International Journal of Fatigue 33 (2011) 842–848

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International Journal of Fatigue

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

Fatigue crack initiation and growth behavior of Ti–6Al–4V undernon-proportional multiaxial loading

H. Nakamura a,⇑, M. Takanashi a, T. Itoh b, M. Wu c, Y. Shimizu c

a Structural Strength Department, Research Laboratory, IHI Corporation, 1, Shin Nakahara-Cho, Isogo-Ku, Yokohama-shi, Kanagawa 235-8501, Japanb Division of Mechanical Engineering, Graduate School of Engineering, University of Fukui, Japanc Graduate School of Engineering, University of Fukui, Japan

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

Article history:Received 31 August 2010Received in revised form 15 December 2010Accepted 29 December 2010Available online 1 January 2011

Keywords:Multiaxial fatigueLow cycle fatigueTitanium alloyCrack nucleationFatigue crack growth

0142-1123/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.ijfatigue.2010.12.013

⇑ Corresponding author. Tel.: +81 45 759 2864.E-mail address: hiroshi_nakamura_2@ihi.co.jp (H.

This paper deals with multiaxial low cycle fatigue crack behavior of Ti–6Al–4V under non-proportionalloading. Strain controlled fatigue tests under proportional loading and non-proportional loading with90� out-of-phase difference between the axial and shear strains e and c were carried out on tubular spec-imens at room temperature. As a result, Mises strain based fatigue lives under non-proportional loadingwere approximately 1/10 of those under proportional loading. The specimen surfaces were observed toevaluate life reduction. These results showed that non-proportional loading caused 10 times more cracksthan those of proportional loading. Non-proportional loading changes directions of the principal stressand strain axes, and high shear stresses acting on many planes stimulate more slip planes. Consequently,non-proportional loading resulted in many cracks. In addition, the high shear stresses also resulted in lar-ger misorientation under non-proportional loading. The crack initiation life defined as the length 2a of30 lm did not differ significantly between proportional loading and non-proportional loading. However,the fatigue cracks under non-proportional loading propagated faster those under proportional loading.This is because the fatigue cracks under non-proportional loading are subjected to more severe strainfield than those under proportional loading, even though the crack size and applied strain are the same.Thus, the significant reduction in fatigue life under non-proportional loading is caused by the acceleratedcrack growth due to a higher strain intensity factor.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Ti–6Al–4V has been used for aero engine components becauseof its excellent specific strength and corrosion resistance. Mechan-ical structures such as the rotating aero engine components areroutinely subjected to thermal stress and body force, and thesecyclic loading conditions cause multiaxial low cycle fatigue (LCF).It has been reported that the fatigue lives under the multiaxialLCF non-proportional loading, in which the directions of the prin-cipal axes are changed within a cycle, are seriously reduced [1–15]. In general, fatigue life reduction under non-proportional load-ing is accompanied by an additional strain hardening. Several stud-ies have reported the material dependence between life reductionand additional strain hardening [4,5,7,8,13,15]. Among them, onlya few papers deal with the multiaxial fatigue of Ti alloys [16–18],and the fatigue behavior under the non-proportional loading hasnot been clarified yet.

In this study, strain controlled fatigue tests were conducted onTi–6Al–4V using proportional loading and non-proportional load-

ll rights reserved.

Nakamura).

ing. Observations on microcrack initiation and propagation werecarried out by replicating the specimen surface during the fatiguetests. Since electron back-scattering diffraction (EBSD) measure-ments could identify crystal misorientation which relates to fati-gue damage [19], the EBSD measurements were used to evaluatethe damage to microstructure under proportional loading andnon-proportional loading. The fracture mechanism under non-pro-portional loading will be discussed in view of additional hardening,microcrack initiation and propagation and crystal misorientation.

2. Materials and experimental procedure

2.1. Material and specimen

Table 1 shows the chemical composition of the Ti–6Al–4V used.The material was forged and subjected to the following heat treat-ment: 960 �C for 1 h with water-cooling, and 705 �C for 2 h. withair-cooling. The bimodal a and b microstructure in Fig. 1 was ob-tained after the heat treatment. Table 2 lists the mechanical prop-erties of the material at room temperature. The shape of thespecimen for fatigue tests is shown in Fig. 2. The dimensions of

Table 1Chemical composition of Ti–6Al–4V (in wt.%).

C V Al N Fe O H Ti

0.002 4.16 6.30 0.004 0.16 0.20 0.006 Bal.

Fig. 1. Microstructure of Ti–6Al–4V.

Table 2Mechanical properties of Ti–6Al–4V alloy.

Elastic modulus (GPa) 0.2% Proof stress (MPa) Tensile strength (MPa)

118 943 1017

Fig. 2. Geometry and sizes of specimen (in mm).

Type of loadingsProportional loadin

PP

Time step

Strain path

Fig. 3. Strain path

H. Nakamura et al. / International Journal of Fatigue 33 (2011) 842–848 843

the specimen were 6.8 mm for the parallel part, 9 mm inner diam-eter and 11 mm outer diameter in the gage part.

2.2. Fatigue test and multiple step test

Fig. 3 shows strain paths in this study. e and c are axial andshear strains, respectively. There are two types of proportionalpaths. PP is the push–pull path and TP is the torsion path. CP rep-resents 90� out-of-phase loading which is a non-proportional path.The Mises equivalent strain controlled tests at the strain rate of0.1%/s or 0.5%/s were carried out at room temperature. The strainrate of 0.1%/s was used for short fatigue life. On the other hand,0.5%/s was used for higher cycle region to reduce testing time.All of the tests were conducted at fully reversed conditions, i.e.strain ratio R of �1. The number of cycles to failure, Nf, was deter-mined as the cycle at which the axial or shear stress amplitude de-creased by 25% from its cyclically stable value. The multiple steptests with increasing strain levels were conducted to examinethe additional hardening effect due to non-proportional loading.Strain sequence for the test is illustrated in Fig. 4. Stress–strainrelationship was obtained after 10 cycles at each strain level. Strainpaths for the multiple step tests were PP and CP at the strain rate of0.1%/s.

2.3. Replicating fatigue crack

Information on microcrack initiation and propagation was ob-tained by replicating the specimen surface with RepliSet (StruersA/S). Replication was performed for the specimen with Mises strainrange Deeq = 1.2% at selected cycles. The replicas were observedwith a laser microscope (manufactured by Keyence Corp., VK-9700), and were examined at a magnification of 400 times. Crackinitiation was defined as a crack length 2a of 30 lm because acrack of 30 lm was identifiable at this magnification. The observedarea was 2.5 � 2.2 mm2 centered on a critical crack leading to fati-gue failure.

2.4. Surface preparation and conditions for EBSD measurements

After the fatigue test, specimens of Deeq = 1.2% and 1.7% wereused for the EBSD measurements. The samples were observed attheir uncracked portions. The surface of the sample was polishedusing 1 lm aluminum oxide followed by colloidal silica (Struers

g Non-proportional loadingTP CP

s employed.

Fig. 4. Strain sequence for multiple step test.

844 H. Nakamura et al. / International Journal of Fatigue 33 (2011) 842–848

A/S, OP-S) with hydrogen peroxide in order to achieve flat surfacesfree from damage. The EBSD measurements were carried out on afield emission scanning electron microscope (Royal Philips Elec-tronics, XL30) equipped with an orientation imaging microscopy(OIM) system developed by TSL. The crystal orientation maps wereobtained from EBSD measurement with an acceleration voltage of20 kV. The observed area was 50 � 100 lm2 with a step size of0.2 lm. Three areas were measured for each sample. Since the a-phase is generally assumed to support mainly the material defor-

:PP:TP:CP

Ti-6Al-4V, RT, Δ eq=1.7%ε

(a) Δεeq=1.7%

Number of cycles

Mis

es s

trai

n ra

nge

Δeq

(MP

a)σ

100 101 102 103 1040

500

1000

1500

2000

2500

:PP:TP:CP

Ti-6Al-4V, RT, Δ eq=1.2%ε

(b) Δε eq=1.2%

Number of cycles100 101 102 103 105104

Mis

es s

trai

n ra

nge

Δeq

(MP

a)σ

0

500

1000

1500

2000

2500

Fig. 5. Variation of Mises stress ranges with number of cycles.

mation for bimodal Ti–6Al–4V, the investigations were focusedon this phase. Kernel average misorientation (KAM) was used toevaluate the fatigue damage. The KAM is the arithmetic mean ofscalar misorientations between groups of pixels and representslocalized misorientation [20].

3. Experimental results

3.1. Low cycle fatigue life and cyclic stress–strain relationship

The variation of Mises stress ranges during the fatigue tests areshown in Fig. 5. Applied strain of Deeq = 1.7% caused plastic strain,and the material indicates cyclic softening behavior under each ofPP, TP and CP. For Deeq = 1.2%, the material deformed elastically sothat the stress is stabilized during the fatigue tests. These resultssuggest that the cyclic behaviors do not differ seriously betweenproportional loading and non-proportional loading.

Fig. 6 shows the relationship between the Mises equivalentstrain range and Nf. The solid line is the regression line based onthe data of fatigue lives for PP and the dashed lines represent a fac-tor of 2 scatter band. The fatigue lives for TP were within the factorof 2 scatter band. On the other hand, the fatigue lives for CP wereapproximately 1/10 of those for PP.

Fig. 7 indicates the cyclic stress–strain relationships for themultiple step tests under PP and CP. The two axes represent Misesstress and strain amplitudes from the hysteresis loop. A monotoniccurve is also shown in this figure. There was no significant differ-ence between the stress–strain relationships under PP and CP. In

Ti-6Al-4V, RT

PPTPCP

Factor of 2

Mis

es s

trai

n ra

nge

Δeq

(%)

ε

Number of cycles to failure Nf

102 103 104 105

1

0.5

0.6

0.70.80.9

2

3

Fig. 6. Mises equivalent strain range and number of cycles to failure.

Ti-6Al-4V, RT

:PP:CP:Monotonic curve

Mises strain amplitude eq (%)ε

Mis

es s

tres

s am

plit

ude

eq

(MP

a)σ

0.5 1 1.50

200

400

600

800

1000

1200

Fig. 7. Cyclic stress–strain relationship in PP and CP.

H. Nakamura et al. / International Journal of Fatigue 33 (2011) 842–848 845

general, fatigue life reduction under non-proportional loading isaccompanied by additional strain hardening [1–15]. Although Ti–6Al–4V under non-proportional loading had not shown any addi-tional hardening effect, the fatigue lives were significantly reduced.Similar phenomenon has been reported for other titanium alloys aswell [18].

3.2. Microcrack initiation and propagation behavior

The replicated cracks at the later stage of the fatigue life areshown in Fig. 8. The PP specimen exhibited a few microcracksaround the critical crack in Fig. 8a. The number of microcrackswas eight in the observed area. On the other hand, Fig. 8b showsthat there were about 40 microcracks around the critical one forTP. Moreover, the CP specimen had much more microcracks thanthose of PP and TP in Fig. 8c. The number of cracks was more than200.

Fig. 9 indicates the number of microcracks, which are classifiedby angle and length, as a function of the life ratio N/Nf. The totalnumber of cracks for PP increased with the life ratio in Fig. 9a. Mostcracks were oriented at angles in the range between ±22.5� and 0�.The number of long cracks augmented with the life ratio in Fig. 9b.Each microcrack had grown independently.

Fig. 9c shows that the total number of cracks for TP increasesuntil N/Nf = 0.91, and then the number decreases slightly. Thenumber of cracks longer than 200 lm in length increased withthe life ratio, but the number of cracks from 100 to 200 lm de-creased at N/Nf = 0.97 in Fig. 9d. This result indicates that crackcoalescence had occurred at the later stage of the fatigue life.Fig. 9c shows that microcracks propagate primarily at 0�, and thatthey grow at other angles as well.

m1Axi

al d

irec

tion

(a) PP: N=10000 (N/Nf =0.99) (b) TP: N=204

Fig. 8. Critical cracks lead

The total number of cracks for CP indicates the local maximumat N/Nf = 0.65 in Fig. 9e. After that, the number decreased and thenincreased again. This tendency could be seen particularly in thenumber of cracks shorter than 100 lm in Fig. 9f. Thus, the crackcoalescence for CP had started in the middle stage of the fatiguelife. Fig. 9e shows that most cracks are oriented at angles in therange between ±22.5� and 0�.

The critical crack length versus the number of cycles is shown inFig. 10. For PP, the critical crack longer than 30 lm in length wasobserved on the replica at 500 cycles (N/Nf = 0.05). The crack forPP propagated slowly during most of the fatigue life. The crackfor TP longer than 30 lm was seen at 2000 cycles (N/Nf = 0.1).The crack growth behavior was almost the same as that of PP.For CP, a crack longer than 30 lm was also observed at 500 cycles(N/Nf = 0.22). These results suggest that the crack initiation lives donot differ significantly between proportional loading and non-pro-portional loading. However, the critical crack for CP propagatedfaster than those of PP and TP.

3.3. Kernel average misorientation

Fig. 11 shows KAM for axial section versus applied total strainrange. The KAM for PP with Deeq = 1.7% increased compared to thatof pristine, but the KAM with Deeq = 1.2% indicated almost thesame magnitude as that of pristine. This suggests that Deeq = 1.2%for PP may be too small to cause enough change in misorientationto be detected by EBSD measurements. On the other hand, the KAMfor CP increased with the applied strain range and was larger thanthat of PP. These tendencies could also be seen in a round section.Therefore, non-proportional loading caused larger misorientationin the microstructure than proportional loading.

4. Discussion

The variation of normal and shear stress ranges acting on thecrack plane is shown in Fig. 12. These stresses are normalized withrespect to the applied axial or shear stress ranges. For PP, the max-imum normal stress, i.e. the principal stress, acts on a plane at anangle of 0� and maximum shear planes are on ±45� in Fig. 12a.The maximum normal stress planes for TP are on ±45� and themaximum shear stresses act on the 0� and ±90� planes inFig. 12b. For CP, it has been pointed out that both maximum nor-mal stress and maximum shear stress act on the same plane atan angle of 0� in Fig. 12c. Moreover, relatively greater shear stres-ses act on many planes for CP compared to PP and TP. Previousstudies have shown that non-proportional loading activates moreslip planes than proportional loading because the principal stressand strain axes are rotated during a cycle [4,7,11,13,15]. Therefore,high shear stresses acting on many planes stimulate more slipplanes so that CP caused 10 times more cracks than those of PP.

m

00(N/Nf =0.97) (c) CP: N=2100 (N/Nf =0.91)

ing to fatigue failure.

Ti-6Al-4VStrain path:PP

eq=1.2%εΔN

umbe

r of

cra

cks

Life ratio N/Nf

0 0.2 0.4 0.6 0.8 1

10

20

30

40

50

Num

ber

of c

rack

s

Life ratio N/Nf

0 0.2 0.4 0.6 0.8 1

10

20

30

40

50

Num

ber

of c

rack

s

Life ratio N/Nf

0 0.2 0.4 0.6 0.8 1

(a) Crack for PP classified by angle.

Ti-6Al-4VStrain path:TP

eq=1.2%εΔ

(c) Crack for TP classified by angle.

Ti-6Al-4VStrain path:CP

eq=1.2%εΔ

(e) Crack for CP classified by angle.

(b) Crack for PP classified by length.

(d) Crack for TP classified by length.

(f) Crack for CP classified by length.

50

100

150

200

250

Fig. 9. Number of fatigue cracks as a function of life ratio.

846 H. Nakamura et al. / International Journal of Fatigue 33 (2011) 842–848

In addition, these stresses also resulted in larger misorientation forCP.

For elastic–plastic loading conditions, a strain intensity factor isoften used. Socie et al. suggested the equivalent strain intensityfactor based on the strain energy release rate as follows [21]:

DKeqðeÞ ¼ Y1EDen

ffiffiffiffiffiffipap� �2 þ Y2GDcm

ffiffiffiffiffiffipap� �2

h i12 ð1Þ

where Y1 and Y2 are the Modes I and II geometry factors, E is elasticmodulus, G is shear modulus, Den is total tensile strain range normalto the maximum shear strain plane, Dcm is the total maximumshear strain range, a is the half surface crack length. Fig. 13 showsequivalent strain intensity factor DKeq(e) with respect to the surfacecrack direction. The crack length 2a = 100 lm and applied strainDe ¼ Dc=

ffiffiffi3p¼ 1:2% were used for the calculation. The aspect ratio

Number of cycles

:PP:TP:CP

Ti-6Al-4V

eq=1.2%Su

rfac

e cr

ack

leng

th 2

(μ

m)

aεΔ

102 103 104 1050

500

1000

1500

2000

Fig. 10. Critical crack propagation behavior.

Ker

nel a

vera

ge m

isor

ient

atio

n (d

egre

e)

Ti-6Al-4V, RTAxial section

: PP: CP

Strain range eq (%)εΔ0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

Fig. 11. Kernel average misorientation with respect to applied strain range.

(a) PP

Normal stress

Shear stressTP

(b) TP

Normal stress

Shear stressPP

Nor

mal

ized

str

ess

Angle (degree)θ-90 -60 -30 0 30 60 900

0.5

1

1.5

Nor

mal

ized

str

ess

Angle (degree)θ-90 -60 -30 0 30 60 900

0.5

1

1.5

Normal stress

Shear stressCP

(c) CP

Nor

mal

ized

str

ess

Angle (degree)θ-90 -60 -30 0 30 60 900

0.5

1

1.5

Fig. 12. Variation of normal and shear stress ranges with respect to crack plane.

H. Nakamura et al. / International Journal of Fatigue 33 (2011) 842–848 847

of the crack was assumed to be 0.5. For PP, the maximum strainintensity factor occurs at 0�, and this agrees with the crack growthdirection (Fig. 9a). For TP, the constant strain intensity factor acts onevery plane. Therefore, fatigue cracks for TP propagated not only at0� but also at other angles (Fig. 9c). For CP, the maximum strainintensity factor is caused on the plane at 0� and this correspondsto the crack propagation direction (Fig. 9e). Moreover, CP showshigher strain intensity factor than those of PP and TP. This suggeststhat even though the crack size and applied strain are the same, thefatigue crack for CP is subjected to a more severe strain field. Con-sequently, the crack for CP propagated faster than those of PP andTP.

Fig. 14 indicates the relationship between the crack growth rateda/dN and the equivalent strain intensity factor DKeq(e). The corre-lation is represented by Paris law as follows:

dadN¼ C DKeqðeÞ

� �m ð2Þ

The material constants C and m in Paris law are explained in Fig. 14.It shows good correlation among crack propagation behavior for PP,TP and CP.

Fig. 15 shows fatigue life prediction based on crack growthanalysis. The Paris equation was integrated for the analysis asfollows:

Np ¼Z af

ai

da

C DKeqðeÞ� �m ð3Þ

where Np is the fatigue crack propagation life, ai is initial crack size,af is the final crack size. The material constants C and m were used

for PP, TP and CP shown in Fig. 14 respectively. Fatigue life Nf isgiven by

Nf ¼ Nc þ Np ð4Þ

Here, Nc is the fatigue crack nucleation life defined as 500 cyclesfrom the crack propagation behavior at Deeq = 1.2% shown inFig. 10. The initial and final crack sizes are defined as 2ai = 30 lm,and 2af = 1 mm. The aspect ratio of the crack was assumed to be0.5. To simplify the crack growth analysis, it was assumed thatthe cracks were fully open during the entire compressive part ofloading cycle. The dashed lines represent a factor of 2 scatter bandin fatigue life in Fig. 15. Most calculated lives are within the band sothat the correlation of the test results appears good. At short fatiguelives, Eqs. (3) and (4) overestimate because crack nucleation lives

:PP

:TP

:CP

eq(

)(M

Pa

m)

K

εΔ

Angle (degree)θ-90 -60 -30 0 30 60 90

5

10

15

20

Fig. 13. Variation of equivalent strain intensity factor with respect to crack plane.

:PP:TP:CP

Ti-6Al-4V

eq( ) (MPa m)K εΔ

eq=1.2%εΔ

(m/c

ycle

)a

d /d

N

1 10 10010-9

10-8

10-7

10-6

C m PP 9.79×10-11 2.04 TP 8.71×10-11 1.92 CP 9.86×10-11 2.37

Fig. 14. Surface crack growth rate versus equivalent strain intensity factor.

Experimental life

Pre

dict

ed li

fe

: PP: TP: CP

Ti-6Al-4V, RT

Factor of 2

102 103 104 105 106102

103

104

105

106

Fig. 15. Life prediction based on crack growth analysis.

848 H. Nakamura et al. / International Journal of Fatigue 33 (2011) 842–848

would be shorter than 500 cycles. On the contrary, fatigue lives athigher cycle region are underestimated because crack nucleationlives would be longer. Therefore, the significant life reductionshown in Ti–6Al–4V under non-proportional loading is attributed

to the accelerated crack growth due to higher strain intensity fac-tors than those under proportional loading.

5. Conclusion

This paper focused on the fatigue crack behavior of Ti–6Al–4Vunder non-proportional loading. The fatigue lives under non-pro-portional loading were approximately 1/10 of those under propor-tional loading. Observations of microcracks showed that therewere 10 times more cracks for CP than those of PP. The non-proportional loading changes directions of the principal stress andstrain axes, and high shear stresses acting on many planes stimu-late more slip planes. As a result, CP caused 10 times more cracksthan those of PP. In addition, high shear stresses also resulted inlarger misorientation for CP. The crack initiation lives did not differamong PP, TP and CP, but the fatigue crack for CP has propagatedfaster. This is because the fatigue cracks for CP are subjected tomore severe strain field than those of PP and TP, even though thecrack size and applied strain are the same. Thus, the significantreduction of fatigue life under non-proportional loading is causedby the accelerated crack growth due to higher strain intensity fac-tor than those under proportional loading.

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