Multiaxial fatigue of extruded AZ61A magnesium alloy

International Journal of Fatigue 33 (2011) 437–447

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Multiaxial fatigue of extruded AZ61A magnesium alloy Qin Yu a, Jixi Zhang a, Yanyao Jiang a,⇑, Qizhen Li b a b

Department of Mechanical Engineering, University of Nevada, Reno, Reno, NV 89557, United States Department of Chemical and Metallurgical Engineering, University of Nevada, Reno, Reno, NV 89557, United States

a r t i c l e

i n f o

Article history: Received 4 May 2010 Received in revised form 8 September 2010 Accepted 30 September 2010 Available online 20 October 2010 Keywords: Extruded AZ61A magnesium alloy Multiaxial fatigue Cyclic deformation

a b s t r a c t Strain-controlled multiaxial fatigue experiments were conducted on extruded AZ61A magnesium alloy using thin-walled tubular specimens in ambient air. The experiments included fully reversed tension– compression, cyclic torsion, proportional axial-torsion, and 90° out-of-phase axial-torsion. For the same equivalent strain amplitude, fatigue life under proportional loading was the highest and the nonproportional loading resulted in the shortest fatigue life. Detectable kinks were identified in the strain–life curves for all the loading paths. Fatigue experiments subjected to fully reversed strain-controlled torsion with a static axial load were also conducted. A positive static axial stress reduced the fatigue life and a compressive static axial stress was found to significantly enhance the fatigue life. Two critical plane multiaxial fatigue criteria were evaluated in terms of fatigue life predictions based on the experimental results. The Fatemi–Socie criterion correlated well with the fatigue life in the low-cycle fatigue regime which was characterized by shear cracking. The fatigue life predictions made by the Fatemi–Socie criterion did not agree well with the experimental results in the high-cycle fatigue regime. A modified Smith–Watson–Topper (SWT) criterion was found to be able to predict fatigue lives well for all the loading paths conducted in the current investigation. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction Due to low density, high specific strength and stiffness, and good recyclability, magnesium (Mg) alloys are becoming attractive structural materials in automobile and aircraft industries [1–3]. Structural components are inevitably subjected to cyclic loading, and fatigue failure often occurs. Therefore, it is vital to understand the fatigue behavior and the damage mechanisms of Mg alloys. Mg alloys are usually classified into cast and wrought (rolled, extruded, and forged) alloys. Cast Mg alloys have defects such as casting pores and inclusions, whereas wrought alloys are essentially free of casting defects. Therefore, wrought Mg alloys exhibit superior fatigue properties and are appropriate for the study of the intrinsic fatigue mechanisms [4–7]. Most of the earlier studies were concentrated on the fatigue resistance of cast Mg alloys. Due to low strength, cast Mg alloys have limited applications. Wrought Mg alloys have found their applications in critical loadbearing components in transportation vehicles. As a result, significant research has been conducted on the cyclic deformation and fatigue of wrought Mg alloys in the last decade [8–21]. Due to the effects of texture and microstructure, fatigue properties of wrought Mg alloys are anisotropic [22,23].

⇑ Corresponding author. Tel.: +1 775 784 4510; fax: +1 775 784 1701. E-mail address: [email protected] (Y. Jiang). 0142-1123/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2010.09.020

The fatigue behavior of wrought Mg alloys is very different from that of cast Mg alloys. For cast Mg alloys, fatigue cracks typically initiate at casting pores and propagate through precipitate particles, interdentritic regions, or coalescence of small cracks [24–31]. Fatigue crack initiation life is short and the majority of fatigue life is spent on propagating the fatigue crack. For wrought Mg alloys, the fatigue crack preferentially initiates at slip bands, inclusions, or deformation twins, depending on microstructure and loading amplitude. Fatigue crack propagation is mainly along the persistent slip bands or deformation twins [8,16,18,19,23,32–35]. Most of the fatigue life is spent on crack initiation and microcrack propagation. In addition to manufacturing process, many other factors including heat treatment [36–38], microstructure [5,33,34,39], environmental effect [21,29,40–42], and loading ratio [9,11,24] have significant influences on the fatigue properties of Mg alloys. For example, corrosive environment and elevated temperature drastically reduce the fatigue strength of Mg alloys [3,29,42,43]. At the same strain amplitude, the fatigue life is increased with decreasing strain ratio [9,11,24]. The strain–life fatigue curve of both cast and wrought Mg alloys was reported to exhibit a smooth transition from the low-cycle fatigue regime to the high-cycle fatigue regime [13,44] and the fatigue curve can be described well by the Basquin and Manson–Coffin equations [7,10,15,25,44,45]. Hasegawa et al. [15] found that the Manson–Coffin relationship nicely described the fatigue lives of extruded AZ31 Mg alloy obtained under stress-controlled loading

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but failed to describe the fatigue lives obtained from the straincontrolled experiments. Matsuzuki and Horibe [45] observed that the fatigue data of extruded AZ31 alloy exhibited a bilinear tendency in the Manson–Coffin curve, implying that the twinning– untwinning process was dominant at plastic strains of greater amplitude whereas dislocation slips were dominant at lower plastic strains. Similar bilinear Manson–Coffin curves were observed in cast AZ91-T4 alloy [46]. A recent study of extruded AZ61A alloy under fully reversed strain-controlled tension–compression indicated that the strain–life fatigue curve displayed a detectable transition from lower cycle fatigue regime to higher cycle fatigue regime in the vicinity corresponding to a strain amplitude of 0.5% [16]. When the strain amplitude was higher than 0.5%, shear cracking and significant twinning were observed. When the strain amplitude was lower than 0.5%, tensile cracking and marginal twinning were observed [16]. Most studies on the fatigue properties of Mg alloys employed uniaxial tension–compression loading. Very limited work has been done on the fatigue of Mg alloys under multiaxial loading. Bentachfine et al. [47,48] compared the fatigue lives of a magnesium–lithium alloy tested in tension and compression, alternating torsion, combined in-phase axial-torsion, and out-of-phase axialtorsion. It was found that the fatigue life depended on the outof-phase angle between the axial strain and the shear strain, and the maximum life was obtained with in-phase loading. Amadi and Zenner [49] revealed an increase of fatigue life in die casting alloy AZ91 under out-of-phase axial-torsion loading as compared to that under the in-phase loading at the same equivalent strain amplitude. It was noticed that the definitions of the equivalent strain amplitude by Bentachfine et al. [47,48] and that by Amadi and Zenner [49] were different. The mean stress effect on fatigue life of Mg alloys was also studied. Goodenberger and Stephens [24] analyzed the mean stress effect using the Smith, Watson, and Topper (SWT), Morrow, and Lorenzo–Laird models based on the experimental data of cast AZ91E-T6 Mg alloy. It was found that these models frequently overestimated the mean stress effect on the fatigue life of the material. Hasegawa et al. [15] conducted uniaxial fatigue experiments with different mean stresses on extruded AZ31 Mg alloy and concluded that various mean stress correction models workable for cubic metals were not effective for Mg alloys. A similar conclusion was reached by Park et al. [50]. In addition, Park et al. [50] developed an energy-based model to consider both the elastic strain energy density and the plastic strain energy density. A good agreement was achieved between the predictions and the experimental fatigue lives of rolled AZ31 Mg alloy. It should be pointed out that all these evaluations of the mean stress effect on fatigue life were based on the experiments conducted under uniaixal tension–compression. Bentachfine et al. [48] examined several multiaxial fatigue criteria using the fatigue data of a magnesium–lithium alloy tested under combined axial-torsion loading. It was concluded that the model predictions agreed with the experimental fatigue lives reasonably well. Based on the in situ SEM observations of fatigue fracture process, microstructure-based multistage fatigue modeling technique was developed to predict the fatigue life of cast Mg alloys [31,51]. The technique partitions the fatigue damage into three distinct stages: crack incubation, microstructurally small crack growth, and long crack growth. Accounting for the operant micromechanisms of each stage, the predicted fatigue lives agreed well with the experimental results. A general multiaxial fatigue model for Mg alloys is still unavailable due to the lack of the multiaxial experimental results. In the current investigation, extensive fatigue experiments were carried out using thin-walled tubular specimen of extruded AZ61A Mg alloy under fully reversed strain-controlled tension–compression, cyclic torsion, proportional axial-torsion, and nonproportion-

al axial-torsion loading at room temperature. Two critical plane multiaxial criteria were evaluated with the experimental results. 2. Experiments 2.1. Material and specimens The material used in the current investigation is extruded AZ61A Mg alloy. The material has a chemical composition in weight percentage of 6.5 Al, 0.95 Zn, 0.325 Mn, 0.1 Si, 0.05 Cu, 0.005 Fe, 0.005 Ni, and 0.3 other impurities, balanced by Mg [52]. No heat treatment was made on the material before the test. The tubular specimen was machined from commercially acquired extruded tubing with an outer diameter of 25.91 mm and a thickness of 2.29 mm. The geometry and dimensions of the thin-walled tubular specimen are shown as in Fig. 1. For those tested at an equivalent strain amplitude of 1.0%, the specimens had an inner diameter of 21.33 mm and an outer diameter of 24.60 mm. Specimens with a larger wall thickness were used for higher strain amplitude experiments in order to avoid possible buckling during cyclic loading. Before testing, the outer surface of the gage section was finely polished using sanding paper with grit numbers starting from 600 to 1200. Optical microstructure observations reveals that the extruded AZ61A Mg alloy consists of equiaxed grains with an almost identical average grain size of 20 microns, measured using the Mean Lineal Intercept Method. There are precipitates at the grain boundaries, most probably the intermetallic compounds b-phase Mg17Al12. Twins were not observed in the undeformed state [52]. 2.2. Experiments and results The static mechanical properties of the material are summarized in Table 1. The elastic limit is approximately 50 MPa under both tension and compression. The material displays significant  2g twintension–compression asymmetry associated with f1 0 1  2g ning: due to the pole nature of twinning deformation, f1 0 1 twinning occurs under compression but does not occur under tension. 0.2% offset proof stress under tension is 192 MPa and 0.2% offset proof stress under compression is 120 MPa. A significant difference in the yield stress between tension and compression in wrought Mg alloys was frequently reported in literature [53–56]. Fatigue experiments were conducted using an Instron Servohydraulic tension–torsion load frame. The testing system is equipped with Instron 8800 electronic control, computer control, and data acquisition. It has a capacity of ±222 kN in the axial load and ±2800 N m in torque. A modified MTS extensometer was attached to the gage section of the specimen to measure the axial, shear, and diametral strains. The extensometer has a range of ±5% in the axial strain, a range of ±3° in the torsion deformation, and a range of 0.25 mm in the diametral direction. All the experiments were conducted in ambient air.

Fig. 1. Geometry and dimensions of thin-walled tubular specimen used in the fatigue experiments (all dimensions in mm).

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Q. Yu et al. / International Journal of Fatigue 33 (2011) 437–447 Table 1 Static material properties of extruded AZ61A magnesium alloy. Elasticity modulus, E Shear modulus, G 0.2% Offset proof stress under tension, ryt 0.2% Offset proof stress under compression, ryc Ultimate strength under tension, Sut Fracture stress under tension, rft Elongation under tension, eft Reduction in area under tension, RA

43.3 GPa 16.4 GPa 192 MPa 120 MPa 279 MPa 293 MPa 8.94% 12.7%

Axial Stress, MPa

300 200

Extruded AZ61A Tension-Compression (Path a) Δε/2 = 0.137 %

100 1%

0

0.15 %

0.7 %

-100

0.2 % 0.25 % 0.3 %

0.6 %

-200

0.5 % 0.45 %

0.4 %

0.35 %

-300 -0.010

ε

ε

γ/ 3

γ/ 3

(a)

(b)

ε

ε

γ/ 3

(c)

γ/ 3

(d)

σ

γ/ 3

(e) Fig. 2. Loading paths used for the fatigue experiment.

0.010

(a) 150

Shear Stress, MPa

Five axial-torsion loading paths were used in the fatigue experiments (Fig. 2): fully reversed strain-controlled tension–compression (Fig. 2a), fully reversed strain-controlled cyclic torsion (Fig. 2b), fully reversed strain-controlled proportional (in-phase) pffiffiffi axial-torsion (Fig. 2c) with e ¼ c= 3 at any loading moment (where e denotes the axial strain and c is the shear strain), fully reversed strain-controlled nonproportional (circular shaped, 90° outpffiffiffi of-phase) axial-torsion with De=2 ¼ Dc=2= 3 (where De/2 is the axial strain amplitude and Dc/2 is the shear strain amplitude) (Fig. 2d), and fully reversed strain-controlled torsion with pffiffiffia static tensile or compressive axial stress (Fig. 2e). In the e  c= 3 strain space, an equivalent strain amplitude (De/2)eq is defined as the radius of the minimum circle circumscribing the loading path. For uniaxial tension–compression, the equivalent strain amplitude is identical to the axial strain amplitude. The equivalent strain of the circular loading path shown in Fig. 2d is equal to the radius of the circle. It should be noted that the equivalent strain defined does not possess physical significance and it is only used to facilitate the presentation of the fatigue results under different loading paths. The stress–strain hysteresis loops at half fatigue life at different strain amplitudes under tension–compression (Path a) and pure torsion (Path b) are summarized in Fig. 3. When the strain ampli-

0.000

Axial Strain

0.443% 0.52%

Δγ/2 = 2.6%

100

0.606%

0.312%

2.425%

50

0.346%

1.732% 1.524%

0 1.212%

-50

1.04%

0.693%

-100 0.887%

-150 -0.03

-0.02

0.83%

-0.01

0.779%

Extruded AZ61A Cyclic Torsion (Path b)

0.00

0.01

0.02

0.03

Shear Strain

(b) Fig. 3. Stabilized stress–strain hysteresis loops under fully reversed strain-controlled (a) tension–compression; (b) torsion.

tude was larger than 0.45% for Path a (tension–compression), the stress–strain hysteresis loops exhibited an asymmetric sigmoidal shape with a positive mean stress. This is associated with the initial strong basal texture of the extruded AZ61A which results in mechanical twinning in the compression phase and untwining in the subsequent tension phase. The mean stress was less than 10 MPa when the strain amplitude was lower than 0.4%. The mean stress increased significantly when the strain amplitude increased from 0.4% to 0.6%. When the strain amplitude was in the range between 0.7% and 1.0%, the mean stress was almost a constant value of 34 MPa regardless of the value of the strain amplitude. This indicates that cyclic deformation is dominated by mechanical twins when the strain amplitude is in the range between 0.5% and 1% but by dislocation slips when the strain amplitude is lower than 0.5%. Under cyclic torsion, the stress–strain hysteresis loops are almost symmetric (Fig. 3b) although mechanical twinning was observed at high shear strain amplitudes [52]. When the material is subjected to combined axial-torsion loading, the alternative occurrence of twinning and untwining processes under the axial stress results in asymmetric shear stress–strain hysteresis loops [52]. The twinning/untwining processes in each loading cycle and the resulting asymmetric stress–strain hysteresis loops have a significant influence on the fatigue fracture process and fatigue life, particularly in the lowcycle fatigue regime. Detailed characteristics of cyclic deformation of the material under strain-controlled paths (a–d) shown in Fig. 2 were discussed elsewhere [16,52]. All the fatigue experiments are tabulated in Tables 2–5. The fatigue life was determined corresponding to the moment when the maximum stress in a loading cycle displayed a 5% reduction from the stabilized or peak value, or a visible crack was found on the outer surface of the specimen. For most cases, there was a visible crack on the specimen outer surface when the test was terminated. The stabilized stress and strain values listed in Tables 2–5

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Table 2 Fatigue results under fully reversed tension–compression (Path a). Spec ID

De/2 (%)

rm (MPa)

Dr/2 (MPa)

Nf cycles

Spec ID

De/2 (%)

rm (MPa)

Dr/2 (MPa)

Nf cycles

A10 A17 A27 A29 A05 A28 A16

1 0.7 0.6 0.52 0.5 0.45 0.4

34.5 33.6 31.6 27.9 22.2 17.8 7.8

190.2 178.5 169.1 163.9 161.9 155.0 142.8

400 1020 1520 2620 4500 5860 8740

AII01 A04 AII02 A08 A15 A20

0.35 0.3 0.25 0.2 0.15 0.137

2.8 4.0 3.1 2.5 3.4 1.2

130.1 115.6 98.6 83.7 63.9 58.4

16,000 16,750 39,600 75,000 370,000 >1191,000

De/2 = axial strain amplitude; rm = mean axial stress; Dr/2 = axial stress amplitude; Nf = number of cycles to failure.

Table 3 Fatigue results under fully reversed cyclic torsion (Path b). Spec ID

Dc/2 (%)

sm (MPa)

Ds/2 (MPa)

Nf cycles

Spec ID

Dc/2 (%)

sm (MPa)

Ds/2 (MPa)

Nf cycles

A14 A30 A37 A60 A02 A58 A51

1.73 1.39 1.21 1.04 0.89 0.83 0.78

0.7 0.4 0.5 0.7 0.1 0.3 0.9

83.1 79.1 75.3 70.1 68.6 66.5 66.8

280 520 800 1060 2300 3580 6340

A11 A54 A03 A61 A06 A07 A12

0.69 0.61 0.53 0.52 0.44 0.36 0.31

0.5 0.9 0.6 0.9 0.6 0.8 0.6

64.2 59.3 57.2 56.5 53.2 49.5 47.4

7720 11,150 27,300 37,400 52,200 261,500 >976,000

Dc/2 = shear strain amplitude; sm = mean shear stress; Ds/2 = shear stress amplitude; Nf = number of cycles to failure.

Table 4 Fatigue results under proportional (Path c) and nonproportional loading (Path d). Load path

Spec ID

De/2 (%)

Dc/2 (%)

rm (MPa)

Dr/2 (MPa)

Ds/2 (MPa)

Nf cycles

c

AT3 A31 A35 A43 A40 A33 A48 A45 A57 A39 A49 A53 A50 A47

1.06 0.71 0.49 0.42 0.37 0.35 0..32 0.28 0.25 0.21 0.18 0.16 0.14 0.12

1.84 1.22 0.86 0.73 0.64 0.61 0.55 0.49 0.43 0.37 0.31 0.27 0.24 0.21

21.0 23.1 18.8 13.8 5.6 0.7 2.2 3.1 5.3 7.1 4.4 6.5 3.9 3.1

211.1 186.5 170.6 157.8 148.0 148.9 136.9 127.8 119.6 110.2 98.9 91.6 85.9 74.1

sm (MPa) 5.5 5.7 5.2 3.0 2.2 1.9 1.2 1.1 1.1 0.7 1.4 1.8 1.3 0.3

121.9 107.6 98.5 91.1 85.5 86.0 79.1 73.8 69.0 63.6 57.1 52.9 49.6 42.8

206 580 1280 2360 4329 5040 8680 12,639 17,426 34,859 76,819 179,136 573,040 >1485,960

d

A23 A24 A25 A56 A32 A52 A22 A26 A44

0.7 0.5 0.4 0.35 0.3 0.25 0.2 0.15 0.12

1.21 0.87 0.69 0.61 0.52 0.43 0.35 0.26 0.21

22.4 15.3 5.9 1.2 1.7 3.0 2.8 3.2 2.3

187.8 150.7 134.2 127.6 116.5 101.3 88.9 70.1 57.1

3.2 2.1 2.1 2.2 0.3 0.0 0.7 1.0 0.3

108.4 87.0 77.5 73.7 67.3 58.5 51.3 40.5 33.0

273 727 1460 3356 6779 14,350 27,280 100,200 641,840

De/2 = axial strain amplitude; Dc/2 = shear strain amplitude; rm = mean axial stress; Dr/2 = axial stress amplitude; sm = mean shear stress; Ds/2 = shear stress amplitude; Nf = number of cycles to failure.

Table 5 Fatigue data under fully reversed strain-controlled shear with static axial stress (Path e). Spec ID

Dc/2 (%)

rstat (MPa)

sm (MPa)

Ds/2 (MPa)

Nf cycles

AII12 AII14 AII18 AII13 AII15 AII17

0.52 0.52 0.52 0.52 0.52 0.52

50 100 115 50 100 115

0.1 0.5 1.0 1.8 2.0 2.3

58.4 59.1 58.6 55.5 61.8 63.9

18,400 17,800 10,600 34,600 85,300 130,380

Dc/2 = shear strain amplitude; rstat = static axial stress; sm = mean shear stress; Ds/ 2 = shear stress amplitude; Nf = number of cycles to failure.

were taken from the representative hysteresis loop at approximately 50% of the fatigue life. The axial strain, the axial stress, and the shear stress were assumed to distribute uniformly over the wall thickness and the shear strain was assumed to have a linear distribution over the wall thickness of the specimen. The shear strain on the outer specimen surface was used for the fatigue analysis. Using the equivalent strain amplitude concept defined previously, the strain–life curves under fully reversed tension–compression, cyclic torsion, proportional loading, and 90° out-of-phase nonproportional loading were presented and compared in Fig. 4. Each point represents one fatigue test. The data point with a

(Δε/2)eq , Equivalent Strain Amplitude

Q. Yu et al. / International Journal of Fatigue 33 (2011) 437–447

0.01 8 7 6 5 4 3 2

Extruded AZ61A Path a Exp Path b Exp Path c Exp Path d Exp Eq.(1) Fitting Eq.(1) Fitting

0.001 10

2

10

3

10

4

10

5

10

6

N f , Number of Cycles to Failure Fig. 4. Strain–life curves under fully reversed strain-controlled tension–compression (Path a), cyclic torsion (Path b), proportional loading (Path c), and nonproportional loading (Path d).

horizontal arrow denotes a run-out fatigue test. It can be found in Fig. 4 that for the same equivalent strain amplitude, the fatigue life under proportional loading (Path c) is the highest and the nonproportional loading path (Path d) results in the shortest fatigue life. The fatigue lives under fully reversed tension–compression (Path a) and cyclic torsion (Path b) come between that under proportional loading and that under nonproportional loading. The following three-parameter equation is used to describe the strain–life curve by fitting the experimental data,

!n   De  e0 Nf ¼ C 2 eq

ð1Þ

where (De/2)eq is the equivalent strain amplitude and Nf is the number of loading cycles to failure. The remaining three symbols e0, n, and C are the constants identified by best fitting the experimental data. Eq. (1) can be used to describe the strain–life curves of a variety of metallic materials [57–63]. Results shown in Fig. 4 clearly indicate that a single Eq. (1) cannot well describe the strain–life curve for a given loading path. In fact, a distinguishable kink in each strain–life curve can be identified for a loading condition. A strain–life curve can be adequately described by two three-parameter equations and the intersection point forms a kink in the strain–life curve. The equivalent strain amplitudes corresponding to the kink points are 0.5% for fully reversed strain-controlled tension–compression (Path a), 0.46% for pure torsion (Path b), 0.47% for proportional loading (Path c), and 0.29% for nonproportional loading (Path d). The fatigue lives corresponding to the kink points are approximately 4500 cycles under tension–compression (Path a), 5740 cycles under cyclic torsion (Path b), 6730 cycles under proportional loading (Path c), and 9680 cycles under nonproportional loading (Path d). The kink points in the strain–life curves may represent the degree of involvement of mechanical twinning in the cyclic deformation. For the fully reversed strain-controlled tension–compression (Path a), significant twinning was observed when the strain amplitude was above 0.5% and the material was almost twinning free after the fatigue experiment when the strain amplitude was less than 0.5% [16]. The existence of significant mean stress in the fully reversed strain-controlled loading is an indicator of the involvement of twinning–detwinning process. The twinning–detwinning process resulted in the development of the mean stresses during cyclic loading when the equivalent strain amplitudes were above the kink points under all the fully reversed loading paths except for the cyclic torsion (Refer to Tables 2–5). Under fully reversed ten-

441

sion–compression, when the strain amplitudes were higher than 0.5%, significant tensile mean stresses were detected while small tensile mean stresses were observed when the strain amplitudes were lower than 0.5%. Under proportional loading, when the equivalent strain amplitudes were higher than 0.47%, tensile axial mean stresses within 20 MPa and positive shear mean stresses in the range of 2 MPa and 6 MPa were observed. When the equivalent strain amplitudes were lower than 0.47%, small compressive axial mean stresses higher than 10 MPa and small positive shear mean stresses within 2 MPa were detected. Under nonproportional loading, similar axial mean stress behavior relevant to the kink point in the strain–life curve as that under proportional loading was observed. The difference was that for the same equivalent strain amplitude, the magnitude of the axial mean stress was lower than that under the proportional loading. The shear mean stresses under nonproportional loading were generally negative and became more remarkable when the strain amplitude was higher than that at the kink point. Also, for the same equivalent strain amplitude, the magnitudes of the shear mean stress under nonproportional loading was a slightly lower than that under proportional loading. Under cyclic torsion, very small positive mean shear stresses with an average value of 0.6 MPa were found at all the equivalent strain amplitudes. For the extruded AZ61A tubular specimen, the caxis of most grains is aligned in the radius direction perpendicular to the extrusion direction (axial direction). Under cyclic torsion, there is no bulk axial stress and the tensile twins cannot be activated in these grains. However, there still exist some preferentially oriented grains with their c-axis parallel to the maximum tensile stress, which can trigger the mechanical twinning process. Although mechanical twinning is involved under cyclic torsion at large strain amplitude, the symmetry of shear stress–strain hysteresis loop is not influenced and the developed shear mean stress is not significant. One reason is that only a small portion of grains experiencing mechanical twinning. A more important reason is that at a positive shear strain and at a negative shear strain with the identical absolute value, the amount of mechanical twinning (and detwinning) deformation was statistically equal. Under fully reversed strain-controlled tension–compression, a detailed examination of fatigue crack revealed a transition of the cracking behavior associated with the kink in strain–life curve [16]. When the strain amplitudes were below 0.5%, tensile cracking was dominated and the cracking plane was found to be perpendicular to the loading direction. At the strain amplitude of 0.5%, fatigue crack was initiated from the outer specimen surface and the crack plane was found to be perpendicular to the loading direction. Approximately half through the tubular wall, the crack started to grow in a shear fashion. When the strain amplitudes were higher than 0.5%, a more three-dimensional crack profile was formed preliminarily dominated by Case B shear cracking [16]. In contrast to the transition of cracking behavior under fully tension–compression, consistent shear cracking with a cracking plane along the axial direction was observed for all the tubular specimens under cyclic torsion in the full range of shear strain amplitudes ranging from 0.31% to 2.6%. In light of the fatigue crack profiles under fully reversed tension–compression and cyclic torsion, it is concluded that the material exhibits a mixed cracking behavior when the equivalent strain amplitude was lower than 0.5% and a shear cracking behavior when the equivalent strain amplitude was higher than 0.5% according to Jiang [58]. Under axial-torsion, the cracking behavior was complicated, depending on the specific straining path and the strain amplitude. Fig. 5 shows the influences of static axial stress on the fatigue lives under fully reversed strain-controlled cyclic torsion at a shear strain amplitude of 0.52%. It can be found that a positive static axial stress reduces the fatigue life and a negative static axial stress significantly enhances the fatigue life. Such an influence of the static

Q. Yu et al. / International Journal of Fatigue 33 (2011) 437–447

150

Extruded AZ61A Δγ/2 = 0.52%

100 50

0 -50 -100 -150 8 9

10

2

4

3

4

5

6 7 8 9

10

2 5

N f , Number of Cycles to Failure Fig. 5. Influence of the static axial stress on fatigue life for the loading with static axial stress and a fully reversed shear strain amplitude of 0.52%.

axial stress in the fully reversed strain-controlled torsion on the Mg alloy is different from that on aluminum alloy 7075T651 where the tensile static stress had a very significant influence while the compressive axial static stress had much less significant influence on fatigue life [61]. 3. Multiaxial fatigue criteria A number of multiaxial fatigue criteria have been developed to predict the fatigue life under general multiaxial stress state for a variety of metallic materials [58,64,65]. Critical plane approaches are widely accepted now. A critical plane approach relates fatigue damage to the stress and strain associated with a material plane where fatigue crack initiation and early growth take place [64]. In this study, two critical plane multiaxial fatigue criteria are evaluated based on the results of the fatigue experiments conducted on the extruded AZ61A alloy. 3.1. Fatemi–Socie criterion Fatemi and Socie [66] developed a shear-strain based multixial fatigue criterion which can be expressed in the following mathematical form,

FP ¼

  Dc rn max 1þK 2 Sy

ð2Þ

where FP denotes ‘‘fatigue parameter.” The symbols Dc/2 and rnmax represent the shear strain amplitude and the maximum normal stress, respectively, associated with the critical plane. Sy is the yield stress of the material and K is a material constant to be identified from the baseline experimental data. The original fatigue criterion proposed by Fatemi and Socie [66] defined the critical plane as the plane associated with the maximum shear strain amplitude. In the current study, the critical plane is determined as the material plane on which the fatigue parameter (FP) expressed in Eq. (2) is a maximum. As discussed by Jiang [58], the critical plane deviated from the maximum shear planes when the material constant K was not equal to zero. Arguments can be raised with regard to the choice of the critical plane. However, as shown by Jiang [58] and Jiang et al. [60], the choice of definitions of the critical plane does not alter the conclusion that the fatigue criterion expressed in the form of Eq. (2) cannot predict tensile cracking orientation. Although the two different definitions provide practically identical fatigue life predictions, they do result in different predictions of the cracking orientation.The identification of the material constant K in the fatigue criterion is based on the experimental data under fully reversed strain-controlled tension–compression and cyclic torsion. For the extruded AZ61A,

the 0.2% offset proof stress under tension and compression are different, and therefore, the elastic limit is used for Sy (50 MPa). Since the combination of K and Sy can be treated as a single material constant, the selection of the value for Sy does not alter the prediction using the criterion. The material constant K in Eq. (2) is determined iteratively until the FP  Nf curves under fully reversed tension– compression and cyclic torsion are converged to a single curve. As shown in Fig. 6, the experimental data of tension–compression and torsion with fatigue lives ranging from 10 cycles to approximate 2  104 cycles can be brought together with a single curve using K = 0.1. However, when the fatigue life is higher than 2  104 cycles, divergence can be found between the experimental data under fully reversed tension–compression and those under cyclic torsion.For a large number of metallic material, the relationship between the fatigue parameter (FP) of a given fatigue model such as that of Eq. (2) and the fatigue life in terms of the number of loading cycles to failure (Nf) can be described by using a threeparameter equation that is mathematically identical to Eq. (1),

ðFP  FP0 Þm Nf ¼ C

ð3Þ

where FP denotes ‘‘fatigue parameter” and Nf represents the fatigue life. The symbols FP0, m, and C are the constants obtained by best fitting the baseline experimental data. For the Fatemi and Socie criterion used for the extruded AZ61A Mg alloy, two baseline curves in the form of Eq. (3) were attempted to best fit the tension–compression and torsion experimental data as shown in Fig. 6. The first three-parameter baseline curve was obtained through fitting most of the experimental data under cyclic torsion and the constants in Eq. (3) are FP0 = 0.00285, m = 2.18, and C = 0.0326 (solid line in Fig. 6). The second three-parameter baseline curve was attempted to fit most of the experimental data under fully reversed tension– compression. The material constants are FP0 = 0.00184, m = 2.34, and C = 0.0215 (dotted line in Fig. 6). Under a given loading path, the maximum fatigue parameter, FP, can be obtained following Eq. (2) through the stress and strain transformations. Once the FP is determined, Eq. (3) is used to predict fatigue life. Fig. 7 shows the comparison of the experimentally obtained fatigue lives with the predictions calculated by the Fatemi–Socie criterion. The abscissas in the plots are for the observed fatigue lives and the ordinates are the predicted fatigue lives made by using the fatigue criterion. The thick solid diagonal lines represent perfect predictions, and the two dotted lines are the factor-of-two boundaries. A vertical arrow represents a predicted fatigue life beyond the scope of the coordinate and a horizontal arrow denotes a run-out experiment. It can be seen from Fig. 7 that most of predictions by the Fatemi–Socie criterion in the low-cycle regime with life less than 2  104 cycles are within the factor-of-two boundaries. However, in the

FP=Δγ/2(1+0.1σnmax /Sy)

σstat, Static Axial Stress, MPa

442

Extruded AZ61A Baseline Data Fatemi-Socie

0.1 6 4

Ten-comp (Path a) Cyclic torsion (Path b)

2

(FP-0.00285) (FP-0.00184)

0.01

2.18

Nf=0.0326

2.34

Nf=0.0215

6 4 2

10

1

10

2

10

3

10

4

10

5

10

6

10

7

N f , Number of Cycles to Failure Fig. 6. Fully reversed strain-controlled tension–compression and torsion for the determination of the material constant K in the Fatemi–Socie criterion.

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Predicted Life, Cycles

10

10

10

7

FP = 2bΔε<σmax >+ΔτΔγ(1-b)/2

10

Extruded AZ61A 6

5

4

2

Extruded AZ61A Baseline Data Modified SWT

10 6 4

Ten-comp (Path a) Cyclic torsion (Path b)

2

1.918

(FP-0.116)

6 4 2

0.1 10

10

10

10

Fatemi-Socie 2.18 (FP-0.00285) Nf =0.0326

3

1

10

2

10

3

10

4

10

5

10

6

10

10

7

FP ¼ 2bDehrmax i þ

(a)

Predicted Life, Cycles

10

10

10

10

7

Extruded AZ61A 6

5

4

Fatemi-Socie

3

2.34

Nf=0.0215 Ten-comp (Path a) Cyclic torsion (Path b) Proportional (Path c) Nonprop (Path d) Path e

(FP-0.00184) 10

10

2

1

10

1

10

2

10

3

10

4

10

5

10

6

10

10

3

10

4

10

5

10

6

10

7

and Sehitoglu [69] modified the SWT parameter to consider general cracking behavior. The modified SWT parameter can be expressed as

Observed Life, Cycles

10

2

Fig. 8. Baseline experimental data for determining the material constants for the modified SWT criterion.

1

10

1

Nf , Number of Cycles to Failure

Ten-comp (Path a) Cyclic torsion (Path b) Proportional (Path c) Nonprop (Path d) Path e

2

Nf =2453.0

1

7

Observed Life, Cycles

(b) Fig. 7. Comparison of observed fatigue lives with predictions using the Fatemi– Socie criterion. (a) FP0 = 0.00285, m = 2.18, and C = 0.0326; (b) FP0 = 0.00184, m = 2.34, and C = 0.0215.

high-cycle regime beyond 2  104 cycles, the predictions using the baseline curve accounting for most cyclic torsion data can only correlate well the fatigue experiments under cyclic torsion and proportional loading. The criterion overestimates those under the other loading paths (Fig. 7a). In the high-cycle regime beyond 2  104 cycles, the predictions using the material constants best fitting the fully reversed tension–compression data can only correlate well the fatigue results under nonproportional loading path. The predictions are outside the factor-of two boundaries for the other loading paths (Fig. 7b). 3.2. Modified SWT criterion The Smith, Watson, and Topper (SWT) parameter [67] was originally developed to consider the mean stress effect under uniaxial loading. The criterion was extended to multiaxial fatigue by Socie [68] with a critical plane interpretation. The parameter was designed for materials displaying tensile cracking behavior [68]. Jiang

1b Ds Dc 2

ð4Þ

where again FP denotes the ‘‘fatigue parameter.” r and s are the normal stress and shear stress on a material plane, respectively. e and c are the normal strain and shear strain corresponding to the normal stress and shear stress, respectively. The symbol D denotes the range of the stress or strain and rmax is the maximum stress in a loading cycle. The symbol hi is the MacCauley bracket which is defined as hxi = 0.5(x + |x|). The use of the MacCauley bracket is to ensure that no negative damage can be produced. The symbol b in Eq. (4) is a material constant ranging from 0 to 1.0. Clearly when b = 1.0, the modified SWT parameter is reduced to the original SWT model. The critical plane is defined as the plane where FP reaches a maximum. The modified SWT criterion has the capability of predicting different cracking behavior with a proper choice of the value for the material constant b in Eq. (4) [61]. When 0 6 b 6 0:37, the criterion can predict shear cracking behavior. When b P 0:5, tensile cracking behavior is predicted. Mixed cracking behavior can be taken account by choosing b between 0.37 and 0.5. Theoretically, the correct choice of the material constant b is to satisfy two requirements. First, it should make the baseline fatigue parameters under fully reversed tension–compression and cyclic torsion coincide on a single curve. Second, the material constant b should reflect the observed cracking behaviors under uniaxial tension–compression and cyclic torsion. For the metallic materials that display mixed cracking behavior, a dependence of cracking behavior on the load magnitude is usually observed. The material constant b can be related to the equivalent stress magnitude, such as that in the cases of 7075T651 aluminum alloy [61] and AL6XN stainless steel [62]. Due to strong texture influence, the cyclic stress–strain curves of the extruded AZ61A Mg alloy under fully reversed tension–compression and cyclic torsion do not coincide [52]. It becomes difficult to find a single relationship between the material constant b and the equivalent stress magnitude. With respect to the other physical quantities of the stabilized stress–plastic strain hysteresis loop, a dependence of b on the plastic strain energy density per cycle, DWp, was found to successfully bring the FP  Nf curves of fully reversed tension–compression and cyclic torsion on a single curve. In addition, the cracking behavior under tension–compression and cyclic torsion can be predicted satisfactorily. The following function was obtained for the material constant b,

b ¼ expð2:5DW p  0:88Þ

ð5Þ

where DWp is the plastic strain energy density per loading cycle. Fig. 8 shows the baseline FP  Nf relationship by using Eq. (5) for b under fully reversed tension–compression and cyclic torsion. It

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10

Predicted Life, Cycles

10

10

10

10

10

10

7

Extruded AZ61A 6

5

4

3

Modified SWT Ten-comp (Path a) Cyclic torsion (Path b) Proportional (Path c) Nonprop (Path d) Path e

2

1

10

1

10

2

10

3

10

4

10

5

10

6

10

7

Observed Life, Cycles Fig. 9. Comparison of observed fatigue lives and predictions made using the modified SWT criterion.

can be seen that the experimental data of tension–compression and torsion are converged on a single curve which can be described by Eq. (3) with FP0 = 0.116, m = 1.918, and C = 2453. For any loading case, the maximum fatigue parameter, FP, is obtained using Eq. (4) with the help of stress and strain transformations for the stresses and strains associated with any possible material planes. Eq. (3) is used to determine the predicted life. Fig. 9 shows the comparison of the experimental fatigue lives and the predictions obtained using the modified SWT criterion. It can be found that most of the predictions by the modified SWT criterion fall within the factor-of-two lines. The model underestimates the fatigue lives for the experiments of torsion with large compressive static axial mean stresses.

4. Discussions A kink in the strain–life curve is a characteristic of the fatigue of the Mg alloy under investigation. The kink in the strain–life curve under fully reversed strain-controlled tension–compression was found to be associated with the formation of mechanical twins before fatigue crack initiation [16]. Occurrence of mechanical twinning was also observed at high equivalent strain amplitudes for the other loading paths shown in Fig. 4, including the fully reversed cyclic torsion path under which the hysteresis loops were almost symmetric (Fig. 3b). A recent study reveals that the ZK60 Mg alloy also displays a distinguishable kink in the strain–life curve when the material is subjected to fully reversed strain-controlled tension–compression [70]. However, occurrence of mechanical twinning may not be the only contributor to the formation of the kink in the strain–life curve. It was found that for the AZ61A Mg alloy under investigation the kink disappeared when the strain R ratio (minimum strain over maximum strain in a loading cycle) was zero under strain-controlled uniaxial tension–compression where mechanical twinning was observed at high strain amplitudes [71]. In addition, a similar kink was observed in 7075T651 aluminum alloy under fully reversed strain-controlled torsion [61]. Mechanical twinning deformation does not occur in an aluminum alloy. In the aluminum case, the kink point in the strain–life curve was found to be associated with the transition of cracking behavior. However, for any other axial-torsion loading paths conducted for the aluminum alloy, no kink points were found in the strain–life curves [61]. As a comparison, AZ61A under fully re-

versed pure torsion exhibited a kinked strain–life curve, but a consistent shear cracking behavior was observed at all the tested strain amplitudes. More work is needed to explore the microscopic mechanisms associated with the kink in the strain–life curves of the Mg alloy. The Fatemi–Socie criterion was found to be able to predict fatigue life for various materials under multiaxial loading [62,66,73– 79]. As a shear based fatigue parameter, the criterion is appropriate for the fatigue life prediction of the materials displaying shear cracking. The Mg alloy under investigation exhibits loading magnitude-dependent cracking behavior. It displays shear cracking when the equivalent strain amplitude is larger than 0.5% but mixed cracking (early crack growth is found on the material plane of maximum shear under torsion and on the maximum tensile plane under tension–compression) when the equivalent strain amplitude is smaller than 0.5%. This may explain partly why the Fatemi–Socie criterion can predict well the low-cycle fatigue life but unable to do a good prediction for the high-cycle fatigue regime (Fig. 7). The modified SWT criterion is able to correlate the fatigue experiments of the extruded AZ61A Mg alloy very well. An important consideration is the relationship of the material constant, b, in the model and a quantity reflecting loading magnitude of a cyclic loading condition. This is because the material under investigation displays loading magnitude-dependent cracking behavior. It is noticed that 7075T651 aluminum alloy also displayed magnitudedependent cracking behavior, and the modified SWT parameter correlated well the multiaxial fatigue experiments of the material [61]. Most multiaxial fatigue criteria are designed for isotropic materials. The extruded AZ61A Mg alloy has a strong texture and the cyclic deformation in tension–compression and pure shear shows a clear anisotropy of the material [52]. A comparison of the two fatigue criteria suggests that the modified SWT model may have a capability to correlate fatigue experiments of a metallic material with strong textures. It may be worthy of mentioning that the fatigue criterion developed by Jiang [58] has a good similarity to the modified SWT criterion [61]. The Jiang criterion can be used for fatigue life predictions for single crystals [72] and polycrystalline materials with strong textures [59]. Both the Fatemi–Socie and modified SWT models can predict the fatigue lives well for loading Path e (Fig. 2e, fully reversed shear with static axial stress) when the static axial stress is tensile or slightly compressive. However, it is noticed that neither fatigue model can do a good prediction of the fatigue lives for Path e when the compressive static axial stresses were high (Figs. 7 and 9). This is possibly related to the capability of the models to predict cracking behavior. Theoretically, a multiaixial fatigue criterion can predict fatigue life and at the same time identify the cracking behavior [58–63]. A general conclusion from earlier studies of multiaxial fatigue of several engineering materials is that the critical plane multiaxial fatigue criteria can reasonably predict the fatigue lives if the stress–strain quantities are either obtained directly from experiments or obtained through numerical analysis by employing an accurate cyclic plasticity model. However, the prediction of the early cracking behavior is not as desirable [60–63]. An examination of the early crack growth behavior of the AZ61A Mg alloy under fully reversed shear with a static axial load (Path e) can offer a comparison of the two multiaxial fatigue criteria in terms of their capability to predict cracking behavior. As shown in Fig. 10, when the axial static stress changed from a high tensile stress (100 MPa) to a high compressive stress (115 MPa), the cracking behavior changed from shear cracking to tensile cracking. At an axial static tensile stress of 100 MPa, the Fatemi–Socie criterion predicts the cracking orientation correctly, and the predicted cracking orientations by the modified SWT criterion are close to

Q. Yu et al. / International Journal of Fatigue 33 (2011) 437–447

445

Fig. 10. Observed early cracking and predictions for specimens subjected to cyclic torsion with static applied axial stresses (Path e): (a) r = 100 MPa; (b) r = 50 MPa; (c) r = 100 MPa; (d) r = 115 MPa.

the experimental observation (Fig. 10a). With an axial tensile static stress of 50 MPa, the observed early cracking plane has its normal being perpendicular to the axial direction (Fig. 10b) while the Fatemi–Socie criterion predicts a cracking plane whose normal is parallel to the axial direction. The modified SWT criterion is able to predict correct cracking behavior (Fig. 10b). With a large static compressive axial stress, the early cracks are found to form on the materials planes with their normal directions are ± 45from the axial axis (Fig. 10c and d) which correspond to the material planes of maximum alternating tensile stresses. Both criteria fail to correctly predict the fatigue cracking orientations for the cases with large compressive static axial stresses (Fig. 10c and d), and neither fatigue model predicted the fatigue life well for these two specimens (Figs. 7 and 9). It should be noticed that the predicted cracking orientations can be multiple valued due to the fact that multiple peaks of a given fatigue parameter can exist with respect to the orientation of the material plane [60–63]. A common disadvantage of any multiaxial fatigue criterion making use of the maximum or the range/amplitude of the stress and strain is the requirement of the definition of a loading cycle. While the rain-flow counting method is widely acceptable for random loading, it is limited to uniaxial loading. There is still no physically significant and widely acceptable cycle counting method for general multiaxial loading. Therefore, it would be desirable to have a multiaxial fatigue criterion that does not require the definition of a loading cycle. The multiaxial fatigue criterion developed by Jiang [58] was aimed at such a direction. However, due to the use of plastic deformation in the model, the Jiang criterion is limited to ductile materials which display measurable plastic deformation for the range of fatigue lives. For the Mg alloy under investigation, the macroscopic plastic strain is too small to be accurately and reli-

ably determined for the loading conditions with fatigue lives being higher than 5  104 cycles. Further investigations will require an understanding of the local deformation within a grain in the polycrystalline materials. Yang et al. [33] pointed out that deformation twins could contribute to fatigue damage from three aspects. First, the sharp edge of existing twin lamellas could result in local stress concentration [80] and twin boundaries could serve as preferential sites for crack initiation due to the strong and complicated interaction between slip and twinning [18,81]. Second, the accumulation of residual twins could lead to cyclic deformation irreversibility which is an essential mechanism for fatigue damage. Finally, in addition to the direct shear strain contributed by deformation twins, the reorientation of mechanical twins with respect to the parent grains may favor the activation of new slip systems within twins to provide further plastic deformation [55]. Furthermore, the twinning/ untwining process under axial cyclic loading introduces a positive axial mean stress, which is deleterious to fatigue resistance. However, mechanical twins may not be the only contributor to the fatigue crack initiation at high strain amplitudes. Other metallurgical factors, such as inclusions and precipitates which may serve as stress raisers and induce clusters of slip bands, can significantly influence the fatigue crack initiation behavior. Further investigations are needed to study the fatigue crack initiation mechanisms related to mechanical twins.

5. Conclusion An experimental study was performed on the fatigue behavior of an extruded AZ61A magnesium alloy under fully reversed

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strain-controlled tension–compression, torsion, and combined axial-torsion. The effect of static axial stress on the fatigue life under cyclic torsion was also investigated. The following conclusions are obtained: (1) Under each fully reversed strain-controlled loading path, the strain–life curve displays a detectable kink. The strain amplitude at the kink point depends on the specific strain path. When the strain amplitude is higher than the value at the kink point, significant mean stresses develop for all the loading paths except the pure torsion path, indicating significant involvement of mechanical twinning in cyclic plasticity. (2) At the same equivalent strain amplitude, the fatigue life is the shortest under 90° out-of-phase nonproportional loading and the longest under proportional axial-torsion loading. (3) Based on the fatigue cracking observations under tension– compression and cyclic torsion, the material is identified to exhibit shear cracking behavior when the equivalent strain amplitude is higher than 0.5% and mixed cracking behavior when the equivalent strain amplitude is lower than 0.5%. (4) Under fully reversed strain-controlled torsion, a tensile static axial stress reduces the fatigue life while a compressive static axial stress significantly enhances fatigue life. (5) The Fatemi–Socie criterion predicts the fatigue life well in the low cyclic fatigue regime for all the loading paths investigated but does not predict fatigue life well for the highcycle fatigue regime. The modified SWT criterion combining the normal and shear components of the stresses and strains on material planes can predict the fatigue lives of the material satisfactorily. Both models fail to predict fatigue life and early cracking behavior for the loading condition under cyclic torsion with a high compressive axial stress.

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