non-isothermal oscillations of pseudoelastic devices

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International Journal of Non-Linear Mechanics 38 (2003) 1297–1313

Non-isothermal oscillations of pseudoelastic devicesDavide Bernardini, Fabrizio Vestroni ∗

Dipartimento di Ingegneria Strutturale e Geotecnica, Universit�a di Roma “La Sapienza”, Via Eudossiana 18, 00184 Roma, Italy

Abstract

The non-linear dynamic response of a pseudoelastic oscillator embedded in a convective environment is studied taking intoaccount the temperature variations induced, during oscillations, by the latent heat of transformation and by the heat exchangewith the surroundings. The asymptotic periodic response under harmonic excitation is characterized by frequency–responsecurves in terms of maximum displacement, maximum and mean temperature. The periodic thermomechanical response iscomputed by a multi-component harmonic balance method implemented within a continuation algorithm that enables to traceout multivalued frequency–response curves. The accuracy of the results is checked by comparison with the results of thenumerical integration of the basic equations governing the dynamics of the system. The response is investigated for variousexcitation amplitude levels and in various material parameters ranges. The resulting picture of the mechanical responseshows, in some cases, features similar to other hysteretic oscillators, while, in other cases, points out peculiar behaviors. Itturns out that the temperature variations induced by the phase transformations in1uence the mechanical response and thatthe results obtained under the simplifying assumption of isothermal behavior can be rather di2erent from those obtained ina fully thermomechanical setting. ? 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Shape memory alloys; Hysteretic models; Non-linear dynamics; Harmonic balance

1. Introduction

The pseudoelastic behavior of shape memory ma-terials (SMM) can be pro8tably exploited to realizeinnovative vibration reduction devices characterizedby the absence of residual displacements after un-loading, satisfactory energy dissipation capability andhigh durability [1–3]. In such applications, the materialbehavior is hysteretic and involves a strong couplingbetween thermal and mechanical aspects. Pseudoelas-ticity is due to the occurrence of solid phase transfor-mations between Austenite (A) and multiple variantsof Martensite (M). In particular, the mechanical

∗ Corresponding author. Tel.: +39-0644-585198; fax: +39-0648-84852.E-mail addresses: [email protected] (D. Bernardini),[email protected] (F. Vestroni).

loading and unloading can induce, respectively, theexothermic A → M and the endothermic M → Atransformations. At the macroscopic level such trans-formations give rise to an hysteresis loop in the stress–strain plane [4]. The heat released and absorbed duringthe transformations is partly exchanged with the sur-rounding environment and partly contributes to changethe temperature of the material. The magnitude ofthese temperature variations increases with the load-ing rate and may signi8cantly a2ect the plateau slopesand the hysteresis loop shape (Fig. 1) enough to be-come a relevant aspect in the application design [2].When dealing with dynamic applications temperaturevariations have thus to be taken into account in themodels.The modelling of the macroscopic behavior of

SMM is the subject of an active research activity

0020-7462/03/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved.PII: S0020 -7462(02)00065 -3

1298 D. Bernardini, F. Vestroni / International Journal of Non-Linear Mechanics 38 (2003) 1297–1313

Fig. 1. Schematic isothermal and non-isothermal pseudoelasticity with indication of the start and 8nish of the phase transformations:A → M (points Ms; Mf ) and M → A (points As; Af ).

since the last two decades and several models areavailable [5–8]. However, most of them neglect ther-mal aspects and focus on isothermal response. Amongthe few exceptions, the hysteretic model proposed byIvshin and Pence [9,10] enables to reproduce the fullthermomechanical behavior under arbitrary loadingpaths. A variant of the Ivshin and Pence’s model,more directly focused on the SMM devices, has beengiven in [11]. The latter work uses a di2erent thermo-dynamic formulation that leads to a richer descriptionof the thermal aspects and introduces di2erent mate-rial parameters more suited for devices modelling.Since most of the application of SMM have been

so far essentially static, the dynamic response of sys-tems based on SMM have received little attentionin the literature, whereas it is important in view ofthe dynamic applications. In the works [12,13], thenon-linear dynamic behavior of pseudoelastic oscilla-tors has been studied by means of an isothermal con-stitutive model for SMM hence neglecting the e2ectsof the thermal aspects. In [14] a more re8ned consti-tutive model is used but only a few aspects of the dy-namic response are studied. An attempt to discuss thenon-isothermal non-linear dynamics through a ther-momechanical model was made in [15] by using asimpli8ed model and discussing a few aspects.In this work, the non-linear dynamic non-isothermal

response of a pseudoelastic oscillator embedded in aconvective environment is studied by the model de-scribed in [11]. The response of the system is charac-terized by the displacement and the temperature thatare subjected to proper balance equations. The consti-tutive model allows for the latent heat of transforma-tion and the heat exchange with the environment and

thus enables the prediction of the temperature vari-ations observed during oscillations. After a transientwith features dependent also on the heat exchangemodalities, the asymptotic response of the system un-der harmonic excitation turns out to be periodic in awide range of material parameters. The observed pe-riodicity of the asymptotic response enables one tocharacterize the dynamics by the frequency–responsecurves in terms not only of maximum displacement butalso of maximum and mean temperature. The compu-tation of the curves is performed by a multiple compo-nent harmonic balance technique [16,17] in which thecoeHcients of the Galerkin approximation of the func-tions involving hysteretic components are obtained nu-merically by taking the fast Fourier transform of thecorresponding time histories [18,19]. A numerical pro-cedure for the construction of the frequency–responsecurves has been implemented on the basis of a con-tinuation algorithm.

2. Basic equations

The dynamical system under consideration is apseudoelastic oscillator of mass m embedded in a con-vective environment at temperature #E and excitedby an harmonic force F(t) = F cos�t (Fig. 2).The restoring force is provided by a pseudoelastic

device (fpe) and a viscous dashpot (fv). During os-cillations, the mass and dashpot are assumed to remainat constant temperature #E, whereas the pseudoelasticdevice can change its temperature # due to the latentheat of transformation and the heat exchange with theenvironment. Since the thermomechanical response of

D. Bernardini, F. Vestroni / International Journal of Non-Linear Mechanics 38 (2003) 1297–1313 1299

Fig. 2. Arrangement of the pseudoelastic device (a) and layout of the pseudoelastic oscillator (b).

the system is of concern, the equation of motion, thatexpresses the linear momentum conservation, has tobe complemented by an additional equation express-ing the energy balance [11]

m Kx = F cos�t − fpe − fv; (1a)

#�= Q + : (1b)

Where Q is the rate of heat exchange with the en-vironment, positive if absorbed by the system, � theentropy and the rate of energy dissipation. Ac-cording to Clausius–Duhem inequality, must bealways non-negative. In order to have a completeset of equations, constitutive relations are needed forfpe; fv; �; ; Q that are speci8ed in the following.

A pseudoelastic device is understood as an ar-rangement of shape memory material components,e.g. wires or bars, that connects two points A and B,whose relative displacement uA − uB is denoted byx. All the components are assumed to be at the sametemperature # (Fig. 2).The fraction of martensitic phase in the whole de-

vice �∈ [0; 1] is used to describe the evolution of thephase transformations. When � = 0 the device is ina fully austenitic state (A), whereas, when � = 1, ina fully martensitic state (M). Peculiar feature of themechanically-induced phase transformations in SMMis that they involve length variations. In particular, alength increase is observed when � increases (A →M, Forward transformation) and vice versa. This ef-fect is inherent in the material microstructure and istaken into account by the material parameter

�:=x(�=1) − x(�=0); (2)

which expresses the displacement variation due to acomplete phase transformation.

The constitutive equations of the device areassumed in the following form:

fpe = Mf(x; #; �); �= M�(x; #; �); �= M�(x; #; �);

�=M(x; #); (3)

where Mf; M�; M� are almost everywhere smoothmappings R× R+ × [0; 1] → R and M an hysteresisoperator [20] that describes the evolution of the phasetransformations. Standard thermodynamic argumentscan be used to show that constitutive equations forforce and entropy cannot be arbitrary in the classde8ned by (3) but must be in the following reducedform:

Mf =@ M�@x

; M�=−@ M�@#

: (4)

Moreover, the rate of energy dissipation turns outto be determined by M� and �

=−@ M�@�

�¿ 0: (5)

The derivatives of M� have to be understood, whennecessary, in the sense of distributions. Taking intoaccount (4) and (5), the constitutive description iscomplete as soon as the free energy function M� andthe hysteresis operator M are given. According to thegeneral structure of the free energies for SMM [25],the following function is taken as basic ingredient forthe device modelling:

M�(x; #; �) :=K2(x − sgn (x)��)2

+ c(#− #0 − # ln

##0

)

+(#− #0)b��+ a0 − b0#; (6)


where K ¿ 0 is the elastic sti2ness, �¿ 0 the max-imum transformation displacement de8ned in (2),c¿ 0 the heat capacity, b¿ 0 the slope in thetemperature-transformation force plane (see the ap-pendix), a0 and b0 the internal energy and entropyof the device in fully austenitic state at the referencetemperature #0.On the basis of expression (6), the constitu-

tive equations for force and entropy directly followfrom (4)

fpe = K(x − sgn (x)��); (7a)

�= c ln##0

− �b�+ b0; (7b)

where sgn (x)�� is the pseudoelastic part of the to-tal displacement and the signum function has beenintroduced to reproduce a symmetric response that,although the material itself commonly shows strongasymmetry, is often exhibited by the devices. Asshown in [11], the evolution of the phase transforma-tions can be described by the di2erential equation

�=− b�G(�; �; sgn �)

1 + K�2G(�; �; sgn �)� (8)

in which the dependence on x and # takes placethrough the thermomechanical loading parameter

�(x; #):=#− bK |x|: (9)

The solution of (8) de8nes a Duhem hysteresis oper-ator M with � as input and � as output. The constitu-tive function G describes the features of the forward(A → M) and reverse (M → A) phase transforma-tions and, in general, can be determined experimen-tally or expressed in various analytical forms. Here,the following expression is adopted in analogy to [10]:

G(�; �; sgn �):=

k1(1− �)(1 + tanh[k2 − k1�]) if sgn � =−1;

k3�(1− tanh[k4 − k3�]) if sgn � = 1;

(10)

where �(�; �):=b�(�+ K��=b− #0) and the ki’s arematerial parameters governing the hysteresis loopsshape. Since k1; k3 ¿ 0 then G¿ 0 and 1+K�2G¿ 0.

The heat exchange with the environment is de-scribed by

Q = h(#E − #); (11)

where h¿ 0 is the heat exchange coeHcient. The vis-cous restoring force is assumed, as usual, linear in thevelocity, i.e. fv = Cx.All the elements of the model have been now speci-

8ed. By means of (7a)–(11), the two di2erential equa-tions (1b) and (8) can be rearranged in the form

c#+ Z2�= h(#E − #); (12a)

[− sgn (x)(K=b)Z1]x + Z1#+ �= 0; (12b)

where, for the sake of simplicity, the following nota-tion has been introduced:

Z1(�; �; sgn �):=b�G

1 + K�2G;

Z2(�; �; sgn �):=− K�(|x| − ��)− #0b�: (13)

Eqs. (12a) and (12b) govern, respectively, the evo-lution of the temperature and of the phase fraction.They have to be added to the equation of motion(1a) in order to obtain the system of equations thatdescribes the non-isothermal dynamics of thepseudoelastic device. Since the equations are cou-pled, temperature variations during oscillations canoccur, as expected. After a further rearrangement, thesystem may be conveniently reduced to a 8rst-orderexplicit form x = f(x; #). More particularly,

x = v;

v= (F=m) cos�t − (K=m)(x − sgn (x)��)− (C=m)v;

#=1

c − Z1Z2

[− sgn (x)KZ1Z2

bv+ h(#− #E)

];

�=− Z1c − Z1Z2

[− sgn (x)Kc

bv+ h(#− #E)

]; (14)

where c − Z1Z2 ¿ 0 due to the sign of G and of theother parameters.In order to better capture the essential features

of the response, system (14) has been written ina non-dimensional form by assuming, as referencecondition, the onset of A → M transformation atthe temperature #R. In such conditions force and


displacement are denoted by fMs and xMs (Fig. 1),with fMs =KxMs . The temperature #R may be chosenamong the values at which the device exhibits pseu-doelastic response and should not be confused with thetemperature #0 in the free energy (6). Accordingly,the non-dimensional counterpart of (14) is

x = v;

v= F cos %#− x + sgn (x)&�− 2'v;

#=1

1− Z1Z2[− sgn (x)Z1Z2v+ h(#− #E)];

�=− Z1

1− Z1Z2[− sgn (x)v+ h(#− #E)]; (15)

where the non-dimensional variables

x:=xxMs

; #:=##R

; v:=xMsv; #:=Km

t; (16)

parameters

&:=�xMs

; J :=fMs

b#R; L:=

b�c; h:=

hmcK

;

%:=�mK

; ':=C2K

; F :=FfMs

; k1:=b�#Rk1;

k3:=b�#Rk3 (17)

and quantities

G:=Gb�#R ; Z1:=G

1 + J&G;

Z2:=− LJ (|x| − &�)− L#0 (18)

have been introduced. In (15) the superposeddot denotes di2erentiation with respect to thenon-dimensional time #. Note that the parameters k2and k4 are already nondimensional.

Although useful in the de8nition of G (Eq. (10)),the ki’s do not have a direct physical interpretation andtherefore it is useful to express them in terms of thephysically more representative parameters introducedin [11] and recalled below. Denoting by fMs , fMf ,fAs and fAf the force levels at which the A → Mand M → A transformations, respectively, start and8nish at the temperature #R (Fig. 1), the following

non-dimensional ratios can be de8ned:

q1:=fMf

fMs

; q2:=fAf

fAs

; q3:=fAs

fMs

: (19)

They give a synthetic characterization of the basichysteresis loop features: in particular, q1 and q2 cor-relate with the slope of the upper and lower pseu-doelastic plateaus (q1¿ 1 and 06 q26 1), while q3with the hysteresis loop width. Since the width of theloop have to be bounded by the limit case of zerohysteresis (fAs =fMf that corresponds to a non-linearelastic behavior) and by the fact that the value of theforce at the end of the cycle, fAf , cannot be nega-tive (because otherwise pseudoelastic e2ect would belost due to residual displacements), then q3 must obeyq1−16 q36 q1. Taking into account (19) and intro-ducing r:=arctanh(1−2%r) with %r being the residualAustenite at the completion of the A → M transfor-mation:

k1 =2r

J (q1 − 1); k2 = k1(1− #0)− r

q1 + 1q1 − 1

;

k3 =2r

(1− q2)q3J; k4 = k3(1− #0)− r

1 + q21− q2

(20)

so that the ki’s can be replaced by the qi’s and %r asparameters describing the hysteresis loop shape.Since all the subsequent development will be fo-

cused on the non-dimensional formulation, the under-line will be omitted henceforth.

3. Harmonic balance solution

The asymptotic periodic response of the pseudoe-lastic oscillator is sought by the harmonic balancemethod [16,17]. Given displacement and temperaturetime-histories x(#) and #(#) and an initial conditionfor �, the fourth equation in (15) can be solved for�(#) and (15) can be rewritten as follows

Kx = f1:=F cos %#− x + sgn(x)&�− 2'x;

#= f2:=h(#E − #)− Z2� (21)

provided �(#) = M(�0; x(#); #(#)). The thermome-chanical response of the pseudoelastic oscillator isthen described by x(#) = [x(#); #(#)]T. Let x be anelement of the space P2

T of the pairs of continuous


periodic functions of period T=2-=%,H an integer ande2N , with N=2H+1, the ordered set of trigonometricfunctions

e2N :=[eN ; eN ]T; eN :=[1=2; cos k%#; sin k%#]T

k = 1; : : : ; H: (22)

The trigonometric polynomial

xN :=qTe2N (23)

formed with the vector q of the 8rst N Fourier coeH-cients of x(#)

q:=[a; b]T ∈R2N (24)

is calledH th order Galerkin approximation of x∈P2T .

The right-hand side of (21) depends on the time notonly directly, due to the forcing excitation, but alsothrough x(#). Since the excitation is periodic by hy-pothesis, if x∈P2

T is substituted in f :=[f1; f2]T, thena periodic function of time f(#):=f(xN (#); #)∈P2

T isobtained. Accordingly, the trigonometric polynomial

fN :=QT(xN )e2N =QT(q)e2N (25)

formed with vectorQ of the 8rstN Fourier coeHcientsof f(#)

Q(q):=[A(q);B(q)]T ∈R2N ; (26)

i.e.

Ai:=%-

∫ T

0f 1(#)ei(#) d#;

Bi:=%-

∫ T

0f 2(#)ei(#) d#; i = i; : : : ; N; (27)

is called H th order Galerkin approximation of f . Bymeans of (23), the left-hand side of (21) can be writtenas{

KxN

#N

}= (Tq)Te2N (#); (28)

where T is a 2N × 2N matrix

T:=

[SS 0

0 SS

](29)

and S is a N × N matrix de8ned by

eN (#) = STeN (#): (30)

Substituting (28) and (25) into (21), the H -orderGalerkin approximation of the system [Q(q) −Tq]Te2N (#) = 0 is obtained. Since it must hold forany # the harmonic balance equations

Q(q)− Tq = 0 (31)

are obtained. Eq. (31) is a system of 2N non-linearalgebraic equations in the 2N unknowns q de8ningthe Galerkin approximation of the periodic solution of(21).The analysis of the eigenvalues of the Jacobian of

system (31) can be used to investigate the local stabil-ity of the periodic solution in the space of the Fouriercomponents.

4. Algorithm for the frequency–response curvescomputation

The harmonic balance procedure above describedprovides an approximation method for the computa-tion of the periodic solution of the pseudoelastic os-cillator due to a given harmonic excitation. Since themain goal of this work is the characterization of thedynamic response by varying the excitation frequency,% is regarded as a parameter. The solution of (31) isthus a one-parameter family of vectors q whose graphis a surface S in the (q; %) space R2N+1. Since Smay exhibit strong non-linearities and turning points,an adaptive continuation method is used to trace it outnumerically. Let us recall that, in general, continuationmethods [26] lead to the pointwise computation of Sthrough the parametric representation of the solution

u(1):=[q(1); %(1)]T (32)

and the addition of a constraint equation

h(u; %; 1) = 0: (33)

For any value of the path parameter 1, the intersec-tions between S and the graph of h are determinedby solving the augmented system

G(u; 1):=

{Q(q; %)− T(%)q

h(q; %; 1)

}=

{0

0

}: (34)

by a suitable iterative method

u(i+1) = u(i) +'(u(i)); (35)


which produces a sequence {u(i)} converging to u(1).Consequently, given an initial estimate u(0)m lying inthe convergence domain of the iterative method (pre-dictor phase), each point of S can be computed by(35) (corrector phase). Where, the subscriptm denotesthe incremental step, the parenthesized superscript (i)the iterative step and um the converged approximationof the solution u(1m). Since S is expected to haveturning points, a reduction of the computational e2ortcan be obtained by selecting shorter increments of 1in the regions of greater non-linearity and viceversa.Any speci8c continuation algorithm is thus charac-terized by the choice of four ingredients: the pathparameter 1, the constraint equation h, the predictor–corrector algorithm and the step-length determinationrule.In this work, frequency–response curves are ob-

tained by a continuation algorithm based on the fol-lowing choices. A spherical arclength constraint isused as parameterization

h(u; 1; um; 1m) = (u − um)T(u − um)− (1 − 1m)2:(36)

In literature, several kinds of predictor algorithmshave been proposed, either on the basis of Taylor ap-proximations of various order or based on higher or-der Lagrangian interpolation formulas. However, asclaimed in [26,27], higher-order predictors are rarelyconvenient. For this reason, a simple secant predictor

u(0)m+1 = um + sm(um − um−1) (37)

has been implemented, where sm is a step-dependentparameter. Once the predictor u(0)m+1 has been com-puted by (37), the corrector phase is performed by a2-norm trust region algorithm with a 8nite di2erenceapproximation of the Jacobian [28,29]. The implemen-tation of the iterative algorithm bears on the evaluationof Q(q), i.e. the Fourier coeHcients of the functionsf 1(#) and f 2(#) corresponding to the Fourier coef-8cients of the response q. To this end, for any givenq, the inverse fast Fourier transform (FFT) is used toretrieve discrete samplings of x(#) and #(#) that, bynumerical integration of (12b), are used to determine�(#). Finally, with x(#), #(#) and �(#) at hand, (21)yields the functions f 1(#) and f 2(#) and then, viadirect FFT, the components of Q associated to q. Anadaptive step-length determination scheme has been

implemented according to the criterion suggested in[30]

Q1m =

Q1min (−∞;Q1min]

3 if 3∈ [Q1min ;Q1max]

Q1max [Q1max;∞)

with 3=Q1m−1

(N iteropt

N iterm−1

)0:5; (38)

where Q1m and Q1m−1 are the step-lengths at the cur-rent and previous incremental step, N iter

m−1 the numberof iterations at the step m−1, N iter

opt an optimal numberof iterations (e.g. 3–10) and Q1min ;Q1max cautelativebounds for Q1m. The criterion (38) yields incrementsQ1m smaller near the turning points and larger else-where.

5. Asymptotic response analysis

First, a reference set of material parameters phys-ically meaningful for typical devices is selected andthe asymptotic response of the system analyzed, then,the in1uence of the parameter variation with respectto the reference case is investigated.

5.1. Reference case

As described in appendix, the following set of in-dependent parameters, denoted as reference model pa-rameters (RMP), can be representative of a devicebased on commercial grade NiTi wires in a convectiveenvironment

&= 7:0; q1 = 1:05; q3 = 0:6; %r = 0:1; J = 0:315;

L= 0:124; h= 0:08: (39)

The asymptotic response of the system with RMPto excitations of various amplitudes has been com-puted via harmonic balance and numerical integrationof (15). The analysis of the results leads to distin-guish three ranges, each giving rise to responses withsimilar qualitative features: low amplitude (F = 0:1–0.6),medium amplitude (F=0:7–0.8), high amplitude(F¿ 0:9). For di2erent choices of material parame-ters the three kinds of behaviors are still observed butat di2erent amplitude levels.


Fig. 3. Response for RMP, F = 0:6 and % = 0:6: displacement (a) and temperature (b) histories (time axis in number of periods),force–displacement curve (d) and phase plane portrait (e). In (c) detail of the last 5 periods of (a) and (b) (displacement: thin line andleft axis, temperature: thick line and right axis).

5.1.1. Low amplitudeThe features of the response in the range F=0:1–0.6

are described in Fig. 3 with reference to F=0:6, %=0:6and initial conditions x0 =[0; 0; 1; 0]T. It is found that,after a transient of few periods, the displacement tendsto a symmetric T -periodic solution with zero meanvalue. On the other hand, after a slightly longer tran-sient, the temperature tends to a T=2-periodic solutionwith non-zero mean value. The mean temperature isgreater than one indicating that, in this case, the over-all energy balance leads to an overheating of the de-vice with respect to the environment temperature. Asexpected, the frequency of the temperature is doublesince two phase transformations cycles, one in ten-sion and one in compression, correspond to a singledisplacement cycle. Phase transformations contributeto the temperature variations via the term Z2� in (12a)that depends on the absolute value of the displacementand causes the temperature to be characterized by theeven harmonics. Repeating the analysis for di2erentexcitation frequencies and amplitudes, it turns outthat the temperature variations during the oscillationsare in1uenced by two factors: the amount of phasefraction transformed during the cycle (shortly thetransformation amount), which correlates with themaximum displacement, and the loading rate, whichcorrelates with the frequency. In particular, temper-

ature variations are found to increase with the max-imum displacement and with the frequency and viceversa.The periodicity of the asymptotic solution enables

to give a synthetic characterization of the system dy-namics in terms of frequency–response curves (FRC).However, while for the displacement a plot of themaximum value versus the excitation frequency con-tains the basic information, the non-zero mean of thetemperature requires the consideration of FRCs notonly for the maximum but also for the mean value.Furthermore, the curves have to be understood as ofperiod T and T=2 for displacement and temperature,respectively. FRC have been computed via the har-monic balance procedure described in the previoussections using 12 components.The maximum displacement-frequency curves for

excitation amplitudes from 0.1 to 0.6 and RMP arereported in Fig. 4a. Multivalued curves with qual-itative features similar to those of other kinds ofhysteretic oscillators such as the elastic–plastic,Masing and Bouc-Wen ones [21–24] are observed.Here, however, the response is characterized also bythe temperature evolution and therefore the maxi-mum and mean temperature FRCs are reported inFigs. 4b and c. Each FRC in Fig. 4 is characterizedby four regions delimited by the points A–E. In the


Fig. 4. FRCs for RMP in the low amplitude range (F = 0:1–0.6): maximum displacement (a), maximum (b) and mean temperature (c).

zones AB and DE the maximum displacement isless than 1 hence there is no phase transformation,the behavior is linear elastic and the temperature isconstant and unitary, i.e. the device remains at theenvironment temperature. In the zones BC and CDphase transformations and hysteresis loops occur. InBC, the maximum and mean temperature increase inspite of the frequency decrease due to the signi8cantmaximum displacement rise. It turns out that, in thiscase, the e2ect of the transformation amount prevailson that one of the loading rate and hence governs thetemperature variations. A similar e2ect is observedalong CD where the temperature decreases whilethe frequency increases. It is worth recalling that thedimensional temperature is expressed in Kelvin sothat, for example, if the environment temperature is20◦C (293 K) then #= 1:05 corresponds to 34:65◦C,which means a notable temperature variation.The observed coexistence of multiple (two stable

and one unstable) periodic solutions with the same

frequency, points out the occurrence of saddle-nodebifurcations with analogies with other classes of hys-teretic systems [24,21,31].

5.1.2. Medium amplitudeIf the excitation amplitude is increased above 0.6,

di2erent FRCs are observed (Figs. 5a and b) and arecharacterized by six zones delimited by the points A toG. The attention is focused 8rst on the displacementcurves drawn in Fig. 5a for F = 0:7 and 0.8. Untilthe maximum displacement is less than the thresholdvalue xMf (zones BC and EF), the curves exhibitfeatures similar to the low amplitude ones. However,when the maximum displacement overcomes xMf , thecurves undergo a strong slope change that leads toa sudden deviation of the peak (zone CDE). Thisphenomenon is related to the fact that, for x¿ xMf ,the phase transformation is complete, the device is infully martensitic state and behaves in a linear elasticway. This can be appreciated in Figs. 5c and e that


Fig. 5. FRCs for RMP in the medium amplitude range (F =0:7; 0:8): maximum displacement (a) and temperature (b). Force–displacementcurves (c, d) and phase plane portraits (e, f) corresponding respectively to points p and q.

show the force–displacement cycles correspondingto the points q and p lying, respectively, above andbelow xMf .The maximum temperature response, illustrated in

Fig. 5b, also exhibits an upper resonant branch sud-denly bent to the right (zone CDE). Along CD thetransformation amount is constant as all points corre-spond to complete phase transformation cycles. Dif-ferently from the previous cases, the temperature vari-ations are now driven by the loading rate e2ect, sothat the temperature increases with the frequency. Inthe range DE, two solutions associated with com-plete phase transformation cycles are found for thesame frequency (i.e. those lying to CD and DE). Suchsolutions are characterized by the same transformedamount and by the same loading rate but, in spite ofthis, they exhibit di2erent maximum temperature anda hook is observed in the curve. This e2ect is due tothe fact that the solutions di2er for the length of thelinear elastic segments. During elastic loading and un-loading heat exchange with the environment withoutany heat production occurs; hence the solution withgreater maximum displacement (i.e. the one lying toCD) can exchange with the environment more heatthan the other solution and then shows a lower maxi-mum temperature.

Even if with di2erent qualitative features, also inthis amplitude range multiple solutions (two-stableand one-unstable) are found. Moreover, in the re-gion of sudden bending of the FRC (e.g. closeto point C of Fig. 5a) a more complex dynamicbehavior is expected with the likely occurrenceof non-periodic oscillations. However, the analy-sis of such dynamical phenomena is beyond thepotentialities of the harmonic balance method;in order to clarify such behavior, a further re-search is presently ongoing by resorting to timedomain methods based on the PoincarTemap.

5.1.3. High amplitudeFor higher excitation amplitudes, the FRC’s still

show six di2erent regions (Fig. 6). The qualita-tive features of the zones BC; CD;DE; EF are sim-ilar to those obtained for the medium amplituderange, while the various e2ects are ampli8ed by thegreater displacements. The main di2erence arisesin the zone AB where phase transformations arealready present. The e2ect of superharmonic reso-nances becomes relevant and the hysteresis cyclesinvolve also subloops internal to the outer one asshown in Fig. 7. Each superharmonic peak in the


Fig. 6. FRCs for RMP in the high amplitude range (F = 0:9; 1:0): maximum displacement (a) and temperature (b).

Fig. 7. Responses for RMP, F = 1:0 and % = 1:0; 0:2: displacement (a, e), temperature (b, f) histories (time axis in number of periods),force–displacement curves (c, g) and phase plane portraits (d, h).

displacement curve 8nds its counterpart in the tem-perature one as the loading rate is low and the tem-perature variations are driven by the transformationamount.The above results have been obtained by means of

the harmonic balance based on a 12 components ex-pansion, which has been found to ensure a good ap-proximation both on the FRCs and on the historiesof the various response quantities. Fig. 8 shows theimprovement of the approximation of the pseudoelas-tic force and temperature histories obtained by tak-ing into account 1, 4 and 12 harmonics. The FRCscomputed by the harmonic balance are then comparedwith the results of the numerical integration in Fig.9 that points out a very good agreement both fordisplacement and temperature at medium and highamplitudes.

5.2. In:uence of material parameters variations

By varying the seven independent parameters (39),various device behaviors can be modeled. However,the parameter &, that correlates with the length of thepseudoelastic plateaus, and the parameter %r , that cor-relates with the smoothness of the transition betweenlinear elastic zones and plateaus, may be expected toassume similar values for most devices. Therefore,their values are taken as in (39) and the attention isfocused on the in1uence of the other parameters.The parameter variations considered in the

following are aimed to study: (i) devices with dif-ferent hysteresis loop shapes, i.e. characterized bydi2erent values of q1 and q3, (ii) devices with dif-ferent heat production and exchange modalities, i.e.characterized by di2erent values of J , L and h. In


Fig. 8. Comparison among the pseudoelastic force (a–c) and temperature (d–f) histories computed by numerical integration (thin lines)and their Galerkin approximations with di2erent numbers of harmonics (thick lines) for RMP, F = 0:8 and % = 0:5.

Fig. 9. Comparison of the FRCs computed, for RMP, by 12 components-harmonic balance (thin lines) and by numerical integration (thickpoints): maximum displacement and maximum temperature; in the low amplitude range F = 0:6 and in the high amplitude range F = 1:0.

each case the results are presented with reference toexcitation amplitudes F = 0:1–0.6.

5.2.1. Hysteresis loop shapeAs the RMP refer to a rather narrow hysteresis loop,

it is interesting to evaluate the behavior of deviceswith wider loops, characterized by smaller values of

q3, e.g. q3 = 0:3 as opposed to q3 = 0:6 of the RMP.From the maximum displacement FRCs shown in Fig.10a, it emerges that, while the curves keep the samequalitative features of those with RMP, the increaseof the hysteresis loop width and the correspondinggreater energy dissipation, yields a strong reduction(by almost one-half) of the maximum displacements.


Fig. 10. Comparison of FRCs for RMP (thin line) and q3 = 0:3 (thick line) in the low amplitude range (F = 0:1–0.6): maximumdisplacement (a) and temperature (b).

Due to the reduced displacements also reduced tem-perature variations with respect to RMP are observed(Fig. 10b).The parameter q1 regulates the pseudoelastic

plateaus slopes and, in particular, its increase causeshysteresis loops with steeper plateaus. Since the ef-fect of the variation of q1 on the frequency–responsecurves is similar to the one of L, which is describedlater, for the sake of conciseness, the FRCs are notreported.

5.2.2. Thermal aspectsDuring phase transformations a certain heat quan-

tity is produced and exchanged with the externalenvironment. The net heat that remains in the systemin1uences the response and is determined by the bal-ance Eq. (12a), where the latent heat of transforma-tion rate is given by the term Z2� and the rate of heatexchange by h(#E − #). Since Z2 depends, besides &and #0 that here are 8xed, on J and L, the parametersthat mainly in1uence the thermal e2ects are J , Land h.The e2ect of L is considered 8rst. In the limit case

L= 0 the device would have, ideally, zero latent heatand therefore would behave isothermally. As shown inFig. 11, in isothermal conditions the frequency rangein which the curves are multivalued and the peak dis-placements are increased with respect to RMP. Thisincrease also produces an anticipation of the peak de-viation associated to phase transformation completion.If, on the other hand, the parameter L is increased,then the latent heat and the temperature variations aregreater. As a consequence, also the slope of the pseu-

Fig. 11. Comparison of maximum displacement FRCs for RMP(thin line) and L = 0 (thick line) in the low amplitude range(F = 0:1–0.6).

doelastic plateau increases and the hysteresis loop be-comes thinner [11]. The FRCs reported in Fig. 12 forthe case of L=0:6, exhibit smaller displacements, re-ductions of the multivaluedness region (Fig. 12a) andevident temperature increases (Fig. 12b).As increase of the parameter J produces e2ects sim-

ilar to a decrease of L and vice versa. For the sake ofbrevity, the FRCs are not reported.The third parameter that in1uences thermal aspects

is the heat exchange coeHcient h. In the limit case ofh=∞ the device behaves isothermally and the curvesof Fig. 11 are recovered. On the other hand, a decreaseof h with respect to the RMP is indicative of less heatexchange; the FRCs obtained in the case of h = 0:01are reported in Fig. 13. From Fig. 13b it turns outthat the temperature variations are remarkably higherthan those for RMP. The corresponding e2ect on the


Fig. 12. Comparison of FRCs for RMP (thin line) and L=0:6 (thick line) in the low amplitude range (F=0:1–0.6): maximum displacement(a) and temperature (b).

Fig. 13. Comparison of FRCs for RMP (thin line) and h = 0:01 (thick line) in the low amplitude range (F = 0:1–0.6): maximumdisplacement (a) and temperature (b).

displacement response again consists in smaller valuesand smaller multivaluedness region (Fig. 13a), as foran increase of L but with ampli8ed e2ect due to thegreater temperature rises.

6. Conclusions

A multi-component harmonic balance method hasbeen implemented within a continuation algorithm tocharacterize, via the frequency–response curves, theperiodic thermomechanical response of pseudoelasticdevices subjected to harmonic excitation. The com-parison between the numerical integration and the har-monic balance method shows good agreement alreadywith a low number of harmonics for displacements

FRC, whereas for the temperature FRC, as well as forthe computation of the time histories of the solution,more harmonics are needed. The asymptotic dynamicsof the system shows di2erent features depending onthe excitation amplitude and material parameters. Atlow amplitudes, multivalued displacement frequency–response curves are observed with similarities withother hysteretic systems [21–24,31], while at mediumamplitudes a peculiar e2ect of deviation of the peak ofthe curve is observed and it is found to be correlatedwith the linear elastic behavior of the device in fullymartensitic state. At high amplitudes the e2ects of su-perharmonics becomes relevant in the low frequencyrange.In this thermomechanical framework, the response

of the device is described also by the temperature that


oscillates at a frequency double than the displace-ment. The FRCs in terms of temperature shows qual-itative features that resemble the displacement onesand include multivalued resonance curves and lowfrequency peaks at high amplitudes. Some di2erencesarise in the deviated peak at medium and high ampli-tudes. On the quantitative side, in all the investigatedcases, the increase of the temperature variations in-duces a reduction of the multivaluedness regions ofthe mechanical FRCs and produces smaller maximumdisplacements. Moreover, the comparison of the dis-placement FRCs of the isothermal and non-isothermalcases shows that, if an isothermal model is used todescribe the device, signi8cant di2erences can arisein the quantitative description of the mechanicalresponse.More complex non-linear dynamic behaviors, such

as non-periodic oscillations, are expected in some re-gions of the FRCs and therefore further research ispresently ongoing with the aim to study in detail suchnon-regular motions and to provide a complete anal-ysis of bifurcations and stability.

Acknowledgements

The support of the Italian Ministry for the Uni-versity and the Scienti8c Research (MURST) in theframework of the Project COFIN 1999–2000 is grate-fully acknowledged.

Appendix Model parameters

According to the discussion in Section 2, thenon-isothermal response of the pseudoelastic de-vice is characterized by nine non-dimensionalparameters:

&; q1; q2; q3; %r ; J; L; #0; h;

whereas the isothermal response is characterized bythe 8rst 8ve. However, two of them, namely q2 and#0, have been constrained to the others on the basisof the following considerations.The attention is focused on the case of hysteresis

loops with upper and lower plateaus with equal slopes.Thus, the condition fMf −fMs =fAs −fAf is enforced

Table 1Typical values of material properties of SMA

Elastic modulus in austenitic phase E = 5–10× 104 MPaTransformation stress at room 7Ms = 400–800 MPatemperatureSpeci8c heat at constant stress cs = 450–600 J=kg KSlope in transformation stress– b7 = 5–8 MPa=Ktemperature planeMaximum transformation strain 8tr = 0:05–0.08Density 9 = 6500 kg=m3

CoeHcient of convective heat hc = 10–100 W=m2 Kexchange

and q2 can be computed from q1 and q3 by

q2 =1 + q3 − q1

q3: (A.1)

Moreover, according to the common practice of ex-pressing the equilibrium temperature #0 as the meanvalue of the stress-free transformation temperatures[3], i.e.

#0 =Ms +Mf + As + Af

4; (A.2)

taking into account that

Ms = 1− J; Mf = 1− Jq1;

As = 1− Jq3; Af = 1− Jq2q3 (A.3)

and (A.1), #0 can be computed form J and q3

#0 = 1− J1 + q3

2: (A.4)

By virtue of (A.1) and (A.4) seven parameters needsto be identi8ed. In the following, reasonable estimatesof such parameters are given with reference to a pseu-doelastic vibration isolation device made of commer-cial grade NiTi wires in a convective environment.Most of the device parameters can be computed fromthe knowledge of the properties of the shape memorymaterial; common values for typical NiTi wires aregiven in Table 1 [3,4].In addition to the data of Table 1, some hypotheses

on the devices are needed to identify the parameters. Itis assumed that the device is made of an arrangementof wires of total cross-sectional area A and e2ectivelength l. Then

K = EA=l; fMs = 7MsA; �= 8trl; m= 9Al;

c = csm: (A.5)


Using the de8nition (17), recalling thatfMs=KxMs andusing (A.5) with intermediate shape memory materialproperties in the ranges of Table 1

&= 7:0:

From the analysis of the literature [e.g. 3, 4] commonvalues of q1 and q3 for isothermal mechanical on wiresare

q1 = 1:05; q3 = 0:6:

However, for the devices lower values of q3 are alsopossible.The parameter %r actually governs the smoothness

of the phase transformation start and 8nish zones inthe force–displacement plane. After a speci8c analy-sis, %r = 0:1 has been found well suited for commondevices curves.From the de8nition (17), J = fMs =b#ref , where b

is the slope in the transformation force–temperatureplane which is related to the slope in the transforma-tion stress plane b7 by b = b7A. Considering #ref =293 K it turns out that

J = 0:315:

From the de8nition (17) and (A.5) then

L= 0:124:

The device heat exchange coeHcient h is related tothe one of the wires hc by h= hcAlat where Alat is thesurface of the area exposed to the heat exchange. Ifthe device is made of Nw wires each one with areaand lateral surface A1; Alat;1, then, introducing the ratio;=Alat;1=A1, Alat =;NwA1 =;A. Taking into account(17), h is given by h=(hc;)=(cspec9l!2) where! is thefrequency of the small amplitude oscillations. Makingreference to circular wires of 1 mm diameter and 0:1 mlength and to an oscillator with natural period of 0:8 s8nally

h= 8:2761× 10−2 ≈ 0:08:

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non-isothermal oscillations of pseudoelastic devices

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