Crystallographic Attributes of a Shape-Memory Alloy

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1 Kushik Bhttchry Division of Engineering & Applied cience, Cliforni Institute of Technology, Psden, CA Crystllogrphic Attributes of hpe-memory Alloy hpe-memory lloys re ttrctive for mny potentil pplictions. In n ttempt to provide ides nd guidelines for the development of new shpe-memory lloys, this pper reports on series of investigtions tht exmine the resons in the crystllogrphy tht mke (i) shpe-memory lloys specil mongst mrtensites nd (ii) Nickel-Titnium specil mong shpe-memory lloys. 1 Introduction hpe-memory effect (ME) is the bility of certin metllic lloys to recover, on heting, pprently plstic deformtion sustined below criticl temperture. hpe-memory lloys (MAs) lso disply other relted phenomen such s superor pseudo-elsticity. All these mke MAs very ppeling for mny potentil pplictions. Though sizble number of MAs re known, pplictions hve essentilly been limited to nickeltitnium (t compositions close to equitomic) due to vriety of resons. The high cost of Ni-Ti, s well s the nrrow temperture rnge in which it cn be used, limit these pplictions. Therefore, it is importnt to improve known mterils nd develop new MAs. This pper summrizes the results of line of reserch motivted by these concerns. The gol is to understnd if there re resons in the crystllogrphy tht mke 1. MAs specil mong mrtensites nd 2. Ni-Ti specil mongst MAs. In doing so, it hopes to provide ides nd guidelines for the development of new MAs. The phenomenology of ME is well understood in quuttive fshion, see for exmple buri nd Nenno (1981) nd Wymn (1992). The hert of this effect lies in the reversible or "thermoelstic" mrtensitic trnsformtion tht these crystlline solids undergo. A mrtensitic trnsformtion is temperture-induced first-order diffusionless phse trnsformtion between the high temperture ustenite phse nd the low temperture mrtensite phse. In typicl MAs, the lttice of the ustenite hs higher symmetry thn tht of the mrtensite. This gives rise to more thn one vrint of mrtensite; vrints re identicl crystl lttices which re oriented differently with respect to the ustenite. Consider specimen of given shpe in Fig. 1(). It is in the ustenite phse. uppose momentrily tht our specimen is single crystl. On cooling, the ustenite trnsforms to mrtensite. In prticulr, it trnsforms to coherent fine-scle microstructure involving the different vrints in such mnner tht there is no mcroscopic chnge in shpe (Fig. l{b)). This is known s self-ccommodtion. When lods re pplied to the mrtensite, it deforms by converting one vrint to nother nd forming new coherent fine-scle microstructure (Fig. 1(c)). On heting, ech vrint trnsforms bck to ustenite. ince there is only one vrint of ustenite, ll the strin is recovered nd the specimen returns to its originl shpe (Fig. 1()). Notice tht the strins re recoverble becuse the deformtion below the trnsformtion temperture is not due to slip, but rther due to the rerrngement of mrtensitic vrints. Notice lso tht only certin strins re recovered: those tht cn be Contributed by the Mterils Division for publiction in the JOURNAL OF ENGI NEERING MATERIAL AND TECHNOLOGY. Mnuscript received by the Mterils Division Februry 11, 1998; revised mnuscript received My 2, Guest Editors; H. ehitoglu nd Y. Chumlykov. chieved by the rerrngement of mrtensitic vrints. Lrger strins introduce stress, giving rise to lttice defects nd nonrecoverbility. If our specimen is polycrystl, the sitution is more complicted. At the high temperture it consists of number of grins of ustenite. As it is cooled, ech grin trnsforms to selfccommodted microstructure of vrints. As the polycrystl is deformed, ech grin tries to ccommodte the strin by djusting its microstructure of mrtensitic vrints. However, ech grin is cpble of sustining different clss of microstructures due to the vrying orienttion. The recoverble strins in polycrystl re the mcroscopic verges of those inhomogeneous strin fields tht my be ccommodted in ech grin by the rerrngement of mrtensite vrints. In summry, ME cn only be observed in mrtensitic mterils tht re self-ccommodting, tht cn form lrge clss of coherent microstructures nd tht cn deform by chnging microstructure t reltively smll stresses. Our tsk is to find quntittively ny restrictions tht these requirements impose on the crystllogrphy of the mteril. 2 Theory of Mrtensite Microstructure Ericksen (198, 1984, 1986), Jmes (1984), Bll nd Jmes (1987, 1992) nd others hve developed theory in the frmework of finite thermoelsticity to describe the behvior of mrtensitic mterils. ee Bll nd Jmes (1998) or Bhttchry (1998) for detiled explntion. We ssume tht the behvior of the mteril is described by (Helmholtz) free energy density which depends on temperture nd deformtion grdient. Above the trnsformtion temperture, the free energy density hs n bsolute minimum t the ustenite stte; below the trnsformtion temperture, it hs n bsolute minimum t the mrtensite stte. The energy density stisfies ll requirements of mteril frme-indifference nd mteril symmetry. Consider single crystl of ustenite t the trnsformtion temperture. Choose this s the reference configurtion nd describe ll other configurtions of the crystl s deformtions of this reference configurtion. For exmple, the trnsformtion my be described by the homogeneous deformtion y = t/*"x. [/"' is known s the trnsformtion or the Bin mtrix. However, due to the chnge in symmetry, there re k vrints of mrtensite with trnsformtion mtrices [/'", t/*^',..., f/**'. We note tht k nd the mtrices t/*", [/'^',..., [/*'* re known for ny given mteril: they my be clculted from the chnge in symmetry nd the chnge in lttice prmeters during trnsformtion. ee Tble 1 for few importnt specil cses (only [/<" is shown; t/"', t/"\..., [/<'* my be obtined from it by symmetry). Therefore, the identity mtrix / describes the ustenite stte while the mtrices [/*", t/'^',..., t/'*' describe the mrtensite stte. Further, frme-indifference sys tht rigid rottion does not chnge the energy of crystl (or in other words, rigid rottion does not chnge the stte of the crystl). Therefore, we conclude tht the deformtion grdients corresponding to the Journl of Engineering Mterils nd Technology Copyright 1999 by AME JANUARY 1999, Vol. 121 / 93 Downloded From: on 1/8/213 Terms of Use:

2 Fig. 1 The shpe-memory effect Fig. 2 chemtic view of wedge-lil<e microstructure 1. Austenite sttes re R for ny rottion mtrix R; 2. Mrtensite sttes re /?f/<", RU^^\..., or «!/<*' for ny rottion mtrix R. (1) Bll nd Jmes (1987) s well s Chipot nd Kinderlehrer (1988) showed tht energy minimiztion with such n energy density leds to fine-scle microstructure. Roughly, the ide is the following. When subjected to certin boundry conditions, the mteril tries to minimize its energy by mking mixtures of the different vrints of mrtensite while trying to stisfy ll the coherence requirements. This leds to "minimizing sequences" which re interpreted s fine-scle microstructure. Bll nd Jmes (1987) lso showed tht the well-known phenomenologicl or crystllogrphic theory of mrtensite follows s consequence of this theory. ince then, there hs been much progress in understnding mrtensitic microstructure, nd nlyticl tools Uke the "Young mesure" nd verge comptibility conditions like the "minors reltions" hve been developed. 3 The Wedge-Like Microstructure It is very common to observe wedge-like or sper-like microstructure in MAs. When the lloy is cooled from bove the trnsformtion temperture, wedge-shped regions of mrtensite grow into the ustenite. As shown in Fig. 2, the wedge consists of two sets of fine mrtensitic twins (fine lternting bnds of two mrtensite vrints) seprted by midrib. This microstructure provides n esy wy for the initition of trnsformtion nd is thus importnt for thermoelsticity nd reversibility of trnsformtion (Otsuk nd himizu, 1969). To check whether mteril cn form wedge, it is necessry to enforce two conditions: (/) the deformtion grdients within the wedge Tble 1 ome trnsformtions nd trnsformtion mtrices Trnsformtion Cubic to tetrgonl Eg: Ni-Al Cubic to orthorhombic Eg: y! Cu-Al-Ni Cubic to monoclinic-i Eg: Ni-Ti Cubic to monoclinic-ii Eg: Cu-Zn-Al Trnsformtion mtrix ' ^ ' \e ' fi 6 6 e 7 13 The xis of monoclinic symmetry is (11)cubic ((1>cubic) in monoclinic- I (monoclinic-ii). P e P /9 should correspond to mrtensite vrints while the deformtion grdients outside the wedge should correspond to the ustenite nd (ii) comptibility conditions hve to be stisfied t the five interfcestwo twin interfces, two ustenite-mrtensite interfces nd one midrib. Bhttchry (1991) showed tht the five comptibility conditions re too restrictive for ny rbitrry mteril to stisfy. For exmple, mteril tht undergoes cubic to tetrgonl trnsformtion cn form wedge if nd only if the mteril prmeters nd P (cf. Tble 1) stisfy the condition (1 - py + 4/3^(1 -t- p^) (1 - P^Y + 8/3" This describes curve in -P spce (Fig. 3), nd only those mterils whose mesured prmeters lie on the curve cn form wedge. imilrly, in mterils tht undergo cubic to orthorhombic trnsformtion, the mteril cn form wedge if nd only if the mesured mteril prmeters, P nd 6 lie on certin fmily of surfces in -p spce. Furthermore, the theory predicts vrious geometricl fetures of the wedge. Mterils which re known to form wedge stisfy these conditions very closely. We highlight Ni-Al nd Cu-Al-Ni in Tble 2; lso see Fig. 3. Further, the geometric detils of the observed wedges gree very well with the predictions; for exmple the theory predicts tht wedges with Type I twins in Cu- Al-Ni resemble the bottom-right of Fig. 2 while those with Type II twins the top-right in greement with observtions. This clcultion shows tht microstructure often depends criticlly on the lttice prmeters: smll chnge in lttice prmeters cn result in significntly different clss of microstructures nd consequently significntly ffect mcroscopic properties. Therefore, it suggests tht the bility of mteril to disply P d Wedge possible No volume chnge (elf-ccommodting) NiMn b-- NiZnCu c NiAl d- NiZni e- InTl f- FeAlC g-- FePt h-- FeCrC i - FeNiC Fig. 3 The specil reltions on the trnsformtion strin for wedge nd self-ccommodtion in cubic to tetrgonl trnsformtion. The mesured lttice prmeters of some lloys re lso shown. Wedges hve been observed in lloys b, d, f nd i while lloys -e nd g re selfccommodting (see Bhttchry (1991, 1992) for detils nd references). (2) 94 / Vol. 121, JANUARY 1999 Trnsctions of the AME Downloded From: on 1/8/213 Terms of Use:

3 Tble 2 Wedge nd self-ccommodtion: Comprison of theory nd experiment. Wedges re seen in Ni-Al nd Cu- Al-Ni; ll three lloys re self-ccommodting (see Bhttchry (1991,1992) for detils nd references) Mteril Ni-Al Cu-Al-Ni Ni-Ti Observed prmeters =.9392 = = P = =.194 Prmeters for wedge =.9445 = = = =.194 Observed volume chnge.35%.297%.23% the ME my depend criticlly on the lttice prmeters nd consequently on composition. 4 elf-accommodtion elf-ccommodtion is the bility of MA to trnsform from the ustenite to the mrtensite with no mcroscopic chnge in shpe. Aprt from being n inherent prt of ME, Wymn (1992) nd others hve emphsized the role of self-ccommodtion s necessry condition for ME. Not every mteril tht undergoes mrtensitic trnsformtion is self-ccommodting. Consider, for exmple, mteril where the volume of the mrtensite is smller thn tht of the ustenite. Clerly, the trnsformtion leds to chnge in volume (unless the mteril is subjected to lrge stresses) nd the mteril is not self-ccommodting. o, wht re the conditions tht gurntee tht mteril is self-ccommodting? From the rguments bove, it is cler tht volume preserving trnsformtion is necessry condition for self-ccommodtion. Bhttchry (1992) showed tht if the symmetry of the ustenite is cubic, then this condition is lso sufficient for self-ccommodtion. If on the other hnd, the symmetry of the ustenite is not cubic, the lttice prmeters of the mteril hve to stisfy dditionl restrictions which re extremely stringent nd "nongeneric." Therefore, mterils with cubic ustenite hve to stisfy rther mild constrint, while mterils with non-cubic ustenite hve to stisfy very restrictive non-generic conditions in order to be self-ccommodting. It is highly unlikely tht relistic mterils will stisfy such nongeneric conditions. Consequently, only mterils with cubic ustenite which undergo lmost volume preserving trnsformtion re self-ccommodting nd hence shpe-memory mterils. This conclusion is in greement with experimentl observtions s highlighted in Tble 2 nd Fig. 3; lso see Bhttchry (1992) for extensive comprison. There is lso very interesting nd probbly importnt coincidence. Consider cubic to tetrgonl trnsformtion. Volume preserving trnsformtion corresponds to ^p = 1 which describes curve in -P (trnsformtion strin) spce. As shown in Fig. 3, this curve is very close to the curve on which mterils cn form wedge (see Eq. (2)) when nd /? re close to 1, the rnge of experimentl interest. imilrly, in cubic to orthorhombic trnsformtion, the volume preserving surfce is close to the wedge forming surfces in the -p spce. Thus, mny mterils with cubic ustenite undergoing volume preserving trnsformtions my lso disply wedge! Another interesting nd importnt consequence of self-ccommodtion is tht it is not possible to induce the two-wy ME by mking textured poly crystls. Finlly, word bout the theoreticl methods. In the cse of the wedge, we were nlyzing given microstructure. In contrst, here we re sking broder question: is there ny microstructure tht is self-ccommodting? Therefore, the method used to study the erlier problemswriting down nd trying Tble 3 ummry of experimentl observtions of tensile recoverble strins (see Bhttchry nd Kohn (1996) for detils nd references) Mteril Ni-Al CuAlNi/Cu-Zn-Al Ni-Ti ingle crystl Polycrystl -13% Almost none 2-9% Typiclly -2%; up to i % in specil ribbons 3-1% 5-8% in wires/sheets to solve ll the comptibility conditionscnnot be used here. No mtter how mny microstructures we check, there re lwys others. Therefore, we need methods to nlyze generl clsses of microstructures. It is here tht tools like the "Young mesure" nd verge comptibility conditions like the "minors reltions" re very useful. 5 Geometriclly Liner Theory of Mrtensite Microstructure We now turn to clculting the recoverble strins tht re inherent in the fundmentl crystllogrphy of given mteril. This question turns out to be extremely difficult nd t this time we re unble to crry through such clcultion in the geometriclly nonliner theory tht we hve been using so fr. However, it is possible to write "geometriclly liner" version of this theory by ssuming infinitesiml displcements. This is similr to the theory used by Roitburd (1978) nd Khchturyn (1983) mong others (see Bhttchry (1993) for detiled comprison). I believe tht in the problems discussed below, this pproximte theory gives resonble results though the exct quntittive detils my be different. In this geometriclly liner theory, microstructures correspond to continuous displcements u such tht the infinitesiml strins e = 5(VM -I- VM'^) "essentilly" tke vlues 1. (Austenite sttes) or 2. e<", e"',..., e'*> (Mrtensite sttes). (3) Here, e*** = [/'** - / re the trnsformtion strins. 6 Recoverble trins Mny lloys re known to be good MAs s single crystls, but the extent to which they retin their ME in polycrystlline form is widely vried s highlighted by three lloys in Tble ingle Crystl. We define the set of recoverble strins in single crystl,, s the set of ll possible strins tht cn be chieved by mixing the different vrints of mrtensite in (grin 1) Fig. 4 The set of recoverble strins for mteril undergoing lrge chnge in symmetry (left) nd smll chnge in symmetry (right). The recoverble strins () in single crystl is determined by the trnsformtion strins of the vrints while in polycrystl it Is determined by the interction between the grins nd Is estimted by the set T. 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4 coherent microstructure. We cn clculte it for mny interesting cses bsed on the following result of Bhttchry (1993). If the trnsformtion strins re pirwise comptible or twin relted then is simply their convex hull, i.e., the set of ll possible verges s shown in Fig. 4: c = [e\\e = X 6l<'>e<''with6'' OndXfi""^ 1}. (4) This holds when the phse trnsformtion is cubic to tetrgonl, cubic to trigonl or cubic to orthorhombic, nd Bhttchry nd Kohn (1996) hve explicitly clculted this set. The cubic tq monoclinic trnsformtions re different, becuse some pirs of vrints re not comptible. The exct sets re unknown but it is possible to obtin inner nd outer estimtes (Bhttchry nd Kohn, 1996 nd hu s Bhttchry, 1998). 6.2 Polycrystl. A polycrystl is n ssemblge of grins, ech composed of the sme mteril in different orienttion. The texture of polycrystl (i.e., the shpes of the grins nd their orienttions) cn be represented by rottion-vlued function R{x) which gives the orienttion of the grin t x reltive to reference single crystl. A deformtion of the polycrystl below the trnsformtion strin is recoverble if it cn be ccommodted in ech grin by the rerrngement of the mrtensite vrints. The set of recoverble strin in given grin, (x), is obtined by suitble rottion of the set of recoverble strins of the reference single crystl: {x) = R(x)R(xy (see Fig. 4). Therefore, if we ssume tht (i) the microstructure size is much smller thn the grin size, which in turn is much smller thn the specimen size, nd (ii) the grins re perfectly bonded, then the set of recoverble strins in polycrystl, f, is the set of verges of strin fields tht tke vlues in (x) t ech jc. This set is rther difficult to clculte; however it is possible to obtin n esy inner bound or Tylor bound. uppose strin e is recoverble in ech grin, then it is certinly recoverble in the polycrystl. Therefore, f D fl where 57= r^mx). (5) In other words, the polycrystl cn recover ll strins tht re common to ll the sets {x). The polycrystl cn possibly recover lrger strins by the coopertive effects between the grinsbut, the Tylor bound is surprisingly good estimte of the ctul recoverble strins (Bhttchry nd Kohn, 1996 nd 1997, Bhttchry et l., 1997, hu nd Bhttchry, 1998). Therefore we use the inner estimte to study recoverble strins. Bhttchry nd Kohn (1996) found tht the recoverble strins cn depend criticlly on the chnge of symmetry during trnsformtion. Briefly, polycrystls of mterils which undergo smll chnge of symmetry (cubic to tetrgonl/trigonl) disply no recoverble strin while polycrystls of mterils which undergo lrge chnge of symmetry (cubic to orthorhombic/monoclinic) lwys disply some chnge of symmetry. This result is in good greement with experimentl observtions in Tble 3: Ni-Al undergoes cubic to tetrgonl while Cu-Al-Ni/Cu-Zn- Al/Ni-Ti undergo cubic to orthorhombic/monoclinic trnsformtion. The ide is explined in Fig. 4. A mteril undergoing smll chnge of symmetry hs few vrints, so the set & does not spn devitoric strin spce (shown schemticlly s line on the right). Therefore, if we rotte this set nd intersect it ccording to (5), we find tht the intersection reduces to single point s shown on the right of Fig. 4. On the other hnd, in mterils undergoing lrge chnge in symmetry we hve sufficient number of vrints so tht the set spns the five dimensionl devitoric strin spce (shown schemticlly s tringle on the left). Consequently, when we rotte it nd intersect it ccording to (5) we obtin non-trivil set fs shown on the left of Fig. 4. Tble 4 The predicted unixil recoverble extension for vrious textures Texture rndom (11) wire/rod (111) wire/rod (1) film/ribbon {11) film (111) film/sheet (^'R TiNi CR Recoverble strins (%) <:« CuZnAl e« While the effect of symmetry explins much experimentl observtion, it fils to mke ny distinction within lloys which undergo lrge chnge in symmetry. In prticulr, it is unble to explin why TiNi is so much better thn Cu-bsed MAs since both undergo cubic-monoclinic trnsformtion nd hve similr trnsformtion strins. hu nd Bhttchry (1998) hve systemticlly explored the role of texture to understnd this difference. They develop n inner bound e'n nd inner estimte R for the recoverble strin for polycrystl with given texture. Tble 4 compres the predicted recoverble strin in TiNi nd CuZnAl for vrious textures. Notice tht for specimens with rndom texture, the recoverble strin in either lloy is predicted to be rther smll. Wires nd rods mde of these mterils hve either (11) or (111) texture: notice tht in either of these cses the strins re much lrger in TiNi compred to CuZnAl. imilrly, common rolling textures re lso good textures for lrge recoverble strins in TiNi, but not in CuZnAl. All these predictions re in good greement with experimentl observtion. We conclude by commenting on some recent efforts to produce MA thin-fllms by sputtering. It is possible to derive n inner bound for the mximum recoverble strins in very thin films e'i (hu nd Bhttchry, 1998). puttering TiNi produces films with either rndom or {11}-film texture. Tble 4 shows tht both these textures yield only very smll strins, in greement with the limited strins observed in films. Finlly, lrge strins re expected in TiNi films with {111} texture nd CuZnAl films with {1} texture. 7 Conclusions These results provide the following brod principles. The microstructure of mrtensitic mteril, nd consequently its mcroscopic properties, depend criticlly on the lttice prmeters nd consequently on composition. Only mterils with cubic ustenite tht undergo volume preserving trnsformtion cn be shpe-memory mterils. It is not possible to induce the two-wy shpe-memory effect by mking specilly textured polycrystls. Mterils with monoclinic mrtensite will disply the best shpe-memory effect in polycrystlline form. Polycrystls of mterils with tetrgonl (or trigonl) mrtensite will disply the shpe-memory effect only in pplictions where the required recovery is limited to unixil tension or compression. Even in such pplictions it is necessry to process the mteril, perhps by rolling, drwing or by repeted deformtion, followed by nneling, to endow it with (1) texture (or <111) texture). Despite ll this, it is nturl to expect some unrecoverble strins. Ni-Ti is such n outstnding MA s wires nd sheets becuse it undergoes lrge chnge of symmetry during trnsformtion nd the texture tht develops during roll- 96 / Vol. 121, JANUARY 1999 Trnsctions of the AME Downloded From: on 1/8/213 Terms of Use:

5 ing, extrusion nd drwing is extremely fvorble from the point of view of lrge recoverble strins. The rndom or {11} sputtering texture is not good thin-film texture for Ni-Ti or Cu-Zn-Al. Lrge strins re expected in TiNi films with {111} texture nd CuZnAl films with {1} texture. Acknowledgments This reserch strted when I ws grdute student t the University of Minnesot nd continued through my post-doctorl sty t the Cournt Institute. I m deeply indebted to Prof. Richrd D. Jmes for introducing me to this subject nd for continuing to inspire me, nd Prof. Robert V. Kohn for his continuing support nd encourgement. I grtefully cknowledge finncil support from AFOR (t Minnesot nd Cltech), ARO (t Cournt), NF (t Minnesot, Cournt nd Cltech), nd ONR (t Cltech). References Bll, J. M., nd Jmes, R. D., 1987, "Fine Phse Mixtures s Minimizers of Eneigy," Archive for Rtionl Mechnics nd Anlysis, Vol. 1, pp Bll, J. M., nd Jmes, R. D., 1992, "Proposed Experimentl Tests of Theory of Fine Microstructure," Philosophicl Trnsctions of the Royl ociety of London, Vol. 338A, pp Bll, J. M., nd Jmes, R. D., 1996, Fine Microstructures, in preprtion. Bhttchry, K., 1991, "Wedge-Like Microstructure in Mrtensite," Act Metllurgic et Mterili, Vol. 39, pp Bhttchry, K., 1992, "elf-accommodtion in Mrtensites," Archive for Rtionl Mechnics nd Anlysis, Vol. 12, pp Bhttchry, K., 1993, "Comprison of the Geometriclly Nonliner nd Liner Theories of Mrtensitic Trnsformtion," Continuum Mechnics nd Thermodynmics, Vol, 5, pp Bhttchry, K., 1998, "A Theory of Mrtensitic Microstructure nd its Implictions for the hpe-memory Effect," to pper in hpe-memory Effect, ed. Airoldi, G., Miyzki,. nd Muller, I, eds.. Trns. Tech. Pub. Bhttchry, K., nd Kohn, R. V., 1996, "ymmetry, Texture, nd the Recoverble trin of hpe-memory Polycrystls," Act Mterili, Vol. 44, pp Bhttchry, K., nd Kohn, R. V., 1997, "Elstic Energy Minimiztion nd the Recoverble trins of Polycrystlline hpe-memory Mterils," Archive for Rtionl Mechnics nd Anlysis, Vol. 139, pp Bhttchry, K., Kohn, R. V., nd hu, Y. C, 1997, "The Tylor Estimte of Recoverble trin in hpe-memory Polycrystls," The Proceedings of the lu- TAM ymposium on Trnsformtion Problems in Composite nd Active Mterils, Bhei-el-Din, Y. A. nd Dvork, G. J., eds., Kluwer, in press. Chipot, M., nd Kinderlehrer, D., 1988, "Equilibrium Configurtions of Crystls," Archive for Rtionl Mechnics nd Anlysis, Vol. 13, Ericksen, J. L., 198, "ome Phse Trnsitions in Crystls," Archive for Rtionl Mechnics nd Anlysis, Vol. 73, pp Ericksen, J. L., 1984, "The Cuchy nd Born hypothesis for crystls," Phse Trnsformtions nd Mteril Instbilities in olids, Gurtin, M. E., ed.. Acdemic Press, pp Ericksen, J. L., 1986, "tble Equilibrium Configurtions of Elstic Crystls," Archive for Rtionl Mechnics nd Anlysis, Vol. 94, pp Jmes, R. D., 1986, "Displcive Phse Trnsformtions in olids," Journl of Mechnics nd Physics of olids, Vol. 34, pp Khchturyn, A. G., 1983, Theory of tructurl Trnsformtions in olids, Wiley, New York. Otsuk, K., nd himizu, K., 1969, "Morphology nd Crystllogrphy of Thermoelstic y Cu-Al-Ni Mrtensite," Jpnese Journl of Applied Physics, Vol. 8, pp Roitburd, A., 1978, "Mrtensitic Trnsformtion s Typicl Phse Trnsformtion in olids," olid tte Physics, Vol. 33, pp buri, T., nd Nenno,., 1981, "The hpe Memory Effect nd Relted Phenomen," Proc. Interntionl Conference on olid-olid Phse Trnsformtions, H. I. Aronson et l., eds.. The Metllurgicl ociety AIME, New York, pp hu, Y. C, nd Bhttchry, K., 1998, "The Influence of Texture on the hpe-memory Effect in Polycrystls," Act Mterili, in press. Wymn, C 1992, "hpe Memory nd Relted Phenomen," Progress in Mterils cience, Vol. 36, pp Journl of Engineering Mterils nd Technology JANUARY 1999, Vol. 121 / 97 Downloded From: on 1/8/213 Terms of Use: