Dynamic Fracture Theory

  • F. Nilsson
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 310)


Dynamic (inertia and strain rate) effects influence crack growth processess if the applied loading rate is sufficiently high or if the crack tip moves with a speed that is a significant fraction of the wave velocities. In the paper basic results as well as solutions to boundary value problems incorporating dynamic effects are discussed. The integral expression for the energy flow to a moving crack tip is derived and related path-area integrals are discussed. It is shown that these quantities are non-trivial only if the energy density behaves as O(r−½). The J-integral emerges as a special case of the general formulations. For linear dynamic problems a line integral defined in the Laplace transform space is introduced. The asymptotic field of a crack growing dynamically in a linear elastic material is derived and the stress-intensity factors are defined. KI-solutions are discussed for a number of problems involving both stationary cracks under dynamic loading as well as moving tips. Asymptotic solutions are given for both stationary and moving tips in different non-linear materials with either rate-independent or rate-dependent elasto-plastic behaviour. These solutions provide the basis for a discussion of dynamic crack growth criteria. Finally, some aspects on numerical modelling of dynamic crack problems are briefly discussed.


Stress Intensity Factor Crack Surface Energy Flow Process Zone Dynamic Fracture 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Pugh, C. E., Bass, B. R., Naus, DJ., Nanstad, R. K., deWit, R., Fields, R. J. and S. R. Low IE: Crack run-arrest behaviour in wide SEN plates of a LWR pressure vessel Material, in: Trans. 9th Int. Conf. on Structural Mechanics in Reactor Technology, (Ed. F. H. Wittman), Vol. G, 1987, 21–26.Google Scholar
  2. 2.
    Strifors, H.: Thermomechanical theory of fracture, kinematics and physical principles, Department of Strength of Materials, Royal Institute of Technology, Stockholm, Sweden, Report 27, 1980. 1Google Scholar
  3. 3.
    Strifors, H.: Thermomechanical theory of fracture based upon the linear strain tensor, Department of Strength of Materials, Royal Institute of Technology, Stockholm, Sweden, Report 28, 1980.1Google Scholar
  4. 4.
    Lidström, P.: Equations of balance and related problems in continuum fracture mechanics, Lund University, Lund, Sweden, LUTFD2/(TFME-1001)/1–11/, (1985), 1985.Google Scholar
  5. 5.
    Barenblatt, G. I.: The formation of equilibrium cracks during brittle fracture. General ideas and hypotheses. Axially-symmetric cracks, J. Appl. Math. Mech. PMM, 23 (1959), 622–635.CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Broberg, K. B.: Mathematical methods in fracture mechanics, in: Trends in Applications of Pure Mathematics to Mechanics,Vol III, (Ed. H. Zorski), Pitman Publ. Ltd, 1979, 57–78.Google Scholar
  7. 7.
    Griffith, A. A.; The phenomena of rupture and flow in solids, Phil. Trans. Roy. Soc., A221 (1921), 163–173.CrossRefGoogle Scholar
  8. 8.
    Kostrov, B. V. and L. V. Nikitin: Some general problems of mechanics of brittle fracture, Arch. Mech. Stos., 22 (1970), 749–776.MATHGoogle Scholar
  9. 9.
    Strifors, H. C.: A generalized force measure of conditions at crack tips, Int. J. Solids and Structures, 10 (1974), 1389–1404.CrossRefMATHGoogle Scholar
  10. 10.
    Kishimoto, K., Aoki, S. and M. Sakata: Dynamic stress intensity factors using Ĵ-integral and finite element method, Eng. Fract. Mech., 13 (1980), 387–394.CrossRefGoogle Scholar
  11. 11.
    Atluri, S. N.: Path-independent integrals in finite elasticity and inelasticity, with body force, inertia and arbitrary crack-face conditions, Eng. Fract. Mech., 16 (1982), 341–364.CrossRefGoogle Scholar
  12. 12.
    Moran, B. and C. F. Shih: Crack tip and associated domain integrals from momentum and energy balance, Eng. Fract. Mech., 27 (1987), 615–642.CrossRefGoogle Scholar
  13. 13.
    Nilsson, F.: A path-independent integral for transient crack problems, Int. J. of Solids and Structures, 9 (1973), 1107–1115.CrossRefMATHGoogle Scholar
  14. 14.
    Gurtin, M. E.: On a path-independent integral for elastodynamics, Int. J. of Fracture, 12 (1976), 643–644.Google Scholar
  15. 15.
    Gol’dstein, R. V. and RX. Salganik: Brittle fracture of solids with arbitrary cracks, Int. J. of Fract., 10 (1974), 507–523.CrossRefGoogle Scholar
  16. 16.
    Achenbach, J. D.: Wave Propagation in Elastic Solids, North-Holland, Amsterdam, 1973.MATHGoogle Scholar
  17. 17.
    Sih, G. C.: Dynamic aspects of crack propagation, in: Inelastic Behaviour of Solids (Ed. Jaffee, R. I. and M. F. Kanninen), McGraw-Hill, New York, 1970.Google Scholar
  18. 18.
    Nilsson, F.: A note on the stress-singularity at a non-uniformly moving crack tip, J. of Elasticity, 4 (1974), 73–75.CrossRefMATHGoogle Scholar
  19. 19.
    Clifton, R. J. and L. B. Freund: On the uniqueness of plane elastodynamic solutions for running cracks, J. of Elasticity, 4 (1974), 293–299.CrossRefMATHGoogle Scholar
  20. 20.
    Achenbach, J. D. and Z. P. Bazant: Elastodynamic near-tip stress and displacement fields for rapidly propagating cracks in orthotropic materials, J. Applied Mechanics, 42 (1975), 183–189.CrossRefMATHGoogle Scholar
  21. 21.
    Maue, A. W.: Die Beugung Elastischer Wellen an der Halbebene, ZAMM, 33 (1953), 1–10.CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Sih, G. C., Embley, G. T. and R. S. Ravera: Impact response of a finite crack in plane extension, Int. J. Solids and Structures, 7 (1971), 731.CrossRefGoogle Scholar
  23. 23.
    Thau, S. A. and T. H. Lu: Transient stress intensity factors for a finite crack in an elastic solid caused by a dilatational wave, Int. J. Solids and Structures, 7 (1971), 731–750.CrossRefMATHGoogle Scholar
  24. 24.
    Kim, K. S.: Dynamic crack propagation of a finite crack, Int. J. Solids and Structures, 15 (1979), 685–699.CrossRefMATHGoogle Scholar
  25. 25.
    Chen, Y. M.: Numerical computation of dynamic stress intensity factors by a Lagrang-ian finite-difference method(the HEMP code), Eng. Fract. Mech., 7 (1975), 653–660.CrossRefGoogle Scholar
  26. 26.
    Aberson, J. A., Anderson, J. M. and W. W. King: Dynamic analysis of cracked structures using singularity finite elements, in: Mechanics of Fracture, 4, (Ed. G. C. Sih) Noordhoff, Leyden 1977.Google Scholar
  27. 27.
    Brickstad, B.: A FEM-analysis of crack arrest experiments, Int. J. Fracture, 21 (1983), 177–191.CrossRefGoogle Scholar
  28. 28.
    ABAQUS, User’s Manual, version 4. 5, Hibbitt, Karlsson and Sorensen Inc., Providence R. I., 1985.Google Scholar
  29. 29.
    Nilsson, F.: Dynamic stress intensity factors for finite strip problems, Int. J. Fract. Mech., 8 (1972), 403–411.Google Scholar
  30. 30.
    Dahlberg, L., Nilsson, F. and B. Brickstad: Influence of specimen geometry on crack propagation and arrest toughness, in: Crack Arrest Methodology and Applications, ASTM STP 711(ed. G. T. Hahn and M. F. Kanninen), ASTM, Philadelphia (1980), 9–108.Google Scholar
  31. 31.
    Willis, J. R.: Self-similar problems in elastodynamics, Phil. Trans Roy. Soc., Series A, 274 (1973), 435–491.CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Cherepanov, G. P. and E. F. Afas’anev: Some dynamic problems of the theory of elasticity- a review, Int. J. Eng. Science, 12 (1974), 665–690.CrossRefMATHGoogle Scholar
  33. 33.
    Broberg, K. B.: The propagation of a brittle crack, Arkiv för Fysik, 18 (1960), 739–750.MathSciNetGoogle Scholar
  34. 34.
    Freund, L. B.: Crack propagation in an elastic solid subjected to general loading-I. Constant rate of extension, J. Mech. Phys. Solids, 20 (1972), 129–140.CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Eshelby, J. D.: The elastic field of a crack extending non-uniformly under general anti-plane loading, J. Mech. Phys. Solids, 17 (1969), 177–199.CrossRefMATHGoogle Scholar
  36. 36.
    Freund, L. B.: Crack propagation in an elastic solid subjected to general loading-II. Non-uniform rate of Extension, J. Mech. Phys. Solids, 20 (1972), 141–152.CrossRefGoogle Scholar
  37. 37.
    Freund, L. B.: Crack propagation in an elastic solid subjected to general loading-Ill. Stress-wave loading, J. Mech. Phys. Solids, 21 (1973), 47–61.CrossRefMATHGoogle Scholar
  38. 38.
    Freund, L. B.: Crack propagation in an elastic solid subjected to general loading-IV. Obliquely incident stress pulse, J. Mech. Phys. Solids, 22 (1974), 137–146.CrossRefMATHGoogle Scholar
  39. 39.
    Burridge, R.: An influence function for the intensity factor in tensile fracture, Int. J. Eng. Science, 14 (1976), 725–730.CrossRefMATHGoogle Scholar
  40. 40.
    Kostrov, B. V.: On the crack propagation with variable velocity, Int. J. of Fracture, 11 (1975), 47–56.CrossRefGoogle Scholar
  41. 41.
    Nilsson, F.: Steady crack propagation followed by nonsteady growth-Mode I solution, Im. J. Solids and Structures, 13 (1977), 1133–1139.CrossRefMATHGoogle Scholar
  42. 42.
    Brickstad, B. and F. Nilsson: Numerical evaluation by FEM of crack propagation experiments, Int. J. of Fract., 16 (1980), 71–84.CrossRefGoogle Scholar
  43. 43.
    Gudmundsson, P.: Validity of asymptotic solutions for plastic materials, To be presented at ICF7, Houston, 1989.Google Scholar
  44. 44.
    Hult, A. J. and F. McClintock: Elastic-plastic stress and strain distributions around sharp notches under repeated shear, Proc. 9th Int. Congr. Appl. Mech., Brussels, 8 1956), 51–58.Google Scholar
  45. 45.
    Rice, J. R.: Mathematical Analysis in the Mechanics of Fracture, in: Fracture: An Advanced Treatise II (Ed. H. Liebowitz), Academic Press, New York 1968, 191–308.Google Scholar
  46. 46.
    Hutchinson, J. W.: Singular behaviour at the end of a tensile crack in a hardening material, J. Mech. Phys. Solids, 16 (1968), 13–31.CrossRefMATHGoogle Scholar
  47. 47.
    Rice, J. R. and G. Rosengren: Plane strain deformation near a crack tip in a power-law hardening material, J. Mech. Phys. Solids, 16 (1968), 1–12.CrossRefMATHGoogle Scholar
  48. 48.
    Slepyan, L. I.: Crack dynamics in an elastic plastic body, Izv. Akad. Nauk SSSR MTT, 11 (1976), 126.Google Scholar
  49. 49.
    Freund, L. B. and A. S. Douglas: The influence of inertia on elastic-plastic antiplane-shear crack growth, J. Mech. Phys. Solids, 30 (1982), 59–74.CrossRefMATHGoogle Scholar
  50. 50.
    Drugan, W. J., Rice, J. R. and T. L. Sham: Asymptotic analysis of growing plane strain tensile cracks in elastic-ideally plastic solids, J. Mech. Phys. Solids, 30 (1982), 447–473.CrossRefMATHGoogle Scholar
  51. 51.
    Leighton, J. T., Champion C. R. and L. B. Freund: Asymptotic analysis of steady dynamic crack growth in an elastic-plastic material, J. Mech. Phys. Solids 35 (1987), 541–564.CrossRefMATHMathSciNetGoogle Scholar
  52. 52.
    Achenbach, J. D. Kanninen, M. F., and C. H. Popelar: Crack-tip fields for fast fracture of an elastic-plastic material, J. Phys. Solids, 29 (1981), 211–225.CrossRefMATHGoogle Scholar
  53. 53.
    Östlund, S. and P. Gudmundson: Asymptotic crack-tip fields for dynamic fracture of linear strain-hardening solids, Department of Strength of Materials, Royal Institute of Technology, Stockholm, Sweden, Report 83, 1987.Google Scholar
  54. 54.
    Nilsson, F. and P. Ståhle: Crack growth criteria and crack tip models, to appear in SM Archives, (1988).Google Scholar
  55. 55.
    Perzyna P.: The constutive equations for rate sensitive plastic materials. Quarterly of Appl Math., 20(1963), 321.MATHMathSciNetGoogle Scholar
  56. 56.
    Hui, C. Y. and H. Riedel The asymptotic stress and strain field near the tip of a growing crack under creep conditions, Int. J. Fract., 17 (1981), 409–425.CrossRefGoogle Scholar
  57. 57.
    Lo, K. K.: Dynamic crack tip fields in rate sensitive solids, J. Mech. Phys. Solids, 31 (1983), 287–305.CrossRefMATHGoogle Scholar
  58. 58.
    Yang, W. and L. B. Freund: An analysis of anti-plane shear crack growth in a rate sensitive elastic-plastic material, Int. J. Fract., 30 (1986), 157–174.CrossRefGoogle Scholar
  59. 59.
    Nilsson, F.: Crack growth initiation and propagation under dynamic loading, Proc. 3rd Conference on Mechanical Properties of High Rates of Strain, Institute of Physics, ser no 70, Oxford, 1984, 185–204.Google Scholar
  60. 60.
    Nilsson, F., Ohlsson, P., Sjöberg, F. and P. Ståhle: A strain-rate dependent strip yield model for rapid loading conditions, to appear in Eng. Fract. Mech., (1989).Google Scholar
  61. 61.
    Wilson, M. L., Hawley, R. H. and. J. Duffy: The effect of loading and temperature on fracture initiation in 1020 hot-rolled steel, Eng. Fract. Mech., 13, 371–385Google Scholar
  62. 62.
    Brickstad, B.: A viscoplastic analysis of rapid crack propagation in steel, J. Mech. Phys. Solids, 31 (1983), 307–327.CrossRefGoogle Scholar
  63. 63.
    Freund, L. B. and A. S. Douglas: The influence of inertia on elastic-plastic antiplane-shear crack growth, J. Mech. Phys. Solids, 30 (1982), 50–74.CrossRefGoogle Scholar
  64. 64.
    Freund, L. B., Hutchinson, J. W. and P. S. Lam: Analysis of high-strain-rate elastic-plastic crack growth, Eng. Fract Mech., 23 (1986), 119–129.CrossRefGoogle Scholar
  65. 65.
    Mataga, P., Freund L. B. and J. W. Hutchinson: Crack tip plasticity in dynamic fracture, J. Phys. Chem. Solids, 48 (1987), 985–1005.CrossRefGoogle Scholar
  66. 66.
    Henshell, R. D. and K. G. Shaw: Crack tip elements are unnecessary, Int J. Num. Meth. Eng., 9 (1975), 495–507.CrossRefMATHGoogle Scholar
  67. 67.
    Barsoum, R. S.: On the use of isoparametric finite elements in linear fracture mechanics, Int J. Num. Meth. Eng., 10 (1976), 25–37.CrossRefMATHGoogle Scholar
  68. 68.
    Akin, J. E.: The generation of elements with singularities, Int. J. Num. Meth., Eng., 10 (1976), 1249–1259.CrossRefMATHMathSciNetGoogle Scholar
  69. 69.
    Thesken, J. C. and P. Gudmundson: Application of a variable order singular element to dynamic fracture mechanics, Comp. Mech., 2 (1987), 307–316.MATHGoogle Scholar
  70. 70.
    Rydholm, G., Fredriksson, B. and F. Nilsson: Numerical investigation of rapid crack propagation, in: Numerical Methods in Fracture Mechanics, Pineridge Press, 1978, 660–672.Google Scholar
  71. 71.
    Malluck, J. F. and W. W. King: Fast fracture simulated by a finite element analysis which accounts for crack. tip energy dissipation, in: Numerical Methods in Fracture Mechanics, Pineridge Press, 1978, 648–659.Google Scholar
  72. 72.
    Bazant, Z. P., Glazik Jr., J. L. and J. D. Achenbach: Elastodynamic fields near running cracks by finite elements”, Comput. and Struct, 8 (1978), 193–198.CrossRefMATHGoogle Scholar
  73. 73.
    Östlund, S. and P. Gudmundson: The application of moving finite elements for the study of crack propagation in linear elastic solids, Comput. and Struct., 25 (1987), 765–774.CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Wien 1990

Authors and Affiliations

  • F. Nilsson
    • 1
  1. 1.University of UppsalaUppsalaSweden

Personalised recommendations