Journal of Elasticity

, Volume 76, Issue 2, pp 163–189 | Cite as

On the Path of a Quasi-static Crack in Mode III

  • Gerardo E. Oleaga


A method for finding the path of a quasi-static crack growing in a brittle body is presented. The propagation process is modelled by a sequence of discrete steps optimizing the elastic energy released. An explicit relationship between the optimal growing direction and the parameters defining the local elastic field around the tip is obtained for an anti-plane field. This allows to describe a simple algorithm to compute the crack path.


crack propagation discrete model singular fields linear elasticity 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. Amestoy and J.B. Leblond, Crack paths in plane situations II: Detailed form of the expansions of the stress intensity factors. Internat. J. Solids Struct. 29(4) (1992) 465–501. MATHMathSciNetGoogle Scholar
  2. 2.
    C.M. Bender and S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, Internat. edn. McGraw-Hill, Singapore (1978). MATHGoogle Scholar
  3. 3.
    M. Brokate and A. Khludnev, On crack propagation shapes in elastic bodies. Z. Angew. Math. Phys. 55 (2004) 318–329. CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    M. Buliga, Energy minimizing brittle crack propagation. J. Elasticity 52 (1999) 201–238. MATHMathSciNetGoogle Scholar
  5. 5.
    M. Buliga, Brittle crack propagation based on an optimal energy balance. Rev. Roum. Math. Pures Appl. 45(2) (2001) 201–209. MathSciNetGoogle Scholar
  6. 6.
    B. Cotterell and J.R. Rice, Slightly curved or kinked cracks. Internat. J. Fracture 16(2) (1980) 155–169. Google Scholar
  7. 7.
    G. Dal Maso and R. Toader, A model for the quasi-static growth of brittle fracture: existence and approximation results. Arch. Rational Mech. Anal. 162(2) (2002) 101–135. MATHADSGoogle Scholar
  8. 8.
    G.A. Francfort and C.J. Larsen, Existence and convergence for quasi-static evolution in brittle fracture. Comm. Pure Appl. Math. LVI (2003) 1465–1500. MathSciNetGoogle Scholar
  9. 9.
    G.A. Francfort and J.J. Marigo, Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46(8) (1998) 1319–1342. MATHADSMathSciNetGoogle Scholar
  10. 10.
    L.B. Freund, Dynamic Fracture Mechanics, 2nd edn. Cambridge Univ. Press, Cambridge (1998). Google Scholar
  11. 11.
    R.V. Goldstein and R.L. Salganik, Brittle fracture of solids with arbitrary cracks. Internat. J. Fracture 10 (1974) 507. Google Scholar
  12. 12.
    A.A. Griffith, The phenomenon of rupture and flow in solids. Phylosoph. Trans. Roy. Soc. London A 221 (1920) 163–198. ADSGoogle Scholar
  13. 13.
    G.R. Irwin, Analysis of stresses and strains near the end of a crack transversing a plate. J. Appl. Mech. 24 (1957) 361–364. Google Scholar
  14. 14.
    J.B. Leblond, Crack paths in plane situations I: General form of the expansion of the stress intensity factors. Internat. J. Solids Struct. 25(11) (1989) 1311–1325. MATHMathSciNetGoogle Scholar
  15. 15.
    Z. Nehari, Conformal Mapping. Dover, New York (1975). Google Scholar
  16. 16.
    J.R. Rice, Mathematical analysis in the mechanics of fracture. In: H. Liebowitz (ed.), Fracture, An Advanced Treatise, Vol. 2. Academic Press, New York (1968) pp. 191–311. Google Scholar
  17. 17.
    G.C. Sih, Stress distribution near internal crack tips for longitudinal shear problems. J. Appl. Mech. (March 1965) 51–58. Google Scholar
  18. 18.
    T.J. Stone and I. Babuska, A numerical method with a posteriori error estimation for determining the path taken by a propagating crack. Comput. Methods Appl. Mech. Engrg. 160 (1998) 245–271. MATHMathSciNetGoogle Scholar
  19. 19.
    C.H. Wu, Elasticity problems of a slender Z-crack. J. Elasticity 8(2) (1978) 235–257. MATHGoogle Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Departamento de Matemática Aplicada, Facultad de MatemáticasUniversidad Complutense de MadridMadridSpain
  2. 2.Max Planck Institute for Mathematics in the SciencesLeipzigGermany

Personalised recommendations