Functional fatigue of Ni–Ti shape memory wires under various loading conditions

International Journal of Fatigue xxx (2012) xxx–xxx

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Functional fatigue of Ni–Ti shape memory wires under various loading conditions G. Scirè Mammano, E. Dragoni ⇑ Department of Engineering Sciences and Methods, University of Modena and Reggio Emilia, Reggio Emilia, Italy

a r t i c l e

i n f o

Article history: Received 24 January 2012 Received in revised form 5 March 2012 Accepted 13 March 2012 Available online xxxx Keywords: SMA wires Functional fatigue Constant-stress Constant-strain Limited maximum strain

a b s t r a c t The rational design of smart actuators based on shape memory alloys requires reliable strength data from the thermo-mechanical cycling of the material (functional fatigue). Functional tests results do not abound in the technical literature and the few data available are mostly limited to the condition of constant applied stress, which is hardly achieved in operation. The disagreement between actual working conditions and laboratory conditions leads to suboptimal designs and undermines the prediction of the life of the actuator. To bridge the gap between experiment and reality, this paper envisions four cyclic tests spanning the range of loadings which can occur in practice: constant-stress, constant-strain, constantstress with limited maximum strain and linear stress–strain cycle. Commercial NiTi wires (0.15 mm diameter) are tested under constant-stress, constant-strain and constant stress with limited maximum strain conditions using a custom machine and the disclosed results are critically discussed. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Shape memory alloys (SMAs) are increasingly employed to build solid state actuators, characterized by sleek design and outstanding power density. For the rational design of shape memory actuators, a thorough knowledge of the fatigue behavior of the alloy is needed [1]. Since the actuation is always provided by thermal activation of the alloy, the fatigue incurred by the material is called ‘‘thermomechanical’’ or ‘‘functional’’, implying that material performs mechanical work against the external applied load when heated during the actuation cycle. Although the purely mechanical fatigue (i.e. at constant temperature) of shape memory alloys is well established in the technical literature [2], only scanty data exist on the response of these materials to functional fatigue. The reason for this unbalance is due to the fact that shape memory alloys so far have been predominantly exploited for their superelastic behavior (stents, orthodontics, springs, dampers, etc.) rather than for the shape memory effect (actuation). Furthermore, the few papers available on functional fatigue of shape memory alloys are limited to the constant-stress loading condition and cover a very limited number of cycles. In [3,4] the functional fatigue of commercial SAES Getters SmartFlexÒ wires is presented. The wires are loaded by a constant stress (weight) and are cycled thermally by a suitable current supply. During the heating phase, the current is cut-off when the recovered displacement of the wire from the low-temperature position has

⇑ Corresponding author. E-mail address: [email protected] (E. Dragoni).

achieved a pre-established value (partial transformation). The results of the fatigue tests in reference [3] are summarized in a diagram with applied stress and response strain plotted along the axes. Areas of the diagram are labelled with the minimum life expected for the corresponding combination of stress and strain levels. In [5], Bertacchini et al. investigated the functional fatigue under constant stress of shape memory wires by taking into account possible corrosion of the material. Eggeler [2] tested helical springs under constant external tensile forces, showing that both the spring length and the useful stroke increase with the number of cycles. In [6] Lagoudas et al. examined the functional fatigue of ternary NiTiCu wires with particular reference to the effect on the fatigue response of applied stress, annealing of the material and the degree of transformation of the alloy upon heating. In [7] Bigeon and Morin compared the functional fatigue behavior of binary (NiTi) and ternary (CuZnAl) wires, showing that the shape memory effect is affected both by the applied stress and by the number of cycles. In particular, the effect of the number of cycles is beneficial (training) and is more significant for the CuZnAl alloy. Bigeon and Morin followed up their functional fatigue tests in [8], highlighting a direct relationship between the stressassisted-two-way-memory-effect and the applied stress. They found that maximum recovery of the NiTi alloy was obtained for an applied stress of 200 MPa. In [9], a similar study is presented for TiNiCu wires and a Woehler-like curve linking applied stress to the number of cycles is proposed. Test methods different from the constant-stress concept were carried out in [10,11]. Mertmann et al. [10] investigated the stability of the shape memory effect under constant-strain conditions. They observe that memory effect is quickly lost under these

0142-1123/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijfatigue.2012.03.004

Please cite this article in press as: Scirè Mammano G, Dragoni E. Functional fatigue of Ni–Ti shape memory wires under various loading conditions. Int J Fatigue (2012), http://dx.doi.org/10.1016/j.ijfatigue.2012.03.004

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Nomenclature Af As b, c E Mf Ms N Nf SME S0

ee ea eap eA e0f elim

austenite finish temperature austenite start temperature empirical constants in Coffin–Manson equation modulus of elasticity of the austenitic wire martensite finish temperature martensite start temperature number of cycles number of cycles to failure shape memory effect (De = eM–eA) initial stress level in the JSME fatigue test method wire strain amplitude of total strain cycle amplitude of plastic strain cycle wire strain in the austenitic state fatigue ductility coefficient in Coffin–Manson equation maximum wire strain in the constant-stress cycle with limited maximum strain

conditions accompanied by a general elongation of the specimen. Demers et al. [11] tested the functional fatigue of a NiTi alloy after various thermomechanical training processes based on three steps: ‘‘stress-free shape recovery’’, ‘‘constrained recovery’’ (constantstrain) and ‘‘constant-stress recovery’’ (assisted-two-way-shapememory-effect). The difficulties faced by the designer of shape memory actuators are twofold. First, the commercial alloys available are unlikely to be covered by the test results retrieved from the literature. Second, the constant-stress loading condition behind the rare test results available are poorly representative of the true working conditions of the alloy in a real SMA actuator (see below). This discrepancy leads to suboptimal design and to the practical impossibility to predict the fatigue life of the device. This paper aims at filling the gap between test conditions and working conditions of the shape memory materials in the context of functional fatigue. To this purpose, a set of four different loading conditions are defined so as to meet closely the typical working conditions of real actuators: constant-stress, constant-strain, constant-stress with limited maximum strain and linear stress–strain cycle. The paper presents briefly the custom equipment designed to achieve these test conditions [12] and illustrates the fatigue results for commercial NiTi wires (SmartflexÒ 150) tested under constant-stress, constant-strain and constant-stress with limited maximum strain conditions. The main contributions this article brings to the scientific community in quantitative terms are the high number of cycles covered in the constant-stress tests, the variety of test condition examined (the constant-strain cyclic tests and the constant stress tests with limited strain are hardly found in the literature) and the comparison of behavior between them.

2. Materials and methods 2.1. Characteristic stress–strain tests Typically, a shape memory actuator comprises an active SMA element (for example a wire) and a backup element (for example a spring) used to restore the reference position when the SMA element is deactivated. The particular load acting on the active element strongly depends on the backup element and on the external load applied to the actuator. Apart from the very particular case of constant backup force and constant conservative external load, the force on the active SMA element is not constant over the oper-

eM emax emin De Dr D0 r D00 r DS

r

r0f

rmax rmin rw

wire strain in the martensitic state maximum wire strain recorded in the cycle (eM) minimum wire strain recorded in the cycle (eA) Shape memory effect (SME = eM–eA) percent stress variation recorded in the constant-stress cycle [=(rmaxrmin)/r] stress variation recorded in the constant-strain cycle (=rmax–rmin) percent stress variation recorded in the constant-stress cycle with limited maximum strain [=(rmax–r)/r)] stress increment used in the staircase method nominal stress applied to the wire fatigue strength coefficient in Coffin–Manson equation maximum wire stress recorded in the cycle minimum wire stress recorded in the cycle fatigue limit

ating cycle but oscillates between a minimum and a maximum value, depending on the particular situation. If the end positions of the actuator are ensured by hard stops, the force on the active element is still variable but its strains are confined between a lower and an upper limit determined by the mechanical restraints. A limit case of this condition occurs when the SMA element is activated in a fully constrained position so that the stress varies at constant strain. This situation typically occurs in SMA devices used as static force generators [11], but it can arise also in actuators used to actuate grippers [10] or to displace heavy masses subject to high accelerations. In the latter case, the loading conditions of the SMA element are close to a fixed-strain situation if the actuation time of the SMA is lower than the characteristic time of the inertial load [13]. Another situation of fixed-strain loading can occur when an external constraint locks the actuator in a fixed position during the active stroke of the actuator. Fig. 1 illustrates four test conditions in which the stress and the strain in the SMA element vary along specific paths representative of what can occur in the operation of an actuator as described above. Fig. 1a describes the classical constant-stress test, in which the SMA element undergoes variable strain under prescribed stress, provided for example by an applied constant load. This test reproduces the situation occurring in an actuator backed up by a constant force and operated under no external load (positioning device). Given the material, this test is fully defined by the stress applied to the SMA element. Fig. 1b describes the constant-strain test, in which the SMA element undergoes variable stress under prescribed strain, provided for example by rigid restraints fixing the ends of the sample. This test reproduces the situation in which the actuator is operated in a locked position. Given the material, this test is fully defined by the strain imposed to the SMA element. Fig. 1c describes a constant-stress test with limited maximum strain, in which the SMA element undergoes variable strain under prescribed stress but the maximum strain achievable is limited by an external restraint. This test reproduces the situation in which an actuator backed up by a constant force works between hard stops. Given the material, this test is fully defined by two parameters: the applied stress and the maximum strain. Fig. 1d describes a cyclic-stress test, in which the SMA element undergoes a linear variation of stress and strain, with the stress level increasing during strain recovery. Given the material, this test can be performed in several ways, the simplest of which is defined by two parameters: the stress–strain slope and the initial martensite strain.

Please cite this article in press as: Scirè Mammano G, Dragoni E. Functional fatigue of Ni–Ti shape memory wires under various loading conditions. Int J Fatigue (2012), http://dx.doi.org/10.1016/j.ijfatigue.2012.03.004

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3

maximum strain) are implemented. The test conditions in Fig. 1d (linear stress–strain cycle) can be obtained by attaching a linear spring (not shown here) to the lower end of the wire. This condition was not applied in the present investigation. 2.3. Properties of the tested shape memory alloy

(a)

To the aim of gathering fatigue data on shape memory materials readily available to the designer, the NiTi wire SmartflexÒ 150 marketed by SAES Getters was used throughout the experimental campaign. The wire has a diameter of 0.150 mm, a Ni content of 54% by weight and the following transformation temperatures: As = 86 °C, Af = 94 °C, Ms = 65 °C, Mf = 57 °C. The stress–strain tensile properties of the wire in the martensitic and austenitic states are presented in [12].

(b)

2.4. Constant-stress tests 2.4.1. Testing procedure Based on exploratory experiments, the constant-stress tests on the SMA wire were performed according to the following procedure:

(c)

(d)

Fig. 1. Specific tests proposed to characterize the SMA wires under functional fatigue: (a) constant-stress; (b) constant-strain; (c) constant-stress with limited maximum strain and (d) linear stress–strain cycle.

2.2. Experimental equipment The custom test machine used to apply the four stress–strain conditions of Section 2.1 is described in detail in [12]. Basically, the machine comprises a C-shaped aluminum chassis to the upper part of which one end of the tested SMA wire is clamped. The lower end of the SMA wire is loaded or constrained as shown in Fig. 2, according to the test to be performed. Heating of the wire is provided by electric current supplied by an electronic power board. Cooling of the wire is enhanced by an electronically-controlled variable-speed fan. In the configuration detailed in Fig. 2a, with a loading basked appended to the wire, the machine performs the constant-stress test of Fig. 1a. By removing the basket and locking the bottom end of the rod as in Fig. 2b, constant-strain test conditions as in Fig. 1b are achieved. By placing a polymer spacer on top of the displacement sensor to function as a hard stop for the loading basket (Fig. 2c), the test conditions in Fig. 1c (constant-stress with limited

(a)

(b)

1. An undeformed wire length of about 130 mm is cut from the coil and electrical-type cord end ferrules (http://www.partex.co.uk/cefs.html) are crimped to both ends giving a net test length of 100 mm. (In this and in the other types of tests, the use of cord ends helped avoid fractures of the wire under the clamps of the machine, see Section 4). 2. One end of the wire is attached to the load cell and the reading of the cell is set to zero. 3. The free wire (no applied load) is heated above Af to recover the memorized shape. 4. The basket is attached to the lower end of the wire (which is maintained above temperature Af). 5. The reading of the displacement transducer is set to zero. 6. The current supply is cut-off and the wire is let to cool down to room temperature. 7. The basket is filled with lead beads until the desired stress in the wire (read by the load cell and displayed by the user interface) is reached. 8. The test is started by supplying the wire with a sinusoidal current oscillating from zero to the least peak value (by determined by trial and error) capable of producing the full transformation of the alloy under the cooling conditions

(c)

Fig. 2. Close-ups of the loading devices used to apply three of the loading conditions shown in Fig. 1: (a) constant-stress; (b) constant-strain and (c) constant-stress with limited maximum strain.

Please cite this article in press as: Scirè Mammano G, Dragoni E. Functional fatigue of Ni–Ti shape memory wires under various loading conditions. Int J Fatigue (2012), http://dx.doi.org/10.1016/j.ijfatigue.2012.03.004

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adopted (no direct measurement of the wire temperature was performed). 9. The test is terminated automatically at fracture of the SMA wire (no current flowing in the sample) or when a prescribed number of cycles (5  105) have been endured with no failure. By acting on the peak supply current and on the speed of the cooling fan, the test frequency in Step 8 above was kept as high as possible to reduce the test time. The maximum achievable frequency was limited by the force ripples in the wire produced by the inertia of the moving basket. All the tests were performed so that the actual stress in the wires was contained within 6% of the nominal stress (i.e. the stress produced by the total weight of the basket). This requirement led to test frequencies ranging from 0.31 to 0.42 Hz and peak supply currents ranging from 0.83 to 1.04 A. The maximum strain rate measured during the tests ranged between 0.200 s1 and 0.260 s1 during activation (wire shortening) and between 0.100 s1 and 0.105 s1 during relaxation (wire lengthening). 2.4.2. Stress levels There is a strong similarity between the present constant-stress tests on SMA wires and the classical fatigue tests on ordinary construction materials. Both tests aim at identifying the number of cycles to failure for given applied stress (Woehler’s strength-life curve). Due to this analogy, the fatigue data on the SMA wires were collected following the standard test method developed by the Japanese Society of Mechanical Engineers (JSME) [14]. The JSME method identifies Woehler’s curve with as few as 14 tests, with 8 tests used to trace the inclined part of the curve and 6 used to calculate the horizontal part (fatigue limit). Applied to the SMA wire under investigation, the JSME protocol produced the following stress levels needed to identify the inclined part of the Woehler’s curve: 225, 200, 175, 150, 125, 100, 75 MPa. The tests begin at level S0 = 150 and go on by stepping down one level at a time until one sample survives the test (no failure observed after 5  105 cycles). Further tests are then performed by stepping up one level at a time until two life results are obtained at each of the four levels above the observed survival. Using the fatigue data from the inclined part of the curve, the method identifies a stress increment, DS, equal to the standard deviation of the stress data for the failed specimens. This increment is used in a final staircase method, in which the stress level of the next test is increased or decreased by DS according to whether the current test has produced a survival or a failure. The fatigue limit (for a probability of 50%) is computed as the stress average of the last six tests of the staircase.

2.5. Constant-strain tests 2.5.1. Testing procedure After several trial tests, the constant-strain characterization was performed according to the following procedure: 1. An undeformed wire length of about 130 mm is cut from the coil and cord end ferrules are crimped to both ends giving a net test length of 100 mm. 2. One end of the wire is attached to the load cell and the reading of the load cell is set to zero. 3. The free wire (no load applied) is heated above Af to recover the memorized shape. 4. The other end of the wire (which is maintained above Af) is attached to the lower cross-head (see Fig. 2b) 5. The wire is slightly stretched (about 5 MPa) by means of the lower adjusting screws (Fig. 2b) and the reading of the displacement transducer are set to zero

6. The current supply is cut-off and the wire is let to cool down to room temperature 7. The prescribed strain (read by the displacement transducer and displayed by the user interface) is applied to the wire by lowering the cross-head in Fig. 2b 8. The test is started by supplying the wire with a sinusoidal current oscillating from zero to the least peak value (determined by trial and error) capable of producing the full transformation of the alloy under the cooling conditions adopted (no direct measurement of the wire temperature was performed) 9. The test is terminated automatically at fracture of the SMA wire (no current flowing in the sample) or when a prescribed number of cycles (5  105) have been survived. For the constant-strain tests, high test frequencies were easier to achieve than for constant-stress tests because there was no inertia load on the wire due to the moving ballast. However, for similarity with the constant-stress tests, all the constant-strain experiments were performed at test frequencies ranging from 0.31 to 0.38 Hz. The supply currents needed to sustain these frequencies ranged from 1.12 to 1.22 A. 2.5.2. Strain levels Similarly to the constant-stress tests, the constant-strain tests were aimed at determining the fatigue-life curve of the SMA in terms of cycles to failure as a function of the applied strain. Since generally recognized test schemes are not available for strain-controlled fatigue tests, the experimental plan adopted in the test campaign included five strain values (1%, 2%, 3%, 4% and 5%) with three samples per level. These five test strains were chosen so as to encompass the strain range that would occur in typical applications of the wire for actuation purposes. 2.6. Constant-stress tests with limited maximum strain 2.6.1. Testing procedure The constant-stress tests with limited maximum strain were performed following the same procedure adopted for the constant-stress tests (Section 2.4). The only departure from that procedure consisted in adding, between steps 4 and 5, the following step: 4a. the position sensor is adjusted vertically so that a gap is formed between the hard stop and the basket (Fig. 2c), corresponding to the maximum strain allowed in the wire. Also in this case, the maximum achievable frequency was limited by the overloads induced in the wire by the inertia forces on the accelerating basket. All the tests were performed with a maximum target deviation of 6% between the peak stress and the nominal stress. This requirement led to test frequencies ranging from 0.36 to 0.46 Hz and peak supply currents ranging from 0.86 to 1.28 A. The maximum strain rate measured during the tests was similar for both limit strains (3% and 4%) and ranged between 0.150 s1 and 0.300 s1 during activation (wire shortening) and between 0.065 s1 and 0.150 s1 during relaxation (wire lengthening). 2.7. Stress and strain levels The constant-stress tests with limited maximum strain were carried out for two levels of limit strain: elim = 3% and 4%. At each strain level, the fatigue data were collected and processed following the standard JSME method [14] as for the constant-stress tests described in Section 2.4. The characteristic stress levels for the inclined part of the Woehler’s curves were 458.3, 391.7, 325, (S0 =) 258.3, 191.7, 125, 58.3 MPa (for elim = 3%) and 430, 365, 300, (S0 =) 235, 170, 105, 40 MPa (for elim = 4%).

Please cite this article in press as: Scirè Mammano G, Dragoni E. Functional fatigue of Ni–Ti shape memory wires under various loading conditions. Int J Fatigue (2012), http://dx.doi.org/10.1016/j.ijfatigue.2012.03.004

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3. Results 3.1. Constant-stress tests Table 1 reports the experimental results obtained from the test campaign under constant-stress conditions. Each row in Table 1 corresponds to one of the 14 samples required by the JSME

fatigue scheme described in Section 2.4. The nine tests 1–9 belong to the first part of the method, devoted to the identification of the inclined part of Woehler’s curve. Test number 3 (applied stress of 100 MPa) corresponds to the lowest stress level during the first part of the method at which no failure (after 5  105 cycles) was detected. This test is the starting point of the final staircase method, including tests 10–14 as following

Table 1 Experimental results from the constant-stress fatigue tests.

a

Test

r (MPa)

Nf

1 2 3 4 5 6 7 8 9 10 11 12 13 14

150 125 100 175 200 125 150 175 200 111 100 111 100 89

9688 25 901 500 000a 5783 4940 22 449 7731 5354 3509 51 802 500 000a 32 923 389 614 500 000a

Second cycle after start

Last cycle before termination

Dr (%)

emin  eA (%)

emax  eM (%)

De  SME (%)

Dr (%)

emin  eA (%)

emax  eM (%)

De  SME (%)

3.8 4.4 2.7 3.6 4.5 4.3 3.7 5.7 5.4 3.8 4.6 4.1 3.2 5.7

0.50 0.47 0.50 0.60 0.79 0.71 0.56 0.81 0.68 0.41 0.60 0.36 0.47 0.43

4.93 4.64 4.51 5.15 5.47 4.94 5.03 5.33 5.29 4.47 4.54 4.49 4.40 4.41

4.43 4.17 4.01 4.55 4.68 4.23 4.47 4.52 4.61 4.06 3.94 4.13 3.93 3.98

4.4 4.5 3.2 4.4 4.0 4.0 4.5 4.3 3.8 4.0 3.0 4.3 3.4 2.5

1.01 0.72 0.95 1.48 2.34 0.97 1.01 1.70 2.59 0.63 1.02 0.53 0.84 0.74

5.39 4.84 4.61 5.96 6.92 5.15 5.39 6.17 7.06 4.61 4.64 4.50 4.45 4.39

4.38 4.12 3.65 4.49 4.58 4.18 4.39 4.47 4.47 3.98 3.62 3.97 3.61 3.65

No failure observed.

450

log (N) = 4.89 − 4.080×10−3σ

Failure

400

Interpolation

4% limited strain

350

Stress, σ (MPa)

Survival

Constant-stress

3% limited strain

300 250

log (N) = 4.87 − 4.069×10−3σ

σw = 127.3 MPa

200

σw = 108.7 MPa

150 log (N) = 5.53 − 9.919×10−3σ

100

σw = 101.8 MPa

50 0 1,0E+03

1,0E+04

1,0E+05

1,0E+06

Cycles, N Fig. 3. Woehler’s diagram with the fatigue results for the constant-stress tests and the tests at constant stress with limited maximum strain (3% and 4%).

8

8 2nd cycle

7

7 6

5

εeA

4

εeM

3

SME

ε (%)

6

ε (%)

Last cycle before end of test

5

εeA

4

ε

3

SME

2

2

1

1 0

0 0

50

100

σ

150 (MPa)

(a)

200

250

0

50

100

σ

150

200

250

(MPa)

(b)

Fig. 4. Strains in the wire under constant-stress loading measured in: (a) the second cycle after test start and (b) the last cycle before test termination.

Please cite this article in press as: Scirè Mammano G, Dragoni E. Functional fatigue of Ni–Ti shape memory wires under various loading conditions. Int J Fatigue (2012), http://dx.doi.org/10.1016/j.ijfatigue.2012.03.004

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8

8

ε eA

εeM

SME SME

εeA

7

6

6

5

5

ε (%)

ε (%)

7

4

εeM

4

σ = 200MPa

3

3

σ = 100MPa

2

SME

2 1

1

0

0 0

125000

250000

375000

0

500000

1250

2500

Ν

Ν

(a)

(b)

3750

5000

Fig. 5. Evolution of wire strains during the constant-stress tests: (a) 100 MPa and (b) 200 MPa.

1.6

0.25

89MPa

0.20

100MPa

0.15

111MPa

0.10

Drift (%)

Drift (%)

0.30

0.05 0.00

1.4

125MPa

1.2

150MPa

1.0

175MPa

0.8

200MPa

0.6 0.4

-0.05 -0.10

0.2

-0.15

0.0 -0.2

-0.20 0

100000 200000 300000 400000 500000

0

5000

10000 15000 20000 25000

Ν

Ν

(a)

(b)

Fig. 6. Drift of martensitic wire length during the constant-stress tests: (a) low applied stresses and (b) high applied stresses.

steps. It is seen that the stress increment for the staircase method was DS = 11 MPa. The columns in Table 1 contain the applied stress (r), the number of cycles survived (Nf), the percent stress oscillation measured during the test (Dr), the minimum (emin  eA), maximum (emax  eM) and differential (SME  De = emax–emin) strains measured in the wire. The last four variables (from the stress oscillation, Dr, to the differential strain, SME) are listed both for the second cycle after the start of the test and for the last cycle before test termination. The data of applied stress (r) and cycles to failure (Nf) from Table 1 are plotted in the semi-log Woehler diagram of Fig. 3 (circles) together with the bilinear interpolation (solid line) obtained from the statistical treatment of the results (Section 2.4). The equations of the interpolating lines are also provided. Fig. 4 shows the relationship between applied stress and corresponding strains (eA, eM and De = SME induced in the wire in the second cycle afters test start (Fig. 4a) and in the last cycle before test termination (Fig. 4b). For a selection of stress levels (r = 100 and 200 MPa), Fig. 5 shows the trend of the characteristic strains as a function of the number of cycles endured by the wire. Fig. 6 shows the drift in the length of the martensitic wire as a function of the number of cycles. The drift is defined as the difference between the martensitic strain (eM) at the current cycle N and the martensitic strain measured in the second cycle. The drift parameter is a measure of the plastic strain accumulated by the wire during the tests.

3.2. Constant-strain tests Table 2 reports the experimental results obtained from the test campaign under constant-strain condition. The structure of this table is similar to Table 1 for the constant-stress tests, with the rows identifying the sample tested (three samples per strain level) and the following variables placed in the next columns: applied Table 2 Experimental results from the constant-strain fatigue tests. Test

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

e (%)

1 1 1 2 2 2 3 3 3 4 4 4 5 5 5

Nf

16 987 18 792 16 621 12 978 10 532 10 612 6406 8141 8238 8846 9702 9607 7306 7056 7159

Second cycle after start

Last cycle before termination

rmin

rmax

rmax

(MPa)

D0 r (MPa)

rmin

(MPa)

(MPa)

(MPa)

D0 r (MPa)

0 0 0 0 0 0 3 3 0 0 8 0 57 27 31

550 563 566 870 836 819 983 1044 975 932 960 962 1025 1133 1034

550 563 566 870 836 819 981 1041 975 932 951 962 968 1106 1003

0 0 0 0 0 0 1 1 0 0 3 1 37 24 4

433 393 400 461 463 473 512 475 468 417 466 457 490 425 415

433 393 400 461 463 473 511 474 468 417 463 456 453 401 412

Please cite this article in press as: Scirè Mammano G, Dragoni E. Functional fatigue of Ni–Ti shape memory wires under various loading conditions. Int J Fatigue (2012), http://dx.doi.org/10.1016/j.ijfatigue.2012.03.004

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6

Strain, ε (%)

5

4

3

log(N)=4.362 -1.207x10 -1 · ε% 2

1

0 1.0E+03

1.0E+04

1.0E+05

Cycles, N

1200

1200

1000

1000

800

800

σ max (MPa)

σ max (MPa)

Fig. 7. Woehler’s diagram with the fatigue results for the constant-strain test.

600

600

400

400 2nd Cycle

200

Last cycle before failure

200

σ− ε response austenite state

1%

2%

3%

4%

5%

0

0 0

1

2

3

ε

4

5

1

6

10

100

1000

10000

100000

Ν

(%)

Fig. 8. Maximum stresses in the wire under constant-strain loading measured for different applied strains in the second cycle after test start and in the last cycle before test termination.

Fig. 9. Evolution of the maximum stress in the wire under constant-strain loading.

strain (e), the number of cycles survived (Nf), the minimum (rmin), maximum (rmax) and differential (Dr = rmax–rmin) stresses measured in the wire during the tests. The last four variables (from

the minimum stress, rmin, to the differential stress, Dr) are listed both for the second cycle after the start of the test and for the last cycle before test termination.

Table 3 Experimental results from the constant-stress fatigue tests with maximum strain limited to 3%. Test

1 2 3 4 5 6 7 8 9 10 11 12 13 14 a

r (MPa) 258.3 191.7 125.0 325.0 391.7 191.7 258.3 325.0 391.7 138.7 125.0 111.3 125.0 138.7

Nf

6411 13 131 500 000a 3097 2159 14 975 6438 3294 2230 40 733 209 700 500 000a 500 000a 47 245

Second cycle after start

Last cycle before termination

D0 0 r (%)

emin  eA (%)

emax  eM (%)

De  SME (%)

D00 r (%)

emin  eA (%)

emax  eM (%)

De  SME (%)

3.81 6.64 3.70 3.92 3.77 3.64 3.47 4.34 5.28 5.74 3.81 4.53 8.97 4.24

0.46 0.31 0.29 0.94 0.90 0.31 0.47 0.58 0.77 0.25 0.18 0.20 0.21 0.25

3.06 3.05 2.94 3.02 3.01 3.03 3.00 3.01 3.06 3.04 3.01 3.05 3.01 2.97

2.60 2.74 2.65 2.08 2.11 2.72 2.53 2.43 2.29 2.79 2.83 2.85 2.80 2.72

5.59 3.01 2.63 2.10 0.00 2.12 2.70 3.32 0.77 3.41 3.03 3.86 7.03 2.63

1.11 0.59 0.68 2.03 2.86 0.62 1.05 1.63 2.74 0.40 0.42 0.48 0.60 0.45

3.05 3.02 2.99 3.03 3.01 3.05 2.98 3.00 3.06 3.03 3.02 3.00 3.05 2.99

1.94 2.43 2.31 1.00 0.15 2.43 1.93 1.37 0.32 2.63 2.60 2.52 2.45 2.54

No failure observed.

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G. Scirè Mammano, E. Dragoni / International Journal of Fatigue xxx (2012) xxx–xxx

The data of applied strain (e) and cycles to failure (Nf) from Table 2 are plotted in the semi-log Woehler diagram of Fig. 7. The straight line drawn through the test points corresponds to the equation shown in the diagram, which was obtained by least-squares best fitting of the experimental points.

Fig. 8 displays the maximum stress recorded in the second cycle after start and in the last cycle before test termination as a function of the applied strain. Finally, Fig. 9 plots the maximum stress in the wire as a function of the number of cycles for all the 5 strain levels applied in the

Table 4 Experimental results from the constant-stress fatigue tests with maximum strain limited to 4%. Test

1 2 3 4 5 6 7 8 9 10 11 12 13 14 *

r (MPa)

235 170 105 300 365 170 235 300 365 127.2 105 82.8 105 127.2

Nf

Second cycle after start

8899 17 693 500 000* 3579 2790 15 665 6972 3301 2977 36 444 339 280 500 000* 500 000* 36 458

Last cycle before termination

D0 0 r (%)

emin  eA (%)

emax  eM (%)

De  SME (%)

D00 r (%)

emin  eA (%)

emax  eM (%)

De  SME (%)

2.64 4.3 4.9 3.7 4.5 4.1 3.4 4.5 3.8 4.7 7.9 2.9 5.0 4.2

0.35 0.346 0.218 0.566 0.755 0.282 0.388 0.532 0.726 0.155 0.11 0.075 0.117 0.117

4.00 3.988 4.008 4.099 3.935 4.036 4.03 4.048 4.057 4.022 4.035 3.953 4.058 4.063

3.65 3.642 3.79 3.533 3.18 3.754 3.642 3.516 3.331 3.867 3.925 3.878 3.941 3.946

3.40 2.4 4.5 3.3 2.6 4.4 4.1 3.7 1.7 2.8 4.4 2.3 3.7 3.0

1.15 0.81 0.70 2.08 2.86 0.80 1.24 1.99 2.92 0.472 0.57 0.393 0.635 0.44

4.00 4.00 4.00 4.08 4.08 4.04 4.05 4.05 4.06 4.044 4.038 3.97 4.052 4.072

2.85 3.19 3.30 2.00 1.22 3.24 2.81 2.06 1.14 3.572 3.468 3.577 3.417 3.632

No failure observed.

5.0

5.0 4.5

2nd cycle

4.0

4.0

3.5

3.5

3.0

3.0

ε (%)

ε (%)

4.5

2.5 2.0

ε eA

1.5 1.0

Last cycle before end of test

2.5 2.0

εeA

ε eM

1.5

εeM

SME SME

1.0

SME SME

0.5

0.5

0.0

0.0 0

100

200

σ

300

0

400

100

300

400

(MPa)

(b)

(a) 5.0

5.0

4.5

2nd cycle

Last cycle before end of test

4.5

4.0

4.0

3.5

3.5

3.0

3.0

ε (%)

ε (%)

200

σ

(MPa)

2.5

εeA

2.0

εeA

2.0

εeM

1.5

εeM

1.5

SME SME

1.0

SME SME

1.0

2.5

0.5

0.5

0.0

0.0 0

100

200

σ

(MPa)

(c)

300

400

0

100

200

σ

300

400

(MPa)

(d)

Fig. 10. Strains in the wire under constant-stress loading with limited maximum strain measured: (a) in the second cycle after test start for 3% limit strain; (b) in the last cycle before test termination for 3% limit strain; (c) in the second cycle after test start for 4% limit strain; and (d) in the last cycle before test termination for 4% limit strain.

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5.0

5.0

εeA

εeM

SME SME

4.0

4.0

3.5

3.5

3.0

3.0

2.5

εeM

SME SME

2.5 2.0

2.0

σ = 125MPa

1.5

1.5

1.0

1.0

0.5

0.5

0.0

0.0 0

σ = 325 MPa 0

100000 200000 300000 400000 500000

1000

2000

Ν

Ν

(a)

(b)

3000

4000

3000

4000

5.0

5.0

εeA

4.5

εeM

SME SME

εeA

4.5

4.0

4.0

3.5

3.5

3.0

3.0

ε (%)

ε (%)

εeA

4.5

ε (%)

ε (%)

4.5

2.5

εeM

SME SME

2.5 2.0

2.0

σ = 127.2MPa

1.5

1.5

1.0

1.0

0.5

0.5

σ = 300MPa

0.0

0.0 0

10000

20000

30000

40000

0

1000

2000

Ν

Ν

(c)

(d)

Fig. 11. Evolution of wire strains during the constant-stress tests with limited maximum strain: (a) 125 MPa up to 3% strain; (b) 325 MPa up to 3% strain; (c) 127.2 MPa up to 4% strain; and (d) 300 MPa up to 4% strain.

tests. The curves refer to the first sample tested at each level (see Table 2). 3.3. Constant-stress tests with limited maximum strain Tables 3 and 4 report the experimental results obtained from the test campaign under constant-stress with limited maximum strain for the strain limit of 3% and 4%, respectively. The structure of these tables is similar to Table 1 (constant-stress tests) with the nine tests 1–9 belonging to the first part of the JSME method and test number 3 (125 and 105 MPa for 3% and 4% limit strain, respectively) being the lowest stress level at which a survival (5  105 cycles) was detected. Launched by test 3, the final staircase method includes tests 10–14 with stress increments DS = 13.7 and 22.2 MPa for the limit strains of 3% and 4%, respectively. The column headings in Tables 3 and 4 have the same meaning as in Table 1 with the only exception of the stress oscillation D00 r, which now measures the percent difference between the maximum stress during the reference cycles and the nominal applied stress r. The data of applied stress (r) and cycles to failure (Nf) from Tables 3 and 4 are plotted in the semi-log Woehler’s diagram of Fig. 3 (triangles and diamonds for limit strains of 3% and 4%, respectively) together with the interpolating lines (dotted and dashed lines for limit strains of 3% and 4%, respectively) obtained from the statistical elaboration of the results (Section 2.6). The equations of the interpolating lines are also provided.

Fig. 10 shows the relationship between applied stress and corresponding strains (eA, eM and De = SME induced in the wire in the second cycle afters test start (Fig. 10a and c) and in the last cycle before test termination (Fig. 10b and d) for the limit strains elim = 3% (Fig. 10 a,b) and elim = 4% (Fig. 10c and d). For a selection of stress levels (r = 125 and 325 MPa for elim = 3% and r = 127.2 and 300 MPa for elim = 4%), Fig. 11 shows the trend of the characteristic strains as a function of the number of cycles endured by the wire.

4. Discussion 4.1. Constant-stress tests Table 1 shows that the target of maximum stress oscillation (Dr within 6% of the nominal stress) was achieved for all specimens, indicating a good accuracy of the tests performed. The very low scatter of the experimental points (circles) in Fig. 3 is also a remarkable accomplishment of this work, given the regular shape and the very smooth surface of the wire samples under test (repeatability of fatigue lives are more common for highly notched specimens). Finally, an important quality achievement was the occurrence of all failures in the free span of the wire samples with the fractures taking place at a distance never less than 10 mm from the grips. This good results is attributable to two reasons. On one hand, the use of soft electric crimps at the end of the wires,

Please cite this article in press as: Scirè Mammano G, Dragoni E. Functional fatigue of Ni–Ti shape memory wires under various loading conditions. Int J Fatigue (2012), http://dx.doi.org/10.1016/j.ijfatigue.2012.03.004

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G. Scirè Mammano, E. Dragoni / International Journal of Fatigue xxx (2012) xxx–xxx

which mediate the contact with the grips and reduce the mechanical stress concentration. On the other hand, the lower temperature of the wire under the grips, which prevents the complete martensite-austenite transformation thus reducing locally the actual strain cycle. Fig. 3 shows that the inclined part of Woehler’s curve is reasonably well represented by a straight line. The slope of the interpolating line is rather flat, indicating that the fatigue life of the SMA wire decreases very quickly for increasing applied stress. From the equation of the inclined portion reported in Fig. 3, a slope r/ log(Nf)  101 MPa is calculated, a figure that correlates favorably with the slope of 117 MPa reported by Araujo et al. [9] for an alloy Ti.45.0Ni–5Cu (%at). The fatigue limit calculated from Fig. 3 for the chosen life (5  105 cycles) is rw = 101.8 MPa, corresponding to a confidence level of 50%. The distribution of the experimental points in Fig. 3 along the horizontal part of the Woehler’s curve suggests that the existence of a true fatigue limit of the material under constant-stress conditions is very likely. To verify this conclusion, one sample that survived the staircase method in Table 1 (specimen 3) was mounted on the machine and cycled again up to one million cycles without any sign of failure. More systematic longterm tests are now being planned to strengthen the evidence of a genuine fatigue limit in these test conditions. It is worth noting that the life results under constant-stress cycles reported by the manufacturer of the wires tested here (SAES Getters) are much higher than the life data displayed in Table 1. For example, the manufaturer’s data sheet gives a life Nf = 105 cycles and a length drift of about 0.17% for an applied stress r = 170 MPa. These results compare poorly with the present outcome, giving a life Nf = 8710 cycles (average of tests 1 and 7 in Table 1) and a drift of about 0.45% (Fig. 6b) under an applied stress r = 150 MPa. This big disagreement can perhaps be explained by the difference in test conditions under which the fatigue data were obtained. Starting from the wire undergoing the maximum elongation in the martensitic state (low temperature), SAES Getters test conditions supply heat to the wire until a differential strain SME = 3.5% is achieved, testifying only a partial transformation of the alloy. In the present investigation, the alloy is transformed completely, giving for r = 150 MPa a differential strain SME  4.4% (Table 1, tests 1 and 7). The beneficial effects on the fatigue life associated to partial austenite transformation has been reported by several authors [5,6] and could explain the difference in fatigue performance observed here. Exploring systematically the effect of partial transformation on the fatigue response of shape memory alloys undergoing constant-stress loading would be a fruitful research field to investigate. The wire strains reported in Fig. 4a for the second test cycle show that both maximum (martensitic) and minimum (austenitic) strains increase almost linearly with the applied stress. This behavior is coherent with the monotonic stress–strain curve of the alloy [12], which shows that in the range of applied stresses covered in Fig. 4a (100–200 MPa) both martensitic and austenitic stress– strain curves are fairly linear. As a consequence, also the differential strain (SME in Fig. 4a) increases almost linearly with the applied stress. The small increase in differential strains seen in Fig. 4a when the applied stress passes from r = 175 MPa to r = 200 MPa supports the conclusion by Bigeon and Morin [8] that the shape memory effect (measured by the differential strain SME) achieves a maximum in NiTi alloys for an applied stress around 200 MPa. When the behavior of the alloy after a considerable number of cycles is examined (Fig. 4b), a different strain pattern emerges. For the last cycle before test termination, Fig. 4b shows that the differential strain (SME) is lower than for the second cycle when the applied stress r stays below 125 MPa. By contrast, the perfor-

mance remains more or less stable (in comparison to the second cycle) for stresses r P 125 MPa. Apparently, the shape memory alloy becomes less and less efficient as the material accumulates strain cycles under relatively low applied stresses. This remark is confirmed by the strain evolution with the number of cycles plotted in Fig. 5. Fig. 5a shows that for r = 100 MPa the martensite strain is nearly constant across the cycles while the austenitic strain increases with the cycles so that the differential strain progressively decreases. Macroscopically, this behavior is equivalent to a decrease of the stiffness of the austenitic wire as the cycles accumulates and translates into a poorer actuation performance. For higher applied stresses (r = 200 MPa), Fig. 5b shows that martensite and austenite strains increase with the same rate over the cycles so that the differential strain SME remains unchanged. For the sake of the actuator’s efficiency (high strokes) the above results would suggest using the SMA wire under relatively high backup stresses to maximize the differential strain SME. However, Fig. 6b shows that for high applied stresses (r P 125 MPa) the length drift of the wire is considerable, indicating that the alloy accumulates permanent plastic strains as the cycles go on. The length drift (hence the permanent plastic strain) of the wire is much more contained for the lower stress levels (r 6 111 MPa) as shown in Fig. 6a. From a practical standpoint, a trade-off between the advantages (higher stroke) and disadvantages (high drift, hence poor repeatability) of the stress on the wire is to be found. Overall, the data from the present research suggest that a stress level of about 125 MPa would ensure appreciable differential strains (large actuator strokes) with acceptable length drift (less than 0.2% from Fig. 6b).

4.2. Constant-strain tests Since the failures of all the samples tested under constant-strain conditions occurred outside the grips of the machine (see explanation above), the fatigue data in Table 2 are quite meaningful for the true response of the alloy. Table 2 shows that, irrespective of the strain applied, all samples failed after a relatively low number of cycles (7000 < Nf < 19 000). This outcome confirms that the constant-strain loading condition is much more demanding on the material than the constant-stress test, even though the maximum strains induced in the alloy are of the same order of magnitude in the two tests. Woehler’s semi-log diagram in Fig. 7 shows that the fatigue points are reasonably well interpolated by a straight line. A small departure from the linear trend is represented by lives at 3% strain, which are lower than those at 4% strain, but these uncertainties are typical of the behavior of most materials even under purely mechanical fatigue. The slope of the interpolating line in Fig. 7 is e/ln(N)  8.28 (e expressed in percent). From Fig. 8 it is seen that the maximum stress induced in the wire during the second loading cycle (hollow circles) for each strain level is consistent with the monotonic stress–strain curve of the austenitic alloy [12]. During the last cycle before failure (crosses in Fig. 8) the fatigue data fall quite below the monotonic stress–strain curve and failures occurs at a terminal stress between 400 and 500 MPa, regardless of the applied strain. The decrease of the maximum stress during the tests would perhaps have warranted a decreasing supply current to transform completely the alloy as the cycles accumulated. However, since the temperature of the wires was not measured directly (see testing procedures in Section 2), the supply current chosen at the beginning of each test was maintained throughout the test. This condition may have produced overheating of the alloy with respect to the diminishing transformation temperature, resulting in a lower fatigue life with respect to a pinpointed heat supply.

Please cite this article in press as: Scirè Mammano G, Dragoni E. Functional fatigue of Ni–Ti shape memory wires under various loading conditions. Int J Fatigue (2012), http://dx.doi.org/10.1016/j.ijfatigue.2012.03.004

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The departure of the stress–strain response from the monotonic curve for increasing cycles is confirmed by Fig. 9, showing a progressive decrease of the maximum stress with the life. Since the trend of the interpolating curves in Fig. 9 is approximately linear in the semi-log chart, the stress decay is exponential. Although starting from very different maximum stresses at cycle 2, the curves in Fig. 9 terminate when the stress falls in the range 400–500 MPa. Dropping of the maximum stress into this range could be adopted as a predictor of incipient fracture of the material undergoing constant-strain functional cycling of whatever amplitude. The constant-strain tests have shown that the shape memory wire under examination exhibits to some degree a two-way shape memory effect. This became clear in step 6 of the test procedure (see Section 2.5) when the wire loosened slightly between the grips after cooling down from the restraining step carried out in the (hot) austenitic state. This two-way shape memory effect, which tended to disappear over the cycles, could explain the peculiar trend of the wire drift detected in some constant-stress tests. For the lowest applied stresses (r = 89 and 100 MPa) Fig. 6a shows that the drift decreases over the first cycles, denoting a shortening of the wire that could be due to the second memory of the material fading away. 4.3. Constant-stress tests with limited maximum strain Tables 3 and 4 show that the target of maximum stress oscillation (D00 r within 6% of the nominal stress) was achieved for most specimens. As for the constant-stress tests, also in these tests with limited maximum strain all failures occurred in the free span of the wire samples at a good distance from the grips (see explanation in Section 4.1). Fig. 3 confirms minor scatter of the fatigue lives across the descending leg of the Woehler’s curves and of the stress levels about the horizontal leg. The comparison of the descending legs of all the curves in Fig. 3 demonstrates that putting a limit, however small, to the maximum strain during constant-stress tests is extremely beneficial in terms of fatigue life. For example, for an applied stress of 175 MPa, Table 1 shows that the maximum free strain induced in the wire is just above 5% and the life is about 5570 cycles. For a slightly lower applied stress of 170 MPa and the maximum strain limited to 4%, Table 4 indicates a threefold life increase to 16 680 cycles. It is worth noting that limiting the maximum strain to 3% does not produce further advantages because the sloped legs of the Woehler’s curves in Fig. 3 for 3% (dashed line) and 4% maximum strain (dotted line) are very close to each other. This occurrence suggests that a threshold limit strain could exist above which the described life increase is obtained with respect to constant-stress tests with unlimited strain. Further tests under constant-stress with limit strains greater than 4% are being conducted to further explore this behavior. From the descending legs of the Woehler’s curves in Fig. 3, a common slope r/log(Nf)  245 MPa is calculated for the tests with limited strain, regardless of the specific limit. This slope is higher than for the constant-stress tests (101 MPa, Section 4.1) so that the life for all tests is nearly the same (22 000 cycles) when the applied stress is close to 125 MPa. This behavior can be explained by observing from Table 1 that the maximum strain produced by an applied stress of 125 MPa in the constant-stress tests is not much higher than 4%. The greater severity of the strain cycle seems to be compensated by the fact that in the tests with limited strain the stress drops to zero when the basket comes in contact with the hard stop, thus producing a cyclic stress condition, which does not occur in the regular constant-stress tests. For increasing applied stresses, the maximum strain in the constant-stress tests increases more and more with respect to the tests with limited strain and this severer strain condition leads to shorter and shorter lives.

11

The trend of the experimental points in Fig. 3 for the constantstress loading with limited strain suggests the existence, as for the regular constant-stress tests, of a genuine fatigue limit. This conclusion will be checked by prolonging the tests for the survived specimens in Tables 3 and 4. As expected, the fatigue limits estimated by the JSME elaboration is greater here than for the constant-stress conditions (101.8 MPa) and takes up for decreasing limit strain (108.7 and 127.3 MPa for limit strains of 4% and 3%, respectively). Quantitatively, the difference between the fatigue limits with limited strain and the fatigue limit at constant-stress is proportional to the difference in strains achieved in the tests under those stresses. For the constant-stress tests, Table 1 shows an average maximum strain of 4.45% (tests 3, 11 and 13) for an applied stress of 100 MPa, close to the 101.8 MPa fatigue limit. From this reference strain and the above fatigue limits, the following ratios are calculated: (108.7–101.8)/(0.0445–0.04)  1533 MPa (for elim = 4%) and (127.3–101.8)/(0.0445–0.03)  1759 MPa (for elim = 3%). Fig. 10 shows that the minimum strain in the austenitic wire, eA, increases linearly with the applied stress, confirming the nearlyelastic behavior observed for the constant-stress tests. Given the stress, the austenitic strain eA is greater at the end of the test (Fig. 10b and d) than at the beginning (Fig. 10a and c), leading to a lesser efficiency of the alloy for actuation purposes. This effect is greater for increasing applied stress with a twofold increase of the austenitic strain for an applied stress of about 100 MPa and a four-time increase for an applied stresses close to 400 MPa. The same behavior is emphasized in Fig. 11, which shows that the austenitic strain eA (hollow circles) builds up progressively with the accumulated cycles. Since the maximum strain is limited (solid circles), the increase in eA reflects directly in a reduction of the differential strain, or shape memory effect, SME (stars in Fig. 11). As observed in the constant-strain tests, also the constant-stress tests with limited strain gave evidence of a marked two-way shape memory effect promoted by the test cycling. When heated under no load, the survived wires in Tables 3 and 4 recovered almost entirely their original length. Upon subsequent cooling, they lengthened by an amount corresponding to the maximum strain (3% or 4%) endured during the test. A last remark is worth making by comparing the results of the constant-strain tests for applied strains of 3% and 4% (Table 2) with the constant-stress tests for maximum strain limited to the same values (Tables 3 and 4). It is seen that, despite the achievement of maximum stresses always exceeding 900 MPa in the former case (Table 2), the lives Nf are greater than in the latter case whenever the applied stress equals or exceeds 235 MPa (Tables 3 and 4). For example, tests 10, 11 and 12 in Table 2 average a life of 9385 cycles and a maximum stress of 951 MPa for a constant applied strain of 4%. By contrast, tests 1 and 7 in Table 4 average a life of 7935 cycles for an applied stress of only 235 MPa and tests 5 and 9 in the same table average a life of 2883 cycles for an applied stress of 365 MPa. This data indicate that the damage in the wire cannot be explained solely by the maximum stress and maximum strain attained in the thermomechanical cycling and that the applied strain range plays an important role. 4.4. Fitting the fatigue data with the CoffinManson equation The low-cycle fatigue data for engineering materials involving high plastic strains are usually described by the Coffin–Manson equation [15,16].

eap ¼ e0f ð2Nf Þc

ð1Þ

where eap is the plastic strain amplitude (one half of the peakto-peak plastic strain), e0f is the fatigue ductility coefficient (the failure strain for a single reversal), Nf is the number of cycles to failure and c is an empirical constant known as the fatigue ductility

Please cite this article in press as: Scirè Mammano G, Dragoni E. Functional fatigue of Ni–Ti shape memory wires under various loading conditions. Int J Fatigue (2012), http://dx.doi.org/10.1016/j.ijfatigue.2012.03.004

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G. Scirè Mammano, E. Dragoni / International Journal of Fatigue xxx (2012) xxx–xxx

exponent (commonly ranging from 0.7 to 0.5 for metals in timeindependent fatigue). When augmented with the Basquin term [17] for classical fatigue, the standard Coffin-Manson Eq. (1) becomes [18]:

ea ¼

r0f E

ð2Nf Þb þ

e0f ð2Nf Þc

ð2Þ

where ea is the total (elastic and plastic) strain amplitude (one half of the peak-to-peak total strain range), r0f is the fatigue strength coefficient (the failure stress for a single reversal), E is the modulus of elasticity and b is an empirical constant known as the fatigue strength exponent (commonly ranging from 0.14 to 0.06 for metals in time-independent fatigue). The generalized Coffin–Manson law (2) is a comprehensive model which fits the strain-life data for both low-cycle and highcycle fatigue with a single curve. Conceptually, this is an improvement over the JSME approach adopted above, which uses two straight lines to interpolate the same set of data. In order to apply Eqs. (1) and (2). to the fatigue data of the previous sections, parameters e0f , r0f and E were approximated by the fracture strain, the fracture strength and the elastic modulus, respectively, derived from the monotonic tensile tests of the austenitic wire. The following values were obtained: e0f = 0.09, r0f = 1300 MPa, E = 42 900 MPa. Eq. (1) was fitted to the strain-life data for the constant-strain tests (eap = e/2 in Table 2 and Fig. 7), obtaining c = 0.1802 (with a standard error of 0.0113). Likewise, Eq. (2) was fitted to the strain-life data for the constant-stress tests (ea = SME/2 for the ‘‘last cycle before termination’’ in Table 1) obtaining b = 0.0375 (standard error 0.0006) and c = 0.4894 (standard error 0.0223). Fitting of Eq. (2) to the strain-life data for the constant-stress tests with limited maximum strain (ea = SME/2 in Tables 3 and 4) was not possible because the strain amplitude decreases with the number of cycles. (This would give positive values for b and c.) Although the calculated values for b and c do non agree with the typical values for ordinary metals (see above), they fit well the fatigue data where applicable (constant-strain and constant-stress tests). In the next step of the research, focused on the fatigue loading of the wires under linearly dependent waveforms of stress and strain (Fig. 1b), this approach will be given due consideration for the comprehensive description of the endurance of the alloy. 5. Conclusions Four types of tests have been envisioned to characterize shape memory wires under functional fatigue loading: constant-stress test, constant-strain test, constant-stress test with limited maximum strain, linear stress–strain cyclic test. An experimental campaign carried out on commercial NiTi wires (SAES Getters SmartflexÒ) under the fist three test conditions (constant-stress, constant-strain, constant stress with limited maximum strain) has produced the following results. Constant-stress loading: – The stress-life (Woehler’s) curve on a semi-log diagram is similar to the ordinary metals’, with an initial inclined line followed by a final horizontal line. For the SmartflexÒ wire the inclined line has a slope r/log(Nf)  101 MPa while the horizontal line identifies a fatigue limit (at 5  105 cycles) rw  102 MPa; – for applied stresses below the fatigue limit, the martensitic length drift of the wire is negligible (less than 1%) and the differential strain (shape memory effect) in the wire is acceptable for practical purposes; – for applied stresses above the fatigue limit, the differential strain increases but is accompanied by greater and greater length drifts.

Constant-strain loading: – The strength-life (Woehler’s) curve on a semi-log diagram features only the inclined part (the wire failed for all strain levels tested between 1% and 5%) and there is no indication of a strain threshold (strain fatigue limit); – the functional life can be as low as 7000 cycles (for applied strain of 5%) and never exceeds 19 000 cycles (applied strain of 1%) in the strain range investigated; – the stress induced in the wire decreases exponentially with the number of cycles. The residual stress just before failure falls in the range 400–500 MPa, regardless of the strain applied. Constant-stress loading with limited maximum strain (3% and 4%): – The stress-life (Woehler’s) curves confirm the existence of an initial inclined line followed by a final horizontal line. For the SmartflexÒ wire the inclined line has a slope r/log(Nf)  245 MPa for both the limit strains investigated. The the horizontal line identifies a fatigue limit (at 5  105 cycles) rw  127 MPa for 3% limit strain and rw  109 MPa for 4% limit strain; – the increase of the fatigue limit with respect to the regular constant-stress test is proportional to the difference between the maximum strains achieved in the corresponding tests; – for any applied stresses the differential strain (shape memory effect) is reduced as the cycles go on and the loss is greater for increasing applied stresses. In summary, NiTi shape memory wire should be used in actuators under loading conditions as close as possible to a constantstress situation. Exposure to constant-strain conditions due to over-restraint of the wire or pseudo-restraint conditions promoted by dynamic loading is very detrimental to the fatigue life and should be avoided.

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Please cite this article in press as: Scirè Mammano G, Dragoni E. Functional fatigue of Ni–Ti shape memory wires under various loading conditions. Int J Fatigue (2012), http://dx.doi.org/10.1016/j.ijfatigue.2012.03.004