Hysteretic transformation behaviour of shape memory alloys

Transcription

1 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 Appliquee, 1988, 23 (4), pp < /rphysap: >. <jpa > HAL Id: jpa Submitted on 1 Jan 1988 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Revue Phys. Appl. 23 (1988) AVRIL 1988, Classification Physics Abstracts 81.30K L 557 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) 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 déformation 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 applique une déformation, la transformation martensitique sera induite à une contrainte critique. Cette on 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 traction (03C3-03B5 ), présente une boucle d hytérés is bien fermée. est aussi liée à la croissance des variantes de marten- 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 transformation. La déformation, apparemment plastique, site sélectionnées. Mécaniquement, la connaissance du comportement pseudoélastique est importante pour calculer les caractéristiques 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

3 558 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 endtemperatures 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 thermoelastic. 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 , 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 betamartensite, 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 a 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

4 559 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 temperaturestrain 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 (Temperaturedisplacement) 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 thermomechanical history and temperature oscillations have an influence on the creep behaviour. Transformation creep is observed and related with isothermal transformation as determined by electrical 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 transformation induced creep can be recovered by a 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 temperaturetime 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. REVUE DE PHYSIQUE APPLIQUÉE. - T. 23, N 4, AVRIL 1988

5 560 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 on 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 potential. In this way, an associated yield-criterion is obtained describing the behaviour of the monocrystal. 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 different parameters used in each model should be physically identified on the basis of thermomechanical 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 transformation 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 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 K. Tanaka, Res. Mech, 18 (1986), p D. Favier, P. Guelin, N.K. Nowacki, P. Pegan, IUTAM SYMP ON "Thermomechanical couplings in solids", Jean Mandel Memorial Symp., Paris- September E. Aernoudt and Lu Li, in Proc. of Int. Symp. on SME-Alloys, sept. 6-9, 1986, Guilin, China, p , Ed. China Academic Publishers, 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 aug. 1985, p , Ed. Pergamon Press, Oxford-New York, Figure 1 : Influence of stress. Set of isostress temperature-strain curves a (0 MPa) to g (150 MPa). Between temperature cycles 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. 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 C at 0 MPa, temperature-time profile E, between profiles D and F.

6 561 Figure 4 : Influence of loading and. unloading, in cold condition. Set of isostress (0 MPa) temperature-strain curves. Between thermal cycles 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 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 5 : Influence of partial unloading and reloading in warm condition. Set of isostress (150 MPa) temperature-strain curves. between thermal cycles 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.

7 562 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 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 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.

8 563 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 martensite side (A and C) or on beta side (B and D) within the transformation temperature range.

9 564 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 figures. The number of transformed éléments during loading (dark cuve) is represented on the ordinate axis (6).