Non-proportional multiaxial whole-life transformation ratchetting and fatigue failure of super-elastic NiTi shape memory alloy micro-tubes

International Journal of Fatigue 80 (2015) 372–380

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Non-proportional multiaxial whole-life transformation ratchetting and fatigue failure of super-elastic NiTi shape memory alloy micro-tubes Di Song a,b, Guozheng Kang a,b,⇑, Qianhua Kan b, Chao Yu b, Chuanzeng Zhang c a

State Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu, Sichuan 610031, PR China School of Mechanics and Engineering, Southwest Jiaotong University, Key Laboratory of Advanced Technologies of Materials, Ministry of Education of China, Chengdu, Sichuan 610031, PR China c Department of Civil Engineering, University of Siegen, Siegen 57068, Germany b

a r t i c l e

i n f o

Article history: Received 2 March 2015 Received in revised form 21 June 2015 Accepted 29 June 2015 Available online 3 July 2015 Keywords: Shape memory alloys Non-proportional loading Multiaxial fatigue Low cycle fatigue

a b s t r a c t Experimental observations are performed to investigate the non-proportional multiaxial whole-life transformation ratchetting and fatigue failure of superelastic NiTi SMA micro-tubes in stress-controlled loadings at human-body temperature (310 K). The effects of axial mean stress and stress hold on the whole-life transformation ratchetting and fatigue life are investigated with uniaxial, torsional and five different multiaxial loading paths. The results show that the stress holds on the upper or lower transformation plateaus will both promote forward and reverse transformation, and lead to shorter fatigue life. The multiaxial fatigue lives of NiTi shape memory alloy depend significantly on loading paths and applied stress levels. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction NiTi shape memory alloy (SMA) has been widely used as medical implants, such as dental orthodontic wire and endovascular stent, due to its excellent biological compatibility, unique super-elasticity and shape memory effect. However, unexpected fractures of NiTi implants in vivo mainly caused by fatigue failure may harm human’s life. Thus, it is important to investigate the cyclic deformation and fatigue failure of NiTi SMA. In the last two decades, many experimental observations were carried out to investigate the cyclic deformation and fatigue failure of super-elastic NiTi SMA. For examples, Eggeler et al. [1], Wagner et al. [2], Matsui et al. [3], Bahia et al. [4], Yan et al. [5], Cheung et al. [6], Figueiredo et al. [7], Bernard et al. [8] and Chan et al. [9] performed some strain-controlled cyclic bending and rotating–bending tests, and discussed the effects of the loading rate, temperature and pre-processing method on the cyclic deformation of fatigue failure of NiTi SMA; Moumni et al. [10], Predki et al. [11], Kang et al. [12,13], Chatellet and Moumni [14], Mammano et al. [15], and Song et al. [16,17] conducted some stress-controlled uniaxial and torsional cyclic loading tests and investigated the effects ⇑ Corresponding author at: State Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu, Sichuan 610031, PR China. Tel.: +86 28 87603794; fax: +86 28 87600797. E-mail addresses: [email protected], [email protected] (G. Kang). http://dx.doi.org/10.1016/j.ijfatigue.2015.06.028 0142-1123/Ó 2015 Elsevier Ltd. All rights reserved.

of the peak stress, mean stress and ambient temperature on the cyclic deformation (including the so-called transformation ratchetting (Kang et al. [12])) and fatigue life of the NiTi SMA. It should be noted that the previous experimental observations to the cyclic deformation and fatigue failure of super-elastic NiTi SMA were focused on the uniaxial, pure torsional, bending and rotating– bending tests, but the multiaxial cyclic deformation and fatigue failure are insufficiently investigated, and the effect of non-proportional multiaxial loading with different load paths on the fatigue life is still not well understood yet. Recently, Wang et al. [18] and Song et al. [19] performed some multiaxial strain-controlled and stress-controlled cyclic tests with different non-proportional load paths to observe the multiaxial cyclic deformation of the NiTi SMA, respectively. A significant pathdependence of non-proportional multiaxial cyclic deformation was observed. However, the prescribed number of cycles in these two works was all no more than 50, and only the transformation ratchetting of NiTi SMA was addressed. To the best knowledge of the authors, the whole-life multiaxial transformation ratchetting (i.e., the ratchetting evolving in the whole fatigue test) and the fatigue failure life of the NiTi SMA micro-tubes have not been reported and discussed yet in the literature, including the previous work done by the authors (i.e., Song et al. [19]). As commented by Robertson et al. [20], a complicated stress condition, e.g., multiaxial one is often encountered in the service process of implanted stents at the human’s joints. Thus, it is extremely necessary to perform detailed

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non-proportional multiaxial tests to observe the cyclic deformation and fatigue failure of super-elastic NiTi SMA at human’s body temperature (310 K). Therefore, as a substantial extension of our previous work (Song et al. [19]), some stress-controlled multiaxial fatigue tests are performed at 310 K by using five non-proportional load paths (i.e., square, hourglass-typed, butterfly-typed, rhombic, and octagonal ones), and the multiaxial whole-life transformation ratchetting and fatigue life of super-elastic NiTi SMA as well as their dependence on the multiaxial load paths are investigated in this paper. Moreover, some uniaxial and torsional cyclic tests with or without peak/valley stress holds are conducted to discuss the effect of the stress holds on the transformation ratchetting and fatigue life by comparing with those obtained in the multiaxial fatigue tests. Some new findings are obtained, which are useful to construct appropriate models predicting the multiaxial whole-life transformation ratchetting and fatigue life of super-elastic NiTi SMA with different multiaxial load paths.

Fig. 1. It should be noted that, for each non-proportional path, asymmetrical axial stress-controlled cyclic loading (i.e., axial stress is non-zero) is prescribed, but the torsional mean equivalent stress pffiffiffi is set as zero and the peak equivalent stress (i.e., 3s ¼ 283 MPa, s is the shear stress) is lower than the torsional martensite transforpffiffiffi mation start stress (i.e., 3ssAM ¼ 450 MPa), which implies that the multiaxial transformation ratchetting presents mainly in the axial direction by referring to Song et al. [19]. In the non-proportional multiaxial fatigue tests, some load cases are set to be with an identical axial stress amplitude of 283 MPa and four axial mean stresses (i.e., 240, 283, 400 and 450 MPa) for all five prescribed load paths; while others are performed with three axial peak stresses (i.e., 566, 637 and 683 MPa) and zero axial valley stress only for the rhombic and octagonal paths. Before the multiaxial fatigue tests are performed, some uniaxial and pure torsional cyclic tests are conducted with or without peak/valley stress holds (hold time is 28.3 s) in order to aid the investigation of multiaxial transformation ratchetting and fatigue failure of the NiTi SMA, since there are peak/valley stress holds (simplified as P/V holds) in some multiaxial load paths as shown in Fig. 1. From the typical stress–strain data of uniaxial and torsional loading–unloading tests, it is obtained that the transformation start/finish stresses in the two load modes are close, i.e., the uniaxial forward/reverse transformation start/finish stresses are

2. Experimental procedures Super-elastic NiTi SMA micro-tubes (Ni, 55.9% at mass, from Jiangyin Fasten-PLT Materials Science Co., Ltd, China) with an outer diameter of 2.5 mm and inner diameter of 2.2 mm are employed in the tests. The transformation temperatures As, Af, Ms and Mf are measured to be 271, 302, 285 and 238 K, respectively, by differential scanning calorimetry (DSC), and it meets the requirements of medical NiTi alloys (i.e. with Af temperature between 298 K and 303 K) [20]. The total specimen length is 60 mm and the gauge length is 20 mm. All the tests are performed under the stress-controlled cyclic loading conditions by the test machine of MTS858-5KN, and the ambient temperature is kept being 310 ± 0.5 K by a constant temperature and humidity chamber of SDH4004 made by Chongqing Inborn Experiment Instrument Co., Ltd, China. The applied stress rate is set as 20 MPa/s in each loading direction. The axial and torsional strains are obtained by the same method as used in Song et al. [19]. The tension–torsion fatigue tests with different non-proportional load paths are shown in

3τ

f f rsAM ¼ 472, rAM ¼ 610, rsMA ¼ 148, and rMA ¼ 65 MPa, the torsional pffiffiffi pffiffiffi f pffiffiffi 3ssAM ¼ 450, 3sAM ¼ 581, 3ssMA ¼ 216, and ones are pffiffiffi f 3sMA ¼ 84 MPa, respectively. It means that the equivalent for-

ward transformation stresses in the torsional direction are lower, but the equivalent reverse transformation stresses are higher than those in the axial direction. In this work, seven representative cases (including two axial and five multi-axial ones) are tested repeatedly (two or three times), respectively; and the obtained results demonstrate a good repeatability and the largest error of the recorded fatigue lives is less than 6%. Therefore, for the remained load cases, only one test for each case is conducted, and the obtained data can illustrate the fatigue failure of the NiTi SMA micro-tubes reasonably.

3τ

3τ

σ

σ

σ

(a)

(b) 3τ

(c) 3τ

σ

(d)

σ

(e)

Fig. 1. Schematic diagrams of multiaxial loading paths: (a) square; (b) hourglass-typed; (c) butterfly-typed; (d) rhombus; (e) octagon.

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stresses higher than the martensite transformation finish stress

3. Experimental results

f rAM (i.e., the cases with applied mean stresses of 400 MPa and

3.1. Whole-life transformation ratchetting 3.1.1. Uniaxial cyclic tests with or without P/V stress holds The uniaxial whole-life transformation ratchetting of the super-elastic NiTi SMA micro-tubes observed in the fatigue tests with an identical stress amplitude of 283 MPa and four different mean stresses (i.e., 240, 283, 400 and 450 MPa, respectively) and with or without P/V stress holds are shown in Fig. 2. It can be seen that, as reported by Song et al. [17], the uniaxial whole-life transformation ratchetting of the NiTi SMA depends greatly on the applied mean stresses and peak stress holds if the stress amplitude is identical. For the load cases with peak stresses lower than the f initial martensite transformation finish stress rAM ¼ 610 MPa (i.e., those with mean stresses of 240 MPa and 283 MPa), a much wider stress–strain hysteresis loops are obtained in the first beginning of the cyclic tests with the P/V stress holds than those without any P/V stress hold, especially for the cases with a mean stress of 240 MPa as shown in Fig. 2b (where, the peak strain in the first cycle without P/V stress hold is only 4%, but it increases to approximately 8% after the peak stress hold for 28.3 s). Therefore, the uniaxial transformation ratchetting of the NiTi SMA is more apparent in the cyclic tests with P/V stress holds than those without P/V stress holds, when the applied peak stress is lower than the initial f transformation finish stress rAM . Since the valley stresses in the two cases are lower than the initial reverse transformation finish f ¼ 65 MPa, the responded valley strains are not so stress rMA remarkably different in the beginning of the cyclic tests, especially for that in the first cycle, as shown in Fig. 2d. The further increase of the valley strain in the subsequent cycles is caused by the accumulation of plastic deformation. For the load cases with peak

(a)

Axial stress σ, MPa

600

1st 50th

2nd 5th Nf (1516th)

3.1.2. Pure torsional cyclic tests with or without P/V stress holds At first, a symmetrical stress-controlled pure torsional cyclic test of the super-elastic NiTi SMA micro-tubes is performed with an applied equivalent shear stress of 566 MPa, and the obtained stress–strain curves are shown in Fig. 3. It is seen from Fig. 3 that a nearly symmetrical stress–strain curve is obtained, and no apparent transformation ratchetting is observed even if the martensite transformation and its reverse occur during the pure torsional cyclic test similar to Song et al. [19]. Moreover, it can be also seen that the stress–strain curves hardly change till the fatigue failure

700

10th

500 400 300 200 100

0

1

2

3

4

5

1st 50th

(b)

600

300 200 100

240± 283MPa (B)

0 6

-100

0

2

4

Mean stress

14

240MPa (A) 283MPa (A) 400MPa (A) 450MPa (A)

(B) (B) (B) (B)

Axial valley strain ε v, %

Axial peak strain ε p, %

16

12 10 8 6 4 100

101

102

6

8

10

Axial strain ε, % 14

(c)

10th

400

Axial strain ε, % 18

2nd 5th Nf (1146th)

500

240± 283MPa (A)

0 -100

f start and finish stresses (i.e., rsMA ¼ 148 MPa and rMA ¼ 65 MPaÞ, which causes an obvious decrease of the valley strain during the valley stress hold due to the continuous reverse transformation. For the load case with a mean stress of 450 MPa, the valley stress (167 MPa) is higher than the initial reverse transformation start stress (i.e., rsMA ¼ 148 MPaÞ, and then nearly no reverse transformation occurs during the cyclic tests. Thus, the valley stress hold also hardly causes an apparent variation of the valley strain, as shown in Fig. 2d.

Axial stress σ, MPa

700

450 MPa), the forward martensite transformation has finished, and the peak stress hold hardly causes an apparent increase of the peak strain due to the weak viscosity of the NiTi SMA at 310 K, as shown in Fig. 2c (where the responded peak strains are nearly the same in the tests with and without P/V stress holds). However, for the load case with a mean stress of 400 MPa, the valley stress hold greatly promotes the reverse transformation and then leads to an obvious decrease of the valley strain in the first beginning of the cyclic test (Fig. 2d), because its valley stress (117 MPa) just locates between the initial reverse transformation

10 3

Number of cycles N, Cycle

10 4

12

(d) Mean stress

10

240MPa (A) 283MPa (A) 400MPa (A) 450MPa (A)

(B) (B) (B) (B)

8 6 4 2 0 10 0

10 1

10 2

103

104

Number of cycles N, Cycle

Fig. 2. Uniaxial whole-life transformation ratchetting with identical stress amplitude of 283 MPa and various mean stresses: (a) stress–strain curves with mean stress of 240 MPa; (b) stress–strain curves with mean stress of 240 MPa and P/V stress holds; (c) curves of axial peak strain vs. number of cycles; (d) curves of axial valley strain vs. number of cycles.

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It can be concluded from Fig. 4 that:

1/2

Equivalent shear stress 3 τ, MPa

800

1st 50th

600

2nd 5th Nf (104th)

10th (1) An apparent transformation ratchetting occurs during the pure torsional cyclic tests of the NiTi SMA micro-tubes, the peak and valley strains increase with the increasing number of cycles and become saturated after certain cycles, as shown in Fig. 4c and d. (2) Comparison of Fig. 4a and b with Fig. 2a and b shows that somewhat higher transformation hardening occurs during the pure torsional cyclic deformation than that during the uniaxial cyclic one, but the difference of the transformation hardening moduli observed in the cyclic tension–unloading and pure torsional tests is not so remarkable as that discussed by Sun and Li [21] and Runciman et al. [22], where the hardening modulus in the pure torsional test was much higher than that in the uniaxial ones. The higher transformation modulus in the pure torsional cyclic tests results in a weaker transformation ratchetting than the uniaxial ones; if the applied equivalent stress level is the same in both cases, as shown in Figs. 2 and 4. (3) The P/V stress holds in the torsional direction also prompt the transformation ratchetting greatly, and the ratchetting strain rate (i.e., the increment of peak/valley strains per cycle) increases with the increasing number of cycles and then results in relatively large peak and valley strains within fewer cycles which cause a quick fatigue failure as shown in Fig. 4c and d. The saturated cyclic stress–strain response cannot be observed during the pure torsional cyclic tests with P/V stress holds, which is different from that observed during the pure torsional cyclic tests without P/V stress holds (shown in Fig. 4c and d) and the uniaxial cyclic tests with or without P/V stress holds (shown in Fig. 2c and d), where the saturated response is obtained within 100 cycles.

400 200 0 -200

Peak stress=566MPa fully reversed

-400 -600 -10

-8

-6

-4

-2

0

2

4

6

8

10

Equivalent shear strain γ/31/2, % Fig. 3. Stress–strain curve of the NiTi alloy obtained in a symmetrical stresscontrolled cyclic pure torsional test with an equivalent shear stress amplitude of 566 MPa.

700

(a)

600

1st 50th

2nd 5th Nf (535th)

10th

500 400 300 200

283±283MPa (A)

100 0

0

1

2

3

4

5

6

7

8

9

Equivalent shear stress 31/2τ, MPa

Equivalent shear stress 31/2τ, MPa

occurs. This feature is different from that observed in the symmetrical cyclic tension–compression test of the NiTi SMA reported by Kang et al. [12], where an obvious transformation ratchetting occurred due to the anisotropic responses of the alloy in the tensile and compressive directions, even if the applied mean axial stress was zero. Then, some stress-controlled torsional fatigue tests of the NiTi micro-tubes are performed with identical equivalent shear stress amplitude of 283 MPa and four different mean stresses (i.e., 240, 261, 283 and 300 MPa, respectively) and with or without P/V stress holds, which were not addressed in Song et al. [19]. The obtained results are shown in Fig. 4.

700

(b)

600

300 200

283±283MPa(B)

100 0

0

Mean stress

10

(B) (B) (B) (B)

8 6 4 2 100

101

102

Number of cycles N, Cycle

1

2

3

4

5

6

7

8

9

Equivalent shear strainγ/31/2, %

103

Equivalent shear valley strain 3 1/2τv, %

Equivalent shear peak strain 31/2τp, %

12

240MPa (A) 261MPa (A) 283MPa (A) 300MPa (A)

10th

400

Equivalent shear strainγ/3 , %

(c)

2nd 5th Nf (61st)

500

1/2

14

1st 50th

4

3

(d) Mean stress

240MPa (A) 261MPa (A) 283MPa (A) 300MPa (A)

(B) (B) (B) (B)

2

1

0

100

101

102

103

Number of cycles N, Cycle

Fig. 4. Whole-life transformation ratchetting of the NiTi micro-tubes in pure torsional cyclic tests: (a) stress–strain curves with 283 ± 283 MPa; (b) stress–strain curves with 283 ± 283 MPa and P/V stress holds; (c) curves of peak strain vs. number of cycles; (d) curves of valley strain vs. number of cycles.

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(2) The multiaxial whole-life transformation ratchetting of the NiTi alloy depends greatly on the prescribed load paths, and the accumulated peak and valley strains obtained in the cyclic tests with the load paths involving certain P/V stress holds (e.g., square, butterfly and hourglass-typed ones) are higher than that with the load paths containing no P/V stress holds (e.g., rhombic and octagonal ones) as shown in Fig. 6a and b, in which the larger equivalent peak stresses and P/V stress holds both take effects. The maximum evolution rates of the responding peak and valley strains are observed in the multiaxial cyclic loading test with the square path, since both P/V stress holds are involved in the axial and torsional directions. Although the evolution rate of the peak/valley strains decreases with the increasing number of cycles, it does not become zero even at the last stage of the fatigue tests, which is similar to that obtained in the pure torsional ones with P/V stress holds (shown in Fig. 4c and d), but different from that observed in the uniaxial fatigue tests (shown in Fig. 2c and d) and the pure torsional ones without any P/V stress holds (shown in

3.1.3. Multiaxial whole-life transformation ratchetting Five types of non-proportional multiaxial load paths shown in Fig. 1 are employed in the multiaxial stress-controlled cyclic tests. It should be noted here that, to compare the obtained results more directly, some non-proportional polygonal paths, such as square, rhombic and octagon ones are used in this work, rather than the classical out-of-phase circular path. The responding curves of the equivalent shear strain vs. the axial strain for five prescribed load paths with an axial mean stress of 400 MPa are shown in Fig. 5, and the evolution curves of the peak/valley strains vs. the number of cycles are illustrated in Fig. 6 for five prescribed load paths with various axial mean stresses. It can be observed here that:

5

(a)

4

1st 50th

2nd 5th Nf (463rd)

3

Equivalent shear strain γ/31/2,%

Equivalent shear strain γ /31/2,%

(1) An apparent transformation ratchetting also occurs mainly in the axial direction during the prescribed multiaxial cyclic loading of the NiTi SMA alloy, and the transformation ratchetting in the torsional direction is very slight as shown in Fig. 5, since the applied equivalent shear stress is prescribed to be zero in all five load paths.

10th

2 1 0 -1 -2 -3 -4 0

2

4

6

8

10

12

14

16

2.0

1st 50th

(b)

1.5

0.5 0.0 -0.5 -1.0 -1.5 0

18

2

4

2.5

2nd 5th Nf (512th)

10th

2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 0

2

4

6

8

6

8

10

12

14

Axial strain ε, % Equivalent shear strain γ/31/2,%

Equivalent shear strain γ/31/2,%

1st 50th

(c)

10th

1.0

Axial strain ε, % 3.0

2nd 5th Nf (406th)

10

12

14

3

1st 50th

(d) 2

10th

1 0 -1 -2 0

2

4

Axial strainε, % Equivalent shear strain γ/31/2,%

2nd 5th Nf (442nd)

6

8

10

12

14

Axial strain ε, % 4

1st 50th

(e)

3

2nd 5th Nf(415th)

10th

2 1 0 -1 -2 -3 0

2

4

6

8

10

12

14

Axial strain ε, % Fig. 5. Curves of equivalent shear strain vs. axial strain for five non-proportional multiaxial load paths with an axial mean stress of 400 MPa: (a) square; (b) hourglass-typed; (c) butterfly-typed; (d) rhombic; (e) octagonal.

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12

(a) Mean stress=283MPa Square Hourglass-typed Butterfly-typed Rhombic Octagonal

16 14

Axial valley strain ε v, %

Axial peak strain εp, %

18

12 10 8 6 100

10 1

10 2

(b) Mean stress=283MPa Square Hourglass-typed Butterfly-typed Rhombic Octagonal

10 8 6 4 2 0 100

10 3

16

20

(c) Mean stress

14 12 10 8 6

Square 240MPa Square 283MPa Square 400MPa Square 450MPa

4 2 0 100

10 1

Rhombic 240MPa Rhombic 283MPa Rhombic 400MPa Rhombic 450MPa

10 2

Axial valley strain ε v, %

Axial peak strain ε p, %

18

10 1

10 2

10 3

Number of cycles N, Cycle

Number of cycles N, Cycle

(d) Mean stress

18

Square 240MPa Square 283MPa Square 400MPa Square 450MPa

16 14 12 10 8 6 4 2 0 100

10 3

Rhombic 240MPa Rhombic 283MPa Rhombic 400MPa Rhombic 450MPa

Number of cycles N, Cycle

10 1

10 2

10 3

Number of cycles N, Cycle

Fig. 6. Curves of axial peak and valley strains vs. number of cycles in the multiaxial cyclic tests: (a) peak strain with different load paths and an axial mean stress of 283 MPa; (b) valley strain with different load paths and an axial mean stress of 283 MPa; (c) peak strain with different axial mean stresses and square and rhombic paths; (d) valley strain with different axial mean stresses and square and rhombic paths.

Fig. 4c and d). It means that no shake-down of the transformation ratchetting occurs in the multiaxial cyclic tests, especially for those with load paths containing P/V stress holds. The prompting effect of P/V stress holds on the evolution of the transformation ratchetting has been proved by the uniaxial and pure torsional tests in Sections 3.1.1 and 3.1.2. (3) Comparing the multiaxial whole-life transformation ratchetting shown in Figs. 5 and 6 with the uniaxial and pure torsional ones shown in Figs. 2 and 4, we can conclude that the multiaxial loading prompts the evolution of the transformation ratchetting by the martensite re-orientation occurred in a multiaxial stress state, especially for the load paths with certain P/V stress holds, as discussed by Song et al. [19] for the initial stage of the cyclic loading, i.e., within 50 cycles.

(4) The multiaxial whole-life transformation ratchetting also depends greatly on the applied stress level, and the accumulated peak and valley strains increase with the increasing axial mean stress (shown in Figs. 6c and d) and peak stress (shown in Fig. 7), respectively. However, it should be noted that the most remarkable transformation ratchetting (especially for the accumulated valley strain) for the square path occurs in the test with an axial mean stress of 400 MPa, rather than with 450 MPa as shown in Fig. 6c and d, which is corresponding to the uniaxial transformation ratchetting shown in Fig. 2c and d with variable axial mean stress and P/V stress holds. It is mainly caused by the stress holds at the forward and reverse transformation plateaus as discussed in Section 3.1.1 for the uniaxial ones with a mean stress of 400 MPa and P/V stress holds.

16

10

(b) Peak stress

14 12 10 8 6 4 0 10

Rhombic 566MPa Rhombic 637MPa Rhombic 683MPa

101

Octagonal 566MPa Octagonal 637MPa Octagonal 683MPa

102

Number of cycles N, Cycle

103

Axial valley strain ε v , %

Axial peak strain εp , %

(a) Peak stress 8 6 4 2

0 100

Rhombic 566MPa Rhombic 637MPa Rhombic 683MPa

101

Octagonal 566MPa Octagonal 637MPa Octagonal 683MPa

102

103

Number of cycles N, Cycle

Fig. 7. Curves of axial peak and valley strains vs. number of cycles in the multiaxial cyclic tests: (a) peak strain with different load paths and axial peak stresses; (b) valley strain with different load paths and axial peak stresses.

D. Song et al. / International Journal of Fatigue 80 (2015) 372–380

100

3

(a) Mean stress=283MPa

90

Dissipation energy W, MJ/m

Dissipation energy W, MJ/m

3

378

Square Hourglass-typed Butterfly-typed 30 Rhombic Octagonal 20

80 70 60 50 40

10

30

0

2

5x10

20

3

10

10 0 0 10

1

10

10

2

10

3

120

(b) Square

100

Mean stress 240MPa 283MPa 400MPa 450MPa

80 60

30 20 10

40

0 5x10

20 0 0 10

Number of cycles N, Cycle

1

10

10

2

10

3

2

10

3

Number of cycles N, Cycle

Fig. 8. Curves of the energy dissipation per cycle vs. number of cycles in the multiaxial cyclic tests: (a) with different load paths and an axial mean stress of 283 MPa; (b) with square path and different axial mean stresses.

From the obtained curves of the dissipation energy per cycle vs. the number of cycles in the multiaxial cyclic tests shown in Fig. 8, the following conclusions can be drawn:

fatigue life vs. various applied mean stresses, peak stresses and load paths are investigated in this section. 3.2.1. Uniaxial and pure torsional cyclic tests with or without P/V stress holds Fig. 9 gives the results of the fatigue lives obtained in the uniaxial and pure torsional cyclic tests with or without P/V stress holds. It can be concluded that:

(1) The dissipation energy per cycle obtained in the multiaxial cyclic tests with each load path decreases quickly with the increasing number of cycles in the first beginning of the cyclic loading, and then gradually saturates after certain cycles and keeps unchanged till the fatigue failure as shown in Fig. 8a, which is similar to that observed in uniaxial cyclic tests by Moumni et al. [10], Kang et al. [13] and Song et al. [17]. (2) For the multiaxial cyclic tests with the same load path, the decrease of the dissipation energy with the increasing number of cycles is more remarkable with higher axial mean stresses than that with lower ones, and the saturated values of the dissipation energy are also much lower in the cyclic tests with higher axial mean stresses, as shown in Fig. 8b. However, the highest one is obtained in the cyclic test with an axial mean stress of 283 MPa, rather than with 240 MPa as shown in Fig. 8b, which is corresponding to the uniaxial case discussed by Song et al. [17] and mainly caused by an incomplete forward transformation and a complete reverse transformation.

(1) The fatigue life decreases obviously with the increasing mean stress (where the stress amplitude is set to be identical) and peak stress (where the valley stress is set to be zero) in both uniaxial and pure torsional cyclic tests. However, in the pure torsional cases, the fatigue life decreases almost linearly with the increasing stress level, but it decreases somewhat nonlinearly in the uniaxial ones, as shown in Fig. 9. (2) The P/V stress holds can promote the transformation ratchetting (shown in Figs. 2 and 4), and hence cause a shorter fatigue life than that without P/V stress holds, especially for the pure torsional cyclic tests and the cases with lower mean stress and peak stresses, as shown in Fig. 9. It demonstrates the degeneration of the fatigue life resulted from the transformation ratchetting. (3) Though the transformation ratchetting of the NiTi SMA micro-tubes in the pure torsional cyclic tests is lower than that in the uniaxial ones, the pure torsional fatigue life is lower than the uniaxial one, if the equivalent stress level is the same in both cases, especially for the pure torsional cases with P/V stress holds, as shown in Fig. 9a. It is mainly caused by the anisotropic thermo-mechanical responses of

3.2. Fatigue life To discuss the fatigue life and its dependence on the applied stress level and load path more directly, the variations of the

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the NiTi SMA micro-tubes due to the preferable texture formed in the manufacture process, and the different modes of the martensite nucleation and extension in NiTi SMA during the tension and torsion loadings, as observed by Peng et al. [23]. 3.2.2. Multiaxial fatigue life Fig. 10 shows the relationships of the fatigue lives with different axial mean stresses and peak stresses obtained in the multiaxial cyclic tests with various multiaxial loading paths. It is seen from Fig. 10 that: (1) No evident relationship of the fatigue life with the applied axial mean stress can be found in the multiaxial cyclic tests with five multiaxial loading paths. Moreover, even for a certain applied axial mean stress, the fatigue lives of the NiTi SMA micro-tubes obtained in various loading paths do not possess a determinate relationship with the multiaxial loading path, as shown in Fig. 10a. In general, the fatigue lives obtained in the multiaxial cyclic tests with the loading paths containing certain P/V stress holds (e.g., square, hourglass-typed and butterfly-typed ones) are shorter than that with the paths containing no P/V stress holds (e.g., rhombic and octagonal ones). It is very difficult to precisely predict such complicated fatigue behavior by the existing failure models as suggested by Maletta et al. [24,25], Moumni et al. [10] and Kan et al. [26] for the uniaxial ones. As shown in Fig. 10b, the fatigue lives of the NiTi SMA micro-tubes obtained in the multiaxial cyclic tests with certain loading paths (e.g., rhombic and octagonal paths) decrease monotonically with the increasing axial peak stress. However, it does not illustrate that the fatigue lives in the rhombic path are always higher than that in the octagonal one for all the three prescribed axial peak stresses. (2) Comparing with the results shown in Fig. 9, it can be concluded that the multiaxial fatigue life of the NiTi SMA micro-tubes is much shorter than the uniaxial one when the applied axial stress level is identical, since the multiaxial stress state can prompt the transformation ratchetting. However, the multiaxial fatigue life is obviously longer than that obtained in the pure torsional fatigue tests with P/V stress holds (where the fatigue lives are shorter than 110 cycles, as shown in Fig. 9a). It implies that the fatigue resistance of the NiTi SMA micro-tubes in the torsional direction is much lower than that in the uniaxial direction, and the fatigue failure in the multiaxial cyclic test is controlled by the applied stress level and P/V stress holds in the torsional direction to a relatively higher extent.

3.2.3. Discussions It can be concluded from the above-mentioned experimental observations that the multiaxial fatigue life of super-elastic NiTi SMA micro-tubes depends greatly on the applied stress level and loading paths simultaneously, and is too complicated to be reasonably predicted by the existing failure models mainly based on the dissipation energy per cycle and its evolution (Moumni et al. [10]; Kan et al. [26]), or the modified Manson–Coffin models (Maletta et al. [24,25]). Furthermore, in the stress-controlled cyclic loading test with a non-zero mean stress, the occurrence of the transformation ratchetting can degenerate the fatigue life of the NiTi SMA micro-tubes and should be reasonably considered in the construction of the corresponding fatigue failure model. It is seen from Figs. 4, 6 and 7 that, in general, the larger the transformation ratchetting (i.e., accumulated peak and valley strains) is, the shorter the fatigue life will present. Moreover, the multiaxial fatigue lives are shorter than the corresponding uniaxial ones due to the quicker evolution of the multiaxial transformation ratchetting. From Fig. 8a and b, it can be concluded that for the different multiaxial loading paths, the multiaxial fatigue life decreases generally with the increasing saturated dissipation energy, as shown in Fig. 8a, but it keeps almost the same in the multiaxial cyclic tests with an identical loading path (e.g., square one) and different axial mean stresses, except for the case with a mean stress of 400 MPa as shown in Fig. 8b. It implies that although the saturated dissipation energy is an important factor which determines the fatigue life of the NiTi SMA micro-tubes, the whole-life transformation ratchetting (i.e., final accumulated peak and valley strains) should be also considered in the construction of adequate fatigue failure models. The stress-level and path- dependence of the multiaxial fatigue life of the NiTi SMA micro-tubes can be considered by the assumed evolution rules of the transformation ratchetting and saturated dissipation energy in the fatigue failure model to be built in the future work. 4. Conclusions From the experimental results of this paper, the following essential conclusions can be drawn: (1) The peak/valley (P/V) stress holds can prompt the evolution of the transformation ratchetting, and the accumulated peak and valley strains are much larger in the uniaxial and pure torsional cyclic tests with the P/V stress holds than that without any holds. No saturation is reached during the pure torsional transformation ratchetting with P/V stress holds till the fatigue failure occurs. The transformation ratchetting also occurs in the multiaxial stress-controlled cyclic tests, and the multiaxial transformation ratchetting evolves more

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quickly than the uniaxial one because the martensite re-orientation and the re-orientation-induced plasticity can be more easily activated in the multiaxial stress state. The multiaxial whole-life transformation ratchetting depends greatly on the multiaxial loading paths, and is more remarkable in the tests with the loading paths containing certain P/V stress holds than that without any holds. No shake-down of the transformation ratchetting occurs in the multiaxial cyclic tests, especially for those with load paths containing P/V stress holds. (2) The multiaxial fatigue lives of the NiTi SMA micro-tubes are much shorter than the corresponding uniaxial ones due to the quicker evolution of the multiaxial transformation ratchetting, and depend greatly on the multiaxial loading paths and the applied stress levels simultaneously. In general, the fatigue lives with the loading paths containing certain P/V stress holds are shorter than that with the paths containing no P/V holds. The larger the transformation ratchetting and the saturated dissipation energy, the shorter the fatigue lives are. The effects of the stress level and loading path on the multiaxial fatigue life should be reflected by the evolution rules of the transformation ratchetting and energy dissipation in the failure models to be constructed in the future work.

Acknowledgements Financial supports by the National Natural Science Foundation of China (11025210; 11202171) and the project for Sichuan Provincial Youth Science and Technology Innovation Team, China (2013TD0004) are gratefully acknowledged. D. Song appreciates the China Scholarship Council for its support to her research stay at the Chair of Structural Mechanics, Department of Civil Engineering, Germany, from September 2012 to September 2013. References [1] Eggeler G, Hornbogen E, Yawny A, Heckmann A, Wagner M. Structural and functional fatigue of NiTi shape memory alloys. Mater Sci Eng, A 2004;378(1– 2):24–33. [2] Wagner M, Sawaguchi T, Kausträter G, Höffken D, Eggeler G. Structural fatigue of pseudoelastic NiTi shape memory wires. Mater Sci Eng, A 2004;378(1– 2):105–9. [3] Matsui R, Tobushi H, Furuichi Y, Horikawa H. Tensile deformation and rotating–bending fatigue properties of a highelastic thin wire, a superelastic thin wire, and a superelastic thin tube of NiTi alloys. J Eng Mater-T ASME 2004;126(4):384–91.

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