Embed Size (px)
Citation preview
Article
Journal of Intelligent Material Systemsand Structures23(14) 1593–1611� The Author(s) 2012Reprints and permissions:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/1045389X12448446jim.sagepub.com
Non-linear dynamic thermomechanicalbehaviour of shape memory alloys
Mohamed O Moussa1, Ziad Moumni1, Olivier Doare1,Cyril Touze1 and Wael Zaki2
AbstractThe non-linear dynamic thermomechanical behaviour of superelastic shape memory alloys is investigated. To this end,the Zaki–Moumni model, initially developed for quasi-static loading cases, is extended to simulate the uniaxial forcedoscillations of a shape memory alloy device. First, the influence of loading rate is accounted for by considering the ther-momechanical coupling in the behaviour of NiTi shape memory alloy. Comparisons between simulations and experimen-tal results show good agreement. Then, the forced response of a shape memory alloy device is investigated atresonance. Both isothermal and non-isothermal conditions are studied, as well as non-symmetric tensile-compressiverestoring force. In the case of large values of forcing amplitudes, simulation results show that the dynamic response isprone to jumps, bifurcations and chaotic solutions.
Keywordsdynamic behaviour, hysteresis, shape memory alloy, pseudoelasticity, chaos, symmetry breaking, vibrations, thermome-chanical coupling, natural frequencies
Introduction
The unusual behaviour of shape memory alloys(SMAs) is due to their ability to undergo martensitictransformation (Funakubo, 1987; Patoor andBerveiller, 1990). Beyond a certain temperature Af ,mechanical loading may lead to high inelastic deforma-tion of the material that can be recovered by unloading.This so-called pseudoelastic behaviour is usuallyaccompanied by considerable dissipation of energy andby stiffness variation, which can be used advanta-geously for vibration damping according to Saadatet al. (2002). Strain rate effects on the pseudoelasticcharacteristics have been the focus of a number of pub-lished articles from theoretical and experimental pointsof view. It is well established that increasing the strainrate leads to an increase in the slopes of the plateaus offorward and reverse martensitic transformations andleads to a change in the area of the hysteresis loop,which measures the dissipated energy.
Humbeeck and Delaey (1981) appear to be the firstto underline the exothermic/endothermic characteristicof forward/reverse transformations and its influence onthe hysteresis loop. Experiments carried out at differentstrain rates and initial temperatures show an increase instress thresholds with respect to initial temperature andstrain rate in Mukherjee et al. (1985). A relationship
between phase transformation stress and temperaturewas given by Leo et al. (1993). It allowed establishing athermomechanical diagram by Leo et al. (1993). Thedependency of stress thresholds was also highlighted inthe experimental work by Shaw and Kyriakides (1995).However, according to the study done by Lin et al.(1996), variations in both dissipated energy and phasechange thresholds occur only above a certain strainrate. From those studies emerges the question of distin-guishing the most prominent effect between tempera-ture and strain rate. On the one hand, Nemat-Nasserand Wei-Guo (2006) showed that pseudoelastic beha-viour of NiTi SMAs is influenced by temperature varia-tion induced by increasing the strain rate. On the otherhand, Helm and Haupt (2002) argue that the depen-dency of pseudoelastic behaviour on strain rate is dueto the viscosity of the SMAs.
1Unite de Mecanique (UME), ENSTA-ParisTech, Palaiseau Cedex, France2Khalifa University of Science, Technology and Research (KUSTAR), Abu
Dhabi, UAE
Corresponding author:
Ziad Moumni, Unite de Mecanique (UME), ENSTA-ParisTech, Chemin de
la Huniere, 91761 Palaiseau Cedex, France.
Email: [email protected]
This divergence on the origin of strain rate effectsremains in literature until recent works in this area. Inparticular, the experimental results shown by Grabe inGrabe and Bruhns (2008), obtained by finely control-ling material temperature at different strain rates, allowto discard the assumption of the viscosity of SMAs.Morin et al. (2011) developed a model that accountsfor the increase of the slope of transformation plateausand the change in the dissipated energy with respect tostrain rate by taking into account the influence of ther-momechanical coupling. In addition, Chrysochooset al. (1996) presented a thermomechanical model, afterinvestigating temperature measurement using infraredthermography, which considers only the generatedlatent heat as a heating source. Since we are interestedin the dynamic behaviour of SMAs, thermomechanicalcoupling must be considered when investigatingdynamic response of a SMA device.
Regarding inertia effect and dynamics of SMAsstructures, Machado et al. (2003) and Savi andPacheco (2002) studied the dynamic behaviour of one-and two-degree-of-freedom SMA oscillators. A consti-tutive model is derived considering a non-convex poly-nomial free energy potential for the SMA, whichdepends on total strain and temperature as the onlystate variables. For lower temperatures, the polynomialpossesses two minima representing stable martensitestates, whereas for higher temperatures, a unique mini-mum exists corresponding to stable austenite. Theexplicit dependence on temperature allows the model tocapture the salient features of the pseudoelastic andshape memory responses of the SMA in a simple andstraightforward way. However, the model does notaccount for intrinsic mechanical dissipation, which con-tributes to the overall damping capacity of the system.The same SMA constitutive model is used by Savi et al.(2002) in order to analyse free and forced vibrations ofa two-bar Von Mises frame structure. Constitutive andstructural non-linearities are shown to have significantinfluence on the dynamic response of the frame, whichis found to be highly non-linear with variable numberof equilibrium states depending on temperature.Lagoudas et al. (2005) investigated a SMA-based pas-sive damping system consisting of a mass attached toSMA wires. The experiments were carried out at ahigher temperature where the material response waspseudoelastic. The authors measured vibration trans-missibility and found it to decrease with increasingexcitation frequency. The resonant frequency of thesystem was also found to decrease because of the soft-ening behaviour due to the apparent stiffness of theSMA during phase transformation. Numerical simula-tions were performed using a constitutive model byQidwai and Lagoudas (2000) specialized to the uniaxialSMA response, where temperature variations measuredin the SMA wires are used as input. Strong thermome-chanical coupling was considered in a subsequent
article by Machado et al. (2009). Similar analysis wasundertaken by Sitnikova et al. (2010) who consideredan impact oscillator system consisting of an oscillatorand a separate SMA support. When the vibrationamplitude of the oscillator exceeds a certain threshold,impact against the support leads to energy dissipationby means of phase transformation in the SMA. Saviet al. (2008) used a uniaxial version of a constitutivemodel in order to simulate the dynamic response of asimple SMA oscillator. The model features four dissi-pative state variables that account for phase transfor-mations between austenite, twinned martensite andmartensite variants detwinned in tension and in com-pression. The use of different martensite volume frac-tions for the tensile and compressive martensitevariants allows the model to account for tensile-compressive asymmetry. The equations of motion aresolved using a Runge–Kutta iterative scheme, and thedissipative variables are updated using closest-pointprojection to enforce consistency with the phase trans-formation functions. Bernardini and co-workers inves-tigated thoroughly the non-linear dynamic responses ofa SMA device in a series of articles (Bernardini, 2001;Bernardini and Pence, 2005; Bernardini and Rega,2009; Bernardini and Vestroni, 2003; Lacarbonaraet al., 2004). Based on the Ivshin–Pence thermomecha-nical model and using robust path-following continua-tion methods, they draw out a complete picture of theresonant behaviour of the SMA device from small tomedium amplitude range. The isothermal and non-isothermal conditions were investigated, and numerousnon-linear phenomena (bifurcations, jumps, chaoticmotions) and the complete picture of the influence ofthe model parameters were reported. Other works bySeelecke (2002) investigate free and forced vibrations ofa SMA torsional pendulum, while chaotic responses oftwo coupled SMA oscillators were reported inMachado et al. (2003). Lagoudas et al. also investigatedthe thermomechanical dynamic behaviour of a SMAdevice in Hartl et al. (2010).
All these investigations and issues show that thenon-linear frequency–response functions (FRFs) ofSMAs exhibit a softening behaviour, as a consequenceof the decrease of effective stiffness at the starting pointof martensitic transformation. Jump phenomenon wasalso predicted. On the experimental side, softeningeffect was independently reported in many publications(e.g. Collet et al., 2001; Feng and Li, 1996; Schmidtand Lammering, 2004). However, jump phenomenonwas almost not observed, being hidden by the increaseof damping capacity. Recently, measurements on a tor-sional pendulum clearly evidenced the jump (Doareet al., 2011; Sbarra et al., 2011).
The main objective of the current contribution is toextend the model developed by Zaki (2010) and ZakiandMoumni (2007a) in quasi-static conditions, in orderto study the general dynamic behaviours of structures
1594 Journal of Intelligent Material Systems and Structures 23(14)
made of SMAs. As already underlined, thermomecha-nical coupling is a key issue for correctly reproducingexperimental observations. Section ‘Model equations’ isdevoted to an introduction of the thermomechanicalcoupling in the Zaki–Moumni (ZM) model. Simulationof a tensile test on a cylinder shows that the main effectsare well reproduced.
In order to validate the model in dynamic situations,as a necessary step before using it for full three-dimensional (3D) simulations of structures made ofSMAs, the 3D model (ZM) is reduced to a singledegree of freedom, allowing for exhaustive compari-sons with the numerical results given in the articles byBernardini (2001), Bernardini and Pence (2005),Bernardini and Rega (2009), Bernardini and Vestroni(2003) and Lacarbonara et al. (2004). One main differ-ence with their work lies in the fact that the thermody-namic admissibility is ensured in the ZM model thanksto the framework of generalized standard materials(GSMs) with internal constraints, developed byMoumni et al. (2008), used to derive the equations ofthe model. The GSM framework ensures the verifica-tion of the second law of thermodynamics.
Section ‘Reduced model for a uniaxial SMA oscillator’explains the model reduction to a SMA device.Numerical results are then compared to those given byLacarbonara et al. (2004), showing good agreement. Newresults, such as asymmetric restoring force in tensile-compression loadings, are also studied. All these resultsshow the useful properties of the non-linear dynamicbehaviour of SMAs in vibration control for instance.
Model equations
ZM 3D thermomechanical model for SMAs
Equations in isothermal conditions. The ZM model forSMAs is cast within the framework of GSMs with inter-nal constraints, presented in Moumni et al. (2008). Itwas first introduced by Moumni (1995) for pseudoelas-tic behaviour and later generalized to take into account,in a unified way, all features associated with SMA(shape memory effect and reorientation) by Zaki andMoumni (2007b), cyclic SMA behaviour and trainingby Zaki and Moumni (2007a), tension–compressionasymmetry by Zaki (2010) and irrecoverable plasticdeformation of martensite by Zaki et al. (2010). Theoriginal ZM model uses the following expression of theHelmholtz free energy for the derivation of constitutiverelations
W iso = 1� zð Þ 1
2eA : S�1
A : eA
� �
+ z1
2eM � eorið Þ : S�1
M : eM � eorið Þ+C Tð Þ� �
+Gz2
2+
z
2az+b 1� zð Þ½ � 2
3eori : eori
� �ð1Þ
In the above equation, eA and eM are the local straintensors of austenite and martensite, respectively; T isthe temperature; z is the volume fraction of martensiteand eori is the orientation strain tensor (note that thetensors and vectors are written with bold font in equa-tions). SA and SM are the compliance tensors of auste-nite and martensite. G, a and b are material parametersthat influence the shape of the superelastic hysteresisloop and the slopes of the stress–strain curve duringphase change and martensite orientation. C(T ) is anenergy density that depends on temperature as follows
C Tð Þ= j T � T0� �
+ k ð2Þ
where j and k are material parameters associated withthe density of the generated latent heat.
The state variables obey the following physicalconstraints:
� The macroscopic strain tensor e is an averageover the representative elementary volume(REV) of the strain within the austenite andmartensite phases. By construction, e is given by
1� zð ÞεA + zεM � ε= 0 ð3Þ
� The equivalent orientation strain cannot exceeda maximum detwinning strain emax
εmax �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2
3εori : εori
r� 0 ð4Þ
The above constraints derive from the followingpotential
Wl =� l : 1� zð ÞεA + zεM � ε½ �
�m εmax �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2
3εori : εori
r !� n1z� n2 1� zð Þ ð5Þ
where the Lagrange multipliers n1, n2 and m are suchthat
n1 � 0, n1z= 0 ð6Þn2 � 0, n2 1� zð Þ= 0 ð7Þ
and
m � 0, m emax �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2
3εori : εori
r != 0 ð8Þ
The sum of the Helmholtz energy density (equation(1)) and the potential Wl (equation (5)) gives theLagrangian L, which is then used to derive the stateequations. With some algebra, the following stress–strain relation is obtained
s=S�1 : ε� zεorið Þ ð9Þ
Moussa et al. 1595
where S is the equivalent compliance tensor of thematerial, given by
S= 1� zð ÞSA + zSM ð10Þ
One can also derive the state equations for two ther-modynamical forces Az and Aori associated with z andεori, respectively, and given by
Az =�∂L∂z
=�1
2S�1
A : s : s +1
2S�1
M : s : s +s : eori
� C Tð Þ � Gz� a� bð Þz+ b
2
� �+
2
3eori : eori
� �ð11Þ
and
Aori =�∂L∂eori
= zs � z az+ 1� zð Þbð Þeori �2
3m
eoriffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi23eori : eori
qð12Þ
Then, the thermodynamic forces are taken as subgra-dients of a pseudopotential of dissipation D defined by
D= a 1� zð Þ+ bz½ � _zj j+ z2Y
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2
3_εori : _εori
rð13Þ
where a and b are positive material parameters and Y isthe orientation yield stress in tension. This allows defi-nition of yield functions for phase change (F 1
z and F 2z )
and for martensite orientation (F ori). The evolution ofthe state variables z and εori is governed by the consis-tency conditions associated with the yield functions. Ifthe orientation-finish stress is lower than the criticalstress for forward phase change (i.e. if stress finish srf
of the detwinning process is lower than the stress startsms of the phase change), the model is such that thestress-induced martensite is completely oriented as soonas forward phase change begins.
Thermomechanical coupling. It is worth mentioning thatin the classical model previously presented, theHelmholtz free energy does not account for the heatcapacity of SMA. The current section aims at includinga constant heat capacity CP, assuming that it is thesame in austenite and martensite phases. Therefore, theaugmented expression of the Helmholtz free energyreads
W=W iso + rCP T � T 0 � T lnT
T0
� �� �ð14Þ
where r is the mass density. Besides, considering a con-stant heat capacity involves changes only in the stateequation of the entropy. The latter now reads
h= � ∂W∂T
= � jz+ rCP lnT
T0
� �
Then, the first law of thermodynamics can be writtenin terms of the Helmholtz free energy as
_W+ T _h+h _T =s : _e� div qð Þ+ Sv ð16Þ
where q is the heat influx and Sv is the density of theinternal heat generation.
The second law of thermodynamics gives
D= T _h+ div qð Þ � qrT
T� Sv � 0 ð17Þ
Using the first and the second laws simultaneously,one deduces that
D=s : _e� _W+ _Th� �
� qrT
T� 0 ð18Þ
where Dth = � qrT=T is the heat dissipation.Using the differentiated expression ofW and h
_W=∂W∂e
: _e+∂W∂z
_z+∂W∂eori
: _eori +∂W∂z
_z+∂W∂T
_T
ð19Þ
and
_h=∂h
∂e: _e+
∂h
∂z_z+
∂h
∂eori
: _eori +∂h
∂z_z+
∂W∂T
_T ð20Þ
the dissipation can be given, thanks to h= � ∂W=∂T ,Az = � ∂W=∂z and Aori = ∂W=∂eori, in terms of theHelmholtz free energy as
D=Dint � T∂2W∂T∂e
: _e� T∂2W∂T∂eori
: _eori
� T∂2W∂T∂z
_z� T∂2W∂T2
_T + div qð Þ � qrT
T� Sv
ð21Þ
The second right-hand term accounts for thermoe-lastic effects, the third for the dissipation generated bythe orientation of martensite, the fourth for the latentheat and the fifth for the heat capacity.
Assuming that the intrinsic dissipation Dint is posi-tive, one can write
Dint =Az _z+Aori : _eori � 0 ð22Þ
Moreover, using Fourier’s law of heat conduction,q= � KvrT , where Kv is the thermal conductivityassumed to be the same in both phases, allows one to
1596 Journal of Intelligent Material Systems and Structures 23(14)
derive the heat equation after neglecting the thermoe-lastic effects
rCP_T �div KvrTð Þ= �T
∂Aori
∂T: _εori�T
∂Az
∂T_z+Sv+Dint
ð23Þ
Besides, complete martensite orientation is assumedat the beginning of forward phase change, leading tothe following expression of the martensite orientationstrain, in the case of proportional loading
eori = emax
s9
sVM
� �ð24Þ
where emax is the maximum orientation strain, s9 thedeviatoric stress tensor and sVM the Von Mises equiva-lent stress.
Indeed, while only the phase change phenomenoninvolves dissipation, heat generation due to martensiteorientation is null and heat equation reads now
rCP_T � div KvrTð Þ= � T
∂Az
∂T_z+Az _z ð25Þ
Natural convection is assumed at the boundary (∂O)such that
Q∂O = h Tex � Tð Þ ð26Þ
where Tex is the temperature of the surroundingmedium and h is the heat convection coefficient.
Thermomechanical coupling leads to the increase ofthe slope of the transformation plateaus in stress–strainspace with respect to strain rate: the completion of thephase transformation becomes more difficult at highstrain rate (Figure 1), and the material temperatureincreases, which influences its mechanical response. Itis well known, in addition, that the area of the
hysteretic loop changes while increasing strain rate(Grabe and Bruhns, 2008; Hartl et al., 2010; He andSun, 2011; Lexcellent and Rejzner, 2000; Morin et al.,2011).
In order to verify the ability of the model to recoverproperly, these two effects, a finite element (FE) modelof a wire of length 10�1 m and a circular cross sectionof radius 10�3 m m, have been implemented with abehaviour law dictated by ZM model. The discretiza-tion is performed, thanks to 500 cubic elements with 20nodes (Gauss’s points) where mechanical fields areevaluated. There are three degrees of freedom by nodeand, therefore, 30,000 degrees of freedom in total.
Thus, at any node N1 of the meshing O (Figure 2),one solves the following set of equations:
Mechanical equilibrium and heat equations
div sð Þ= 0, 8N1 2 O
rCP_T � div KvrTð Þ= � T
∂Az
∂T_z+Az _z, 8N1 2 O
ZM model
s=S�1 : ε� zεorið Þ, 8N1 2 O
Az 2 ∂_zD, 8N1 2 O
Mechanical boundary conditions
ux = 0, 8N1 2 ST
umaxx = 16310�3 m 8N1 2 SU
Thermal boundary conditions:
q=Q∂O = h Tex � Tð Þ, N1 2 ∂O
where ST and SU are the boundary areas at x= 0 andx= 0:15, respectively (see Figure 2). ∂O stands for thewhole boundary area, that is, including ST , SU and thelateral surface. Fifty loading increments and physicalthree time steps (dt = 40, 0.4 and 0.004 s) are used in
TwinnedMartensite
DetwinnedMartensite
σ σ σ σσσ
σ
θ
εε
Loadingαe ∼ 10−8
ε
Austenite
Mf Ms AfAs Te
ε εε ε
Figure 1. Stress–temperature diagram showing that anychange in the temperature for different strain rates involvesdifferent mechanical behaviours.
ux
max
ST
SU
Figure 2. Meshed structure O with boundary zones.
Moussa et al. 1597
the FE simulations, conducted for increasing strainrates as follows: 4310�5 s�5, 4310�2 s�2 and4310�1 s�1.
Table 1 gives the values of material parameters usedfor simulations as already identified in Morin et al.(2011). The selected node for showing simulation resultsis located far from the edges of the specimen in order toavoid boundary effects (according to the Saint-Venantprinciple). Using the above expression for eori allows toavoid the computation of the consistency condition onthe orientation yield function F ori.
The results of the simulation are shown in Figures 3and 4. Figure 3(a) and (b) shows the simulated
behaviour at a selected node within the post-processingzone for the three increasing values of the strain rateand for two heat exchange situations (h = 50 and 800Wm22 K21). The thermal hardening is well reproducedby the model for both the values of h. However, smallerthe value of h, greater the effect of thermomechanicalcoupling: the slope variation of the martensitic trans-formation plateaus is increased about seven timesbetween the responses at _e= 4310�5 s�1 and_e= 4310�2 s�1 for h = 50 Wm22 K21 (Figure 3(b)).Physically, this effect is explained by the increase of thetemperature with increasing strain rate, hence makingthe nucleation and growth of martensite more difficult.The size of the hysteresis loop is also shown to decreasein the FE simulation, which recovers the main observedfeatures brought by the thermomechanical couplingwith a fair qualitative accuracy.
Figure 4(a) and (b) shows the corresponding tem-perature variations during the simulated experiment,for two different strain rates: 4310�2s�1 and4310�5 s�1. The temperature variations are governedby the competition between mechanical dissipationand latent heat. One can observe that in zone I inFigure 4(a) and (b), the temperature remains constantbecause of the absence of thermoelastic coupling in thecurrent numerical investigation. However, along zoneII, the beginning of the exothermic phase transforma-tion generates the temperature variations. Zone III cor-responds to stabilization of the temperature betweenforward transformation finishing point and the reversetransformation starting one. The endothermic charac-teristic of the reverse transformation involves a decreasein the temperature evolution as evidenced in zone IV,where the reverse phase change begins. Finally, zone Vshows no temperature evolution because of the
0 0.01 0.02 0.03 0.04 0.05 0.06 0.070
200
400
600
800
(a)ε
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080
1
2
3
4
5
6
7
8
9 x 108
σ (Pa)σ (Pa)
(b)ε
Figure 3. Influence of the strain rate on the stress–strain response: stress–strain curves at different strain rates _e= 4310�5 s�1
(solid line), _e= 4310�2 s�1 (square) and _e= 4310�1 s�1 (dashed line): (a) for h = 800 and (b) for h = 50.
Table 1. ZM parameters corresponding to the identifiedparameters in Morin et al. (2011).
Symbol Value
EA 6153108 PaEM 2:43109 Pam 0:3a 2753107 PaY 113107 Paj 0:29143106 Pa=8Cemax 4:0%Cp 440 J kg�1 K�1
r 6500 kg m�3
b 6.90913106 PaG 4:65563106 Pab 553108 PaAf 408Ck 6:89203106 Pakv 18 W m�1 K�1
a 6:89203106 Pahwater 800 W m�2 K�1
hair 50 W m�2 K�1
ZM: Zaki–Moumni.
1598 Journal of Intelligent Material Systems and Structures 23(14)
vanishing of all heat sources in the current problem.However, one can observe that at the end, the tempera-ture can reach a value below the initial temperaturedue to the competition between the latent heat and theheating exchange rate.
A quantitative comparison is now drawn out byusing the data of the experiments shown in Shaw andKyriakides (1995), at two different strain rates andwhere the external environment is air. Figure 5(a)and (b) shows the stress–strain responses at 431025s21
and 4310�2 s�1 for simulation and experiment. Onecan highlight that the material parameters used are thesame for both figures; hence, only the strain rate effectis under consideration currently. The comparison,although with little mismatches, reveals the ability ofour model to reproduce both the quasi-static behaviourand thermomechanical coupling effects.
Since the 3D thermomechanical behaviour is vali-dated on a simple quasi-static simulation, the remain-der of the article is devoted to extending the model tohandle the case of vibrations. For the purpose of vali-dation, the 3D model will first be reduced to a singledegree of freedom oscillator displaying pseudoelasticity(SMA device) in order to compare numerical resultswith the literature.
Reduced model for a uniaxial SMAoscillator
The 3D model presented above is now particularized tomodel a non-linear spring included in an oscillator. Aschematic view of the system under consideration isgiven in Figure 6. More precisely, our goal is to derive
0 0.01 0.02 0.03 0.04 0.05 0.06 0.070
1
2
3
4
5
6 x 108σ (Pa)
(a)ε
0 0.02 0.04 0.06 0.080
1
2
3
4
5
6
7
8
9x 10
8σ (Pa)
(b)ε
Figure 5. Stress–strain responses with ZM model (solid line) and Shaw and Kyriakides’s experiment (dotted line) at (a)_e= 4310�5 s�1 and (b) _e= 4310�2 s�1.ZM: Zaki–Moumni.
0 20 40 60 80 100335
340
345
350
0 20 40 60 80342.98
342.99
343
343.01
T (K)T (K)
ττ
II
II
II IIIIII IV
IV
V
V
(a) (b)
Figure 4. Temperature variation with time: (a) at _e= 4310�2 s�1 and (b) at _e= _e= 4310�5 s�1.
Moussa et al. 1599
a non-linear spring law reproducing the SMA beha-viour, so as to include it in a zero-dimensional oscilla-tor equation. To that end, a 1D bar of cross-section S
and length l, clamped at x= 0 and attached to the massM at x= l, is considered, only for derivation of thenon-linear spring behaviour law. One has to keep inmind that although the bar is 1D, eventually the oscil-lator model is zero-dimensional. This remark will guidethe retained choice for the spatial dependence of thetemperature one has to account for when reducing themodel. A damper C is mounted in parallel to modelother sources of dissipation. An external force F =Fex
is exerted on x= l. Under the assumption that l � D,D being a typical length characterizing the cross sec-tion, the stress in the SMA bar can be considered to be1D, s =sex � ex, with
s =F
Sð27Þ
One defines the total strain in the x-direction as
e=X
lð28Þ
where X is the displacement of the system at x= l.Besides, the 1D detwinning strain reads
eori, xx =Xori
l
where Xori is an internal displacement due the martensi-tic detwinning process.
Equation (9) gives in 1D version
s =Eeq zð Þ e� zeorið Þ ð30Þ
where Eeq(z) is the equivalent Young’s modulus, givenby
Eeq = 1� zð ÞEA + zEM ð31Þ
Using equation (27), equation (28) and
eori =Xori
lð32Þ
one can derive the relationship between the forceexerted on the cylinder and its elongation as
F =K zð Þ X � zXorið Þ ð33Þ
K(z) is a non-linear stiffness depending on theamount of martensitic phase z. Its expression in the sin-gle-degree-of-freedom case is analogous to equation(10) for the 3D case and reads
K zð Þ= 1� z
Ka
+z
Km
� ��1
ð34Þ
where Ka =EaS=l and Km =EmS=l are the stiffnessesof the austenitic and martensitic phases, respectively. Inthe context of 1D deformations, the thermodynamicforce Az now reads
Az =F
S
� �21
2Em
� 1
2Ea
� �+
FXori
Sl� C Tð Þ
�Gz� a� bð Þz+ b
2
� �Xori
l
� �2ð35Þ
In this simplified approach, the detwinning criterionis disregarded, and the displacement Xori has the follow-ing form
Xori =Xmax sgn Fð Þ ð36Þ
where sgn(F) denotes the sign of F. This assumptionimplies that the detwinning process is fully achieved atany time during the transformation. Actually, the initialtemperature is such that the detwinning finish force issmaller than the starting force of phase change: accord-ing to the phase diagram in Figure 1, the level of thedetwinning process becomes maximum at such a tem-perature. Criteria functions have the following form inthe 1D model
F cri1, 2 =6Az � a 1� zð Þ � bz ð37Þ
where plus and minus signs apply to F cri1 and F cri
2 ,respectively. The evolution of z is then governed byconsistency conditions. The dynamic equation of thenon-linear mass–spring oscillator on which a harmonicforcing amplitude Emax and frequency v is exerted hasthe following expression
M€X + C _X +K zð Þ X � z � Xorið Þ=Emax cos vtð Þ ð38Þ
whereM and C are the mass and damping coefficients,respectively. Temperature evolution is now addressedconsidering that a spatially homogeneous heat flowexists between the SMA device and the outer domainat temperature Tex. Consequently, in the 1D equivalentform of equation (25), the term involving the diver-gence of the temperature cancels, and the temperaturein the material is characterized by its time evolutiononly. This choice of modelization is motivated by thefact that the final oscillator model is zero-dimensional;thus, spatial dependence of temperature is disregarded.
Heat exchanges
M
C
K(z )
H
Emax cos(ωt )
Figure 6. Schematic view of a SMA mass–spring oscillator withan external forcing.SMA: shape memory alloy.
1600 Journal of Intelligent Material Systems and Structures 23(14)
The heat exchange is modelled by a natural convectionwith a heat exchange coefficient H , so that the tempera-ture of the SMA device is now governed by the follow-ing equation
rCP_T � Az _z� jT _z=H Tex � Tð Þ ð39Þ
In this last equation, the intrinsic dissipation has asimpler form than in the general 3D case because thedetwinning process is expected to be fully achievedevery time and _X
ori
= 0.The following non-dimensional parameters are now
introduced
O=v
vn
, t =vnt ð40Þ
x=X
Xms
, g =Emax
Fms
ð41Þ
u=T
Tref
, uex =Tex
Tref
ð42Þ
h=H
rCPvn
, z =C
2vnMð43Þ
In these expressions, vn ¼DffiffiffiffiffiffiffiffiffiffiffiffiffiffiKa=M
pis the natural
frequency of the oscillator in the austenitic phase, Fms
is the force at the beginning of the martensitic phasetransformation and Tref is a reference temperature, inthe austenitic phase. Finally, it is assumed thatKa =Km for simplicity. Finally, the dynamics of thereduced model for the SMA oscillator is governed bythe following system
€x+ 2z _x+ f x, z, xorið Þ)= g cos Ot
where f x, z, xorið Þ= x� zxori
and xori = xmax sgn fð Þ ð44aÞ
F crit1 <0, _z � _F crit
1 = 0 ð44bÞ
F crit2 <0, _z � _F crit
2 = 0 whereF crit1
=Az � a 1� zð Þ � bz and F crit2
= �Az � a 1� zð Þ � bz
ð44cÞ
_u� j
rCP
� �u_z� Az
rTref CP
� �_z= h uex � uð Þ ð44dÞ
Parameters identification
The reference article we consider for comparison withour results is that of Lacarbonara et al. (2004) where athermomechanical SMA model is also used. In the non-dimensional version of this model, only four materialparameters are considered and denoted by q1, q2, q3
and J . Their dependency with respect to the parametersin this article is
q1 =smf
sms
, q2 =saf
sas
, q3 =sas
sms
, J = 1�Mms
Tref
ð45Þ
where sms, smf , sas, saf , Mms, Mmf , Aas and Aaf arestress and temperature thresholds. Subscripts ms andmf and as and af refer to forward and reverse transfor-mations, respectively. Our model needs dimensionalparameters. Some of them can be chosen arbitrarily,the other ones being calculated from the values of q1,q2, q3 and J . The radius of the cylinder under test ischosen equal to 1 mm, so that S = 3:14310�6 m2; itslength (l) is chosen equal to 0.1 m and the referencetemperature (Tref) is chosen equal to 296 K. Some mate-rial properties are also fixed to realistic quantities ofreal materials, sms = 83108 Pa, ssori = 83107 Pa (det-winning stress start) and sfori = 1653106 Pa (detwin-ning stress finish). Young’s modulus and maximumdetwinning strain are set to Ea =Em = 53109 Pa andemax = 0:112. Once these quantities are known, all otherparameters of the model may be calculated using thefollowing relationships
Xms =smsl
Ea, Mms = 1� Jð ÞTref , Mmf = 1� Jq1ð ÞTref
ð46ÞAms = 1� Jq3ð ÞTref , Amf = 1� Jq3 � q2ð ÞTref ð47Þ
a=1
2
1
Em
� 1
Ea
� �s2
ms � s2af
2+ emax sms � saf
� �" #
ð48Þ
b=1
2
1
Em
� 1
Ea
� �s2
mf � s2as
2+ emax smf � sas
� �" #
ð49Þ
b=sfori
emax
, a=b� ssori
sfori
eori ð50Þ
G=1
2
1
Em
� 1
Ea
� �s2
mf � s2af � s2
ms +s2as
2
"
+ smf +sas � sms � saf
� �emax � a� bð Þe2
max
�ð51Þ
xmax =emaxl
Xms
, T0 = Tref , k= a� be2max
2ð52Þ
C T0ð Þ=1
Em
� 1
Ea
� �s2
ms +s2af
2+ emax sms +saf
� �)
�bemax, j =C u0ð Þ � k
T0 � Amf
ð53Þ
Comparison is made, for isothermal oscillations,with two cases of Lacarbonara et al.’s parameter val-ues. Next section will provide the numerical methodused for the time integration of equations.
Moussa et al. 1601
Numerical scheme
Time integration is numerically achieved using a non-linear implicit Newmark scheme adapted to the treat-ment of the criteria functions for the evolution of thevolumic fraction z. The main idea is to use a classicalNewmark scheme for the mechanical part of the dynamicsystem. Inside the Newton–Raphson loops, iterations areperformed on the two unknowns z and u, so as to achieveconvergence of metallurgic and temperature variables.The main steps are here briefly explained with emphasison the peculiar treatments needed by the present model.The reader is referred to Geradin and Rixen (1994) andHughes (2000) for more details on the Newmark scheme.Let us assume that at time tn, the state variables (xn, _xn,€xn, zn, xori, n and un) are known. The first step is to predictthe displacement, velocity and accelerations at timetn+ 1 = tn +Dt with the classical formula
xn+ 1 = xn + _xnDt +Dt2
21� 2bNð Þ€xn ð54aÞ
_xn+ 1 = _xn + 1� gNð ÞDt€xn ð54bÞ€xn+ 1 = 0 ð54cÞ
with (gN ,bN ) the classical parameters of the Newmarkfamily. The three remaining variables are kept constantso that zn+ 1 = zn, xori, n+ 1 = xori, n and un+ 1 = un. Theresidue Rn+ 1 is introduced
Rn+ 1 =€xn+ 1 + 2z _xn+ 1
+ xn+ 1 � zn+ 1 � xori, n+ 1ð Þ � g cos Otn+ 1
ð55Þ
Table 2 gives an overview of the main steps of thenumerical integration. In the numerical steps, we focuson the case of forward transformation for the sake ofsimplicity. In addition, to ensure Kuhn–Tucker condi-tions at the discrete level, simultaneous test of the posi-tivity of F cri
1 at time n: F cri, n1 .0 and of _F cri
1 viaF cri, n+ 1
1 .F cri, n1 is computed. Enforcing 1� zn+ 1.0,
an update of zn+ 1 is obtained by expanding the criteriafunction at time tn+ 1 at first orderF cri, n+ 1
1 =F cri, n1 + ∂F cri
1 =∂z� �
n� dzn+ 1.
As cancellation of F cri, n+ 11 is searched, one gets the
increment dzn+ 1 for correcting the first prediction. Inthe case of the reverse transformation, one has to sub-stitute F cri
2 for F cri1 and to enforce zn+ 1.0 in the previ-
ous reasoning. In the numerical calculations, theNewmark parameters have been set to their classicalvalues gN = 1=2 and bN = 1=4, while a tolerance eeq of10210 has been used. Results are shown in the next sec-tion for isothermal and non-isothermal oscillations.
Isothermal oscillations
Case 1. In this section, the numerical results providedby our model are compared to those presented by
Lacarbonara et al. (2004). In this first case, the follow-ing values are considered for the material parameters:q1 = 1:3, q2 = 0:667, q3 = 0:9 and J = 0:315.Following the identification procedure explainedbefore, the ZM parameters are deduced and given inTable 3. Comparison of results are reported on FRFs,in order to get the most complete picture of thedynamics, in the vicinity of the normalized eigenfre-quency. A simple and direct method for obtainingFRFs is used: for each value of the forcing frequencyO 2 ½0, 2�, a time simulation is performed. Neglectingthe transient phase, the maximum amplitude of abso-lute value of the displacement jxjmax is recorded. Forthe simulations, the time awaited for the transient todie away has been set to 80 periods of the forcing, while
Table 2. Numerical scheme for time integration.
1. The convergence criterion is set to Rn+ 1j j<eeq Rnj j2. Updating of mechanical variables is performed after
determining the displacement increment( ∂R
∂dx
� �ndxn+ 1 = �Rn, where ∂R
∂dx
� �n= 1+ 1
bDt2 + 2z gbDt
),
using Geradin and Rixen (1994)3. Computation and test of criteria functions corresponding to
equation (37)4. Computation of the increment dzn+ 1
5. Updating value of zn+ 1 = zn+ 1 + dzn+ 1
6. Computation of the thermodynamic force Az, n+ 1
7. Temperature can be predicted from the heat equationdiscretized by an Euler implicit scheme:
un+ 1 =un+ 1 + huexdt +
Az, n+ 1rTref CP
�zn+ 1�znð Þ
1+ hdt� jrCp
�zn+ 1�znð Þ
8. Computation of the latent heat density C un+ 1ð Þ and criteriafunctions F cri, n+ 1
1, 29. Repeat the process until mechanical equilibrium and criteria
are fulfilled
Table 3. ZM parameters for case 1, corresponding to thefollowing parameters in Lacarbonara et al. (2004): q1 = 1:3,q3 = 0:9, q2 = 0:667 and J = 0:315.
Symbol Value
a 17:9203106 Pab 17:9203106 Pab 1:47323109 Paa 1:47323109 PaG 26:883106 PaY 1643106 Pak 8:683106 Paj 0:531143106 Pa= 8Cemax 0.112Af 38.5945 KEa 531010 PaEm 531010 Pau0 233.3498 K
ZM: Zaki–Moumni.
1602 Journal of Intelligent Material Systems and Structures 23(14)
the settle time used to pick out the maximum amplitudeis 120 periods. Time integration is performed with theadapted Newmark scheme described in section‘Numerical scheme’, where the time step is selected asdt = 2p=1000O, corresponding to 1000 points per peri-ods. In order to get the complete branches of solutionsin the non-linear range, the procedure is repeated forincreasing and decreasing frequencies. For each forcingfrequency O, the initial condition selected for the timesimulation is taken as the end point of the previousrun, in order to minimize the transient time. Figure 7(a)shows the FRFs obtained for four increasing values ofthe forcing amplitude: g = 0.1, 0.2, 0.5 and 0.8. Forg = 0.1, the amplitude of the response never exceeds 1,so that no phase transformation occurs: a linearresponse is observed. For g = 0.2, non-linear responseof the material is excited. As the effective stiffness ofthe material severely decreases when martensitic trans-formation begins, a softening behaviour is observedwith appearance of jump phenomena, which areenhanced for g = 0.5. In that case, a first jump isobserved at point A when increasing excitation fre-quency from 0 and the amplitude of the response sud-denly increases as the response jumps to the higherbranch. On the other hand, when decreasing the excita-tion frequency and following the higher branch, a jumpis observed at point B where the response suddenly goesto very small values. For g = 0.8, levels of responseamplitudes are attained such that the martensitic trans-formation is completed. This results in the appearanceof a third branch of solutions. When increasing O from0, a first jump is observed at point C where the solutiongoes to the third branch, which is left at point D wherea second jump occurs. Decreasing O from 2, jumps arenow observed at points E and F. The FRFs obtained
with our model are completely in the line of previousresults reported elsewhere, where the softening-typebehaviour and the appearance of the third branch witha hardening-type non-linearity have already beenobserved by Lacarbonara et al. (2004). A quantitativecomparison with their results is shown in Figure 7(b),for g = 0.5 and 0.8. They used a refined continuationmethod so that stable and unstable responses areobtained. This is in contrast with our result where thedirect integration only leads to the stable solutions. Thequantitative comparison for g = 0.5 shows a very goodagreement, the only noticeable difference being thesaddle-node bifurcation at point A occurring later inour model, for O = 0.7 instead of 0.6 in their work.The second noticeable difference is the maximumamplitude of the main branch, near point B, which is alittle bit less in our model. However, all the other fea-tures are qualitatively and quantitatively found. Thesame comparison gives satisfactory results for g = 0.8,where the saddle-node bifurcation points and theamplitudes of the solution responses are the same.However, a main difference is found in the low-frequency range, where the model of Lacarbonara et al.predicts occurrence of 1/3 and 1/5 superharmonic reso-nance whereas our model do not. In our simulations,the amplitude of the response never exceeds 1 for Ofrom 0 to point C, hence no phase transformationoccurs and no superharmonic resonance can be excited.
The response is further investigated by increasing theamplitude of the forcing to g = 1:2. Figure 8(a) showsthat the hardening-type behaviour now dominates theresponse. Superharmonic responses are also nowexcited in the low-frequency range, resulting in a disor-dered cloud of points. To get insight into the responsein that frequency range, Poincare section (stroboscopy
0 0.5 1 1.50
2
4
6
8
10
12
0 0.5 1 1.50
2
4
6
8
10
12
0.4 0.6 0.8 1 1.20
2
4
6
8
Ω[adim]
Ω[adim]
Ω [adim]
max
[adi
m]
|x||x| m
ax[a
dim
]
|x| m
ax[a
dim
]γ = 0.8
(a) (b)
A
B
C
D
EF
A
B
C
D
EF
ΩΩ
= 0.2γ
= 0.5γ
= 0.5γ
= 0.8γ
= 0.1γ
Figure 7. (a) Maximum amplitude of displacement jxjmax of the device for varying excitation frequencies O and for four amplitudesof forcing: g = 0.1, 0.2, 0.5 and 0.8. (b) Comparison of the results provided by our model (solid line) and those by Lacarbonara et al.(circles), for g = 0:8 and g = 0:5 (inset).adim: adimensional.
Moussa et al. 1603
at the forcing frequency) is computed to distinguish per-iodic from chaotic response, the result is shown inFigure 8(b). The term ‘chaos’ is used here in its classicaldefinition from dynamical systems theory that can befound from many textbooks (e.g. Guckenheimer andHolmes, 1983; Manneville, 1990; Schuster and Just,2005), that is, a non-periodic permanent state, charac-terized by a strange attractor displaying fractal dimen-sion and (at least) one positive Lyapunov exponent.Increasing the excitation frequency from 0, one canobserve clearly the successive appearance of 1/9 and 1/7superharmonic resonance. Below O= 0:2, periodicresponse persists, as indicated by the single point givenby the Poincare section. In the region where 1/5 and 1/3superharmonic resonance should be excited, a chaoticregion is found. At the limiting values of the chaoticregion, pitchfork symmetry-breaking bifurcations areobserved and followed by period-doubling scenario.The symmetry-breaking bifurcation gives rise to unsym-metrical responses with appearance of even harmonicsin the response, whereas the internal force is symmetric.This feature has been first observed in the Duffing oscil-lator by Parlitz and Lauterborn (1985) and is alsoreported in the work by Lacarbonara et al. (2004) butfor other parameter values, as they did not test suchhigh values of forcing amplitude. The chaotic responseis illustrated in Figure 9 for O= 0:23 and g = 1:2. Thechaotic behaviour is here assessed by the continuouscomponent in the Fourier spectrum (Figure 7(b)) aswell as in the self-similar (fractal) appearance of thephase portrait (Figure 7(c)). The period-doubling routeobserved in Figure 9(b) is also a clear indicator statingthe presence of chaotic oscillators at the end of theFeigenbaum scenario (Guckenheimer and Holmes,1983; Schuster and Just, 2005). Note that a completecharacterization of the chaotic nature of the dynamics,which would be ascertained by the presence of (at least)
one positive Lyapunov exponent, was not the primarygoal of that article. The interested reader can find suchdevelopments in Machado et al. (2009) as well as inLacarbonara et al. (2004) and Sitnikova et al. (2008,2010). The symmetry-breaking bifurcation is clearly evi-denced by the appearance of even harmonics in theresponse, as shown in Figure 9(b). The chaotic motionis characterized by a chaotic amplitude modulation,whereas a strong persistency of the excitation frequencyis observed. The phase portrait underlines the fractalnature of the attractor as well as its asymmetry betweenpositive and negative values of the displacements,resulting from the symmetry-breaking bifurcation.Finally in the restoring force, one can observe an accu-mulation of straight lines near xMf , highlighting the factthat chaotic responses are observed for oscillationamplitudes in the vicinity of the completed martensiticphase.
Case 2. In this section, our model is further confrontedto the results presented in Lacarbonara et al. (2004),where an hysteresis loop having nearly flat pseudoelas-tic plateaus and a larger area is considered by setting,for their model: q1 = 1:05, q3 = 0:3, q2 = 0:833 andJ = 0:315. The corresponding ZM parameters are thenfixed to the values given in Table 4. Figure 10 showsthe frequency–response obtained in that case, for fourincreasing values of the forcing amplitude, g = 0.1,0.2, 0.5 and 0.8. Results have been quantitatively com-pared to theirs and show once again very good agree-ment. In comparison to case 1, all the essential featuresare found back. The larger area of the hysteresis loopin case 2 with comparison to case 1 induces that moredissipation is at work in the non-linear responses sothat the peak amplitudes for each value of g is smallerthan those found in case 1. Once again, no
0.05 0.1 0.15 0.2 0.25 0.33
4
5
6
7
8
9
10
0 0.2 0.4 0.6 0.8 10
2
4
6
8
10
12
14
16
Ω [adim] [adim]
|x| max
[adi
m]
|x| strob
[adi
m]
(a) (b)
1/9
1/7
1/5 1/3
ΩΩ
Figure 8. (a) Maximum amplitude of displacement jxjmax of the device for varying excitation frequencies O and for g = 1:2. (b)Poincare section for O 2 ½0, 0:35�, showing the appearance of chaotic solutions.adim: adimensional.
1604 Journal of Intelligent Material Systems and Structures 23(14)
superharmonic responses are found below the normal-ized eigenfrequency for g = 0.8, which is also the resultshown in Lacarbonara et al. (2004). The appearance ofa third branch corresponding to completed martensitictransformation is also depicted as in case 1.Interestingly, this branch splits into two parts, which isin line with the results they presented where chaoticresponse were exhibited on the low-frequency part ofthis third, upper branch. A Poincare section is realizedfor the corresponding frequency and amplitude rangeand is shown in the inset of Figure 10, clearly eviden-cing the fact that chaotic responses are also predictedby our model. The chaotic response is shown in Figure
11 for O= 0:22. Once again, the chaotic character ofthe response is completely enclosed in the temporal var-iations of the envelope, see Figure 11(a). The timeseries of the volumic fraction as well as the behaviourof the internal force, shown in Figure 11(d) and (e),respectively, highlights the fact that during the oscilla-tions, the system either fully reached the completedtransformation, so that z saturates to 1, or missed thecomplete transformation so that z starts to decreasejust before reaching 1. These results show that chaotic
200 400 600 800−10
−5
0
5
10
0 0.5 1 1.5 2 2.5 3
10−10
10−5
100
−10 −5 0 5 10−4
−2
0
2
4
−10 −5 0 5 10−3
−2
−1
0
1
2
3
x [a
dim
]
τ [adim] Ω [adim]
FFT(x)
(a) (b)
(c) (d)
x [adim]
v [a
dim
]
x [adim]f
[ad
im]
Ω2Ω 3Ω
4Ω 5Ω6Ω 7Ω
pc1
pc2pc3
pc4 pc5pc6 pc7
pc8pc9 pc10
pc11
Ω
Ω
2Ω3Ω
4Ω 5Ω6Ω
7Ω8Ω
Figure 9. Chaotic response for case 1, with O= 0:23 and g = 1:2: (a) time response in the permanent regime, (b) Fouriertransform of displacement x, (c) phase portrait displacement x versus velocity v = _x and (d) restoring force of the device as afunction of the displacement.adim: adimensional; FFT: Fast Fourier Transform
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80
1
2
3
4
5
6
7
8
9
10
0.2 0.22 0.24 0.26
7.8
8
8.2
8.4
Ω [adim]
Ω [adim]
|x| m
ax[a
dim
]
xst
rob
[adi
m]γ = 0.8
γ
γ
γ
Ω
Ω= 0.5
= 0.2
= 0.1
Figure 10. Maximum amplitude of displacement jxjmax for case2 and for increasing values of the forcing, g = 0.1, 0.2, 0.5 and0.8. Inset: Poincare section for O 2 ½0:18, 0:26� and g = 0:8,corresponding to the solutions found on the upper (third)branch.adim: adimensional.
Table 4. ZM parameters for case 2, corresponding to thefollowing parameters in Lacarbonara et al., (2004): q1 = 1:05,q3 = 0:3, q2 = 0:833 and J = 0:315.
Symbol Value
a 33:603106 Pab 33:603106 Pab 0:714293109 Paa 0:714293109 PaAf 306:7763 K KY 163107 Pak 29:1203106 Pau0 264:8183 KG 4:48003106 Pa
ZM: Zaki–Moumni.
emax , Ea and Em have not been reported in the table as they do not
change with respect to case 1.
Moussa et al. 1605
responses are generally observed at the limit pointwhere martensitic transformation is completed. Phaseportrait and Fourier transformation can be comparedto the ones in Lacarbonara et al., showing an impres-sive agreement between the predictions given by thetwo models.
Asymmetric restoring force. In this paragraph, thedynamic analysis is carried out in the case of asym-metric constitutive law. Indeed, according to experi-ments, SMAs clearly exhibit an asymmetric behaviourin tension and in compression (Lexcellent and Rejzner,2000; Orgeas and Favier, 1998). The ZM model cantake this asymmetry into account by introducing thethird invariant in the expression of the equivalent det-winning strain (Morin et al., 2011; Zaki, 2010). Thisasymmetric 3D behaviour law can be reduced to a 1Dlaw following the procedure explained in section‘Reduced model for a uniaxial SMA oscillator’. Afterreduction, the parameters defining the asymmetricbehaviour are now given by two values for emax, one fortensile loading et
max and one for compression ecmax. In
this section, the model parameters of case 1 have beenselected, except the values of emax that have been set toet
max = 0:112 and ecmax = 0:064, respectively. As com-
pared to the symmetric case studied in previous sec-tions, the compression loop has been shortened and theplateaus made more stiff. The behaviour of this asym-metric device is shown in Figure 12(b). The frequency–response curve for an excitation amplitude of g = 0:5 is
shown in Figure 12(a) where the maximum amplitude iscompared to the symmetric case. As the behaviour hasbeen selected more stiff in compression, the softeningbehaviour is less important than in the symmetric case.Asymmetry is also observable by the fact that maxi-mum and minimum values of the displacement are nowdifferent. Inset in Figure 12(a) shows minus the mini-mum displacement (-xmin) for comparison. Figure12(c) shows the time series of the displacement forO= 0:6, where this asymmetry is also clearly evidenced:one can see different values of maximum displacementin tensile loading and compression, respectively.Consequently, the Fourier spectrum of the displace-ment now shows odd and even harmonics, whereas thedisplacement corresponding to symmetric restoringforce contains only odd harmonics, as shown in Figure12(d). This shows that the model is able to reproducethe more realistic case of an asymmetric restoring forceand reproduce the expected features corresponding tothat case.
Non-isothermal oscillations
As shown previously, SMAs display thermomechanicalcoupling, which is responsible for numerous importanteffects that have already been underlined (Bernardiniand Rega, 2009; Bernardini and Vestroni, 2003; Heand Sun, 2011; Morin et al., 2011). Hence, it is neces-sary to take into account this phenomena, when deal-ing with the dynamic behaviour of the material. The
−10 −5 0 5 10−3
−2
−1
0
1
2
3
1000 1100 1200
0
0.2
0.4
0.6
0.8
1
−5 0 5
−1
0
1
0 0.5 1 1.5 2 2.510
−10
10−5
100
500 1000 1500 2000 2500−10
−5
0
5
10
ω [adim]
FFT
x [adim]
v [a
dim
]
z
t [adim] x [adim]
f [ad
im]
(b)
(d) (e)(c)
(a)x
[adi
m]
t [adim] Ω
Figure 11. Chaotic response of the device for case 2, O= 0:22 and g = 0:8. (a) Time series of the displacement x in thepermanent regime, (b) FFTof x, (c) phase portrait , (d) time series of the volumic fraction z and (e) internal force versusdisplacement in the permanent regime.adim: adimensional; FFT: Fast Fourier Transform
1606 Journal of Intelligent Material Systems and Structures 23(14)
equations of motions now include the temperature var-iations as shown in section ‘Reduced model for a uni-axial SMA oscillator’. Heat sources are the sum of aterm proportional to _z, describing heat release andabsorption due to the martensitic forward and reversetransformations (latent heat), together with a term pro-portional to h, and describing the heat exchange withexternal environment. It is worth noting that the cur-rent contribution neglects thermoelastic coupling effect.Moreover, choosing the value of the material convec-tive coefficient h is inspected in Figure 13, showing theforced response of the non-isothermal system for a for-cing amplitude of g = 0:8 and a frequency O= 0:8.Material parameters, unless specified, are those fromcase 1 of the previous section (see Table 5 for addi-tional parameters). In particular, a symmetric internalrestoring force is selected. For h = 0, an adiabatic sys-tem is at hand. As no thermal energy is transferred tothe environment, the heat generated by the martensitictransformation remains in the device that continuouslywarms up. Hence, no stable state solutions exist for thetemperature, so that this non-realistic case is not proneto a study on permanent states. As soon as h.0, energyexchange with environment makes possible the exis-tence of a permanent regime where the temperature sta-bilizes at a mean constant value. For increasing values
of h, one can observe that (a) the transient regime isshorter (for h = 0.01, permanent regime is obtainedafter time t = 1000); (b) the mean temperature stabi-lizes at a smaller value and (c) amplitude of oscillationsaround this mean value also decreases. These threeobservations are completely in the lines of the thermalbehaviour and the equilibrium between the device andthe external environment, governed by h. The fre-quency of the oscillations in u is also seen to be twicethat of the displacement x, since for one cycle of thedevice, including tension and compression, two cyclesfor the temperature are realized. Finally, Figure 13clearly highlights the fact that the behaviour of u isalmost completely driven by that of the volumic frac-tion of martensite z.
As highlighted in section ‘Model equations’, thetemperature is prone to show important variations,especially during a simulation for constructing an FRFwhere the initial condition of a new run is selected asthe last values of the precedent (see section ‘Isothermaloscillations’). Before showing the numerical frequency–response curves obtained in the non-isothermal case,the most important effects of initial temperature andthermomechanical coupling are shown in Figure 14.
Figure 14(a) highlights the fact that for two increas-ing values of the initial temperature (u0
1 = 408C \
0.2 0.4 0.6 0.8 1 1.2 1.40
1
2
3
4
5
6
7
0.5 1 1.50
2
4
6
140 160 180 200 220 240 260−4
−2
0
2
4
0 1 2 3 4 5 6
10−10
100
0 2 4 6 80
0.5
1
1.5
Ω [adim]
x max
Ω [adim]
Ω [adim] |f| [a
dim
]
|x| [adim]
t [adim]
x [a
dim
]
FFT(x)
min−x
(d)
(b)(a)
(c)
compression
tractionTensile
Ω
Ω
Ω
Figure 12. Asymmetric behaviour of the pseudoelastic device. (a) Frequency–response curve xmax (inset: xmin) versus excitationfrequency O, for g = 0:5, (b) asymmetric behaviour (solid line) is compared to the corresponding symmetric behaviour (dash-dottedline), (c) time response x in the permanent regime for O= 0:6, upper branch and (d) Fourier transform of the displacement x (solidline), compared to the symmetric case (dash-dotted line).adim: adimensional; FFT: Fast Fourier Transform
Moussa et al. 1607
u02 = 538C), force thresholds and loop area increase. Inaddition, the already underlined features (thermalhardening and loop area variation) are retrieved in thebehaviour of the device, as shown in Figure 14(b) that
compares isothermal and non-isothermal stress–strainrelationships, for an initial temperature ui = 408C.
Finally, frequency–response curves are shown inFigure 15 for two different values of h. Because of ther-momechanical coupling, as illustrated in Figure 13,transients are now longer as compared to the isother-mal case. Hence, the number of periods awaited for thetransient to die away has been set to 320, and the settletime for recording the maximum amplitude is set to 80periods. This results in simulation times significantlylonger. As compared to the isothermal case, softeningeffect is less pronounced for h= 0:008. For decreasingvalues of h, one can see the disappearance of the third
0 100 200 300 400 5001
1.2
1.4
1.6
1.8
2
2.2
400 405 410 415 4200
0.1
0.2
400 405 410 415 4201.06
1.08
1.1
1.12
400 405 410 415 420
−2
0
2[a
dim
]θ [a
dim
]θ
[adim]
(c)
(b)
(d)
(a)
x [a
dim
]z
h = 0
h = 0.01
h = 0.02
h = 0.08
[adim]τ τFigure 13. Non-isothermal behaviour of the shape memory device. (a) Time series of the non-dimensional temperature u whenthe device is forced harmonically with an amplitude g = 0:8 and a frequency O= 0:8, for four increasing values of h: 0 (adiabatic),0.01, 0.02, and 0.08; (b) displacement x versus time for h= 0:008 in the permanent regime; (c) temperature u and (d) volumicfraction z at the same instants of time.adim: adimensional.
0 1 2 30
0.5
1
1.5
0 0.5 1 1.5 2 2.50
0.2
0.4
0.6
0.8
1 Non-isothermal behavior
Isothermal behavior
(a) (b)
x [adim]
x [adim]
f[a
dim
]
f[a
dim
]
θ0 =53°C2
θ0 =40°C2
Figure 14. Non-linear restoring force in non-isothermal conditions (only positive part is shown and negative part is taken assymmetric). (a) Initial temperature effect on mechanical response and (b) comparison between non-isothermal and isothermalbehaviours (h = 0.08).adim: adimensional.
Table 5. Additional material parameters (specific heat capacityCP, density r and external medium temperature Text)
Symbol Value
CP 440 J K21 m23 kg21
r 6500 kg m23
Text 333 K
1608 Journal of Intelligent Material Systems and Structures 23(14)
upper branch occurring for h= 0:001. Small values ofh mean that the material does not get enough time toevacuate all heat and lead to an increase of the tem-perature, and as highlighted above, the phase transfor-mation does not complete.
Figure 16(a) and (b) shows the frequency–responseof the mean value (umoy = 1=N (
PNi= 1 u(ti)), where N is
the remainder periods in permanent regime) of the tem-perature and the temperature variation (Du=umax � umoy), respectively. One can observe similar fea-tures as that of maximum displacement: jump phenom-enon, bifurcation at the frequency phase change onsetand the emergency of a third stable branch.
Dissipation as a function of loading frequency
This section investigates the variations of the dissipatedenergy in the SMA device, with respect to loading
0 0.5 1 1.5 20
0.02
0.04
0.06
0.08
0.1
0.12
0 0.5 1 1.5 21
1.05
1.1
1.15
1.2
1.25
1.3
θmoy [adim] ∆θ [adim]
Ω [adim]Ω [adim]
(a) (b)
Figure 16. Frequency–response of material temperature in non-isothermal conditions for h= 0:08 and g = 0:8: (a) mean valueumoy and (b) temperature variation Du.adim: adimensional.
0 0.5 1 1.5 20
1
2
3
4
5
6x 10
−3
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
x 10−8HH
(a) (b)Ω Ω
Figure 17. Area loop (J m�3) at g = 0:8: (a) isothermal case and (b) non-isothermal case (h= 0:01).
0 0.5 1 1.50
2
4
6
8
10
12
[adim]Ω
|x|
[adi
m]
max
h = 0.08
h = 0.01
isothermal
Ω
Figure 15. Maximum amplitude of displacement jxjmax of thedevice for varying excitation frequencies O. Isothermalconditions (dash-dotted line) is compared to non-isothermalcase with h= 0:008 and h= 0:01.adim: adimensional.
Moussa et al. 1609
frequency in both isothermal and non-isothermal cases.The dissipated energy is defined by the area of the hys-teresis loop under one cycle (
Hf _xdt). Figure 17(a) shows
a constant value of dissipated energy (area of hysteresisloop) remaining in the frequency range 0:2 0:57½ �.That is due to the accomplishment of phase change andthe absence of viscosity. Indeed, the loop area does notchange with respect to loading frequency.
In the case of non-isothermal oscillations (Figure 17(b)), the area of the hysteresis loop does not reach athird stabilized branch in frequency–response and wit-nesses a decrease as compared to the isothermal case inthe same frequency range. One can highlight that forsmall values of h, increasing loading amplitude couldallow to reach a stabilized value of the dissipatedenergy.
Conclusion
The model was extended to account for thermomecha-nical coupling as a means for describing strain ratedependency. A reduced version was then used to simu-late the response of a single-degree-of-freedom SMAoscillator. The simulations were shown to fit experi-mental data taken from Lacarbonara et al. (2004).Time integration of the equations of motion using thereduced model was accomplished using a Newmarkscheme. Deviation from experimental data was dis-cussed, and new results were obtained for higher forcingand for asymmetric restoring force. Non-linear featuresof the damper response were observed, including jumps,period-doubling, symmetry-breaking bifurcations andchaotic responses. The influence of thermomechanicalcoupling on the frequency–response and on dissipationis studied.
The ZM model, as extended in this article, can nowbe used to simulate the dynamic response of SMAstructures. Future work will focus on modelling a tor-sional pendulum system, like the one in Doare et al.(2011) and Sbarra et al. (2011).
Acknowledgements
The authors would like to thank P. Riberty for his help inmechanical experiments and Lahcene Cherfa for designingexperiment specimens.
Funding
This work is funded by the FNR (Fonds National deRecherche) via CRP HENRI TUDOR center atLuxembourg.
References
Bernardini D (2001) On the macroscopic free energy func-
tions for shape memory alloys. Journal of the Mechanics
and Physics of Solids 49: 813–837.
Bernardini D and Pence TJ (2005) Uniaxial modeling ofmulti-variants shape memory materials with internal sub-
looping using dissipation functions. Meccanica 40:339–364.
Bernardini D and Rega G (2009) The influence of model para-
meters and of the thermomechanical coupling on the beha-vior of shape memory devices. International Journal of
Non-Linear Mechanics 45: 933–946.Bernardini D and Vestroni F (2003) Non-isothermal oscilla-
tions of pseudoelastic devices. International Journal of
Non-Linear Mechanics 38: 1297–1313.Chrysochoos A, Pham H and Maisonneuve O (1996) Energy
balance of thermoelastic martensite transformation understress. Nuclear Engineering 162(1): 1–2.
Collet M, Foltete E and Lexcellent C (2001) Analysis of the
behavior of a shape memory alloy beam under dynamicalloading. European Journal of Mechanics-A/Solids 20: 615–
630.Doare O, Sbarra A, Touze C, et al. (2011) Experimental anal-
ysis of the quasi-static and dynamic behavior of shape
memory alloys. International Journal of Solids and Struc-
tures 49: 32–42.Feng ZC and Li DZ (1996) Dynamics of a mechanical system
with a shape memory alloy bar. Journal of Intelligent
Material Systems and Structures 7(4): 399–410.Funakubo H (1987) Shape Memory Alloys. Precision Machin-
ery and Robotics, vol. 1. CRC Press. Gordon and Breach
Science Publishers.Geradin M and Rixen D (1994) Mechanical Vibrations: The-
ory and Application to Structural Dynamics. Chichester:
Wiley.Grabe C and Bruhns OT (2008) On the viscous and strain rate
dependent behavior of polycrystalline NiTi. InternationalJournal of Solids and Structures 45(7–8): 1876–1895.
Guckenheimer J and Holmes P (1983) Nonlinear Oscillations,
Dynamical Systems and Bifurcations of Vector Fields. NewYork: Springer-Verlag.
Hartl DJ, Mooney JT, Lagoudas DC, et al. (2010) Use of a
Ni60Ti shape memory alloy for active jet engine chevronapplication. I. Thermomechanical characterization. Smart
Materials and Structures 19(1): 015020 (14 pp).He YJ and Sun QP (2011) On non-monotonic rate depen-
dence of stress hysteresis of superelastic shape memory
alloy bars. International Journal of Solids and Structures
48: 1688–1695.Helm O and Haupt P (2002) Thermomechanical representa-
tion of the multiaxial behavior of shape memory alloys. In:
Lynch CS (ed.) Proceedings of SPIE: Smart Structure and
Materials 2002: Active Materials: Behavior and Mechanics,
vol. 4699. Society of Photo Optical, pp. 343–354.Hughes T (2000) The Finite Element Method: Linear Static
and Dynamic Finite Element Analysis. Dover Publications.Humbeeck JV and Delaey L (1981) The influence of strain-
rate, amplitude and temperature on the hysteresis of apseudoelastic Cu-Zn-Al single crystal. Journal de PhysiqueIV 42: 1007–1011.
Lacarbonara W, Bernardini D and Vestroni F (2004) Non-
linear thermomechanical oscillations of shape memorydevices. International Journal of Solids and Structures 41:
1209–1234.Lagoudas D, Machado L and Lagoudas M (2005) Nonlinear
vibration of a one-degree of freedom shape memory alloy
1610 Journal of Intelligent Material Systems and Structures 23(14)
oscillator: a numerical-experimental investigation. In:Structural dynamics and materials conference. Austin, TX,18–22 April.
Leo PH, Shield TW and Bruno OP (1993) Transient heattransfer effects on the pseudoelastic behavior of shapememory wires. Acta Metallurgica and Materialia 41(8):2477–2485.
Lexcellent C and Rejzner J (2000) Modeling of the strain rateeffect, creep and relaxation of a NiTi shape memory alloyunder tension (compression)-torsional proportional load-ing in the pseudoelastic range. Smart Materials and Struc-
tures 9: 613–621.Lin P, Tobushi H, Tanaka K, et al. (1996) Influence of strain
rate on deformation properties of TiNi shape memoryalloy. JSME International Journal Series A-Mechanics and
Material Engineering 39(1): 117–123.Machado LG, Lagoudas DC and Savi MA (2009) Lyapunov
exponents estimation for hysteretic systems. InternationalJournal of Solids and Structures 46: 1269–1286.
Machado LG, Savi MA and Pacheco PMCL (2003) Non-linear dynamics and chaos in coupled shape memory oscil-lators. International Journal of Solids and Structures
40(19): 5139–5156.Manneville P (1990) Dissipative structures and weak turbu-
lence. Boston: Academic Press.Morin C, Moumni Z and Zaki W (2011) A constitutive
model for shape memory alloys accounting for thermome-chanical coupling. International Journal of Plasticity 27:748–767.
Moumni Z (1995) Sur la modelisation du changement de phase
a l etat solide. PhD Thesis, Ecole Nationale Superieure desPonts et Chaussees, Paris, France.
Moumni Z, Zaki W and Nguyen QS (2008) Theoretical andnumerical modeling of solid-solid phase change: applica-tion to the description of the thermomechanical behaviorof shape memory alloys. International Journal of Plasticity24: 614–645.
Mukherjee K, Sircar S and Dahotre NB (1985) Thermaleffects associated with stress-induced martensitic transfor-mation in Ti-Ni alloy. Materials Science and Engineering
74: 75–84.Nemat-Nasser S and Wei-Guo G (2006) Superelastic and cyc-
lic response of NiTi SMA at various strain rate and tem-peratures. Mechanics of Materials 38(5–6): 463–474.
Orgeas L and Favier D (1998) Stress-induced martensitictransformation of a NiTi alloy in isothermal shear, tensionand compression. Acta Materialia 46: 5579–5591.
Parlitz U and Lauterborn W (1985) Superstructure in thebifurcation set of the Duffing equation. Physics Letters A107(8): 351–355.
Patoor E and Berveiller M (1990) Les Alliages a Memoire de
Forme. Paris: Hermes.
Qidwai MA and Lagoudas DC (2000) On thermomechanics
and transformation surfaces of polycrystalline shape mem-
ory alloy materials. International Journal of Plasticity 16:
1309–1343.Saadat S, Salichs J, Noori M, et al. (2002) An overview of
vibration and seismic applications of NiTi shape memory
alloy. Smart Materials and Structures 11(2): 218–229.Savi MA and Pacheco PMCL (2002) Chaos and hyperchaos
in shape memory systems. International Journal of Non-
Linear Mechanics 12: 645–657.Savi MA, Pacheco PMCL and Braga AMB (2002) Chaos in a
shape memory two-bar truss. International Journal of Non-
Linear Mechanics 37: 1387–1395.Savi MA, Sa MAN, Paiva A, et al. (2008) Tensile-compres-
sive asymmetry influence on shape memory alloy system
dynamics. Chaos, Solitons and Fractals 36: 828–842.Sbarra A, Doare O, Moussa MO, et al. (2011) Dynamical
behaviour of a shape memory alloy torsional pendulum.
In: Proceedings of ENOC 2011, Rome, Italy, 24–29 July.Schmidt I and Lammering R (2004) The damping behaviour
of superelastic NiTi components. Materials Science and
Engineering A 378: 70–75.Schuster HG and Just W (2005) Deterministic Chaos: An
Introduction. WILEY-VCH Verlag GmbH & Co. KGaA,
Weinheim, Fourth revised and augmented edition.Seelecke S (2002) Modeling the dynamic behavior of shape
memory alloys. International Journal of Non-Linear
Mechanics 37(8): 1363–1374.Shaw JA and Kyriakides S (1995) Thermomechanical aspects
of NiTi. Journal of the Mechanics and Physics of Solids
43(8): 1243–1281.Sitnikova E, Pavlovskaia E and Wiercigroch M (2008)
Dynamics of an impact oscillator with SMA constraint.
European Physical Journal Special Topics 165: 1269–1286.Sitnikova E, Pavlovskaia E, Wiercigroch M, et al. (2010)
Vibration reduction of the impact system by an SMA
restraint: numerical studies. International Journal of Non-
linear Mechanics 45: 837–849.Zaki W (2010) An approach to modeling tensile-compressive
asymmetry for martensitic shape memory alloys. Smart
Materials and Structures 19: 025009 (7 pp).Zaki W andMoumni Z (2007a) A 3D model of the cyclic ther-
momechanical behavior of shape memory alloys. Journal
of the Mechanics and Physics of Solids 55(11): 2427–2454.Zaki W and Moumni Z (2007b) A three-dimensional model
of the thermomechanical behavior of shape memory
alloys. Journal of the Mechanics and Physics of Solids
55(11): 2455–2490.Zaki W, Zamfir S and Moumni Z (2010) An extension of the
ZM model for shape memory alloys accounting for plastic
deformation. Mechanics of Materials 42: 266–274.
Moussa et al. 1611
