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A Review of Critical Plane Orientations in Multiaxial Fatigue Failure Criteria of Metallic Materials

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The paper presents a review of multiaxial fatigue failure criteria based on the critical plane concept. The criteria have been divided into three groups, according to the fatigue damage parameter used in the criterion, i.e. (i) stress, (ii) strain and (iii) strain energy density criteria. Each criterion was described mainly by the critical plane orientation. Multiaxial fatigue criteria based on the critical plane concept usually apply different loading parameters in the critical plane whose orientation is determined by (a) only shear loading parameters (crack Mode II or III), (b) only normal loading parameters (crack Mode I) or sometimes (c) mixed loading parameters (mixed crack Mode). There are also criteria based on few critical plane orientations and criteria based on critical plane orientations determined by a weighted averaging process of rotating principal stress axes.

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References

  • B. Alfredsson M. Olsson (2001) ArticleTitleApplying multiaxial fatigue criteria to standing contact fatigue International Journal of Fatigue 23 533–548 Occurrence Handle10.1016/S0142-1123(01)00008-1

    Article  Google Scholar 

  • ASTM E 1049-85 (REAPPROVED 1997): Standard practices for cycle counting in fatigue analysis. In: Annual Book of ASTM Standards, Vol. 03.01, Philadelphia (1999), pp. 710–718.

  • A. Banvillet T. Lagoda E. Macha A. Nieslony T. Palin-Luc J.-F. Vittori (2004) ArticleTitleFatigue life under non-Gaussian random loading from various models International Journal of Fatigue 26 349–363 Occurrence Handle10.1016/j.ijfatigue.2003.08.017

    Article  Google Scholar 

  • M. Backstrom G. Marquis (2001) ArticleTitleA review of multiaxial fatigue of weldments: experimental results, design code and critical plane approaches Fatigue of Engineering Materials and Structures 24 279–291

    Google Scholar 

  • W. Bedkowski E. Macha M. Ohnami M. Sakane (1995) ArticleTitleFracture plane of cruciform specimen in biaxial low cycle fatigue – estimate by variance method and experimental verification Journal of Engineering Materials and Technology 117 183–190

    Google Scholar 

  • Berard J.Y., McDowell D.L. and Antolovich, S.D. (1993). Damage observation of a low-carbon steel under tension-torsion low cycle fatigue. In Advances in Multiaxial Fatigue (Edited by McDowell, D.L., R. Ellis, R.). American Society for Testing and Materials STP 1191. Philadelphia pp. 326–344.

  • M.W. Brown K.J. Miller (1973) ArticleTitleA theory for fatigue failure under multiaxial stress–strain conditions Proceedings of the Institute of Mechanical Engineers 187 745–755

    Google Scholar 

  • M.W. Brown K.J. Miller (1979) ArticleTitleInitiation and growth of cracks in biaxial fatigue Fatigue of Engineering Materials and Structures 1 231–246

    Google Scholar 

  • A. Carpinteri E. Macha R. Brighenti A. Spagnoli (1999a) ArticleTitleExpected principal stress directions under multiaxial random loading Part I: Theoretical aspects of the weight function method International Journal of Fatigue 21 83–88

    Google Scholar 

  • A. Carpinteri E. Macha R. Brighenti A. Spagnoli (1999b) ArticleTitleExpected principal stress directions under multiaxial random loading. Part II: Numerical simulation and experimental assessment through the weight function method International Journal of Fatigue 21 89–96

    Google Scholar 

  • A. Carpinteri A. Spagnoli (2001) ArticleTitleMultiaxial high-cycle fatigue criterion for hard metals International Journal of Fatigue 23 135–145 Occurrence Handle10.1016/S0142-1123(00)00075-X

    Article  Google Scholar 

  • A. Carpinteri A. Spagnoli A.S. Vantadori (2003) ArticleTitleA multiaxial fatigue criterion for random loading Fatigue of Engineering Materials and Structures 26 515–522

    Google Scholar 

  • X. Chen S. Xu D. Huang (1999) ArticleTitleA critical plane-strain energy density criterion for multiaxial low-cycle fatigue life under non-proportional loading Fatigue of Engineering Materials and Structures 22 679–686

    Google Scholar 

  • A. Constantinescu K. Dang ParticleVan H. Maitournam (2003) ArticleTitleA unified approach for high and low cycle fatigue based on shakedown concepts Fatigue of Engineering Materials and Structures 26 561–568

    Google Scholar 

  • Cox H.L. and Field J.E. The initiation and propagation of fatigue cracks in mild steel pieces of square section. The Aeuronautical Quarterly IV, 1–19.

  • Dang Van, K. (1983). Macro-micro approach in high-cycle multiaxial fatigue. In Advances in Multiaxial Fatigue. (Edited by McDowell, D.L. and Ellis, R.) American Society for Testing and Materials STP 1191. Philadelphia (1993) pp. 120–130

  • K. Dang ParticleVan G. Cailletaud J.F. Flavenot A. Douaron ParticleLe H.P. Lieurade (1989) Criterion for high cycle fatigue failure under multiaxial loading M. Brown K.J. Miller (Eds) Biaxial and Multiaxial Fatigue Mechanical Engineering Publications London 459–478

    Google Scholar 

  • J. Das S.M. Sivakumar (1999) ArticleTitleAn evaluation of multiaxial assessment methods for engineering components International Journal of Pressure Vessels and Piping 76 741–646 Occurrence Handle10.1016/S0308-0161(99)00053-8

    Article  Google Scholar 

  • J. Das S.M. Sivakumar (2000) ArticleTitleMultiaxial fatigue life prediction of a high temperature steam turbine rotor using a critical plane approach Engineering Failure Analysis 7 347–358

    Google Scholar 

  • H. Dietmann T. Bhongbhibhat A. Schmid (1976) Multiaxial fatigue behaviour of steels under in-phase and out-of-phase loading, including different wave forms and frequencies K. Kussmaul D. McDiarmid D. Socie (Eds) Fatigue Under Biaxial and Multiaxial Loading Mechanical Engineering Publications London 449–469

    Google Scholar 

  • A. Fatemi P. Kurath (1998) ArticleTitleMultiaxial fatigue life predictions under the influance of mean-stresses Transaction of ASME Journal of Engineering and Technology 110 380–388

    Google Scholar 

  • A. Fatemi D.F. Socie (1988) ArticleTitleA critical plane approach to multiaxial fatigue damage including out-of-phase loading Fatigue of Engineering Materials and Structures 11 149–165

    Google Scholar 

  • Findley, W.N. (1959). A theory for the effect of mean stress on fatigue of metals under combined torsion and axial load or bending. Journal of Engineering for Industry November, 301–306.

  • Forsyth, P.J.E. (1961). A two-stage process of fatigue crack growth. Proceedings of the Symposium on Crack Propagation, Cranfield, England, 76–94.

  • C. Froustey S. Lasserre (1989) ArticleTitleMultiaxial fatigue endurance of 30NCD16 steel International Journal of Fatigue 11 169–175 Occurrence Handle10.1016/0142-1123(89)90436-2

    Article  Google Scholar 

  • G. Glinka G. Shen A. Plumtree (1995a) ArticleTitleA multiaxial fatigue strain energy density parameter related to the critical fracture plane Fatigue of Engineering Materials and Structures 18 37–46

    Google Scholar 

  • G. Glinka G. Shen A. Plumtree (1995) ArticleTitleMean stress effects in multiaxial fatigue Fatigue of Engineering Materials and Structures 18 755–764

    Google Scholar 

  • Grubisic, V. and Simbürger, A. (1976). Fatigue under combined out-of-phase multiaxial stresses. International Conference on Fatigue, Testing and Design. Society of Environmental Engineers, London, 8 pp.

  • C. Han X. Chen K.S. Kim (2002) ArticleTitleEvaluation of multiaxial fatigue criteria under irregular loading International Journal of Fatigue 24 913–922 Occurrence Handle10.1016/S0142-1123(02)00013-0

    Article  Google Scholar 

  • Hoffmeyer, J., Döring, R., Seeger, T. and Vormwald, M. (2001). Short fatigue crack growth under multiaxial nonproportional loading. In 10th International Conference on Fracture. Elsevier Science, CD, 8 pp.

  • Itoh, T., Karolczuk, A., Lachowicz, C.T. and Macha, E. (2004) Energy models of fatigue life of steels and an aluminium alloy under nonproportional loading, Proceedings of the 7th International Conference on Biaxial/Multiaxial Fatigue and Fracture, DVM, Berlin, pp. 57–62.

  • F.A. Kandil M.W. Brown K.J. Miller (1988) Biaxial low-cycle fatigue fracture of 316 stainless steel at elevated temperatures The Metals Society London 203–210

    Google Scholar 

  • Kenmeugne, B., Weber, B., Carmet, A. and Robert, J.L. (1997). A stress-based approach for fatigue assessment under multiaxial variable amplitude loading. In Proceedings of 5th International Conference on Biaxial/Multiaxial Fatigue & Fracture (Edited by Macha, E. and Mróz, Z.). Technical University of Opole, pp. 557–573.

  • K.S. Kim J.C. Park (1999) ArticleTitleShear strain based multiaxial fatigue parameter applied to variable amplitude loading International Journal of Fatigue 21 475–483 Occurrence Handle10.1016/S0142-1123(98)00091-7

    Article  Google Scholar 

  • M. Küppers C.M. Sonsino (2003) ArticleTitleCritical plane approach for the assessment of the fatigue behaviour of welded aluminium under multiaxial loading Fatigue of Engineering Materials and Structures 26 507–513

    Google Scholar 

  • T. Lagoda E. Macha (1997) ArticleTitleEstimated and experimental fatigue lives of 30CrNiMo8 steel under in- and out-of-phase combined bending and torsion with variable amplitudes Fatigue & Fracture of Engineering Materials & Structures 17 1307–1318

    Google Scholar 

  • Lagoda, T. and Macha. E. (1997). Fatigue life under biaxial stress state with different cross-correlation coefficients of normal stresses. In Proceedings of 9th International Conference of Fracture, Vol. 3. (Edited by Karihaloo, B.L., Mai, Y-W., Ripley, M.I. and Ritchie R.O.) Pergamon, pp. 1371–1377.

  • T. Lagoda E. Macha W. Bedkowski (1999) ArticleTitleA critical plane approach based on energy concepts: Application to biaxial random tension-compression high-cycle fatigue regime International Journal of Fatigue 21 431–443 Occurrence Handle10.1016/S0142-1123(99)00003-1

    Article  Google Scholar 

  • T. Lagoda (2001) ArticleTitleEnergy models for fatigue life estimation under uniaxial random loading. Part II: Verification of the model International Journal of Fatigue 23 481–489

    Google Scholar 

  • B.L. Lee K.S. Kim K.M. Nam (2003) ArticleTitleFatigue analysis under variable amplitude loading using an energy parameter International Journal of Fatigue 25 621–631 Occurrence Handle10.1016/S0142-1123(02)00169-X

    Article  Google Scholar 

  • Liu, K.C. (1993). A method based on virtual strain-energy parameters for multiaxial fatigue life prediction. in Advances in Multiaxial Fatigue (edited by McDowell, D.L. and Ellis, R.). American Society for Testing and Materials STP 1191, Philadelphia, pp. 67–84.

  • K.C. Liu J.A. Wang (2001) ArticleTitleAn energy method for predicting fatigue life, crack orientation, and crack growth under multiaxial loading condition International Journal of Fatigue 23 S129–S134 Occurrence Handle10.1016/S0142-1123(01)00169-4

    Article  Google Scholar 

  • R.D. Lohr E.G. Ellison (1980) ArticleTitleA simple theory for low cycle multiaxial fatigue Fatigue of Engineering Materials and Structures 3 1–17

    Google Scholar 

  • C.D. Lykins S. Mall V. Jain (2000) ArticleTitleAn evaluation of parameters for predicting fretting fatigue crack initiation International Journal of Fatigue 22 703–716 Occurrence Handle10.1016/S0142-1123(00)00036-0

    Article  Google Scholar 

  • Macha, E. (1979). Mathematical models of the life to fracture for materials subjected to random complex stress systems. Scientific Papers of the Institute of Materials Science and Applied Mechanics of Wroclaw Technical University 41, Monographs 13, Wroclaw (1979) 99 pp. (in polish)

  • E. Macha (1989) ArticleTitleSimulation investigations of the position of fatigue plane in materials with biaxial loads Material-wissenschaft und Werkstofftechnik 20 132–136

    Google Scholar 

  • Macha, E. (1998). Generalization of Strain Criteria of Multiaxial Cyclic Fatigue To Random Loadings. Technical University of Opole, Vol. 23, Opole, 89 pp. (in polish)

  • E. Macha C.M. Sonsino (1999) ArticleTitleEnergy criteria of multiaxial fatigue failure Fatigue of Engineering Materials and Structures 22 1053–1070

    Google Scholar 

  • E. Macha (2001) ArticleTitleA review of energy-based multiaxial fatigue failure criteria The Archive of Mechanical Engineering XLVIII 71–101

    Google Scholar 

  • Marquis G.B. and Karjalainen-Roikonen P. Long-life multiaxial fatigue of SG cast iron. In Proceedings of 6th International Conference on Biaxial/Multiaxial Fatigue & Fracture (Edited by Freitas, M.). Instituto Superior Tecnico, Lisboa, pp. 151–158.

  • T. Matake (1977) ArticleTitleAn explanation on fatigue limit under combined stress Bulletin of the The Japan Society of Mechanical Engineers 20 257–263

    Google Scholar 

  • McDiarmid, D.L. (1985). Fatigue under out-of-phase biaxial stresses of different frequencies. In Multiaxial Fatigue. (Edited by Miller, K.M. and Brown, M.W) American Society for Testing and Materials STP 853, 606–621.

  • D.L. McDiarmid (1987) ArticleTitleFatigue under out-of-phase bending and torsion Fatigue of Engineering Materials and Structures 9 457–475

    Google Scholar 

  • D.L. McDiarmid (1990) ArticleTitleA general criterion for high cycle multiaxial fatigue failure Fatigue of Engineering Materials and Structures 14 429–453

    Google Scholar 

  • A. Nitta T. Ogata K. Kuwabara (1989) ArticleTitleFracture mechanisms and life assessment under high-strain biaxial cyclic loading of type 304 stainless steel Fatigue of Engineering Materials and Structures 12 77–92

    Google Scholar 

  • Ogata, T., Nitta, A. and Kuwabara, K. (1989). Biaxial low cycle fatigue failure of type 304 stainless steel under in-phase and out-of-phase straining conditions. In Fatigue under Biaxial and Multiaxial Loading (Edited by kussmaul, K.F., McDiarmid, D.L. and Socie D.S.). MPA Univ. Stuttgart, pp. 377–392.

  • W. Pan C. Hung L. Chen (1997) ArticleTitleFatigue life estimation under multiaxial loadings International Journal of Fatigue 21 3–10

    Google Scholar 

  • I.V. Papadopoulos (2001) ArticleTitleLong life fatigue under multiaxial loading International Journal of Fatigue 23 831–849 Occurrence Handle10.1016/S0142-1123(01)00059-7

    Article  Google Scholar 

  • I.V. Papadopoulos P. Davoli C. Gorla M. Filippini A. Bernasconi (1997) ArticleTitleA comparative study of multiaxial high-cycle fatigue criteria for metals International Journal of Fatigue 19 219–235 Occurrence Handle10.1016/S0142-1123(96)00064-3

    Article  Google Scholar 

  • J. Park D. Nelson (2000) ArticleTitleEvaluation of an energy-based approach and a critical plane approach for predicting constant amplitude multiaxial fatigue life International Journal of Fatigue 22 23–39

    Google Scholar 

  • R. Rolovic S.M. Tipton (1999) An energy based critical plane approach to multiaxial fatigue analysis T.L. Panontin S.D. Shepard (Eds) Fatigue and Fracture Mechanics West Conshohocken PA 599–613

    Google Scholar 

  • M. Sakane M. Ohnami M. Sawada (1987) ArticleTitleFracture modes and low cycle biaxial fatigue life at elevated temperature Journal of Engineering Materials and Technology 109 236–243

    Google Scholar 

  • G. Savaidis T. Seeger (1997) Material behaviour and life evaluation under cyclic multiaxial proportional loading Z. Macha Z. Mróz (Eds) Biaxial/Multiaxial Fatigue & Fracture Technical University of Opole Opole 81–98

    Google Scholar 

  • K.N. Smith P. Watson T.H. Topper (1976) ArticleTitleA stress-strain function for the fatigue of metals Journal of Materials JMLSA 5 767–778

    Google Scholar 

  • D.F. Socie (1987) ArticleTitleMultiaxial fatigu0e damage models Journal of Engineering Materials and Technology 109 292–298 Occurrence Handle10.1115/1.3225980

    Article  Google Scholar 

  • D.F. Socie G.B. Marquis (2001) Multiaxial Fatigue Society of Automotive Engineers Inc. Warrendale Pa 484

    Google Scholar 

  • Socie, D.F., Waill, L.A. and Dittmer, D.F. (1985). Biaxial fatigue of Inconel 718 including mean stress effects. In Multiaxial Fatigue (edited by Miller, K.J. and Brown, M.W.). American Society for Testing and Materials STP 853, Philadelphia, pp. 463–481.

  • C.M. Sonsino (1995) ArticleTitleMultiaxial fatigue of welded joints under in-phase and out-of-phase local strains and stresses International Journal of Fatigue 17 55–70 Occurrence Handle10.1016/0142-1123(95)93051-3

    Article  Google Scholar 

  • A. Spagnoli (2001) ArticleTitleA new high-cycle fatigue criterion applied to out-of-phase biaxial stress state International Journal of Mechanical Sciences 43 2581–2595 Occurrence Handle10.1016/S0020-7403(01)00053-4 Occurrence Handle1077.74043

    Article  MATH  Google Scholar 

  • Stanfield G. (1935). Discussion on “The strength of metals under combined alternating stresses”. by H. Gough and H. Pollard. Proceeding of the Institution of Mechanical Engineers 131, 93.

  • F.B. Stulen H.N. Cummings (1954) ArticleTitleA failure criterion for multiaxial fatigue stresses Proceedings of the ASTM 54 822–835

    Google Scholar 

  • Susmel, L. and Petrone, N. (2001). Fatigue life prediction for 6082-T6 cylindrical specimens subjected to in-phase and out-of-phase bending/torsion loadings. In Proc. 6th International Conference on Biaxial/Multiaxial Fatigue & Fracture (edite by freitas, ed.). Instituto Superior Tecnico, Lisboa pp. 125–132.

  • S. Taira T. Inoue M. Takabashi (1969) ArticleTitleLow-cycle fatigue under multiaxial stresses (in the case of combined cyclic tension-compression and cyclic torsion at room temperature), Trans The Japan Society of Mechanical Engineers 35 525–532

    Google Scholar 

  • S.M. Tipton D.V. Nelson (1997) ArticleTitleAdvances in multiaxial fatigue life prediction for components with stress concentrations International Journal of Fatigue 19 503–515 Occurrence Handle10.1016/S0142-1123(96)00070-9

    Article  Google Scholar 

  • A. Varvani-Farahani (2000) ArticleTitleA new energy-critical plane parameter for fatigue life assessment of various metallic materials subjected to in-phase and out-of-phase multiaxial fatigue loading conditions International Journal of Fatigue 22 295–305 Occurrence Handle10.1016/S0142-1123(00)00002-5

    Article  Google Scholar 

  • Varvani-Farahani, A. and Topper, T.H. (2000). A new multiaxial fatigue life and crack growth rate model for various in-phase and out-of-phase strain paths. In Multiaxial Fatigue Deformation: Testing and Prediction. American Society for Testing and Materials STP 1387, West Conshohocken, PA, pp. 305–322

  • E. Vidal B. Kenmeugne J.L. Robert J. Bahuaud (1996) Fatigue life prediction of components using multiaxial criteria A. Pineau G. Cailletaud T.C. Lindley (Eds) Multiaxial Fatigue and Design, ESIS 21. Mechanical Engineering Publications London 365–378

    Google Scholar 

  • C.H. Wang M.W. Brown (1993) ArticleTitleA path-independant parameter for fatigue under proportional and non-proportional loading Fatigue of Engineering Materials and Structures 16 1285–1293

    Google Scholar 

  • Ying-Yu Wang Wei-Xing Yao (2004) ArticleTitleEvaluation and comparison of several multiaxial fatigue criteria International Journal of Fatigue 26 17–25 Occurrence Handle10.1016/S0142-1123(03)00110-5 Occurrence Handle2106053

    Article  MathSciNet  Google Scholar 

  • Weber, B., Clement, J.C., Kenmeugne, B. and Robert, J.L. (1999a). On a global stress-based approach for fatigue assessment under multuiaxial random loading. In Engineering Against Fatigue (Edited by Beynon, J.H., Brown, M.W., Lindley, T.C., Smith, R.A. and Tomkins, B.). Rotterdam, pp. 407–414.

  • Weber, B., Clement, J.C., Kenmeugne, B. and Robert, J.L. (1999b). A stress-based approach for fatigue assessment under multiaxial variable amplitude loading. In Multiaxial Fatigue and Fracture ESIS Publications 25, Elsevier (Edited by Macha, E., Bedkowski W. and Lagoda T.). Amsterdam, pp. 218–231.

  • A. Zolochevsky Y. Obataya J. Betten (2000) ArticleTitleCritical plane approach with two families of microcracks for modelling of uniteral fatigue damage Forschung im Ingenieurwesen 66 49–56

    Google Scholar 

  • Zenner H., Simbürger A. and Liu J. On the fatigue limit of ductile metals under complex multiaxial loading. International Journal of Fatigue 22, 137–145.

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Karolczuk, A., Macha, E. A Review of Critical Plane Orientations in Multiaxial Fatigue Failure Criteria of Metallic Materials. Int J Fract 134, 267–304 (2005). https://doi.org/10.1007/s10704-005-1088-2

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