Biaxial1 Fracture
1
BIAXIAL/MULTIAXIAL FATIGUE AND FRACTURE
I
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The Behaviour of Short Fatigue Cracks Edited by K.J. Miller and E.R. de 10s Rios The Fmcture Mechanics of Welds Edited by J.G. Blauel and K.H. Schwalbe Biaxial and Multiaxial Fatigue Edited by M.W. Brown and K.J. Miller The Assessment ofCracked Components hv Fracture Mechanics Edited by L.H. Larsson Yielding, Damage, and Failure OfAnisotmpic So1id.s Edited by J.F! Boehler High Temperature Fracture Mechanisms and Mechanics Edited by P. Bensussan and J.P. Mascarell Environment Assisted Fatigue Edited by P Scott and R.A. Cottis Fracture Mechanics Verification by Large Scale Testing Edited by K . Kussmaul Dcfect Assessment in Components Fundamentals and Applications Edited by J.G. Blauel and K.H. Schwalbe Fatigue under Biaxial and Multiaxial Loading Edited by K. Kussmaul, D.L. McDiarmid and D.F. Socie Mechanics and Mechanisms of’ Damage in Composites and MultiMaterials Edited by D. Baptiste High Temperature Structural Design Edited by L.H. Larsson Short Fatigue Cracks Edited by K.J. Miller and E.R. de 10s Rios MixedMode Fatigue and Fracture Edited by H.P. Rossmanith and K.J. Miller Behaviour of Defect.s at High Rmpemtures Edited by R.A. Ainsworth and R.P. Skelton Fatigue Design Edited by J. Solin, G. Marquis, A. Siljander and S. Sipila MisMatching of Welds Edited by K.H. Schwalbe and M. KoCak Fretting Fatigue Edited by R.B. Waterhouse and T.C. Lindley Impnct of Dynamic Fracture of Polymers and Composites Edited by J.G. Williams and A. Pavan Evaluating Material Properties by Dynamic Testing Edited by E. van Walle Multiaxial Fatigue & Design Edited by A. Pineau, G. Gailletaud and T.C. Lindley Fatigue Design of Components. ISBN 0080433 189 Edited by G. Marquis and J. Solin Futigue Design and Reliability. ISBN 0080433294 Edited by G. Marquis and J. Solin Minimum Reinforcement in Concrete Member.s. ISBN 0080430228 Edited by Albert0 Carpinteri Multiaxial Fatigue and Fracture. lSBN 0080433367 Edited by E. Macha, W. Bqdkowski and T. aagoda Fracture Mechanics: Applications and Challenges. ISBN 0080436994 Edited by M. Fuentes, M. Elices, A. MartinMeizoso and J.M. MartinezEsnaola Fracture of Po[vmer.s, Composites andddhesives. ISBN 0080437 109 Edited by J.G. Williams and A. Pavan Fracture Mechanics Testing Method.s,fiirPo1ymer.s Adhesives and Comjmsites. ISBN 0080436897 Edited by D.R. Moore, A. Pavan and J.G. Williams TemperatureFatigue Inteemction. ISBN 0080439829 Edited by L. Remy and J. Petit From Charp.v to Present Impuct Te.sting. ISBN 0080439705 Edited by D. FranCois and A. Pineau
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BIAXIALMULTIAXIAL FATIGUE AND FRACTURE
Editors: Andrea Carpinteri Manuel de Freitas Andrea Spagnoli
ESIS Publication 31 This volume contains 25 peerreviewed papers selected from those presented at the 6thInternational Conference on BiaxialMultiaxial Fatigue and Fracture held in Lisbon, Portugal, 2528 June 2001. The meeting was organised by the Instituto Superior Tecnico and sponsored by the Portuguese Ministerio da Ciencia e da Tecnologia and by the European Structural Integrity Society.
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V
CONFERENCE COMMITTEES International Scientific Committee: P. Bonacuse (USA) M.W.Brown (UK) A. Carpinteri (Italy) K. Dang Van (France) F. Ellyin (Canada) U. Fernando (UK) D. Franqois (France) M. de Freitas (Portugal) Chairman G. Glinka (Canada) S. Kalluri (USA) E. Macha (Poland) G. Marquis (Finland) D.L. McDowell (USA) K.J. Miller (UK) Y. Murakami (Japan) J. Petit (France) A. Pineau (France) L. Pook (UK) V. Shlyannikov (Russia) D. Socie (USA) C. Sonsino (Germany) S. StanzlTschegg (Austria) T. Topper (Canada) V. Troschenko (Uhaine) E. Tschegg (Austria) S. Zamrik (USA) H. Zenner (Germany) Organizing Committee: M. de Freitas (Chairman), M. Fonte, B. Li

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MURAKAMI Metal Fatigue Effects of Small Defects and Nonmetallic Inclusions ISBN: W80440649
FRANPIS and PINEAU From Charpy to prescot Impact Testing. ISBN. 0080439705
RAVICHANDRAN ETAL Small Fatigue Crack Mechpnics. Mechanisms C Applicatims. ISBN: 0084430112
NENTES ETAL. Fracture Mechanics: Applicationsand Challenges.
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TemperatureFatigueInteraction. ISBN: 0080439829
JONES Failure Analysis Care Studies 11. ISBN. WR0439594
TANAKA C DULIKRAVICH Inverse Problems in Engineering Mechanics 11. ISBN: W80436935
MACHA €TAL Muluaxial Fatigue and Fracture. ISBN: 0080433367
UOMOTO NonDestructiveTesting in Civil Engineaing ISBN: w80437174
MARQUIS C SOLIN Fatigue Design of Components.
VOYlADJISETAL Damage Mechanics in hgineering Materials
ISBN: 0080433189
ISBN: 0080433227
MARQUIS & SOLIN Fatigue Design and Reliability.
VOYIADJIS & KATTAN Advances in Damage Mechanics:Metals and Metal Matrix Composites.
ISBN: 0080433294
ISBN. W8043601.3
MOORE E T A L Fracture Mechanics Testing Methods for Polymers. Adhesives and Composites. ISBN: 0080436897
WILLIAMS & PAVAN Fracture of Polymers,Compositesand Adhesives. ISBN: 0080437109
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vii
CONTENTS Preface
xi
1. Multiaxial Fatigue of Welded Structures Assessment of Welded Structures by a Structural Multiaxial Fatigue Approach K. Dung Van,A. Bignonnet and J. L. Fayard Evaluation of Fatigue of Fillet Welded Joints in Vehicle ComponentsUnder Multiaxial Service Loads G. Savaidis, A. Savaidis, R. Schliebner and M. Vormwald Multiaxial Fatigue Assessment of Welded Structures by Local Approach F. LabesseJied, B. Lebrun, E. Petitpas and J.L Robert MicroCrack Growth Behavior in Weldments of a NickelBase Superalloy Under Biaxial LowCycle Fatigue at High Temperature N. Zsobe and S. Sakurai
3
23 43
63
2. High Cycle Multiaxial Fatigue Multiaxial Fatigue Life Estimations for 6082T6 Cylindrical Specimens Under InPhase and OutofPhase Biaxial Loadings L. Susrnel and N. Petrone LongLife Multiaxial Fatigue of a Nodular Graphite Cast Iron G.B. Marquis and P. KarjalainenRoikonen The Influence of Static Mean Stresses Applied Normal to the Maximum Shear Planes in Multiaxial Fatigue R.P. Kaufman and T. Topper
83 105
123
3. NonProportional and VariableAmplitude Loading Fatigue Limit of Ductile Metals Under Multiaxial Loading J. Liu and H . Zenner
147
Sequenced Axial and Torsional Cumulative Fatigue: Low Amplitude Followed by High Amplitude Loading P. Bonacuse and S. Kalluri
165
Estimation of the Fatigue Life of High Strength Steel Under VariableAmplitude Tension with Torsion: Use of the Energy Parameter in the Critical Plane T. Lagoda, E. Macha, A. Niesiony and F. Morel
183
viii Critical PlaneEnergy Based Approach for Assessment of Biaxial Fatigue Damage where the StressTime Axes are at Different Frequencies A. VarvaniFarahani
203
Fatigue Analysis of Multiaxially Loaded Components with the FEPostprocessor FEMFATMAX C. Gaier and H. Dannbauer
223
4. Defects, Notches, Crack Growth The Multiaxial Fatigue Strength of Specimens Containing Small Defects
243
M.Endo An Analysis of ElastoPlastic Strains and Stresses in Notched Bodies Subjected to Cyclic NonProportional Loading Paths A. Buczynski and G. Glinka
265
The Background of Fatigue Limit Ratio of Torsional Fatigue to Rotating Bending Fatigue in Isotropic Materials and Materials with ClearBanded Structure T.Fukuda and H. Nisitani
285
Influence of Defects on Fatigue Life of Aluminium Pressure Diecastings F.J. Lino, R.J. Neto, A. Oliveira and F.M.F. de Oliveira
303
Variability in Fatigue Lives: An Effect of the Elastic Anisotropy of Grains? S. Pommier
32 1
ThreeDimensional Crack Growth: Numerical Evaluations and ExperimentalTests C. Call, R. Citarella and M. Perrella
34 1
The Environment Effect on Fatigue Crack Growth Rates in 7049 Aluminium Alloy at Different Load Ratios M. Fonte, S.StanzlTschegg, B. Holper, E. Tschegg and A. Vasud6van
361
5. Low Cycle Multiaxial Fatigue A Multiaxial Fatigue Life Criterion for NonSymmetrical and NonProportional ElastoPlastic Deformation M. Filippini, S.Foletti, I. V. Papadopoulos and C.M . Sonsino
383
Cyclic Behaviour of a Duplex Stainless Steel Under Multiaxial Loading: Experiments and Modelling V. Aubin, P. Quaegebeur and S. Degallaix
401
A Damage Model for Estimating Low Cycle Fatigue Lives Under Nonproportional Multiaxial Loading T.ltoh and T.Miyazaki
423
Microcrack Propagation Under NonProportional Multiaxial AlternatingLoading M. Weick and J. Aktaa
441
ix
6. Applications and Testing Methods Fatigue Assessment of Mechanical Components Under Complex Multiaxial Loading J.LT. Santos, M.de Freitas, B. Li and T.P. Trigo
463
Geometry Variation and Life Estimates of Biaxial Fatigue Specimens G. Shatil and N.Ersoy
483
Author Index
501
Keyword Index
503
xi
PREFACE The European Structural Integrity Society (ESIS) Technical Committee on Fatigue of Engineering Materials and Structures (TC3) decided to compile a Special Technical Publication (ESIS STP) based on the 115 papers presented at the 6th International Conference on BiaxiaVMultiaxial Fatigue and Fracture. The 25 selected papers included in the STP have been extended and revised by the authors. The Conference was held in Lisbon, Portugal, on 2528 June 2001, and was chaired by Manuel de Freitas, Instituto Superior Tecnico, Lisbon. The meeting was organised by the Instituto Superior Tecnico, and sponsored by the Portuguese Ministerio da Ciencia e da Tecnologia and by the European Structural Integrity Society. It was attended by 151 delegates from 20 countries. The previous International Conferences on BiaxialMultiaxial Fatigue and Fracture were held in San Francisco (1982), Sheffield (1985), Stuttgart (1989), St Germain en Laye (1994), and Cracow (1997). The papers in the present book deal with theoretical, numerical and experimental aspects of the multiaxial fatigue and fracture of engineering materials and structures. They are divided into the following six sections: (1) Multiaxial Fatigue of Welded Structures (4 papers);
(2) High Cycle Multiaxial Fatigue (3 papers); (3) NonProportional and VariableAmplitude h a d i n g (5 papers); (4) Defects, Notches, Crack Growth (7 papers); (5) Low Cycle Multiaxial Fatigue (4 papers); (6) Applications and Testing Methods (2 papers). This book presents recent world advances in the field of multiaxial fatigue and fracture. It is the result of cooperation between many researchers from different laboratories, universities and industries in a number of countries, As is wellknown, most of engineering components and structures in the mechanical, aerospace, power generation and other industries are subjected to multiaxial loading during their service life. One of the most difficult tasks in design against fatigue and fracture is to translate the information gathered from uniaxial fatigue and fracture tests on engineering materials into applications involving complex states of cyclic stressstrain conditions. Numerous people have contributed to the publication of the present book. The editors wish to express their most grateful thanks to the authors of the 25 papers included in the publication, and also to the authors of the papers which the reviewers did not feel able to recommend for inclusion. Further, the editors wish to thank especially the following experts (in alphabetical order) for their valuable contributions as reviewers of manuscripts:
xii
K.Dang Van, &ole Polytechnique, Palaiseau, France D. Frangois, Ecole Centrale de Paris, France G. Glinka, University of Waterloo, Waterloo, Ontario, Canada P. Lazzarin, University of Padua, Vicenza, Italy E. Macha, Technical University of Opole, W l e , Poland Z. Mroz, Instytut Podstawowych Problemow Techniki PAN, Warsaw, Poland Y. Murakami, Kyushu University, Fukuoka, Japan A. Navarro, ETS Ingenieros Industriales, University of Seville, Seville, Spain A. Pineau, Ecole Nationale Superieure des Mines de Paris, Evry, France D.F. Socie, University of Illinois at UrbanaChampaign, Urbana, Illinois, USA C.M. Sonsino, Fraunhofer Institute for Structural Durability LBF, Darmstadt, Germany S.E. StanzlTschegg, University of Agricultural Sciences, Vienna, Austria V.T. Troshchenko, National Academy of Sciences of Ukraine, Kiev, Ukraine H. Zenner, Technical University of Clausthal, ClausthalZellerfeld, Germany Finally, the editors wish to thank all the staff at ESIS and Elsevier, who have made this publication possible. Andrea Carpinteri, University of Parma, Italy Manuel de Freitas, Instituto Superior Tecnico, Portugal Andrea Spagnoli, University of Parma, Italy Novembre, 2002
1. MULTIAXIAL FATIGUE OF WELDED STRUCTURES
BiaxiallMultiaxial Fatigue and Fracture Andrea Carpinteri, Manuel de Freitas and Andrea Spagnoli (Eds.) Q Elsevier Science Ltd. and ESIS. All rights reserved.
3
ASSESSMENT OF WELDED STRUCTURES BY A STRUCTURAL MULTIAXIAL FATIGUE APPROACH Ky DANG VAN', Andr6 BIGNONNET2and JeanLuc FAYARD2 'taboratoire de Mkcanique des Solides, Ecole Polytechnique, 91 128, Palaiseau, France PSA Peugeot Citroen, Route de Gizy, 78943, VdizyVillacoublay Cedex, France
ABSTRACT A structural multiaxial computing method for the fatigue assessment of welded structures is presented. This approach is based on the use of a local equivalent stress, or design stress, derived from the shear stress and the concomitant hydrostatic pressure previously proposed by Dang Van. Associated with a specific shell finite element meshing methodology, the method is successfully used to assess the fatigue resistance of welded automotive structures. The approach also allows the role of the welding process upon the fatigue behaviour to be addressed by a better description of the influence of the local residual stress state. KEYWORDS Welded structures, multiaxial fatigue, residual stresses, computing methods, design stress
INTRODUCTION Although extensive work has already been done, the prediction of the fatigue strength of welded structures is still a widely open subject. Engineers in design offices do not dispose of reliable and accurate methods to evaluate the fatigue life of such structures, with regard to the results provided by the modern structural calculation methods (Finite Element Method). If some propositions exist, they are most of the time inapplicable so that, in practice, engineers use simplified methods of poor accuracy. For example, for the calculation of metallic bridges, a design stress S is evaluated from the nominal stress derived from a beam calculation; then the fatigue life is estimated from SN curves given by the EUROCODE ITI established experimentally by class of structural details. One can imagine that if there are a few changes in the geometry or the loading mode of these details, one can be out of the limits covered by the fatigue tests and therefore have erroneous predictions. Beside the global approaches as the EUROCODE III,several proposals called local approaches exist [ I ] . Among them, one can distinguish those which study the crack initiation and those which consider that microcracks are already formed and only take into account their propagation. The latter uses the Paris law, or derivative laws, which appears to be well founded on the recognised concepts of the Fracture Mechanics, but are nevertheless not so easy to apply on actual structures. As noted recently by D.L. Mc Dowel1 [2], the first cracks initiated in )>
4
K. DANG VAN, A . BIGNONNETAND 1L.FAYARD
fatigue cannot be reduced to plane cracks simply characterised by their length a, submitted to simple in mode I loading and in a linear elastic regime. This makes the calculation of the parameters which are supposed to govern the propagation extremely difficult. The transcription of some parameters which are justified in Fracture Mechanics to fatigue is still poorly founded and therefore ambiguous for a structure. It is the case for instance of the J parameter and the J derived parameter usually invoked to correlate fatigue testing results on specimens. For all these reasons the local approaches based on crack initiation are still preferred by engineers. As for welded structures, the structural method developed for the offshore industry in the 70’ and the 80’s (AWS and API codes) and particularly following Radenkovic’s proposals [ 3 ] , is without any doubt the one which allows the most interesting industrial calculations. In this method, the welded connection is characterised by a design stress S, more or less clearly defined, which better describes the local stress at the l ,
Multiaxial Fatigue Life Estimations for 6082T6 Cylindrical Specimens Under ... P36BTll Ntm = 20730 Cycles
P41BT14 t
97
 N f . 2 =~ 41490 Cycles
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P42BT16  Nf2a,= 1016280 Cycles
Fig. 10. Cracks pattern at 20% stiffness drop on the bidimensional development of the specimen gauge surface: outofphase bendingltorsion tests (h=zxy,dux,a>l, 6~126").
L. SWSMEL AND A! PETRONE
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Fig. 11. Cracks pattern at 20% stiffness drop on the bidimensional development of the specimen gauge surface: inphase bendinghorsion tests ( h = ~ ~ ~ , J 6 u~~~0',). ~ Sint For SAE 1045 steel, BHN 456 and 203, a k value of 0.5 reduced the experimental data into a narrow band for the hard and soft steels. A k value of 0.5 was initially suggested by Findley. The interference free stress (Slnt)is 500 MPa for the hard material and 76 MPa for the as received material. The modified Findley parameter is plotted versus fatigue life for the hard and soft steels in Figs 14 and 15, respectively. The use of the suggested modification to Findley's parameter results in the majority of the experimental parameter values falling within the 2x fatigue life boundaries drawn about the mean curve. The modified Findley parameter was found to be the best static mean stress parameter investigated during this study for condensing the experimental data. Imposing a cut off level, above which further increases in tensile mean stress no longer reduce fatigue life, accounted for the crack face interference free behavior found in this investigation.
R.P KAUFMAN AND TH.TOPPER
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Fig. 14. Modified Findley parameter versus fatigue life for BHN 456.
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Fig. 15. Modified Findley parameter versus fatigue life for BHN 203.
The Influence of Static Mean Stresses Applied Normal to the Maximum Shear Planes in
...
141
CONCLUSIONS For a given fatigue life, data from this test program exhibit an inverse linear relationship between the alternating shear stress and the static mean stress normal to the maximum shear stress amplitude planes for static mean stresses smaller than 500 MPa and 76 MPa for the hard and soft steels, respectively. Increasing the tensile mean stress beyond these values does not result in a further decrease in the alternating shear stress for a given fatigue life. When static mean stresses normal to the plane of maximum alternating shear stress are high, fracture surface asperities are unmarked. Asperity heights increase with increasing tensile mean stress until the stress at which the crack faces no longer touch is achieved. The modified Findley parameter condensed the majority of the experimental results within a 2x band on a parameter versus fatigue life curve. For tensile static mean stresses larger then Sint, the crack faces were found to be fully separated, a condition previously identified as crack face interference free growth. This cracking mechanism is incorporated into the modified Findley parameter by placing a limiting value Sint above which the mean stress value used in the parameter is kept constant. ACKNOWLEDGEMENTS The authors would like to thank NSERC for the funding to complete this program of study and the Canadian Armed Forces. The authors would also like to thank Dr. Bonnen, Dr. Conle, Mr. Barber, Mr. Boldt and Mr. Alberts for their contributionsto this research. REFERENCES I . Bonnen, J., and Topper, T.H., (1999) Multiaxial Fatigue and Deformation: Testing and Prediction, STP 1387 213. 2. VarvaniFarahani, A. and Topper, T.H., (1997) “Short fatigue crack characterization and detection using confocal scanning laser microscopy (CSLM),“ Nontraditional methods of sensing stress, strain and damage in Materials and Structures, ASTM STP 1318 4355. 3. Stulen, F.B., and Cummings, H.N. (1954) “A failure criterion for multiaxial fatigue stresses,” ASTM Proceedings 54,822. 4 . Findley, W.N., (1954) “Experiments in fatigue under ranges of stress in torsion and axial load from tension to extreme compression,” ASTM Proceedings 54, 30 1. 5. Findley, W.N., (1959) “A theory for the effect of mean stress on fatigue of metals under combined torsion and axial loading or bending,” Trans. ASME J. Eng. For Industry 81, 301306. 6. Sines, G. (1961) T h e prediction of fatigue fracture under combined stresses at stress concentrations,”Bulletin of JSME 4,443. 7. Mazelsky, B., Lin, T.H., Lin, S.R. and Yu,C.K., (1969) “Effect of axial compression on lowcycle fatigue of metals in torsion,” J Basic Eng.91,780. 8. Chu, C.C., (1994) “Critical plane analysis of variable amplitude tests for SAE 1045 steels,”SAE Technical Paper #9#0246.
142
R.P KAUFMAN AND ZH. TOPPER
9. Guest, J.J., (1940) “On the strength of ductile materials under combined stresses,” Proc. Inst of Automobile Engrs. 3533. 10. Gough, H.J., and Pollard, H.V., (1935) ‘The effect of specimen form on the resistance of metals to combined alternating stresses,” Proc. Insr Mech Engrs. 131,3. 1 1. Stanfield, G.,( 1935) Discussion of the strength of metals under combined alternating stresses, by Gough, H. and Pollard, H., Proc. Inst Mech Engrs. 131 93. 12. McDiamid, D.L., (1974) “Mean stress effects in biaxial fatigue where the stresses are outofphase and at different frequencies,” Aero J. 78 325. 13. Carpinteri, A., and Spagnoli, A. (2001) “Multiaxial highcycle fatigue criterion for hard metals.” Int J Fatigue 23, 135145. 14. Kandil, F.A., Brown, M.W., and Miller, K.J., (1982) “Biaxial lowcycle fatigue fracture of 316 stainless steel at elevated temperatures,” The Metals Society 280 203. 15. Fatemi, A., and Socie, D. (1988) “A critical plane approach to multiaxial fatigue damage including outofphase loading, “J Far and Fract Eng Mat and Struct.3 149. 16. Socie, D.F., Kurath, P. and Koch, J., “A multiaxial LCF parameter,” (1985) Second International Symposium on Multiaxial Fatigue. 17. Fatemi, A., and Kurath, P., (1988) “Multiaxial fatigue life predictions under the influence of meanstress,” J Eng Mat and Tech 110 380. 18. Sines, G,. PhD Thesis, “Failure of materials under combined repeated stresses with superimposed static stresses,” University of California, USA. 19. Sines, G., and Ohgi, G., (1981) “Fatigue criteria under combined stresses and strains,” Trans ASME, 103 82. 20. Seeger, G., (1 936) Zeitschift d. V.I., 80 698. 21. Smith, M.C., and Smith, R.C., (1988) ‘Towards an understanding of Mode 11 fatigue crack growth,” ASTM STP 924 260. 22. Burns, D.J., and Parry, J.S.C., (1964) “Effect of large hydrostatic pressures on the torsional fatigue strength of two steels,” JMech Eng Sci, 6 293. 23. Socie, D.F., and Waill, L.A., Dittmer, D.F., (1985) “Biaxial fatigue of inconel 718 including mean stress effects,” ASTM STP 853. 24. Socie, D.F., and Shield, T.W., (1984) “Mean stress effects in biaxial fatigue of inconel 71 8,” J Eng Mat Tech 106 227. 25. Bonnen, J., (1998) PhD Thesis, “Multiaxial fatigue response of normalized 1045 steel subjected to periodic overloads: experiments and analysis,” University of Waterloo, Canada. 26. VaravaniFarahni, A., (1998) PhD Thesis, “Biaxial fatigue crack growth and crack closure under constant amplitude and periodic compressive overload histories in 1045 steel,” University of Waterloo, Canada. 27. Kaufman, R., (2002) PhD Thesis, “Static mean stress effects in multiaxial fatigue and their implications on fatigue life predictions of induction hardened shafts,” University of Waterloo, Canada. 28. Havard, D.G., (1970) PhD Thesis, “Fatigue and deformation of normalized mild steel subjected to cyclic loading,” University of Waterloo, Canada. 29. Elkholy, A.H., ( 1 975) Masters Thesis, University of Waterloo, Canada. 30. Snoddy, J., (2001) Work Term Report, “Design, test and use of a confocal laser microscope,” University of Waterloo, Canada.
The Influence of Static Mean Stresses Applied Normal to the Maximum Shear Planes in ___
143
Appendix : NOMENCLATURE
BHN
Brinell hardness number
E
Young's modulus
E
Local strain
f;k
Constant for Findley's parameter
k
Constant for Fatemi and Socie parameter
Y
Engineering shear strain
P
Parameter Local stress Yield stress Shear stress ratio,
Tm$mx
S
Stress
S*
Empirical constant in Modified Kandil, Brown and Miller parameter
z
Local shear stress
h
Ratio between T, and oa
SUBSCRIPTS, SUPERSCRIPTS AND OPERATORS 1, 2, 3
Orthogonal principal stress/strain coordinates
axial
Value of variable in axial direction
HOOP
Value of variable in hoop direction
int
Value of variable when crack faces are interference free
mar
Maximum value of the variable during the load cycle
mean
Value of static variable normal to the maximum shear planes
n
Value of variable normal to the crack plane
no
Value of static variable perpendicular to the maximum shear strain amplitude
static mean
A
Value of static variable normal to the maximum shear planes Range of variable during the loading cycle
3. NONPROPORTIONAL AND VARIABLEAMPLITUDE LOADING
BiaxiaYMultiaxialFatigue and Fracture Andrea Carpinten, Manuel de Freitas and Andrea Spagnoli (Eds.) Q Elsevier Science Ltd. and ESIS. All rights reserved.
147
FATIGUE LIMIT OF DUCTILE METALS UNDER MULTIAXIAL LOADING Jiping LIU' and Harald 2J3NNER2
2
1 Volkswagen AG, 38436 WoEfsburg, Germany TU Clausthal, 38678 ClausthalZellerfeld, Germany
ABSTRACT The furtherdeveloped Shear Stress Intensity Hypothesis (SM)is presented for the calculation of the fatigue limit of ductile materials under multiaxial loading. The fatigue limit behaviour for different cases of multiaxial loading is analysed with SM and experimental results, especially with the effect of mean stresses, phase difference, frequency difference, and wave form. In a statistical evaluation, the furtherdeveloped S M provides a good agreement with the experimental results.
KEYWORDS Fatigue limit, multiaxial loading, multiaxial criteria, weakest link theory INTRODUCTION A multiaxial stress state which varies with time is generally present at the most severely stressed point in a structural component. The stress state is usually of complex nature. The individual stress components may vary in a mutually independent manner or at different frequencies, for instance, if the flexural and torsional stresses on a shaft are derived from two vibrational systems with different natural frequencies. For assessing this multiaxial stress state, the classical multiaxial criteria, such as the von Mjses criterion or the maximum shear stress criterion, are not directly applicable. This is illustrated in Fig. 1 for two load cases. In the first case, an alternating normal stress occurs in combination with an alternating shear stress with a phase shift of 90°, Fig. la. The second case involves a normal pulsating tensile normal stress cr, and a compressively pulsating normal stress oy, Fig. Ib. In both load cases, the principal stresses exhibit the same variation with time. In accordance with the classical multiaxial criteria, the same equivalent stresses are calculated in both cases. The endurance limits are very different, however, as is shown by experiments. This is explained by the fact that the principal direction can vary in the case of multiaxial stress. A variable principal direction is not taken into account by the classical multiaxial criteria.
J. LIU AND H.ZENNER
148
FYY
1
1
*OgOO1+& 90"
0"
0"
90"
90"
o l a ~Endurance limit for smooth specimens, steel 34Cr4: 158MPa 240 MPa Fig. 1. Coordinate stresses, principal stresses and direction of principal stresses for one cycle. Influence of variable principal stress direction on the endurance limit. For calculating the fatigue limit, a number of multiaxial criteria have been developed in the past. The multiaxial criteria can be subdivided as follows: empirical approach, critical plane approach, and integral approach. The empirical theories were derived by extension of the classical criteria or were usually developed for specific load cases in correspondence with test results [I81. With the critical plane approach, the stress components in the critical plane with the maximal value of equivalent stress are considered as relevant for the damage [915]. In the case of the integral approach, the equivalent stress is calculated as an integral of the stresses over all cutting planes of a volume element, for instance, with the hypothesis of effective shear stress [16] or the shear stress intensity hypothesis [17]. The hypothesis of Papadopoulos [ 18,191 is based on the same principle and differs from the shear stress intensity hypothesis by the consideration of the mean stresses.
Fatigue Limit of Ductile Metals Under Multiaxial Loading
149
In the present paper, the furtherdeveloped shear stress intensity hypothesis (SIH)is described. The fatigue limit behaviour of ductile metallic materials is explained with special attention to the effects of the mean stresses, the phase difference, the frequency difference, and the wave form.
SHEAR STRESS INTENSITY HYPOTHESIS The development of the shear stress intensity hypothesis (SLH) can be retraced to the interpretation of the von Mises criterion in accordance with Novoshilov [20].In the past, the von Mises criterion has been interpreted differently:  Distortion energy (Maxwell 1856, Huber 1904, Hencky 1924)  Octahedral shear stress (Nadaj 1939)
 Root mean square of the principal shear stresses (Paul 1968)  Root mean square of the shear stresses for all intersection planes (Novoshilov 1952) Novoshilov proved that the mean square value of the shear stresses over all cutting planes is identical to the von Mises stress:
Simbuerger [ 161 applied this new interpretation according to Novoshilov to cyclic multiaxial loading and developed the hypothesis of the effective shear stresses. Zenner [17,21231 has further developed this multiaxial criterion and designated the result as the shear stress intensity hypothesis (SM). In [24] the classical multiaxial criteria, the maximum shear stress criterion and the von Mises criterion, have been derived as special cases of the weakest link theory. On the basis of this analysis, a general fatigue criterion has been formulated for multiaxial stresses. The existing multiaxial criteria of integral approach and of the critical plane approach can be derived as special cases from the general fatigue criterion. On the basis of this analysis of the weakest link theory in 1241, the shear stress intensity hypothesis SIH is newly formulated and further developed. In the newly developed SM, the equivalent shear stress amplitude and the equivalent normal stress amplitude are evaluated as an integral of the stresses over all cutting planes, Fig. 2:
1 Lru AND H. ZENNER
t' Y
Fig. 2. Integration domain of the SIH and stress components in the intersection plane ~ c p The stress amplitudes zwl? and owl? in each cutting plane are calculated from the time function of the stress components. For loading cases with sinusoidal time functions and same frequencies of stress components explicit equations for zwl? and crw can be derived in dependence on the phase shift [24]. The exponents p 1 and p2 can be chosen between 2 and infinity. If very large values are selected for the exponents, the equivalent stress comesponds to the maximal stress over all the cutting planes according to the maximum norm of the algebra. In order to simplify the calculation, the exponents are selected as p ,=p2=2. The equivalent stress amplitude is calculated by combination of the two equivalent stress amplitudes of the shear stress and normal stress (see Eqs (2) and (3)):
The coefficients a and b are determined from the boundary conditions for pure alternating tensioncompression and pure alternating torsion:
From the conditions a>O and b>o, the ranges of validity for the hypothesis S M are defined by the fatigue limit ratio:
Fatigue Limit of Ductile Metals Under Multiaxial Loading
151
For most ductile materials, the fatigue limit ratio is situated within these limits. For extending the range of validity for the hypothesis S M ,the exponents p 1 and p2 can be selected to be greater than two; thus, the values between 1 and 2 can also be included. For the calculation of the equivalent mean stresses, the mean shear stresses are weighted over the shear stress amplitude, and the mean normal stresses over the normal stress amplitude in all cutting planes:
Generally the sign of the mean shear stress is of no importance. A positive or negative mean shear stress has the same effect. Therefor, VI is selected to be exactly equal to 2 for the equivalent mean shear stress. Thus, for a positive or negative mean shear stress, a positive equivalent value is always obtained; that is, the mean shear stress still exerts a reducing effect. For the evaluation of the normal mean stress, the exponent v2 is selected to be equal to unity; consequently, positive and negative mean normal stresses can be distinguished. For considering the effect of the mean stresses, the failure condition can be formulated in different ways. For example, an equivalent mean stress can be obtained by combining the equivalent mean shear stress and equivalent mean normal stress, CT, = rnovrn,, + The equivalent stress amplitude, Eq. (4),and the equivalent mean stress can be compared with the help of a Haighdiagram. In the following, the failure condition is formulated directly by a combination of the equivalent stresses from Eqs (4),(8), and (9):
The coefficients m and n are determined from the fact that the failure condition is fulfilled in the case of both pulsating tension and pulsating torsion:
152
J;
LIU AND H. Z N N E R
In addition, the characteristic strength values for alternating axial loading aw,pulsating axial loading d&hr alternating torsional loading ZW, and pulsating torsional loading Zsch are necessary. For the pulsating torsional strength, the following value is assumed:
EFFECT OF MEAN STRESS If the time functions of the stress components of a plane stress state are synchronous, the following analytical equation can be deduced for the calculation of the equivalent stress amplitude from Eqs (2) to (6):
and for the equivalent mean stresses from Eqs (8) and (9):
The coefficients Aij depend only on the ratio of the stress amplitudes, , a aya,and zVa. The coefficients are indicated in Table 1 . For the case of an alternating normal stress and an alternating shear stress (cyclic bending and torsion, for instance), the failure condition (Eq. (10)) yields the wellknown elliptical equation for ductile materials:
Fatigue Limit of Ductile Metals Under Multiaxial Loading
153
Table 1. Coefficients AQ for the calculation of equivalent mean stresses, x = O,,, y = Ora and z = 5ya
1
1
2
j 2
1
3
4x2+3y24xy+7z2
3x2+4y24xy+7z2
2 x 2 + 2 y 2  3 ~ + 3 z2
x2 + y 2  xy i 32 2
x2 i y 2  ry +32 2
x2 + y 2  xy i 3z 2
10~z6~~ x2 + y 2  xy i 3 z 2
x2 i y 2 xy
7x2 + 7y 2  6xy + 86r2
2
x iy2xy+3z
2
2xy i 422 5x2 + y 2 i 3x2 i 3y2
+ 5 y 2 + 2xy + 4z2 2 3x2 + 3y2 + 2xy + 42 x2
+ 2xy + 422
 6 x z i l0yz
+ 32 2
k Y )z 3x2+3y2+2xy+4z 2
1
=xyad=w 0.80.6
130 test results
0.40
0.20 0
steel (bending&torsion) steel (tension&torsion) AIalloy ellipse equation
I
0.2
, 0.4
I
1
0.6
0.8
1
axad*w Fig. 3. Fatigue limit under alternating normal and shear stresses If one combines the eIIipticaI equation with coIIected test resuIts [25] to yield a standardised diagram, Fig. 3, Eq. (19) then agrees with the test results. For the case of an alternating n o m 1 stress with a superposed static shear stress, the fatigue limit is decreased by the superposed static shear stress, Fig. 4. Up to a static shear stress zVmwhich is lower than the yield strength RPo.2, the influence of the superposed shear stress is correctly described by the SIH. Beyond this value, however, the influence of the superposed shear stress is overestimated. With rxrm> Rpo.2, severe plastic deformations occur; consequently, this case is defined by a static strength design, and is of no importance for practical applications.
1 LIU AND H. ZENNER
154
1
0.4
0.2 0
I
0
0.2
I
I
0.4
0.6
I
0.8
I 1
ZxydRpO,, Fig. 4. Effect of the mean shear stress on the fatigue limit for cyclic normal stress
Fig. 5. Effect of the mean normal stress on the fatigue limit for cyclic shear stress For the case of an alternating shear stress with a superposed static normal stress, the fatigue limit is decreased by the superposed static positive normal stress (tension). A superposed negative static normal stress (compression) increases the fatigue limit to a limiting value, Fig. 5. If the negative static stress exceeds a certain value, the fatigue limit decreases again with increasing compressive mean stress. The influence of the mean stress essentially comprises the effects of the equivalent mean shear stresses and the equivalent mean normal
Fatigue Limit of Ductile Metals Under Multiaxial Loading
155
stresses. The equivalent mean shear stress is still positive and always decreases the fatigue limit. The equivalent mean normal stress can be positive or negative. A negative mean normal stress increases the fatigue strength, and a positive mean normal stress decreases the fatigue strength. In the case of compression, the two effects result in different behaviour of the mean stress; this depends on the ratio of the coefficients m to n (Eq. (10)). As is shown by the test results [27], the behaviour markedly differs within the compression range for different materials. The effect of a superposed static normal stress on the fatigue limit for cyclic normal stress depends on its direction with respect to the cyclic normal stress. As is shown by experiment and calculation in accordance with SM, the effect of a superposed static normal stress is weaker in the direction perpendicular to the cyclic normal stress than in the direction of the cyclic normal stress, Fig. 6.
1
0.4 Oxm Oym
34 Cr4 [28] St60[29]
0
0.2
B 0
0
II
0
U
I
I
I
I
0.2
0.4
0.6
0.8
Y
U
I
1
Oxn/Rpo,2 and OydRp0,2 Fig. 6. Effect of the mean normal stress on the fatigue limit for cyciic normal stress in two different directions
EFFECT OF PHASE DIFFERENCE A phase shift between an alternating shear stress and an alternating normal stress results in a slight increase in fatigue strength, as is predicted by the shear stress intensity hypothesis. The maximal fatigue limit occurs at a phase shift 8 , of 90", and is higher than the synchronous value by approximately 5 per cent, Fig. 7. The fatigue limit diagram is symmetrical with respect to a phase shift of 90".Therefore, the dependence on the phase shift is plotted only in the range from 0" to 90". As is shown by the test results, the situation is far from uniform. During a phase shift, ,a both an increase and a decrease in strength are observed experimentally, Fig. 7. It has hitherto not been possible to prove the existence of a unique material dependence for the relation o,,D( 6 x , , = 9 0 " ) l ~ x a8'y=O"). ~( The S M is usually applicable.
J LIU AND H.ZENNER
156
1.4 1.3 1.2
SIH 0
1.1 1
34Cr4 [28] 42CrM04[30]
c

n
El
0
0.9
0.8
0
0.7
0
25CrMo4[31] steels [32]
0.6
30"
0"
90"
60"
phaseshift tiw
Fig. 7.Effect of a phase shift between a cyclic normal stress and a cyclic shear stress
1.4 1.3
42 CrMo4 [l 11
0
1.2
1.1 1
'1
0.9 0.8
0.7 0"
30"
60"
90"
phaseshift
120"
150"
180"
6,,
Fig. 8. Effect of a phase shift between two cyclic normal stresses For two cyclic normal stresses, only test results with pulsating loads are available, because between two pulsating normal of experimental difficulties. The effect of a phase shift stresses is small within the range from 0" to W",and results in a significant decrease in the fatigue limit within the range from 90" to 180°, Fig. 8. A slight increase in strength is observed in the vicinity of 60". A phase shift 4 =180" between two normal stresses of equal magnitude
4
Fatigue Limit of Ductile Metals Under Multiaxial Loading
157
corresponds to the case of torsional loading. The minimum of the fatigue limit is situated here. The effect of the phase shift b; is correctly described by the SIH.
EFFECT OF FREQUENCY DIFFERENCE In the case of uniaxial loading, the fatigue limit of metallic materials can usually be regarded as frequencyindependent. In the case of multiaxial loading, however, the frequency difference between the stress components plays an important role. In contrast to the influence of the phase shift, considerably less attention has been paid to the experimental behaviour of the fatigue strength in the presence of differences in frequency of the stress components. The effect of the frequency ratio Axy between a shear stress zXuand a n o m 1 stress oxis illustrated in Fig. 9. The plotted curve is not continuous; that is, it applies only to discrete frequency ratios. The points calculated for discrete values of the frequency ratio have been connected with straight lines. As is shown by test results, the fatigue limit is decreased by a frequency difference between the normal and shear stresses. If z,do,, is equal to 0.5, a frequency ratio Ag of 8 reduces the fatigue limit by about 30 per cent. This behaviour is described well by the SM.For Av > 1 as well as for c 1, the behaviour of the fatigue limit is similar; that is, the behaviour of the fatigue limit is independent of whether the frequency of the normal stress is higher or lower than that of the shear stress. A frequency difference between two pulsating normal stresses also reduces the fatigue limit, Fig. 10. However, only two test results obtained at a frequency ratio of 2 are available. At an initial frequency ratio A,, of 2, the largest portion (by approximately 20 per cent) of decrease in fatigue strength is already achieved, for all practical purposes, as is predicted by the furtherdeveloped shear stress intensity hypothesis SIH.
&
4

 1.4 j 1.3 
7
1%
55.
a . D
3n 0fl
1.2 1.1
SIH

34 Cr4 1281 0
25CrMo4[31]
1 
0.9

0.8 0.7 0.6
0
0
5
I
0.1
1
10
frequency ratio Fig. 9. Effect of a frequency difference between a cyclic normal stress and a cyclic shear stress
1 LEU AND H.ZENNER
158
SIH 1.2
1.1
0.6
0
1
0
34Cr4 [28] St35 [ l l ]
!
I
I
I
I
I
I
I
1
2
3
4
5
6
7
8
frequency ratio
$
Fig. 10. Effect of a frequency difference between two pulsating normal stresses
EFFECT OF WAVE FORM The wave form of the stress components does not generally affect the endurance limit, if loading is uniaxial, or if the stress components oscillate proportionally or synchronously to each other. This result is also predicted by the SM.This conclusion has been experimentally confirmed [11,31]. However, the wave form does affect the fatigue limit if the stress components do not oscillate synchronously to one another. The influence of a phase shift 8 , between a cyclic noma1 and a cyclic shear stress with different wave forms is shown in Fig. 11. In contrast to the sinusoidal wave form, the effect of a phase shift b&, on the endurance limit is more pronounced for the other two wave forms. A phase shift between an alternating normal and an alternating shear stress with a triangular wave form will increase the fatigue limit to a greater extent than with a sinusoidal wave form. If the influence of the phase shift between two normal stresses is compared for different wave forms, it is obvious that the effect of the wave form is not remarkable at a phase shift of 180", Fig. 12. For different wave forms, significant differences exist in the phase shift range between 30" and 150". In the case of a trapezoidal wave form, even a small phase shift of only 30" results in a significant decrease in fatigue limit. With a sinusoidal wave form, in contrast, a remarkable decrease in fatigue limit begins from a phase shift of 60". In the case of cyclic normal stresses with a triangular wave form, the decrease in fatigue limit can be assumed to begin at an even larger value of the phase shift 4.
* Fatigue Limit of Ductile Metals Under Multiaxial Loading
1.4 1.3
1.2 1.1 1 0.9
0.8 0.7 nc
u.u
zxya=o.5axa RFRq1
triavgle
+
sinus
trapez
I
I
0"
30"
60"
I
90"
phaseshift Zxv
Fig. 11. Effect of wave forms and phase shift between an alternating normal stress and an alternating shear stress, test results from [28]
Fig. 12. Effect of the wave form and phase shift between two pulsating normal stresses, test results from [1 11
159
1 LIU AND H. ZENNER
I60
STATISTICAL EVALUATION As a check on the accuracy, the ratio x of the experimentallydetermined value to the calculated
value of the fatigue limit is employed for such an evaluation:
where o ~ , denotes ~ ? the estimated average value of the experimentally determined in accordance with SM, the estimated average fatigue limit. For the calculation of axd.cal values of w a n d OS& are taken as a basis. At x = 1, the result of the calculation is equal to the experimental result. At x>l, the fatigue limit with the SM is underestimated; at 6 1 , the fatigue limit is overestimated. The results of the above statistical evaluation are listed in Table 2 for different load cases: the average value Y. and the standard deviation s of the ratio x are shown. In the case of a correct prediction, the average value of the ratio x should be equal to about the unity and the standard deviation should be very small. 99.99 99.9
182 test series
f
99 95
P
I%?
90
E;
50
30 20 10
5 1
.l
.01
Fig. 13. Comparison between the experimentally determined fatigue limit and the calculated one, as is predicted by the furtherdevelopedSM The results of 182 test series with a maximal von Mises stress lower than 1.1Rpo,2 are considered. The 130 test results from Fig. 3 for the load case where a normal stress and a shear
Fatigue Limit of Ductile Metals Under Multiaxial Loading
161
stress alternate without mean stresses are not considered in the statistical evaluation. As is shown in Fig. 3, the prediction according to S M with the elliptic equation is very good for this simple load case. For each load case, the SIH provides a good prediction of the fatigue limits. The average values X are still near the unity and the standard deviations s are small. A greater scatter is or zVm, and estimated only for the load case where one shear stress alternates with,,a oym for the load case where two normal stresses alternate with phase difference and mean stresses different from zero. The statistical distribution of the ratio x for all 182 test results is plotted in Fig. 13. For the cases considered, the furtherdeveloped shear stress intensity hypothesis SIH provides an accurate prediction of the fatigue limit. The ratio x of the experimental fatigue limit to the calculated fatigue limit has an average value equal to about 1.0, and ranges between 0.8 and 1.2. With these results, 90 per cent of the values are within the range between 0.85 and 1.1. The standard deviation s is equal to 0.067. Table 2. Comparison between experimentally determined fatigue limit and calculated one according to the SIH, for different load cases. Femtic and ferritic perlitic steel, ultimate steel Rm = 400 to 1600 MPa, n  number of test series (maximum von Mises stress < 1,l Rpo,2); 3E mean value, s  standard deviation Load case
n
7
oxalternating orland zxym with om,oym
26
1.003
with oXm, oym or rxym
14
oxand zxyalternating with ox,, or,,or , zVm and with phase difference
xmax
Xmin
0.063
1.088
0.864
0.944
0.077
1.059
0.820
72
0.988
0.050
1.162
0.849
12
0.905
0.055
1.022
0.842
29
0.988
0.092
1.168
0.860
oxand oralternating, with different frequencies
I2
1.031
0.028
1.089
0.970
ox,zq and cyalternating, orland, ,z with o,,, oVm
17
0.993
0.055
1.108
0.912
All results
182
0.984
0.067
1.168
0.820
S
zxyalternating
~~~~~~~
oxand
~~~
zxyalternating,
sinusoidal with different frequencies and nonsinusoidal oxand oyalternating, with ox,,,,oYm and with phase difference
162
J
Lru AND H.ZENNER
CONCLUSIONS The furtherdeveloped shear stress intensity hypothesis (SIH) provides a useful prediction of the fatigue strength for multiaxial loading. It should be pointed out that deviations between experiment and calculation are not due only to the multiaxial criteria [34]. Possible reasons for deviations include anisotropic material behaviour (texture), nonstatistical nature of the fatigue limit, and experimental error. REFERENCES 1. 2. 3.
4. 5.
6. 7. 8. 9.
10. 11. 12. 13. 14. 15.
Marin, J. (1942). Interpretation of Experiments on Fatigue Strength of Metals Subjected to Combined Stresses. Weld. J. 24,245. Sines, G. (1955). Fatigue of Materials under Combined Repeated Stresses with Superimposed Static Stresses. Technical Note 2495, NACA, Washington D. C. Kakuno, K. and Kawada, K. (1979). A New Criterion of Fatigue Strength of a Round Bar Subjected to Combined Static and Repeated Bending and Torsion. Fatigue Fract. Engng. Mater. Struct. 2,229. Lee, S. B. (1985). A Criterion for Fully Reversed Out of Phase Torsion and Bending. In: Multiaxial Fatigue, pp. 553568, Miller, K. J. and Brown, M. W.(Eds). ASTM STP 853, Philadephia Mertens, H. (1988). Kerbgrundund Nennspannungskonzepte zur Dauerfestigkeitsberechnung Weiterentwicklung des Konzeptes der Richtlinie VDI 2226. VDIBerichte Nr. 661, pp. 125 Mertens, H. and Hahn, M. (1993). Vergleichsspannungshypothesezur Schwingfestigkeit bei zweiachsiger Beanspruchung ohne und mit Phasenverschiebungen. Konstruktion 45, 196 Hahn, M. ( 1995). Festigkeitsberechnung und Lebensdauerabschatzung f i r metallische Bauteile unter mehrachsig schwingender Beanspruchung. Diss. TU Berlin Liipfert, H. P. (1994). Beurteilung der statischen Festigkeit und Dauerfestigkeit metallischer Werkstoffe bei mehrachsiger Beanspruchung. Habilitationsschrift TU Bergakademie Freiberg McDiarmid, D. L. (1973). A Criterion of Fatigue Failure under Multiaxial Stress. Research Mem. No. NL54, The City University London Nokleby, J. 0. (1981). Fatigue under Multiaxial Stress Conditions. Report MD81 001, Div. Masch. Elem., The Norw.Institute of Technology, TrondheirdNorwegen Bhongbhibhat, T. (1986). Festigkeitsverhalten von Stiihlen unter mehrachsiger phasenverschobener Schwingbeanspruchung mit unterschiedlichen Schwingungsformen und Frequenzen. Diss. Uni. Stuttgart Troost, A., Akin, 0. and KIubberg, F. (1987). Dauerfestigkeitsverhalten metallischer Werkstoffe bei zweiachsiger Beanspmchung durch drei phasenverschoben schwingende Lastspannungen. Konstruktion 39,479 Sonsino, C. M. and Grubisic, V. (1987). Multiaxial Fatigue Behaviour of Sintered Steels under Combined In and Out of Phase Bending and Torsion. 2. Werkstofftechn. 18, 148 Dang Van, K., Griveau, B., Message, 0.(1989). On a new multiaxial fatigue criterion: Theory and application . In EGF 3, Biaxial and Multiaxial Fatigue, Brown, M.W. and Miller, K.J. (Eds), 459 Carpinteri, A. and Spagnoli, A. (2001). Multiaxial highcycle fatigue criterion for hard metals. Int. J. Fatigue 23 (2), 135
Fatigue Limit of Ductile MetaZs Under Multiaxial Loading
16.
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Simburger, A. ( 1975). Festigkeitsverhalten &er Werkstoffe bei einer mehrachsigen, phasenverschobenen Schwingbeanspruchung mit kerperfesten und veranderlichen Hauptspannungsrichtungen. Diss. TH Darmstadt 17. Zenner, H. and Richter, I. (1977). Eine Festigkeitshypothese fur die Dauerfestigkeit bei beliebigen Beanspmchungskombinationen. Konstruktion 32,11 18. Papadopoulos, I. V. (1994). A New Criterion of Fatigue Strength for OutofPhase Bending and Torsion of Hard Metals. Int. J. Fatigue 16,377 19. Papadopoulos I.V., Davoli, P., Gorla, C., Filippini, M. and Bernasconi, A. (1997). A comparative study of multiaxial highcycle fatigue criteria for metals. Int J Fatigue 19, 219 20. Novoshilov, V. V. (1961). Theory of Elasticity (J. J. Sherrkon trans.). Jerusalem Israel Program for Scientific Translation 21. Zenner, H., Heidenreich, R. and Richter, 1. (1980). SchubspannungsintensitatshypotheseErweiterung und experimentelle Abstutzung einer neuen Festigkeitshypothese fiir schwingende Beanspruchung. Konstruktion 32,143 22. Zenner, H. (1988). Dauerfestigkeit und Spannungszustand. VDIBerichte Nr. 661, pp. 151186 23. Zenner, H. and Liu, J. (1989). Berechnung der Dauerschwingfestigkeit bei mehrachsiger Beanspmchung. WGMKTagung, Clausthal 24. Liu, J. (1997). Weakest Link Theory and Multiaxial Criteria. In: Proc of the 5thInt Conf on BiaxialMultiaxial Fatigue and Fracture, pp, 4562, Cracow 25. Liu, J. (199 1). Beitrag zur Verbesserung der Dauerfestigkeitsberechnung bei mehrachsiger Beanspruchung. Diss. TU Clausthal 26. Baier, F.J. (1970). &it und Dauerfestigkeit bei uberlagerter statischer und schwingender ZugDruck und Torsionsbeanspruchung. Diss. Uni. Stuttgart 27. Bergmann, J. W. ( 1988). Werkstoffdauerfestigkeit 11. FVV Forschungsberichte, Heft 410 28. Heidenreich, R., Zenner, H. and Richter, I. (1983). Dauerschwingfestigkeit bei mehrachsiger Beanspruchung. Forschungshefte FKM, Heft 105 29. ElMagd, E. and Mielke, S. (1977). Dauerfestigkeit bei uberlagerter zweiachsiger statischer Beanspruchung. Konstruktion 29,253 30. Lempp, W. (1977). Festigkeitsverhalten von Stahlen bei mehrachsiger Dauerschwingbeanspruchung durch Norrnalspannungen mit uberlagerten phasengleichen und phasenverschobenen Schubspannungen. Diss. Uni Stuttgart 31. Mielke, S. (1980). Festigkeitsverhalten metallischer Werkstoffe unter zweiachsig schwingender Beanspmchung mit verschiedenen Spannungszeitverlaufen. Diss. RWTH Aachen 32. Nishihara, T. and Kawamoto, M. (1945). The Strength of Metal under Combined Alternating Bending and Torsion with Phase Difference. Mem. of the College of Engng., Kyoto Imperial University, 11,85 33. Issler, L. (1973). Festigkeitsverhalten metallischer Werkstoffe bei mehrachsiger phasenverschobener Schwingbeanspruchung. Diss. Uni. Stuttgart 34. Zenner, H., Simburger, A. and Liu, J. (2000). On the fatigue limit of ductile metals under complex multiaxial loading. Int J Fatigue 22, 137
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Appendix: NOMENCLATURE coefficients of SIH static yield strength of 0.2% plastic strain phase differences to ox frequency ratio to ox exponents of SIH mean stresses stress amplitude fatigue limit for normal stress equivalent stress amplitude equivalent mean stresses fatigue limit (double amplitude) for pulsating tension fatigue limit for alternating tensiontompression normal stress amplitude in the intersection plane p mean normal stress in the intersection plane p fatigue limit for pulsating torsion (double ampIitude) fatigue limit for shear stress fatigue limit for alternating torsion shear stress amplitude in the intersection plane mean shear stress in the intersection plane p
w
BiaxiallMultiaxial Fatigue and Fracture Andrea Carpinteri, Manuel de Freitas and Andrea Spagnoli (Eds.) Published by Elsevier Science Ltd. and ESIS.
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SEQUENCED AXIAL AND TORSIONAL CUMULATIVE FATIGUE: LOW AMPLITUDE FOLLOWED BY HIGH AMPLITUDE LOADING Peter BONACUSE US Army Research Laboratory, NASA Glenn Research Center, Brook Park, OH, USA and Sreeramesh W L U R I Ohio Aerospace Institute, NASA Glenn Research Center, Brook Park, OH, USA
ABSTRACT The experiments described herein were performed to determine whether damage imposed by axial loading interacts with damage imposed by torsional loading, This paper is a follow on to a study [ 11 that investigated effects of loadtype sequencing on the cumulative fatigue behavior of a cobalt base superalloy, Haynes 188, at 538°C. Both the current and the previous study were used to test the applicability of cumulative fatigue damage models to conditions where damage is imposed by different loading modes. In the previous study, axial and torsional two load level cumulative fatigue experiments were conducted, in varied combinations, with the lowcycle fatigue (high amplitude loading) applied first. In present study, the low amplitude fatigue loading was applied initially. As in the previous study, four sequences (axial/axial, torsion/torsion, axial/torsion, and torsion/axial) of two load level cumulative fatigue experiments were performed. The amount of fatigue damage contributed by each of the imposed loads was estimated by both the PalmgrenMiner linear damage rule (LDR) and the nonlinear, damage curve approach (DCA). Life predictions for the various cumulative loading combinations are compared with experimental results. Unlike the previous study where the DCA proved markedly superior, no clear advantage can be discerned for either of the cumulative fatigue damage models for the loading sequences performed. In addition, the cyclic deformation behavior under the various combinations of loading is presented. KEYWORDS Multiaxial, cumulative fatigue, axial loading, torsional loading, tubular specimens.
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INTRODUCTION Many multiaxial fatigue damage models are based on the premise that damage is a tensor, Le., it has both magnitude and direction. These ‘tensorial’ approaches imply that damage imposed by loading in one direction does not readily interact with subsequent damage imposed by loading in another direction. By subjecting thinwalled tubular specimens to fatigue in different loading directions, in sequence, this hypothesis can be tested. In this study, a block of lower amplitude loading, at a given fraction of the estimated life, preceded a second block of higher amplitude loading to failure. This work is a complement to a previous study [I] where higher amplitude cycles were imposed initially, followed by lower amplitude loading to failure. Various combinations of axial and torsional loadtype and loadsequence interactions have been explored in both studies. The most common method of accounting for the fatigue damage accumulated in a material subject to variable amplitude loading is to estimate the fraction of the fatigue life expended for each cycle of a given amplitude and then sum up all the fractions for each of the load excursions. When this sum reaches unity, the material is said to have used up its available life. This model is commonly referred to as the linear damage or PalmgrenMiner rule (LDR) [2]. However, many studies have shown LDR to be inaccurate by as much as an order of magnitude for certain combinations of variable amplitude loading [361. Halford [3] outlines the uniaxial loading combinations where the LDR is likely to break down and where alternative approaches may prove useful. In the present study, two cumulative fatigue damage models are assessed for their ability to predict the remaining cyclic life after prior loading at both different amplitudes and different loading directions. The first model is the LDR [2]. The second is the damage curve approach (DCA) of Manson and Halford [7]. In the previous study (high amplitude loading followed by low amplitude loading), the damage curve approach (DCA) was found to model the load interaction behavior remarkably well for all the cases investigated including the mixed load experiments (axial followed by torsional and torsional followed by axial). This seemed to indicate that the fatigue damage was most likely isotropic, at least when the loading sequence was high amplitude followed by low amplitude. A possible explanation for apparent isotropic nature of damage accumulation under fulIy reversed fatigue loading is that the cyclic hardening behavior of superalloys is, at least to some extent, isotropic in nature. This hardening should influence damage accumulation even when loading is imposed in another direction. The magnitude of plastic deformation is often used as a measure of the accumulated damage in a loading cycle. If the material is in a work hardened state from previous mechanical cycling it should be able to absorb a larger portion of ensuing deformations elastically, resulting in lower plastic strains. Thus, a smaller increment of damage wouId then be accrued in each subsequent cycle. This mechanism would be in competition with propagating cracks that may have initiated in the prior cycling; higher stresses due to work hardening would increase the crack propagation rate in the subsequent loading.
MATERIAL, SPECIMENS, AND TEST PARAMETERS
Specimens used in this study were fabricated from hot rolled, solution annealed, Haynes 188 superalloy, 50.8 mm diameter bar stock (heat number: 1188061714). This is the same heat of material used to perform the experiments described in Ref. [I]. The measured weight
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percent composition of the superalloy was: e 0.002 sulfur, 0.003 boron, < 0.005 phosphorus, 0.052 lanthanum, 0.09 carbon, 0.35 silicon, 0.8 manganese, 1.17 iron, 14.06 tungsten, 22.1 1 chromium, 22.66 nickel, with the balance made up of cobalt. AI1 experiments were performed on thin walled tubes with nominal gage section dimensions of 26 mm outer diameter, 22 mm inner diameter, 41 mm straight section and 25 mm gage length. The interior surfaces of the tubes were honed in an attempt to preclude crack initiation on the inner surface of the specimen. Outer surfaces were polished with final polishing direction parallel to the specimen axis. Further details on the specimen geometry and machining specifications can be found in Ref. [8]. The baseline axial and torsional fatigue lives for this material, specimen geometry, and test temperature can be found in Ref. [I]. The specimens were heated to 538°C with an induction heating system. A11 specimens were subjected to sequential constant amplitude fatigue loading under strain control. A commercially available, watercooled, biaxial, contacting extensometer with a 25 mm gage length, designed specifically for axialtorsion testing, was used. The loading actuator that was not being used for fatigue strain cycling (the torsional actuator during axial cycling or the axial actuator during torsional cycling) was maintained in load control at zero load. This procedure allowed strains to accumulate in the zero load direction. During the axial strain cycling segments, relatively small mean strains in the load controlled torsional axis were observed. However, the torsional strain cycling segments always showed increasing mean axial strains. When torsional strains were applied in the first segment, the magnitudes of these axial strains were recorded and then electronically set to zero prior to commencing the second loading segment. The specimen failure criterion programmed into the testing software was a 10% drop in the measured load in the strain controlled direction. Five experiments were terminated due to a controller interlock. Details on the testing system and test control procedures can be found in Ref. [l].
TEST MATRIX The test matrix for this study is shown in Table 1. Seventeen different combinations of low amplitude followed by high amplitude, two load level experiments were performed. The loading sequences were axial followed by axial (axiallaxial), torsion followed by torsion (torsionltorsion), axial followed by torsion (axialltorsion), and torsion followed by axial (torsionlaxial), with at least four different, first load level life fractions imposed in each combination. A fifth life fraction was imposed in the torsion/torsion subset. One torsional experiment was repeated as a cursory check on the expected specimentospecimen variability in fatigue life. This summed to a total of 18 tests performed for this study. Table 1 also contains the stress range and mean stress at halflife for each load level, the number of cycles imposed, and the final crack orientation.
TABLE 1: Test matrix and interaction fatigue data for Haynes 188 at 538°C.
Specimen
AEI
First Load Level; v = 0.5 Hz AOI Om1 A ~ I A'CI Tml nl (MPa) (MPa) (MPa) (MPa) (Cycles)
Axi al/Ax i a1 ... ... HYII103 0.0067 811 9 ... ... HYII116 0.0066 849 6 ... ... HYII119 0.0066 882 10 HYII114 0.0066 905 9 ... .*. TorsionITorsion ... ... ... 0.0120 515 HYII117 ... ... HYII112 ... 0.0120 536 HYII115 ... ... ... 0.0119 554 ... ... ... 0.0121 521 HYII 109 ... ... HYII118 ... 0.0119 588 ... ... ... 0.0120 579 HYII104 AxiaVTorsion ... ... HYII110 0.0069 841 10 HYIIIll 0.0069 862 ... ... 8 HMIO8 0.0065 892 ... ... 8 HYII105 0.0066 888 9 ... ... Torsion/Axial HYII120 ... ... ... 0.0121 498 HYII 107 ... ... ... 0.0120 523 HYII106 ... ... ... 0.0119 540 HYII113 ... 0.0119 581 ... ... * Angle measured with respect to the specimen axis.
AE2
Second Load Level; v = 0.1 Hz Om2 AYZ ATz 7m2 n2 Crack* (MPa) (MPa) (MPa) (MPa) (Cycles) Orientation
... ... ... ...
3926 7851 15702 23553
0.0203 0.0202 0.0203 0.0205
1254 1247 1244 1267
14 14 11 12
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... ...
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5857 11714 23427 23427 35141 40998
... ...
...
... ... ...
0.0345 0.0349 0.0346 0.0347 0.0347 0.0349
... ... ... ...
3926 7851 15702 23553
...
...
0 1 0 1
5857 11714 23427 35141
0.0201 0.0203 0.0200 0.0204
...
...
... ...
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781 761 794 783
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1400 1414 1432 1452
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CUMULATIVE DAMAGE MODELS The results of the cumulative fatigue experiments performed for this study were compared with predictions of two load interaction models: the LDR (Eq. 1) [23, and the DCA (Eq. 2) [7].
($1 ($1 = I

The LDR assumes that damage accumulated during each load excursion can be simply added to the already accumulated damage in the material. Load sequence and changes in the properties of the material are not taken into account. The LDR has the distinct advantage of being straight forward to implement for virtually all loading histories, provided sufficient baseline fatigue data are available and an adequate cycle counting method is employed. The DCA attempts to model the observed nonlinear interactions between two loadlevel experiments. The underlying assumption of the DCA is that high amplitude loading initiates cracking early in life whereas under lower amplitude loading measurable cracking does not occur until late in life. In the case of the experiments performed in this study, the implication is that the initially imposed lower amplitude loading might not initiate cracks. In the subsequent higher amplitude loading the material might then ‘ignore’ the previous cycling or even derive a benefit from it, allowing the sum of life fractions to be greater than unity.
RESULTS AND DISCUSSION The initially imposed lower amplitude cyclic loading (0.65% axial and 1.24% torsional nominal strain ranges) had sufficient plasticity to cause this solutionannealed material to isotropically harden. The magnitude of the hardening in the first load level was proportional to the number of imposed cycles. The cyclic hardening behavior for the lower amplitude, first load level, loading is presented in Fig. 1. The horizontal lines in each of these figures correspond to stress range for the last cycle of the lower amplitude loading. The vertical drop lines indicate the last cycle of the first load level for each experiment. There was some variation in the first cycle stress range for both the axial and torsional loading. In the first load levels, the average axial first cycle stress range (the left most data point in each of the curves in Fig. I (a) and (c)) was 646 MPa with a standard deviation of 15 MPa, while the torsional first cycle average stress range (the left most data point in each of the curves in Fig. 1 (b) and (d)) was 382 MPa with a standard deviation of 14 MPa. The most likely explanations for these variations include the natural variability in the material properties and discrepancies in the machining andlor measurement of the specimen gage section. A 0.1% error in the measurement of a gage section dimension (inner or outer radius), which is the approximate precision of the micrometers employed, would lead to a 0.2% error in the axial stress and a 0.4% error in the shear stress calculations. This dimensional measurement error would account for only about 10% of the variability observed. The specimen to specimen variation in the hardening rate, however, in all the axial and torsional experiments was small.
I? BONACUSE AND S. KALLURI
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The stress range vs. cycle number plots for the higher amplitude, second load level (Fig. 2) show some interesting behavior. The lower initial life fraction experiments continued to cyclically harden during the second load level. The higher initial life fraction experiments tended to cyclically soften (with exception of the axialaxial where all hardened to failure). The material in each type of load level combination tended to converge to a similar stress range as the cycles accumulated in the second load level, independent of the number of lower amplitude cycles previously imposed. The axiaYaxial and torsiodtorsion experiments tended to stabilize to the same stress levels as the baseline (constant amplitude fatigue tests performed at the second load level) experiments (also plotted in Fig. 2), whereas the mixed loading experiments, axial/torsion and torsionlaxial, stabilized at stress levels 6.5% and 12.0% above the baseline experiments, respectively. The extra hardening observed in the mixed loading experiments may be attributable to the same mechanism that causes additional hardening in mechanically outofphase multiaxial loading [9]. Experiments ending with torsional loading always failed on the maximum shear strain plane, i.e. the final failure cracks were parallel to the specimen axis. Experiments completed under axial loading all failed at or near the plane of maximum tensile stress; perpendicular to the specimen axis. In the tests completed with axial loading, the 10% load drop failure criteria corresponded to an average crack length of 20.5 mm. Somewhat longer surface cracks (27.4 mm average length) occurred in the tests completed with torsional loading. This result is not unexpected in that shear cracks (cracks that form and propagate on maximum shear stress’strain planes) tend to be longer at failure than cracks that form and propagate on maximum normal stress/strain planes [lo]. Five of the eighteen tests were terminated due to controller interlocks. Controller interlocks are preprogrammed limits on: load, strain, and displacement. These limits were typically set to approximately 10%above or below the expected maximum values of the measured variables. An interlock can also occur when the difference between the command and feedback signal in the control loop reaches a preprogrammed threshold, which was 15%of the commanded strain for these experiments. A controller interlock shuts off hydraulic pressure and sends a signal to the control software to indicate that the test has been terminated. Larger final cracks with significant ductile tearing and specimen distortion were associated with the controller interlocks. The length of the cracks that propagated in fatigue, as identified on the fracture surfaces, were of the same order as those where the experiments ended at the 10% load drop. The difference in the number of cycles to failure associated with the interlock events, as compared to those terminated at a 10%load drop, is believed to be small. The results of the axiaYaxial, torsion/torsion, axial/torsion, and torsion/axial experiments are compared with the predictions of the LDR and the DCA in Fig. 3. It’s clear that neither cumulative damage model appears to predict the observed behavior adequately. In most cases, the LDR seems to more closely approximate the observed behavior for lower initially applied life fractions (nl/NI 5 0.4),whereas the DCA model seems to do better with the higher (nl/N1 > 0.4) initially applied life fractions. Figure 4 displays this cumulctive fatigue data with the applied life fraction in the first load level on the horizontal axis and the observed sum of life fractions on the vertical axis. The horizontal line at 1.0 depicts where the data would fall if the LDR perfectly predicted the damage interaction. Again, the deviation from the LDR as the imposed life fraction increases is readily apparent. Figure 5 shows a comparison between the predicted and observed remaining cycle life, n2. for (a) the LDR and (b) the DCA. Both models, in general, predicted fatigue life within the expected experimental scatterband (factors of two of the average life) for LCF. However, the
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Sequenced Axial and Torsional Cumulative Fatigue: ...
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LDR under predicted most of the experimental results, whereas the DCA tended to over predict the life. This finding would seem to favor the LDR for loading conditions similar those imposed in this study. A minimum of 3 experiments for each of the load interaction conditions would be needed to separate the effect of experimental scatter from 'real' load interaction effects. Unfortunately, the resources allotted for this study did not permit an investigation with this level of detail.
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Sequenced Axial and Torsionai Cumulotive Fatigue: ...
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The equivalent axial plastic strain for torsional loading is given in Eq. 5:
Figure 6 tends to support the assertion that the extent of work hardening can affect the subsequent rate of damage accumulation. The equivalent plastic strain range in the second load level is plotted against the sum of life fractions in Fig. 7. The plastic strain ranges in this plot were calculated at the approximately the mid point in the second load level. There seems to be no correlation between the plastic strain range at ‘halflife’ in the second load level and the sum of life fractions. For each combination of load types the ‘stabilized’ equivalent plastic strain range is approximately the same with the exception of the torsion/axial loading where it is slightly lower than the three other loading combinations.
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J 0
a
3
0.0030
Torsion/Torsion AxialITorsion Torsion/Axial
A
c
2
00
i i ‘_ 0.0025 wa 3 0
w
0.0000 0.6
I
0.8
1 .o
1.2
1.4
Sum of Life Fractions (n,/N,
1.6
1.a
+ nJN,)
Fig. 6. Equivalent plastic strain range at the end of the first load level vs. the sum of life fractions. In a literature review of cumulative fatigue modeling by Fatemi and Yang [ I l l , many approaches to account for the observed deviations from the linear damage rule were enumerated. The conclusion of the review is that none of the reviewed models has proven to be universally applicable because each of the models only accounted for, at most, a few of the “phenomenological factors”. Fatemi and Yang grouped the reviewed cumulative damage models into the following six categories: 1) linear damage accumulation, 2) nonlinear damage accumulation, 3) life curve modifications,for load interactions, 4) crack growth approaches, 5) continuum damage models, and 6 ) energy based methods.
f? BONACUSE AND S.KALLURI
180
For many of these model categories, the experimental results discussed in this paper present various difficulties, both in interpretation and implementation. Both the LDR and the DCA have been shown to be less than ideal predictors of the observed interaction behavior. As Haynes 188 has been shown not to have an endurance limit, many of the life curve modification models that incorporate this concept would tend to be inaccurate in many regimes. Crack growth approaches may not prove useful in describing this data set in light of the observed damage accumulation behavior in the mixed loading experiments. The final failure crack direction is only a function of the final loading direction (shear cracking for the axidtorsion experiments and maximum normal stress cracking for the torsiodaxial experiments). The axial/torsion and torsiodaxial experiments seem to closely follow the LDR (with the exception of the highest applied life fraction in the axidtorsion experiments). The mixed loading experiments performed in this study seem to indicate that for Haynes 188 at 538"C, there is no unusual interaction under the conditions imposed. Models that account for damage on the basis of accumulated plastic work implicitly incorporate the effect of strain hardening on the damage induced in the material (strain hardening materials will exhibit more plastic work at similar plastic strain ranges or less plastic work at similar total strain ranges). Because the accumulated plastic work at failure for most materials has been shown to be variable depending on the loading conditions [12], this class of models may not accurately predict the accumulated damage.
a, $
0.0160 I
m
0 I
0.0150
0
a%
0
0.0140
0 0 '
A
w"
a .sf'
*
0.0120
A
CrJ
a .>
s
lz
O
V
0.0130
Gi 0 .c
O
A
Torsionflorsion AxiaVTorsion Torsion/Axial
.. 7
0.0000 0.6
0.8
1.o
1.2
1.4
Sum of Life Fractions (n,/N,
1.6
1.8
+ n.JN2)
Fig. 7. Equivalent plastic strain range in the second load level at the approximate midlife point vs. the sum of life fractions There is a very good chance that competing mechanisms, one linked to the cyclic deformation behavior and the other linked to crack growth, which may be modeled well individually but not in a combined sense, are the root of the discrepancies observed in the literature. Energy and plastic strain based methods do better in modeling crack 'initiation' and
Sequenced Axial and Torsional Cumulative Fatigue: ...
181
early growth and may therefore predict cumulative damage accumulation where observable cracking does not appear until late in the cyclic life. However, methods based on crack propagation arguments would seem to be more likely to produce good correlations where significant cracking develops early in life. Some method of transitioning between these regimes is required for reliable estimates of Cumulative fatigue damage under complex loading conditions.
CONCLUSIONS The following conclusions are drawn from this study of the cumulative fatigue behavior of Haynes 188 at 538°C: 1) Neither the LDA nor the DCA have predicted the interactions in these experiments
particularly well. Predictions by the LDA have been generally conservative (under predicted life), whereas the DCA predictions have been generally nonconservative (over predicted life). 2) There appears to be a transition that occurs at or about an initially imposed life fraction of 0.4 in the set of experiments conducted. Below this level the material adheres to a linear damage rule. Above this level, the damage accumulation appears to be nonlinear. 3) The fatigue life data indicate a transition behavior that is linked to the extent of work hardening imposed by the initial load level: the greater the life fraction expended in lower amplitude loading, the greater the chance of having a sum of life fractions more than unity. 4) The stabilized (halflife) stress response in the second load level does not show any correlation with damage accumulation for the loading conditions imposed.
REFERENCES 1.
2.
3. 4.
5.
6.
Kalluri, S. and Bonacuse, P. J. (2000), “Cumulative Axial and Torsional Fatigue: An Investigation of LoadType Sequencing Effects,” In: STP 1387  Multiaxial Fatigue and Deformation: Testing and Prediction, pp. 281301, S. Kalluri and P. J. Bonacuse (Eds), ASTM, West Conshohocken, PA. Miner, M. A. (1945), “Cumulative Damage in Fatigue,” Journal ofApplied Mechanics 12, No. 3 , (Trans. ASME, Vol. 67), pp. A159A164. Manson, S. S. and Halford, G. R. (1981), “Practical Implementation of the Double Linear Damage Rule and Damage Curve Approach for Treating Cumulative Fatigue Damage,” Int. J. Fracture 17, pp. 169192. BuiQuoc, T., (1982), “Cumulative Damage with Interaction Effect due to Fatigue Under Torsion Loading,” Experimental Mechanics, pp. 180187. McGaw, M. A., et al., (1993). “The Cumulative Fatigue Damage Behavior of MarM 247 in Air and High Pressure Hydrogen,” In: ASTM STP 1211, Advances in Fatigue Lqetime Predictive Techniques: Second Volume, pp. 117131, M.R. Mitchell and R. W. Landgraf, (Eds.), ASTM. Halford, G. R., (1997), “Cumulative fatigue damage modeling  crack nucleation and early growth,” Int. J. Fatigue 19, Supp. No. 1, pp. S253S260.
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7. Manson, S . S . , and Halford, G. R. (1986), “Reexamination of Cumulative Fatigue Damage Analysis  An Engineering Perspective“, Engineering Fracture Mechanics 25, NOS.5’6, pp. 539571. 8. Bonacuse, P. J., and Kalluri, S. (1993), “AxialTorsional Fatigue: A study of Tubular Specimen Thickness Effects,” JTEVA 21,160. 9. Bonacuse, P. J. and Kalluri, S. (1994), “Cyclic AxialTorsional Deformation Behavior of a CobaltBase Superalloy,” Cyclic Deformation, Fracture, and Nondestructive Evaluation of Advanced Materials: Second Volume, ASTM STP 1184, pp. 204229, M. R. Mitchell and 0. Buck (Eds.), ASTM, West Conshohocken, PA. 10. Bonacuse, P. J. and Kalluri, S . (1995), “Elevated Temperature Axial and Torsional Fatigue Behavior of Haynes 188,“ J. of Engineering Materials and Technology 117,pp. 191199. 11. Fatemi, A. and Yang, L. (1998), “Cumulative Fatigue Damage and Life Prediction Theories: a Survey of the State of the Art for Homogeneous Materials,” Int. J. Fatigue 20, NO. 1, pp. 934. 12. Halford, G. R. (1966), “The Energy Required for Fatigue,” J. of Materials 1, No. 1, pp.318. Appendix : NOMENCLATURE Elastic Modulus Shear Modulus Number of cycles to failure for first load level Number of cycles to failure for second load level Number of cycles imposed during first load level Actual number of cycles to failure during second load level Imposed strain ranges for load levels 1 and 2 Total measured axial strain range Plastic axial strain range Equivalent plastic strain range Plastic shear strain ranges for load levels 1 and 2 Plastic shear strain range Total measured shear strain range Axial stress range Axial stress ranges for load levels 1 and 2 Axial mean stresses for load levels 1 and 2 Shear stress range Shear stress ranges for load levels 1 and 2 Mean shear stresses for load levels I and 2
BiaxiallMultiaxiaI Fatigue and Fracture Andrea Carpinten, Manuel de Freitas and Andrea Spagnoli (Eds.) Q Elsevier Science Ltd. and ESIS. All rights reserved.
183
ESTIMATION OF THE FATIGUE LIFE OF HIGH STRENGTH STEEL UNDER VARIABLEAMPLITUDE TENSION WITH TORSION: USE OF THE ENERGY PARAMETER IN THE CRITICAL PLANE Tadeusz EAGODA', Ewald MACHA', Adam NIESEONY' and Franck MOREL* 1 Technical University of Opole, ul.Mikolajczyka 5, 45271 Opole, Poland ENSMA, Futuroscope, France
ABSTRACT
The paper concerns application of the energy parameter, being a sum of the elastic and plastic strain energy density in the critical plane, for describing experimental data obtained in fatigue tests of 35NCD16 steel, subjected to constant amplitude tensioncompression, torsion and variable amplitude tensioncompression, torsion and combined proportional tension with torsion. It has been shown that the normal strain energy density in the critical plane is a suitable parameter for correlation of fatigue lives of 35NCD16 steel under considered kinds of loading. The critical plane is the plane where the normal strain energy density reaches its maximum value.
KEYWORDS Biaxial fatigue, proportional loading, variable amplitude, fracture plane, energy criterion, life time
INTRODUCTION There are three main models of multiaxial fatigue failure criteria applied for reduction of the complex loading state to the equivalent uniaxial state. They are stress, strain and energy  based models. The proposed energy criteria can be classified into three groups, depending on the strain energy density per cycle, assumed as a damage parameter under multiaxial fatigue [l  31. They are:  criteria based on the elastic strain energy for highcycle fatigue,  criteria based on the plastic strain energy for lowcycle fatigue,  criteria based on the sum of the elastic and plastic strain energy for low and highcycle fatigue. At present, the criteria including the strain energy density in the critical plane or in the fracture plane become dominating in energy description of multiaxial fatigue. These criteria seem to be the most promising for future applications. The authors proposed the energy approach to fatigue life estimation under multiaxial random loading [4  91. In the case of uniform stress
I84
T LAGODA ETAL.
distribution (cruciform specimens and thinwalled cylinders), the criterion of the maximum normal strain energy in the critical plane, including the positive energy parameter under tension and the negative energy parameter under torsion was a suitable quantity for describing the test results of lOHNAP and SUS304 steels. All the analyses were based on a parameter of strain energy density which has been presented in details in [IO, 111. In [12  141 the mesoscopic approach to fatigue life estimation was proposed. The papers concern uniaxial tensioncompression with torsion in cylindrical solid specimens made of 35NCD16 steel, and the proposed approach is based on the concepts of Orowan [15], Papadopulos [16] and DangVang [17]. In this paper the authors will reanalyse the former test results of 35NCD16 steel and assess if the proposed energy parameter can correlate these experimental data. As it results from the previous papers [4  91, this parameter seems to be very efficient for correlation of the experimental data obtained for steels IOHNAP, 18G2A, SUS304 and 12010.3 as well as cast irons GGG40 and GGG60. FATIGUE TESTS The tested 35NCDl6 steel has the following chemical composition: Si  0.37%, Mn  0.39%, PO.O1%,S slip systems. The displacements are magnified by a factor 100. This effect is liable to modify the statistic distribution of the shear stress in an individual grain. As a matter of fact, there is a dispersion of both the maximum and the minimum
334
S. POMMIER
principal stress component around their mean values. This dispersion reaches +/ 35% in copper under uniaxial loading conditions. If a biaxial stress state is applied on a sample, with a mean value of the Tresca equivalent stress equal to that applied under uniaxial conditions (Eq. (2)), does it mean that the dispersion around this mean value (Eq. (3)) is also the same? There is no reason to consider that the load percolation networks associated with each direction are uncorrelated. In such a case, the dispersion of the difference between the maximum and the minimum principal stress components is not the sum of the dispersions for each component. The same problem occurs for determining the dispersion on the maximum resolved shear stress on slip systems.
This question is important, since the fatigue life of the material is limited by the "weakest" grains, and consequently by the maximum bound of the distribution of,,,z and not by its mean but different loading value .It is therefore important to check if, with similar , conditions, the maximum bounds for,,,z are similar. To answer this question, the crystalline orientation of one grain at the centre of the model was fixed, while the orientations of the other grains in the model were selected randomly before each computation. The calculations were performed for copper. One hundred and fifty computations have been performed in uniaxial extension and in shear. The data are "measured" at the centre of the grain with a fixed crystalline orientation.
IOOQO
IOOQO
ueq Ttexca
.C
15000
ueq Ttesca
Fig. 12. Probability that the Tresca equivalent stress, in a grain with a given orientation at the centre of the thin sheet, is greater than a given value. Distribution calculated using the FEM, for a given orientation and 150 configurations of the neighbours. Solid symbols: uniaxial extension, empty symbols: shear.
Variability in Fatigue Lives: An Effect of the Elastic Anisotropy of Grains?
335
The statistic distributions of the Tresca equivalent stress calculated in shear and in tension are plotted in Fig. 12. The scatter is larger for shear than for a uniaxial extension. About 7% of the values obtained in shear are over the upper bounds obtained in uniaxial extension. Up to 150 calculations have been performed to build each distribution, in order to check that this effect was not an artefact, and this is not, since it corresponds to 11 points. This effect is interpreted as the effect of the intersections between the links of the load percolation networks. As a matter of fact, in the intensity maps of the Tresca equivalent stress (Fig. 1 1. (d)), the overstressed grains are always found at the intersections between the links of the load percolation networks associated with each principal direction. Therefore, a few grains at the intersections between the two links are heavily loaded, the grains located within the links are moderately loaded, while those located out of the two loads percolation networks sustain very low levels of the Tresca equivalent stress. In comparison, in uniaxial extension, the grains are heavily loaded if they belong to the load percolation network and sustain low values of the Tresca equivalent stress if they are out of the links. Consequently, there is an increase of the scatter on the Tresca equivalent stress in shear as compared with uniaxial loading conditions.
\\ 000
Fig. 13. Probability that the maximum resolved shear stress on FCC slip systems, in a grain with a given orientation at the centre of the thin sheet, is greater than a given value. Distribution calculated using the FEM, for a given orientation and 150 configurations of the neighbours. Solid symbols: uniaxial extension, empty symbols: shear.
In Fig. 13, are plotted the distributions of the maximum resolved shear stress on the FCC slip systems. As for the Tresca equivalent stress, a significant increase of the scatter is found when a shear stress state is applied as compared with a uniaxial stress state. The parameters characterizing the distributions of the Tresca equivalent stress and of the maximum resolved shear stress are gathered in Table 3. The difference between the standard deviation of the distributions obtained in shear and in uniaxial extension is lower for the maximum resolved
S. POMMIER
336
shear stress than for the Tresca equivalent stress. Nevertheless this increase is not negligible, > while it approaches since in tension, the standard deviation for 2, is close to 12 % of < 15%ofinshear. Although the scatter is higher in shear than in uniaxial extension, the maximum bound of the distribution of the Tresca stress is found to be very similar in these two cases (Table 3). As a matter of fact, the boundary conditions, applied on the model of the thin sheet, were not adjusted to obtain similar mean values of the mean Tresca equivalent stress or of in the two cases, but the same equivalent strain. Consequently, with such boundary conditions, the higher level of the scatter in shear, as compared with a uniaxial extension, results in a lower value of the mean Tresca equivalent stress, and not in an increase of the probabilities for high values of the Tresca equivalent stress. Table 3. Values calculated at the centre of the grain located at the centre of the model with a crystalline orientation as follows: (VI, yf, cpz) =(O,O,O). The standard deviations are calculated with 150 random configurations of its neighbours. The model is tested either in uniaxial extension W = O . l %or in shear =0.1%. The standard deviations (6) are given as a percentage of the mean value of the distribution.
150
analyses Uniaxial extension Shear
MPa 101.7 87.6
< Zmax’(1+W
11.2
1+3& ~q)) s( %ax) (MPa) (MPa) (%) 135.8 44.2 12.0
17.2
132.6
48.3
s( OW) (%)
,(
33.5
14.8
Zmax))
(MPa) 60.2
It can be concluded fiom these calculations, that with a similar value of the maximum resolved shear stress on slip systems in a polycrystal, two different loading conditions are not equivalent in terms of the nucleation of microcracks, since the maximum bounds for ‘tmnxcan be different. For example, these calculations show that with a similar mean value amax>. the maximum bound for T,, is higher in torsion as compared with tension. The grains with the highest value of rmnx are located around the intersections between the heavily loaded links associated with each principal direction. These grains are sparse but overstressed. The role in fatigue of the selforganized spatial distribution of stress and strain in the polycrystal. which is described in this paper, should also be important for crack coalescence during subsequent crack growth, since it controls the number of damaged grains per unit surface and their mutual distance.
REFERENCES 1.
2.
3.
4.
5. 6. 7.
8.
Sines, G. and Ohgi, G. (1981). Fatigue criteria under combined stresses or strains. Journal of Engineering Materials and Technology 103,8290. Dang Van, K. (1993). Macromicro approach in highcycle multiaxial fatigue. In: Advances in Multiaxial Futigue, ASTMSTP 1191, pp. 120130, Mc Dowell, D.L. and Ellis. R. (Eds.), ASTM, Philadelphia. Murakami, Y., Toriyama and T.,Coudert, E.M. (1994). Instructions for a New Method of Inclusion Rating and Correlations with the Fatigue Limit. Journal of Testing & Evaluation 22,3 18326. Hild, F.. Billardon, R. and Beranger, A S . (1996). Fatigue failure maps of heterogeneous materials. Mechanics of Materials 22, 1121. Beretta, S. (2001). Analysis of multiaxial fatigue criteria for materials containing defects. In: ICB/MF&F, pp. 755762, de Freitas, M., (Eds.), ESIS, Lisboa. Murakami, Y. and Endo, M. (1994). Effects of defects, inclusions and inhomogeneities on fatigue strength. Int. J. Fatigue 16, 163182. Susmel, L. and Petrone, N. (2001). Fatigue life prediction for 6082T6 cylindrical specimens subjected to inphase and out of phase bendingkorsion loadings. In: ICB/MF&F, pp. 125142, de Freitas, M., (Eds.), ESIS, Lisboa. Guyon, E. and Troadec, J.P., (1994). Du suc de hilles au tas de sable, Odile Jacob (Eds), Paris.
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9.
Savage, S.B. (1997). Problems in the static and dynamics of granular materials. In: Powder and grains, pp. 185194, Behringer, R.P. and Jenkins, J.T. (Eds.), Balkema, Rotterdam. 10. Dantu, P. (1968). Etude statistique des forces intergranulaires dans un milieu pulvkrulent. Giotechnique. 18.5055 11. Radjai, F., Wolf, D.E., Roux, S., Jean, M. and Moreau, J.J. (1997). Force networks in dense granular media. In: Powder and grains, pp. 21 1214, Behringer, R.P. and Jenkins, J.T. (Eds.), Rotterdam. 12. Roux, J.N. (1997). Contact disorder and nonlinear elasticity of granular packings: A simple model. In: Powder and grains, pp. 215218, Behringer, R.P. and Jenkins, J.T. (Eds.), Balkema, Rotterdam. 13. Le Biavant, K., Pommier, S. and Prioul, C. (1999), Ghost structure effect on fatigue crack initiation and growth in a Ti6A14V alloy. In : Titane 99:Science and technology, pp 48 1487, Goryin, LV. and Ushkov, S.S. (Eds), Saint Petersburg, Russia. 14. Le Biavant, K., Pommier, S. and Prioul, C. (2002). Local texture and fatigue crack initiation in a Ti6A14V Titanium alloy. Fat. Fract. Engng. Mater. Struct 25,527545. 15. Pommier, S. (2002). “Arching” effect in elastic polycrystals. Fat. Fract. Engng. Mater. Struct. 25,331348.
Appendix : NOMENCLATURE Oxygen Free High Conductivity Copper Finite element method Coordinate systems attached to the grain and to the model Stress tensor as calculated using the FEM in (1,2,3) Schmid factor Maximum resolved shear stress on the slip systems of a crystal Tresca equivalent stress Principal stress Maximum, minimum principal stress components Mean value off Standard deviation off Face centered cubic, body centered cubic, hexagonal closepacked
BiaxiaVMultiaxial Fatigue and Fracture Andrea Carpinkri, Manuel de Freitas and Andrea Spagnoli (Eds.) 0 Elsevier Science Ltd.and ESIS. A11 rights reserved.
34 1
THREEDIMENSIONAL CRACK GROWTH. NUMERICAL EVALUATIONS AND EXPERIMENTAL TESTS Calogero CALi, Roberto CITARELLA, Michele PERRELLA Department of Mechanical Engineering, University of Salerno via Ponte don Melillo I , 84084 Fisciano (SA), Italy
ABSTRACT Experimental observations of three and twodimensional fatigue crack growth are compared to numerical predictions from the computer code BEASY. The two dimensional propagation occur in a Multiple Site Damage (MSD) scenario created on a prenotched specimen, undergoing a traction fatigue load as defined by a general load spectrum. Experimental analyses on a fatigue machine were carried out in order to validate the numerical simulation and to provide the necessary material fatigue data for the aluminium plates. The numerical code adopted (BEASY) is based on Dual Boundary Element Method (DBEM). General modelling capabilities are allowed by this approach, with the allowance for general crack front shape and a fully automatic propagation process. By means of a nonlinear regression analysis, applied on in house obtained experimental data, the material parameters for the NASGRO 2.0 crack propagation law were defined, capable to effectively keep into account the threshold effect and the unstable final propagation (the crack closure option was switched off). A satisfactory agreement between numerical and experimental crack growth rates was obtained, even starting from a complex MSD scenario, created by the presence of three holes in the plate. Moreover the load introduction to the specimen was monitored by strain gauge equipment. The numerical simulation include also the through the thickness propagation, corresponding to 3D partthrough cracks; in this case some specimen were prenotched by a comer crack on one of the holes and the 3D experimental crack propagation monitored. KEYWORDS MSD, Partthrough crack, Load Spectrum, D ~ aBEM, l NASGRO 2.0, BEASY INTRODUCTION Damage Tolerance is used in the design of many types of structures, such as bridges, military ships, commercial aircraft, space vehicle and merchant ships. Damage tolerant design requires accurate prediction of fatigue crack growth under service conditions and typicaIly this is accomplished with the aid of a numerical code. Many aspects of fracture mechanics are more complicated in practice than in twodimensional laboratory tests, textbook examples, or overly
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C. CALi R. CITARELLA AND M.PERRELLA
simplified computer programs. Load spectrum, threshold effects, environmental conditions, microstructural effects, small crack effects, Multiple Site Damage (MSD) conditions, material parameters scatter, mixed loading conditions (material flaws or precracks, which may have been introduced unintentionally during the manufacturing process, can have an arbitrary orientation with respect to the loading applied to the component) and complex three dimensional geometry, all complicate the process of predicting fatigue crack growth in real word applications. This paper focuses on some of these complications (see also [I]): load spectrum influence, complex three dimensional geometry, fatigue material parameters assessment, mixed loading conditions and MSD conditions. A series of laboratory tests have been designed and implemented to assess fatigue material parameters and to evaluate three dimensional fatigue life prediction capabilities for a numerical life prediction code (BEASY), based on Dual Boundary Element Method (DBEM). With such code the geometry of the test specimen and the shapes of evolving crack fronts, are not restricted to the simplified configurations found in the libraries of many commercial codes. Many complications were purposely minimised: initial crack sizes and loading were such that small crack and threshold effects had little consequence on the propagation phenomena; environmental and microstructural effects were considered part of the experimental scatter. EXPERIMENTS Experimental fatigue tests were performed on complex geometry notched plates undergoing cyclic axial load. The crack initiation process and the crack propagation were monitored and a general traction load spectrum was applied on the specimen. Experimental crack growth rate and crack path were compared with those obtained with a numerical procedure based on DBEM and the correct simulation of the load introduction to the specimen was checked by strain gauge measurements.
Simple geometry crackedplates and twodimensional crack propagation The fatigue data necessary for the numerical analysis were previously obtained by experimental crack growth tests on simple geometry specimens [2]. All of the test aluminium alloy specimens described in this paper were manufactured from a single lot. The first part of the fatigue experimental tests was carried out on 3 simple notched (hole/slut) specimens (Fig. l), in such a way to work out, with statistical significance (for R=O. I), the material fatigue parameters for crack growth simulation. The initial through the thickness notch was cut by a thin saw. The plate geometry as well as the whole testing procedure was consistent with the ASTM E 64791 specifications. A constant amplitude fatigue traction load (P=14 kN R=P,,JP=O. I) was applied by a servohydraulic machine (Instron 8502), with a frequency f=5 Hz, at ambient temperature. During the experimental fatigue test on simple specimens, crack length data, measured by optical systems, were recorded and used to work out, by analytical formula, the corresponding SIF’s (Stress Intensity Factors) and crack growth rates. The overall specimen size was chosen sufficiently large in order to get a mainly elastic behaviour, with plasticity effects confined to the final part of crack propagation, whose monitoring can start only after a precracking phase, as prescribed by the ASTM E 647 standard. The observed rectilinear crack propagation path turned out to be consistent with symmetric
ThreeDimensional Crack Growth: Numerical Evaluations and Experimental Tests
343
boundary conditions but, whenever out of tolerance deviations from the ideal rectilinear path came out, the corresponding specimens were discarded from the analysis. In the final part of crack propagation, due to high plasticity effects, a 45” deflection of the crack surface preceded the ductile failure [3].
Fig. 1. Simple specimen (N. 57) geometry adopted for material fatigue parameters assessment. For each valid specimen (N” 57), according to the mentioned standard, a chart with crack length against the number of cycles was plotted, in order to assess crack growth rates da/dN, calculated by the secant method and SIF’s, calculated by analytical formulas. Crack growth rate values were then plotted against AK in a bilogarithmic chart and a linear regression was performed to estimate the Paris law material constants C and n (Fig. 2).
7n
28,5 Crack 1

Crack 2 Crack 4
Crack 3
353
I
Fig. 2. Regression curve using Paris law.
Fig. 3. Notched complex specimens (N.12): initial MSD scenario (dimensions mm).
344
c. C A L ~R., CITARELLA AND M. PERRELLA
Complex geometry cracked plates: two and threedimensional crack propagation Two dimensional. For that concern the twodimensional long crack propagation, a complex MSD initial scenario was artificially created on a rectangular plate (Fig. 3), by means of notched holes (a triangular notch was cut in correspondence of each hole) and, after the precracking process, four different cracks simultaneously propagating were obtained. The propagating crack lengths were monitored on both sides of the specimen in order to check the correctness of the load introduction on the specimen: any misalignment between the machine grips could create a bending condition for the specimen and consequently an elliptical propagating crack front [4]. This check, together with the strain gauge measurements, allowed assessing the validity of the 2D hypothesis for such long crack propagation analysis. Three dimensional. To study the through the thickness crack propagation a triangular notch emanating from hole 2 (corresponding to crack 2 in Fig. 3) was cut. After the crack initiation and precracking period, a quarter elliptical comer crack started to propagate. Such crack was monitored by an optical measuring system based on a moving camera (Fig. 4), in order to alternatively follow the two break points of the elliptical crack front. The images obtained by the (manually) moving camera (Fig. 5) were elaborated by an image analysis software in order to obtain the two ellipse semiaxes measures. For long crack initiation period it is possible, with an in house made software, to automatically make periodic photo that can be recorded and postprocessed, avoiding the need for a continuous surveillance of the specimen by the operator. In the latter case two camera are needed for measuring the two semiaxes and only a short propagation length can be monitored before the crack tip get out of focus (but this drawback can be overcome by using special lenses or with an automatically moving camera).
Fig. 4. Equipment adopted for monitoring the through the thickness crack propagation.
ThreeDimensional Crack Growth: Numerical Eualuations and Experimental Tests
345
Fig. 5. Images of the propagating crack: size A (left) and size C (right). In the latter only the crack tip at the internal hole is visible, few cycle before getting a through the thickness crack. NUMERICAL ANALYSIS Two and threedimensional analysis are respectively needed to simulate the through the plate and through the thickness crack propagation. The material fatigue parameters, obtained by the experimental analysis previously described, are useful to perform a crack growth simulation on a complex geometry specimen made of the same material. The results of such numerical analysis were compared with those from the experimental tests, in order to validate and improve the numerical procedure, based on DBEM [591. Two dimensional simulation on MSD plates
Two loading  conditions were considered on different complex specimens: Cyclic load with constant amplitude (PmXP~"=12.6KN) and stress ratio (R=O.I), on specimen N.l. The Paris law was adopted for numerical crack growth assessment (the same values had been used for the simple notched specimens). Crack paths (Figs. 67) and propagation times (Figs. 811), obtained by BEASY code [lo], were compared with the experimental ones, getting a satisfactory agreement; Cyclic load with variable amplitude and stress ratio on specimen N.2. With the experimental data coming from simple notched specimens and from the previous complex specimen (N.l), a nonlinear regression was attempted, in order to model the threshold phenomena and the fracture toughness for the final unstable crack propagation. To this aim the NASGRO 2.0 law was chosen for numerical crack growth assessment (for such case the Paris law did not give satisfactory results). The crack paths are the same as for the previous case (Figs. 67) and the propagation times (Figs. 811), obtained by BEASY code, were compared with the experimental ones, getting a satisfactory agreement especially in the first part of the propagation. The differences, however limited, in the final part suggest the need of an improved correlation; this can be done by increasing the experimental data coming from simple specimens (it is necessary to test the simple specimens with different R values). The parameter values for the NASGRO 2.0 Eq.(l), with no crack closure effect, are
346
c. C A L ~R. CITARELLA AND M. PERRELLA indicated in Table 1. Such parameters were extracted fiom the NASGRO database, in correspondence of AI 22 19T87 which was reputed the most similar to the aluminium used, in order to get the unknowns C and n h m the nonlinear regression (made with MATHEMATICA 4) which is based on in house obtained experimental data:
In table 2 the load spectrum adopted for the specimen No2 is illustrated. From Fig. 8 which is related to the first appearing crack 1, it is possible to note the relatively little difference between initiation times for the two complex geometry specimens. Table 1. Material parameters adopted for crack growth simulation Youngs Modulus [MPa] P o i s s o n s ratio Yield s t r e s s ( Y s) [M P a ] U l t i m a t e tensile s t r e n g t h ( U T S ) [ M P a ] Plane strain f r a c t u r e t o u g h n e s s I M P a mm1’21 E m p iric a1 c o n s t a n t Exponent ‘ T h r e s h o l d S I F a t R=O f M P a mm1’21 ICut off s t r e s s ratio IIntrinsic c r a c k length [ m m ] ,
I
E V OYS
OUTS
K IC Ak,
Bk
P 9 AKO Rrl a0
7.2OEtO4 0.3 2 83 . 309 900 1 1 120 1 0.7 0.102
1
.
I
The NASGRO 2.0 form of the equation for AK&is given by:
if R < R , if R > R , The fracture toughness & for 2D plane stress and 3D crack m o d s that have ,een identified as “allthrough”, is given by:
K, f r 4
[
 e 4;) ’)&,
ThreeDimensional Crack Growth: Numerical Evaluations and Experimental Tests
Fig. 6. Experimental crack propagation paths (specimen N02).
341
Fig. 7. Numerical crack propagation paths at 774000 (upper) and 1070000 cycles (lower) for specimen N02.
Table 2. Load spectrum applied to specimen N. 2. Number of cycles
p m (KN)
R
581000 774000 866000 968000 1046000 1076000
14 16.3 21.5 19.2 17.2 19.2
0.10 0.22 0.41 0.58 0.77 0.58
c. C A L ~R. CITARELLA AND M.PERRELLA
348

15
Crack 1
 NumericnO1 0
 t    t  17i+ 0
,
570000
Experimental n O 1
~
  ' , 
670000
820000
tlA
t
620000
cci
720000
770000
+ , I
870000
920000
 
I
970000 I020000 1070000 IlZOOOO
N [cycles] Fig. 8. Crack length versus number of cycles for crack 1, as obtained by simulation with Paris law (applied to specimen 1) and NASGRO 2.0 law (applied to specimen 2), together with related experimental data.
18
Crack 2
A
____ __ Experimental n"1 FL. Experimental n"2 1
a i 570000
630000
690000
750000
810000
870000
930000
990000
1050000
1110000
N [cycles]
Fig. 9. Crack length versus number of cycles for crack 2, as obtained by simulation with Paris law (applied to specimen 1) and NASGRO 2.0 law (applied to specimen 2), together with related experimental data.
ThreeDimensional Crack Growth: Numerical Evaluations and Experimental Tests
349
Crack 3 _
7
_
_
~
I
~
~


7
6
5
1
A
Experimental n"2
TE 4 Y
a
3 2 1 I
I 0 570000 620000 670000 720000 770000 820000 870000 920000 970000 1020000 1070000 1120000
N [cycles] Fig. 10. Crack length versus number of cycles for crack 3, as obtained by simulation with Paris law (applied to specimen 1) and NASGRO 2.0 law (applied to specimen 2), together with related experimental data.
Crack 4
570000
620000
670000
720000
770000
820000
870000
920000
970000
1020000
1070000
N [cycles] Fig. 11. Crack length versus number of cycles for crack 4, as obtained by simulation with Paris law (applied to specimen 2) and NASGRO 2.0 law (applied to specimen 2), together with related experimental data.
c. C A L ~R. CITARELLA AND M PERRELLA
350
Three dimensional simulation: theoretical aspects and results. Surface crack solutions are widely used in applications of fracture mechanics to fatigue and monotonic loading. A semielliptical surface crack lying perpendicular to the surface and subject to applied stresses with no shear component parallel to the crack, experience mode I conditions around its edge, as in the problem presented in the following. Generally this orientation is the critical one, also in mixed mode conditions if the fatigue crack threshold is governed by (AG I)W, where G is the energy release rate.
The dual boundary element method. BEASY uses dual elements for 3D crack growth analysis. The dual boundary element method (DBEM) incorporates two independent boundary integral equations: the displacement equation applied at the collocation point on one of the crack surfaces and the traction equation on the other surface. Application of the DBEM to threedimensional crack growth analysis has been presented in [111 31. In the absence of body force, the displacement boundary equation can be written as: cIJ(xI).u,(xI)+ ~ ~ , ( x ~ , x ) . u , ( x ' ) . ( r r (=x )l u , ( . ' , x ) . t , ( x ' ) . ~ ( x )
r
where iJ denote Cartesian components,
r
r, (x 'J)and Uv(x'J)represent the Kelvin traction and
displacement fundamental solutions at a boundary point x, respectively. The symbol
stands
for the Cauchy principal value integral. Assuming continuity of both strains and traction at x ' on a smooth boundary, the stress components G,, are given by:
where Tkv (x 'J) and uk,(x'J)contain derivatives of z, (x f ) and The symbol
uv(x'T),respectively.
stands for the Hadamard principal value integral. The traction components 6
are given by:
where n,(x ') denotes the component of outward unit normal to the boundary at x '. Equations (2) and (4) constitute the basis of DBEM. A more complete description is given in u11.
ThreeDimensional Crack Growth: Numerical Evaluations and Experimental Tests
35 1
Stress Intensity Factors. Since Irwin [ 141 demonstrated the importance of the stress intensity factor (SIF) in determining cracktip stress fields in two dimensional problems, many different methods have been devised for obtaining SIF's [15]. The stress intensity factors (SIF's) are computed using crack opening displacement method. When one point formulae are employed, the Mode I, I1 and I11 SIF's are evaluated as:
where the displacement u p is evaluated at point P as in Fig. 12; ubp,unPand u r are projections of up on the coordinate directions of the local crack front coordinate system and 6=n and 6=n denote upper and lower crack surfaces respectively. K p , KIP and KIIF'are approximations of SIF's corresponding to the point P' along the crack front and on a normal line to the front as in Fig. 12.
Fig. 12. Calculation of SIF using crack opening displacement.
During the crack growth, instabilities may cause peaks in the SIF values along the crack front. It is advisable to enable the BEASY option that allows these peaks to be smoothed by either smoothing the stress intensity factors or by smoothing the crack growth increment or by performing both smoothing operations. This should also be used if the values computed at the crack breakout points are not very accurate and give excessively high stress intensity factor values, as it happens in the problem under study. As a matter of fact the asymptotic stress field at the crack tip is usually obtained by performing analytical twodimensional plane stress or plain strain calculations. This yields the well known singularity which prevails along the crack front within the body. However, at the intersection of the crack front with a free boundary, the stress state is a genuine threedimensional one for which two dimensional approximations are not a plicable: the mode I stress singularity decreases at the free boundary and takes the value r0.45 whereas the mode I1 and mode I11 singularity increase in strength
P',
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352
and attain a value of [16]. At the end, high resolution accuracy at the comer may not be worth pursuing, in any case, because the geometry where the crack intersects the surface is expected to adjust under loading so as to ameliorate the stronger singularity [171.
The incremental direction. The crack growth direction and SIF equivalent are computed by the minimum strain energy density criterion. The criterion for three dimensional problems can be found in [18]. The explicit expression of strain energy density around the crack front can be written as: =dW
dV
s(0) +0(1)
r.cosq5
where S(8) is given by

S(0) = a,, K:
+ 2a,,  K,
K,
+ a22 Ki + aY3.Ki!
(9)
and
1 a,, = . (3  4v  [email protected]).(1 + COSO) 16rp 1 aI2= sine. (cos0  1 + 4v)
8 v
a22= .
1
16 n ,
[4(1 v ) (1  [email protected])+ (3cos0  1). (1 + cos 011
in which pstands for the shear modulus of elasticity and v is the Poisson ratio. S/cos( represents the amplitude of intensity of the strain energy density field and it varies with the angle 4 and 0. It is apparent that the minimum of S/cosb always occurs in the normal plane of the crack front curve, namely +=O. S is known as strain energy density factor and plays a similar role to the stress intensity factor. The theory is based on three hypotheses: 1 . The direction of the crack growth at any point along the crack front is toward the region with the minimum value of strain energy density factor S as compared with other regions on the same spherical surface surrounding the point. 2. Crack extension occurs when the strain energy density factor in the region determined by hypothesis S = &in, reaches a critical value, say SCr. 3. The length, r,, of the initial crack extension is assumed to be proportional to Sm,,such that Smi,,/r0 remains constant along the new crack front. It can be seen that the minimum strain energy density criterion can be used both in two and three dimensions. Note that the direction evaluated by the criterion in three dimensional cases is insensitive to KII~.The crack growth direction angle 8, is obtained by minimising the strain energy density factor S(8) 6f Eq.(9) with respect to 8. The minimum strain density factor S(0,)
ThreeDimensional Crack Growth: Numerical Evaluations and Experimental Tests
353
is denoted by Smin.Locating the minimum of S(e) with respect to 8 is carried out numerically using the bisection method by solving:
The equivalent stress intensity factor, Keq, can be calculated from the minimum strain density factor as:
The incremental size. Determination of the incremental size posses two problems: the first is the determination of the amount of each increment in terms of a reference size, the second is the relationship between the maximum incremental size and other incremental sizes along the crack front. It is stated in the hypothesis (3) of the strain energy density criterion that the length ro of the initial crack extension is assumed to be proportional to Sminsuch that Sm,,,lroremains constant along the crack front. Since the strain energy factor Sminis proportional to the square power of the equivalent stress intensity factor Kes, the incremental size at the crack front point under consideration is given by:
where mar{K,,} is the maximum SIF equivalent evaluated at a set of discrete point along the front, and Aa is the incremental size at the point corresponding to the mar{Keq} which is chosen beforehand as being the maximum distance from the crack front to the opposite side of the element containing the crack front as in Fig. 13 .The above expression is used for standard crack growth. Crack increment Crack front
Fig. 13. Maximum incremental distance.
354
c. C A L ~R. CITARELLAAND M . PERRELLA
Fatigue Growth Calculation.During fatigue crack growth, the relation between the incremental size and the number of load cycles may be represented by a number of crack growth laws, such as PARIS, FORMAN, RHODES or NASGRO. Alternatively, a tabulated form can be used to supplyDuring the fatigue analysis, there are options on the method of computing the dN values: 1. the SIF’s are constant over the step; 2. the SIF’s at the previous step and the current step are used to compute the dN value over the last crack growth step. This requires two analysis to be performed before the first dN value is computed (backwardcorrection) 3. the previous two results are used to predict a guess for the dN over the next step. This may be inaccurate as the SIF’s may change significantly over the step cforwardprediction). Crack modelling. In this section, the modelling and discretization strategy of implementation of the DBEM for three dimensional crack problems is presented. Because of the continuity requirements of the displacements and tractions for the existence of traction boundary integral equations and coplanar characteristic of crack surfaces, special consideration has to be taken for modelling discretization. In order to maintain efficiency and simplicity of the boundary elements, the present formulation uses discontinuous quadratic element for the crack modelling. The general modelling strategy can be summarised as follows: Crack surfaces are modelled with discontinuous quadratic quadrilateral elements; Surfaces intersecting a crack surface are modelled with edgediscontinuous quadrilateral or triangular elements; The displacement integral equation is applied for collocation on one of the crack surfaces (say the upper surface G+); The traction integral equation is applied for collocation on the opposite crack surface (say the upper surface GJ; The displacement integral equation is applied for collocation on all other surfaces. The requirement of the continuity on udx) and t,(x) for the existences of the displacement and traction boundary integral equations is satisfied by the fact that discontinuous elements are used on crack surfaces. The above strategy is robust as it maintains the consistency with the theory and, at the same time, allows effective modelling of general edge or embedded crack problems. Edge crack is defined here when the crack front intersects the boundary surface, while in the embedded crack the crack front is positioned in the interior of the problem domain. Note that the increment of an edge crack requires remesh of the boundary surfaces intersecting the crack surface. Results related to threedimensional crack propagation This time the complex specimen undergoes a fatigue load with P,,=27.7 KN and R=O.1 and the frequency adopted on the fatigue machine is 10 Hz. The analysis has been divided in two part: 1. the propagation of the elliptical part through crack up to a condition that immediately precede the through crack appearance; for such analysis the partthrough fracture toughness was assumed K1~=1320 MPa.mm”* (from NASGRO database and correspondingly to A1 22 19T87) 2. the initial part of the through crack propagation, when the phenomena is still threedimensional because of the differences between the two crack front sizes.
ThreeDimensional Crack Growth: Numerical Evaluations and Experimental Tests
355
Partthrough crack propagation. Even if the notch introduced on the specimen hole is triangular, for the simulation a quarter circular comer crack has been modelled (because already available in the BEASY database) but checking that, after the precracking phase, two identical initial conditions were obtained between the numerical and experimental crack front. Starting from this point the comparisons between numerical and experimental crack shapes and propagation times are presented (Fig. 14): the simulated propagation proceed faster than the experimental one and this turns out to be conservative even if not strongly accurate. The SIF’s effective along the crack front for each propagation step are presented in Fig. 15. Total Number of Cycles vs Crack Size 5.00 4.50
.......
4.00
7 3.50 E 3.00
2 5
e!
2.50
. . . . . . . . . . . . . . . . .
2.00
. . . . . . . .
........ ............... ..................
1.50 1
.oo
OSO 0.00
. . . . . . . . . . . . . . . . . . . . . . .
2 0
5000
loo00
15000
2oooO
25000
30000
35000
40000
Number of Cycles
Fig. 14. Crack propagation data (C is the break point inside the hole, whilst A is on the visible surface): an exponential regression line was drawn through the experimental data. 3,6E+02 3,4E+02
,.
+STEPS
+STEP7
+STEP6
*STEP8
3.ZEi02
2 3.0E+02 Y
a
2; ;
2,6E+02
.
_L
0,7
0,8
.’
’

b
2,4E+02
_ _ _ _ _ ~
2,2E+02 2.OEi02
0.0
0,I
0,2
0,3
0,4
0.5
0,6
0,9
1.0
Local Position on Crack Front
Fig. 15. SIF’s effective along the crack front for the elliptical partthrough crack.
356
C. CALI, R. CITARELLA AND M.PERRELLA
Fig. 16. Experimental and numerical front shapes for partthrough cracks, at different stages.
A
5.71429et001 2.85714et001 0.00000et000
Von Mises effective stress
Max= 446.70 Min= 0.17642
Fig. 17. Von Mises effective stresses and boundary element mesh on the overall plate. Even if for that concern the simulation on crack propagation times some further refinement are necessary, it is possible to foresee very accurately the crack front shapes (Fig. 16). The final part of the propagation is not under analysis, because BEASY simulation relies on the
ThreeDimensional Crack Growth: Numerical Evaluations and Experimental Tests
357
hypotheses of linear elastic fracture mechanics (LEFM), which is not anymore applicable due to the strong plasticity effects. The automatically created and updated mesh is based on a variable number of quadratic elements: 512 initially and 758 at the last crack propagation step; it is visible, together with Von Mises stresses in Figs. 1718.
i
Fig. 18. Von Mises effective stresses on an magnified deformed plot of the plate (the partthrough crack opening is evident). Throughcrackpropagation. After the crack breaksout on the opposite side (with respect to the notch position) of the plate, an initial scenario obtained from experimental measurements was considered for the simulation. The related results are presented in Figs. 1921. Again the simulated propagation proceed a bit faster than the experimental one. Again the crack front shapes foreseen by the simulation are well in agreement with the experimental ones (Fig. 22).
Fig. 19. Von Mises effective stresses on an magnified deformed plot of the plate (the through crack opening is evident).
358
C. CALI. R. CITARELLA AND M. PERRELLA
Total Number of Cycles vs Crack Size
+CRACK +CRACK

0;o 500
0
1000
1500
2000

SIZE A (expenm ) SIZE C (experim.)
1 i
CRACK SIZE C CRACKSIZEA
~~
2500

II_._
lll____._
I
~~~
3000
3500
4000
4500
cycles
Fig. 20: Experimental and numerical crack sizes on the two sides of the plate (A and C are at the extremity of the crack front).
6.5E+02
6.OE+02

5,5E+02
2
E
5 5,OE+02
&
E Y
{ 4,5E+02
z
4,OE+02
3,5E+02
3,OE+02 0,OO
0,IO
0,20
0,30
0,40
0,50
0,60
0,70
0,80
0,90
Local Position on Crack Front
Fig. 21. SIF effective along the crack front for the through crack.
1,OO
ThreeDimensional Crack Growth: Numerical Evaluations and Experimental Tests
359
Fig. 22. Experimental and numerical crack shapes for through cracks at different stages. CONCLUSIONS
With reference to twodimensional MSD crack propagation, a satisfactory agreement was obtained between numerical and experimental crack propagation rates on specimen 1 when using the Paris formula, with the related constants provided by in house made experimental tests. Such formula was not anymore accurate for variable amplitude load cycles, as applied to specimen 2, because unable to keep in account the load ratio variability. That is why a more complex correlation, based on an enriched set of experimental data and on information from NASGRO database (without the need to model crack closure effect), was attempted getting a satisfactory agreement between numerical and experimental results. The later approach could be improved by increasing the experimental data by cycling some simple notched specimen with different R values. For that concern the threedimensional crack propagation again a very interesting correlation between numerical and experimental results was obtained even if some further refinement are necessary with regards to the crack propagation times. It is to point out the extreme flexibility
C. CALt R. CITARELLA AND M.PERRELU
360
and efficiency of the methodology adopted, because, apart from the accuracy typical of the Boundary Element Method applied to fracture mechanics, the three dimensional crack propagation proceed in a fully automatic way. REFERENCES 1.
2. 3. 4.
5. 6.
z 8.
9. 10. 11.
12. 13.
14.
15. 16.
17. IS.
19. 20.
Riddell, W.T., Ingraffea, A.R., Wawrzynek, P.A. (1997), Experimental observations and numerical predictions of threedimensional fatigue crack propagation, Engineering Fracture Mechanics, 58,2933 10. Schijve, J. (1998), Fatigue Specimens for Sheet and Plate Material, Fatigue & Fracture of Engineering Materials & Structures, 21, 347358. Hsien Yang Yeh, Chang H. Kim (1995), Fracture Mechanics of the Angled Elliptic Crack under Uniaxial Tension, Engineering Fracture Mechanics, 50, 103110. Fawaz, S. A. (1997), Fatigue Crack Growth in Riveted Joints, MSc Thesis, Air Force Institute of Technology, Ohio. Apicella, A., Citarella, R., Esposito, R. (1994), Sulla previsione della propagazione per fatica di cricche multiple tramite elementi di contorno discontinui, Proceedings of the 261d AIAS national conference, Italy. Apicella, A., Amentani, E., Cali, C., Citarella, R., Soprano, A. (1999), Crack propagation in Multi Site Damage conditions for a riveted joint, Proc. of International Conference AMME 99, Poland. Apicella, A., Citarella, R., Esposito, R., Cariello, G. (1997), Crack problems by FEMBEM coupled procedures, The Int. Journal of Boundary Element Methods, 8,22Z229. Apicella, A., Citarella, R., Esposito, R., Soprano, A. (1998), Some SIF’s evaluations by Dual BEM for 3D cracked plates, Proceedings of the International Conference AMME, Poland. Apicella, A,, Citarella, R., Soprano, A. (1999), 3D stress intensity factor evaluation by Dual BEM, Conference proceedings “Fracture and Damage Mechanics”, UK. Beasy User Guide, Computational Mechanics Beasy, Southampton, England, 1994. Mi,Y., Aliabadi, M.H. (1992), Dual Boundary Element Method for Three Dimensional Fracture Mechanics Analysis, Engineering Analysis with Boundary Elements, 10,161171. Mi,Y., Aliabadi, M.H. (1994), Threedimensional crack growth simulation using BEM, Computers & Structures, Computers & Structures, 52. 87 1878. Mi,Y., (1996), Threedimensional analysis of crack growth, Topics in Engineering, 28, Computational Mechanics Publ., Southampton, U.K. Irwin, G.R., (1957), Analysis of stresses and strains near the end of a crack traversing a plate, Trans. ASME J. Appl. Mech., 24, 361364. Aliabadi, M.H., Rooke, D.P. (1991), Solid Mechanics and its applications, 8, Computational Mechanics Publ ., Southampton, U.K.. Dhondt, G., Chergui. A., Buchholz, F.G. (2001), Computational fracture analysis of different specimens regarding 3D and mode coupling effects, Engineering Fracture Mechanics, 68,383401. He, M.Y., Hutchinson, J.W. (2000), Surface crack subject to mixed mode loading, Engineering Fracture Mechanics, 65, 1 14. Sih, G.C., Cha, B.C.K. (1974). Journal of Engineering Fracture Mechanics, 6,699732. Forman, R.G., Shivakumar, V., Newman, J.C., (1993), Fatigue Crack Growth Computer Program “NASMLAGRO’ Version 2.0, National Aeronautics and Space Administration Lyndon B. Johson Space Center, Houston, Texas. Paris, P.C., (1962). PhD. Thesis, Lehigh University, Bethlehem.
BiaxiayMultiaxialFatigue and Fracture Andrea Carpinten, Manuel de Freitas and Andrea Spagnoli (Eds.) 0 Elsevier Science Ltd. and ESIS. All rights reserved.
361
THE ENVIRONMENT EFTECT ON FATIGUE CRACK GROWTH RATES IN 7049 ALUMINIUM ALLOY AT DIFFERENT LOAD RATIOS Manuel FONTE’, Stefanie STANZLTSCHECG’, Bemd HOLPER’, Elmar TSCHEGG3 and Asuri VASLJDEVAN4 Nautical School, 2780572, Portugal. ’University of Agricultural Sciences, Wien,Austria. Technical University of Wien, Austria. Naval Research Laboratory, Arlington, VA 2221 7, USA.
’
ABSTRACT The influence of environment and microstructure is investigated on a high strength 7049 aluminium alloy sheet cold rolled and heattreated. This aluminium alloy was artificially aged to underaged (UA) and overaged (OA) conditions, resulting in approximately the same yield strength, but different mode of slip deformation. The UA alloy deforms by planar slip while the OA alloy by wavy slip. Crack growth measurements were performed at constant load ratios between 1 and 0.8 in ambient air and vacuum. The influence of load ratio is discussed in terms of slip deformation mechanisms, microstructure and environmental effects using the two intrinsic parameters, dK and K. The two parameters lead to two intrinsic thresholds that must be simultaneously exceeded for a fatigue crack growth. Mechanisms of nearthreshold crack growth are briefly discussed for several concurrent processes involving environmentally assisted cracking with intrinsic microstructural effects.
KEYWORDS Fatigue crack growth, nearthreshold fatigue, environment, microstructure, AK,K, slip mode, load ratio effects, 7049 and 7075 aluminium alloys.
INTRODUCTION Microstructure and environment strongly influence the fatigue crack growth resistance of aluminium alloys and have been investigated for the last two decades [I91. However, the nature of the underlying interactions between microstructure and environment is still not clearly understood. Synergetic effects of environment and loading make the understanding of the underlying mechanisms between microstructure and environment difficult. Since the mid1970s, increasing demands for failsafe designs and damagetolerant constraints have given importance to the fracture toughness and fatigue crack growth resistance properties [10,1 I]. The significance of grain boundaries (GB) precipitations on
362
M.FONTE ET AL.
toughness is clearly seen in commercial alloys when inadequate heat treatments are applied [12141. The microstructure/aging condition is known to have a significant influence. The underaged (UA) microstructure has the maximum susceptibility and the overaged (OA) microstructure a susceptibility, which is decreasing with aging. The heat treatment clearly influences many metallurgical parameters and since the 1960s it was hypothesised that dislocationprecipitate interactions play an important role during stress corrosion cracking of the AIZnMgCu alloys 1151. Stress corrosion cracking of aluminium alloys is a complex phenomenon involving timedependent interactions between alloy microstructure, mechanical deformation, and local environment conditions [16,171. Environment effects are timedependent and K,, is recognised as the characterising parameter. The mechanical behaviour of materials depends strongly on its microstructure and environment effect [ 1824]. It is wellknown that an aluminium alloy exhibits very different properties depending on whether it is cold rolled or heat treated under different temper conditions. Although a combination of the local microstructural features and the applied stress intensity range (AK) primarily governs the slip characteristics and the growth mechanisms, the resulting cyclic crack advance can be substantially changed by the presence of an environment. Kirby and Beevers [25], for example, demonstrated that even the seemingly innocuous environment of laboratory air can lead to a marked increase of crack propagation rates in the nearthreshold fatigue regime of 7XXX series aluminium alloys, if compared to vacuum. Lin and Starke [26] showed that microstructureenvironment interactions at low stress intensities could be completely different from those at higher growth rate levels. It has been recognised [27] that environment effects on slow fatigue crack growth in highstrength aluminium alloys strongly depend on alloy composition, heat treatment, moisture content of the surrounding air and the presence of certain embrittling species. The aim of this study is to examine the mechanisms governing the fatigue behaviour of commercial AI 7049 alloy under controlled microstructural and environment conditions, specifically involving an underaged (VA) and overaged (OA) alloy, having the same chemical composition, crystallographic texture and yield stress, but different precipitate features. The experimental work was designed to obtain the fatigue crack growth thresholds. Then, several mechanical tests were performed on both material conditions and both macroscopic and microscopic responses are compared. Nearthreshold fatigue behaviour in room temperature environments is contrasted with that for vacuum for a range of load ratio values. Micromechanisms of fatigue crack growth are discussed in terms of the specific role of several concurrent processes involving crack closure, environmentally assisted crack growth, and intrinsic microstructural effects. Results are discussed on the basis of the main deformation mechanisms and microstructure, the embrittling influence of environment (ambient air and vacuum) and the two intrinsic parameters of crack growth: A&,, Kmx. ENVIRONMENT AND MICROSTRUCTURE INTERACTIONS Crack closure efects The nearthreshold fatigue properties quite often are discussed in light of crack closure mechanisms. The concept of plasticityinduced closure was introduced by Elber [28], to explain decreasing crack growth rates with increasing crack length as a result of plastic deformation at the crack tip. Later, roughness of fatigue surfaces, oxides, etc. have been identified as additional reasons for reduced crack growth rates [29321. The load ratio dependence was considered not as an intrinsic material property, but arising from changes in
The Environment Effect on Fatigue Crack Growth Rates in 7049 Aluminium Alloy at
___
363
the stress intensity amplitude AK due to premature contact of crack surfaces. Vasudkvan and Sadananda claimed, however, that plasticity in the wake of a crack has a minor influence on crack closure [33351 and argued that fatigue propagation in vacuum does show loadratio effects. It has experimentally been observed that da/dN near the threshold up to higher growth rates is nearly independent of the Rratios in a high vacuum [36,37], which leads to conclude that plasticity does not induce closure. They found that crack closure effects were less important when compared with microstructure and environment effects and do not influence the overall crack growth behaviour. Thus, crack growth rate data generated in laboratory should not be used to predict crack growth rates in a structure unless one has measured crack closure of the structure in service, which depends on the component geometry, load, environment, and crack length. Since crack closure is usually assumed as one of the major causes for retardation effects, the examination of the magnitude of crack closure and its relative role in crack growth processes is very important. The existence of plasticityinduced crack closure has been severely questioned by Vasudevan and Sadananda [33351, while asperityinduced closure, which includes roughness due to crack tortuosity, oxide or chemical reaction debris, etc. is more accepted. Paris [38] have provided a theoretical justification for modified criteria for crack closure. According to Sadananda et al. all methods used to date for measuring crack closure, tend to overestimate the crack closure levels. Therefore they proposed a crack closure measurement based on the shape of the loaddisplacementcurve[39]. Keeping in mind that the role of environment in the nearthreshold regime is strongly more significant than any mechanical contributions such as plasticity, roughness, oxide, etc. effects, one may conclude according to Vasudevan and Sadananda that AKth decreasing with R is an intrinsic fatigue property of the material for that environment. Overload effects have predominantly been attributed to either plasticity induced crack closure behind the crack tip, residual stresses ahead of a crack tip, or a combination of both. The mechanisms such as crack tip blunting, crack deflection, branching and secondary cracking, as well as crack tip strain hardening or residual stresses ahead of the crack tip, involve mainly transient conditions at or ahead of the crack tip [40]. Mechanisms such as plasticityinduced closure and roughnessinduced closure operate behind the crack tip indirectly affecting the crack driving force. Vasudevan and Sadananda presented an 'Unified Approach to Fatigue" which considers closure as a minor factor for crack advance [40]. They modelled fatigue crack propagation for a wide variety of materials by assuming two stress intensity parameters, AK and Kmax,as the relevant crack tip driving forces. They showed that for most situations, the description in terms of AK and Kmx i s necessary and sufficient for fatigue crack growth without the need of crack closure. According to this Unified Approach to Fatigue, K,, and AK are two intrinsic parameters simultaneously required for quantifying fatigue crack growth data. These two driving forces are intrinsic parameters of each material and are valid for short or long cracks, having no anomalies between these two regimes [41]. The apparent different behaviour of short and long cracks is related to residual stresses. They claim that the residual stress cannot be crack closure. Crack closure exists only behind the crack tip and is induced by roughness, oxides, plastic deformation, etc. Crack tip plasticity can only produce compressive residual stresses at the crack tip and not crack closure effects according to Vasudkvan and Sadananda as consequence of overloads for instance. The two parameters AK and Kmx lead to two intrinsic thresholds that must be simultaneously exceeded for a fatigue crack to grow [41]. Crack retardation owing to residual stresses ahead of the crack tip is reported in [42,43]. Ling and Schijve [44] confirmed that residual stresses play a major role in the retardation by demonstrating that the effects can be eliminated by annealing after the overloads.
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Magnitude of AK*and K* to a crack advance Schmidt and Paris [45]were the first who plotted AK& and K, in terms of the Rratio identified the two thresholds and interpreted this result with the crack closure concept. Later, Doker and Marci [46] plotted A K t h vs K,, to [email protected] the two critical thresholds (AK'th and K*) as minimum condition for crack growth. It was recognised that AK and K provide two crack tip driving forces [47]. These two driving forces are required to specify the loads unambiguously. In the fatigue literature, the load ratio, R,is normally specified in addition to AK as the second parameter. But this is considered as a not appropriate load parameter since one does not have a critical Rratio below which crack growth does not occur. Since crack closure contributions are considered as small or negligible for most cases, AK and K alone can adequately explain the material response to fatigue loading, and crack closure is unnecessary. Figure 1 shows the interrelation between parameters mapping the regime where crack growth is permissible, according to Vasud6van and Sadananda [48]. 'l?x m a e t u d e of the limiting values for a given material, microstructure and environment, AK and K , in Fig. 1 (a), depends on the material resistance to fatigue crack growth. The curve in e g . [email protected])can be path is considered as a trajectory corresponding to crack growth mechanisms; the AK = K characteristicof the purecycle controlled fatigue crack growth phenomenon.
dK*U
L
......... K*x.
th
Nonpropagation regime
b
(4
L a x
(b)
K*max
Fig.1. (a) Schematic illustration showing two limiting values, dK' and KOrnmfor each crack growth rate in the Unified Approach. (b) Trajectory map showing the variation of AK*and K* with increasing crack growth rate. AK* = KOm, line represents ideal fatigue behaviour. represent the two limiting For a given crack growth rate, the two values, hlyI and K'values in terms of the two parameters, AK and K, required for fatigue crack growth. According to this approach [49], these parameters, AK and K,,, are simultaneously required for qualifying fatigue crack growth data. Of the two, Kmax is the dominant parameter for all fracture phenomena. An additional parameter AK arises due to cyclic nature of the fatigue damage. Correspondingly there are two thresholds that must simultaneously be exceeded for a fatigue crack to grow. In addition, environmental interactions being time atd strefsdependent process affect fatigue crack growth through the Kmm parameter. The AK =K,, line, Fig. 1
The Environment Effect on Fatigue Crack Growth Rates in 7049 Aluminium Alloy at _ _ .
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(b), represents pure or ideal fatigue crack growth, and this forms the reference line for the ideal inert behaviour which is the basis for the environment contributions [48]. This ideal behaviour appears only if the vacuum is very high or impurities in the socalled inert environments are very low or the material is nonreactive to a given environment. Deviation from the ideal line occurs if the crack p w t h mechanism changes. In principle, K,,, or its nonlinear equivalent is essential for all fracture process involving creation of two new surfaces. For monotonic fracture, this parameter reduces to KIC.For time dependent crack growth process involving stress corrosion, sustained load crack growth or creep crack growth, Kmax is the governing parameter [50,51]. However, due to the cyclic nature of loading in fatigue, an additional parameter is needed, which describes the amplitude. This is M. Therefore the requirement of two parameters is intrinsic to fatigue. Of the two parameters, it has been shown that the magnitude of K is much larger than AK for crack growth, and hence is the more dominant parameter of the two. Thus, there are two corresponding thresholds that must be exceeded for a crack to grow, Fig. 2 (a). At the low end of R=Km,JKm, and especially when R is negative, Kmsx controls fatigue crack growth. Similarly at high R (as R approaches 1) AK controls the growth [52,53].
Fig. 2. Schematic illustration of A K t h Kmx versus Rratio for controlled region (a) and the two parametric crack driving force hK*thK* requirement for fatigue damage, with respective definition of the parameters. Threshold data, when represented in terms of a AK versus Kmx curve, typically show an Lshaped curve with two limiting values corresponding to two fundamental thresholds, Fig. 2(b). At any other crack growth rates, the Lshaped curve shifts with the asymptotic limiting values, AK and K* increasing with crack growth rate, as shown in Fig. 1 (a). Taking into account to the minor effect of the mechanical contributions to the crack advance than the environment effect, Vasudkvan and Sadananda [40] have graphically systematised a wide variety of materials by assuming two independent loading parameters as the relevant crack tip driving forces: AK and K. These critical threshold parameters should be satisfied simultaneously for a crack to grow and can be identified by plotting A&, (cyclic) and K,, (static). The threshold line resulting of graphic construction can map the region where crack growth is only possible for each material under fatigue conditions. These two parameters depend on the alloy microstructure (an intrinsic property of the material), slip mode and environment.
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MATERIAL AND EXPERIMENTAL PROCEDURES The chemical composition of A1 7049 alloy is shown in Table 1. This material is essentially similar to the 7075 alloy, but is more susceptible to corrosion fatigue due to a higher Zn content (7.1% instead of 5.16.1%). The specimens were obtained from a cold rolled sheet of a 7049 aluminium alloy with 10 mm thickness. The 10 mm thick sheets were underaged (UA) and overaged (OA) such that approximately the same yield strengths resulted. The overaged heat treatment was the standard temper treatment (T7351), and the UA treatment was carefully done to match the OA yield strength. The heat treatments are shown in Table 2. Table 1. Composition (wt pct) of 7049 aluminium alloy plate material Zn 7.1
Mg 2.8
Cu 1.7
Cr 0.06
Fe 0.3
Si 0.1
Mn 0.06
Ti 0.05
Ga 0.01
Zr 0.1
A1 Bal.
Table 2. Heat treatment properties of the AI 7049 alloy Solution heat treatment Temper
4 7 0 W 45mid water quench overage (OA) T735 1: 107 "C/8 hours + 163 "C I65 hours
underage (UA) liquid nitrogedl 5 min + S O "C/lOmin + 117 'CY90 min
Table 3. Room temperature mechanical properties of 7049 aluminium alloy alloy/ temper 7049UA 7049OA
yield strength (MPa) Rp0..2 445 44 1
UTS (MPa) 578 497
elongation % 17.2
8.8
area reduction I9 % 23 %
kc (MPaJm)
32.0
The mechanical properties of the two materials are listed in Table 3. Their yield strengths are identical ( 4 4 0 MPa); the tensile strength of the UA material is 16% higher and its ductility is 100% higher. The underaged material contains extremely fine GP zones and q' intermetallic precipitates, whereas the overaged structure contains predominantly coarse q as well as q' precipitates. The 7049UA and 7049OA are two materials exhibiting the same crystallographic texture and grain morphology, but differing in precipitate microstructure. Any difference in the mechanical behaviour of 7049UA and 7049OA can then be attributed to these different precipitate microstructures. The UA alloy specimens were stored at 20 "C prior to fatigue testing to prevent further room temperature aging. After the heat treatments, 6 2 . 5 ~ 6 0 ~ 1mm 0 compact tension (CT) specimens were machined, and notches were introduced parallel to the longitudinal direction so that crack propagation took place in the rolling direction.
The Enuironment Effect on Fatigue Crack Growth Rates in 7049 Aluminium Alloy at
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The fatigue experiments were performed on a servohydraulic testing machine under constant load control with 30 Hz sine wave loading, in ambient air at 20 "C, 50% HR or in vacuum (2.6~10"Pa). The fatigue crack growth curves were obtained by shedding the load in steps of 7% until a crack growth rate approximately 1.3xlO" dcycle was obtained. In the threshold regime the load was reduced 3.5 % only and Aa was less than 0.04 mm during 3x106 cycles at the crack growth rate of 1.3x10" dcycle. After reaching the threshold, the load was increased again to obtain higher crack growth rates. This procedure allowed to get the entire crack growth curve from one specimen. Crack propagation was detected by observing the polished specimen surfaces with a microscope at a magnification of 50x. Calculations of the SIF and dddN were performed according to the ASTM E674 standard.
EXPERIMENTAL RESULTS Fatigue crack propagation rates (dddN) versus the stress intensity factor ranges (AK) are shown in Fig. 3. for the underaged (7049UA) and overaged (7049OA) alloy, respectively, in ambient air (20 "C, 50% HR) and in Fig. 4 under vacuum conditions (2.6~10'~Pa) at R valuesof1, 0.5,0.05, 0.5 andO.8.
Fig. 3. Influence of Rratio on FCGR in overaged (OA) and underaged (UA) A1 7049 alloy in air.
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uAwo.05
0
lo"*,
2
3
i
0
0
lo"'
4
I 1
5 6 7,# 913
AK [MPam ]
2 0 3 0
Fig. 4. Influence of Rratio on FCGR in (OA) and (UA) A1 7049 alloy in vacuum. The threshold values, Kmx, and C and m of Paris regime are summarised in Table 1 and Table 5, respectively, for different Rratios.
~
AL7049 [MPam] R 1 0.5 +0.05 +0.50 +0.80
AL 7049 [dcycle] R 1  0. 5 +0.05 +0.50 +o.80
UA
OA
K,,
x i h
air 8.24 10.53 7.45 4.46 2.80
vacuum 11.0 10.67 6.07 4.35
air 4.12 7.02 7.84 8.92 15.10
vacuum
5.5
air 9.04
vacuum 14.90
air 4.52
vacuum 7.45
11.2 12.14 21.75
4.50 2.72 1.90
5.40 3.04 2.91
4.90 5.44 9.49
5.68 6.08 14.55
dddN =C (AK) rn air vacuum air vacuum 1 ~ 1 0 . ~ ' l x l 0 ~ l y 7.0 8.9 3x10" 7,3 4 ~ 1 0 " ~ 1 ~ 1 0 ~7.3~ 9.0 4x10" SxlO'* 8.0 9.0 3xio'3 ix1ol7 8.1 9.0 UA
C
K,,
M t h
da/dN = C (AK) m air vacuum air vacuum 4x101~ i x ~ o  ~ ~5.0 6.2 OA
C
iX10l3 lxlO" 4x10"
~ ~ 1 0 5.2 . ~ ~ 4x10" 5.1 I X ~ O  ~ ' 5.0
6.3 6.2 6.2
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Figure 5 shows the dependence of thresholds h K t h on R and the h K t h and K, relationship at different Rratios for UA and OA 7049 A1 alloy in ambient air and in vacuum, respectively. An Lshape curve is obtained for the OA alloy over the entire R range for air as well as for vacuum. An increasing h K t h with decreasing R is necessary in order to accomplish crack growth, even at a negative Rratio = 1, whereas no further increase of h K t h is observed for the UA alloy below R=O.O5.
tc,[MPamlR] Fig. 5. Threshold h K t h versus Rratio (a) and K,,,,,. (b) for the UA and OA A1 7049 alloy in air and vacuum.
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In Figures 6 (a) and (b), the threshold values of and K ,, are plotted for different Rratios in ambient air and vacuum, respectively. They show that at low (and negative) Rratios IC is the crack growth governing process, whereas it is A&, at higher Rvalues. This is the case for both environments, ambient air as well as vacuum.
Fig. 6. Thresholds d K t h  KmaXvalues of UA and OA alloy for different stress ratios R in ambient air (a) and in vacuum (b).
The Environment Effect on Fatigue Crack Growth Rates in 7049 Aluminium Alloy at
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Figures 7 (a) and (b) show the dependence of on R for specified constant fatigue crack growth rates in vacuum for OA and UA A1 7049 alloy, respectively. These figures again demonstrate the different deformation features of the two microstructures.
32 30 28
26 24
22
F a E 18
B ;:
3:: 8
6 4
2 1.0
0.5
R
0.0
1.o
3
Fig. 7. h K t h values for different FCGR at different Rratios for (a)OA alloy and (b) UA alloy, in vacuum. Vasudevan and Sadananda [48,49] have pointed out that two critical threshold values, are the necessary requirements for a crack to advance. Likewise two namely A&, and K, critical values above the threshold regime are necessary to obtain specified constant fatigue crack growth rates. This is shown in principle in Fig. 1 (a). In order to obtain defined crack growth rates of (dalcW),, (daldN)~. ..the critical values & , I , & h a . . .and Kmx,l,Kmax,z...are needed. As a result, the values, which have been reconstructed from data of the OA alloy as in
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Fig. 3 and Fig. 4 are plotted in Fig. 8 (a) and (b). The A K ~..., J etc. values have been obtained for several specified constant crack growth rates from the R=0.8 curve (assuming that 0.8 approximates R=l well enough) and the Kmax,lK,z ...values from the R=0.05 curve. This has been done for both environments. The results (Fig. 8 (a)) show that for the same K, higher AK are needed to obtain the same dddN in vacuum than in air. For comparison, the data of Kirby and Beevers for 7075 alloy [25] are plotted as dashed lines. The plot shows that crack propagation took place at lower K,, values than in the present study. The vacuum curve of Kirby et al. results shows a slope of 1, and lower K, are needed (for identical AK) in order to obtain identical dddN values. In Fig. 8 (b) are plotted the results only for UA alloy in ambient air and vacuum. The deviation from slope of 1 may be explained by not high vacuum. 9
6
7
F6 €5
g 4
5 3 2
1 0 0
2
4
6
8
6
1 0 1 2 1 4
L [ N p a mls 9
7 6
F
E5
2
1pJ
0 , . 1 3 0
/.Vacuum1
1. .
a
6
.
1
,
9
b
,
I
,
i
I
Air .
I
,
I
1 2 1 5 1 6 2 1 2 4 2 7
L [ M p a mls
Fig. 8. dK vs. K, results, plotted according to Fig. 1 (a), for OA 7049 alloy (this study) and 7075 alloy (Kirby et al.) in air; (b) for UA 7049 alloy in vacuum
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FRACTOGRAPHY Characterisations of the fracture surface morphology were made by scanning electron microscopy (SEM). Fig. 9 shows the different crack growth surfaces for two UAOA alloys at R=1. The (a)(b) and (c)(d) pictures represent the OA and UA alloys, respectively, under ambient air and vacuum conditions. The (e)(f) pictures show the different microstructures for UA and OA alloys in vacuum, respectively. The OA alloy shows a transcrystalline fracture mode, homogeneous and wavy slip, but more brittle in ambient air than in vacuum, probably induced by hydrogen. The UA alloy shows a planar and localised slip microstructure, with crack branching.
(b) OA, in vacuum
(a) OA, in ambient air
a
I
I
(c) UA, in ambient air
(d) UA, in vacuum
(e) UA, in vacuum
(t) UA, in vacuum
Fig. 9. SEM fracture surfaces after fatigue loading of UAOA alloy, in air and vacuum.
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DISCUSSION Studies of the mechanisms governing fatigue behaviour in aluminium alloys rationalised accelerated crack growth rates in moist media (as compared to those in vacuum or inert environments) in terms of conventional corrosion fatigue processes such anodic dissolution and/or hydrogen embrittlement [26,50,]. Apart from environmental effects, certain intrinsic metallurgical phenomena, in particular those related to slip characteristics, are also considered to cause pronounced differences in nearthreshold crack growth behaviour between different alloys. In addition to environment and microstructurally influenced growth mechanisms, crack closure processes can significantly affect fatigue behaviour in the nearthreshold regime [8,18,32]. Microstructural features directly influence material properties. The toughness, for example, may be reduced by large fractions of GB precipitates produced by inefficient quenching and by aging [14,18,19,22]. The concept of strain localisation in planar slip bands appears to be significant in both monotonic and fatigue testing. Environment and microstructure also strongly influence the fatigue crack growth resistance of high strength aluminium alloys [57] with crack deflection and branching leading to important consequences for the mechanical behaviour [ 101. Local microstructure and the applied AK primarily control the slip mode being responsible for crack propagation. In addition, crack advance can be significantly altered by the presence of the environment [Sl]. As a result, both microstructural and environmental factors have a strong effect on the near threshold fatigue crack growth behaviour. Aiming to contribute for the understanding of these phenomena, the discussion will lay in these two areas: (a) environment and (b) microstructure via slip characteristics. First of all one needs to compare the fatigue results in ambient air to the results in vacuum, in order to distinguish the role of microstructure and environment. In vacuum, the planar slip alloy exhibits a significant fatigue resistance in comparison with to the wavy slip OA alloy microstructure shown by the increased threshold in both d K * , h and x ",. Moreover, due to slip reversibility in the UA alloy, both m * t h and , d can have independently different contributions to the crack growth process: crack branching e.g. can occur in planar slip materials and the crack path can be tortuous, in zigzag, with crystallographic facets. Figures 5 (a) shows the AKh versus Rratio relationship of the OA and UA alloy with decreasing of Rratios. The resulting curve is similar to the systematic curve in Fig. 2 (a) [49] for the OA alloy. In compression (R=1), the UA alloy looses its fatigue resistance in contrast to the OA alloy. This anomalous behaviour of the UA alloy could be due to compressive parts of loading, causing shear loads that induce tensile stresses, which result in secondary cracks parallel to the compression axis. The hKth versus Kmax plot in Fig.5 (b) shows the expected Lshaped curves [52,53] according to Fig. 1 (a) and Fig. 2 (b). It may be seen again that the UA alloy looses the expected Lshape under compression loading probably due to shear loads which can induce tensile stresses. Figures 6 (a) and (b), AK versus R, show that dala is mainly controlled by K,, at Rvalues up to 0.5 and by AK above R=0.5, see also Fig. 2 (a). However, this behaviour is not as pronounced for the UA alloy, with a fatigue resistance at negative Rratios, which results in almost constant AK values and slightly increasing K values at negative Rratios. These results are in principle similar for both environments, although the magnitudes differ. Figures 7 (a) and (b) show a similar dependence of AK on R of the OA alloy for specified constant crack growth rates as in the threshold regime for tests in vacuum. Therefore, again Lshaped curves result, which points to a class IIIa behaviour according to Vasud6van and
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Sadananda [40,48]. The UA alloy shows an Lshape at positive Rratios too, whereas a loss of fatigue crack propagation resistance may be recognised at negative Rratios again. AK reaches a plateau for R