Multiaxial behaviour of wrought magnesium alloys – A review and suitability of energy-based fatigue life model

Theoretical and Applied Fracture Mechanics 73 (2014) 97–108

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Multiaxial behaviour of wrought magnesium alloys – A review and suitability of energy-based fatigue life model H. Jahed a, J. Albinmousa b,⇑ a b

Mechanical and Mechatronics Engineering Department, University of Waterloo, Waterloo, ON, Canada Mechanical Engineering Department, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia

a r t i c l e

i n f o

Article history: Available online 27 August 2014 Keywords: Wrought magnesium Multiaxial fatigue Strain energy Fatigue modeling

a b s t r a c t Different wrought magnesium alloys from AM, AZ, and ZK family in the form of extrusion, rolled sheet and rolled plate have been selected for this study. Monotonic and cyclic behaviours are presented and compared. In particular, multi axial behaviours under proportional and non-proportional loadings are discussed. Despite the differences between the investigated alloys, it has been found that these alloys exhibit similar monotonic and cyclic characteristics. The similarity is attributed to the limited slip system in HCP magnesium, and the dominant role of deformation twinning in causing yield and hardening asymmetry. With strain energy density merit as a suitable fatigue parameter, it is therefore hypothesized that a simple two-parameter energy-based fatigue model is capable of correlating fatigue life of wrought magnesium alloys irrespective of material process, loading conditions and loading orientations. The hypothesis is then tested over a large number of fatigue results (354 tests). It is shown that fatigue lives predicted using the energy-life model are in good agreement with experimental results. Such simple model may prove to be useful in the early design stages lightweight components out of magnesium alloys. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Energy crises and environmental concerns have led to strict transportation laws asking for aggressive reduction in fuel consumption and pollutant gas emission. The demand on fueleconomic vehicles has forced transportation industry to boost light-weighting research and seriously consider its applications. Owing to its high specific strength and excellent machinability, magnesium alloys have recently been in the front and center of structural lightweight research. Cyclic loading is the prime loading of vehicles structural body and suspension parts. In particular these loads have a multiaxial nature and hence study of multiaxial behaviour of Mg alloys is of significant importance. The predominant manufacturing method for magnesium is casting. Many researchers have studied fatigue of cast magnesium alloys, mainly AZ91 and AM60. Research focused on high cycle fatigue behaviour of these alloys has demonstrated that fatigue cracks initiate at porosities [1–3], and that crack initiation life is primarily

⇑ Corresponding author. E-mail addresses: [email protected] (H. Jahed), [email protected] (J. Albinmousa). http://dx.doi.org/10.1016/j.tafmec.2014.08.004 0167-8442/Ó 2014 Elsevier Ltd. All rights reserved.

controlled by pore size [2,4]. Lu et al. [3] and Horstemeyer et al. [4] reported that fatigue crack initiation occurs below the surface. The fatigue crack propagation path in AM60B depends on the local microstructure and casting defects. Fatigue cracks grow through a-magnesium dendrites under low porosity conditions, and through interdendritic regions under high porosity conditions [3]. El Kadiri et al. [5] investigated crack propagation mechanisms in AM50 cast alloy and found that fatigue cracks initially propagate along the interface of a-magnesium dendrites and at the aluminum-rich boundary. Cracks then coalesce into a small, main fatigue crack that advances interdendritically. In the long crack regime, the crack advances in a mixed transdendritic–interdendritic mode. However, due to their low strength and ductility, cast alloys are not yet good candidates for load-bearing structural components. Wrought alloys have shown superior strength and ductility when compared to cast alloys [6,7]. Extrusion, rolled sheet and plates, and warm-forge are the material processes producing wrought alloys [8]. Fatigue behaviour of these alloys has in the past decade been of interested of many research centers. The focus of research, among others, has been on fatigue crack initiation, crack propagation, process induced anisotropy and asymmetry, uniaxial cyclic behaviour under push–pull and shear strains, and multiaxial loading.

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AZ31B is the most common wrought magnesium alloy in the industry. The AZ31B magnesium sheet, similar to other wrought magnesium alloys, has a strong texture [9,10], which results in different modes of deformation under in-plane tension and compression. Cyclic in-plane loading causes twinning and de-twinning deformation in the two consecutive reversals. Twin deformation accumulated during compressive loading is not fully recovered in subsequent tension reversals [11]. Therefore, the AZ31B sheet exhibits cyclic hardening during compression, and cyclic softening during the tension reversal [11,12]. An unusual asymmetric shape of the hysteresis loop is the key feature of the cyclic behaviour of wrought magnesium alloys, which is more pronounced at high strain amplitudes. The strain-life curve shows a kink at the strain amplitude above which hardening behaviour is asymmetric in tension and compression reversals [13,14]. AZ31B has superior fatigue strength in the transverse direction, as compared with the rolling direction, under both stress- and strain-control conditions [12]. Fatigue crack initiation occurs in the transgranular mode, while crack growth occurs in the intergranular mode [15]. Although hardening behaviour of AZ31B for strain-controlled cyclic axial loading tests is asymmetric, stress-control loading produces symmetric hysteresis [16]. Refining grain size in extruded AZ31 alloys makes the asymmetry less pronounced under monotonic and cyclic axial loading [17,18]. Grain size refinement also improves fatigue strength, especially in the high cycle regime [19]. Extruded AZ31B exhibits cyclic hardening behaviour due to residual twins accumulated during cycling [20]. Unlike cast magnesium alloys, Masing behaviour is not observed in extruded magnesium alloys due to the strong tension–compression asymmetry [21]. The S–N curve for extruded AZ31B has a sharp bend [19,22,23]. Twin bands are the preferable locations for the initiation of fatigue cracks in extruded AZ31 [24]. Ishihara et al. [25] reported that fatigue crack initiation life is negligible compared to the total life, and fatigue life can be reasonably estimated by a fracture mechanics approach. Twinning is the predominant mechanism of plastic deformation at high plastic strain amplitude whereas slip prevails at low plastic strain amplitude [7]. Lin and Chen [26] reported that the Bauschinger effect in extruded AZ31 is more pronounced at high strain amplitudes. There have been few comprehensive multiaxial studies of wrought magnesium alloys. The study on AZ31B extrusion by Albinmousa et al. [27–29] and on AZ61A extrusion by Zhang et al. [30] that discusses the effect of multi-axiality on fatigue behaviour of wrought alloys are among the most comprehensive multiaxial studies. In this paper, the results of these works are compared with other multiaxial work on Mg alloys. First, the asymmetry possess by wrought alloys is reviewed and causes are discussed. Then multiaxial loading behaviour is reviewed and results for different wrought alloys are compared. A two-parameter energy-based fatigue model is then proposed. It is shown that a large number of fatigue results for Mg wrought alloys can be correlated through the proposed model.

2. Microstructure Three types of wrought magnesium are considered in this review: extrusion, sheet and rolled plate. Chemical compositions and microstructural characteristics of the considered alloys are listed in Tables 1 and 2. It is seen from Table 2 that grain size varies depending on the manufacturing process. The effect of grain size on the mechanical behaviour of magnesium alloys have been investigated by many researchers. Koike et al. [31] did monotonic tensile tests on fine-grained AZ31B extrusion with an average grain size of 6.5 ± 0.4 lm and found substantial non-basal slip activities at 2% strain. Uematsu et al. [19] studied the effect of

extrusion conditions on grain refinement and fatigue behaviour of several magnesium extrusions. For AZ31B, experiments by Uematsu et al. showed that grain size decreases with decreasing working temperature. They were able to achieve grain size of 2.1 lm at extrusion rates of 67 and outlet temperatures of 625°K. Fatigue strength improvement was found to be associated with smaller grain size, especially in the high cycle regime. The fatigue strengths at 107 cycles for the samples with grain sizes of 7.2 and 2.1 lm were 90 and 130 MPa, respectively. Like Uematsu et al. [19], Chino et al. [32] observed the same relation between extrusion temperature and grain size. In addition, they found that monotonic tension–compression anisotropy became less pronounced with fine-grained samples compared to other samples with larger grain sizes. Zhu et al. [18] performed cyclic tension–compression tests on ultrafine-grained AZ31 extrusion with an average grain size of 5.6 lm and compared it with a conventional one that had an average grain size of 30 lm. Comparisons of the cyclic behaviour showed that, while conventional extrusion exhibits stress–strain asymmetry and cyclic hardening, the ultrafine-grained extrusion exhibits symmetric stress–strain behaviour and cyclic softening.

3. Monotonic behaviour Monotonic tension, compression and torsion stress–strain curves are compared for several alloys, including, AZ31 [33], AZ31B [28,34], AM30 [35], AM60 [36], AZ61A [30] and ZK60 [37]. Stress–strain curves for different orientations are presented. Definition of orientations is shown in Fig. 1. Monotonic axial behaviours, tension and compression, of different magnesium extrusion at different orientations are compared in Fig. 2. Considering the tensile curves for extrusion direction in Fig. 2a it can be seen that the post-yielding behaviour of AM30 [35], AM60 [36], AZ31B [34], AZ61A [30] and ZK60 [37] appears to be of a power-law nature. The post-yielding behaviour of AZ31 [33] and AZ31B [28] is plateau. Conversely, the tensile behaviour for AZ31B along normal direction [28] is distinctly different compared to other tensile as it shows concave upward post-yielding behaviour. It should be noted that 45° specimens [28] were machined in the ED-ND plane. Monotonic compressive stress–strain curves of AZ31B [28,34,38], AZ61A [30] and ZK60 [37] are compared in Fig. 2b. It is seen from this figure the three alloys exhibit concave upward post-yielding behaviour when loaded along the extrusion or transverse directions. Such unusual behaviour is commonly attributed to twinning deformation activated due to loading orientation with respect to the texture. The compressive responses for AZ31B along the 45° and normal directions show different behaviours compared to other curves. Their post-yielding behaviour is concave downward. Monotonic tensile and compressive behaviours for AZ31 [33] and AZ31B [39,40] magnesium rolling at different orientations are compared in Fig. 3a and b. Stress–strain curves were plotted in two graphs to better show the difference. Fig. 3a shows that the post-yielding behaviour of the tensile curves is of a power-law nature while the compressive curves exhibit sigmoidal-type behaviour with concave upward characteristics due to deformation twinning. On the other hand, the tensile and compressive behaviours for both transverse and normal directions are different as shown in Fig. 3b. This figure shows that tensile loading along the normal direction results in sigmoidal-type behaviour similar to that observed in compression. In contrast, compressive loading along the normal direction results in usual concave downward behaviour. Similarly, tensile loading along the transverse direction results in concave downward behaviour.

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Material

Al

Zn

Mn

Mg

Ca

Albinmousa et al. SB Behravesh

AZ31B Extrusion AZ31B-H24 Sheet

3.1 2.73

1.05 0.915

0.54 0.375

Bal Bal

– –

Lugo et al.

AZ31 Sheet AZ31 Extrusion AZ31 Rolling AZ61A Extrusion ZK60 Extrusion AM60 Extrusion AZ31B Rolling AZ31B-H24 Rolling AZ31 Sheet

2.8 2.6 2.9 6.5 – 5.6–6.4 3.6 3 3

– – – 0.95 5.2 – 1 1 1

0.44 0.63 0.55 0.325 – 0.26–0.5 0.5 0.2 –

Bal Bal Bal Bal Bal Bal Bal Bal Bal

Zhang et al. Yu et al. Zeng et al. Park et al. Wu et al. Yan et al.

<0.01 <0.01 <0.01 – – – – – –

Cu 0.0008 – <0.01 <0.01 <0.01 0.05 – 60.008 – – –

Fe 0.0035 – <0.005 <0.005 <0.005 0.005 – 60.004 – – –

Ni 0.0007 – <0.005 <0.005 <0.005 0.005 – 60.001 – – –

Si – – <0.01 <0.01 <0.01 0.1 – 60.05 – – –

Zr

Other impurities

– –

– –

– – – – 0.47 – – – –

– – – – 0.3 – – – –

Table 2 Grain size and microstructural characteristics for different magnesium alloys. Author

Material

Plane

Size (lm)

Shape

Comments

Intermetallic

Intermetallic compound

Albinmousa et al.

AZ31B Extrusion

58.8 17.8 34.6 5

– – – Equiaxed

Some larger grains Fine with some large gains Fine, elongated and large grains Annealed

– – – –

Mg17Al12 Mg17Al12 Mg17Al12 –

5.7 28.8 10.7 20 10 56 50 15 80

Elongated – Equiaxed Equiaxed – – – – –

– Some grain has size of 140 – – Fibrous structure of elongated grains Grain size range from 10 to 150 – – –

3.7 5.8–40 3.3 – – – – – –

– – – Mg17Al12 – Mg17Al13 & Al–Mn Particles – – –

SB Behravesh

AZ31B-H24 Sheet

ED-ND TD-ND ED-TD RD-TD

Lugo et al.

AZ31 Sheet AZ31 Extrusion AZ31 Rolling AZ61A Extrusion ZK60 Extrusion AM60 Extrusion AZ31B Rolling AZ31B-H24 Rolling AZ31 Sheet

RD-TD ED-TD RD-TD Average TD-ND Average Average Average Average

Zhang et al. Yu et al. Zeng et al. Park et al. Wu et al. Yan et al.

Fig. 1. Definition of material orientation.

In general, the monotonic behaviours of rolled alloys are comparable to that of extruded alloys. Fig. 4 compares monotonic axial stress–strain curves for AZ31 [33] and AZ31B-H24 [13] sheets at different orientations. It is seen from Fig. 4 that the post-yielding behaviour is independent of the orientation of loading. Similar to extrusion, the concave upward post-yielding behaviour in compression is observed in all orientations. The 45° specimens were machined from the RD-TD plane. In general, the tensile behaviour of AZ31 sheet is similar to that of extrusion shown in Fig. 2. Sigmoidal stress–strain relation is a major characteristic of the mechanical behaviour of wrought magnesium alloys. In monotonic loading such behaviour is only observed at certain orientation. For example, applying compressive loading along the working

direction, such as extrusion or rolling, as seen in Figs. 2–4 causes the sigmoidal shape behaviour. Another example is applying tensile loading along the normal direction as seen in Figs. 2a and 3b. Such behaviour does not depend on the mode of loading such as tension or compression. Rather, it depends on whether or not the applied loading is orientated such that it causes extension of the c-axis of magnesium hexagonal lattice. Metals with hexagonal crystal structure, such as magnesium, deform plastically under different mechanisms: slipping, twinning and detwinning. Wrought magnesium alloys develop strong texture due to different manufacturing processes such as extrusion or rolling. Such processes result in aligning the basal plane with the extrusion or rolling direction and with the c-axis perpendicular to it. Extension along the c-axis activates extension twins resulting in the observed concave upward post-yielding behaviour in Figs. 2–4. First, twinning occurs leading to low stress yielding. As the loading continues, the twinning process ends and detwinning starts, resulting in concave upward hardening. Finally, a slip mechanism starts causing rapid increase in the hardening rate. This rapid hardening rate is attributed to the contribution of pyramidal slip [41]. Monotonic shear behaviour of AZ31B extrusion along the extrusion [28] and normal [38] directions are compared in Fig. 5. Unlike axial loading, the monotonic torsional behaviour shows a linear hardening behaviour. Ideally, the crystals are oriented such that the basal plane is parallel to the extrusion direction. In this case, shear loading is not expected to cause extension along the c-axis and hence no twinning deformation is anticipated. Observations of mechanical twins had been reported [30,42] at large shear strains, however, their existence was found to cause no influence on the mechanical behaviour. Summaries of monotonic tensile, compressive and torsional properties for different magnesium alloys are listed in Tables 3–5.

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400

400

(a)

300

Stress (MPa)

Stress (MPa)

300

(a)

AZ31B-ED [A] AZ31B-45 [A] AZ31B-ND [A] AZ31-ED [B] AM60-ED [C] ZK60-ED [D] AM30-ED [E] AZ31B-ED [F] AZ61A-ED [G]

200

100

200

AZ31B-RD [A] AZ31B-RD [B] AZ31-RD [C] AZ31-RD [C] AZ31B-RD [C] AZ31B-RD [C]

100

0

0 0

5

10

15

20

25

-30

-20

500

0

-10

Strain (%)

10

20

30

Strain (%) 400

(b)

(b)

400

Stress (MPa)

Stress (MPa)

300 300

200

100

0 -18

AZ31B-ED [F] AZ61A-ED [G] AZ31B-TD [H] ZK60-ED [D] AZ31B-ED [A] AZ31B-45 [A] AZ31B-ND [A]

200

AZ31B-ND [A] AZ31B-TD [A] AZ31B-ND [B] AZ31B-ND [A] AZ31B-TD [A] AZ31B-ND [B]

100

0 -16

-14

-12

-10

-8

-6

-4

-2

0

-30

-20

Fig. 2. Monotonic axial stress–strain curves for different magnesium extrusions at different orientations. (a) Tensile and (b) compression. In legend: [A] Albinmousa et al., 2011, [B] Lugo et al., 2013, [C] Zeng et al. 2010 at 0.0053/s, [D] Yu et al. 2012, [E] Jiang et al. 2006, [F] Xiong et al. 2012, [G] Zhang et al. 2011 and [H] MahmoudiAsl, 2011. Curves from [C] and [E] are for true stress and strain.

0

-10

Strain (%)

10

20

30

Strain (%) Fig. 3. Monotonic axial true stress–strain curves for rolled magnesium plates at different orientations. (a) rolling direction and (b) transverse and normal direction. In legend: [A] Wu et al., 2010, [B] Park et al., 2010 and [C] Lugo et al., 2013.

350 300

4.1. Hysteresis loops

250

Cyclic hysteresis loops for several magnesium alloys are compared in Figs. 6–8. In Fig. 6a, cyclic axial hysteresis loops for four magnesium extrusions, AZ31 [33], AZ31B [28], AZ61A [43] and ZK60 [37], are compared. These loops show the cyclic behaviour at axial strain amplitude of 0.6% along the extrusion direction. All hysteresis loops exhibit asymmetric behaviour due to twinning deformation. Comparing the tensile part of the hysteresis with the compressive one can recognize such asymmetry. On the other hand, Fig. 6b shows the cyclic shear behaviour of two magnesium extrusions, AZ31B [28,34] and AZ61A [43] at shear strain of 0.35%. The shear hysteresis loops in Fig. 6b are symmetric and no sign of twinning deformation is observed. Summary of cyclic axial and shear properties for different magnesium alloys is listed in Table 6. Hysteresis loops for proportional, and 90° out-of-phase are presented in Figs. 7 and 8, respectively. Two magnesium extrusions are compared in these figures: AZ31B [28,34] and AZ61A [30]. These figures suggest that when the axial strain amplitude is as high as 0.5%, as shown in Fig. 8, axial hysteresis loops become asymmetric as a result of twinning deformation. This is not observed in the hysteresis loops of the shear mode. Wrought magnesium alloys have strong texture, with the majority of the basal planes parallel to the working direction [44–46].

Stress (MPa)

4. Cyclic behaviour

200 150 AZ31B-24H-RD [A] AZ31B-24H-TD [A] AZ31-RD [B] AZ31B-24H-RD [A] AZ31B-24H-45 [A] AZ31B-24H-TD [A]

100 50 0 -10

-5

0

5

10

15

20

25

Strain (%) Fig. 4. Tensile and compressive stress–strain curves for magnesium AZ31B-H24 and AZ31 sheets at different orientations. In legend: [A] SB Behravesh, 2013 and [B] Lugo et al., 2013. Curve from [B] is for true stress and strain.

Only loading that causes extension along the c-axis can activate tension twinning [42,44,47,48]. Tensile loading that is parallel to the extrusion direction does not activate twinning [48]. Also, pole figure analyses on AZ31B extrusion [47] and AZ31B-O sheet [42] indicate that tensile loading and unloading to zero stress amplitude preserve the texture orientation. Twinning is the dominant deformation mechanism in compression. The twinning deformation

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180 160

Stress (MPa)

140 120 100 80 60 40

Albinmousa et al (AZ31B-ED) Mahmoudi-Asl (AZ31B-ND)

20 0 0

10

20

30

40

Strain (%) Fig. 5. Monotonic shear stress–strain curves for AZ31B magnesium extrusion at extrusion and normal directions.

causes a re-orientation of the lattice to about 86.6° [44,47–49]. In this case, the lattice orientation is in a favourable position to detwin through subsequent tensile loading [42,47–50]. After the

exhaustion of the detwinning process, slip deformation becomes operative in order to accommodate the applied strain, which is seen as a rapid increase in the strain hardening rate after the inflection point [51,52]. Detailed experimental observations on the texture evaluation of magnesium under monotonic and cyclic loadings can be found in [42]. The cyclic shear behaviour is symmetric as seen from Fig. 6b. As explained earlier, crystals are oriented such that their basal planes are aligned with the working direction, with their c-axis perpendicular to it. Pure shear stress, which produces no normal stress along and/or perpendicular to the c-axis, yields symmetric behaviour. This means that slip is the dominant plastic deformation mechanism. Although twins were observed in cases where large shear strain amplitudes were applied, their effect on the cyclic shear behaviour was insignificant [30,42]. Also, due to the fact that the basal planes are generally oriented parallel to the extrusion direction, the basal and the prismatic planes come under direct shear loading. Basal slip is the dominant slip system in AZ31B at room temperature with the lowest CRSS among other slip systems [42,48,53]. Therefore, basal slip could be the dominant slip system in shear loading. It is seen from Fig. 8 that the hysteresis loops for axial mode show sign of twinning-detwinning deformation. This is not observed in axial hysteresis loops for proportional loading, Fig. 7,

Table 3 Monotonic tensile properties for different magnesium alloys. Author

Material

Direction

E (GPa)

ry,0.2% (MPa)

rUTS (MPa)

%EL

K (MPa)

n

Albinmousa et al.

AZ31B Extrusion

ED 45 ND

43.7 43.5 40.6

213.3 65.7 55.67

227.5 209 246.6

10 17.51 9.38

260 497.72 –

0.0283 0.386 –

SB Behravesh

AZ31B-H24 Sheet

RD TD

45

224 281

292 320

14 22

347 348

0.067 0.035

Lugo et al.

Zhang et al. Yu et al. Zeng et al.

AZ31 Sheet AZ31 Extruded AZ31 Rolling AZ61A Extrusion ZK60 Extrusion AM60 Extrusion

40.3 40.3 40.3 43.3 45 40

250 224 141 192 279 253

342 297 305 279 364 139

16.8 13.8 17 8.94 16.7 15.8

Park et al.

AZ31B Rolling

RD ND

153 53

281 330

12.4 14.5

Wu et al.

AZ31B-H24 Rolling

RD TD ND

230 180 85

340 339 371

19.4 24 14.5

Table 4 Monotonic compressive properties for different magnesium alloys. Author

Material

Direction

ry,0.2% (MPa)

rUCS (MPa)

Deformation (%)

Albinmousa et al.

AZ31B extrusion

ED 45 ND

108 94 127

364 230.3 316.7

12.6 16.8 5.3

SB Behravesh

AZ31B-H24 sheet

RD 45 TD

162 167 181

211 213 223

5.2 4.8 4.6

Lugo et al. Zhang et al. Q. Yu et al. Park et al.

AZ31 Extruded AZ31 Rolling AZ61A Extrusion ZK60 Extrusion AZ31B Rolling

– – – – RD ND

65 77 120 171 66 128

335.1 378.6 308 190 292 273

51.2 12.1 8 5 17.5 18.5

Wu et al.

AZ31B-H24 Rolling

Mahmoudi-Asl

AZ31B Extrusion

RD TD ND TD

100 86 150 50

354.3 323.2 331.4 307

9.7 10 3.7 1.2

102

H. Jahed, J. Albinmousa / Theoretical and Applied Fracture Mechanics 73 (2014) 97–108

Table 5 Monotonic torsional properties. Author

Material

Direction

G (GPa)

sy,0.2% (MPa)

sMax (MPa)

Albinmousa et al. Mahmoudi-Asl Zhang et al. Xiong et al.

AZ31B extrusion AZ31B extrusion AZ61A extrusion AZ31B extrusion

ED ND ED ED

16.47 15 16.4 16.9

46.9 40 – –

162.3 – – –

400

60

(a)

Axial

(b)

Shear

40

Stress (MPa)

Stress (MPa)

200

0

-200

Yu et al (ZK60) Yu et al (AZ61A) Albinmousa et al (AZ31B) Lugo et al (AZ31)

0.2

0.4

0 -20 -40

Albinmousa et al (AZ31B) Xiong et al (AZ31B) Yu et al (AZ61A)

-60 -0.4

-400 -0.6 -0.4 -0.2 0.0

20

0.6

-0.2

Strain (%)

0.0

0.2

0.4

Strain (%)

Fig. 6. Hysteresis loops for pure cyclic loading. (a) At ea = 0.6% and (b) at ca = 0.35%.

200

80

(a)

Axial

60

Shear

40

Stress (MPa)

100

Stress (MPa)

(b)

0

20 0 -20 -40

-100 Albinmousa et al (AZ31B) Xiong et al (AZ31B) Zhang et al (AZ61A)

-200 -0.4

-0.2

0.0

0.2

Albinmousa et al (AZ31B) Xiong et al (AZ31B) Zhang et al (AZ61A)

-60 -80 -0.6

0.4

-0.4

-0.2

Strain (%)

0.0

0.2

0.4

0.6

Strain (%)

Fig. 7. Hysteresis loops for multiaxial proportional loading. (a) Axial mode hysteresis and (b) shear mode hysteresis. Albinmousa et al.: ea = 0.28% and ca = 0.36%, Xiong et al. and Zhang et al.: ea = 0.28% and ca = 0.49%.

300

(a)

Axial

100

(b)

Shear

Stress (MPa)

Stress (MPa)

200 100 0 -100 -200 -300 -0.6

Albinmousa et al (AZ31B) Xiong et al (AZ31B) Zhang et al (AZ61A)

-0.4

-0.2

0.0

0.2

Strain (%)

0.4

50

0

-50 Albinmousa et al (AZ31B) Xiong et al (AZ31B) Zhang et al (AZ61A)

-100 0.6

-1.0

-0.5

0.0

0.5

1.0

Strain (%)

Fig. 8. Hysteresis loops for 90° out-of-phase multiaxial loading. (a) Axial mode hysteresis and (b) shear mode hysteresis. Albinmousa et al. and Zhang et al.: ea = 0.5% and ca = 0.76%. Xiong et al.: ea = 0.5% and ca = 0.87%.

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Material

Direction

Zhang et al. Yu et al. Wu et al.

AZ31B Extrusion AZ31B-H24 AZ31 Sheet AZ31 Extruded AZ31 Rolling AZ61A Extrusion ZK60 Extrusion AZ31B-H24 Rolled

Xiong et al.

AZ31B Extrusion

ED RD RD ED RD ED ED RD TD ND ED

Shear Albinmousa et al. Yu et al. Xiong et al.

AZ31B Extrusion AZ61A Extrusion AZ31B Extrusion

ED ED ED

Axial Albinmousa et al. SB Behravesh Lugo et al.

K0

n0

383 494 622 387

0.07 0.14 0.15 0.15

375.2 393.04 731.48

0.11 0.13 0.25

226.9

0.2442

because the applied axial strain for the proportional loading case is 0.28% while it is 0.5% for the 90° out of phase loading case. It has been observed that activation of twin deformation due to the application of axial mode influences the shear stress–strain response [29,34]. Depending on the applied axial strain amplitude, axial hysteresis loops become symmetric and the characteristic features of twinning deformation vanish with cycling. This behaviour was observed in multiaxial loading and the influence of axial mode on the shear stress–strain response was seen to vanish accordingly [29,34].

r0f (MPa)

b

e0f

c

723.5 405 543 538 575 670.54 448.8 693.04 686.12 1361 369.6

0.159 0.093 0.12 0.12 0.16 0.168 0.09 0.15 0.16 0.26 0.089

0.252 0.487 0.04 0.08 0.05 1.527 0.606 1.54 0.59 0.36 0.466

0.718 0.659 0.5 0.62 0.44 0.806 0.808 0.9 0.78 0.7 0.702

142.82 132 149.15

0.11 0.076 0.084

0.131 0.19 0.0788

0.427 0.434 0.352

4.2. Hardening and mean stress development Cyclic hardening and mean stress development for AZ31B magnesium extrusion [29,54] are presented in Figs. 9 and 10. The cyclic axial stress response shown in Fig. 9a and b indicates that AZ31B extrusion exhibits pronounce cyclic hardening and mean stress development. This is not observed in the cyclic shear stress response shown in Fig. 9c and d. Relatively, the same observation applies in multiaxial cyclic test as shown in Fig. 10. Figs. 10a–d show that cyclic hardening and mean stress development are more

Axial

(a)

80

Axial

(b)

200 Mean stress (MPa)

Stress ampliutde (MPa)

250

150

100 0.6% 0.5% 0.4% 0.3% 0.2%

50

0 100

101

102

103

104

60

40

20

0

105

100

101

Number of cycles (N)

102

103

104

105

104

105

Number of cycles (N)

(d)

Shear

80 10

Mean stress (MPa)

Stress amplitude (MPa)

20

Shear

(c)

60

40 1.50% 1.26% 0.90% 0.70% 0.60% 0.35%

20

0

0

-10

-20 100

101

102

103

Number of cycles (N)

104

105

100

101

102

103

Number of cycles (N)

Fig. 9. Stress amplitude and mean stress variations with cycling for AZ31B extrusion at different strain amplitudes. (a) and (b) Pure axial loading (c) and (d) pure shear loading.

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40

(a)

(b)

Axial

140

Mean stress (MPa)

Stress amplitude (MPa)

160

120 100 80 60 40

Phase = 0 Phase = 45 Phase = 90

20 0

0

10

1

10

10

2

3

30

20

10

0

4

10

10

Axial

100

Number of cycles (N)

101

102

103

104

Number of cycles (N) 20

(c)

(d)

Shear

Mean stress (MPa)

Stress amplitude (MPa)

80

60

40

20

0 100

Phase = 0 Phase = 45 Phase = 90

15

10

5

0

-5 101

102

103

104

Shear

1

100

3 Total energy (MJ/m )

103

104

Number of cycles (N)

Number of cycles (N) 1.4

102

10

(e)

1.2 1.0 0.8 0.6 0.4 Phase 0 Phase 45 Phase 90

0.2 0.0 0.1

1

10

100

1000

10000

Number of cycles (N) Fig. 10. Stress amplitude, mean stress and total energy variations with cycling for AZ31B extrusion tested with three phase angles at ea = 0.3% and ca = 0.8%. (a) and (b) Axial mode, (c) and (d) shear mode and (e) total energy.

pronounced in axial mode than shear mode. Unlike stress, total strain energy density, defined as the sum of plastic and positive elastic energies as illustrated in Fig. 11, is seen to be constant over the entire life as shown in Fig. 10e. Similar cyclic hardening and mean stress development seen in Figs. 9 and 10 were observed in several magnesium alloys [16,30,33,37,40,50,55,56]. Zhang et al. [30] attributed the decrease in the mean stress to the fact that the capacity of the material to forming mechanical twins was saturated after 10% of fatigue life. It is clear from Fig. 10 that the stress response from the 90° outof-phase test is highest among the multiaxial tests. The 45° out-ofphase is next and finally the proportional test. Therefore, it can be said that AZ31B extrusion exhibits additional hardening due to non-proportionality. Out of phase loading causes the principal strain axes to rotate during cyclic loading. In some materials, this rotation results in additional hardening development which is different than that observed during uniaxial or multiaxial proportional cyclic loading [57]. The additional hardening associated with the application of non-proportional loading is caused by dis-

location–dislocation interaction; they are forced to move along all possible slip planes [58]. Although the numbers of slip systems in magnesium are limited, however, the stress response from the 90° out of phase test is highest among the multiaxial tests. The 45° out of phase is next and finally the proportional test. Therefore, it can be said that AZ31B extrusion exhibits a low additional hardening due to non-proportionality. Zhang et al. [30] performed multiaxial tests on AZ61A extrusion; however, they found that non-proportionality of loading effect on hardening development was insignificant and attributed this to limited number of slip systems and the formation of mechanical twin. 4.3. Fatigue Life There are three approaches to predicting fatigue life: stress, strain and energy. Critical plane concept has widely been used in both strain- and stress-based methods because it predicts both fatigue life and fatigue cracking plane. Critical plane stress- based methods have been used to estimate multiaxial fatigue life for

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100

(a)

354 data points

3 Total energy (MJ/m )

10

1

0.1

0.01 10-1

100

101

102

103

104

105

106

107

Cycles to failure (N f ) 107

(b)

Fig. 11. Plastic and positive elastic strain energy densities [59].

DW t ¼ CNm f

ð1Þ

where DWt, similar to Ellyin et al. [58] is the total strain energy density as defined in Fig. 11, and m and C are fitting constants. Nf is the fatigue life. In a general multiaxial loading, the total strain energy density is defined as

DW t ¼

I

rde þ

r2max 2E

þ

I

sdc þ

s2max 2G

ð2Þ

105

Predicted life

welded joints [60] and notched components [61,62]. Similarly, critical plane strain based methods have been employed to analyze various materials and loading conditions [63–68]. However, researchers [13,69–71] have shown that strain- and stress-based fatigue models, such as Coffin-Manson, are insufficient to explain the fatigue damage in magnesium. As discussed earlier the strain control, and stress control behaviours of magnesium are quite different. The difference is mainly due to asymetric behaviour of magnesium caused by deformation twinning. Hence, incorporating either of the two leads to an incomplete representation of behaviour. Many researchers [13,27–29,40,69–72] proposed the use of a parameter that takes both stress and strain into account. Hence, they used strain energy density per cycle as the fatigue parameter. It was shown (e.g., [29]) that unlike stress and strain, strain energy density remains constant through out the life over a wide range of strain and stress amplitudes as shown in Fig. 10e. Moreover, although the stress response in strain control tests of different magnesium alloys with the same fatigue lives, as shown above, are different, however, the strain energy density of different magnesium alloys at the same life level remain fairly close (see Fig. 12). Further, Albinmousa et al. [29,54] has shown that nonproportionality and load phase angle effects are insignificant on life of magnesium alloys especially in LCF. This may be attributed to the limited number of slip systems in HCP magnesium. Such limitation results in other readily available deformation mechanisms (twinning). Hence, rotation of principal axes due to nonproportional loading, which would otherwise lead into activation of other slip systems, does not facilitate any extra deformation systems. It is therefore hypothesized that a simple energy model is capable of correlating fatigue life of different magnesium alloys under variety of load histories. In this paper, a simple two parameters energy based model was used to correlate fatigue damage and predicted lives for 354 different test results such as

106

104 103

354 data points 102 101 100 10 0

10 1

10 2

10 3

10 4

10 5

10 6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

10 7

Experimental life Fig. 12. Fatigue life modeling. (a) Fatigue damage parameter-life correlation and (b) fatigue life prediction. References for the data are listed in Table 7. Long dashed lines for ±3x and short dashed lines for ±2x.

The integral parts of Eq. (2) consider the contribution of plasticity effect from both axial and shear loadings. The effect of mean stress is included by the maximum stress terms in Eq. (2). Fatigue life-damage correlation is presented in Fig. 12a. This figure combines experimental data for several wrought magnesium alloys that were tested at various loading conditions. These conditions include: axial, shear, axial with mean strain or mean stress, stress and strain controlled, and multiaxial loading. In addition, this figure includes test that were performed on specimens machined at different orientation such as extrusion or rolling, transverse and normal directions. Fig. 12a shows that total strain energy density, sum of plastic and positive elastic energies, combines all data into a single scatter band. Strain energy densities were calculated using experimentally obtained stabilized hysteresis loops. Fitting these data with generalized energy-life power relation, i.e., Eq. (1), fatigue data were predicted as shown in Fig. 12b. The fitting constants were found to be C = 44.35 MJ/m3 and m = 0.51. Considering the diversity of loading conditions, materials, manufacturing method and loading orientation, Fig. 12 suggests that strain energy density is a potential fatigue damage for wrought magnesium alloys. The legends of Fig. 12 are listed in Table 7.

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Table 7 References for the experimental data presented in Fig. 12. Re: mean strain, Rr: mean stress, SC: stress control. No. 1

2 3 4 5 6 7 8 9 10 11

12 13 14 15

16 17 18

19 20

21 22

23 24 25

26 27

28 29 30 31 32 33 34 35 36 37 38 39 40 41

References

Material

Type

Direction

Loading

Jafar Albinmousa, Hamid Jahed, Steve Lambert. ‘‘Cyclic axial and cyclic torsional behaviour of extruded AZ31B magnesium alloy.’’ International Journal of Fatigue 33.11 (2011): 1403–1416

AZ31B

Extrusion

ED

Axial

J. Albinmousa, H. Jahed, Multiaxial Effects on LCF Behaviour and Fatigue Failure of AZ31B Magnesium Extrusion. International Journal of Fatigue, 2014

AZ31B

Extrusion

ED

Shear 0 Phase

RD

45 Phase 90 Phase Re = 1

Hyuk Park, Sung, et al. Low-cycle fatigue characteristics of rolled Mg–3Al–1Zn alloy. International Journal of Fatigue 32.11 (2010) 1835–1842

AZ31

Rolled

Chen, Lijia, et al. Low-cycle fatigue behaviour of an as-extruded AM50 magnesium alloy. Metallurgical and Materials Transactions A 38.13 (2007) 2235–2241 Begum, S., et al. Strain-controlled low-cycle fatigue properties of a newly developed extruded magnesium alloy. Metallurgical and Materials Transactions A 39.12 (2008) 3014–3026 Luo, T.J., et al. Fatigue deformation characteristic of as-extruded AM30 magnesium alloy. Materials & Design 31.3 (2010) 1617–1621. Huppmann, Michael, et al. Fatigue properties of the hot extruded magnesium alloy AZ31. Materials Science and Engineering: A 527.21 (2010) 5514–5521

AM50

Extrusion

ED

Re = 0.5 Rr = 0 Rr = 1 Axial

AM30

Extrusion

ED

Axial

AM30

Extrusion

ED

Axial

AZ31

Extrusion

ED

Axial (SC)

Wu, L., et al. The effects of texture and extension twinning on the low-cycle fatigue behaviour of a rolled magnesium alloy, AZ31B. Materials Science and Engineering: A 527.26 (2010) 7057–7067

AZ31B

Extrusion (Counterpressure) Rolled

ED RD

Axial (SC) Axial

Axial Axial Axial

Shiozawa, K., et al. Low-cycle fatigue deformation behaviour and evaluation of fatigue life on extruded magnesium alloys. Procedia Engineering 10 (2011) 1244– 1249

AZ31

Extrusion

TD ND ED

Shiozawa, K., et al. Low-cycle fatigue deformation behaviour and evaluation of fatigue life on extruded magnesium alloys. Procedia Engineering 10 (2011) 1244– 1249

AZ61

Extrusion

ED

Axial (SC) Axial

Chen, D.L., Emami, A.R., Luo, A.A. Cyclic deformation of extruded AM30 magnesium alloy in the transverse direction. Journal of Physics: Conference Series. Vol. 240. No. 1. IOP Publishing, 2010 Lv, F., et al. Effects of hysteresis energy and mean stress on low-cycle fatigue behaviours of an extruded magnesium alloy. Scripta Materialia 65.1 (2011) 53–56

AM30

Extrusion

TD

Axial (SC) Axial

AZ31

Extrusion

ED

Axial

Jordon, J.B., et al. Effect of twinning, slip, and inclusions on the fatigue anisotropy of extrusion-textured AZ61 magnesium alloy. Materials Science and Engineering: A 528.22 (2011) 6860–6871

AZ61

Extrusion

TD ED

Axial Axial

Kwon, S.H., et al. Low cycle fatigue properties and an energy-based approach for asextruded AZ31 magnesium alloy. Metals and Materials International 17.2 (2011) 207–213 Chen, Cheng, et al. Study on cyclic deformation behaviour of extruded Mg–3Al–1Zn alloy. Materials Science and Engineering: A 539 (2012) 223–229 Wang, Fenghua, et al. Cyclic deformation and fatigue of extruded Mg–Gd–Y magnesium alloy. Materials Science and Engineering: A 561 (2013) 403–410 Zhang, Jixi, et al. An experimental study of cyclic deformation of extruded AZ61A magnesium alloy. International Journal of Plasticity 27.5 (2011) 768–787

AZ61

Extrusion

TD ED

Axial Axial

AZ31

Extrusion

ED

Axial

GW83

Extrusion

ED

Axial

AZ61A

Extrusion

ED

Axial

Yu, Qin, et al. Multiaxial fatigue of extruded AZ61A magnesium alloy. International Journal of Fatigue 33.3 (2011) 437–447

AZ61A

Extrusion

ED ED

Shear 0 Phase

Xiong, Ying, Qin Yu, Yanyao Jiang. Multiaxial fatigue of extruded AZ31B magnesium alloy. Materials Science and Engineering: A 546 (2012) 119–128

AZ31B

Extrusion

ED ED

90 Phase Axial

ED

Shear 0 Phase 90 Phase Re = 1

ED

Re = 0 Re = inf Axial

Yu, Qin, et al. Effect of strain ratio on cyclic deformation and fatigue of extruded AZ61A magnesium alloy. International Journal of Fatigue 44 (2012) 225–233

Yu, Qin, et al. An experimental study on cyclic deformation and fatigue of extruded ZK60 magnesium alloy. International Journal of Fatigue 36.1 (2012) 47–58

AZ61A

ZK60

Extrusion

Extrusion

H. Jahed, J. Albinmousa / Theoretical and Applied Fracture Mechanics 73 (2014) 97–108

5. Conclusions and remarks This paper reviews experimental work on several wrought magnesium alloys. Monotonic tensile, compressive and shear behaviours at different loading orientations were compared and discussed. It was found that the investigated alloys exhibit similar monotonic characteristics. Tensile or compressive loading that causes extension along c-axis activates tension-twining resulting in sigmoidal-type behaviour. Cyclic axial loading produces asymmetric hysteresis loops due to the activation of deformation twining in the compressive reversal. Such asymmetry was not observed in cyclic shear hysteresis because shear loading activates slip deformation mechanism. Pronounced cyclic hardening and mean stress development were observed in axial loading and axial mode of multiaxial tests. Fatigue damage and fatigue lives for several wrought magnesium alloys were correlated using a simple twoparameter energy model. This model quantifies fatigue damage using the sum of plastic and positive elastic strain energy densities. The aforementioned model was shown to correlate a total of 354 data points obtained from testing specimens at various loading conditions and at different loading orientations in a single scatter band. These conditions include: axial, shear, axial with mean strain or mean stress, stress and strain controlled, and multiaxial loading. Acknowledgments The authors acknowledge the financial support of AUTO21 Network Center of Excellence, the Natural Science and Engineering Research Council of Canada (NSERC), and the Canada foundation for Innovation (CFI). General Motors Research & Development Center, Warren, MI is acknowledged for making the AZ31B extrusion material available. The corresponding author would like to acknowledge the supported of King Fahd University of Petroleum & Minerals (KFUPM) for supporting this work under Junior Faculty Research Grant, an internally funded project from DSR (Project No. JF111007). References [1] H. Mayer, M. Papakyriacou, B. Zettl, S. Stanzl-Tschegg, Influence of porosity on the fatigue limit of die cast magnesium and aluminium alloys, Int. J. Fatigue 25 (2003) 245–256. [2] M. Horstemeyer, N. Yang, K. Gall, D. McDowell, J. Fan, P. Gullett, High cycle fatigue of a die cast AZ91E-T4 magnesium alloy, Acta Mater. 52 (2004) 1327– 1336. [3] Y. Lu, F. Taheri, M. Gharghouri, Study of fatigue crack incubation and propagation mechanisms in a HPDC AM60B magnesium alloy, J. Alloy. Compd. 466 (2008) 214–227. [4] M. Horstemeyer, N. Yang, K. Gall, D. McDowell, J. Fan, P. Gullett, High cycle fatigue mechanisms in a cast AM60B magnesium alloy, Fatigue Fract. Eng. Mater. Struct. 25 (2002) 1045–1056. [5] H.E. Kadiri, Y. Xue, M. Horstemeyer, J.B. Jordon, P.T. Wang, Identification and modeling of fatigue crack growth mechanisms in a die-cast AM50 magnesium alloy, Acta Mater. 54 (2006) 5061–5076. [6] Z.B. Sajuri, Y. Miyashita, Y. Hosokai, Y. Mutoh, Effects of Mn content and texture on fatigue properties of as-cast and extruded AZ61 magnesium alloys, Int. J. Mech. Sci. 48 (2006) 198–209. [7] M. Matsuzuki, S. Horibe, Analysis of fatigue damage process in magnesium alloy AZ31, Mater. Sci. Eng., A 504 (2009) 169–174. [8] C.S. Roberts, Magnesium and Its Alloys, Wiley, 1960. [9] S.R. Agnew, Ö. Duygulu, Plastic anisotropy and the role of non-basal slip in magnesium alloy AZ31B, Int. J. Plast. 21 (2005) 1161–1193. [10] E. Yukutake, J. Kaneko, M. Sugamata, Anisotropy and non-uniformity in plastic behavior of AZ31 magnesium alloy plates, Mater. Trans. 44 (2003) 452–457. [11] L. Wu, S. Agnew, Y. Ren, D. Brown, B. Clausen, G. Stoica, H. Wenk, P. Liaw, The effects of texture and extension twinning on the low-cycle fatigue behavior of a rolled magnesium alloy, AZ31B, Mater. Sci. Eng., A 527 (2010) 7057–7067. [12] F. Lv, F. Yang, Q. Duan, Y. Yang, S. Wu, S. Li, Z. Zhang, Fatigue properties of rolled magnesium alloy (AZ31) sheet: influence of specimen orientation, Int. J. Fatigue 33 (2011) 672–682. [13] S.B. Behravesh, Fatigue Characterization and Cyclic Plasticity Modeling of Magnesium Spot Joints, PhD Thesis, University of Waterloo, Canada, 2013.

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