International Journal of Fracture

, Volume 189, Issue 1, pp 77–101 | Cite as

A new basis for the application of the \(J\)-integral for cyclically loaded cracks in elastic–plastic materials

  • W. OchensbergerEmail author
  • O. Kolednik
Original Paper


Fatigue crack propagation is by far the most important failure mechanism. Often cracks under low-cycle fatigue conditions and, especially, short fatigue cracks cannot be treated with the conventional stress intensity range \(\Delta K\)-concept, since linear elastic fracture mechanics is not valid. For such cases, Dowling and Begley (ASTM STP 590:82–103, 1976) proposed to use the experimental cyclic \(J\)-integral \(\Delta J^{\exp }\) for the assessment of the fatigue crack growth rate. However, severe doubts exist concerning the application of \(\Delta J^{\exp }\). The reason is that, like the conventional \(J\)-integral, \(\Delta J^{\exp }\) presumes deformation theory of plasticity and, therefore, problems appear due to the strongly non-proportional loading conditions during cyclic loading. The theory of configurational forces enables the derivation of the \(J\)-integral independent of the constitutive relations of the material. The \(J\)-integral for incremental theory of plasticity, \(J^{\mathrm{ep}}\), has the physical meaning of a true driving force term and is potentially applicable for the description of cyclically loaded cracks, however, it is path dependent. The current paper aims to investigate the application of \(J^{\mathrm{ep}}\) for the assessment of the crack driving force in cyclically loaded elastic–plastic materials. The properties of \(J^{\mathrm{ep}}\) are worked out for a stationary crack in a compact tension specimen under cyclic Mode I loading and large-scale yielding conditions. Different load ratios, between pure tension- and tension–compression loading, are considered. The results provide a new basis for the application of the \(J\)-integral concept for cyclic loading conditions in cases where linear elastic fracture mechanics is not applicable. It is shown that the application of the experimental cyclic \(J\)-integral \(\Delta J^{\exp }\) is physically appropriate, if certain conditions are observed.


Configurational force concept Crack driving force  Cyclic \(J\)-integral Incremental theory of plasticity Low-cycle fatigue 



The authors would like to express their gratitude to Prof. R. Pippan and Prof. F.D. Fischer for helpful discussions. 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 projects A4.11-WP4 and 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– 221Google 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 40:355–370 Google Scholar
  5. 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
  6. 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
  7. Döring R, Hoffmeyer J, Seeger T, Vormwald M (2006) Short fatigue crack growth under nonproportional multiaxial elastic–plastic strains. Int J Fatigue 28:972–982CrossRefGoogle Scholar
  8. Dowling NE, Begley JA (1976) Fatigue crack growth during gross plasticity and the \(J\)-integral. ASTM STP 590:82–103Google Scholar
  9. Dowling NE (1976) Geometry effects and the \(J\)-integral approach to elastic–plastic fatigue crack growth. ASTM STP 601:19–32Google Scholar
  10. Dowling NE (1977) Crack growth during low-cycle fatigue of smooth axial specimens. ASTM STP 637:97–121Google Scholar
  11. Eftis J, Liebowitz H (1975) On fracture toughness evaluation for semi-brittle fracture. Eng Fract Mech 7:101–135CrossRefGoogle Scholar
  12. Elber W (1970) Fatigue crack closure under cyclic tension. Eng Fract Mech 2:37–45CrossRefGoogle Scholar
  13. Elber W (1971) The significance of fatigue crack closure. ASTM STP 486:230–242Google Scholar
  14. Eshelby JD (1951) The force on an elastic singularity. Philos Trans R Soc A 244:87–112CrossRefGoogle Scholar
  15. 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
  16. ESIS P2–92 (1992) ESIS procedure for determining the fracture behavior of materials. European Structural Integrity Society, Delft, The NetherlandsGoogle Scholar
  17. Griffith AA (1920) The phenomena of rupture and flow in solids. Philos Trans R Soc A 221:163–198CrossRefGoogle Scholar
  18. Gurtin ME (1995) The nature of configurational forces. Arch Ration Mech Anal 131:67–100CrossRefGoogle Scholar
  19. Gurtin ME (2000) Configurational forces as basic concepts of continuum physics. Springer, New YorkGoogle Scholar
  20. Hutchinson JW (1968) Singular behavior at the end of a tensile crack tip in a hardening material. J Mech Phys Solids 16:13–31CrossRefGoogle Scholar
  21. Hutchinson JW, Paris PC (1979) Stability analysis of \(J\)-controlled crack growth. ASTM STP 668:37–64Google Scholar
  22. Kienzler R, Herrmann G (2000) Mechanics in material space. Springer, BerlinCrossRefGoogle Scholar
  23. Kolednik O (1991) On the physical meaning of the \(J-\Delta a\)-curves. Eng Fract Mech 38:403–412CrossRefGoogle Scholar
  24. Kolednik O (1993) A simple model to explain the geometry dependence of the \(J\)\(a\)-curves. Int J Fract 63:263–274CrossRefGoogle Scholar
  25. Kolednik O, Stüwe HP (1985) The stereophotogrammetric determination of the critical crack tip opening displacement. Eng Fract Mech 21:145–155CrossRefGoogle Scholar
  26. Kolednik O, Schöngrundner R, Fischer FD (2014) A new view on \(J\)-integrals in elastic–plastic materials. Int J Fract 187:77–107Google Scholar
  27. Laird C (1967) The influence of metallurgical structure on the mechanisms of fatigue crack propagation. ASTM STP 415:131–168Google Scholar
  28. Laird C (1979) Mechanisms and theories of fatigue. ASM, Metals Park, OH, pp 149–203Google Scholar
  29. Lamba HS (1975) The \(J\)-integral applied to cyclic loading. Eng Fract Mech 7:693–703CrossRefGoogle Scholar
  30. 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
  31. Maugin GA (1995) Material forces: concepts and applications. ASME J Appl Mech Rev 48:213–245CrossRefGoogle Scholar
  32. McClung RC, Chell GG, Russell DA, Orient GE (1997) A practical methodology for elastic–plastic fatigue crack growth. ASTM STP 1296:317–337Google Scholar
  33. McMeeking RM (1977) Path dependence of the \(J\)-integral and the role of \(J\) as a parameter characterizing the near tip field. ASTM STP 631:28–41Google Scholar
  34. McMeeking RM, Parks DM (1979) On criteria for \(J\)-dominance of crack-tip fields in large-scale yielding. ASTM STP 668:175–194Google Scholar
  35. 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
  36. Mueller R, Gross D, Maugin GA (2004) Use of material forces in adaptive finite element methods. Comput Mech 33:421–434CrossRefGoogle Scholar
  37. Newman JC (1976) A finite element analysis of fatigue crack closure. ASTM STP 590:281–301Google Scholar
  38. Ochensberger W, Kolednik O (2014) Physically appropriate characterization of fatigue crack propagation rate in elastic–plastic materials using the \(J\)-integral concept. Int J Fract (submitted)Google Scholar
  39. Paris PC, Gomez MP, Anderson WP (1961) A rational analytic theory of fatigue. The trend in Eng 13:9–14Google Scholar
  40. Paris PC, Erdogan F (1963) A critical analysis of crack propagation laws. J Basic Eng 85:528–534CrossRefGoogle Scholar
  41. Parks DM (1977) The virtual crack extension method for nonlinear material behavior. Comput Methods Appl Mech Eng 12:353–364CrossRefGoogle Scholar
  42. Pippan R, Grosinger W (2013) Fatigue crack closure: from LCF to small scale yielding. Int J Fatigue 46:41–48CrossRefGoogle Scholar
  43. 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
  44. Rice JR (1967) Mechanics of crack tip deformation and extension by fatigue. ASTM STP 415:247–309Google Scholar
  45. 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
  46. 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
  47. 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
  48. 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
  49. Rice JR, Paris PC, Merkle JG (1973) Some further results of \(J\)-integral analysis and estimates. ASTM STP 536:231–245Google Scholar
  50. Rice JR, Rosengren GF (1968) Plane strain deformation near a crack tip in a power-law hardening material. J Mech Phys Solids 16:1–12CrossRefGoogle Scholar
  51. Schweizer C, Seifert T, Riedel H (2010) Simulation of fatigue crack growth under large-scale yielding conditions. J Phys Conf Ser 240(1):012043Google Scholar
  52. Siegmund T, Kolednik O, Pippan R (1990) Direkte Messung der Rissspitzenverformung bei wechselnder Belastung. Z Metallkde, Bd. 81 H.9, 677–683Google Scholar
  53. 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
  54. 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 56:2876–2895 Google Scholar
  55. Suresh S (1998) Fatigue of materials, 2nd edn. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  56. Tanaka K (1983) The cyclic \(J\)-integral as a criterion for fatigue crack growth. Int J Fract 22:91–104CrossRefGoogle Scholar
  57. Tanaka K (1989) Mechanics and micromechanics of fatigue crack propagation. ASTM STP 1020:151–183Google Scholar
  58. Turner CE, Kolednik O (1994) Application of energy dissipation rate arguments to stable crack growth. Fatigue Fract Eng Mater Struct 17:1089–1107CrossRefGoogle Scholar
  59. Wüthrich C (1982) The extension of the \(J\)-integral concept to fatigue cracks. Int J Fract 20:R35–R37CrossRefGoogle Scholar

Copyright information

© 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

Personalised recommendations