On the Computation of a Generalised Dynamic J-Integral and its Application to the Durability of Steel Structures



A theoretical description and a computational method are presented to calculate the J-integral in the context of the finite element method. In the derivation, we use the theory of configurational forces where the fully three-dimensional case and large deformations for non-linear elastic materials under dynamic loading are taken into account. Analogue to the local balance of momentum, the so-called Eshelby stress holds a configurational force balance, where configurational forces correspond to the volume forces in the physical space. A discretised finite element description is obtained by the weak form of the configurational force balance. Thus, the configurational forces acting on the finite element nodes may be computed as the physical boundary value problem is solved. For the static case and small deformations, the configurational force corresponds to the well known J-integral in fracture mechanics, though not restricted to the crack-mode I state. As a practical example, we show how the J-integral, combined with Paris’ equation, can be used to predict the ultimate life time of a steel structure containing components with cracks.


Fatigue Crack Stress Intensity Factor Crack Length Crack Growth Rate 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. 1.
    Rice J.R. (1968) A path independent integral and the approximate analysis of strain concen-tration by notches and cracks. J. Appl. Mech., 35: 379-386.Google Scholar
  2. 2.
    Paris P.C., Erdogan F. (1960) A Critical Analysis of Crack Propagation Laws, Journal of Basic Engineering, 85: 528-534.Google Scholar
  3. 3.
    Braun M. (1997) Configurational forces induced by finite-element discretization. Proc. Es-tonian Acad. Sci. Phys. Math., 46(1/2): 24-31.MATHMathSciNetGoogle Scholar
  4. 4.
    Steinmann P. (2000) Application of material forces to hyperelastostatic fracture mechanics I: Continuum mechanical setting. International Journal of Solids and Structures, 37: 7371-7391.MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Steinmann P., Ackermann D., Barth F.J. (2001) Application of material forces to hyperelasto-static fracture mechanics II: computational setting. International Journal of Solids and Struc-tures, 38: 5509-5526.MATHCrossRefGoogle Scholar
  6. 6.
    Mueller R., Kolling S., Gross D. (2002) On configurational forces in the context of the fi-nite element method. International Journal of Numerical Methods in Engineering, 53: 1557-1574.MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Mueller R., Maugin G.A. (2002) On material forces and finite element discretizations. Computational Mechanics 29: 52-60.MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Eshelby J.D. (1951) The force on an elastic singularity. Phil. Trans. Roy. Soc. A (244): 87-112.Google Scholar
  9. 9.
    Eshelby J.D. (1970) Energy relations and the energy-momentum tensor in continuum mechanics. In Kanninen M.F. (editor) Inelastic behaviour of solids. McGraw Hill. New York.Google Scholar
  10. 10.
    Maugin G.A. (1993) Material Inhomogeneities in Elasticity. Chapman & Hall.Google Scholar
  11. 11.
    Gurtin M.E. (2000) Configurational forces as a basic concept of continuum physics. Springer Verlag.Google Scholar
  12. 12.
    Kienzler R., Herrmann G. (2000) Mechanics in material space. Springer Verlag.Google Scholar
  13. 13.
    Maugin G.A. (2000) Geometry of material space: its consequences in modern computa-tional means. Technische Mechanik, 20(2): 95-104.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Chair in Steel StructuresBundeswehr University Munich85577 NeubibergGermany
  2. 2.Laboratory of MechanicsGiessen University of Applied Sciences35390 GiessenGermany

Personalised recommendations