International Journal of Fracture

, Volume 192, Issue 1, pp 25–45 | Cite as

Physically appropriate characterization of fatigue crack propagation rate in elastic–plastic materials using the \(J\)-integral concept

  • W. OchensbergerEmail author
  • O. Kolednik
Original Paper


The current paper discusses the physically correct evaluation of the driving force for fatigue crack propagation in elastic–plastic materials using the \(J\)-integral concept. This is important for low-cycle fatigue and for short fatigue cracks, where the conventional stress intensity range (\(\Delta K\)) concept cannot be applied. Using the configurational force concept, Simha et al. (J Mech Phys Solids 56:2876–2895, 2008) , have derived the \(J\)-integral for elastic–plastic materials with incremental theory of plasticity, \(J^{\mathrm{ep}}\), which is applicable for cyclic loading and/or for growing cracks, in contrast to the conventional \(J\)-integral. The variation of this incremental plasticity \(J\)-integral \(J^{\mathrm{ep}}\) is studied in numerical investigations conducted on two-dimensional C(T)-specimens with long cracks under cyclic Mode I loading. The crack propagates by an increment after each load cycle. The maximum load is varied so that small- and large-scale yielding conditions prevail. Three different load ratios are considered, from pure tension to tension-compression loading. By theoretical considerations and comparisons with the variation of the crack tip opening displacement \(\delta _{\mathrm{t}}\), it is demonstrated that the cyclic, incremental plasticity \(J\)-integral \(\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}\), which is computed for a contour around the active plastic zone of the growing crack, is physically appropriate to characterize the growth rate of fatigue cracks. The validity of the experimental cyclic \(J\)-integral, \(\Delta J^{\mathrm{exp}}\), proposed by Dowling and Begley (ASTM STP 590:82–103, 1976), is also investigated. The results show that \(\Delta J^{\mathrm{exp}}\) is correct for the first load cycle, however, not fully appropriate for a growing fatigue crack.


Configurational force concept Crack driving force  Cyclic \(J\)-integral Low-cycle fatigue Fatigue crack growth Overload effect 



Financial support by the Austrian Federal Government and the Styrian Provincial Government within the research activities of the K2 Competence Center on “Integrated Research in Materials, Processing and Product Engineering”, under the frame of the Austrian COMET Competence Center Programme, is gratefully acknowledged (strategic Project A4.20-WP3).


  1. Anderson TL (1995) Fracture mechanics. CRC Press, Boca Raton, FLGoogle Scholar
  2. ASTM E1820-05 (2005) Standard test method for measurement of fracture toughness. In: Annual book of ASTM standards, vol 03.01. ASTM International, West Conshohocken, PA, USAGoogle Scholar
  3. Atkins AG, Mai YW (1986) Residual strain energy in elastoplastic adhesive and cohesive fracture. Int J Fract 30:203–221CrossRefGoogle Scholar
  4. Banks-Sills L, Volpert Y (1991) Application of the cyclic \(J\)-integral to fatigue crack propagation of Al 2024–T351. Eng Fract Mech 40:355–370CrossRefGoogle Scholar
  5. Bichler C, Pippan R (1999) Direct observation of the residual plastic deformation caused by a single overload. In: McClung RC, Newman JC Jr (eds) Advances in fatigue crack closure measurement and analysis, vol. 2, ASTM STP, West Conshohocken, PA. 1342:191–206Google Scholar
  6. Bichler C, Pippan R (2007) Effect of single overloads in ductile metals: a reconsideration. Eng Fract Mech 74:1344–1359CrossRefGoogle Scholar
  7. Brocks W, Cornec A, Scheider I (2003) Computational aspects of nonlinear fracture mechanics. In: de Borst R, Mang HA (eds) Comprehensive structural integrity, numerical and computational methods, vol 3. Elsevier, New York, pp 127–209CrossRefGoogle Scholar
  8. Christensen RH (1959) Metal fatigue. McGraw-Hill, New YorkGoogle Scholar
  9. Denzer R, Barth FJ, Steinmann P (2003) Studies in elastic fracture mechanics based on the material force method. Int J Numer Methods Eng 58:1817–1835CrossRefGoogle Scholar
  10. Dougherty JD, Padovan J, Srivatsan TS (1997) Fatigue crack propagation and closure behavior of modified 1071 steel: finite element study. Eng Fract Mech 56(2):189–212CrossRefGoogle Scholar
  11. Dowling NE, Begley JA (1976) Fatigue crack growth during gross plasticity and the \(J\)-integral. ASTM STP 590:82–103Google Scholar
  12. Dowling NE (1976) Geometry effects and the \(J\)-integral approach to elastic–plastic fatigue crack growth. ASTM STP 601:19–32Google Scholar
  13. Elber W (1970) Fatigue crack closure under cyclic tension. Eng Fract Mech 2:37–45CrossRefGoogle Scholar
  14. Elber W (1971) The significance of fatigue crack closure. ASTM STP 486:230–242Google Scholar
  15. Eshelby JD (1951) The force on an elastic singularity. Philos Trans R Soc A 244:87–112CrossRefGoogle Scholar
  16. Eshelby JD (1970) Energy relations and the energy-momentum tensor in continuum mechanics. In: Kanninen M, Adler W, Rosenfield A, Jaffee R (eds) Inelastic behavior of solids. McGraw-Hill, New York, pp 77–115Google Scholar
  17. ESIS P2-92 (1992) ESIS procedure for determining the fracture behavior of materials. European Structural Integrity Society, DelftGoogle Scholar
  18. Fischer FD, Simha NK, Predan J, Schöngrundner R, Kolednik O (2012) On configurational forces at boundaries in fracture mechanics. Int J Fract 174:61–74CrossRefGoogle Scholar
  19. Fleck NA (1988) Influence of stress state on crack growth retardation. In: Fong JT, Fields RJ (eds) Basic questions in fatigue. ASTM STP 924, vol 1. ASTM:157–83Google Scholar
  20. Forsyth PJE, Ryder DA (1960) Fatigue fracture. Aircr Eng 32:96–99CrossRefGoogle Scholar
  21. Gurtin ME (1995) The nature of configurational forces. Arch Ration Mech Anal 131:67–100CrossRefGoogle Scholar
  22. Gurtin ME (2000) Configurational forces as basic concepts of continuum physics. Springer, New YorkGoogle Scholar
  23. Heerens J, Cornec A, Schwalbe K-H (1988) Results of a round robin on stretch zone width determination. Fatigue Fract Eng Mater Struct 11:19–29CrossRefGoogle Scholar
  24. Kienzler R, Herrmann G (2000) Mechanics in material space. Springer, BerlinCrossRefGoogle Scholar
  25. Kolednik O (1981) A contribution to stereophotogrammetry with the scanning electron microscope. Pract Metall 18:562–573Google Scholar
  26. Kolednik O (1991) On the physical meaning of the \(J\!-\!\Delta a\)-curves. Eng Fract Mech 38:403–412Google Scholar
  27. Kolednik O (1993) A simple model to explain the geometry dependence of the \(J-a\)-curves. Int J Fract 63:263–274CrossRefGoogle Scholar
  28. Kolednik O, Schöngrundner R, Fischer FD (2014) A new view on \(J\)-integrals in elastic–plastic materials. Int J Fract 187:77–107CrossRefGoogle Scholar
  29. Kolednik O, Stüwe HP (1985) The stereophotogrammetric determination of the critical crack tip opening displacement. Eng Fract Mech 21:145–155CrossRefGoogle Scholar
  30. Kolednik O, Stüwe HP (1987) A proposal for estimating the slope of the blunting line. Int J Fract 33:R63–R66CrossRefGoogle Scholar
  31. Kolednik O, Kutlesa P (1989) On the influence of specimen geometry on the critical crack-tip-opening displacement. Eng Fract Mech 33:215–223CrossRefGoogle Scholar
  32. Krupp U, Floer W, Lei J, Hu Y, Christ HJ, Schick A, Fritzen CP (2002) Mechanisms of short crack initiation and propagation in a beta-titanium alloy. Phil Mag A82:3321–3332Google Scholar
  33. Kuna M (2008) Numerische Beanspruchungsanalyse von Rissen. FEM in der Bruchmechanik. Vieweg-Teubner, WiesbadenGoogle Scholar
  34. Laird C (1967) The influence of metallurgical structure on the mechanisms of fatigue crack propagation. ASTM STP 415:131–168Google Scholar
  35. Laird C (1979) Mechanisms and theories of fatigue. ASM, Metals Park, OH, pp 149–203Google Scholar
  36. Lamba HS (1975) The \(J\)-integral applied to cyclic loading. Eng Fract Mech 7:693–703CrossRefGoogle Scholar
  37. Lambert Y, Saillard P, Bathias C (1988) Application of the \(J\) concept to fatigue crack growth in large-scale yielding. ASTM STP 969:318–329Google Scholar
  38. Maugin GA (1995) Material forces: concepts and applications. ASME J Appl Mech Rev 48:213–245CrossRefGoogle Scholar
  39. Metzger M, Seifert T, Schweizer C (2014) Does the cyclic \(J\)-integral describe the crack-tip opening displacement in the presence of crack closure? Eng Fract Mech. doi: 10.1016/j.engfracmech.2014.07.017
  40. Mueller R, Kolling S, Gross D (2002) On configurational forces in the context of the finite element method. Int J Numer Methods Eng 53:1557–1574CrossRefGoogle Scholar
  41. Mueller R, Gross D, Maugin GA (2004) Use of material forces in adaptive finite element methods. Comput Mech 33:421–434Google Scholar
  42. Newman JC (1976) A finite element analysis of fatigue crack closure. ASTM STP 590:281–301Google Scholar
  43. Nguyen TD, Govindjee S, Klein PA, Gao H (2005) A material force method for inelastic fracture mechanics. J Mech Phys Solids 53:91–121CrossRefGoogle Scholar
  44. Ochensberger W, Kolednik O (2014) A new basis for the application of the J-integral for cyclically loaded cracks in elastic-plastic materials. Int J Fract 189:77–101CrossRefGoogle Scholar
  45. Özenç K, Kaliske M, Lin G, Bhashyam G (2014) Evaluation of energy contributions in elasto-plastic fracture: a review of the configurational force approach. Eng Fract Mech 115:137–153CrossRefGoogle Scholar
  46. Ohji K, Ogura K, Yoshiji O (1975) Cyclic analysis of a propagating crack and its correlation with fatigue crack growth. Eng Fract Mech 7:457–464CrossRefGoogle Scholar
  47. Paris PC, Gomez MP, Anderson WP (1961) A rational analytic theory of fatigue. Trend Eng 13:9–14Google Scholar
  48. Paris PC, Erdogan F (1963) A critical analysis of crack propagation laws. J Basic Eng 85:528–534CrossRefGoogle Scholar
  49. Parks DM (1977) The virtual crack extension method for nonlinear material behavior. Comput Methods Appl Mech Eng 12:353–364CrossRefGoogle Scholar
  50. Pippan R, Bichler C, Tabernig B, Weinhandl H (2005) Overloads in ductile and brittle materials. Fatigue Fract Eng Mater Struct 28:971–981CrossRefGoogle Scholar
  51. Pippan R, Zelger C, Gach E, Bichler C, Weinhandl H (2010) On the mechanism of fatigue crack propagation in ductile metallic materials. Fatigue Fract Eng Mater Struct 34:1–16CrossRefGoogle Scholar
  52. Rice JR (1968a) A path independent integral and the approximate analysis of strain concentration by notches and cracks. ASME J Appl Mech 35:379–386CrossRefGoogle Scholar
  53. Rice JR (1968b) Mathematical analysis in the mechanics of fracture. In: Liebowitz H (ed) Fracture—an advanced treatise, vol 2. Academic Press, New York, pp 191–311Google Scholar
  54. Rice JR (1979) The mechanics of quasi-static crack growth. In: Kelly RE (ed) Proceedings of the eighth U.S. National congress of applied mechanics. ASME, New York, pp 191–216Google Scholar
  55. Rice JR, Johnson MA (1970) The role of large crack tip geometry changes in plane strain fracture. In: Kanninen MF (ed) Inelastic behavior of solids. McGraw-Hill, New York, pp 641–672Google Scholar
  56. Rice JR, Paris PC, Merkle JG (1973) Some further results of \(J\)-integral analysis and estimates. ASTM STP 536:231–245Google Scholar
  57. Sadananda K, Vasudevan AK, Holtz RL, Lee EU (1999) Analysis of overload effects and related phenomena. Int J Fatigue 21:S233–S246CrossRefGoogle Scholar
  58. Schijve J (1960) Fatigue crack propagation in light alloy sheet material and structures. NRL report 195 MP National Aeronautical Research Institute, Amsterdam, HollandGoogle Scholar
  59. Siegmund T, Kolednik O, Pippan R (1990) Direkte Messung der Rissspitzenverformung bei wechselnder Belastung. Z Metallkde 81 H.9:677–683Google Scholar
  60. Simha NK, Fischer FD, Kolednik O, Chen CR (2003) Inhomogeneity effects on the crack driving force in elastic and elastic–plastic materials. J Mech Phys Solids 51:209–240CrossRefGoogle Scholar
  61. Simha NK, Fischer FD, Kolednik O, Predan J, Shan GX (2005) Crack tip shielding due to smooth and discontinuous material inhomogeneities. Int J Fract 135:73–93CrossRefGoogle Scholar
  62. Simha NK, Fischer FD, Shan GX, Chen CR, Kolednik O (2008) \(J\)-integral and crack driving force in elastic? Plastic materials. J Mech Phys Solids 56:2876–2895CrossRefGoogle Scholar
  63. Sistaninia M, Kolednik O (2014) Effect of a single soft interlayer on the crack driving force. Eng Fract Mech 130:21–41CrossRefGoogle Scholar
  64. Skorupa M (1998) Load interaction effects during fatigue crack growth under variable amplitude loading. A literature review. Part I: empirical trends. Fatigue Fract Eng Mater Struct 21:987–1006CrossRefGoogle Scholar
  65. Solanki K, Daniewicz SR, Newman JC Jr (2003) Finite element modeling of plasticity-induced crack closure with emphasis on geometry and mesh refinement effects. Eng Fract Mech 70:1475–1489CrossRefGoogle Scholar
  66. Solanki K, Daniewicz SR, Newman JC Jr (2004) Finite element analysis of plasticity-induced fatigue crack closure: an overview. Eng Fract Mech 71:149–171CrossRefGoogle Scholar
  67. Suresh S (1983) Micromechanisms of fatigue crack growth retardation following overloads. Eng Fract Mech 18:577–593CrossRefGoogle Scholar
  68. Suresh S (1998) Fatigue of materials, 2nd edn. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  69. Tanaka K (1983) The cyclic \(J\)-integral as a criterion for fatigue crack growth. Int J Fract 22:91–104CrossRefGoogle Scholar
  70. Tanaka K (1989) Mechanics and micromechanics of fatigue crack propagation. ASTM STP 1020:151–183Google Scholar
  71. Turner CE, Kolednik O (1994) Application of energy dissipation rate arguments to stable crack growth. Fatigue Fract Eng Mater Struct 17:1109–1127CrossRefGoogle Scholar
  72. Tvergaard V (2005) Overload effects in fatigue crack growth by crack-tip blunting. Int J Fatigue 27:1389–1397CrossRefGoogle Scholar
  73. von Euw EFJ, Hertzberg RW, Roberts R (1972) Delay effects in fatigue crack propagation. In: Stress analysis and growth of cracks, ASTM STP 513, part I. Philadelphia ASTM:230–259Google Scholar
  74. Wüthrich C (1982) The extension of the \(J\)-integral concept to fatigue cracks. Int J Fract 20:R35–R37CrossRefGoogle Scholar
  75. Zappfe CA, Worden CO (1951) Fractographic registrations of fatigue. Trans Am Soc Met 43:958–969Google Scholar

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© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Erich Schmid Institute of Materials ScienceAustrian Academy of SciencesLeobenAustria
  2. 2.Materials Center Leoben Forschung GmbHLeobenAustria

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