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Continuum shape sensitivity and reliability analyses of nonlinear cracked structures

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Abstract

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.

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References

  • H.O. Madsen S. Krenk N.C. Lind (Eds) (1986) Methods of Structured Safety Prentice-Hall Inc. Englewood Cliffs, New Jersey

    Google Scholar 

  • M. Grigoriu M.T.A. Saif S. EI-Borgi A. Ingraffea (1990) ArticleTitleMixed-mode fracture initiation and trajectory prediction under random stresses International Journal of Fracture 45 19–34

    Google Scholar 

  • J.W. Provan (Eds) (1987) Probabilistic Fracture Mechanics and Reliability Martinus Nijhoff Publishers Dordrecht, The Netherlands Occurrence Handle0692.73066

    MATH  Google Scholar 

  • G.H. Besterfield W.K Liu M.A. Lawrence T. Belytschko (1991) ArticleTitleFatigue crack growth reliability by probabilistic finite elements Computer Methods in Applied Mechanics and Engineering 86 297–320 Occurrence Handle0764.73078

    MATH  Google Scholar 

  • G.H. Besterfield M.A. Lawrence T. Belytschko (1990) ArticleTitleBrittle fracture reliability by probabilistic finite elements ASCE Journal of Engineering Mechanics 116 IssueID3 642–659

    Google Scholar 

  • S. Rahman (1995) ArticleTitleA stochastic model for elastic--plastic fracture analysis of circumferential through-wall-cracked pipes subject to bending Engineering Fracture Mechanics 52 IssueID2 265–288

    Google Scholar 

  • S. Rahman J-S. Kim (2000) ArticleTitleProbabilistic fracture mechanics for nonlinear structures International Journal of Pressure Vessels and Piping 78 IssueID4 9–17

    Google Scholar 

  • S. Rahman (2001) ArticleTitleProbabilistic fracture mechanics by Jestimation and finite element methods Engineering Fracture Mechanics 68 107–125

    Google Scholar 

  • S.C. Lin J. Abel (1988) ArticleTitleVariational approach for a new direct-integration form of the virtual crack extension method International Journal of Fracture 38 217–235

    Google Scholar 

  • H.G. deLorenzi (1982) ArticleTitleOn the energy release rate and the J-integral for 3-D crack configurations International Journal of Fracture 19 183–193

    Google Scholar 

  • H.G. deLorenzi (1985) ArticleTitleEnergy release rate calculations by the finite element method Engineering Fracture Mechanics 21 129–143

    Google Scholar 

  • R.B. Haber H.M. Koh (1985) ArticleTitleExplicit expressions for energy release rates using virtual crack extensions International Journal of Numerical Methods in Engineering 21 301–315 Occurrence Handle0551.73091

    MATH  Google Scholar 

  • E.J. Barbero J.N. Reddy (1990) ArticleTitleThe Jacobian derivative method for three-dimensional fracture mechanics Communications in Applied Numerical Methods 6 507–518 Occurrence Handle0709.73058

    MATH  Google Scholar 

  • X.Z. Suo A. Combescure (1992) ArticleTitleDouble virtual crack extension method for crack growth stability assessment International Journal of Fracture 57 127–150

    Google Scholar 

  • C.G. Hwang P.A. Wawrzynek A.K. Tayebi A.R. Ingraffea (1998) ArticleTitleOn the virtual crack extension method for calculation of the rates of energy release rate Engineering Fracture Mechanics 59 521–542

    Google Scholar 

  • D.J. Keum B.M. Kwak (1992) ArticleTitleEnergy release rates of crack kinking by boundary sensitivity analysis Engineering Fracture Mechanics 41 833–841

    Google Scholar 

  • R.A. Feijóo C. Padra R. Saliba E. Taroco M.J. Vénere (2000) ArticleTitleShape sensitivity analysis for energy release rate evaluations and its application to the study of three-dimensional cracked bodies Computational Methods in Applied Mechanics and Engineering 188 649–664 Occurrence Handle0965.74049

    MATH  Google Scholar 

  • G. Chen S. Rahman Y.H. Park (2001b) ArticleTitleShape sensitivity and reliability analyses of linear-elastic cracked structures International Journal of Fracture 112 IssueID3 223–246

    Google Scholar 

  • G. Chen S. Rahman Y.H. Park (2001a) ArticleTitleShape sensitivity analysis in mixed-mode fracture mechanics Computational Mechanics 27 IssueID4 282–291 Occurrence Handle1068.74596 Occurrence Handle2001CompM..27..282C

    MATH  ADS  Google Scholar 

  • G. Chen S. Rahman Y.H. Park (2002) ArticleTitleShape sensitivity analysis of linear-elastic cracked structures under mode-I loading ASME Journal of Pressure Vessel Technology 124 IssueID4 476–482

    Google Scholar 

  • E. Taroco (2000) ArticleTitleShape sensitivity analysis in linear elastic cracked structures Computational Methods in Applied Mechanics and Engineering 188 697–712 Occurrence Handle0964.74047 Occurrence Handle1784105

    MATH  MathSciNet  Google Scholar 

  • W.H. Fleming (Eds) (1965) Functions of Several Variables Addison-Wesley Reading, Massachusetts Occurrence Handle0136.34301

    MATH  Google Scholar 

  • J.P. Zolesio (Eds) (1979) Identification de Domains par Deformations Thése d État Université de Nice

    Google Scholar 

  • E.J. Haug K.K. Choi V. Komkov (Eds) (1986) Design Sensitivity Analysis of Structural Systems Academic Press New York, NY Occurrence Handle0618.73106

    MATH  Google Scholar 

  • R.A. Adams (Eds) (1975) Sobolev Spaces Academic Press New York, NY Occurrence Handle0314.46030

    MATH  Google Scholar 

  • ABAQUS (1999). User’s Guide and Theoretical Manual, Version 5.8, Hibbitt, Karlsson, and Sorenson, Inc., Pawtucket, RI.

  • W.F. Chen D.J. Han (Eds) (1988) Plasticity for Structural Engineers Springer-Verlag New York, NY Occurrence Handle0666.73010

    MATH  Google Scholar 

  • T.L. Anderson (Eds) (1995) Fracture Mechanics: Fundamentals and Applications EditionNumber2 CRC Press Inc. Boca Raton, Florida Occurrence Handle0999.74001

    MATH  Google Scholar 

  • G. Chen (Eds) (2001) Shape Sensitivity and Reliability Analysis of Linear and Nonlinear Cracked Structures The University of Iowa Iowa City, Iowa

    Google Scholar 

  • J.R. Rice (1968) ArticleTitleA path independent integral and the approximate analysis of strain concentration by notches and cracks Journal of Applied Mechanics 35 379–386

    Google Scholar 

  • J.W. Hutchinson (1983) ArticleTitleFundamentals of the phenomenological theory of nonlinear fracture mechanics ASME Journal of Applied Mechanics 50 1042–1051 Occurrence Handle10.1115/1.3167187 Occurrence Handle1983nyap.book.....H

    Article  ADS  Google Scholar 

  • P.L. Liu A.D. Kiureghian (1991) ArticleTitleOptimization algorithms for structural reliability Structural Safety 9 161–177

    Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical Engineering, The University of Iowa, Iowa City, IA, 52242

    Sharif Rahman & Guofeng Chen

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  1. Sharif Rahman
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  2. Guofeng Chen
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Correspondence to Sharif Rahman.

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Rahman, S., Chen, G. Continuum shape sensitivity and reliability analyses of nonlinear cracked structures. Int J Fract 131, 189–209 (2005). https://doi.org/10.1007/s10704-004-3948-6

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  • DOI: https://doi.org/10.1007/s10704-004-3948-6

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