International Journal of Fracture

, Volume 131, Issue 2, pp 189–209 | Cite as

Continuum shape sensitivity and reliability analyses of nonlinear cracked structures

  • Sharif Rahman
  • Guofeng Chen


A new method is proposed for shape sensitivity analysis of a crack in a homogeneous, isotropic, and nonlinearly elastic body subject to mode I loading conditions. The method involves the material derivative concept of continuum mechanics, domain integral representation of the J-integral, and direct differentiation. Unlike virtual crack extension techniques, no mesh perturbation is required in the proposed method. Since the governing variational equation is differentiated before the process of discretization, the resulting sensitivity equations are independent of any approximate numerical techniques. Based on the continuum sensitivities, the first-order reliability method was employed to perform probabilistic analysis. Numerical examples are presented to illustrate both the sensitivity and reliability analyses. The maximum difference between the sensitivity of stress-intensity factors calculated using the proposed method and the finite-difference method is less than four percent. Since all gradients are calculated analytically, the reliability analysis of cracks can be performed efficiently.


Crack energy release rate J-integral nonlinear fracture mechanics probabilistic fracture mechanics reliability shape sensitivity analysis 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Madsen, H.O.Krenk, S.Lind, N.C. eds. 1986Methods of Structured SafetyPrentice-Hall Inc.Englewood Cliffs, New JerseyGoogle Scholar
  2. Grigoriu, M., Saif, M.T.A., EI-Borgi, S., Ingraffea, A. 1990Mixed-mode fracture initiation and trajectory prediction under random stressesInternational Journal of Fracture451934Google Scholar
  3. Provan, J.W. eds. 1987Probabilistic Fracture Mechanics and ReliabilityMartinus Nijhoff PublishersDordrecht, The NetherlandsMATHGoogle Scholar
  4. Besterfield, G.H., Liu, W.K, Lawrence, M.A., Belytschko, T. 1991Fatigue crack growth reliability by probabilistic finite elementsComputer Methods in Applied Mechanics and Engineering86297320MATHGoogle Scholar
  5. Besterfield, G.H., Lawrence, M.A., Belytschko, T. 1990Brittle fracture reliability by probabilistic finite elements ASCE Journal of Engineering Mechanics116642659Google Scholar
  6. Rahman, S. 1995A stochastic model for elastic--plastic fracture analysis of circumferential through-wall-cracked pipes subject to bendingEngineering Fracture Mechanics52265288Google Scholar
  7. Rahman, S., Kim, J-S. 2000Probabilistic fracture mechanics for nonlinear structuresInternational Journal of Pressure Vessels and Piping78917Google Scholar
  8. Rahman, S. 2001Probabilistic fracture mechanics by Jestimation and finite element methodsEngineering Fracture Mechanics68107125Google Scholar
  9. Lin, S.C., Abel, J. 1988Variational approach for a new direct-integration form of the virtual crack extension methodInternational Journal of Fracture38217235Google Scholar
  10. deLorenzi, H.G. 1982On the energy release rate and the J-integral for 3-D crack configurationsInternational Journal of Fracture19183193Google Scholar
  11. deLorenzi, H.G. 1985Energy release rate calculations by the finite element methodEngineering Fracture Mechanics21129143Google Scholar
  12. Haber, R.B., Koh, H.M. 1985Explicit expressions for energy release rates using virtual crack extensionsInternational Journal of Numerical Methods in Engineering21301315MATHGoogle Scholar
  13. Barbero, E.J., Reddy, J.N. 1990The Jacobian derivative method for three-dimensional fracture mechanicsCommunications in Applied Numerical Methods6507518MATHGoogle Scholar
  14. Suo, X.Z., Combescure, A. 1992Double virtual crack extension method for crack growth stability assessmentInternational Journal of Fracture57127150Google Scholar
  15. Hwang, C.G., Wawrzynek, P.A., Tayebi, A.K., Ingraffea, A.R. 1998On the virtual crack extension method for calculation of the rates of energy release rateEngineering Fracture Mechanics59521542Google Scholar
  16. Keum, D.J., Kwak, B.M. 1992Energy release rates of crack kinking by boundary sensitivity analysisEngineering Fracture Mechanics41833841Google Scholar
  17. Feijóo, R.A., Padra, C., Saliba, R., Taroco, E., Vénere, M.J. 2000Shape sensitivity analysis for energy release rate evaluations and its application to the study of three-dimensional cracked bodiesComputational Methods in Applied Mechanics and Engineering188649664MATHGoogle Scholar
  18. Chen, G., Rahman, S., Park, Y.H. 2001bShape sensitivity and reliability analyses of linear-elastic cracked structuresInternational Journal of Fracture112223246Google Scholar
  19. Chen, G., Rahman, S., Park, Y.H. 2001aShape sensitivity analysis in mixed-mode fracture mechanicsComputational Mechanics27282291MATHADSGoogle Scholar
  20. Chen, G., Rahman, S., Park, Y.H. 2002Shape sensitivity analysis of linear-elastic cracked structures under mode-I loadingASME Journal of Pressure Vessel Technology124476482Google Scholar
  21. Taroco, E. 2000Shape sensitivity analysis in linear elastic cracked structuresComputational Methods in Applied Mechanics and Engineering188697712MATHMathSciNetGoogle Scholar
  22. Fleming, W.H. eds. 1965Functions of Several VariablesAddison-WesleyReading, MassachusettsMATHGoogle Scholar
  23. Zolesio, J.P. eds. 1979Identification de Domains par DeformationsThése d ÉtatUniversité de NiceGoogle Scholar
  24. Haug, E.J.Choi, K.K.Komkov, V. eds. 1986Design Sensitivity Analysis of Structural SystemsAcademic PressNew York, NYMATHGoogle Scholar
  25. Adams, R.A. eds. 1975Sobolev SpacesAcademic PressNew York, NYMATHGoogle Scholar
  26. ABAQUS (1999). User’s Guide and Theoretical Manual, Version 5.8, Hibbitt, Karlsson, and Sorenson, Inc., Pawtucket, RI. Google Scholar
  27. Chen, W.F.Han,  D.J. eds. 1988Plasticity for Structural EngineersSpringer-VerlagNew York, NYMATHGoogle Scholar
  28. Anderson, T.L. eds. 1995Fracture Mechanics: Fundamentals and Applications2CRC Press Inc.Boca Raton, FloridaMATHGoogle Scholar
  29. Chen, G. eds. 2001Shape Sensitivity and Reliability Analysis of Linear and Nonlinear Cracked StructuresThe University of IowaIowa City, IowaDoctoral dissertationGoogle Scholar
  30. Rice, J.R. 1968A path independent integral and the approximate analysis of strain concentration by notches and cracksJournal of Applied Mechanics35379386Google Scholar
  31. Hutchinson, J.W. 1983Fundamentals of the phenomenological theory of nonlinear fracture mechanicsASME Journal of Applied Mechanics5010421051CrossRefADSGoogle Scholar
  32. Liu, P.L., Kiureghian, A.D. 1991Optimization algorithms for structural reliabilityStructural Safety9161177Google Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringThe University of IowaIowa City

Personalised recommendations