International Journal of Fatigue 59 (2014) 170–175

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International Journal of Fatigue

journal homepage: www.elsevier .com/locate / i j fa t igue

Multiaxial fatigue life prediction for titanium alloy TC4 underproportional and nonproportional loading

0142-1123/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijfatigue.2013.08.028

⇑ Corresponding author. Tel.: +86 15062224253; fax: +86 25 84893666.E-mail address: wuzhirongnuaa@gmail.com (Z.-R. Wu).

Zhi-Rong Wu a,⇑, Xu-Teng Hu a, Ying-Dong Song a,b

a College of Energy and Power Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu 210016, Chinab State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu 210016,China

a r t i c l e i n f o a b s t r a c t

Article history:Received 13 June 2013Received in revised form 27 August 2013Accepted 30 August 2013Available online 12 September 2013

Keywords:Multiaxial fatigueLife predictionCritical plane approachNonproportional loadingAdditional cyclic hardening

Both proportional and nonproportional tension–torsion fatigue tests were conducted on titanium alloyTC4 tubular specimens. Six multiaxial fatigue parameters are reviewed and evaluated with life dataobtained in the tests. It is found that the effective strain, the maximum shear strain and the Smith–Wat-son–Topper (SWT) criteria tend to give non-conservative results under nonproportional loading. Theshear strain-based critical plane approaches, especially Wu–Hu–Song (WHS) approach show better lifeprediction abilities. The prediction results based on WHS parameter are all within a factor of two scatterband of the test results.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Fatigue failure, which occurs in many engineering componentsand structures in service, is actually attributed to the multiaxialloads. Multiaxial stresses often exist at notches even under uniax-ial loads for the geometrical complexity. Some structures could besubjected to proportional or nonproportional multiaxial loads. Thechanging of the principle stress and strain axes under nonpropor-tional cyclic loading leads to additional hardening, which is consid-ered to be closely related to the reduction of fatigue life [1–3].Currently, multiaxial fatigue life prediction approaches can be clas-sified into three categories, namely equivalent stress strain criteria,energy criteria and critical plane criteria. Early multiaxial fatiguelife prediction methodologies focused on finding equivalent fatiguedamage parameters, which were assumed to produce the samefatigue damage as the uniaxial load. Von Mises criteria and Trescacriteria are the two representative approaches in this category. Oneof the main shortcomings for these approaches is that they givenonconservative life for nonproportional loading conditions [1,4].To overcome the shortcomings of the equivalent parameters crite-ria, the energy-based approach and critical plane approach weredeveloped. Some researchers believed that the fatigue damage pro-cess is closely related to cyclic plastic deformation or plastic strainenergy. Garud [5] and Jordan et al. [6] used a weight factor 0.5 mul-tiplied on the shear strain energy to account for shear plastic effect

and a good correlation of multiaxial fatigue data could be obtainedin the case of axial-torsional fatigue loading conditions. To over-come the problem that no significant amount of plasticity in thehigh cycle fatigue can be dealt with, some researchers [7,8] addedan elastic energy term into energy parameter. Critical plane criteriaare based on the physical observations. Cracks initiate and grow onspecific planes. Brown–Miller(BM) [9], Fatemi–Socie (FS) [10], andSmith–Watson–Topper (SWT) [11] make a significant contributionto this category.

Due to high strength and stiffness to weight ratio of titanium al-loys, they are widely used from aerospace to many industries. Thecomponents and structures made by this material such as turbineengine blades and rotors are always subjected to multiaxial loads.The objective of this paper is to study the multiaxial fatigue behav-ior of titanium alloy TC4 and find some suitable multiaxial fatiguemodels to predict fatigue life of this material. TC4 is the titaniumalloy mark in China. The similar material in America is Ti–6Al–4V. The proportional and nonproportional tension–torsion fatiguetests conducted on titanium alloy TC4 are presented firstly. Sixexistent multiaxial fatigue models (the effective strain, the maxi-mum shear strain, the Kandil–Brown–Miller parameter [12], theSmith–Watson–Topper parameter [11], the Fatemi–Socie parame-ter [10], the Wu–Hu–Song parameter [13]) are reviewed. Then,these multiaxial fatigue models are evaluated with life dataobtained in the tests.

Table 1The chemical composition of TC4 (wt%).

Al V Fe C N H O Ti

6.4 4.1 0.2 0.01 0.01 0.002 0.16 Balance

Fig. 1. Typical microstructure of titanium alloy TC4 (400�).

Fig. 2. Tubular specimen configuration and dimensions (unit: mm).

Z.-R. Wu et al. / International Journal of Fatigue 59 (2014) 170–175 171

2. Experiments

The material used in this investigation is titanium alloy TC4.The chemical composition of the material is given in Table1. Thematerial was subjected to the following heat treatment: 730 �Cfor 1.5 h and then air cooling. The typical microstructure of the al-loy is shown in Fig. 1. Solid bars with 35 mm diameter were ma-chined to solid specimens with 5 mm diameter/30 mm gaugelength for monotonic tests and 6 mm diameter/15 mm gaugelength for axial fatigue tests. The same bars were also machinedto tubular specimens with 17 mm outside diameter, 14 mm insidediameter, and 32 mm gauge length based on ASTM standard E2207for pure torsional and multiaxial fatigue tests. The configurationand dimensions of tubular specimens are given in Fig. 2. Fatiguetests were conducted on a servo-hydraulic MTS Model 809 axial–torsion testing system. All fatigue tests including axial, torsion,in-phase, 45� out-of-phase, and 90� out-of-phase were carriedout under fully reversed sinusoidal waveforms with frequency of0.5–1.0 Hz. Displacement and angle were used as control modefor axial and torsion respectively. An axial–torsion extensometerwas used to measure axial and shear strains. Axial load and torquewere also recorded. Failure criterion was considered as 10–15%load or torque drop (whichever occurred first) compared withthe stable values obtained at midlife.

Axial and shear strain amplitudes, De/2 and Dc/2, were mea-sured from the extensometer directly. Axial and shear stress ampli-tudes, Dr/2 and Ds/2, were uniform across the tubular specimengauge section and calculated from:

ra ¼Dr2¼ DP

2Að1Þ

sa ¼Ds2¼ DT

2rmAð2Þ

where DP/2 and DT/2 are axial load amplitude and torque ampli-tude respectively, A is specimen cross-section area, and rm is mid-section radius. The experimental results including axial/shearstrain and stress amplitudes as well as failure life Nf are summa-rized in Table 2. The fatigue lives under constant-amplitude axialloading and torsional loading were correlated by Manson–Coffinequations [14]:

ea ¼r0fEð2Nf Þb þ e0f ð2Nf Þc for axial loading ð3Þ

ca ¼s0fGð2Nf Þb0 þ c0f ð2Nf Þc0 for torsional loading ð4Þ

where ea and ca are Axial and shear strain amplitudes respectively,Nf is the failure life, E is Young’s modulus and G is the shear modu-lus. r0f , e0f , b and c are the axial fatigue properties. s0f , c0f , b0 and c0 arethe shear fatigue properties. Monotonic properties as well as axialand shear fatigue properties for titanium alloy TC4 are listed in Ta-ble 3. The stable cyclic stress–strain relation was correlated byRamberg–Osgood relation. The cyclic stress–strain properties arealso listed in Table 3. All data in Tables 2 and 3 were obtained orcalculated at the cycle of 0.5Nf.

3. Evaluation of multiaxial fatigue models

3.1. The effective strain method

The fatigue life for multiaxial loading is postulated to depend onthe value of effective strain amplitude in this approach. Eq. (3) isused here and can been written as [15]

�et;a ¼r0fEð2Nf Þb þ e0f ð2Nf Þc ð5Þ

�et;a ¼ �ee;a þ �ep;a ð6Þ

where �et;a is the total effective strain amplitude, �ee;a and �ep;a are theelastic and plastic effective strain amplitudes respectively. Foraxial–torsion loading situations, �ee;a and �ep;a are obtained usingthe von Mises effective strain equations:

�ee;a ¼1ffiffiffi

2pð1þ meÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2e2

e;að1þ meÞ2 þ32c2

e;a

rð7Þ

�ep;a ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie2

p;a þ13c2

e;p

rð8Þ

where ee,a and ep,a are axial elastic and plastic strain respectively, ce,a

and cp,a are shear elastic and plastic strain respectively, and me iselastic Poisson’s ratio. The stable effective cyclic stress–strain curveis plotted in Fig. 3 in comparison with the monotonic stress–straincurve. It is observed that the material softens under cyclic loadings.The Ramberg–Osgood relationship was used to fit the axial stablecyclic stress–strain curve. It shows that significant additional hard-ening for non-proportional loading tests.

Fig. 4 shows the fatigue life prediction based on the effectivestrain parameter. In Fig. 4 we found that non-proportional pathscaused more fatigue damage in the region of short life. It isobserved that most fatigue data can be correlated well. Severaldata in the region of long life are out of the bound of factor twoon the non-conservative side.

Table 2The experimental results for titanium alloy TC4.

Phase angle (�) ea (%) ca (%) ra (MPa) sa (MPa) Nf Phase angle (�) ea (%) ca (%) ra (MPa) sa (MPa) Nf

– 0.55 – 610.2 – 60,048 – – 1.302 – 431.2 2691– 0.6 – 655.2 – 25,069 – – 1.645 – 417.8 951– 0.7 – 728.6 – 8457 – – 1.942 – 413.5 459– 0.8 – 738.9 – 4135 – – 2.309 – 404.5 345– 0.8 – 766.4 – 2544 0 0.345 0. 648 388.8 278.5 47,195– 0.9 – 772.5 – 1708 0 0.427 0.710 466.4 296.0 20,611– 0.9 – 746.4 – 1730 0 0. 576 0. 938 490.6 292.8 4141– 1.1 – 755.2 – 1007 0 0. 687 1.111 532.1 312.7 1795– 1.1 – 746.7 – 822 0 0.863 1.371 538.8 299.4 868– 1.3 – 782.2 – 510 0 1.391 2.038 530.5 261 351– 1.3 – 787.6 – 529 45 0.391 0.643 435.6 276.9 20,953– 1.5 – 815.8 – 339 45 0.418 0.702 472 303.2 9478– 1.7 – 819.2 – 221 45 0.496 0.831 545.2 342.6 4898– 2.0 – 856.5 – 124 45 0.620 1.043 592 340.9 1563– 2.0 – 861.6 – 134 45 0.772 1.255 629 341.3 683– 2.3 – 869.3 – 89 45 1.224 1.756 679.8 353.8 185– 2.3 – 861.7 – 127 90 0.349 0.639 392.8 279.6 45,138– – 0.798 – 345.6 69,269 90 0.418 0.704 475.7 307.8 37,273– – 0.833 – 359.8 51,146 90 0.499 0.821 562.6 356.4 11,152– – 0.848 – 374.6 37,449 90 0.556 0.934 623.6 401.2 2332– – 0.889 – 390.3 17,887 90 0.632 1.079 703.2 427.7 1017– – 1.038 – 398.1 7218 90 1.229 1.700 678.6 382.3 233

Table 3Fatigue and cyclic stress–strain properties of titanium alloy TC4.

Monotonic properties E (GPa) G (GPa) ry (MPa) me K (MPa) n108.4 43.2 942.5 0.25 1054 0.0195

Uniaxial properties r0f (MPa) b e0f c K0

(MPa) n0

1116.9 �0.049 0.579 �0.679 1031 0.0478

Torsional properties s0f (MPa) b0 c0f c0 K 00 (MPa) n00716.9 �0.06 2.24 �0.8 446.7 0.016

0.000 0.003 0.006 0.009 0.012 0.015 0.0180

200

400

600

800

1000

1200

Eff

ectiv

e st

ress

am

plitu

de

Effective strain amplitude

Uniaxial Torsional Inphase

45oOut-of-phase

90oOut-of-phase

Montonic Axial Ramberg-Osgood

Fig. 3. Effective cyclic stress–strain relations.

172 Z.-R. Wu et al. / International Journal of Fatigue 59 (2014) 170–175

3.2. The maximum shear strain method

It is widely accepted that fatigue crack initiation is due to local-ized plastic deformation in persistent slip bands of materials. Thedirections of these persistent slip bands are always aligned withthe maximum shear strain direction [1,10,16]. Fatigue cracks arealways found to initiate on the maximum shear strain planes underdifferent loading conditions [16–19]. Based on this mechanisticobservation, the methodology that multiaxial fatigue life was pos-tulated to depend on the maximum shear strain was proposed. Eq.(4) is used here and can been expressed as follows:

ca;max ¼s0fGð2Nf Þb0 þ c0f ð2Nf Þc0 ð9Þ

where ca,max is the maximum shear strain amplitude.

Fig. 5 shows the fatigue life prediction based on the maximumshear strain amplitude. The correlation is poor for the most fatiguedata especially for the non-proportional fatigue data. From Fig. 5(a)we find that the torsional data almost all lie above the uniaxial databased on the maximum shear strain parameter. This phenomenonis the base of the following models.

3.3. Critical plane models

Many research results as well as this paper show that thetorsional data always lie above the uniaxial data based on the max-imum shear strain parameter [10,17,18,20]. Several researcherssuggest that a second parameter must be involved in the multiaxialfatigue damage parameter. Brown and Miller [9] took the maxi-mum shear amplitude as the main damage parameter. The normalstrain on the maximum shear plane was proposed as the seconddamage parameter. A convenient form of the Brown and Millermodel was suggested by Kandil, Brown and Miller (KBM) [12]:

ca;max þ sDen ¼ ½1þ me þ sð1� meÞ�r0fEð2Nf Þb þ ½1þ mp þ sð1

� mpÞ�e0f ð2Nf Þc ð10Þ

where Den is the normal strain range on the maximum shear plane,s is the material constant that can be determined by fitting the uni-axial data against the pure torsion data. For the titanium alloy TC4,the value of s is 0.2. The comparison of the predicted and experi-ment lives are shown in Fig. 6. The prediction results are almostwithin a factor of two scatter band of the test results except severaldata in the region of long life.

Fatemi and Socie [10] considered that the KBM criterion basedonly on strain values was not enough to explain the effect of

102 103 104 105 106

0.01

0.02

0.03

Eff

ectiv

e st

rain

εt,a

Observed life Nf

Uniaxial Torsional Inphase

45oOut-of-phase

90oOut-of-phase

(a)

102 103 104 105 106102

103

104

105

106(b)

Uniaxial Torsional Inphase

45oOut-of-phase

90oOut-of-phase

Pred

icte

d lif

e N

f

Observed life Nf

Fig. 4. Fatigue life prediction based on the effective strain: (a) effective strain amplitude vs. Nf correlation; (b) comparison of predicted and experimental lives.

102 103 104 105 106

0.01

0.02

0.03

0.04(a)

Max

imum

she

ar s

trai

n γ a,

max

Uniaxial Torsional Inphase

45oOut-of-phase

90oOut-of-phase

Observed life Nf

102 103 104 105 106102

103

104

105

106

(b)

Pred

icte

d lif

e N

f

Observed life Nf

Uniaxial Torsional Inphase

45oOut-of-phase

90oOut-of-phase

Fig. 5. Fatigue life prediction based on the maximum shear strain: (a) maximum shear strain amplitude vs. Nf correlation; (b) comparison of predicted and experimental lives.

102 103 104 105 106

0.01

0.02

0.03

0.04(a)

KB

M P

aram

eter

Observed life Nf

Uniaxial Torsional Inphase

45oOut-of-phase

90oOut-of-phase

102 103 104 105 106102

103

104

105

106(b)

Pred

icte

d lif

e N

f

Observed life Nf

Uniaxial Torsional Inphase

45oOut-of-phase

90oOut-of-phase

Fig. 6. Fatigue life prediction based on the KBM parameter: (a) KBM parameter vs. Nf correlation; (b) comparison of predicted and experimental lives.

Z.-R. Wu et al. / International Journal of Fatigue 59 (2014) 170–175 173

additional hardening occurring under non-proportional loading. Inorder to take into account this effect, they proposed to replace thevalue of normal strain en with the maximum normal stress rn,max

on the maximum shear plane. The critical plane is also themaximum shear plane and the Fatemi and Socie (FS) criterion isexpressed as [21]:

Dcmax

21þ k

rn;max

ry

� �¼

s0fGð2Nf Þb0 þ c0f ð2Nf Þc0 ð11Þ

where ry is the yield strength, k is an experimental coefficientfound by fitting uniaxial and torsion fatigue data. For the titaniumalloy TC4, the value of k is 0.3. The prediction results (Fig. 7) basedon FS parameter are similar to the KBM parameter. The reason forthis is that the non-proportional paths have less effect on the fati-gue damage in the region of long life for the titanium alloy TC4 usedin this paper.

Socie [2] suggested that a fatigue life prediction model shouldbe dependent on the cracking modes. Smith, Watson and Topper(SWT) [11] parameter was proposed to evaluate multiaxial loadingfor materials with tensile crack. The maximum normal strain planewas taken as the critical plane. The form of SWT parameter can beexpressed as follows:

rn;maxDe1

2¼

r02fEð2Nf Þ2b þ r0f e

0f ð2Nf Þbþc ð12Þ

where De1 is the maximum normal strain range, rn,max is the max-imum stress on the critical plane.

The fatigue life correlation based on the SWT parameter isshown in Fig. 8. It has relatively poor performance for pure torsionand multiaxial fatigue tests. Figs. 9 and 10 show the Fatigue crackdirections for in-phase and 45� out-of-phase tests respectively. Thedirections of these cracks are aligned with the maximum shear

102 103 104 105 106

0.01

0.02

0.03

0.04(a)

FS P

aram

eter

Observed life Nf

Uniaxial Torsional Inphase

45oOut-of-phase

90oOut-of-phase

102 103 104 105 106102

103

104

105

106(b)

Pred

icte

d lif

e N

f

Observed life Nf

Uniaxial Torsional Inphase

45oOut-of-phase

90oOut-of-phase

Fig. 7. Fatigue life prediction based on the FS parameter: (a) FS parameter vs. Nf correlation; (b) comparison of predicted and experimental lives.

102 103 104 105 1061

10

100(a)

SWT

Par

amet

er

Observed life Nf

Uniaxial Torsional Inphase

45oOut-of-phase

90oOut-of-phase

102 104 106 108102

104

106

108(b)

Pred

icte

d lif

e N

f

Observed life Nf

Uniaxial Torsional Inphase

45oOut-of-phase

90oOut-of-phase

Fig. 8. Fatigue life prediction based on the SWT parameter: (a) SWT parameter vs. Nf correlation; (b) comparison of predicted and experimental lives.

Fig. 9. Fatigue crack direction for in-phase test with strains: ea = 0.345%,ca = 0.648%.

Fig. 10. Fatigue crack direction for 45� out-of-phase test with strains: ea = 0.391%,ca = 0.643%.

174 Z.-R. Wu et al. / International Journal of Fatigue 59 (2014) 170–175

strain direction. It is known that the SWT parameter is more suit-able for normal fracture materials and it is not surprising to obtainthe poor prediction results based on this parameter.

It is noted that the critical plane may be planes of either shearstrain or tensile strain plane depending on cracking modes. Cracksof most materials may initiate on the maximum shear plane andthen propagate on the plane of maximum normal strain plane. Itis assumed that the main fatigue damage parameter is the maxi-mum shear strain. And the normal stress and strain on the maxi-mum shear plane assist the fatigue damage. A model based onthe FS and SWT models proposed by Wu et al. (WHS) was first

mentioned in Ref. [13]. This model contains the advantages of FSand SWT models. The parameter is given by

Dcmax

2þ k

rn;maxDen

E

� �0:5

¼s0fGð2Nf Þb0 þ c0f ð2Nf Þc0 ð13Þ

where Dcmax2 is the maximum shear strain amplitude, rn,max and Den

are the maximum normal stress and the normal strain range on themaximum shear plane respectively, k is a material constant foundby fitting uniaxial and torsion fatigue data. The form of k isexpressed as:

102 103 104 105 106

0.01

0.02

0.03

0.04(a)

WH

S Pa

ram

eter

Observed life Nf

Uniaxial Torsional Inphase

45oOut-of-phase

90oOut-of-phase

102 103 104 105 106102

103

104

105

106(b)

Pred

icte

d lif

e N

f

Observed life Nf

Uniaxial Torsional Inphase

45oOut-of-phase

90oOut-of-phase

Fig. 11. Fatigue life prediction based on the WHS parameter: (a) WHS parameter vs. Nf correlation; (b) comparison of predicted and experimental lives.

Z.-R. Wu et al. / International Journal of Fatigue 59 (2014) 170–175 175

k ¼s0

f

G ð2Nf Þb0 þ c0f ð2Nf Þc0 � ð1þ meÞr0

f

E ð2Nf Þb � ð1þ mpÞe0f ð2Nf Þc

ð1� meÞr02

f

2E2 ð2Nf Þ2b þ ð1� mpÞr0

fe0

f

2E ð2Nf Þbþc� �0:5

ð14Þ

The parameter k theoretically is not a constant but vary with fatiguelife. But in the region of the short lives the fatigue life is less sensi-tive to small changes in the strain amplitude. Therefore, k is as-sumed to be the mean value from the intermediate fatigue life tolong fatigue life. For the titanium alloy TC4, the value of k is 0.35.Fig. 11 shows the fatigue life prediction based on the WHS param-eter. The comparison of the predicted and experiment lives are alsoshown in Fig. 8. All predicted lives are within a factor of two scatterband of the test lives. From these it can be concluded that the WHSparameter is the proper parameter for titanium alloy TC4.

4. Conclusions

Multiaxial fatigue tests as well as uniaxial and pure torsionaltests were conducted on titanium alloy TC4 specimens. Six multi-axial fatigue prediction models were evaluated based on the testdata. The major conclusions are as follows:

1. There is additional hardening observed for titanium alloyTC4 under 45o and 90o out-of-phase loading conditions.The nonproportional loading paths have a little effect onthe fatigue damage in the region of short life.

2. The effective strain, the maximum shear strain and theSWT criteria tend to give non-conservative results.

3. The shear strain-based critical plane approaches (KBMparameter and FS parameter) provide similar life predictionabilities. The prediction results based on these two param-eters are almost within a factor of two scatter band of thetest results except several data in the region of long life.For the additional hardening has less effect on fatiguedamage in the region of long life, the advantage that FSparameter can account for additional hardening is notshown for titanium alloy TC4.

4. The WHS parameter proposed based on the FS model andSWT model has better prediction ability than others. Theresults show that the WHS parameter is suitable for tita-nium alloy TC4.

Acknowledgement

This work was supported by National Defense Basic ScientificResearch Project of China.

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