Multiaxial fatigue life prediction for titanium alloy TC4 under proportional and nonproportional loading

International Journal of Fatigue 59 (2014) 170–175

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International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue

Multiaxial fatigue life prediction for titanium alloy TC4 under proportional and nonproportional loading Zhi-Rong Wu a,⇑, Xu-Teng Hu a, Ying-Dong Song a,b a b

College of Energy and Power Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu 210016, China 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

Article history: Received 13 June 2013 Received in revised form 27 August 2013 Accepted 30 August 2013 Available online 12 September 2013 Keywords: Multiaxial fatigue Life prediction Critical plane approach Nonproportional loading Additional cyclic hardening

a b s t r a c t Both proportional and nonproportional tension–torsion fatigue tests were conducted on titanium alloy TC4 tubular specimens. Six multiaxial fatigue parameters are reviewed and evaluated with life data obtained in the tests. It is found that the effective strain, the maximum shear strain and the Smith–Watson–Topper (SWT) criteria tend to give non-conservative results under nonproportional loading. The shear strain-based critical plane approaches, especially Wu–Hu–Song (WHS) approach show better life prediction abilities. The prediction results based on WHS parameter are all within a factor of two scatter band of the test results. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Fatigue failure, which occurs in many engineering components and structures in service, is actually attributed to the multiaxial loads. Multiaxial stresses often exist at notches even under uniaxial loads for the geometrical complexity. Some structures could be subjected to proportional or nonproportional multiaxial loads. The changing of the principle stress and strain axes under nonproportional cyclic loading leads to additional hardening, which is considered to be closely related to the reduction of fatigue life [1–3]. Currently, multiaxial fatigue life prediction approaches can be classified into three categories, namely equivalent stress strain criteria, energy criteria and critical plane criteria. Early multiaxial fatigue life prediction methodologies focused on finding equivalent fatigue damage parameters, which were assumed to produce the same fatigue damage as the uniaxial load. Von Mises criteria and Tresca criteria are the two representative approaches in this category. One of the main shortcomings for these approaches is that they give nonconservative life for nonproportional loading conditions [1,4]. To overcome the shortcomings of the equivalent parameters criteria, the energy-based approach and critical plane approach were developed. Some researchers believed that the fatigue damage process is closely related to cyclic plastic deformation or plastic strain energy. Garud [5] and Jordan et al. [6] used a weight factor 0.5 multiplied on the shear strain energy to account for shear plastic effect ⇑ Corresponding author. Tel.: +86 15062224253; fax: +86 25 84893666. E-mail address: [email protected] (Z.-R. Wu). 0142-1123/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijfatigue.2013.08.028

and a good correlation of multiaxial fatigue data could be obtained in the case of axial-torsional fatigue loading conditions. To overcome the problem that no significant amount of plasticity in the high cycle fatigue can be dealt with, some researchers [7,8] added an elastic energy term into energy parameter. Critical plane criteria are based on the physical observations. Cracks initiate and grow on specific planes. Brown–Miller(BM) [9], Fatemi–Socie (FS) [10], and Smith–Watson–Topper (SWT) [11] make a significant contribution to this category. Due to high strength and stiffness to weight ratio of titanium alloys, they are widely used from aerospace to many industries. The components and structures made by this material such as turbine engine blades and rotors are always subjected to multiaxial loads. The objective of this paper is to study the multiaxial fatigue behavior of titanium alloy TC4 and find some suitable multiaxial fatigue models to predict fatigue life of this material. TC4 is the titanium alloy mark in China. The similar material in America is Ti–6Al– 4V. The proportional and nonproportional tension–torsion fatigue tests conducted on titanium alloy TC4 are presented firstly. Six existent multiaxial fatigue models (the effective strain, the maximum shear strain, the Kandil–Brown–Miller parameter [12], the Smith–Watson–Topper parameter [11], the Fatemi–Socie parameter [10], the Wu–Hu–Song parameter [13]) are reviewed. Then, these multiaxial fatigue models are evaluated with life data obtained in the tests.

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

171

Table 1 The 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

2. Experiments The material used in this investigation is titanium alloy TC4. The chemical composition of the material is given in Table1. The material was subjected to the following heat treatment: 730 °C for 1.5 h and then air cooling. The typical microstructure of the alloy is shown in Fig. 1. Solid bars with 35 mm diameter were machined to solid specimens with 5 mm diameter/30 mm gauge length for monotonic tests and 6 mm diameter/15 mm gauge length for axial fatigue tests. The same bars were also machined to tubular specimens with 17 mm outside diameter, 14 mm inside diameter, and 32 mm gauge length based on ASTM standard E2207 for pure torsional and multiaxial fatigue tests. The configuration and dimensions of tubular specimens are given in Fig. 2. Fatigue tests 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 carried out under fully reversed sinusoidal waveforms with frequency of 0.5–1.0 Hz. Displacement and angle were used as control mode for axial and torsion respectively. An axial–torsion extensometer was used to measure axial and shear strains. Axial load and torque were also recorded. Failure criterion was considered as 10–15% load or torque drop (whichever occurred first) compared with the stable values obtained at midlife. Axial and shear strain amplitudes, De/2 and Dc/2, were measured from the extensometer directly. Axial and shear stress amplitudes, Dr/2 and Ds/2, were uniform across the tubular specimen gauge section and calculated from:

ra

Dr DP ¼ ¼ 2A 2

sa

Ds DT ¼ ¼ 2r m A 2

ð1Þ

ca ¼

r0f E

s0f G

ð2Nf Þb þ e0f ð2Nf Þc

ð2Nf Þb0 þ c0f ð2N f Þc0

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

3. Evaluation of multiaxial fatigue models 3.1. The effective strain method The fatigue life for multiaxial loading is postulated to depend on the value of effective strain amplitude in this approach. Eq. (3) is used here and can been written as [15]

et;a ¼

r0f E

ð2Nf Þb þ e0f ð2Nf Þc

et;a ¼ ee;a þ ep;a ð2Þ

where DP/2 and DT/2 are axial load amplitude and torque amplitude respectively, A is specimen cross-section area, and rm is midsection radius. The experimental results including axial/shear strain and stress amplitudes as well as failure life Nf are summarized in Table 2. The fatigue lives under constant-amplitude axial loading and torsional loading were correlated by Manson–Coffin equations [14]:

ea ¼

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

for axial loading

for torsional loading

ð3Þ

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 modulus. r0f , e0f , b and c are the axial fatigue properties. s0f , c0f , b0 and c0 are the shear fatigue properties. Monotonic properties as well as axial and shear fatigue properties for titanium alloy TC4 are listed in Table 3. The stable cyclic stress–strain relation was correlated by Ramberg–Osgood relation. The cyclic stress–strain properties are also listed in Table 3. All data in Tables 2 and 3 were obtained or calculated at the cycle of 0.5Nf.

ð5Þ ð6Þ

where et;a is the total effective strain amplitude, ee;a and ep;a are the elastic and plastic effective strain amplitudes respectively. For axial–torsion loading situations, ee;a and ep;a are obtained using the von Mises effective strain equations:

ee;a

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2e2e;a ð1 þ me Þ2 þ c2e;a ¼ pffiffiffi 2 2ð1 þ me Þ

ð7Þ

ep;a

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ¼ e2p;a þ c2e;p 3

ð8Þ

1

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 is elastic Poisson’s ratio. The stable effective cyclic stress–strain curve is plotted in Fig. 3 in comparison with the monotonic stress–strain curve. It is observed that the material softens under cyclic loadings. The Ramberg–Osgood relationship was used to fit the axial stable cyclic stress–strain curve. It shows that significant additional hardening for non-proportional loading tests. Fig. 4 shows the fatigue life prediction based on the effective strain parameter. In Fig. 4 we found that non-proportional paths caused more fatigue damage in the region of short life. It is observed that most fatigue data can be correlated well. Several data in the region of long life are out of the bound of factor two on the non-conservative side.

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Table 2 The 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 0.6 0.7 0.8 0.8 0.9 0.9 1.1 1.1 1.3 1.3 1.5 1.7 2.0 2.0 2.3 2.3 – – – – –

– – – – – – – – – – – – – – – – – 0.798 0.833 0.848 0.889 1.038

610.2 655.2 728.6 738.9 766.4 772.5 746.4 755.2 746.7 782.2 787.6 815.8 819.2 856.5 861.6 869.3 861.7 – – – – –

– – – – – – – – – – – – – – – – – 345.6 359.8 374.6 390.3 398.1

60,048 25,069 8457 4135 2544 1708 1730 1007 822 510 529 339 221 124 134 89 127 69,269 51,146 37,449 17,887 7218

– – – – 0 0 0 0 0 0 45 45 45 45 45 45 90 90 90 90 90 90

– – – – 0.345 0.427 0. 576 0. 687 0.863 1.391 0.391 0.418 0.496 0.620 0.772 1.224 0.349 0.418 0.499 0.556 0.632 1.229

1.302 1.645 1.942 2.309 0. 648 0.710 0. 938 1.111 1.371 2.038 0.643 0.702 0.831 1.043 1.255 1.756 0.639 0.704 0.821 0.934 1.079 1.700

– – – – 388.8 466.4 490.6 532.1 538.8 530.5 435.6 472 545.2 592 629 679.8 392.8 475.7 562.6 623.6 703.2 678.6

431.2 417.8 413.5 404.5 278.5 296.0 292.8 312.7 299.4 261 276.9 303.2 342.6 340.9 341.3 353.8 279.6 307.8 356.4 401.2 427.7 382.3

2691 951 459 345 47,195 20,611 4141 1795 868 351 20,953 9478 4898 1563 683 185 45,138 37,273 11,152 2332 1017 233

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

Torsional properties

E (GPa) 108.4

G (GPa) 43.2

ry (MPa)

me

942.5

0.25

r0f (MPa)

b

e0f

c

K (MPa)

n

1116.9

0.049

0.579

0.679

1031

0.0478

K 00 (MPa) 446.7

n00

s0f (MPa)

b0

c0f

c0

716.9

0.06

2.24

0.8

Effective stress amplitude

1000 800 Uniaxial Torsional Inphase o 45 Out-of-phase o 90 Out-of-phase Montonic Axial Ramberg-Osgood

400 200 0 0.000

0.003

0.006

0.009

0.012

0.015

0.018

3.2. The maximum shear strain method It is widely accepted that fatigue crack initiation is due to localized plastic deformation in persistent slip bands of materials. The directions of these persistent slip bands are always aligned with the maximum shear strain direction [1,10,16]. Fatigue cracks are always found to initiate on the maximum shear strain planes under different loading conditions [16–19]. Based on this mechanistic observation, the methodology that multiaxial fatigue life was postulated to depend on the maximum shear strain was proposed. Eq. (4) is used here and can been expressed as follows:

G

ð2Nf Þb0 þ c0f ð2Nf Þc0

where ca,max is the maximum shear strain amplitude.

0.016

Many research results as well as this paper show that the torsional data always lie above the uniaxial data based on the maximum shear strain parameter [10,17,18,20]. Several researchers suggest that a second parameter must be involved in the multiaxial fatigue damage parameter. Brown and Miller [9] took the maximum shear amplitude as the main damage parameter. The normal strain on the maximum shear plane was proposed as the second damage parameter. A convenient form of the Brown and Miller model was suggested by Kandil, Brown and Miller (KBM) [12]:

Fig. 3. Effective cyclic stress–strain relations.

s0f

0

3.3. Critical plane models

Effective strain amplitude

ca;max ¼

0

n 0.0195

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

1200

600

K (MPa) 1054

ð9Þ

ca;max þ sDen ¼ ½1 þ me þ sð1  me Þ  mp Þe0f ð2Nf Þc

r0f E

ð2Nf Þb þ ½1 þ mp þ sð1 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 uniaxial data against the pure torsion data. For the titanium alloy TC4, the value of s is 0.2. The comparison of the predicted and experiment lives are shown in Fig. 6. The prediction results are almost within a factor of two scatter band of the test results except several data in the region of long life. Fatemi and Socie [10] considered that the KBM criterion based only on strain values was not enough to explain the effect of

173

0.03 0.02

(b) 10

Uniaxial Torsional Inphase 45oOut-of-phase 90oOut-of-phase

Predicted life Nf

(a) Effective strain εt,a

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

0.01

102

103

104

105

6

105 104 10

102 102

106

Uniaxial Torsional Inphase 45oOut-of-phase 90oOut-of-phase

3

103

104

105

106

Observed life Nf

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.

0.04 Uniaxial Torsional Inphase 45oOut-of-phase 90oOut-of-phase

0.03 0.02

0.01

102

103

104

105

(b) 106

Predicted life Nf

Maximum shear strain γa,max

(a)

105 104 103 102 102

106

Uniaxial Torsional Inphase 45oOut-of-phase 90oOut-of-phase

103

104

105

106

Observed life Nf

Observed life Nf

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.

(a) 0.04 0.02

Predicted life Nf

0.03

KBM Parameter

(b) 10

Uniaxial Torsional Inphase o 45 Out-of-phase o 90 Out-of-phase

0.01

102

103

104

105

106

6

105 104 Uniaxial Torsional Inphase o 45 Out-of-phase o 90 Out-of-phase

103 102 102

Observed life Nf

103

104

105

106

Observed life Nf

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

additional hardening occurring under non-proportional loading. In order to take into account this effect, they proposed to replace the value of normal strain en with the maximum normal stress rn,max on the maximum shear plane. The critical plane is also the maximum shear plane and the Fatemi and Socie (FS) criterion is expressed as [21]:

  s0f Dcmax rn;max ¼ ð2Nf Þb0 þ c0f ð2Nf Þc0 1þk 2 ry G

ð11Þ

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

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

rn;max

De1 r02 f ¼ ð2Nf Þ2b þ r0f e0f ð2Nf Þbþc 2 E

ð12Þ

where De1 is the maximum normal strain range, rn,max is the maximum stress on the critical plane. The fatigue life correlation based on the SWT parameter is shown in Fig. 8. It has relatively poor performance for pure torsion and multiaxial fatigue tests. Figs. 9 and 10 show the Fatigue crack directions for in-phase and 45° out-of-phase tests respectively. The directions of these cracks are aligned with the maximum shear

174

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

Uniaxial Torsional Inphase 45oOut-of-phase 90oOut-of-phase

0.03

FS Parameter

(b)

0.04

0.02

Predicted life Nf

(a)

0.01

102

103

104

105

106 105 104

10

Uniaxial Torsional Inphase 45oOut-of-phase 90oOut-of-phase

3

102 102

106

103

104

105

106

Observed life Nf

Observed life Nf

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

Uniaxial Torsional Inphase o 45 Out-of-phase o 90 Out-of-phase

10

1 102

103

104

105

106

(b)

108

Predicted life Nf

SWT Parameter

(a) 100

106 Uniaxial Torsional Inphase o 45 Out-of-phase o 90 Out-of-phase

104

102 102

104

106

108

Observed life Nf

Observed life Nf

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%.

strain direction. It is known that the SWT parameter is more suitable for normal fracture materials and it is not surprising to obtain the poor prediction results based on this parameter. It is noted that the critical plane may be planes of either shear strain or tensile strain plane depending on cracking modes. Cracks of most materials may initiate on the maximum shear plane and then propagate on the plane of maximum normal strain plane. It is assumed that the main fatigue damage parameter is the maximum shear strain. And the normal stress and strain on the maximum shear plane assist the fatigue damage. A model based on the FS and SWT models proposed by Wu et al. (WHS) was first

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

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

Dcmax þk 2



rn;max Den E

0:5 ¼

s0f G

ð2Nf Þb0 þ c0f ð2Nf Þc0

ð13Þ

where Dc2max is the maximum shear strain amplitude, rn,max and Den are the maximum normal stress and the normal strain range on the maximum shear plane respectively, k is a material constant found by fitting uniaxial and torsion fatigue data. The form of k is expressed as:

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Uniaxial Torsional Inphase o 45 Out-of-phase o 90 Out-of-phase

WHS Parameter

0.03 0.02

0.01

102

103

104

105

106

(b) 10 Predicted life Nf

(a) 0.04

6

105 104

10

Uniaxial Torsional Inphase o 45 Out-of-phase o 90 Out-of-phase

3

102 102

Observed life Nf

103

104

105

106

Observed life Nf

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

s0f

k¼

G

r0

Acknowledgement

ð2Nf Þb0 þ c0f ð2Nf Þc0  ð1 þ me Þ Ef ð2Nf Þb  ð1 þ mp Þe0f ð2N f Þc  0:5 r02 r0f e0f ð1  me Þ 2Ef 2 ð2Nf Þ2b þ ð1  mp Þ 2E ð2N f Þbþc

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

ð14Þ The parameter k theoretically is not a constant but vary with fatigue life. But in the region of the short lives the fatigue life is less sensitive to small changes in the strain amplitude. Therefore, k is assumed to be the mean value from the intermediate fatigue life to long 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 parameter. The comparison of the predicted and experiment lives are also shown in Fig. 8. All predicted lives are within a factor of two scatter band of the test lives. From these it can be concluded that the WHS parameter is the proper parameter for titanium alloy TC4. 4. Conclusions Multiaxial fatigue tests as well as uniaxial and pure torsional tests were conducted on titanium alloy TC4 specimens. Six multiaxial fatigue prediction models were evaluated based on the test data. The major conclusions are as follows: 1. There is additional hardening observed for titanium alloy TC4 under 45o and 90o out-of-phase loading conditions. The nonproportional loading paths have a little effect on the fatigue damage in the region of short life. 2. The effective strain, the maximum shear strain and the SWT criteria tend to give non-conservative results. 3. The shear strain-based critical plane approaches (KBM parameter and FS parameter) provide similar life prediction abilities. The prediction results based on these two parameters are almost within a factor of two scatter band of the test results except several data in the region of long life. For the additional hardening has less effect on fatigue damage in the region of long life, the advantage that FS parameter can account for additional hardening is not shown for titanium alloy TC4. 4. The WHS parameter proposed based on the FS model and SWT model has better prediction ability than others. The results show that the WHS parameter is suitable for titanium alloy TC4.

References [1] Kanazawa K, Miller KJ, Brown MW. Low-cycle fatigue under out-of-phase loading conditions. Trans ASME J Eng Mater Technol 1977;99:222–8. [2] Socie DF. Multiaxial fatigue damage models. J Eng Mater Technol 1987;109:293–8. [3] Xiong Y, Yu Q, Jiang Y. Multiaxial fatigue of extruded AZ31B magnesium alloy. Mater Sci Eng A 2012;546:119–28. [4] Tipton SM, Nelson DV. In: Miller KJ, Brown MW, editors. Multiaxial fatigue. ASTM STP 853. Philadelphia: ASTM; 1985. p. 514–50. [5] Garud YS. A new approach to the evaluation of fatigue under multiaxial loadings. J Eng Mater Techol Trans ASME 1981;103:118–25. [6] Jordan EH, Brown MW, Miller KJ. In: Miller KJ, Brown MW, editors. Multiaxial fatigue, ASTM STP 853. Philadelphia: ASTM; 1985. p. 569–85. [7] Lee BL, Kim KS, Nam KM. Fatigue analysis under variable amplitude loading using an energy parameter. Int J Fatigue 2003;25:621–31. [8] Noban M, Jahed H, Winker S, Ince A. Fatigue characterization and modeling of 30CrNiMo8HH under multiaxial loading. Mater Sci Eng A 2011;528:2484–94. [9] Brown MW, Miller KJ. A theory for fatigue failure under multiaxial stress– strain conditions. Proc Inst Mech Eng 1973;187(65):745–55. [10] Fatemi A, Socie DF. A critical plane approach to multiaxial fatigue damage including out-of-phase loading. Fatigue Fract Eng Mater Struct 1988;11(3):149–66. [11] Smith RN, Watson P, Topper TH. A stress–strain parameter for the fatigue of metals. J Mater 1970;5(4):767–78. [12] Kandile FA, Brown MW, Miller KJ. Biaxial low cycle fatigue fracture of 316 stainless steel at elevated temperatures, vol. 280. London: The Metal Society; 1982. p. 203–10. [13] Wu ZR, Hu XT, Song YD. Multiaxial fatigue life prediction model based on maximum shear strain amplitude and modified SWT parameter. Chin J Mech Eng 2013;49(2):59–66 [in Chinese]. [14] Han C, Chen X, Kim KS. Evaluation of multiaxial fatigue criteria under irregular loading. Int J Fatigue 2002;24:913–22. [15] Dowling NE. Mechanical behavior of materials. 3rd ed. USA: Prentice; 2007. p. 736–7. [16] Taira S, Inoue T, Yoshida T. Low cycle fatigue under multiaxial stresses (in the case of combined cyclic tension-compression and cyclic torsion) at room temperature. In: Proc 12th Japan Cong Test. Mater 1969;2:50–55. [17] Brown MW, Miller KJ. High temperature low cycle biaxial fatigue of two steels. Fatigue Fract Eng Mater Struct 1979;1:217–29. [18] Socie DF, Waill LA, Dittmer DF. In: Miller KJ, Brown MW, editors. Multiaxial Fatigue, ASTM STP 853. Philadelphia: ASTM; 1985. p. 463–81. [19] Pascoe KJ, Devilliers JWR. Low cycle fatigue of steel under biaxial straining. J Strain Anal 1967;2:117–26. [20] Shamsaei N, Gladskyi M, Panasovski K, Shukaev S, Fatemi A. Multiaxial fatigue of titanium including step loading and load path alteration and sequence effects. Int J Fatigue 2010;32:1862–74. [21] Socie DF, Marquis GB. Multiaxial fatigue. USA: Warrendale; 2000.