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, J. Ortin

To cite this version:

J. van Humbeeck, E. Aernoudt, L. Delaey, Lu Li, H. Verguts, et al.. Hysteretic transformation

behaviour of shape memory alloys. Revue de Physique Appliquée, Société française de physique /

EDP, 1988, 23 (4), pp.557-564. �10.1051/rphysap:01988002304055700�. �jpa-00245802�

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)

Classification

Physics ^Abstracts

81.30K - 81.40L

Résumé - Après ^un traitement thermomécanique approprié ^certains alliages ^{de cuivre et de Ni-Ti} présentent ^un changement ^{de forme} (effet ^de mémoire) lors d’un cyclage thermique ^{dans une zone} d’environ 40 degrés.Deux conditions importantes pour obtenir cet effet sont nécessaires : il faut une 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-dessus de 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. ^Cette contrainte dépend linéairement de la température. ^{Lors de cette} transformation induite, ^une grande déformation de l’échantillon est possible. ^{En inversant} la direction de la déformation, ^{le matériau}

retrouve 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 par

déformation, ^{et la} transformation inverse montrent une hystérèse. L’origine ^{de cette} hystérèse peut être 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 about 40 degrees. ^Two important conditions for this effect are that a thermoelastic martensitic transformation occurs in this temperature range and that the total strain of its shape change does not exceed more than a few percent. ^At temperatures ^above the martensitic transformation temperatures, the material behaves pseudoelastic : by applying ^a strain, the martensitic transformation will start at a critical stress, dependent ^{on the} temperature. ^After complete transformation a considerable strain can be obtained which disappears during unloading. ^The

stress 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 thermodynamic considerations 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} shape change 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

558

Introduction

Shape memory alloys ^are characterised by ^a reversible shape change during heating ^and cooling in a small (± ⁴⁰ degrees) temperature region.

This effect is directly related to a thermoelastic martensitic transformation occurring ^{in this} temperature ^zone. A material deformed in the martensitic state will recover its original ^form

when heated into the parent phase ^or beta state.

When the straining - heating (recovering) - cooling - straining cycle ^is repeated ^{a sufficient}

number of times, also the deformed state will be created spontaneously ^on coolin.g : the material is said to have been "trained". The forward and

reverse shape change ^{start at} the same

temperatures ^{as - those where the} martensitic transformation starts and ends :Ms, Mf, As and Af.

Ms and Mf ^are respectively the start and end- temperatures ^{of the} forward transformation (beta

to martensite) while As and Af are the start and the 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 and end of the shape change.

As and Af being ^{shifted to somewhat} higher temperatures relative to Ms and Mf a hysteretic cycle ^is described. The hysteresis ^{occurs thus in}

the transforming volume as well as in the shape change.

Another reversible shape change ^{linked to}

martensite is the deformation behaviour at a

constant temperature above Af. Relative large

strains (10 ^{to 15 % for} single crystals, 4 to 6 % for polycrystals) ^{obtained after} loading

annihilate completely during unloading although

the loading ^curve looks very similar to a plastic stress-strain curve.

During loading, the material will transform into these martensite variants that most extensively

relax the applied ^{stress while} during unloading

the material will retransform to the beta-phase

since above Af martensite is unstable at zero

stress.

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

experimentally ^{shown and} theoretically proven that the critical stress- or strain is (to ^{a first} approximation) linearly dependent ^on temperature.

Finally ^a special ^{case of} hysteretic loops ^should

be mentioned here, although ^it is not related to the beta to martensite transformation. When a

fully martensitic sample (below Mf) ^{is submitted}

to a stress-strain cycle ^a hysteretic closed loop

can also _appear. This is in this case due to reorientation of the variants to those variants that 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 the thermoelastic martensitic transformation

If one considers the free energy of the beta- and the martensitic phase ^{as a function} of temperature, ^an equilibrium temperature ^{To exists} where both energies ^are equal ^or A G

^{0. However}

due to nucleation, the forward transformation will start at a temperature Ms, lower than To where a

critical value AGc ^exists. Once 0394G ~ 0394Gc, ^the martensitic transformation starts. A similar

hypothesis ^{is valid for the reverse} transformation : As thus being above To. One can

consider this as a part of the hysteresis although

it is probably ^{a minor} contribution to it.

Therefore we omit the problem of nucleation and consider the transformation as purely ^thermo-

elastic. During ^this transformation, ^the

difference in free energy between beta and martensite should be ^{zero :}

where T is the temperature, ^{x the} transformed volume 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 ^such

as the elastic strain energy, induced in the material by ^local shape change ^and AGirrev which

can be seen as a frictional force due to the movement of the martensitic interface in the lattice. The value of4YGirr determines mainly ^the hysteretic 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 ⁱⁿ

order to describe ^{in a general and} mathematical way the observed behaviour (1-7). ^{In the} present paper ^we will therefore review the experimental

observations and mention the important conclusions when 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-strain

curves 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 ^at

the low temperature. ^{The increase of the} transformation temperatures ^{is shown as a function} of the applied ^stress.

The difference in slope ^of heating ^and cooling legs ^{at the} loop ends indicates that further transformation is possible by ^further cooling ^or heating. ^{So the curves are} taken within the transformation 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 thermomechanical treatment and the response of the material to this treatment. Instead of using ^a simple ^linear

stress-time profile, ^the specimen ^was subjected ^to special stress-time profiles in the form of a saw

tooth with different heights ^{to show also the form}

of isothermal hysteresis subloops.

After cold loading ^the specimen ^and heating up to 61° 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 the strain at 150 MPa decreases each time, points ⁶

b c d e. Then, profile ^{C is} applied. ^The hysteresis subloops ^are closed with common point

at 0 MPa.

Then, profile ^D (the ^{same as} B) ^is applied.

The hysteresis loops ^are closed with common point

at 150 MPa. Note that point 6 in B is more to the

martensite side than point ^{6 in} D, although ^both

are at the same temperature and stress. This is due to the different way point 6 was reached in both cases before applying ^{B or D.}

Next, ^a small heating and cooling cycle ^at 0 MPa

was performed. ^The specimen ^{is at 61° C more to}

the beta side than before this small cycle. ^This

is a normal hysteresis effect, due to the transformation temperature hysteresis.

So point 0 in F is at a lower strain than point ⁰

in D.

Then, profile ^{F is} applied (the ^{same as} C). ^After

each 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 loops

closed at 150 MPa. The form _{is very} similar to these of D, but their position ^is different. The whole stress-strain loops ^{are more} to the beta side in G than in D. Applying profile ^H (same ^as

F and C), gives ^closed loops ^{at 0} MPa, ^{but here} also they ^are shifted to the beta side when

compared ^{with C.}

The occurrence of open ^or closed loops, ^{and shifts}

in position ^{of the} loops within the general transformation 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 cold} condition induces more martensite, ^{and increases} the starting strain (move ^to the martensite side)

for a subsequent heating cycle. ^The higher ^the

cold 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} changes

the starting strain at the high temperature. The lower the stress to which is unloaded, the more

martensite reverts to beta, ^{and the} greater ^{is the} shift to the beta side. So the cooling ^{curves at}

150 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 the} outer hysteresis loop ^can be reached.

Creep related to transformational plasticity Creep experiments have been performed ^{on Cu-base} shape memory alloys ^{around and within the} transformation region. ^Not only ^{stress and} temperatures, ^{but also the} previous ^thermo-

mechanical history and temperature oscillations have an influence on the creep behaviour.

Transformation _creep is observed and related with isothermal transformation as determined by ^elec- trical resistivity. Indeed, ^since the appearance of martensite increases the electrical resistivity by 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 isothermal transformation. 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 cycle

or an unloading cycle). ^But also small temperature variations cause a drastic elongation

of the specimen. Fig. 8 is a typical example ^of

transformation creep. It occurs even at very low

loads, e.g. at 5 MPa. In fact the creep strain and _creep strain rate are not only influenced by small changes in temperature, but are also

strongly dependent ^on previous thermomechanical hystory, ^{i.e. the described} hysteresis loop. ^This

can lead to partial recovery of transformation creep ^on temperature cycling ^and negative creep strain rates.

Since transformation _creep is also present ^at low temperatures ^and at very low loads, ^some experiments ^were performed ^{on load-free} specimens

in the transformation temperature region which

were subjected ^to carefully ^chosen temperature- time profiles, ^{which are} presented ⁱⁿ fig. ^9.

During isothermal holding, ^the resistivity increases (fig. 9A, B, ^C and D). ^{This can be} explained if further transformation from beta to martensite occurs during isothermal holding. ^The isothermal rate of transformation is different from fig. 9A, B, ^{C and D. There is an} increase in resistivity, ^{hence further} transformation, after each 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 a} small temperature cycle. ^{So the} temperature cycles ^of fig. ^{9A and B cause an} increase, ^and the cycles of fig. 9C and D decrease in overall tranformation rate.

Modelling ^the transformation plasticity

Several models have been developed to describe

mathematically the observed mechanical and transformational behaviour.

Müller proposed ^a model for phase transitions in

pseudoelastic bodies in 1980 (1). Later the model

was used simulating stress-strain hysteresis ^at

different temperatures ^and simulating creep (2-3).

The model relies basically ^on statistical mechanics.

The basic features of the model are :

a. the specimen consists of parallel layers

which are at 45° with the tensile or

compression axis;

b. several small parts of the metallic lattice

are arranged regularly ^{and can be at three}

different equilibrium configurations;

c. the potential energy is ^a function of the atom deviation 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 ^so that the nucleation and the growth ^{of the}

martensite 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.

560

Favier and Guelin (5) ^{use a discrete} memory scheme whose form is strongly suggested by symbolic models consisting ^{of an} infinite number of springs ^and friction sliders. The periodically

restored thermomechanical properties ^{and the} possibility ^{of not} using the purely conventional elastic-plastic strain decomposition ^{are two of}

the most interesting features of such discrete memory schemes. The model can be easily adapted

to experimental observations.

Lu Li ^{and E. Aernoudt} (6) ^{use a} multi-element

methud. Thé basic feature of the model is that the specimen ^is divided into a set of parallel

loaded pseudoelastic ^elements. Each element has a

different critical stress so that the martensitic transformation occurs succesively ^{in the different} éléments

loading. Further in order to keep ^the model simple, ^{it is} assumed that the reverse

transformation takes place ^{after full elastic} unloading and that the modulus in the parent ^and the martensitic phase ^are the same. Extension to

more complex boundary conditions is under _way.

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

Patoor Eberhardt and Berveiller (7) ^defined

a transformation criterion based on thé Gibbs free energy change during transformation and put ⁱⁿ evidence the existence of a pseudoelastic poten-

tial. In this _way, an associated yield-criterion is obtained describing ^{the behaviour} of the mono-

crystal.

The results for single crystals ^{can be} extended to polycrystals using ^an analogous criterion as the one of Drücker-Prager (8). ^The

results obtained with this model are in excellent agreement 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 miss

a 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 experimental conditions. Only ^then quantitative predictions ^of hysteresis behaviour will be possible.

At the other side, ^the existing ^{models are} already

very useful to calculate qualitatively the trans- formation behaviour of real systems ^without performing ^the experiments. ^{In this} respect ^the model 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 isostress} temperature-strain ^{curves a} (0 MPa) to g (150 MPa). ^Between temperature cycles 30-83-30°C, ^{a stress increment}

of 25 MPa is given ^{at the low} temperature. The increase of technical transformation temperatures is shown for increasing ^stress.

Curves i and h (both ⁰ Mpa) ^{are taken} afterwards.

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 7th Int. 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} of isothermal stress-strain curves in

Fig. ^{3. Note} especially ^the equal stress-time profiles B, ^{D and} G, ^and

the equal profiles C, ^F and H. Note

also the small heating cycle ^61-82-

61°C at 0 MPa, temperature-time

profile E, ^between profiles ^{D and F.}

Figure 3 : Influence of previous history. ^Sets

of 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 different stress-strain curves due to a

different history before the profile

was applied. ^{The same} is true for C, F and H.

Figure 4 : ^Influence of loading and. unloading, ⁱⁿ

cold condition. Set of isostress (0

MPa) 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 between} thermal cycles from 0 to 150 MPa with

an increment of 25 MPa.

Figure ^{5 : Influence of} partial unloading ^and reloading ^{in warm} condition. Set of isostress (150 MPa) temperature-strain

curves. between thermal cycles 82-30-

82°C, ^a partly unloading-loading

stress cycle ^is given ^{at the} high temperature. ^The peak unloading

stress decreases between thermal

cycles ^{from 150 to} 0 MPa with a

decrement of 25 MPa.

562

Figure 6a-b-c : ^{The effect of a} "force-history" ^on the force displacement (cr-e)- pseudoelastic characteristic of a beta Cu-Zn-Al alloy. Remind that "force"

can be replaced by "temperature" ⁱⁿ

case of trained sample.

Figure ^{7 : Influence} of stress and temperature ^on

creep in the martensite side of the transformation temperature range.

Creep in martensite at RT and in martensite plus beta at 79 and 84°C.

Figure ^{8 : Influence of small} temperature deviations on creep ^on the beta side of the transformation temperature range. Creep ^{in beta} plus martensite,

below room temperature.

Fig. 9 : Influence of isothermal holding and small periodic temperature deviations on electrical resistivity

and 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.

564

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).