Science China Physics, Mechanics and Astronomy

, Volume 54, Issue 7, pp 1309–1318 | Cite as

Transient near tip fields in crack dynamics

  • Vladimir Bratov
  • Yuri PetrovEmail author
  • Alexander Utkin
Research Paper


Transient effects of stress-strain fields in the vicinity of a stationary crack tip under high rate loads are discussed. Exact analytical solutions to near tip stresses are compared to fields prescribed by leading terms (one or several) of Williams asymptotic expansion. Influence of load application mode, time (or, which is the same, distance from a crack tip) and Poisson’s ratio on this discrepancy is extensively examined. Some effects connected with crack tip propagation speed are also discussed. Significant inconsistencies between real (or received in numerical solutions of state equations - e.g. finite element computations) crack tip fields and stress intensity factor (SIF) singular field observed by numerous researchers are explained. The scope of problems where SIF field can be used for correct prediction of dynamic stress-strain fields in the crack tip region is established. Possibility to correctly approximate fields that are not SIF dominated, accounting additional terms of Williams expansion, is studied.


transient crack tip fields dynamic fracture high-rate loads asymptotic expansions 

List of main symbols


components of the Cauchy stress tensor




polar coordinates with origin at the crack tip


the first term of the asymptotic expansion of stresses surrounding the tip of the mode I loaded crack (the mode I stress intensity factor)


the second and the following terms of the asymptotic expansion of stresses surrounding the crack tip


angular functions

x(x1, x2)

Cartesian coordinate


displacement field


components of displacement


shear modulus


Poisson’s ratio


Young’s modulus

\(c_1 = \sqrt {\frac{{1 - v}} {{\left( {1 + v} \right)\left( {1 - 2v} \right)}}\frac{E} {\rho }} = \sqrt {\frac{{2\left( {1 - v} \right)}} {{1 - 2v}}\frac{\mu } {\rho }}\)

longitudinal wave speed

\(c_2 = \sqrt {\frac{\mu } {\rho }}\)

transversal wave speed


Rayleigh wave speed


time dependent load


load amplitude


Heaviside step function equal to 0 if the argument is negative and equal to 1 otherwise


longitudinal and transversal wave potentials


mass density

\(\gamma = {{c_2 } \mathord{\left/ {\vphantom {{c_2 } {c_1 }}} \right. \kern-\nulldelimiterspace} {c_1 }} = \sqrt {\frac{{1 - 2v}} {{2\left( {1 - v} \right)}}}\)

ratio of longitudinal and transverse wave speeds


ratio of Rayleigh and longitudinal wave speeds


the sum of the first i terms of the asymptotic expansion


the critical value for tensile stress (ultimate strength)


the critical value for stress intensity factor


the microstructural time of a brittle fracture process (or fracture incubation time) — a parameter characterizing the response of the studied material to applied dynamic loads. τ is constant for a given material and does not depend on problem geometry, the way a load is applied, the shape of a load pulse and its amplitude


characteristic size of fracture process zone


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Yoffe E H. The moving Griffith crack. Philos Mag, 1951, 42: 739–750zbMATHMathSciNetGoogle Scholar
  2. 2.
    Ang W T. Transient response of a crack in an anisotropic strip. Acta Mech, 1987, 70: 97–109zbMATHCrossRefGoogle Scholar
  3. 3.
    Ang W T. A crack in an anisotropic layered material under the action of impact loading. J Appl Mech, 1988, 55: 120–125zbMATHCrossRefGoogle Scholar
  4. 4.
    Freund L B. The mechanics of dynamic shear crack propagation. J Geophys Res, 1979, 84: 2199–2209ADSCrossRefGoogle Scholar
  5. 5.
    Achenbach J D. Extension of a crack by a shear wave. Z Angew Math Phys, 1970, 21: 887–900zbMATHCrossRefGoogle Scholar
  6. 6.
    Achenbach J D. Crack propagation generated by a horizontally polarized shear wave. J Mech Phys Solids, 1970, 18: 245–259ADSzbMATHCrossRefGoogle Scholar
  7. 7.
    Achenbach J D. Dynamic effects in brittle fracture. Mechanics Today. Nemat-Nasser S, ed. New York: Pergamon, Elmsford, 1974. 1–57Google Scholar
  8. 8.
    Eshelby J D. The elastic field of a crack extending nonuniformly under general anti-plane loading. J Mech Phys Solids, 1969, 17: 177–199ADSzbMATHCrossRefGoogle Scholar
  9. 9.
    Broberg K B. The propagation of a brittle crack. Arch Fysik, 1960, 18: 159–192MathSciNetGoogle Scholar
  10. 10.
    Broberg K B. Cracks and Fracture. London: Academic Press, 1999Google Scholar
  11. 11.
    Kostrov B V. Unsteady propagation of longitudinal shear cracks. Appl Math Mech, 1966, 30: 1241–1248CrossRefGoogle Scholar
  12. 12.
    Kostrov B V, Nikitin L V. Some general problems of mechanics of brittle fracture. Arch Mchaniki Stosowanej, 1970, 22: 749–775zbMATHGoogle Scholar
  13. 13.
    Kostrov B V. On the crack propagation with variable velocity. Int J Fract, 1975, 11: 47–56CrossRefGoogle Scholar
  14. 14.
    Williams M L. On the stress distribution at the base of a stationary crack. J Appl Mech, 1957, 24: 109–114zbMATHMathSciNetGoogle Scholar
  15. 15.
    Irwin G R. Analysis of stresses and strains near the end of a crack traversing a plate. J Appl Mech, 1957, 24: 361–364Google Scholar
  16. 16.
    Ma C C, Freund L B. The extent of the stress intensity factor during crack growth under dynamic loading conditions. J Appl Mech, 1986, 53: 303–310CrossRefGoogle Scholar
  17. 17.
    Filippov A V. Some problems of diffraction of plane elastic waves (in Russian). Appl Math Mech, 1965, 20: 688–703Google Scholar
  18. 18.
    Petrov Y V, Utkin A A. Asymptotic representation of stresses surrounding the crack tip in dynamic problems of elasticity. IPME RAS, St.-Petersburg, 2001Google Scholar
  19. 19.
    Petrov Y V, Morozov N F. On the modeling of fracture of brittle solids. J Appl Mech, 1994, 61: 710–712CrossRefGoogle Scholar
  20. 20.
    Petrov Y V, Morozov N F, Smirnov V I. Structural macromechanics approach in dynamics of fracture. Fatigue Fract Eng Mater Struct, 2003, 26: 363–372CrossRefGoogle Scholar
  21. 21.
    Morozov N, Petrov Y. Dynamics of fracture. Berlin: Springer-Verlag, 2000zbMATHGoogle Scholar
  22. 22.
    Freund L B. Dynamic Fracture Mechanics. Cambridge: Cambridge University Press, 1990zbMATHCrossRefGoogle Scholar
  23. 23.
    Broberg K B. The near-tip field at high crack velocities. Int J Fract, 1989, 39: 1–13CrossRefGoogle Scholar
  24. 24.
    Broberg K B. Intersonic bilateral slip. Geophys J Int, 1994, 119: 706–714ADSCrossRefGoogle Scholar
  25. 25.
    Broberg K B. Intersonic mode II crack expansion. Arch Mech, 1995, 47: 859–871zbMATHGoogle Scholar
  26. 26.
    Rosakis A J, Samudrala O, Coker D. Cracks faster than the shear wave speed. Science, 1999, 284: 1337–1340ADSCrossRefGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Vladimir Bratov
    • 1
    • 2
  • Yuri Petrov
    • 1
    • 2
    Email author
  • Alexander Utkin
    • 1
  1. 1.Institute of Problems of Mechanical EngineeringRussian Academy of SciencesSt.-PetersburgRussia
  2. 2.St.-Petersburg State UniversitySt.-PetersburgRussia

Personalised recommendations