Skip to main content
Log in

An asymptotic finite-deformation analysis of the elastostatic field near the tip of a crack

  • Published:
Journal of Elasticity Aims and scope Submit manuscript

Summary

This paper contains an asymptotic treatment, consistent with the fully nonlinear equilibrium theory of compressible elastic solids, of the stresses and deformations near the tip of a traction-free crack in a slab of all-around infinite extent under conditions of plane strain. The loading applied at infinity is taken to be one of uniform uniaxial tension at right angles to the faces of the crack. For the particular class of elastic materials considered the tensile stress in large homogeneous uni-axial extension is asymptotic to a continuously adjustable power of the corresponding principal stretch. The asymptotic analysis of the foregoing crack problem is reduced to a nonlinear eigenvalue problem, the solution of which is established in closed form, in terms of elementary functions and a transcendental integral of such functions. This solution involves two arbitrary constants, one of which governs the amplitude of the ensuing elastostatic field near the tip of the crack. A precise estimate of the amplitude parameter, valid at sufficiently small load intensities, is deduced with the aid of a known conservation law. The remaining arbitrary constant, which is left indeterminate by the present lowest-order asymptotic analysis, does not affect the dominant behavior of the field quantities of primary physical interest. II-lustrative numerical results, appropriate to both hardening and softening materials, are presented.

Zusammenfassung

Diese Arbeit betrifft die asymptotische Ermittelung, im Rahmen der nichtlinearen Elastizitätstheorie ebener Verformungen, von den Spannungen und Verschiebungen am Ende eines Schlitzes in einer allseitig unendlich ausgedehnten Scheibe. Die Scheibe ist im Unendlichen durch einen gleichförmigen Zug senkrecht zur Schlitzachse belastet. Die asymptotische Behandlung dieses Problems wird auf ein Eigenwertproblem zurückgeführt, dessen Lösung in geschlossener Form durch elementare Funktionen dargestellt wird. Die gefundene Lösung enthält zwei unbestimmte Konstanten von welchen eine die Amplitude der lokalen Feldsingularitäten bestimmt. Diese Konstante wird für kleine Belastungen streng abgeschätzt auf Grund eines Erhaltungssatzes und mit Hilfe der bekannten Lösung des linearisierten Schlitzproblems.

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

Access this article

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

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

Explore related subjects

Discover the latest articles and news from researchers in related subjects, suggested using machine learning.

References

  1. Rice, J. R., Mathematical analysis in the mechanics of fracture, Fracture, Volume II, Chapter 3, Academic Press, New York, 1968.

    Google Scholar 

  2. Rice, J. R. and G. F., Rosengren, Plane strain deformation near a crack tip in a power-law hardening material, Journal of the Mechanics and Physics of Solids, 16, (1968) 1, 1.

    Google Scholar 

  3. Hutchinson, J. W., Singular behaviour at the end of a tensile crack in a hardening material, Journal of the Mechanics and Physics of Solids, 16, (1968) 1, 13.

    Google Scholar 

  4. Hutchinson, J. W., Plastic stress and strain fields at a crack tip, Journal of the Mechanics and Physics of Solids, 16, (1968) 5, 337.

    Google Scholar 

  5. Cherepanov, G. P., Crack propagation in continuous media, Journal of Applied Mathematics and Mechanics (English Translation), 31, (1967) 3, 503.

    Google Scholar 

  6. Atkinson, C., A note on crack problems in power-law elastic materials and contact problems in nonlinear creep, International Journal of Engineering Science, 9, (1971) 729.

    Google Scholar 

  7. Knein, M., Zur Theorie des Druckversuchs, Zeitschrift für angewandte Mathematik und Mechanik, 6, (1926) 414.

    Google Scholar 

  8. Williams, M., On the stress distribution at the base of a stationary crack, Journal of Applied Mechanics, 24, (1957) 1, 109.

    Google Scholar 

  9. Wong, F. S. and R. T., Shield, Large plane deformations of thin elastic sheets of neo-Hookean material, Zeitschrift für angewandte Mathematik und Physik, 20, (1969) 2, 176.

    Google Scholar 

  10. Blatz, P. J. and W. L., Ko, Application of finite elastic theory to the deformation of rubbery materials, Transactions of the Society of Rheology, 6, (1962) 223.

    Google Scholar 

  11. Truesdell, C. and R., Toupin, The classical field theories, Handbuch der Physik, vol. III/1, Springer, Berlin, 1960.

    Google Scholar 

  12. Truesdell, C. and W., Noll, The non-linear field theories of mechanics, Handbuch der Physik, vol. III/3, Springer, Berlin, 1965.

    Google Scholar 

  13. Rivlin, R. S., Some topics in finite elasticity, Structural Mechanics, Proceedings of the First Symposium on Naval Structural Mechanics, Pergamon, New York, 1960.

    Google Scholar 

  14. Knowles, J. K. and E., Sternberg, On a class of conservation laws in linearized and finite elastostatics, Archive for Rational Mechanics and Analysis, 44, (1972) 3, 187.

    Google Scholar 

  15. Eshelby, J. D., The continuum theory of lattice defects, Solia State Physics, vol. 3, Academic Press, New York, 1956.

    Google Scholar 

  16. Rice, J. R., A path-independent integral and the approximate analysis of strain concentration by notches and cracks, Journal of Applied Mechanics, 35, (1968) 2, 379.

    Google Scholar 

  17. Günther, W., Über einige Randintegrale der Elastomechanik, Abhandlungen, Braunschweiger Wissenschafcliche Gesellschaft, 14, (1962) 53.

    Google Scholar 

  18. Coleman, B. D. and W., Noll, On the thermostatics of continuous media, Archive for Rational Mechanics and Analysis, 4, (1959) 2, 97.

    Google Scholar 

  19. Truesdel, C. and R., Toupin, Static grounds for inequalities in finite strain of elastic materials, Archive for Rational Mechanics and Analysis, 12, (1963) 1, 1.

    Google Scholar 

  20. Ogden, R. W., Compressible isotropic elastic solids under finite strain — constitutive inequalities, The Quarterly Journal of Mechanics and Applied Mathematics, 23, (1970) 4, 457.

    Google Scholar 

  21. Baker, M. and J. L., Ericksen, Inequalities restricting the form of the stress-deformation relations for isotropic elastic solids and Reiner-Rivlin fluids, Journal of the Washington Academy of Science, 44, (1954) 33.

    Google Scholar 

  22. Inglis, C. E., Stresses in a plate due to the presence of cracks and sharp corners, Transactions, Institution of Naval Architects, 55, (1913) 219.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

The results communicated in this paper were obtained in the course of an investigation supported under Contract N00014-67-A-0094-0020 of the California Institute of Technology with the Office of Naval Research in Washington, D.C.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Knowles, J.K., Sternberg, E. An asymptotic finite-deformation analysis of the elastostatic field near the tip of a crack. J Elasticity 3, 67–107 (1973). https://doi.org/10.1007/BF00045816

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00045816

Keywords