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C. RemillatDepartment of Aerospace Engineering,

University of Bristol,

Queens Building,

University Walk,

BS8 1TR, UK

M. R. HassanDepartment of Mechanical Engineering,

University of Sheffield,

S1 3JD Sheffield, UK

F. Scarpa1

Department of Aerospace Engineering,

University of Bristol,

Queens Building,

University Walk,

BS8 1TR, UK

e-mail: [email protected]

Small Amplitude DynamicProperties of Ni48Ti46Cu6SMARibbons: Experimental Resultsand ModellingThis work illustrates viscoelastic testing and fractional derivative modelling to describethe thermally induced transformation equivalent viscoelastic damping of NiTiCu SMAribbons. NiTiCu SMA ribbons have been recently evaluated to manufacture novel honey-combs concepts (conventional and negative Poissons ratio) in shape memory alloys forhigh damping and deployable sandwich antennas constructions. The dynamic mechanicalthermal analysis (DMTA) test has been carried out at different frequencies and tempera-tures, with increasing and decreasing temperature gradients. Thermally induced trans-

formations (austenitic and martensitic) provide damping peaks at low frequency rangeexcitations. On the opposite, the storage moduli are not affected by the harmonic pulsa-tion. As the SMA ribbon increases its stiffness, the damping capacity reduces, and the loss

factor drops dramatically at austenite finish temperature. The fractional derivative mod-els provide a compact representation of the asymmetry of the peak locations, as well asthe storage modulus change from martensite to austenite phases.DOI: 10.1115/1.2204949

Introduction

Shape memory alloy SMA materials have always attractedsignificant interest in damping applications. Opposite to classicalpolymer-type materials, SMAs have significant stiffness proper-ties and are relatively insensitive to environmental hazard. One ofthe advantages of using SMAs in austenitic or superelastic state isrelated to the restoring force provided to return to the originalposition after an imposed deformation. From a general point ofview, damping properties of SMAs are due to thermally-inducedand stress-induced transformations, as well as intrinsic dampinggenerated in the coexisting phases. Dissipation originated by tem-

perature variation occurs during heating and cooling. The intrinsicdamping of the martensite phase is significantly high due to thereorientation of the martensite twin variants under stress. In theaustenite phase, intrinsic damping is low, although a martensite-induced phase transformation can be produced by high-appliedstress, leading to high-energy dissipation. Shape memory alloyscan be extremely useful in applications where damping capacity issufficiently elevated and damping stability is also required. Super-elastic damping capacities of SMAs have been studied by De-

jonghe et al.1on copper SMA. Piedboeuf and Gauvin2evalu-ated the dynamic properties of austenitic Nitinol wires underuniform strain rate and sinusoidal loading at different levels ofvibration amplitudes2%, 3%, and 4% strainand four decades offrequency up to 10 Hz, at 25C and 35C. Using an Anova andFratio analysis, they could verify the cross correlation and the

nonlinear viscoelastic properties of the SMA wire, and proposesemiempirical FFT models to describe the dynamic stress re-sponse and loss factor for the material. Gandhi and Wolons 3performed a similar analysis for NiTi wires subjected to a maxi-

mum of 5.48% cyclic peak-to-peak strain, at 50F, 90F, and

130F, with sinusoidal excitation up to 10 Hz. The results werepostprocessed in terms of the complex modulus approach, a for-

mulation extensively used in vibration damping applications toidentify the viscoelastic material and structure combination to in-

crease modal damping ratios close to resonance behaviour 4.While a consistent bulk of literature in damping of SMAs is

devoted to applications of pseudoelasticity or during superelasticphase, less attention has been drawn on the damping behavior at

small amplitude harmonic vibrations with varying temperatureranges. Small amplitude vibration damping in SMAs can be im-portant for example in aerospace applications, where combinedrandom broadband excitationstypical of boundary layer flow andacoustic fatigue loading and thermal excitation spectra are

present in a typical operational environment. Recently, Biscariniet al. 5 have identified extremely low values ofQ 1 factors forthin films of Ni30Ti50Cu20produced by vacuum induction meltingwhen doped with hydrogen, in the range of 0.045 for temperatures

from 275 K to 318 K. The samples were subjected to maximum

strains of 1107 to 3105 at excitation frequencies of

0.48 kHz and 1.5 kHz, respectively using a resonance techniquebased on clamped-free beam testing. A recent work on the small

amplitude dynamic properties of SMA ribbons has been illustrated

by Lu et al. 6, where Ti44Ni47Nb9 have been tested on a DMTAIV type instrument, at harmonic pulsations between 0.1 Hz and

10 Hz, with strain amplitudes ranging from 5106 to 1104.The samples were subjected to heating and cooling between

140C full martensite and 40C at complete austenite phase.A peak loss factor of 13% was measured during the transition

phase between 105C and 80C, with no significant depen-dence over the frequency excitation, in a similar manner recordedfor high frequency ranges in 5.

Several authors have performed the modelling of damping char-acteristics of SMAs. As already mentioned, the determination of

equivalent storage modulus and tan for pseudoelasticity hasbeen considered by Gandhi and Wolons 3, while Piedboeuf andGauvin used a FFT modelling approach 2. Malovrh and Gandhi7 have proposed piecewise linear, multiple friction chains andnonlinear spring models with and without offset to simulate thehysteresis loops at different dynamic strain amplitudes and ther-mal loading of NiTi wires. Oberaigner et al. 8 adapted theirmicromechanical model for NiTi alloys to simulate the behavior

1Corresponding author.

Contributed by the Materials Division of ASME for publication in the J OURNAL OF

ENGINEERING MATERIALS AND T ECHNOLOGY. Manuscript received September 7, 2005;final manuscript received January 24, 2006. Review conducted by Mohammed

Cherkaoui.

260 / Vol. 128, JULY 2006 Copyright 2006 by ASME Transactions of the ASME

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of a vibrating rod between elastic and inelastic regions. After sim-plifications like the presence of a spatial constant stress distribu-tion, combined dissipation, and latent heat and assuming constantseveral parameters related to the thermal conduction and mechani-cal properties, the mechanical energy ratios versus time and tem-perature and therefore the dissipation levels can be determinedusing a nonlinear finite difference scheme.

All the models described so far consider the elastic and inelasticbehaviour of the shape memory material. Classical micromechani-cal models in SMAs like the ones proposed by Tanaka 9, Liangand Rogers10, and Brinson 11 should only be used to predict

the superelastic behavior of NiTinol materials 12, although atfull martensite phase the Brinson model provides more accurateresults. The complex modulus approach3,4 is related strictly tothe equivalent viscoelastic properties of a solid, and its use onpseudoelastic behavior of SMA provides an overall indication ofdamping levels and storage stiffness values of the alloy.

In this work the small amplitude dynamic properties of a

Ni48Ti46Cu6 ribbon storage modulus and loss factor: are mea-sured using a dynamic mechanical thermal analyzer over a tem-

perature range from 10 C to 150 C, subjected to several har-

monic excitation over a decade, from 1 Hz to 60 Hz. The SMAribbons are subjected to a tension-tension loading with a constant

offset and constant dynamic strain of 5106. Both the offsetvalue and the dynamic strain are contained within the elastic rangeof the SMA phases. The type of ribbon tested is used to manufac-

ture a new concept of SMA honeycomb structures13to be usedas high-capacity damping core for sandwich structures and de-ployable antennas. The results are then prostprocessed in terms ofcomplex modulus using a fractional derivative model using a re-duced frequency parameter where the temperature is the solelyvariable. The fractional derivative modelling allows a very com-pact notation of the real and imaginary part of the complex modu-lus for viscoelastic materials. Although closed-form solution mod-els have been proposed in the past for polymers 14, thefractional derivative model allows avoiding the identification oflinear combinations of the Standard model or Prony series 15, aprocedure often used in viscoelastic polymer modelling.

The paper is organized as follows: The first part describes theproperties of the SMA ribbon and the types of mechanical anddynamic testing performed. The second part describes the resultsacquired during heating and cooling process, as well as the stor-

age modulus and loss factor values obtained during the frequencysweep of the mastercurve approach. The third part of the manu-script describes the fractional derivative model developed to simu-late the heating and cooling of the SMA ribbon.

SMA Properties and Testing

Ni46Ti48Cu6Properties.The SMA material used consisted on aribbon manufactured by @Medical Technologies n.v. productcode: SMENTC05OX0.206.40 mm using a thin rollingprocess technique. The transformation temperature of the ribbonwas determined by differential scanning calorimeter DSC andgiven as data specification for the material batch. The martensite

temperatures Mf and Ms were, respectively, 38C and 50C,while the austenite phase was defined by the temperatures As and

Afbeing, respectively, 58C and 76C. At room temperature, thematerial therefore behaved in full martensite structure. Mechani-cal tensile tests conducted with a Testometric M350 machine with

a 5 kN load cell allowed to identify the Youngs modulus andyielding points of the martensite and austenite phases. At room

temperature full martensite the Youngs modulus was 13 GPa,with a yield stress 0 of 170 MPa at 1.1% of strain. At 105C

full austenite phasethe Youngs modulus was 30 GPa, with yieldstress 0 of 320 MPa at 1.8% of strain. The samples were trainedloading up to 8% of strain at room temperature. After load release,the residual strain was around 4%. The ribbons were heated then

until full austenite phase, and the whole process repeated 30times. The training process was performed on the same DMTAused for the equivalent viscoelastic testing.

DMTA Testing. The DMTA machine consisted in a ME-TRAVIB VA2000 Viscoanalyser equipped with a load cell of

100 N, able to support a maximum sinusoidal excitation fre-

quency of 250 Hz ranging from 0.001 Hz. Three types of testwere performed with the DMTA: progressive heating while the

the ribbon was subjected to sinusoidal excitation at 5 Hz, progres-sive cooling at the same cyclic loading, and a master curve test for

frequency range from 1 Hz to 60 Hz.For the first test, the ribbon was tested under tension-tension

loading at the prescribed harmonic pulsation within a temperature

range between 10C and 110C. The ribbon was placed in thetest chamber and fixed with a custom-made aluminium clamp. Toensure an improved fixture and thermal insulation between thealuminium of the clamp and the ribbon itself, a coat of epoxyresin glue has been applied 6. The ribbon was preloaded at roomtemperature with a 5 N tensile load, corresponding to a tensilestrain of 0.03% at full martensite phase. The temperature was then

brought down to 10C at a rate of 5C/min. The data were col-

lected for the test from 10C to 110C maintaining the same con-

stant heating rate of 5C/min. The ribbon was constantly excited

with sinusoidal excitation of 5 Hz and constant dynamic strain of5.6e-4% during the whole test. The dynamic strain was also moni-tored, showing a constant behavior with maximum It has to be

pointed out that the test chamber temperature was kept uniformacross the volume using a fan device, while the temperature au-tomatically monitored and controlled through thermocouple.

The second test was started immediately when the heating pro-

cess stop at 110C. It must be done using the same sample inorder to get the complete cycle of heating and cooling process ofthe ribbon. Also in this case, the data were collected using a cool-

ing rate of 5 C /m from 10C to 110 C. The third test was per-formed to obtain a master curve for the SMA ribbon. From theprevious tests, it was noticed a significant increase in the loss

factor during transition phase, between 50C and 60C. It wastherefore decided to narrow the mastercurve temperature range

between 10 C and 75 C, with discrete frequency values at 1 Hz,

1.5 Hz, 2.26 Hz, 3.41 Hz, 5.14 Hz, 7.74 Hz, 11.66 Hz, 17.56 Hz,

26.4 Hz, 39.84 Hz, and 60 Hz. The values were selected because

frequency was measured in log value. At each temperature step5 C the ribbon was subjected to that frequencies. Between each

sinusoidal loading there was in interval of 1 min between fre-quency steps before the new mechanical loading started.

The values of the storage modulus were then evaluated using:

ES=kFcFf 1

where kis the measured stiffness N/m, Fc is the correction fac-tor,

Fc= cos 2

where is the phase angle. Ffis the stress concentration factor,

Ff= h/Se 3

where h is the specimen height and Se = b tis the specimen cross

section. The loss factor or tangent modulus has been computedusing the classical formulation 4

tan =EI

ES4

where EIis the imaginary part of the SMA complex modulus, and

ESthe storage modulus. The results were postprocessed using theDYNATEST software associated to the DMTA facility.

Results and Discussions

Figures 1 and 2 shows the quasistatic tensile tests performed onthe ribbon at full martensiteFig. 1and austeniteFig. 2phases.

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The linear elastic portion of the curves during the loading phase

shows Youngs moduli of 8.7 GPa 25 C and 23.1 GPa105 Cfor the martensite and austenite of the SMA ribbon con-sidered, respectively. These values are significantly lower com-pared to the ones provided in the data specs, with a percentagedecrease of 42% and 23%, respectively. The stiffness ratio be-tween austenite and martensite phase assumes now the value of2.65, from the value of 2 provided by the data sheet.

Figure 3 shows the storage modulus and loss factors of theribbon during heating process and subjected to a sinusoidal load-

ing of 5 Hz. The storage modulus and loss factors values havebeen calculated using Eqs. 1 and 4; the storage modulus, inparticular, is proportional to the effective Youngs modulus of theSMA material by a factor of 0.64. It can be noticed that the stor-

age modulus tends to decrease after the As temperature, with a

significant drop during the transformation phase. In particular, two

peaks of the storage modulus are present around As and Af tem-peratures. The storage modulus tends to increase further until fullaustenite, with a gradual build up of the real part of the complex

modulus until 150C. The loss modulus features some interestingcharacteristics. The loss factor shows an almost constant dampingvalue of 2%, with a subsequent steep increase at around 6% ataustenite start. In principle, a second smaller peak could be ar-guably identified around Af temperature. The loss factor at fullaustenite has average values around 1%, well below the ones ofthe martensite phase, as illustrated in literature 2,5,6,9. A stepincrease of loss factor up to 6% is recorded during the transition

phase, while a smaller peak could be identified always around Aftemperature. The ratio of storage modulus between austenite andmartensite phase is 2.6, well in line with the ratio of Youngs

Fig. 1 One-cycle quasistatic test on SMA ribbon-full martensite

Fig. 2 One-cycle quasistatic test on SMA ribbon-full austenite

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modulus between austenite and martensite phase recorded duringthe quasistatic tests. Figure 4 shows the same type of test of Fig.3 during cooling. The storage modulus is almost constant for all

the austenite phase, starting to decrease between Af and Ms tem-peratures, with a steep reduction in martensitic transition from Msto Mftemperature. At full martensite phase, the storage modulusis constant, giving a modulus ratio of 5. The loss factor present anaverage value of 1% in the austenite phase, with a gradual de-crease during austenitic transformation and steep peak duringmartensitic one, with maximum value around 6% as in the heatingphase. During martensite form, the loss factor assumes an averagevalue of 2% over the temperature range. It is worth noticing thatthe presence of the peak loss factor coincides with the temperaturepoint approximately corresponding to the zero second derivativeversus temperature of the storage modulus, as it occurs in classicalviscoelasticity 15,16. Figure 5 shows the results of the mastercurve test on the storage modulus of the ribbon during the heatingprocess at different frequencies of excitation. As it can be

noticedand expected to a large extendthe storage modulus isalmost independent of the frequency excitation, also during theaustenitic transition phase. We show for completeness the behav-ior of the loss factor in a master curve approach in Fig. 6. The loss

factors corresponding to the first six frequencies from

1 Hz to 7.74 Hz are shown with a star mark, while the othervalues are represented with a round mark. The behavior of the loss

factor for the first six frequencies of excitation is quite similar,

with a major peak around transition phase, and decreased value at

the austenite form. For the subsequent frequencies, the loss factor

at martensite increases significantly, while not presenting clear

peak behavior during the transformation phase.

The presence of storage modulus and loss factors peaksFigs. 3

and 6 during heating have been recorded as transition peaks PH1and PH2 in other NiTiCu alloys 5, although transition effects

tend to disappear above frequencies of 1 Hz. In this case, not only

the peaks are present at higher frequency of excitation Fig. 3,but also they exist in the storage modulus for higher harmonic

pulsations Fig. 5. The behavior of the loss factor during heatingis quite similar for the first five frequencies of excitation, with the

presence of a peak during austenitic transition. However, for

higher pulsations this behavior is not respected, with increase ofthe loss factor during martensite and no presence of peaks during

transition, while the curves do not show regular patterns as in the

lower frequency range. This behavior would suggest that transient

Fig. 3 Storage modulus and loss factor versus temperature-heating process

Fig. 4 Storage modulus and loss factor versus temperature-cooling process



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effects effectively generate the peaks during transition. However,

above 10 Hz the DMTA used experienced some resonance prob-lems due to the dynamic stiffness of the head designed to testsofter materials. While this resonance behavior does not particu-larly affect the real part of the complex modulus extracted usingEq. 1, the phase shift between real and imaginary part of themechanical response is affected. In fact, the loss factor changessignificantly even during full martensite, opposite to the stable

behavior shown at excitation frequencies up to 7.74 Hz. This fact

suggests that the measurements taken from 11 Hz, especially forthe loss factor, could be prone to experimental error, and shouldbe considered with care.

Although the cooling curve shown in Fig. 4 features all thecharacteristics of regularity and smoothness typical of classicalviscoelastic testing, the final storage modulus corresponding to

full martensite is significantly lower compared to the one present

in Fig. 3, with a decrease of almost 50%, although the dynamicstrain applied was constantly monitored and did not show any signof clamp slip. We are not able to provide explanation of thisphenomenon at this stage. We observe that, considering the con-stant harmonic cycling and the temperature rate applied, the rib-

bon was subjected to a total of 12,000 cycles at the end of themartensite phase during cooling, leading to a possible modifica-tion of residual stresses on the sample at the end of the heating/cooling process. However, the average loss factors at martensiteand austenite phase are substantially the same for the two pro-cesses, indicating that motions of twin boundaries for the highdamping in martensite was still present both after heating andcooling 5. However, the curves shown in Fig. 6 feature anequivalent viscoelastic characteristic of the storage modulus

Fig. 5 Storage modulus versus temperature and frequency-heating process

Fig. 6 Storage modulus and loss factor versus temperature and frequency-heating process. - frequencies from 1 Hz to 7.74 Hz. , frequencies from11.4 Hz to 60 Hz.



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equivalence temperature/frequency, and the presence of a lossfactor peak that would suggest the possible use of modelling tech-

niques used for classical polymeric viscoelastic materials.

Fractional Derivative-Based Model

A basic fractional derivative model used in temperature/frequency equivalence is given by 15

E* =a1+b1i2 fT

1 + c1i2 fT

5

In this relation, E* is a complex number, while a 1, b 1, and c1 are

quantities, not necessarily complex. The quantity fT is the re-duced frequency of the master curve model 15. When the re-duced frequency approaches to zero, the term a1 represents the

lowest fT asymptote of the complex modulus for the lowest

frequency and/or highest temperature. In a similar way, when the

reduced frequency approaches infinity, the ratio b1/c1 approachesthe upper asymptote offTviscoelastic response, correspondingto high frequency and/or low temperature. In the case of theequivalent viscoelastic properties of the SMA ribbon, the fre-

quency dependence is negligible, therefore Eq. 5can be recast interms of pure temperature reliance. Typical values of the coef-

ficient vary between 0.4 and 0.7 15. The standard viscoelasticmodel 15 features a value of equal to 1. Other typical valuesof the coefficients a1 and b1 for polymeric materials range from

200 to 700, while the coefficientc1 is always well below the unity15. To consider the specific loss factor behavior for the SMA,i.e., the nonsymmetric performance at full martensitearound 2%and austenite phase 1%, the fractional derivative model pro-posed is the following:

E* =E01 + 0.02i

1 + mr1 + ima i10T10T0

1 + i10T

10T1

6

In this case,E0

is the value of the storage modulus at full marten-site phase, with a loss of 2% corresponding to that specific crystal

lattice structure. It has to be noticed that the term E01+0.02i

corresponds to a 1 in Eq. 5. The term b 1,

Fig. 7 Fractional derivative representation of storage modulus and loss factorfor the SMA ribbon-cooling process

Fig. 8 Fractional derivative representation of storage modulus and loss factorfor SMA ribbon-heating process



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b1=mr1 + ima

E01 + 0.02i 7

is complex, and allows us to represent the nonsymmetric peak

location of the loss factor. The values for mr and m a are, respec-

tively, 0.042 and 0.01 for the cooling process. The temperatures

T0 and T1 represent the transition temperatures, being, respec-

tively, 38C and 50C. To simulate the heating process, the co-efficient should be decreased by one order of magnitude to

0.0045, and the transition temperatures to 58C and 76C, re-

spectively. The coefficient is equal to 0.15. Figure 7 shows theplot of the storage modulus ratio and loss factor from Eq. 6versus a temperature range for cooling, considering a stiffness

ratio of 2.6 similar to the one from the quasistatic tests. The curvefor heating process is represented in Fig. 8. One can notice thechange of transition temperatures, and the substantial physicalsimilarities that the fractional derivative models allow us to simu-late in a compact form with the experimental results. Figure 9provides a direct comparison between the experimental and ana-lytical values related to the heating process, showing a consistentagreement between the fractional derivative approach and themastercurve data.

Conclusions

In this work small amplitude master-curve-type tests have beenperformed on a NiTiCu ribbon to extract equivalent viscoelasticcharacteristics for broadband low amplitude damping applications

up to 60 Hz of excitation, and heating/cooling from10C to 150C. The storage modulus and loss factors measuredusing DMTA equipment did not show frequency dependence,while changes in stiffness of ratios similar to the ones recordedduring quasistatic tests were observed from full martensite to aus-tenite phase. The loss factors showed significant peak increase

during transition, under harmonic pulsations up to 7.74 Hz. Acompact fractional derivative-based model has been developed tosimulate the equivalent viscoelastic properties of the SMA ribbon.

The storage modulus peaks present during transition were re-corded during the whole frequency sweep in the master curve test.However, the results related to the loss modulus are not conclu-sive, because of the irregularities given by resonance of the test

equipment used above 10 Hz. Further analysis is needed to im-prove the testing at higher frequency ranges, to improve the as-sessment of SMA ribbons to enhance possible implementations inbroadband white noise excitation applications 17,18.

Acknowledgment

This work has been partially supported by the Engineering andPhysical Science Research Council Grant No. EPSRC GR/R/97313. The authors would like to thank Mr. Les Morton for hisassistance during the DMTA testing. Special acknowledgementgoes also to several colleagues: Dr. M. Collet from CNRSFEMTO-ST of University of Besanon, France, Professor F. Maz-

zolai from University of Perugia, Italy, Dr. Farhan Gandhi fromPenn State, and Professor R. Lakes from University of Wisconsin-Madison, for their useful suggestions and discussions.

References1 Dejonghe, W., Delaey, L., De Batist, R., and Van Humbeeck, J., 1977, Tem-

perature and Amplitude Dependence on Internal Friction in Cu-ZnAl Alloys,

Metall. Trans. A, 24A, pp. 21892194.2 Piedboeuf, M. C., and Gauvin, R., 1998, Damping Behaviour of Shape

Memory Alloys: Strain Amplitude, Frequency and Temperature Effects, J.

Sound Vib., 2145, pp. 885901.3 Gandhi, F., and Wolons, R., 1999, Characterization of the Pseudoelastic

Damping Behaviour Of Shape Memory Alloy Wires Using the Complex

Modulus, Smart Mater. Struct., 8, pp. 4956.4 Sun, C. T., and Lu, Y. P., 1995. Vibration Damping of Structural Elements,

Prentice-Hall, Englewood Cliffs, NJ.5 Biscarini, A., Coluzzi, B., Mazzolai, G., Tuissi, A., and Mazzolai, F., 2003,

Extraordinary High Damping of Hydrogen-Doped Niti and Niticu ShapeMemory Alloys, J. Alloys Compd., 355, pp. 5257.6 Lu, X. L., Cai, W., and Zhao, L. C., 2003, Damping Behaviour of a

Ti44Ni47Nb9 Shape Memory Alloy, J. Mater. Sci. Lett., 22, pp. 12431245.7 Malovhr, B., and Gandhi, F., 2001, Mechanism-Based Phenomenological

Models for the Pseudoelastic Hysteresis Behaviour of Shape Memory Alloys,

J. Intell. Mater. Syst. Struct., 12, pp. 2130.8 Oberaigner, E. R., Tanaka, T., and Fischer, F. D., 1996, Investigations of the

Damping Behaviour of a Vibrating Shape Memory Alloy Rod Using a Micro-

mechanical Model, Smart Mater. Struct., 3, pp. 456463.9 Tanaka, K., 1986, A Thermomechanical Sketch for Shape Memory Alloy

Effect: One-Dimensional Tensile Behaviour, Res. Mech., 18, pp. 251263.10 Liang, C., and Rogers, C. A., 1990, One Dimensional Thermomechanical

Constitutive Relations for Shape Memory Material, J. Intell. Mater. Syst.

Struct., 1, pp. 207234.11 Brinson, L. C., 1993, One-Dimensional Constitutive Behaviour of Shape

Fig. 9 Comparison between fractional derivative model and experimental re-sults during the heating process. , experimental values; , analytical model.



5/22/2018 260_1

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Memory Alloys: Thermomecanical Derivation With Non-Constant MaterialFunctions, J. Intell. Mater. Syst. Struct., 4, pp. 229242.

12 Zak, A. J., Cartmell, M. P., Ostachowicz, W. M., and Wiercigroch, M., 2003,One-Dimensional Shape Memory Alloy Models for Use With Reinforced

Composite Structures, Smart Mater. Struct., 12, pp. 338346.13 Hassan, M. R., Scarpa, F., and Mohamed, N. A., 2004, Shape Memory Alloy

Honeycomb: Design and Properties, Proceedings of the SPIE Smart MaterialStructure Conference, paper 5387-99, San Diego, 1118 March, CA.

14 Havriliak, S., and Negami, S., 1966, A Complex Plane Analysis of

-Dispersion in Some Polymer Systems, J. Polym. Sci., Part C: Polym.

Symp., 14, pp. 99117.

15 Jones, D. I., 2002, Handbook of Viscoelastic Vibration Damping, Wiley,

Chichester, UK.

16 Lakes, R. S., 1999, Viscoelastic Solids, CRC Press, London.

17 Metravib, V. A., 2000, Viscoanalyser Manual, 2001, Metravib Spa, Lyons, FR.

18 Mazzolai, F. M., Biscarini, A., Campanella, R., Coluzzi, B., Mazzolai, G.,

Rotini, A., and Tuissin, A., 2003, Internal Friction Spectra of the

Ni40Ti50Cu10 Shape Memory Alloy Charged With Hydrogen, Acta Mater.,

51, pp. 573581.



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