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Ya

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Materials Science and Engineering A 546 (2012) 119– 128

Contents lists available at SciVerse ScienceDirect

Materials Science and Engineering A

jo ur n al hom epage: www.elsev ier .com/ locate /msea

ultiaxial fatigue of extruded AZ31B magnesium alloy

ing Xionga, Qin Yub, Yanyao Jiangb,∗

Key Laboratory of E&M, Ministry of Education & Zhejiang Province, Zhejiang University of Technology, College of Mechanical Engineering, Hangzhou, Zhejiang 310032, ChinaUniversity of Nevada, Reno, Department of Mechanical Engineering (312), Reno, NV 89557, USA

r t i c l e i n f o

rticle history:eceived 4 September 2011eceived in revised form 11 March 2012ccepted 13 March 2012vailable online 23 March 2012

a b s t r a c t

Multiaxial fatigue behavior of extruded AZ31B magnesium alloy was experimentally studied in ambientair under strain-controlled axial-torsion loading using tubular specimens. Four fully reversed loadingpaths were employed: tension–compression, pure torsion, proportional axial-torsion, and 90◦ out-of-phase nonproportional axial-torsion. Tension–compression asymmetry in cyclic plastic deformation was

eywords:ultiaxial fatigue

xtruded AZ31B magnesium alloy

observed for tension–compression and combined axial-torsion loading. For the same equivalent strainamplitude, the 90◦ out-of-phase nonproportional loading path resulted in the shortest fatigue life and theproportional loading path produced the longest fatigue life. Two critical plane multiaxial fatigue models,namely, the modified Smith–Watson–Topper (SWT) model and the Jiang multiaxial fatigue criterion,were evaluated based on the experimentally obtained fatigue results. Both models were found to be ableto correlate the fatigue experiments reasonably well.

. Introduction

Wrought magnesium (Mg) alloys are potential structuralaterials due to the excellent properties such as high strength-to-eight ratio, high specific stiffness, and superior fatigue resistance.

ignificant research has been conducted to explore the physicalnd mechanical properties of the Mg alloys. Extruded and rolledZ31 (or AZ31B) are the most widely studied wrought Mg alloys

1–13].A great concern in designing a structural component is the

atigue behavior of the material. Typically, a strain–life fatigueurve serves as a baseline for the fatigue strength of the mate-ial. For wrought Mg alloys, it was found that the strain–life curveould be reasonably described by the well-known Basquin andanson–Coffin equations [2,4,7,12,14–17]. Hasegawa et al. [2]

ound that the Manson–Coffin equation nicely described the fatigueives of extruded AZ31 Mg alloy obtained under stress-controlledoading but failed to describe the fatigue lives obtained from thetrain-controlled experiments because of the tensile mean stressesroduced. Matsuzuki and Horibe [4] observed that the fatigueata of extruded AZ31 alloy exhibited a bilinear tendency in theanson–Coffin curve, implying that the twinning–detwinning pro-

ess was dominant at larger strain amplitudes whereas dislocationlips were dominant at lower strain amplitudes. A recent studyf extruded AZ61A alloy under fully reversed strain-controlled

∗ Corresponding author. Tel.: +1 775 784 4510; fax: +1 775 784 1701.E-mail addresses: yxiong@zjut.edu.cn (Y. Xiong), yjiang@unr.edu (Y. Jiang).

921-5093/$ – see front matter © 2012 Elsevier B.V. All rights reserved.oi:10.1016/j.msea.2012.03.039

© 2012 Elsevier B.V. All rights reserved.

tension–compression indicated that the strain–life fatigue curvedisplayed a detectable transition from the lower cycle fatigueregime to higher cycle fatigue regime in the vicinity correspond-ing to a strain amplitude of 0.5% [18,19]. When the strain amplitudewas higher than 0.5%, shear cracking and significant twinning wereobserved. When the strain amplitude was lower than 0.5%, tensilecracking and marginal twinning were observed. A similar phe-nomenon was observed on ZK60 Mg alloy where a plateau occurredin the strain–life curve at a strain amplitude of 0.52% [20]

Structural components are often subjected to multiaxial stressstate. However, most studies of Mg alloys employed uniaxialtension–compression loading. Very limited work has been con-ducted on multiaxial fatigue of Mg alloys. Bentachfine et al. [21,22]compared the fatigue lives of a magnesium–lithium alloy tested intension and compression, alternating torsion, combined in-phaseaxial-torsion, and out-of-phase axial-torsion. It was found thatfatigue life depended on the out-of-phase angle between the axialstrain and the shear strain, and the maximum life was obtained within-phase loading. Recently, an extruded AZ61A Mg alloy was exper-imentally studied for fatigue behavior under different combinedaxial-torsion loading using tubular specimens [19,23]. A modifiedSmith–Waltson–Topper fatigue model [24–26] was found to cor-rect the experiments well. Albinmousa et al. [27] experimentallyinvestigated the multiaxial fatigue behavior of an extruded AZ31BMg alloy. It was found that that the Fatemi–Socie critical plane mul-tiaxial fatigue model [28] and an energy-based model [29] provided

good fatigue life estimates for the AZ31B Mg alloy under propor-tional and nonproportional loading.

The current work is a further study of cyclic deformationand fatigue of an extruded AZ31B Mg alloy under combined

120 Y. Xiong et al. / Materials Science and Engineering A 546 (2012) 119– 128

Table 1Chemical composition of extruded AZ31B magnesium alloy, wt.%.

Al Zn Mn Si

2.5–3.5 0.7–1.3 0.2 0.3

aa

2

fcstoss

of ±3◦ in the torsion deformation, and a range of 0.25 mm in the

Fig. 1. Thin-walled tubular testing specimen (all dimensions in mm).

xial-torsion loading. Two critical plane multiaxial fatigue criteriare evaluated based upon the experimental observations.

. Materials and experiment

AZ31B magnesium (Mg) alloy was commercially acquired in theorm of extruded solid rods with a diameter of 46 mm. The chemicalomposition of the material is summarized in Table 1. Dog-bonehaped thin-walled tubular testing specimens were machined fromhe rod along its extrusion direction. The geometry and dimension

f the tubular specimen are shown in Fig. 1. Before testing, the outerurface in the gage section of the specimen was polished by usingilicon carbide papers with grit No. 400 up to 1200.

Fig. 2. Initial microstructure and texture of the extruded AZ31B Mg alloy: (a) th

Cu Ni Fe Mg

0.05 0.005 0.005 Balance

The optical microstructure of the undeformed extruded AZ31BMg alloy is represented by a three-dimensional stereographyshown in Fig. 2a. On the plane perpendicular to the extrusion direc-tion (ED), an inhomogeneous microstructure consisting of bothlarge and small equiaxed grains was observed. On the planes per-pendicular to the tangential direction (TD) and the normal (orradial) direction (ND), grains were elongated due to the extrusionprocess that resulted in a fibrous grain structure. The average sizesof the large and small grains were approximately 50 �m and 8 �m,respectively. The measurement was made using the Mean LinealIntercept Method. No initial twins were detected in the undeformedstate. The initial texture is characterized as the {0 0 0 2} and {1 0 1 0}pole figures (Fig. 2b). A typical basal texture is revealed: the c-axisof most grains lies on the plane approximately perpendicular to theED direction.

Static and fatigue experiments were carried out using an InstronServo-hydraulic tension–torsion load frame which has a capacityof ±222 kN in the axial load and ±2800 N m in torque. The testingsystem is equipped with computer control and data acquisition. Amodified MTS extensometer was attached to the gage section ofthe specimen to measure the axial, shear, and diametral strains.The extensometer has a range of ±5% in the axial strain, a range

diametral direction.All the experiments were conducted in ambient air employ-

ing the tubular specimens shown in Fig. 1. Monotonic tension and

ree-dimensional stereography; and (b) {0 0 0 2} and {1 0 1 0} pole figures.

Y. Xiong et al. / Materials Science and Engineering A 546 (2012) 119– 128 121

γ/ 3

ε

mtrtt(tnwa

u(sutolSi

3

scaiasts

3

rswwstDaueTTmTt

50 0

40 0

30 0

20 0

10 0

0

Str

ess,

MP

a

0.120.100.080.060.040.020.00

Strain

x

x

xx

AZ31BMonotonic LoadingE=44800 MPa

Tension1.0x10-3

/s

Tension 1.0x10-2

/s

Compression 1.3x10-4

/s

Compression 2.8x10-3

/sx------Fracture

Fig. 4. Stress–strain curves of extruded AZ31B from monotonic tension and mono-tonic compression (compression in absolute values).

Table 2Static material properties of extruded AZ31B.

Elasticity modulus, E 44.8 GPaShear modulus, G 16.9 GPa0.2%-offset yield stress in tension, �yt 244 MPa0.2%-offset yield stress in compression, �yc 151 MPaUltimate strength under tension, Sut 298 MPaUltimate strength under compression, Suc 417 MPaFracture stress under tension, �ft 338 MPaElongation under tension, eft 0.105

tension–compression. When the strain amplitudes were higherthan 0.45%, the stress–strain hysteresis loops exhibited an asym-metric shape. Symmetry here refers to the identical shapes of the

-300

-200

-100

0

100

200

300

Axia

l S

tre

ss,

MP

a

AZ31BTension-compression(Path I)

=2%0.75%

0.4%

0.3%

1%0.5%

0.35%

Fig. 3. Axial-torsion loading paths using tubular specimens.

onotonic compression experiments were conducted to obtainhe static material properties. For fatigue experiments, four fullyeversed strain-controlled axial-torsion loading paths defined inhe ε − �/

√3 strain space (where ε denotes the axial strain and � is

he shear strain) were used (Fig. 3): cyclic tension–compressionpath I), cyclic torsion (path II), proportional (in-phase) axial-orsion with ε = �/

√3 at any loading moment (path III), and the

onproportional (circular shaped, 90◦ out-of-phase) axial-torsionith �ε/2 = (��/2)/

√3 (where �ε/2 is the axial strain amplitude

nd ��/2 is the shear strain amplitude) (path IV).To facilitate presentation and comparison of the fatigue lives

nder different loading paths, an equivalent strain amplitude�ε/2)eq is defined as the radius of the minimum circle circum-cribing the loading path shown in the ε − �/

√3 strain space. For

niaxial tension–compression (path I), the equivalent strain ampli-ude is identical to the axial strain amplitude. For the circular 90◦

ut-of-phase nonproportional loading path (path IV), the equiva-ent strain amplitude is equal to the radius of the loading circle.imply speaking, the equivalent strain amplitudes of the four load-ng paths shown in Fig. 3 are identical.

. Results and discussion

True stress and true strain are reported. The variation of the crossection of the testing specimen during mechanical experiment wasorrected with the consideration of both the elastic deformationnd the plastic deformation incompressibility. For the tubular spec-men, the axial strain, the axial stress, and the shear stress weressumed to distribute uniformly over the wall thickness and thehear strain was assumed to have a linear distribution over the wallhickness of the specimen. The shear strain on the outer specimenurface was used for the fatigue analysis.

.1. Static mechanical properties

Monotonic tension experiments were conducted at two strainates of 10−3/s and ∼10−2/s, respectively. For monotonic compres-ion experiments, two strain rates were also employed and theyere 1.3 × 10−4/s and 2.8 × 10−3/s. All the monotonic experimentsere tested till fracture of the specimen into two pieces. Fig. 4

hows the stress–strain curves in the absolute values obtained fromhe monotonic tension and monotonic compression experiments.espite an order of difference in strain rate, an insignificant discrep-ncy can be found in the tensile or compressive stress–strain curvesnder the two strain rates, indicating that the time-dependentffect is marginal within the range of the strain rates investigated.

he static mechanical properties of the material are summarized inable 2. The material shows significant tension–compression asym-etry in yielding and post-yielding behavior at room temperature.

he anisotropic mechanical property is ascribed to the initial basalexture and the pole nature of mechanical twin.

Elongation under compression, efc 0.093Reduction in area under tension, RA 0.12%

3.2. Stabilized stress–strain hysteresis loops

In the current study, the fatigue life of a specimen is definedas the moment when the maximum axial/shear stress in a load-ing cycle is reduced by 5% from the stabilized or peak value, or avisible crack is found on the outer surface of the testing specimen.The stabilized stress–strain response was taken at the half-fatiguelife.

Fig. 5 shows the stabilized stress–strain hysteresis loops atselected strain amplitudes under fully reversed strain-controlled

-0.02 -0.01 0.00 0.01 0.02

Axial Strain

Fig. 5. Stabilized stress–strain hysteresis loops under tension–compression loading(path I).

122 Y. Xiong et al. / Materials Science and Engineering A 546 (2012) 119– 128

-150

-100

-50

0

50

100

150

Shear

Str

ess, M

Pa

-0.03 -0.02 -0.01 0.00 0.01 0.02 0.03

Shear Strain

AZ31BTorsion (Path II) =2.42%

1.73%

1.3%

0.87%0.69%

0.52%

0.35%0.42%

1.04%

ucrhtdssd

h(tuamphdaissldh

paadtrtifvtftaitNcbiaip

-200

-100

0

100

200a

b

Axia

l S

tress, M

Pa

-0.010 -0.005 0.000 0.005 0.010

Axial Strain

(0.71%, 1.23%)

(0.53%, 0.92%)

(0.35%, 0.61%)

AZ31BProportional (Path III)

( /2=1%, /2=1.73% )

(0.42%, 0.88%)

(0.17%, 0.29%)

(0.28%, 0.49%)

-100

-50

0

50

100

Shear

Str

ess, M

Pa

-0.02 -0.01 0.00 0.01 0.02

Shear Strain

(0.71%, 1.23%)

(0.53%, 0.92%)

(0.35%, 0.61%)

AZ31BProportional (Path III)

( /2=1%, /2=1.73%)

(0.42%, 0.88%)

(0.17%, 0.29%)

(0.28%, 0.49%)

phase nonproportional axial-torsion loading (path IV) at selectedstrain amplitudes are shown in Fig. 8. When the axial and shearstrain amplitudes were higher than 0.30% and 0.52%, respectively,the stress–strain hysteresis loops are found to be asymmetric in

-300

-200

-100

0

100

200

300a

b

Axia

l S

tress, M

Pa

-0.02 -0.0 1 0.00 0.0 1 0.02

Axial Strain

AZ31B90° out-of-phase (Path IV)

(1%, 1.73%)

(0.75%, 1.3%)

(0.5%, 0.87%)

(0.2%, 0.35%)

(0.32%, 0.55%)

( =1.41%, =2.45% )

-200

-100

0

100

200

Shear

Str

ess, M

Pa

-0.03 -0.0 2 -0.0 1 0.0 0 0.01 0.0 2 0.0 3

Shear Strain

AZ31B90° out-of-phase (Path IV)

(1%, 1.73%)

(0.75%, 1.3%)

(0.5%, 0.87%)

(0.2%, 0.35%)

(0.32%, 0.55%)

( =1.41%, =2.45%)

Fig. 6. Stabilized stress–strain hysteresis loops under cyclic torsion.

pper and lower branches of the stress–strain hysteresis loop. Theompressive reversal shows a concave-up shape while the tensileeversal displays a sigmoidal shape. The asymmetric stress–strainysteresis loops at strain amplitude larger than 0.45% is due towinning during the compressive reversal and the detwinninguring the initial portion of the tensile reversal [30]. When thetrain amplitude was lower than 0.45%, symmetric shape of thetress–strain hysteresis loops was observed, and the cyclic plasticeformation is mainly due to dislocation slips.

Under cyclic torsion (path II), a symmetric shape of stress–strainysteresis loop was observed at all shear strain amplitudes testedFig. 6). According to the initial basal texture, pure shear abouthe ED axis will result in most grains being unfavorably stressednder c-axis tension or c-axis compression. However, there exists

small fraction of grains which have their c-axes tilted approxi-ately towards the direction of the principal stress. The {0 0 0 2}

ole figure (refer to Fig. 2b) indicates that the c-axes of these grainsave a rotational symmetry about the ED direction (or the axialirection). Therefore, when a high magnitude of cyclic torsion ispplied, mechanical twinning and detwinning will be activatedn an approximately equal degree under either reversal. Macro-copically, this is reflected by the symmetric shear stress–sheartrain hysteresis loop. Under low shear strain amplitudes, simi-ar to tension–compression, a symmetric operation of dominantislocation slip results in the symmetric shear stress–shear strainysteresis loop.

Fig. 7 shows stabilized stress–strain hysteresis loops under pro-ortional axial-torsion loading (path III). When the axial strainmplitude was higher than 0.4% (the corresponding shear strainmplitude was 0.69%), the stabilized stress–strain hysteresis loopsisplay an asymmetric shape in both the axial and shear direc-ions. Asymmetry in the axial stress–axial strain hysteresis loop isesulted from twinning under compression and detwinning underension. Asymmetry in the shear stress–shear strain hysteresis loops not as significant as that in the axial direction but is detectablerom a careful observation. It is associated with the change of twinolume fraction (or texture) caused by the axial stress-inducedwinning/detwinning. During twinning, the c-axis is rotated 86.3◦

rom the parent grain to orient approximately towards the ED direc-ion. With this twin orientation, under a shear stress about the EDxis, basal slip is more favorable to be operated in twin rather thann the parent grain. As a result, a higher twin volume fraction inhe material leads to a weaker resistance to shear deformation.oting that a higher twin volume fraction was produced at theompressive peak stress than that at the tensile peak stress, it cane expected that the value of the minimum shear stress in a load-

ng cycle is lower than that of the maximum shear stress, resulting

n asymmetry in the shear stress–shear strain hysteresis loop. Anndication of such an asymmetry is reflected by the existence ofositive mean shear stress (Fig. 7b).

Fig. 7. Stabilized stress–strain hysteresis loops under proportional axial-torsionloading (path III): (a) axial stress–axial strain; and (b) shear stress–shear strain.

The stabilized stress–strain hysteresis loops under 90◦ out-of-

Fig. 8. Stabilized stress–strain hysteresis loops under 90◦ out-of-phase nonpropor-tional loading (path IV): (a) axial stress–axial strain; and (b) shear stress–shearstrain.

and En

biasptoaiao[safcddhcftdhidt

lpIif

Fa

Y. Xiong et al. / Materials Science

oth the axial and shear directions. Similar to the proportional load-ng path, the asymmetric shapes of stress–strain hysteresis loopsre induced by the twinning–detwinning process due to cyclic axialtressing. However, unlike positive shear mean stress under pro-ortional loading path, a negative mean shear stress is produced inhe shear stress–shear strain hysteresis loop under the 90◦ out-f-phase nonproportional loading. Due to the 90◦ out-of-phasengle between the axial strain and the shear strain, when the min-mum (or maximum) shear strain is reached, the correspondingxial strain attains zero in the compressive (or tensile) reversal. Asbserved by in situ Electron Backscatter Diffraction (EBSD) tracing3] and in situ neutron diffraction [31], at the moment of zero axialtrain during the tensile reversal, detwinning is not complete and

small fraction of residual twin exists. If the loading path goes onor a half-cycle at which the axial strain reaches zero during theompressive reversal, detwinning is complete and no more twin iseveloped. Therefore, twin volume fraction at the zero axial strainuring the tensile reversal (or at the maximum shear strain) isigher than that at the zero axial strain with proceeding a half moreycle (or at the minimum shear strain). Since a higher twin volumeraction is associated with a weaker shear resistance, it is deducedhat a lower shear stress value is resulted at the zero axial strainuring the tensile reversal (or at the maximum shear strain) and aigher shear stress value is produced at the zero axial strain dur-

ng the compressive reversal (or at the minimum shear strain). Thisiscrepancy of the shear resistance in two shear directions leads tohe observed negative mean stress (Fig. 7b).

The mean stresses in the stabilized stress–strain hysteresisoops in both the axial and shear directions under the four loading

aths are summarized in Fig. 9. Under tension–compression (path

), positive mean stresses are identified when the strain amplitudes higher than 0.3% (Fig. 9a). It is noticed that the stabilized (halfatigue life) mean stress under fully reversed strain-controlled

60

a

b

40

20

0

-20Axia

l M

ea

n S

tre

ss,

m (

MP

a)

0.0012 3 4 5 6 7 8 9

0.01 2 3

Equivalent Strain Amplitude, eq/2

Path I Path III Path IV

AZ31B

30

20

10

0

-10Sh

ea

r M

ea

n S

tre

ss,

m (

MP

a)

0.0012 3 4 5 6 7 8 9

0.01 2 3

Equivalent Strain Amplitude, eq/2

PathII Path III Path IV

AZ31B

ig. 9. Variation of the mean stress with equivalent strain amplitude (�εeq/2): (a)xial mean stress (�m) versus �εeq/2; and (b) shear mean stress (�m) versus �εeq/2.

gineering A 546 (2012) 119– 128 123

tension–compression increases with increasing strain amplitudeand reaches a maximum value when the strain amplitude is 1.0%.Further increase in strain amplitude results in a reduced meanstress (Fig. 9a). Under proportional loading (path III), the axialmean stress is slightly compressive when the equivalent strainamplitude is lower than 0.40% and is positive when the strainamplitude is higher than 0.40%. A maximum axial mean stress isobserved when the equivalent strain amplitude is 1.0%. For all thestrain amplitudes tested, the proportional loading produces smallpositive shear mean stresses. Under nonproportional loading (pathIV), the axial mean stress is slightly compressive or zero and theshear mean stress is zero when the equivalent strain amplitudeis lower than 0.40%. Above an equivalent strain amplitude of0.40%, the nonproportional loading path produces tensile axialmean stresses and small negative shear mean stresses. For anidentical equivalent strain amplitude, tension–compression (pathI) produces the most significant mean stress and the nonpropor-tional loading path (path IV) results in the smallest value of axialmean stress (Fig. 9a). An insignificant mean stress observed undernonproportional loading suggests that nonproportional hardeningis insignificant for the material under investigation. A similarphenomenon was also reported for extruded AZ61A Mg alloy [23].A reason for the lack of nonproportional hardening is the limitedactivation and interaction of the slip systems of the material atroom temperature together with the relaxation mechanism due tothe twinning/detwinning process [32].

3.3. Fatigue life

All the fatigue results are tabulated in Table 3. The stabilizedstress and strain values listed in Table 3 were taken from the rep-resentative stress–strain hysteresis loop at approximately 50% ofthe fatigue life. It should be reiterated that the fatigue life was cor-responding to the moment when the maximum axial/shear stressin a loading cycle displayed a 5% reduction from its stabilized orpeak value, or a visible crack was found on the outer surface of thetesting specimen.

Using the equivalent strain amplitude concept, the strain–life

curves under the four loading paths are presented and comparedin Fig. 10. An arrow after a data point in the figure designates anexperiment without fatigue failure (run-out).

0.001

2

3

4

5

6

78

0.01

2

3

Eq

uiv

ale

nt

Str

ain

Am

plit

ud

e

101

10 2

10 3

10 4

10 5

10 6

Number of Cycels to Failure, Nf

AZ31B

Path I Exp Eq(1) Fittim g Path II Exp Eq(1) Fittim g Path III Exp Eq(1) Fittim g Path IV Exp Eq(1) Fittim g

Fig. 10. Strain–life curves under different axial-torsion loading paths.

124 Y. Xiong et al. / Materials Science and Engineering A 546 (2012) 119– 128

Table 3Fatigue results.

Load path Spec ID f (Hz) �εeq/2 (%) �ε/2 (%) �m (MPa) ��/2 (MPa) ��/2 (%) �m (MPa) ��/2 (MPa) �WP Nf

I

31BTU83 0.1 2.00 2.00 33.5 225.5 7.031 11031BTU40 0.2 1.00 1.00 57.9 222.8 1.854 32031BTU84 0.2 1.00 1.00 58.3 207.1 1.830 25031BTU81 0.25 0.75 0.75 47.4 221.9 6.7 0.833 72031BTU41 0.5 0.50 0.50 42.3 193.0 0.179 266031BTU82 0.5 0.50 0.50 72.9 196.2 0.144 218031BTU46 0.5 0.47 0.47 44.1 186.2 0.135 262031BTU03 1 0.45 0.45 17.1 174.7 0.143 11,36031BTU02 1 0.40 0.40 22.6 164.9 0.081 14,56031BTU47 1–3 0.35 0.35 13.8 147.7 0.047 29,90031BTU01 1–2 0.30 0.30 −2.0 127.6 0.029 75,84031BTU43 1–5 0.26 0.26 2.2 114.3 0.0067 160,40031BTU09 1/10 0.24 0.24 0.1 103.2 0.0006 >1,185,000

II

31BTU19 0.1 1.41 2.42 0.23 102.50 4.559 4431BTU12 0.2 1.00 1.73 0.36 94.57 2.390 8031BTU06 0.25 0.75 1.30 0.40 90.95 1.269 28031BTU44 0.5 0.60 1.04 −0.06 78.46 0.757 96331BTU08 0.8 0.50 0.87 0.23 74.87 0.532 212031BTU04 1 0.40 0.69 0.23 70.70 0.308 762031BTU10 2 0.30 0.53 0.49 60.49 0.161 24,52031BTU31 1–4 0.24 0.42 0.15 46.20 0.117 71,58031BTU20 1–4 0.20 0.35 0.38 50.61 0.015 556,020

III

31BTU21 0.1 1.41 1.00 28.5 175.3 1.73 3.81 75.49 4.430 29431BTU14 0.2 1.00 0.71 41.2 177.3 1.23 5.22 66.30 1.788 46031BTU22 0.25 0.75 0.50 23.1 162.0 0.92 4.10 69.39 1.036 142031BTU30 0.5 0.60 0.42 20.1 148.6 0.88 4.17 61.78 0.568 259031BTU23 0.5 0.50 0.35 6.1 132.1 0.61 3.95 56.12 0.361 536031BTU42 1 0.40 0.28 −0.4 112.4 0.49 2.32 52.65 0.204 11,60031BTU25 0.5–4 0.30 0.21 −10.7 89.2 0.37 3.31 47.28 0.081 49,67931BTU48 1–5 0.26 0.18 −8.2 78.7 0.32 3.08 43.70 0.043 87,39231BTU29 0.5–5 0.24 0.17 −15.0 74.6 0.29 5.64 43.53 0.0093 >1,057,300

IV

31BTU29e 0.1 1.41 1.41 5.3 237.7 2.45 −7.16 138.67 14.254 3531BTU11 0.1 1.00 1.00 9.4 201.2 1.73 −5.89 117.68 7.006 6831BTU16 0.125 0.75 0.75 11.6 186.8 1.3 −4.74 104.17 3.377 18031BTU13 0.2 0.50 0.50 18.8 166.9 0.87 −5.09 84.45 1.050 58031BTU18 0.5 0.40 0.40 4.0 144.7 0.69 −1.72 68.24 0.530 191031BTU60 0.5 0.35 0.35 2.4 133.8 0.61 −0.87 60.52 0.384 376731BTU28 0.5/1 0.32 0.32 3.1 129.7 0.55 −2.61 60.92 0.284 450031BTU17 1 0.30 0.30 −7.0 121.0 0.52 −0.27 59.87 0.246 15,78031BTU33 0.5/1 0.24 0.24 −5.8 101.5 0.42 −1.11 51.69 0.133 27,02031BTU15 2 0.20 0.20 −7.3 89.6 0.35 −1.30 40.10 0.093 108,80031BTU21 1/4 0.17 0.17 −8.0 74.0 0.29 −0.84 44.20 0.011 >1,012,520

f ude; �� er loa

s

(

wn�d

cIiqte0Iocus

, loading frequency; �εeq/2, equivalent strain amplitude; �ε/2, axial strain amplitm, shear mean stress; ��/2, shear stress amplitude; �WP , plastic energy density p

The following three-parameter equation is used to describe thetrain–life curve by fitting the experimental data,

(�ε

2

)eq

− ε0

)�

Nf = C (1)

here (�ε/2)eq is the equivalent strain amplitude and Nf is theumber of loading cycles to failure. The remaining three symbols ε0,, and C are the constants identified by best fitting the experimentalata.

Results shown in Fig. 10 clearly indicate that a single Eq. (1)annot well describe the strain–life curve for a given loading path.n fact, a distinguishable kink in each strain–life curve can bedentified for a given loading path. A strain–life curve can be ade-uately described by two three-parameter equations, Eq. (1), andhe intersection point forms a kink in the strain–life curve. Thequivalent strain amplitudes corresponding to the kink points are.45% for tension–compression (path I), 0.48% for pure torsion (pathI), 0.56% for the proportional loading (path III), and 0.30% for the 90◦

ut-of-phase nonproportional loading (path IV). The fatigue livesorresponding to the kink points are approximately 11,360 cyclesnder tension–compression (path I), 3800 cycles under cyclic tor-ion (path II), 4000 cycles under proportional axial-torsion loading

m, axial mean stress; ��/2, axial stress amplitude; ��/2, shear strain amplitude;ding cycle; Nf , fatigue life (number of cycles to failure).

(path III), and 15,780 cycles under nonproportional loading (pathIV). Results shown in Fig. 10 suggest that the effect of loading pathon the fatigue life depends on loading magnitude. At an equiv-alent strain amplitude lower than 0.45%, the fatigue lives undertension–compression (path I) and the proportional loading (pathIII) are very close and appear to be the highest compared to theother two loading paths. At an equivalent strain amplitude higherthan 0.45%, the fatigue life under proportional axial-torsion load-ing (path III) is the highest and is clearly longer than that undertension–compression (path I). For a given equivalent strain ampli-tude, 90◦ out-of-phase nonproportional loading (path IV) results inthe shortest fatigue life among the four loading paths tested.

The detrimental influence of the mechanical twin on the fatigueresistance is most aggravated under the nonproportional loadingpath. This is verified by comparing the morphology of residualmechanical twins under tension–compression and that under thenonproportional loading path after the fatigue tests at an equivalentstrain amplitude of 1%. The outer surfaces of the tubular specimensfatigued under path I and path IV were examined in the regions far

away from the fracture surface using optical microscope and theresults are shown in Fig. 11a and b, respectively. At a strain ampli-tude of 1% under fully reversed tension–compression (Fig. 11a),deformation twins were densely distributed in the large elongated

Y. Xiong et al. / Materials Science and En

Fig. 11. Optical microstructure on the outer surface of the testing specimen far awayft(

glgtsggtslntoopi

nrionsii

rom the fracture surface at an equivalent strain amplitude of 1%: (a) fully reversedension–compression (path I); and (b) 90◦ out-of-phase nonproportional loadingpath IV).

rains. Two sets of twins can be observed in some local areas in thearge grains. Almost no deformation twin was detected in the smallrains. For path IV (Fig. 11b), the amount of residual deformationwins in the large grains was significantly increased. Two or moreets of twins were detected frequently in local regions in the largerains. Deformation twin can be also regularly observed in smallrains. It is noticed that during nonproportional loading, the direc-ion of the principal stress rotates with time. Comparing with aimilar loading amplitude under uniaxial loading, nonproportionaloading results in a higher population of grains experiencing twin-ing/detwinning. In addition, for the grain with a fixed orientation,he rotation of the principal stress axis significantly increases thepportunity to activate different sets of twins. A greater amountf deformation twins produced under the 90◦ out-of-phase non-roportional loading path generally serves as the sites for crack

nitiation, leading to a decreased fatigue life.Due to the pole nature of mechanical twin, the twin-

ing/detwinning process under tension–compression loadingesults in asymmetric stress–strain hysteresis loop. Therefore, its reasonable to speculate that an observed asymmetric shapef the stress–strain hysteresis loop reflects persistent twin-

ing/detwinning process. A careful observation of the stabilizedtress–strain hysteresis loops under three loading paths involv-ng axial stresses (Figs. 5, 7 and 8) reveals that the kink pointsn the strain–life curves shown in Fig. 10 nicely match the

gineering A 546 (2012) 119– 128 125

demarcation strain amplitudes above which asymmetricstress–strain hysteresis loops are observed in both the axialand shear directions. It should be mentioned that at a given strainamplitude, twinning/detwinning may occur in the early part offatigue loading but vanishes after certain number of loading cycles.

4. Multiaxial fatigue criteria

Critical plane approaches are widely accepted for fatigue lifepredictions of metallic materials, but the applicability to predictfatigue life of magnesium alloys is still unclear. Two critical planefatigue models will be discussed in terms of the fatigue results ofthe AZ31B magnesium alloy under combined axial-torsion load-ing: a modified Smith–Watson–Topper (SWT) model [24–26] anda model developed by Jiang [33].

4.1. Modified SWT criterion

The modified Smith, Watson, and Topper (SWT) model takes thefollowing mathematic form,

FP = 2b�ε〈�max〉 + 1 − b

2���� (2)

where FP denotes the “fatigue parameter.” � and � are the normalstress and shear stress on a material plane, respectively. ε and �are the normal strain and shear strain corresponding to the nor-mal stress and shear stress, respectively. The symbol � denotesthe range of the stress or strain and �max is the maximum stressin a loading cycle on the material plane. The symbol 〈〉 is the Mac-Cauley bracket which is defined as 〈x〉 = 0.5(x + |x|). The use of theMacCauley bracket is to ensure that no negative damage can beproduced. The symbol b in Eq. (2) is a material-related constantranging from 0 to 1.0. The critical plane is defined as the planewhere FP reaches a maximum. Clearly when b = 1.0, the modifiedSWT parameter is reduced to the original SWT model.

The SWT parameter [24] was originally developed to considerthe mean stress effect under uniaxial loading and the criterion wasextended to multiaxial fatigue by Socie [25] with a critical planeinterpretation. Eq. (2) was a modification of the SWT parameterby Jiang and Sehitoglu [26] to consider general cracking behav-ior. The modified SWT criterion has the capability of predictingdifferent cracking behavior with a proper choice of the value forthe material constant b in Eq. (2) [34]. When 0 ≤ b ≤ 0.37, the cri-terion can predict shear cracking behavior. When b ≥ 0.5, tensilecracking behavior is predicted. Mixed cracking behavior can betaken account by choosing b between 0.37 and 0.5. For the metallicmaterials that display mixed cracking behavior, a dependence ofcracking behavior on the load magnitude is usually observed. Thematerial constant b can be related to the equivalent stress magni-tude, such as that in the cases of 7075-T651 aluminum alloy [34]and AL6XN stainless steel [35]. The modified SWT parameter wasfound to correlate well the multiaxial fatigue experiments of AZ61Amagnesium alloy [19].

Fig. 12 shows the FP–Nf relationship for the uniaxialtension–compression and pure torsion when the original SWTmodel is used (b = 1 in Eq. (2)). Clearly, the original SWTmodel cannot even correlate the fatigue experiments undertension–compression and torsion.

It was found that only when b was taken to be zero would thetwo curves of tension–compression and torsion in the form of FP–Nfcould come together (Fig. 13). The mathematical equation, Eq. (1),

was used to best fit the baseline data from tension–compressionand torsion, and the following relationship was obtained,

(FP − 0.3)1.85Nf = 2200 (3)

126 Y. Xiong et al. / Materials Science and En

0.1

2

4

1

2

4

10

2

FP

=2

ma

x (

b=

1),

MP

a

101

10 2

10 3

10 4

10 5

10 6

Nf, Number of Cycles to Failure

AZ31B

Tension-CompressionTorsion

Fig. 12. Examination of the SWT model (b = 1 in Eq. (2)) for tension–compressionand torsion.

0.1

2

4

1

2

4

10

2

FP

=/2

(b

=0

), M

Pa

101

10 2

10 3

10 4

10 5

10 6

Nf, Number of Cycles to Failure

(FP-0.3)1.85

Nf=2200 AZ31B

Tension-CompressionTorsion Eq.(1) Fitting

Ft

wfo

itmEoopb

Fi

sonably well. Comparing with the modified SWT model, the Jiang

ig. 13. Best fitting using the modified SWT model (b = 0) based onension–compression and torsion results.

here FP is the SWT parameter defined by Eq. (2) and Nf is theatigue life in number of loading cycles to failure. Again, FP wasbtained using Eq. (2) when b = 0.

Eq. (3) was used to predict the fatigue lives. For a given load-ng case, the SWT parameter (FP) can be obtained by using Eq. (2)hrough the search of the maximum FP value in terms of possible

aterial plane orientations. The fatigue life was predicted by usingq. (3) for a given FP. A comparison of the predicted fatigue lives

btained by using the modified SWT model and the experimentalbservations is shown in Fig. 14. The diagonal solid line designateserfect prediction, and the two dotted lines are the factor-of-twooundaries. Results shown in Fig. 14 indicates that the SWT model

101

102

103

104

105

106

107

Pre

dic

ted

Life

, C

ycle

s

101

10 2

10 3

10 4

10 5

10 6

10 7

Observed Life, Cycles

Tenson-CompressionTorsionProportional

Nonproportional

AZ31B Modified SWT

ig. 14. Comparison of observed fatigue lives and predictions made using the mod-fied SWT criterion.

gineering A 546 (2012) 119– 128

with b = 0 can reasonably predict fatigue lives of the loading casesinvestigated. The model underestimates the fatigue lives of the pro-portional axial-torsion loading and overestimates the fatigue livesunder nonproportional loading path.

4.2. Jiang fatigue criterion

The critical plane multiaxial fatigue criterion developed by Jiang[33] is an energy based approach that can be expressed in the fol-lowing incremental form,⎧⎪⎨⎪⎩

dD =⟨

�mr

�0− 1

⟩m(

1 + �

�f

)dY,

dY = b�dεp + 1 − b

2�d�p.

(4)

In the above equations, D represents the fatigue damage and Ystands for plastic strain energy density on a material plane. � and� are the normal and shear stresses on a material plane and εp and�p are the plastic strains corresponding to stresses � and �, respec-tively. The symbols a and m are material constants. �0 and �f arethe correspondingly endurance limit and true fracture stress of thematerial, respectively. �mr is a memory stress reflecting the load-ing magnitude. The use of MacCauley bracket 〈〉 ensures that when�mr ≤ �0 the fatigue damage is zero. The critical plane is definedas the material plane where the fatigue damage accumulation firstreaches a critical value, D0.

The Jiang fatigue criterion is capable of accommodating differ-ent cracking modes under different stress states. This is achievedwith the help of constant, b, in Eq. (4). The constant, b, ranges from0.0 to 1.0. When b ≤ 0.375, the criterion predicts shear cracking.When b ≥ 0.5 tensile cracking is predicted. Mixed cracking mode isrepresented by the range 0.5 ≥ b ≥ 0.375.

For the material under investigation, it was found that b = 0yielded a best fit of the fatigue results of tension–compression andtorsion. Fig. 15 shows a comparison of the predictions by usingthe Jiang criterion and the experimentally obtained fatigue livesunder the four loading paths. Again, the solid diagonal line in thefigure represents a perfect prediction and the two dotted lines arethe factor-of-two boundaries. Overall, the model predicts fatiguelives of the material under the four axial-torsion loading paths rea-

criterion predicts slightly better fatigue lives.Theoretically, a critical plane multiaxial fatigue criterion can

also predict early cracking behavior of the material. However, it

101

102

103

104

105

106

107

Pre

dic

ted

Life

, C

ycle

s

101

10 2

10 3

10 4

10 5

10 6

10 7

Observed Life, Cycles

Tenson-CompressionTorsion Proportional Nonproportional

AZ31B Jiang Model

Fig. 15. Comparison of observed fatigue lives and predictions made using the Jiangcriterion.

Y. Xiong et al. / Materials Science and Engineering A 546 (2012) 119– 128 127

F t straa

hdmmcetstfctcarTtoc[of[

5

ffMT

ig. 16. Observation of early cracking under different loading paths at an equivalenxial-torsion; and (d) 90◦ out-of-phase nonproportional axial-torsion.

as been found that while fatigue lives can be reasonably pre-icted by using a fatigue criterion if the stress and strain are directlyeasured or computed with accuracy, cracking direction is muchore difficult to predict [34–36]. Fig. 16 shows the observed early

racking directions under the four different loading paths at anquivalent strain amplitude of 0.75%. The material was observedo form a crack with its surface normal to the axial direction for thepecimens tested under tension–compression (Fig. 16a) and a ver-ical crack under pure torsion (Fig. 16b). None of the critical planeatigue criteria designed for shear cracking can correctly predictracking behavior under tension–compression (Fig. 16a). Since thewo criteria evaluated in the current work with b = 0 predict shearracking behavior and both predict identical cracking directions for

given case, it is not surprising that neither model can predict cor-ectly the cracking behavior under tension–compression (Fig. 16a).he two criteria can reasonably predict the cracking directions ofhe other three cases (Fig. 16b–d). A similar conclusion was maden AZ61A Mg alloy under axial-torsion loading with respect to theapability of a critical plane approach to predict cracking behavior19]. Elaborated discussion can be found in an earlier investigationn the prediction of early cracking behavior of a material underatigue loading using a critical plane multiaxial fatigue criterion36].

. Further discussion

An earlier study [19] suggested that both the modified SWT

atigue model and the Fatemi–Socie critical plane multiaxialatigue model [28] correlated well the experiments of AZ61A

g alloy under different combined axial-torsion loading paths.he Fatemi–Socie critical plane multiaxial fatigue model had a

in amplitude of 0.75%: (a) tension–compression; (b) pure torsion; (c) proportional

difficulty to describe tension–compression experiments and pureshear experiments simultaneously. A similar conclusion can bemade for AZ31B Mg alloy. The current investigation suggests thatthe modified SWT model predicts reasonably well the fatiguelives of AZ31B Mg alloy under different axial-torsion loadingpaths. For most multiaxial fatigue criteria, the fully reversedtension–compression and pure shear loading cases are used todetermine the baseline data. A general practice is that when thegiven fatigue parameter (FP) is used to plot again the fatigue life (Nf),the FP–Nf curve for tension–compression loading and the FP–Nfcurve for pure torsion loading should merge to form a commonline. Fig. 13 is an example of such a practice. If a fatigue modelcannot bring these two FP–Nf curves together, such as the caseshown in Fig. 12 for the original SWT model, it is very unlikelythat the model can work well for more general loading cases.It was also found without showing the detailed results that theFatemi–Socie criterion [28] cannot bring together the FP–Nf curvesof tension–compression experiments and pure torsion experimentsfor the AZ31B Mg alloy.

It is worthwhile to compare the fatigue properties betweenAZ31B Mg alloy in the current investigation and an AZ61A Mgalloy studied earlier [19]. AZ61A has a higher Al content and lowerSi content than those of AZ31B. In addition, AZ61A consists ofequiaxed grains with an almost identical average grain size of20 �m while AZ31B under the current investigation contains signif-icantly non-uniformly distributed laminated structure with grainsizes ranging from a few microns to as large as 60 �m. There are

several common aspects between the two Mg alloys: both materialsdisplay distinct kinks in the strain–life curves for different loadingaxial-loading paths, significant twinning–detwinning occurs at thestrain levels higher than those of the kink points in the strain–life

1 and En

clwattf1tUcatcslcM

6

aeasetltsa

A

oSSSS

[

[

[[

[

[[

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[[[[

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[

28 Y. Xiong et al. / Materials Science

urves, and nonproportional loading results in the shortest fatigueives and proportional loading paths have the longest fatigue lives

ith the equivalent strain amplitude defined. AZ31B Mg alloy hasn overall better fatigue strength than that of AZ61A with iden-ical strain magnitude for a given loading path. Under uniaxialension–compression, the two materials have similar fatigue livesor a given strain amplitude when the fatigue life is shorter than0,000 cycles, and AZ31B has a much better fatigue strength thanhat of AZ61A when the fatigue life is longer than 10,000 cycles.nder cyclic pure shear loading, the two materials display practi-ally the same strain–life curves. Under the designed proportionalxial-torsion loading path, the two Mg alloys have practically iden-ical strain–life curves when the fatigue life is less than 2 × 105

ycles. Beyond 2 × 105 cycles, AZ31B has a slightly better fatiguetrength than that of AZ61A. Under 90◦ out-of-phase axial-torsionoading, the two materials display practically identical strain–lifeurves for fatigue lives less than 10,000 cycles beyond which AZ31Bg alloy exhibits distinguishable better fatigue strength.

. Conclusions

Fatigue experiments were conducted on an AZ31B magnesiumlloy under four combined axial-torsion loading paths in ambi-nt air. A clear kink in the strain–life curves can be observed forll the four loading paths conducted. With identical equivalenttrain magnitudes defined, nonproportional loading produces low-st fatigue lives and the proportional axial-torsion loading andension–compression loading result in similar and longest fatigueives. The modified Smith, Watson, and Topper fatigue model andhe Jiang critical plane multiaxial fatigue criterion can provide rea-onable fatigue life estimates for the material under the combinedxial-torsion loading paths.

cknowledgments

This research was financially supported by the US Departmentf Energy, Office of Basic Energy Sciences under Grant No. DE-

C0002144. Ying Xiong gratefully acknowledges the support by thecientific Research Foundation for the Returned Overseas Chinesecholars, State Education Ministry, and the support from Naturalcience Foundations of Zhejiang Province (Y107284).

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gineering A 546 (2012) 119– 128

References

[1] H. Zenner, F. Renner, Int. J. Fatigue 24 (2002) 1255–1260.[2] S. Hasegawa, Y. Tsuchida, H. Yano, M. Matsui, Int. J. Fatigue 29 (2007)

1839–1845.[3] S.M. Yin, H.J. Yang, S.X. Li, S.D. Wu, F. Yang, Scripta Mater. 58 (2008)

751–754.[4] M. Matsuzuki, S. Horibe, Mater. Sci. Eng. A 504 (2009) 169–174.[5] S. Begum, D.L. Chen, S. Xu, A.A. Luo, Int. J. Fatigue 31 (2009) 726–735.[6] S. Begum, D.L. Chen, S. Xu, A.A. Luo, Mater. Sci. Eng. A 517 (2009) 334–343.[7] M. Huppmann, M. Lentz, K. Brömmelhoff, W. Reimers, Mater. Sci. Eng. A 527

(2010) 5514–5521.[8] F. Lv, F. Yang, Q.Q. Duan, T.J. Luo, Y.S. Yang, S.X. Li, Z.F. Zhang, Scripta Mater. 61

(2009) 887–890.[9] S.H. Park, S. Hong, W. Bang, C.S. Lee, Mater. Sci. Eng. A 527 (2010) 417–423.10] S.H. Park, S. Hong, B.H. Lee, W. Bang, C.S. Lee, Int. J. Fatigue 32 (2010)

1835–1842.11] F. Lv, F. Yang, Q.Q. Duan, Y.S. Yang, S.D. Wu, S.X. Li, Z.F. Zhang, Int. J. Fatigue 33

(2011) 672–682.12] X.Z. Lin, D.L. Chen, Mater. Sci. Eng. A 496 (2008) 106–113.13] X.Y. Lou, M. Li, R.K. Boger, S.R. Agnew, R.H. Wagoner, Int. J. Plast. 23 (2007)

44–86.14] K. Shiozawa, M. Nagai, T. Murai, T. Takahashi, Jpn. J. Soc. Mater. Sci. 58 (2009)

235–242.15] T.S. Srivatsan, L. Wei, C.F. Chang, Eng. Fract. Mech. 56 (1997) 735–747.16] A.N. Chamos, S.G. Pantelakis, G.N. Haidemenopoulos, E. Kamoutsi, Fatigue Fract.

Eng. Mater. Struct. 31 (2008) 812–821.17] S.H. Park, S. Hong, B.H. Lee, C.S. Lee, Int. J. Mod. Phys. B 22 (2008) 5503–

5508.18] Q. Li, Q. Yu, J. Zhang, Y. Jiang, Scripta Mater. 62 (2010) 778–781.19] Q. Yu, J. Zhang, Y. Jiang, Q. Li, Int. J. Fatigue 33 (2011) 437–447.20] Q. Yu, J. Zhang, Y. Jiang, Q. Li, Int. J. Fatigue, in press.21] S. Bentachfine, G. Pluvinage, L.S. Toth, Z. Azari, Eng. Fract. Mech. 54 (1996)

513–522.22] S. Bentachfine, Z. Azari, G. Pluvinage, Revue Francaise de Mecanique 2 (1993)

371–378.23] J. Zhang, Q. Yu, Y. Jiang, Q. Li, Int. J. Plast. 27 (2011) 768–787.24] R.N. Smith, P. Watson, T.H. Topper, J. Mater. 5 (1970) 767–778.25] D.F. Socie, ASME J. Eng. Mat. Tech. 109 (1987) 293–298.26] Y. Jiang, H. Sehitoglu, Report no. 161, UILU-ENG 92-3602, College of Engineer-

ing, University of Illinois at Urbana-Champaign, 1992.27] J. Albinmousa, H. Jahed, S. Lambert, Int. J. Fatigue 33 (2011) 1127–1139.28] A. Fatemi, D.F. Socie, Fatigue Fract. Eng. Mater. Struct. 11 (1988) 149–165.29] H. Jahed, A. Varvani-Farahani, Int. J. Fatigue 28 (2006) 467–473.30] L. Wu, A. Jain, D.W. Brown, G.M. Stoica, S.R. Agnew, B. Clausen, D.E. Fielden, P.K.

Liaw, Acta Mater. 56 (2008) 688–695.31] O. Muránsky, D.G. Carr, P. Sittner, E.C. Oliver, Int. J. Plast. 25 (2009)

1107–1127.

32] L. Jiang, J.J. Jonas, A.A. Luo, Scripta Mater. 54 (2006) 771–775.33] Y. Jiang, Fatigue Fract. Eng. Mater. Struct. 23 (2000) 19–32.34] T. Zhao, Y. Jiang, Int. J. Fatigue 30 (2008) 834–849.35] S. Kalnaus, Y. Jiang, J. Eng. Mater. Technol. 130 (2008) 031013.36] Y. Jiang, O. Hertel, M. Vormwald, Int. J. Fatigue 29 (2007) 1490–1502.