2006)CrossRefGoogle Scholar
  29. 29.
    Mak, A.F., Lai, W.M., Mow, V.C.: Biphasic indentation of articular cartilage. Part I: theoretical analysis. J. Biomech. 20, 703–714 (1987)CrossRefGoogle Scholar
  30. 30.
    Markert, B.: A constitutive approach to 3-d nonlinear fluid flow through finite deformable porous continua. Transp. Porous Med. 70, 427–450 (2007)CrossRefGoogle Scholar
  31. 31.
    Mow, V.C., Kuei, S.C., Lai, W.M., Armstrong, C.G.: Biphasic creep and stress relaxation of articular cartilage in compression: theory and experiments. J. Biomech. Eng. 102, 73–84 (1980)CrossRefGoogle Scholar
  32. 32.
    McNamee, J., Gibson, R.E.: Displacement functions and linear transforms applied to diffusion through porous elastic media. Q. J. Mech. Appl. Math. 13, 98–111 (1960)CrossRefGoogle Scholar
  33. 33.
    McNamee, J., Gibson, R.E.: Plane strain and axially symmetric problem of the consolidation of a semi-infinite clay stratum. Q. J. Mech. Appl. Math. 13, 210–227 (1960)CrossRefGoogle Scholar
  34. 34.
    Nowinski, J.L.: Bielayev’s point in poroelastic bodies in contact. Int. J. Mech. Sci. 15, 145–155 (1973)CrossRefGoogle Scholar
  35. 35.
    Oyen, M.L.: Poroelastic nanoindentation responses of hydrated bone. J. Mater. Res. 23, 1307–1314 (2008)CrossRefGoogle Scholar
  36. 36.
    Pe\(\tilde{\rm n}\)a, E., Del Palomar, A.P., Calvo, B., Martínez, M.A., Doblaré, M.: Computational modelling of diarthrodial joints. Physiological, pathological and pos-surgery simulations. Arch. Comput. Methods. Eng. 14, 47–91 (2007)Google Scholar
  37. 37.
    Rice, J.R., Cleary, M.P.: Some basic stress-diffusion solutions for fluid saturated elastic porous media with compressible constituents. Rev. Geophys. Space Phys. 14, 227–241 (1976)CrossRefGoogle Scholar
  38. 38.
    Scherer, G.W.: Drying gels VIII. Revision and review. J. Non-Cryst. Solids 109, 171–182 (1989)CrossRefGoogle Scholar
  39. 39.
    Scherer, G.W.: Measurement of permeability I. Theory J. Non-Cryst. Solids 113, 107–118 (1989)CrossRefGoogle Scholar
  40. 40.
    Selvadurai, A.P.S.: The analytical method in geomechanics. Appl. Mech. Rev. 60, 87–106 (2007)CrossRefGoogle Scholar
  41. 41.
    Verruijt, A.: Displacement functions in the theory of consolidation or in thermoelasticity. J. Appl. Math. Phys. (ZAMP) 22, 891–898 (1971)CrossRefGoogle Scholar
  42. 42.
    Yue, Z.Q., Selvadurai, A.P.S.: Contact problem for saturated poroelastic solid. J. Eng. Mech. ASCE 121(4), 502–512 (1995)CrossRefGoogle Scholar

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© 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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