Acta Mechanica

, Volume 183, Issue 3–4, pp 209–230 | Cite as

Analytic parametric solutions for the HRR nonlinear elastic field with low hardening exponents

  • A. Sotiropoulou
  • N. Panayotounakou
  • D. Panayotounakos
Article
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Summary

In this paper, we restore the already constructed approximate asymptotic solutions extracted in [10] concerning the HRR [1] strongly nonlinear fourth-order ordinary differential equation (ODE) for plane strain conditions in nonlinear elastic (plastic) fracture. It is proved that the above equation, for low strain hardening exponents (0 < N « 1), is reduced to a strongly nonlinear ODE of the second order. The method of the total differentials is used so that the last equation is reduced to Abels' equations of the second kind of the normal form, that can be analytically solved in parametric form. In addition, the case of rigid perfect-plasticity (N=0) is extensively investigated and several important results are extracted.

Keywords

Differential Equation Dynamical System Ordinary Differential Equation Normal Form Plane Strain 
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References

  1. Rice, J. R., Rosengren, G. F. 1968Plane strain deformation near a crack tip in a power-law hardening materialJ. Mech. Phys. Solids16112CrossRefGoogle Scholar
  2. Hutchinson, J. W. 1968Singular behavior at the end of a tensile crack in a hardening materialJ. Mech. Phys. Solids161330MATHCrossRefGoogle Scholar
  3. Hutchinson, J. W. 1968Plane stress and strain field in a crack tipJ. Mech. Phys. Solids16337347CrossRefGoogle Scholar
  4. Shih, C. F. 1974Small-scale yielding analysis of mixed mode plane-strain crack problemsASTM STPP500187210Google Scholar
  5. Rice, J. R. 1968Mathematical analysis in the mechanics of fractureLiebowitz, H. eds. Fracture, an advanced treatise, Vol. 2, Mathematical fundamentalsAcademic PressNew York London192311Google Scholar
  6. Hill, R. 1950The mathematical theory of plasticityOxford University PressLondonGoogle Scholar
  7. Prager, W., Hodge, P. G. 1968Theory of perfectly plastic solidsDoverNew YorkGoogle Scholar
  8. Absi, F. 1984La théorie de la plasticité et l' equilibre limite en mécanique des solsAnn. Inst. Tech. Batiment Travaux Publ.42166123Google Scholar
  9. Salençon, J. 1974Théorie de la plasticité pour les applications a la mécanique des solsEyrollesParisGoogle Scholar
  10. Panayotounakos, D. E., Markakis, M. 1990Closed form solutions of the differential equations governing the plastic fracture field in a power-law hardening material with low strain-hardening exponentIng.-Arch.60444462CrossRefGoogle Scholar
  11. Markakis, M.: Analytical techniques for solving nonlinear problems in mechanics of rigid bodies and fluid mechanics. P.H.D. Thesis, National Technical University of Athens, Greece.Google Scholar
  12. Polyanin, A. D., Zaitsev, V. F. 1999Handbook of exact solutions for ordinary differential equationsCRC PressNew YorkGoogle Scholar

Copyright information

© Springer-Verlag Wien 2006

Authors and Affiliations

  • A. Sotiropoulou
    • 1
  • N. Panayotounakou
    • 2
  • D. Panayotounakos
    • 2
  1. 1.A.S.PE.T.E. School of Pedagogical and Technological EducationN. HeraklionGreece
  2. 2.Department of Mechanics, Faculty of Applied Mathematical and Physical SciencesNational Technical University of AthensZographouGreece

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