Computational Mechanics

, Volume 3, Issue 5, pp 331–342 | Cite as

Computational studies on path independent integrals for non-linear dynamic crack problems

  • T. Nishioka
  • M. Kobashi
  • S. N. Atluri


Recently the authors have derived various new types of path independent integrals in which the theoretical limitations of the so-calledJ integral are overcome. First, for elastodynamic crack problems, a path independent integralJ′ which has the physical meaning of energy release rate was derived. Later, more general forms of path independent integralsT* andT were derived, which are valid for any constitutive relation under quasi-static as well as dynamic conditions.

This paper presents the theoretical and computational aspects of these integrals, of relevance in non-linear dynamic fracture mechanics. An efficient solution technique is also presented for non-linear dynamic finite element method in which a factorization of the assembled stiffness matrix is done only once throughout the computation for a given mesh pattern. Finite element analyses were carried out for an example problem of a center-cracked plate subject to a uniaxial impact loading. The material behavior was modeled by three different constitutive relations such as linear-elastic, elastic-plastic, elastic-viscoplastic cases. The applicability of theT* integral to non-linear dynamic fracture mechanics was shown with the numerical results.


Finite Element Method Finite Element Analysis Stiffness Matrix Constitutive Relation Energy Release Rate 
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. Atluri, S. N. (1982): Path indpendent integrals in finite elasticity and inelasticity with body forces, inertia and arbitrary crack-face conditions. Eng. Fract. Mech. 16, 341–364Google Scholar
  2. Atluri, S. N.; Nishioka, T.; Nakagaki, M. (1984): Incremental path independent integrals in inelastic and dynamic fracture mechanics. Eng. Fract. Mech. 20, 209–244Google Scholar
  3. Atluri, S. N.; Nakagaki, M.; Nishioka, T.; Kuang, Z. B. (1986): Crack-tip parameters and temperature rise in dynamic crack propagation. Eng. Fract. Mech. 23, 167–182Google Scholar
  4. Baker, R. B. (1962): Dynamic stresses created by a moving crack. J. Appl. Mech. 29, 449–545Google Scholar
  5. Bodner, S. R.; Symonds, P. S. (1962): Experimental and theoretical investigation of the plastic deformation of cantilever beams subjected to impulse loading. J. Appl. Mech. 29, 719–728Google Scholar
  6. Brust, F. W.; Nishioka, T.; Atluri, S. N.; Nakagaki, M. (1985): Further studies on elastic-plastic stable fracture utilizing theT * integral. Eng. Fract. Mech. 20, 1079–1103Google Scholar
  7. Budiansky, B.; Rice, J. R. (1973): Conservation laws and energy release rates. J. Appl. Mech. 40, 201–203Google Scholar
  8. Kishiomoto, K.; Aoki, S.; Sakata, M. (1980): On the path independent integral J. Eng. Fract. Mech. 13, 841–850Google Scholar
  9. Mondkar, D. P.; Powell, G. H. (1974): Large capacity equation solver for structural analysis. Compt. Structures 4, 699–728Google Scholar
  10. Nishioka, T.; Atluri, S. N. (1980a): Numerical modeling of dynamic crack propagation in finite bodies, by moving singular elements, Part 1: formulation. J. Appl. Mech. 47, 570–576Google Scholar
  11. Nishioka, T.; Atluri, S. N. (1980b): Numerical modeling of dynamic crack propagation in finite bodies, by moving singular elements, Part II: results. J. Appl. Mech. 47, 577–582Google Scholar
  12. Nishioka, T.; Atluri, S. N. (1983a): Path-independent integrals, energy release rates, and general solutions of near-tip fields in mixed-mode dynamic fracture mechanics. Eng. Fract. Mech. 18, 1–22Google Scholar
  13. Nishioka, T.; Atluri, S. N. (1983b): A numerical study of the use of path-independent integrals in elasto-dynamic crack propagation. Eng. Fract. Mech. 18, 23–33Google Scholar
  14. Nishioka, T.; Atluri, S.N. (1984a): On the computation of mixed-mode K-factors for a dynamically propagating crack, using path-independent integralJ′. Eng. Fract. Mech. 20, 193–203Google Scholar
  15. Nishioka, T.; Atluri, S. N. (1984b): A path-independent integral and moving isoparametric elements for dynamic crack propagation. AIAA J. 22, 409–414Google Scholar
  16. Nishioka, T.; Fujihara, H.; Yagami, H. (1986): Finite element analyses of stress intensity factors in dynamic crack propagation using path independentJ′ integral. In: Sih, G. C.; Nishitani, H.; Ishihara, T. (eds.): Role of Fracture Mechanics in Modern Technology, pp. 561–573. North-HollandGoogle Scholar
  17. Perzyna, P. (1963): The constitutive equations for rate sensitive plastic materials. Quarterly of Appl. Mech. 20, 321.Google Scholar
  18. Rice, J. R. (1968): A path-independent integral and the approximate analysis of strain concentration by notches and cracks. J. Appl. Mech. 35, 379–386Google Scholar
  19. Sih, G. C.; Embley, G. T.; Ravera, R. S. (1972): Impact response of a finite crack in plane extension. Int. J. Solids & Struct. 8, 977–993Google Scholar
  20. Thau, S.A.; Lu, T.H. (1971): Transient stress intensity factors for finite crack in an elastic solid caused by a dilatational wave. Int. J. Solids & Struct. 7, 731–750Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • T. Nishioka
    • 1
  • M. Kobashi
    • 1
  • S. N. Atluri
    • 2
  1. 1.Department of Ocean Mechanical EngineeringKobe University of Mercantile MarineKobeJapan
  2. 2.Center for the Advancement of Computational MechanicsGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations