Me Mod Ma

Damping analysis of an ultra thin superelastic NiTi wire

TO: Institut FEMTO ST, D´epartement de M´ecanique Appliqu´ee FROM: Ludek HELLER, Institute of Physics, Academy of Sciences of The Czech Republic DATE: October 17, 2007 SUBJECT: Report of research stay in the Department of Applied Mechanics at FEMTO ST institute. From Monday 18 June to Friday 13 July 2007.

Contents 1 Objectives

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2 Quasi static thermomechanical properties

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3 Damping properties 3.1 Damping in SMA . . . . . . . . . . . . . . . . . . . . 3.2 Damping related to R-phase transformation . . . . . 3.2.1 Prestress effect . . . . . . . . . . . . . . . . . 3.2.2 Effect of mechanical cycling . . . . . . . . . . 3.2.3 Analysis of repeatability . . . . . . . . . . . . 3.2.4 Influence of the sense of frequency sweeping . 3.2.5 Effect of vibration amplitude . . . . . . . . . . 3.2.6 Analysis of the material response non-linearity 3.3 Damping related to the martensitic transformation . 3.3.1 Effect of vibration amplitude . . . . . . . . . . 3.3.2 Analysis of repeatability . . . . . . . . . . . . 3.3.3 Effect of mechanical cycling . . . . . . . . . . 3.3.4 Analysis of the material response non-linearity 4 Conclusion

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1

Objectives

The main goal of presented analysis was to evaluate the damping capacity of an ultra thin superelastic NiTi wire with respect to several parameters. These parameters considered as variables having crucial impact on the damping are represented by the level of prestress determining the state of microstructure and the amplitude and frequency of vibrations conditioning the activation of microstructural processes taking place in NiTi wire. To achieve such an aim a DMA apparatus (BOSE, EnduraTEC) allowing to analyse the damping capacity in term of loss angle within the frequency range 0-200Hz was used. Although this theoretically applicable frequency range, the damping measurements in frequencies up to 20Hz only were realized due to problems linked to the resonance of the whole testing system. Different prestress or prestrain levels were chosen in order to observe the damping capacity due to both the R-phase and the martensitic transformation.

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Quasi static thermomechanical properties

Before studying the damping properties a set of quasi-static experiments had been realized in order to identify the basic thermomechanical properties of the studied NiTi wire having the diameter 100 µ. Microstructural state of the wire was defined by the straight annealing during the wire production. To present the results we use the conventional stress and strain related to the initial length and diameter of the wire.

(a) Tensile test until rupture

(b) stability test - 100 cycles

Fig. 1 Different tensile tests performed on studied wire Standard tensile tests fig.1(a) at room temperature provided the information about its important mechanical properties as follows: ˆ Young modulus of austenite: Ea = 53.6 MPa ˆ Young modulus of martensite: Em = 21.1 MPa ˆ Transformation Yield stress of austenite at room temperature: σFtr = 550 MPa ˆ Maximum recoverable transformation strain: tr = 5.2 % ˆ Ultimate tensile strength: σ U T S = 1590 MPa

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ˆ Yield stress: σ Y = 1330 MPa ˆ Strain at failure: f = 13 % ˆ Pseudoelastic stress hysteresis: ∆hσ = 336 MPa

The fig.1(a) shows also the presence of premartensitic R-phase transformation manifested by a non-linear stress-strain relation at an early stage of deformation up to 1% and by considerable increase of electrical resistivity within this deformation range. The stability of quasi-static properties was evaluated by a cyclic test consisting of 100 complete superelastic cycles fig.1(b). To quantify such a stability the following set of parameters was deduced from the experimental data. ˆ Accumulated nonrecovered strain for N=100 cycles: ac = 0.47 % tr = 26 % ˆ Cyclic accumulated transformation stress decrease for N=100: ∆σac

ˆ Cyclic accumulated pseudoelastic hysteresis decrease for N=100: ∆htr ac = 24 %

Partial mechanical cycles represents an important property of a superelastic wire showing the stress conditions needed for the transformation processes to take place within a mixture state of microstructure. The fig.2 shows clearly that the thermomechanical force associated to these processes is almost the same as that provided the forward/reverse transformation during a complete superelastic cycle.

(a) partial loading – complete unloading

(b) complete loading – partial unloading

Fig. 2 Partial cycles Finally two thermomecanical tests were used in order to construct the stress-temperature (Clausius-Clapeyron) diagram. On one hand several tensile tests at different constant temperatures and on the other hand thermal cyclic tests at different constant applied stresses were performed as illustrated in fig.2.

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Fig. 3 Experimental technique for constructing the stress-temperature diagram These experiments allowed to obtain a partial stress-temperature diagram of the studied wire (fig.2) including the lines A0f , Rs0 , Ms0 defining the effective transformation temperatures. The importance of knowledge of the stress-temperature diagram lies in the fact that it allows to estimate the microstructural state for the known themomechanical conditions established as a result of a defined thermomechanical history. That allowed to realize the damping measurement in approximately known microstructural states of material.

Fig. 4 Experimentally obtained stress-temperature diagram of the studied wire The following important data were deduced from the stress-temperature diagram: ˆ Effective martensite start temperature: Ms0 = −72‰ ˆ Effective austenite finish temperature: A0f = 0‰

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ˆ Effective R-phase start temperature: Rs0 = 11‰ ˆ Temperature dependence of transformation stress for A-R: SA−R = 16.5MPa/‰ ˆ Temperature dependence of transformation stress for A-M: SA−M = 5.5MPa/‰ ˆ Temperature dependence of transformation stress for M-A: SM −A = 7.6MPa/‰

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Damping properties

The damping capacity of the wire was studied in two different microstructural states defined by an appropriate choice of the prestress or prestrain. First the damping in the zone of premartensitic R-phase was investigated and then the dissipation due to the martensitic transformation was studied. In both cases the influence of frequency and amplitude of vibrations on the damping was evaluated. All measurements were done at room temperature by using a DMA apparatus (EnduraTec ELF 3200, BOSE Corporation). To evaluate the damping at R-phase state a basic DMA experimental procedure was used consisting of a sinusoidal excitation under displacement control regime with simultaneous measurement of exerted force. Between two frequency steps the wire was unload and the level of prestress or prestrain was set up again. The evaluation of the damping capacity and other related parameters such as loss modulus, storage modulus etc. was done either using supplied software or by the postprocessing of recorded time signals of the excitation and response. The former method was based uniquely on evaluation of the phase shift between the fundamental harmonics of the force and displacement using fast Fourier transformation. The latter was based on both the fast Fourier transformation and on numeric integration of force-displacement loops allowing to analyse the damping in the case of nonlinear response. Unfortunately this self made post processing is affected by some phase error of electronics which is, as mentioned in manual of the apparatus, corrected during the damping evaluation by the supplied software. The self-made post processing was applied also when analysing the damping capacity at partially martensitic state of microstructure. In this case the testing technique consisted in prestraining the wire at the middle of the plateau and in spreading a given frequency range without setting up the prestrain after each measured frequency. The following notation is used in the result presentation: Values obtained by the supplied software ˆ F ? – dynamic force amplitude ˆ X ? – dynamic displacement amplitude ˆ σ ? = F ? /(πD02 /4) – dynamic stress amplitude (D0 - initial sample diameter) ˆ X ? – dynamic displacement amplitude ˆ ε? = X ? /l0 – dynamic strain amplitude (l0 - initial sample length) ˆ F0 – mean force ˆ σ0 = F0 /(πD02 /4) – mean stress ˆ X0 – mean displacement ˆ ε = X0 /l0 – mean strain

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ˆ K ? = F ? /X ? – dynamic stiffness ˆ δ – loss angle (phase shift between the force and displacement) 0

ˆ K = K ? cos δ – storage stiffness 00

ˆ K = K ? sin δ– loss stiffness ˆ E? = 0

ˆ E = 00

ˆ E =

K ? l0 πD02 /4

– dynamic Young modulus

0

K l0 πD02 /4

– storage Young modulus

00

K l0 πD02 /4

– loss Young modulus

ˆ C = K ? sin δ/ω – damping (ω – frequency) ˆ ∆W = πF ? X ? sin δ – hysteresis, energy lost per cycle (ω – frequency q 2 C 2 ω 2 +K 0 ˆ T r = (K 0 −mω 2 )2 +C 2 ω 2 – transmissibility (m = mean mass = measured mean force / gravitational constant

Values calculated using the parameters obtained by the supplied software √ √ ˆ Q = 1 + tan δ − 1 − tan δ – quality factor ˆ ψ = ∆W = 2π tan δ – specific damping capacity (∆W – energy lost per cycle, W – W energy stored per cycle) ˆ ζeq =

1 ∆W 4π Ekmax

=

ψ 2

– equivalent viscous damping

Values obtained by the self-made postprocessing based on the time force/displacement signals ˆ ∆W – force-displacement loop area calculated by the numerical integration ˆ Fmax /σmax = Fmax /(πD02 /4) – maximal value of the alternating part of force/stress ˆ F0 /σ0 = F0 /(πD02 /4) – mean force/stress value ˆ FA /σA = FA /(πD02 /4) – force/stress amplitude ˆ XA /εA = XA /l0 – displacement/strain amplitude ˆ X(Fmax ) – displacement corresponding to Fmax ˆ Ekmax = 21 Fmax X(Fmax ) – maximal kinetic energy, ˆ ζeq =

1 ∆W 4π Ekmax

– equivalent viscous damping

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3.1

Damping in SMA

The common way to characterise the damping capacity of materials undergoing phase transformation consists in separating the overall internal friction into three parts as follows: −1 −1 −1 Q−1 tot = QT r + QP T + Qint

(1)

, where Q−1 T r is the transient part existing only during cooling or heating (dT /dt 6= 0). It depends on transformation kinetics and hence it is proportional to the volume fraction which is transformed per unit of time. Q−1 P T is related to the mechanism of the phase transformation such as the movement of interfaces between existing phases. Q−1 int represents the internal friction contribution related to each phase. The applied testing procedures involves mainly the isothermal experiments at constant small strain amplitude with several applied prestresses and those at a constant prestress with different applied strain amplitudes. The changes in damping identified during the former experiment is affected mainly by the variation of Q−1 P T due to the fact that by changing the prestress level one modifies the amount of existing interfaces whereas the small amplitude corresponds mainly to the elastic deformation of both phases and only slight changes in the volume fraction of induced phase are produced. However during the tests at a constant prestress level the used amplitudes were sufficiently high so that they could induce the martensitic transformation. Hence these tests allowed to identify the changes in damping due to the transient part Q−1 T r which is affected by the rate of the volume fraction fluctuation within a cycle.

3.2

Damping related to R-phase transformation

The studied NiTi wire exhibit the sequential phase transformations B2-R-B19’. The intermediate transformation to so called R-phase is characterise by a small transformation strain (¡1%) and its presence is manifested during a tensile test by a nonlinear character of the beginning of stress-strain curve. Previous measurements using ultrasonic methods have shown a high damping capacity associated with the R-phase transformation (IFR ) and this fact was the motivation for IFR identification at low frequencies using DMA. 3.2.1

Prestress effect

The influence of the prestress was studied in order to identify the contribution of Q−1 P T to overall R-phase damping capacity. By changing the prestress level one modifies the amount of interfaces between phases present in the material and by this way the energy dissipated due to the irreversible mobility of such interfaces is affected. A DMA analysis in a frequency range up to 20Hz was carried out for different prestresses (see fig.5) using a constant strain amplitude of 0.15%.

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Fig. 5 Applied prestresses during the investigation of R-phase damping Since the R-phase transformation is accompanied by a material softening, the absolute values of both the dynamic and storage Young modulus are related to the degree of transformation progress induced by an applied prestress. By analysing the figures 6, 7 the applied stress levels can be approximately attributed to the beginning and the end of R-phase (94 MPa and 303 MPa respectively) as well as to the middle of the transformation progress where the maximum amount of interfaces is present corresponding to the maximum softening (143-218 MPa).

Fig. 6 Frequency evolution of dynamic Young modulus for different prestress levels

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Fig. 7 Frequency evolution of storage Young modulus for different prestress levels To illustrate visually the dependence of IFR on prestress one can refer to figure 8 showing the largest hysteresis at 218 MPa. Similar comparison displayed in figure 9 shows the dependence of IFR on frequency at a one prestress level where the hysteresis corresponding to 0.5Hz is the largest one compared to those at lower and higher frequencies.

Fig. 8 Evolution of hysteresis loops with increasing prestress

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Fig. 9 Evolution of hysteresis loop with increasing frequency The figures 10,11 show a general character of IFR evolution with respect to both the prestress and the frequency as it was identified. For a given prestress the IFR evolution in term of lost angle posses a local maximum at around 0.5Hz followed by an exponential decrease to a stabilised level. This general IFR frequency evolution is shifted in absolute values depending on the prestress level. The maximal absolute values were observed for the prestresses corresponding to the middle of the transformation progress. Since the mean stress level was kept constant during the frequency sweeping (see fig.12, the identified IFR with respect to the prestress level confirms the strong link between the dissipation and the movement of interfaces.

Fig. 10 Frequency dependence of loss angle for different prestress levels

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Fig. 11 Peaks in loss angle evolution observed within the frequency range 0-2Hz

Fig. 12 Evolution of the mean stress with the frequency 3.2.2

Effect of mechanical cycling

During the thermomechanical cycling some considerable changes in mechanical response of SMA are observed and it tends to be stabilised after certain number of cycles. These changes are generally attributed to the creation of new microstructural defects upon thermomechanical cycling. Moreover the mechanical response due to the B2-R is considered to be more stable compared to the complete superelastic cycle including B2-R-B19’ sequential transformation. Hence the effect of two type of mechanical cycling on the damping capacity was analysed. First the impact of cycling in the complete superelastic range was evaluated. The figure 13 shows the dramatic decrease of IFR peak observed on the damping evolution identified at the same amplitude but after 2000 complete superelastic cycles. However the cycling at the B2-R regime has no apparent impact on the damping frequency evolution as can be seen in fig.14. 12

Fig. 13 Effect of mechanical cycling on IFR measured with 0.3% of amplitude.

Fig. 14 IFR measured by using both the forward and reverse sense of frequency sweeping 3.2.3

Analysis of repeatability

The figure 15 shows the result of an analysis of repeatability which was done by performing several measurements of IFR on virgin samples and the result.

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Fig. 15 Box plot showing the reproducibility of IFR measurement 3.2.4

Influence of the sense of frequency sweeping

The figure 16 shows no significant influence of the sense of frequency sweeping on measured IFR . The observed difference falls within the measurement confidence interval discussed above.

Fig. 16 IFR measured by using both the forward and reverse sense of frequency sweeping 3.2.5

Effect of vibration amplitude

The effect of the strain amplitude on the damping capacity was studied in order to reveal the contribution of transition part of the damping (Q−1 T r ) related to the B2-R transformation. To R investigate such an influence, several IF measurements at different strain amplitudes were performed at a constant prestress corresponding approximately to the maximum damping level as identified from the analysis of the prestress influence (σ0 = 218 MPa). By comparing the hysteretic loops shown in figure 17, it is evident that the amount of 14

dissipated energy increases with increasing strain amplitude. That conclusion is confirmed by plotting the dissipated energy against the frequency for different strain amplitudes (fig.18). The increase of the dissipated energy is affected by two factors. First the enlargement of the movement of excited interfaces leads to an increase of phase transformation damping part (Q−1 P T ). Second influence making the dissipated energy higher is related to a change of volume fraction of the R phase due to proceeding B2-R and R-B2 transformation within a cycle. This R-phase volume fraction rate due to stress assisted B2-R-B2 transformation leads to an increase of the transition part of the internal friction (Q−1 T r ).

Fig. 17 Evolution of the hysteresis loop with increasing vibration amplitude

Fig. 18 Evolution of the dissipated energy with increasing vibration amplitude Although the dissipated energy increases with the strain amplitude the damping shown in term of tan δ (fig.19) doesn’t exhibit such an increase. That is due to the fact that the damping by its definition expresses the ratio of the dissipated energy to the stored 15

energy which also increases with amplitude as can be seen on frequency evolutions of the stress/strain amplitude (fig.20,21) and that of the storage Young modulus (fig.22).

Fig. 19 Frequency dependence of loss angle for different vibration amplitudes

Fig. 20 Frequency evolution of stress amplitude

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Fig. 21 Frequency evolution of strain amplitude

Fig. 22 Frequency evolution of storage Young modulus

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3.2.6

Analysis of the material response non-linearity

As discussed earlier the activation of transformation processes in the material contribute substantially to the overall damping capacity due to increase of transition part of the damping (Q−1 T r ). These microstructural processes can be detected by observing the degree on nonlinearity of the material which is a macroscopic manifestation of the transformation processes. To estimate the degree of nonlinearity of the meatrial response the harmonic amplitudes ratio was introduced expressing the ratio between the nth harmonic amplitude of the force response and displacement excitation as illustrated in fig.23.

Fig. 23 Analysis of the material response non-linearity The analysis of non-linearity was performed for all measured strain amplitudes and for several frequencies as can be seen in figure 24. The most noticeable evolution with respect to these two variables was observed in the case of the second harmonic. The presence of this harmonic as well as the others was revealed at frequencies corresponding to the local maximum of the damping. The evolution of the harmonic ratios was found to be an increasing function of the strain amplitude which confirms that the amplitude affects the amount of wire volume undergoing the transformation B2-R-B2 within a cycle.

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Fig. 24 Presence of higher harmonics in the response of material

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3.3

Damping related to the martensitic transformation

The damping capacity due to the stress induced R-B19’ transformation was analysed only with respect to the frequency and the strain amplitude. Hence the identified changes in the internal friction can be attributed mainly to its transition part (Q−1 T r ). The initial state of microstructure was defined by a prestrain corresponding to the middle of the plateau. The testing procedure was different from that used to analysed the R-phase damping and it consisted in prestraining the wire and in spreading a given frequency band by a sine stepping procedure using a constant displacement amplitude. Once the measurement at a given discrete frequency was terminated the following frequency was set up and several cycles were performed before taking the measured cycles in order to eliminate the transient part of vibrations. The figures 25, 26 illustrates the measurement technique by showing only the last cycles of each frequency which were used to deduced the damping. The identification of the damping and others important parameters was done by analysing the last measured cycles for each frequency. Due to the highly non-linear character of observed force responses, the measurement of a phase shift is not a relevant indicator of the damping and hence the equivalent viscous damping ζeq was used to evaluate the damping. It was calculated as follows: ζeq =

1 ∆W , 4π Ekmax

1 Ekmax = Fmax X(Fmax ) 2

(2)

, where ∆W is dissipated energy during one cycle calculated by a numerical integration of the force-displacement loop. Ekmax represents the maximum kinetic energy calculated by taking the maximum force Fmax and the corresponding displacement X(Fmax ).

Fig. 25 Illustration of the testing procedures - time signals measured during a test The width of the frequency band spread within a measurement was influenced by unstable vibrations occurring when a frequency limit depending on amplitude was exceeded. The figure 27 shows such unstable vibrations at 0.83% of amplitude occuring at 15Hz. The unstable behaviour during vibrations is thought to be caused by insufficient rigidity of the DMA frame resulting in the superposition of its dynamic movement to the wire vibrations.

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Fig. 26 Illustration of the testing procedures - stress-strain loops measured during a test

Fig. 27 Illustration of the dependence of vibration instabilities on the vibration amplitude occurring at 15Hz. 3.3.1

Effect of vibration amplitude

The figure 28 illustrates the role of the strain amplitude in the dissipative process taking place in the wire during the vibrations. For the low amplitudes the stress response involves only elasticity plus the dissipation due to movement of existed phase interfaces (Q−1 P T ). When the vibration amplitude increases the resulting stress is sufficient to induced the martensitic transformation which is reflected by the presence of plateau. The resulting enlargement of force-displacement loops suggests that this mechanism corresponding to the transition part of the internal friction (Q−1 T r ) represents the most important contribution to the overall internal friction.

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(a) f=0.05Hz

(b) f=0.5Hz

(c) f=5.21Hz

(d) f=6.96Hz

Fig. 28 Evolution of force-displacement loops with the amplitude of vibrations

Fig. 29 Equivalent viscous damping at the plateau level with respect to the frequency and the vibration amplitude By analysing the force-displacement loops the damping in term of equivalent viscous damping was deduced with respect to the frequency and the vibration amplitude (fig.29). The general frequency dependence shows an exponential decrease which stabilises at around 10Hz. Ab22

solute values of such a damping evolution is highly dependent on the vibration amplitude. It is apparent that a minimum amplitude lying between 0.33% and 0.5% is needed in order to activate the transformation which leads to a huge increase of damping. The maximum stabilized damping value is as large as 14%. However even the damping corresponding to the minimal tested amplitude is as large as 2% representing a non-negligible damping capacity. Since the testing procedure was based on the strain control regime, the stress response in terms of both the mean level and the amplitude was varying during the frequency spreading as can be seen in figures 30, 31. The observed frequency evolutions are affected by the imposed strain amplitude as well as by the heat processes linked to the dissipative heat and to the martensitic transformation during which the heat is either released (forward) or absorbed (reverse). With increasing frequency this transformation heat can be only partially taken from or absorbed by the surrounding environment and the rest of the heat contributes to the heating or cooling of the wire material which has not been transformed. That leads to an increase or decrease of the transformation temperatures relative to the material which is going to be transformed. This phenomenon is governed by the rate of the volume fraction change which itself is influenced by both the strain amplitude and the frequency. The change in the shape of the stress-strain curves (fig.28) and consequently the increase of stress amplitude (fig.30) might be directly attributed to this heat processes. The frequency evolution of the mean stress level should be also attributed to the heat processes because the substantial changes were observed only at higher strain amplitudes where the stress response of the material is highly affected by the martensitic transformation. The frequency evolution of the stress response might be explained as a set-up of balance between the heat generated/absorbed in the wire and that which can be transferred by the convection to the surrounding space.

Fig. 30 Frequency evolution of the stress amplitude

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Fig. 31 Frequency evolution of the mean stress level 3.3.2

Analysis of repeatability

The analysis of repeatability was performed by measuring 5 virgin samples at the amplitude of 0.5% and it shows (fig.32)a very satisfactory reproducibility of identified frequency evolutions of the equivalent viscous damping.

Fig. 32 Boxplot as a results of an analysis of repeatability showing satisfactory narrow statistical distribution of identified equivalent viscous damping 3.3.3

Effect of mechanical cycling

As it has been mentioned in the section dealing with the damping of the R-phase, the mechanical cycling induces microstructural changes affecting the mechanical response. Since the damping is also affected by these changes an analysis investigating the influence of the mechanical cycling in complete superelastic window was performed. The damping was first identified on a virgin sample and than on the same sample which had undergone several 24

thousands of complete superelastic cycles. The results shows that the most dramatic microstructural changes take place within first 2 thousands cyclic leading to a huge decrease of the damping in a frequency range up to 7Hz. Upon further cycling no other damping decrease was observed. The damping capacity above the 7Hz doesn’t seem to be affected by the mechanical cycling.

Fig. 33 Effect of mechanical cycling on the damping due to the martensitic transformation 3.3.4

Analysis of the material response non-linearity

The material response non-linearity was studied in order to estimate the degree of involvement of the martensitic transformation depending on the vibration amplitude and frequency. The same technique utilising the harmonic amplitudes ratio as that explained in the case of the R-phase damping analysis was applied. The results summarised in figure 34 shows in general higher degree of the material non-linearity commpared to that due to the R-phase transformation. Nevertheless it shows the same tendency to decrease with the frequency and amplitude which corresponds to the decrease of the frequency and amplitude evolution of the damping. That confirms the key role of the martensitic transformation in the dissipative mechanism when cycling at plateau region. The increase of non-linearity with the vibration amplitude illustrates the existence of an amplitude limit needed to activate the transformation processes leading to high damping capacity.

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Fig. 34 Presence of higher harmonics in the response of material

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Conclusion

The damping capacity of an ultra-thin superelastic NiTi wire was evaluated at room temperature using and in two regimes. First the damping due to the R-phase transformation was investigated and then the dissipation caused by the martensitic transformation was studied. In both cases the influence of the excitation frequency and the vibration amplitude were analysed. In addition the effect of prestress level was evaluated in the case of R-phase damping. The results provided the following conclusions: 1. R-phase transformation ˆ a peak in damping observed at around 0.4Hz (ζeq,max = 4.5%) ˆ damping stabilisation and a slight linear decrease as from 2Hz (ζeq ≈ 1.0 − 1.5%) ˆ damping capacity increases with the increasing softening of the Young modulus during the R-phase transformation ˆ higher vibration amplitude leads to a higher dissipated energy and only to a small increase of the damping ˆ large damping reduction due to the mechanical cycling in the complete superelastic range ˆ no impact on the damping due to the mechanical cycling within the R-phase transformation range

2. Martensitic transformation ˆ exponential decrease of the damping with the vibration frequency (ζeq (f = 0.05Hz, εA = 0.8%) = 40%) ˆ damping stabilisation at around 7 Hz (ζeq (7Hz, εA = 0.8%) = 15%) ˆ progressive increase of the damping with the vibration amplitude (ζeq (f = 0.05Hz, εA = 0.8%) = 40%/ζeq (f = 0.05Hz, εA = 0.3%) = 3%) ˆ a minimum vibration amplitude (≈ 0.3 − 0.4%) required in order to activate the martensitic transformation providing the increase of the damping capacity ˆ considerable damping capacity even at lower vibration amplitudes and higher frequencies (ζeq (f = 15Hz, εA = 0.3%) = 2%) ˆ important damping decrease due to the mechanical cycling in the complete superelastic range

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