Skip to main content
SpringerLink
Log in

The Axisymmetric Boussinesq Problem for Solids with Surface Energy

  • Conference paper
Unilateral Problems in Structural Analysis

Part of the book series: International Centre for Mechanical Sciences ((CISM,volume 288))

  • 174 Accesses

Abstract

Besides Young’s modulus E and Poisson ratio v, isotropic elastic solids have a surface energy γ. E and v reflect the behaviour of intermolecular forces for small displacements of atoms around their equilibrium position and 2γ is the work needed to cut these bonds along an imaginary plane of unit area and to reversibly separate the two parts of the solid. Surface energy thus characterises the nature of bonds ensuring the cohesion of the solid through this imaginary plane. Accordingly, metals and covalents have high surface energy (from 1000 to 3000 mJ.m −2), ionic crystals (100 to 500 mJ.m −2) and molecular crystals (γ < 100 mJ.m −2) have lower surface energy. The first to have coupled surface energy and elasticity was Griffith1: to extend the area of a crack by dA the work 2γdA is needed; it is taken from the elastic field and/or the potential energy of the system. Later Irwin2 introduced the strain energy G released when the crack area varies by dA, and stated the Griffith criterion for a crack in (stable or unstable) equilibrium as G = 2γ. The singularity of stresses near a crack tip was pointed out by Sneddon3; Irwin4,5 introducing the stress intensity factor K controlling the intensity of the stress fields showed the relation between the energy approach and that in terms of stress fields.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Griffith, A.A., The phenomena of rupture and flow in solids, Phil. Trans. Roy. Soc. A, 221, 1920.

    Google Scholar 

  2. Irwin, G.R., Kies, J.A., Critical energy rate analysis of fracture strength of large welded structures, Welding Journal (Res. Suppl.), 33, 193, 1954.

    Google Scholar 

  3. Sneddon, I.N., The distribution of stress in the neighborhood of a crack in an elastic solid, Proc. Roy. Soc. A, 187, 229, 1946.

    Article  ADS  MathSciNet  Google Scholar 

  4. Irwin, G.R., Analysis of stresses and strains near the end of a crack traversing a plate, J. Appl. Mech, 24, 361, 1957.

    Google Scholar 

  5. Irwin G.R., Fracture, in Encyclopedia of Physics, vol VI, Flügge, Springer Verlag, 1958, p. 551.

    Google Scholar 

  6. Kendall K, The adhesion and surface energie of elastic solids, J. Phys. D: Appl. Phys. 4, 1186, 1971.

    Article  ADS  Google Scholar 

  7. Johnson, K.L. Kendall, K, Roberts, A.D., Surface energy and the contact of elastic solids, Proc. Roy. Soc. A, 324, 301, 1971.

    Article  ADS  Google Scholar 

  8. Maugis, D, Barquins, M, Fracture mechanics and the adherence of viscoelastics bodies, J. Phys. D: AppZ. Phys. 11, 1989, 1978.

    Google Scholar 

  9. Bavkoor, A.R., Briggs, G.A.C. The effect of tangential force in the contact of elastic solids in adhesion, Proc. Roy. Soc. A, 356, 103, 1977.

    Article  ADS  Google Scholar 

  10. Boussinesq, J, Application des potentiels à l’étude de l’équilibre et du mouvement dessolides élastiques, Gauthiers Villars, Paris ( Blanchard, Paris 1969 ) p. 208.

    Google Scholar 

  11. Sneddon, I.N. The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile, Int. J. Engng Sc. 3, 47, 1965

    Article  MATH  MathSciNet  Google Scholar 

  12. Hertz, H, Uber die Berührung fester Elastischer Korper, J. für die reine und Angewandte Mathematik, 92, 156, 1881

    Google Scholar 

  13. Love, E.E.H., Boussinesq’s problem for a rigid cone, Quat. J. Math (Oxford), 10, 161, 1939.

    Article  ADS  MathSciNet  Google Scholar 

  14. Barquins, M, Maugis, D, Adhesive contact of axisymmetric punches on an elastic half-space: the modified Hertz-Huber’s stress tensor for contacting sphres. J. Meca. Theor. AppZ. 1, 131, 1982.

    Google Scholar 

  15. Mossakovski, V.I, Compression of elastic bodies under condition of adhesion (axisymmetric case), P M M, 27, 418, 1963.

    Google Scholar 

  16. Spence, D.A, Self similar solutions to adhesive contact problems with incremental loading, Proc. Roy. Soc. A, 305, 55, 1968.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  17. Williams, M.L, The stresses around a fault or a crack in dissimilar media, Bulletin of the Seismological Soc. Am, 49, 199, 1959.

    Google Scholar 

  18. Erdogan, F, Stress distribution in an nonhomogeneous elastic plate with cracks, J. AppZ. Mech. 30, 232, 1963.

    Article  MATH  Google Scholar 

  19. Erdogan, F, Stress distribution in bonded dissimilar materials with cracks, J. App Z. Mech., 32, 403, 1965.

    Article  MathSciNet  Google Scholar 

  20. England, A.H., A crack between dissimular media, J. Appl. Mech., 32, 400, 1965.

    Article  ADS  Google Scholar 

  21. Dundurs, J. Discussion, J. App Z. Mech., 36, 650, 1969.

    Article  Google Scholar 

  22. Spence, D.A, Similarity considerations for contact between dissimilar elastic bodies, in The mechanics of the contact between defor-mable bodies, de Pater A.R. and Kalker J.J, Eds, Delft University Press, Delft 1975, p. 67.

    Chapter  Google Scholar 

  23. Spence, D.A., The Hertz contact problem with finite friction, J. Elasticity, 5, 297, 1975.

    Article  MATH  Google Scholar 

  24. Comninou M, The interface crack, J. Appl. Mech. 44, 631, 1977.

    Article  ADS  MATH  Google Scholar 

  25. Comninou, M, Interface crack with friction in the contact zone, J. App Z. Mech, 44, 780, 1977.

    Article  Google Scholar 

  26. Maugis, D., Barquins, M., Fracture mechanics and adherence of viscoélastic solids, in Adhesion and adsorption of polymers, part A, Lee, L.H., Ed, Plenum Publ. Corporation, New York, 1980, p. 203–277.

    Chapter  Google Scholar 

  27. Shield, R.T., Load-displacement relations for elastic bodies, Z. Agnew Math. Phys. 18, 682, 1967.

    Article  MATH  Google Scholar 

  28. Kassir, M.K, Sih, G.C., Three dimensional stress distribution around an elliptical crack under arbitrary loading, J. App Z. Mech, 33, 601, 1966.

    Article  Google Scholar 

  29. Sih, G.C, Liebowitz, H, Mathematical theory of brittle fracture, in Fracture, an advanced treatise, vol. 2, Liebowitz, H., ed, Academic Press, New York, 1968, p 67–190.

    Google Scholar 

  30. Kassir, M.K, Sih, G.C, External elliptic crack in elastic solid, Int. J. Fracture Mech, 4, 347, 1968.

    Article  Google Scholar 

  31. Sneddon, I.N, Boussinesq’s problem for a flat-ended cylinder Proc. Cambridge Phil. Soc., 42, 29, 1946.

    Google Scholar 

  32. Huber, M.T, Zur theory der Berührung fester elastischer Korper, Ann. Physik, 14, 153, 1904.

    Article  ADS  MATH  Google Scholar 

  33. Maugis, D, Barquins, M, Adhesive contact of a conical punch on an elastic half-space, J. Phys. Lettres 42, L95, 1981.

    Article  Google Scholar 

  34. Maugis, D, Barquins, M, Adhesive contact of sectionally smooth-ended punches on elastic half-spaces: theory and experiment, J. Phys. D: AppZ. Phys., 16, 1843, 1983.

    Article  ADS  Google Scholar 

  35. Ejike, U.B.C.0, The stress on an elastic half space due to sectionnally smooth ended punch, J. Elasticity, 11, 395, 1981.

    Article  Google Scholar 

  36. Andrews, E.H. Kinloch, A. J. Mechanics of adhesive failure I, Proc. Roy. Soc.A, 332, 385, 1973.

    Google Scholar 

  37. Barquins, M, Maugis, D, Tackiness of elastomers, J. Adhesion, 13, 53, 1981.

    Article  Google Scholar 

  38. Barquins, M, Influence of the stiffness of testing machine on the adherence of elastomers, J. AppZ. Polym. Sci., 28, 2647, 1983.

    Google Scholar 

  39. Derjaguin, B.V, Muller, V.M, Toporov, Yu. P, Effect of contact deformations on the adhesion of particles, J. Colloid Interface Sci. 53, 314, 1975.

    Article  Google Scholar 

  40. Tabor, D, Surface forces and surface interactions, J. Colloid Interface Sci., 58, 2, 1977.

    Article  Google Scholar 

  41. Derjaguin, B.V, Muller, V.M, Toporov, Yu. P, On the role of molecular forces in contact deformations (critical remarks concerning Dr. Tabor’s report), J. Colloid Interface Sci., 67, 378, 1978.

    Article  Google Scholar 

  42. Tabor, D, On the rele of molecular forces in contact deformation. J. Colloid Interface Sci., 67, 380, 1978.

    Google Scholar 

  43. Derjaguin, B., Muller, V, Toporov, Yu, On different approaches to the contact mechanics, J. Colloid Interface Sci., 73, 293, 1980.

    Article  Google Scholar 

  44. Tabor, D, Rôle of molecular forces in contact deformations, J. Colloid Interface Sci., 73, 294, 1980

    Google Scholar 

  45. Muller, V.M., YUSHENKO, V.S, Derjaguin, B.V, On the influence of molecular forces on the deformation of an elastic sphere and its sticking to a rigid plane, J. Colloid Interface Sci., 77, 91, 1980.

    Article  Google Scholar 

  46. Greenwood, J.A, Johnson, K.L, The mechanics of adhesion of viscoelastic solids, Phil. Mag A, 43, 697, 1981.

    Article  ADS  Google Scholar 

  47. Savkoor, A.R, The mechanics and physics of adhesion of elastic solids, in Microscopic aspects of adhesion and lubrication, Georges, J.M. Ed,Elsevier, Amsterdam, 1982, p. 279.

    Google Scholar 

  48. Hughes, B.D, White, L.R, “Soft” contact problems in linear elasticity, Quat. J. Mech. AppZ. Math, 32, 445, 1979.

    Article  MATH  Google Scholar 

  49. Hughes, B.D, White, L.R, Implications of elastic deformation on the direct measurement of surface forces, J.C.S. Faraday I, 76, 963, 1980.

    Article  Google Scholar 

  50. Muller, V.M, Yushenko, V.S, Derjaguin B.V, General theoretical consideration on the influence of surface forces on contact deformations and the reciprocal adhesion of elastic spherical particles, J. Colloid Interface Sci., 92, 92, 1983.

    Article  Google Scholar 

  51. Mindlin, R.D, Compliance of elastic bodies in contact, J. App Z. Mech. 16, 259, 1949.

    MATH  MathSciNet  Google Scholar 

  52. Comninou, M., Exterior interface cracks, Int. J. Engng. Sci., 18, 501, 1980.

    Article  MATH  MathSciNet  Google Scholar 

  53. Janach, W, Separation bubble at the tip of a shear crack under normal Pressure, Int. J. Fracture, 14, R 235, 1978.

    Google Scholar 

  54. Schallamach, A, How dces rubber slide ?, Wear, 17, 301, 1971.

    Article  Google Scholar 

  55. Barquins, M, Energy dissipation in Schallawach waves, Wear, 91, 103, 1983.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CNRS — LCPC, Paris, France

    D. Maugis

Authors
  1. D. Maugis
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Istituto di Meccanica Teorica ed Applicata, Università di Udine, Viale Ungheria 43, 33100, Udine, Italy

    Gianpietro Del Piero

  2. Facoltà di Ingegneria, II Università di Roma, Via Orazio Raimondo — La Romanina, 00173, Roma, Italy

    Franco Maceri

Rights and permissions

Reprints and permissions

Copyright information

© 1985 Springer-Verlag Wien

About this paper

Cite this paper

Maugis, D. (1985). The Axisymmetric Boussinesq Problem for Solids with Surface Energy. In: Del Piero, G., Maceri, F. (eds) Unilateral Problems in Structural Analysis. International Centre for Mechanical Sciences, vol 288. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2632-5_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-7091-2632-5_8

  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-211-81859-6

  • Online ISBN: 978-3-7091-2632-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation