Journal of Engineering Mathematics

, Volume 107, Issue 1, pp 269–282 | Cite as

On Boussinesq’s problem for a cracked halfspace

  • A. P. S. Selvadurai


This paper examines the axisymmetric elastostatic problem that deals with the action of a concentrated normal force on the surface of an isotropic elastic halfspace containing a penny-shaped crack. The mathematical formulation of the elasticity problem should take into consideration the sense of action of the concentrated force. The paper presents the development of Fredholm integral equations of the second-kind that are associated with this category of problem and indicates the numerical technique that is adopted for their solution. The numerical results are presented for the stress intensity factors generated at the penny-shaped crack experiencing either opening or closure.


Boussinesq’s problem Fredholm integral equations of the second-kind Mixed boundary value problems Penny-shaped crack Stress intensity factors 



The work described in this paper was supported by a Discovery Grant awarded by the Natural Sciences and Engineering Research Council of Canada and the James McGill Research Chairs program. The constructive comments of the reviewers are duly acknowledged. The author is grateful to a former research associate Dr. M.C. Au for assistance with the numerical work.


  1. 1.
    Boussinesq J (1885) Applications des potentials à l’étude de l’équilibre et du mouvement des solides élastique. Gauthier-Villars, PariszbMATHGoogle Scholar
  2. 2.
    Davis RO, Selvadurai APS (1996) Elasticity and geomechanics. Cambridge University Press, CambridgeGoogle Scholar
  3. 3.
    Selvadurai APS (2007) The analytical method in geomechanics. Appl Mech Rev 60:87–106ADSCrossRefGoogle Scholar
  4. 4.
    Selvadurai APS (2000) Partial differential equations in mechanics, vol 2. The biharmonic equation, Poisson’s equation. Springer, BerlinCrossRefzbMATHGoogle Scholar
  5. 5.
    Selvadurai APS (2001) On Boussinesq’s problem. Int J Eng Sci 39:317–322CrossRefzbMATHGoogle Scholar
  6. 6.
    Mindlin RD (1936) Force at a point in the interior of a semi-infinite solid. Physics 7:195–202ADSCrossRefzbMATHGoogle Scholar
  7. 7.
    Lord Kelvin (1848) Note on the integrations of the equations of equilibrium of an elastic solid. Camb Dublin Math J 3:87–89Google Scholar
  8. 8.
    Love AE (1927) A treatise on the mathematical theory of elasticity. Cambridge University Press, CambridgezbMATHGoogle Scholar
  9. 9.
    Westergaard HM (1952) Elasticity and plasticity. Harvard University Press, CambridgeCrossRefzbMATHGoogle Scholar
  10. 10.
    Fröhlich OK (1934) Druckverteilung im baugrunde. Mit besonderer berücksichtigung der plastischen erscheinungen. Julius Springer, WienCrossRefGoogle Scholar
  11. 11.
    Selvadurai APS (2014) On Fröhlich’s solution for Boussinesq’s problem. Int J Numer Analyt Meth Geomech 38:925–934CrossRefGoogle Scholar
  12. 12.
    Cotterell B (2010) Fracture and life. Imperial College Press, LondonCrossRefGoogle Scholar
  13. 13.
    Sneddon IN (1946) The distribution of stress in the neighbourhood of a crack in an elastic solid. Proc R Soc Ser A A228:229–260ADSCrossRefMathSciNetGoogle Scholar
  14. 14.
    Sack RA (1946) Extension of Griffith’s theory of rupture to three dimensions. Proc Phys Soc 58:729–736ADSCrossRefGoogle Scholar
  15. 15.
    Sneddon IN (1972) The use of integral transforms. McGraw Hill, New YorkzbMATHGoogle Scholar
  16. 16.
    Sneddon IN (ed) (1977) Application of integral transforms in the theory of elasticity. International Centre for Mechanical Sciences, Courses and Lectures No. 220, Springer, ViennaGoogle Scholar
  17. 17.
    Hunter SC, Gamblen D (1974) The theory of a rigid circular disc ground anchor buried in an elastic soil either with or without adhesion. J Mech Phys Solids 22:371–399ADSCrossRefzbMATHGoogle Scholar
  18. 18.
    Selvadurai APS (1980) Asymmetric displacements of a ground anchor embedded in a transversely isotropic elastic medium of infinite extent. Int J Eng Sci 18:979–986CrossRefzbMATHGoogle Scholar
  19. 19.
    Keer LM (1975) Mixed boundary value problems for a penny-shaped cut. J Elast 5:89–98CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Selvadurai APS (1993) The axial loading of a rigid circular anchor plate embedded in an elastic halfspace. Int J Numer Anal Meth Geomech 17:43–353CrossRefzbMATHGoogle Scholar
  21. 21.
    Selvadurai APS (1994) On the problem of a detached anchor plate embedded in a crack. Int J Solids Struct 31:1279–1290CrossRefzbMATHGoogle Scholar
  22. 22.
    Selvadurai APS (2002) Mechanics of a rigid circular disc bonded to a cracked elastic halfspace. Int J Solids Struct 39:6035–6053CrossRefzbMATHGoogle Scholar
  23. 23.
    Smith FW, Alavi MJ (1971) Stress intensity factors for a penny-shaped crack in a halfspace. Eng Fract Mech 3:241–254CrossRefGoogle Scholar
  24. 24.
    Srivastava KN, Singh K (1969) The effect of a penny-shaped crack on the distribution of stress in a semi-infinite solid. Int J Eng Sci 7:469–490CrossRefzbMATHGoogle Scholar
  25. 25.
    Erdogan F, Arin K (1971) Penny-shaped crack in an elastic layer bonded to dissimilar halfspaces. Int J Eng Sci 9:213–232CrossRefzbMATHGoogle Scholar
  26. 26.
    Sneddon IN, Lowengrub M (1969) Crack problems in the classical theory of elasticity. Wiley, New YorkzbMATHGoogle Scholar
  27. 27.
    Kassir MK, Sih GC (1975) Mechanics of fracture three-dimensional crack problems, vol 2. Noordhoff International, LeydenzbMATHGoogle Scholar
  28. 28.
    Cherepanov GP (1979) Mechanics of brittle fracture (Translation Editors R. de Witt and W.C. Cooley). McGraw-Hill, New YorkGoogle Scholar
  29. 29.
    Broberg KB (1999) Cracks and fracture. Academic Press, San DiegoGoogle Scholar
  30. 30.
    Green AE, Zerna W (1968) Theoretical elasticity. Oxford University Press, OxfordzbMATHGoogle Scholar
  31. 31.
    Little RW (1973) Elasticity. Prentice-Hall, Upper Saddle RiverzbMATHGoogle Scholar
  32. 32.
    Barber JR (2002) Elasticity. Kluwer Academic Publishers, DordrechtzbMATHGoogle Scholar
  33. 33.
    Podio-Guidugli P (2014) Elasticity for geotechnicians: a modern exposition of Kelvin, Boussinesq, Flamant, Cerruti, Melan and Mindlin problems. Solid mechanics and its applications, vol 204. Springer, SwitzerlandzbMATHGoogle Scholar
  34. 34.
    Baker CTH (1977) The numerical treatment of integral equations. Clarendon Press, OxfordzbMATHGoogle Scholar
  35. 35.
    Delves LM, Mohamed JL (1985) Computational methods for integral equations. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  36. 36.
    Atkinson KE (1997) The numerical solution of integral equations of the second kind. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  37. 37.
    Selvadurai APS (2000) An inclusion at a bi-material elastic interface. J Eng Math 37:155–170CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Selvadurai APS (2003) On the loading of an annular crack by a disc inclusion. J Eng Math 46:377–393CrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    Copson ET (1961) On certain dual integral equations. Proc Glasgow Math Assoc 5:21–24CrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    Obreimoff JW (1930) The splitting strength of mica. Proc R Soc A217:290–297ADSCrossRefGoogle Scholar
  41. 41.
    Selvadurai APS (2000) Fracture evolution during indentation of a brittle elastic solid. Mech Cohesive-Frict Mater 5:325–339CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Department of Civil Engineering and Applied MechanicsMcGill UniversityMontrealCanada

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