Hysteretic transformation behaviour of shape memory alloys

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Hysteretic transformation behaviour of shape memory alloys

J. Van Humbeeck, E. Aernoudt, L. Delaey, Lu Li, H. Verguts and J. Ortin*

Department of Metallurgy and Materials Engineering, de Croylaan 2,3030 Leuven, Heverlee, Belgium

(Reçu le 26 mai 1987, accepté le 8 septembre 1987)

Revue Phys. Appl. 23 (1988) 557-564 AVRIL 1988,Classification

Physics Abstracts81.30K - 81.40L

Résumé - Après un traitement thermomécanique approprié certains alliages de cuivre et de Ni-Tiprésentent un changement de forme (effet de mémoire) lors d’un cyclage thermique dans une zoned’environ 40 degrés.Deux conditions importantes pour obtenir cet effet sont nécessaires : il fautune transformation martensitique et thermoélastique dans la zone de cyclage thermique. La déforma-tion totale du changement de forme sera limitée à quelques pourcents. Dans la phase beta (au-dessusde la zone de transformation) le matériau présente un comportement pseudoélastique : quand on appli-que une déformation, la transformation martensitique sera induite à une contrainte critique. Cettecontrainte dépend linéairement de la température. Lors de cette transformation induite, une grandedéformation de l’échantillon est possible. En inversant la direction de la déformation, le matériauretrouve sa forme originale lors de la transformation inverse (martensite~beta). La courbe de trac-tion (03C3 - 03B5 ), présente une boucle d’ hytérés is bien fermée.

Ainsi, il est clair que la transformation martrnsitique induite, soit par refroidissement, soit pardéformation, et la transformation inverse montrent une hystérèse. L’origine de cette hystérèse peutêtre expliquée par des considérations thermodynamiques et l’induction de défauts par la transforma-tion. La déformation, apparemment plastique, est aussi liée à la croissance des variantes de marten-site sélectionnées.

Mécaniquement, la connaissance du comportement pseudoélastique est importante pour calculer les ca-ractéristiques de l’effet mémoire.

Abstract - After suitable thermomechanical treatment a series of Cu-based and Ni-Ti-alloys can

show a reversible shape change during heating and/or cooling over a temperature region of about40 degrees. Two important conditions for this effect are that a thermoelastic martensitictransformation occurs in this temperature range and that the total strain of its shape changedoes not exceed more than a few percent. At temperatures above the martensitic transformationtemperatures, the material behaves pseudoelastic : by applying a strain, the martensitictransformation will start at a critical stress, dependent on the temperature. After completetransformation a considerable strain can be obtained which disappears during unloading. Thestress strain curve, obtained at constant temperature, appears as a closed loop.

Both reversible transformations, thermally and strain-induced, and so the related shape change,occur in a hysteretic manner. The origin of this hysteresis can be explained by thermodynamicconsiderations and the occurrence of transformation induced defects. The apparent "plastic"deformation is also related to preferential growth of selected martensitic variants.

Mechanistically, the knowledge of the pseudoelastic loops is important to calculate the shapechange during thermal cycling. It has also been shown that pseudoelastic loops for

polycrystalline materials can be calculated using a rigourous mechanical model.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/rphysap:01988002304055700

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Introduction

Shape memory alloys are characterised by a

reversible shape change during heating and coolingin a small (± 40 degrees) temperature region.This effect is directly related to a thermoelasticmartensitic transformation occurring in this

temperature zone. A material deformed in themartensitic state will recover its original formwhen heated into the parent phase or beta state.

When the straining - heating (recovering) -cooling - straining cycle is repeated a sufficientnumber of times, also the deformed state will becreated spontaneously on coolin.g : the material issaid to have been "trained". The forward andreverse shape change start at the same

temperatures as - those where the martensitictransformation starts and ends :Ms, Mf, As and Af.Ms and Mf are respectively the start and end-

temperatures of the forward transformation (betato martensite) while As and Af are the start andthe end-temperature of the reverse transformation(martensite to beta). Since the shape change is

proportional to the transformed volume the same

temperatures are used to indicate the start andend of the shape change.

As and Af being shifted to somewhat highertemperatures relative to Ms and Mf a hystereticcycle is described. The hysteresis occurs thus inthe transforming volume as well as in the shapechange.

Another reversible shape change linked to

martensite is the deformation behaviour at a

constant temperature above Af. Relative largestrains (10 to 15 % for single crystals, 4 to 6 %for polycrystals) obtained after loadingannihilate completely during unloading althoughthe loading curve looks very similar to a plasticstress-strain curve.

During loading, the material will transform intothese martensite variants that most extensivelyrelax the applied stress while during unloadingthe material will retransform to the beta-phasesince above Af martensite is unstable at zero

stress.

It is thus clear that martensite can bestress- or strain-induced. It has been

experimentally shown and theoretically proven thatthe critical stress- or strain is (to a first

approximation) linearly dependent on temperature.Finally a special case of hysteretic loops shouldbe mentioned here, although it is not related to

the beta to martensite transformation. When a

fully martensitic sample (below Mf) is submittedto a stress-strain cycle a hysteretic closed loopcan also appear. This is in this case due to

reorientation of the variants to those variantsthat contribute most to the applied strain mode.The reverse reorientation also occurs hysteretic.Strains in the order of 1 % can be obtained.

Some thermodynamic considérations on thethermoelastic martensitic transformation

If one considers the free energy of the beta- andthe martensitic phase as a function of

temperature, an equilibrium temperature To existswhere both energies are equal or A G = 0. Howeverdue to nucleation, the forward transformation willstart at a temperature Ms, lower than To where acritical value AGc exists. Once 0394G ~ 0394Gc, themartensitic transformation starts. A similar

hypothesis is valid for the reverse

transformation : As thus being above To. One canconsider this as a part of the hysteresis although

it is probably a minor contribution to it.Therefore we omit the problem of nucleation andconsider the transformation as purely thermo-elastic. During this transformation, thedifference in free energy between beta andmartensite should be zero :

where T is the temperature, x the transformedvolume and p the pressure taken as a constant.This means that the chemical deriving force is

completely balanced by a nonchemical force :

aGnchem consists of a reversible part, 0394Grev suchas the elastic strain energy, induced in thematerial by local shape change and AGirrev whichcan be seen as a frictional force due to the

movement of the martensitic interface in thelattice. The value of4YGirr determines mainly thehysteretic strenght. For, this AGirr as well as

the other 6G-values, no quantitative data are

available yet. Therefore, the description of the

hysteretic loops still is purely experimental.However some mechanistic models are developed inorder to describe in a general and mathematical

way the observed behaviour (1-7). In the presentpaper we will therefore review the experimentalobservations and mention the important conclusionswhen they are involved in shape memory

applications.

Hysteretic behaviour : thermomechanical paths of

trained shape memory alloys

1. The influence of stress

Fig. 1 shows a set of isostress temperature-straincurves of a trained Cu-Zn-Al sample with Ast = 48°C and Mst = 46° C. Between temperature cycles 30°-83°-30°, a stress increment of 25 MPa is given atthe low temperature. The increase of thetransformation temperatures is shown as a functionof the applied stress.The difference in slope of heating and coolinglegs at the loop ends indicates that furthertransformation is possible by further cooling orheating. So the curves are taken within thetransformation region. The shift between closed

loops i and a is thus due to thermal cycling.

2. The influence of previous history (9)

Fig. 2 and 3 show repectively the thermomechanicaltreatment and the response of the material to thistreatment. Instead of using a simple linear

stress-time profile, the specimen was subjected tospecial stress-time profiles in the form of a sawtooth with different heights to show also the formof isothermal hysteresis subloops.After cold loading the specimen and heating up to61° C, fully in the two phase region beta-

martensite, profile B was applied. With

progressive unloading and reloading, (Fig. 3B),the specimen transforms further to beta and thestrain at 150 MPa decreases each time, points 6 a

b c d e. Then, profile C is applied. The

hysteresis subloops are closed with common pointat 0 MPa.

Then, profile D (the same as B) is applied.The hysteresis loops are closed with common pointat 150 MPa. Note that point 6 in B is more to themartensite side than point 6 in D, although both

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are at the same temperature and stress. This isdue to the different way point 6 was reached inboth cases before applying B or D.Next, a small heating and cooling cycle at 0 MPawas performed. The specimen is at 61° C more to

the beta side than before this small cycle. Thisis a normal hysteresis effect, due to thetransformation temperature hysteresis.So point 0 in F is at a lower strain than point 0in D.

Then, profile F is applied (the same as C). Aftereach loading-unloading, the strain at 0 MPa

increases, the retained martensite increases, the

specimen moves away from the beta form.Applying profile G (the same as D) gives loopsclosed at 150 MPa. The form is very similar tothese of D, but their position is different. Thewhole stress-strain loops are more to the betaside in G than in D. Applying profile H (same asF and C), gives closed loops at 0 MPa, but herealso they are shifted to the beta side when

compared with C.The occurrence of open or closed loops, and shiftsin position of the loops within the generaltransformation domain, as described here, are

caused by the hysteresis of the transformation.The form and the position of the curves are

dependent on previous history.In fig. 4 and 5, the influence of a simple

stress-strain cycle, in the warm or cold

condition, on the following isostress temperature-strain curve is shown.In fig. 4 a simple loading-unloading cycle in coldcondition induces more martensite, and increasesthe starting strain (move to the martensite side)for a subsequent heating cycle. The higher thecold peak load, the greater the shift in strain.This is clearly a hysteresis effect. At the same

time, an increase in Ast is noted. In fig. 5, a

similar experiment is performed on the beta side.When performing thermal isostress loops (150 MPa),partly unloading-loading in warm condition changesthe starting strain at the high temperature. Thelower the stress to which is unloaded, the moremartensite reverts to beta, and the greater is theshift to the beta side. So the cooling curves at150 PMa, a to g are necessarily different.

Finally, the effect of "partial-cycling" on

the hysteresis of a SME-spring is shown in fig. 6.This way, subloops are created so that in fact no

unique (Force-displacement) or (Temperature-displacement)’relation exist. Dependent on the

previous history almost every point within theouter hysteresis loop can be reached.

Creep related to transformational plasticity

Creep experiments have been performed on Cu-baseshape memory alloys around and within thetransformation region. Not only stress and

temperatures, but also the previous thermo-

mechanical history and temperature oscillationshave an influence on the creep behaviour.Transformation creep is observed and related withisothermal transformation as determined by elec-trical resistivity. Indeed, since the appearanceof martensite increases the electrical resistivityby about 20 % the change in resistivity is a

measure for the change in the transformed volume.It has been observed that large creep rates

occur within the transformation region. (fig. 7).The phenomenon is linked with an isothermaltransformation. As a consequence the transfor-mation induced creep can be recovered by a

REVUE DE PHYSIQUE APPLIQUÉE. - T. 23, N° 4, AVRIL 1988

retransformation to beta (e.g. by a heating cycleor an unloading cycle). But also small

temperature variations cause a drastic elongationof the specimen. Fig. 8 is a typical example oftransformation creep. It occurs even at very low

loads, e.g. at 5 MPa. In fact the creep strainand creep strain rate are not only influenced bysmall changes in temperature, but are also

strongly dependent on previous thermomechanical

hystory, i.e. the described hysteresis loop. Thiscan lead to partial recovery of transformation

creep on temperature cycling and negative creepstrain rates.

Since transformation creep is also present atlow temperatures and at very low loads, some

experiments were performed on load-free specimensin the transformation temperature region whichwere subjected to carefully chosen temperature-time profiles, which are presented in fig. 9.

During isothermal holding, the resistivityincreases (fig. 9A, B, C and D). This can be

explained if further transformation from beta tomartensite occurs during isothermal holding. Theisothermal rate of transformation is differentfrom fig. 9A, B, C and D. There is an increase in

resistivity, hence further transformation, aftereach small temperature cycle in fig. 9A and B. In

fig. 9C and D a decrease in resistivity, hence a

partial reverse transformation, is present after asmall temperature cycle. So the temperaturecycles of fig. 9A and B cause an increase, and thecycles of fig. 9C and D decrease in overalltranformation rate.

Modelling the transformation plasticity

Several models have been developed to describe

mathematically the observed mechanical andtransformational behaviour.Müller proposed a model for phase transitions in

pseudoelastic bodies in 1980 (1). Later the modelwas used simulating stress-strain hysteresis at

different temperatures and simulating creep (2-3).The model relies basically on statisticalmechanics.

The basic features of the model are :

a. the specimen consists of parallel layerswhich are at 45° with the tensile or

compression axis;b. several small parts of the metallic lattice

are arranged regularly and can be at threedifferent equilibrium configurations;

c. the potential energy is a function of the atomdeviation from its original position in a

highly symmetrical parent phase;d. by inducing interfacial energy changes a

hysteresis simulation is obtained;e. when the different layers are sheared they

retain their original shape.

The important features of the Tanaka model (4)are :

a. the specimen is considered as a polycrystal sothat the nucleation and the growth of themartensite plates may be understood to be

fully governed by macroscopic transformation

kinetics;b. the Helmholtz free energy is taken into the

calculation;c. a continuous mechanical method is used to

solve the stress and strain changes.

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Favier and Guelin (5) use a discrete memoryscheme whose form is strongly suggested bysymbolic models consisting of an infinite numberof springs and friction sliders. The periodicallyrestored thermomechanical properties and the

possibility of not using the purely conventional

elastic-plastic strain decomposition are two ofthe most interesting features of such discrete

memory schemes. The model can be easily adaptedto experimental observations.

Lu Li and E. Aernoudt (6) use a multi-elementmethud. Thé basic feature of the model is thatthe specimen is divided into a set of parallelloaded pseudoelastic elements. Each element has adifferent critical stress so that the martensitictransformation occurs succesively in the differentéléments on loading. Further in order to keep themodel simple, it is assumed that the reverse

transformation takes place after full elastic

unloading and that the modulus in the parent andthe martensitic phase are the same. Extension tomore complex boundary conditions is under way.

Fig. 10 and 12 illustrate the basics and theresult of this model.

Patoor Eberhardt and Berveiller (7) defineda transformation criterion based on thé Gibbs freeenergy change during transformation and put inevidence the existence of a pseudoelastic poten-

tial. In this way, an associated yield-criterionis obtained describing the behaviour of the mono-crystal.

The results for single crystals can beextended to polycrystals using an analogouscriterion as the one of Drücker-Prager (8). Theresults obtained with this model are in excellentagreement with the experimental observations on

uniaxial loaded Cu-Zn-Al alloys.

Conclusions

Although all those models look very efficient to

describe the observed behaviour, they almost missa physical background. In this respect the diffe-rent parameters used in each model should be

physically identified on the basis of thermo-mechanical experiments under varying experimentalconditions. Only then quantitative predictions ofhysteresis behaviour will be possible.At the other side, the existing models are alreadyvery useful to calculate qualitatively the trans-formation behaviour of real systems without

performing the experiments. In this respect themodel of ref. (5) is probably the most advanced

describing thermal as well as mechanical

hysteresis, whereas the approach of ref.(6) is

preferable for mechanical cycling because of its

simplicity.

References

1. I. Müller, Il nuovo Cimento, vol. 57B, Nr.

2(1980), p. 283.2. M. Achenbach and I. Müller, Shape Memory as a

Thermally Activated Process, FB9, Herman-

Föttinger-Institut, TU Berlin.3. M. Achenbach, I. Müller and K. Wilmonski, J.

Thermal Stresses 4 (1981), p. 523.4. K. Tanaka, Res. Mech, 18 (1986), p. 251.5. D. Favier, P. Guelin, N.K. Nowacki, P. Pegan,

IUTAM SYMP ON "Thermomechanical couplings in

solids", Jean Mandel Memorial Symp., Paris-September 1916.

Figure 1 : Influence of stress. Set of isostresstemperature-strain curves a (0 MPa) to

g (150 MPa). Between temperaturecycles 30-83-30°C, a stress incrementof 25 MPa is given at the lowtemperature. The increase oftechnical transformation temperaturesis shown for increasing stress.Curves i and h (both 0 Mpa) are takenafterwards.

6. E. Aernoudt and Lu Li, in Proc. of Int. Symp.on SME-Alloys, sept. 6-9, 1986, Guilin, China,p. 23-35, Ed. China Academic Publishers, 1986.

7. E. Patoor, A. Eberhardt, M. Berveiller,accepted for publication by Acta Met.

8. D.C. Drücker, W. Prager, Quart. Appl. Math.,10, 157, (1952).

9. H. Verguts and E. Aernoudt, Proc. of the 7thInt. Conf. on Strength of Metals and Alloys,Montreal 12-16 aug. 1985, p. 563-568, Ed.Pergamon Press, Oxford-New York, 1985.

Figure 2 : Influence pf revious history.Relation between stress and

temperature history for the sets ofisothermal stress-strain curves in

Fig. 3. Note especially the equalstress-time profiles B, D and G, andthe equal profiles C, F and H. Notealso the small heating cycle 61-82-61°C at 0 MPa, temperature-timeprofile E, between profiles D and F.

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Figure 3 : Influence of previous history. Setsof isothermal (61°C) stress-strain

curves, related in time as shown in

Fig. 2. The equal stress-strain

profiles R, D and G lead to differentstress-strain curves due to a

different history before the profilewas applied. The same is true for C,F and H.

Figure 4 : Influence of loading and. unloading, incold condition. Set of isostress (0MPa) temperature-strain curves.

Between thermal cycles 30-82-30°C, a

loading-unloading stress cycle is

given at the low temperature. The

peak loading stress increases betweenthermal cycles from 0 to 150 MPa withan increment of 25 MPa.

Figure 5 : Influence of partial unloading and

reloading in warm condition. Set ofisostress (150 MPa) temperature-straincurves. between thermal cycles 82-30-82°C, a partly unloading-loadingstress cycle is given at the hightemperature. The peak unloadingstress decreases between thermal

cycles from 150 to 0 MPa with a

decrement of 25 MPa.

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Figure 6a-b-c : The effect of a "force-history" onthe force displacement (cr-e)-

pseudoelastic characteristic of a betaCu-Zn-Al alloy. Remind that "force"can be replaced by "temperature" incase of trained sample.

Figure 7 : Influence of stress and temperature oncreep in the martensite side of thetransformation temperature range.Creep in martensite at RT and inmartensite plus beta at 79 and 84°C.

Figure 8 : Influence of small temperaturedeviations on creep on the beta sideof the transformation temperaturerange. Creep in beta plus martensite,below room temperature.

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Fig. 9 : Influence of isothermal holding and small periodic temperature deviations on electrical resistivityand fraction of martensite. No load. Temperature deviations down (A and B) or up (C and D) . Holding on mar-tensite side (A and C) or on beta side (B and D) within the transformation temperature range.

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Figuré 10 : Parallel model of Lu Li and Figure 11 : Loading-unloading behaviour (6).E. Aernoudt (6).

Figure 12: Sets of simulated stress-strain curves related to the saw-tooth profiles on the right of the fi-gures. The number of transformed éléments during loading (dark cuve) is represented on the ordina-te axis (6).