Falk - Three Dimensional Landou Theory Describing the Martensitic Phase Transformation of Shape...

8/3/2019 Falk - Three Dimensional Landou Theory Describing the Martensitic Phase Transformation of Shape Memory Alloys

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Three-dimensional Landau theory describing the martensitic phase transformation of shape-

memory alloys

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1990 J. Phys.: Condens. Matter 2 61

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J . Phys.: Conden s. Matter 2 (1990) 61-77. Pr int ed in th e UK

Three-dimensional Landau theory describing themartensitic phase transformation of shape-memoryalloysF Falki and P KonopkaFB Physik, Universi tat-GH-Paderborn. D-4790 Paderborn, Federal R epublic of Germany

Rece ived 13 De cem ber 1988, in final form 3 July 1989

Abstract. In shap e-me mory alloys a first-order martensitic ph ase transition is responsible forpseudo-elastic and for ferro-el astic stress-strain relations. To describe this behaviour amodified Landa u the ory is proposed in which the free energy of the crystal depen ds on th etemperature and on the full strain tensor. T he ene rgy is invariant with respect to the cubicpoint group O hof the high-temperature phase. To predict the cubic-to-monoclinic phaset ransi t ionofb-phaseshape-memory alloysanexpansion uptosixth order instrain isnecessaryfor which , for the class of alloys considered, odd terms may be neglected. For a CuAlN i alloythe expansion coefficients are determined by comparison with experimental results . Incontrast to classical Landau theory of second-order phase transitions, not only a singlesecond-order but also a fourth-ord er expansion coefficient depe nd on temperature.

1. IntroductionTh e shape- me mo ry effect in certain m etallic systems called shape-memory alloys hasbeen the sub ject of considerable experim ental effort . In som e tem pera ture ranges thesealloys show pseudo-elastic a nd ferro-elastic stress-strain curves. It transpires th at thes epeculiarities ar e th e con sequence of a first-order martensitic phase transition which isconnected with a nearly-volume-conserving spontaneous deformation of the crystallattice (Delaey et a1 1974). In addition to tem peratu re, extern al stress may induce thephase transition from the high-temperature austenitic phase to the low-temperaturemartensitic phase. At low temperature stress causes deformation twinning betweencrystallographically equ ivalen t ma rtens ite varia nts.In spite of so m e interesting applications of thes e alloys, up to now ther e has been noconclusive theo retical description of their mechanical or thermomechanical behaviou r.We propose here to use a modified Landau theory to derive a free-energy functiondepend ent on strain and o n tem peratu re which, as a therm odyna mic potential, enablescalculation of the equilibrium prop erties. F or som e alloys the re have been attem pts inthis direction. Nittono an d K oyama (1982) and K oyama an d Nittono (1982) derived afree-energy function for indium alloys such as InT l, InC d, InPb an d InSn undergoing acubic-to-ortho rhom bic or cu bic-to-tetragonal phase transition. G rou p theory was usedin orde r to guaran tee the correct symm etry of the free energy. Later on Barsch andt Present address : Bat te lle- Ins t i tu t e .V . , A m Roemerhof 35 , D-6000 Frankfurt 90, Federal Republic ofGermany.0953-8984/90/010061 + 17 $03.50 @ 1990 IO P Publishing L td 61


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62 F Falk and P K o n o p k aKrumhansl (1984) included strain-gradient terms to calculate th e struc ture of movingtwin bo undaries between tetragonal m artensite variants in InTl. NiTi or copper-basedshap e-m em ory alloys, which are m ore im por tant for applications, undergo a cubic-to-monoclinic phase trans ition. W ithin a one-dimensional model Falk (1980) pro pos ed afree-energy expansion of sixth ord er in a single shear strain comp onent. A n expansionup to four th ord er in th e full strain ten sor invariant with respect to the cubic gro up hasbeen given by Liakos and Sau nders (1982). A s will be seen below, for shape-mem oryalloys an expansion up t o sixth or de r is necessary. It is the aim of this pape r to cons tructsuch a free-e nerg y function invariant with respect to the cubic symm etry grou p Ohofthe high-temperature phase describing the first-order phase transition to monoclinicmartensite. Th e expansion coefficients are d etermined for a CuAlNi alloy. Furthermo rethe stress-strain relations and th e hea t of transformation are derived and com pare d withexperiments .

2. The martensitic phase transition in shape-memory alloysBefore constructing the free -energy function some remark s on the m artensitic phasetransition in shape-memory alloys are appropriate. In this section we deal with thetemperature -induced transition, i .e. both the phases are assumed to b e in a stress-freestate. In NiTi or copper-b ased alloys such as Cu Zn , CuA lNi, CuAlZ n and Cu AlG a andother /?-phase alloys such as NiAl, A uC d and A gC d, the austenite shows, at tem peraturesnot much above the martensitic transition temperature, an ordered BC C structure ofpoint gro up Oh (m 3m ). Th e space groups differ in the various alloys but they a re notrelevant for the symmetry of the strain tenso r.When austenite transforms to m artensite on cooling a spontaneo us strain appears.O n the scale of the lattice cell of aus ten ite, th at is ov er a few Angstrom, the Bain straindeforms the cubic unit cell to an orthorhombic one. As a consequence the followingmisfit problem arises. Imagine an inclusion of orthorhombic martensite in a cubicaustenitic matrix. Even i f the re is no volum e change then across the interface b etweenboth th e phas es a lattice misfit occurs leading t o a long-rang e stress field of high energy .O ne way of reducing this stress is to build an incoh eren t interf ace, i.e. an am orp hou sbou nda ry laye r. This is not the way s hap e-m em ory alloys solve the problem since it isobserved that the phase transition proceeds by rather rapid interface motion which ispossible only for coh ere nt int erf ace s. T he only possibility of vanishing lattice misfit stressacross a coh eren t interface occurs if i t is invariant with respect to th e strain . This meansthat any materia l vector in the interface must not be d efor me d. In orde r that an invariantplane exists at least on e eigenvalue of the strain tenso r has to vanish. In a dditio n, theremaining two eigenvalues must be of opp osite sign. Ho we ver, the ortho rhom bic Bainstrain does not fulfil this condition. As a consequence, in shape-memory alloys mar-tensite f orm s as an internally microtwinned o r faulted p hase , that is, layers of twins oforthorhombic martensite alternate on a scale of a few atomic distances. In the alloysmen tioned the twinning or basal plane is (110) if referred to cubic axes. Th e relativeam oun t of twins is such tha t the total strain resulting from Bain and m icrotwinning straindoes not deform the so called habit plane which therefore acts as the contact planebetween the austen ite and m artensite. Th e orientation of twinning and habit plane, theBain strain, the total strain an d the am oun t of twins are related by W LR theory (Wechslereta1 1953). A review on this topic is given in the bo ok by Waym an (1964).In the shap e-m em ory alloys me ntion ed the microtwinning occurs in a regular pattern


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Land au theory of shape-memory alloys 63on a scale of a few cubic unit cells, so that a monoclinic lattice containing many atom sper unit cell results for min g a m arten site varian t. Typically, tho se variants show a 2H ,3R , 9R o r 18R structu re, depending on the alloy u nder co nsideration. On a length scaleroughly fo ur order s of ma gnitu de bigger it is observ ed by light microscopy th at shap e-memo ry alloys form a rnacrotwinned struc ture in the marten site state. This is because aplate-like inclusion of m arten site variant in an auste nite matrix ge nera tes, due to a shap emisfit, a stress field at its circum ference eve n i f the re is no lattice misfit across the habitplan e. This stress field de creas es almost to ze ro i f an arra nge me nt of twins of mo noclinicmartensite variants forms (self-accommodating groups: see, e. g. , Tas et a1 (1973) andde Vos et a1 (1978)).Th e free-energy function to be con structed in the present pap er deals with thedeformation of single crystals on a scale between the monoclinic lattice and the self-accommod ating gr oup s. T he ap pr op riat e strain is there fore defined on a scale of a fewnanometres . As a con sequ ence the Bain strain is below the resolution of the theory anddoes not occur here. On the other hand the deformation of a macroscopic sampleconsisting of self-accommodating gro ups of mar tensite is not dealt with eith er. Inste adthe theory describes the phase-transition phenom ena between cubic austenite singlecrystals and monoclinic single-crystalline ma rtens ite variants o n a m esoscale. In or derto describe the macroscopic beh aviour of shap e-mem ory alloys one could proceed bysom e averaging over th e different m arten site variants forming the macroscopic samplein the way sketc hed by Falk (1989) for a one-dimensional mod el. How ever , this is notthe objective of the pres ent pap er.T o define m athematically the ap pro priate mesoscale strain me asure consider a triadof materially fixed vectors u1which in stress-free austenite are chosen to be orthog ona land of the sam e mesoscale length a. of a few n m . Fo r simplicity take vectors parallel tothe cubic axes. As a result of def orm ation which may be due to th e martensitic phasetransition o r to an applied load the vectors change to U. Th e symmetric strain tensor isdefined by

(1 ), = f (U * U, - a . l ) / a ; = +(cl a / a ; - 811).Th e product phase on the mesoscale is called a martensite variant. In the shape-m emo ryalloys m ent ion ed , a ma rtens ite variant show s monoclinic symmetry of point gro up C2(2)whereas th e space grou ps are different for th e various alloys. Since the o rd er of th e cubicsymmetry group 0 , f th e parent phase is 48 and that of th e monoclinic group Czof theproduct phase is two, 24 possible orie ntatio ns of th e product occ ur, i.e. the re exist 24martensite variants. All of them follow from a single one by applying symm etry op era -tions of th e cubic gro up . Since they a re equivalent crystallographically their energiescoincide.

3. Free-energy expansionTh e equilibrium properties of a system ar e determined by the free-energy function. InLandaus theory of sec ond -ord er phase transitions on e restricts the variables on whichF depends to the temp erature and to the so called orde r param eter which characterisesthe difference betw een the phas es. Furthe rmo re, i t is argued that the ord er parametertransforms as one of the irreducible representations ( I R S ) of the mo re symmetric high-temperature ph ase (L andau and Lifshitz 1980). In ord er to describe not only the first-ord er martensitic phase transition b ut the comp lete thermomechanical behaviour of the


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64 F Falk and P Konopkasystem, one has to tak e into account th e full strain tensor. Th e dependen ce of F onstrain is governed by the point gro up Oh of the high-temperature phase. If a symmetryoperation in three-dim ensional space is represented by the 3 X 3 matrix D with com-ponents D , ith respect to th e cubic ortho gon al base vectors u i , hen the strain matrixor strain tensor transforms according to

Symmetry requires thatF ( e , T ) = F ( e ' ,T ) ( 3 )

for each of the 48 matrices D. Necessary group-theoretical tools for exploiting thesymmetry require men ts are fou nd in Lyubarsky (1960). Fo r the purpos e of generatingsymmetry-adapted free-energy functions it is mo re convenient to look at the sym metricstrain tens or as a vector in a six-dimensional strain space S span ned by the dyadsQ1 1 = a1 @a12 q 4 = a 2 @ a 3 + a3 @ a 22 9 , = a l @ a 3 + a 3 @ a l2 q 6 = a , @ a 2 + a 2 @ a , .

4 7 2 = a2 @a2 Q1 3 = a3 @a3(4)

Th e com pon ents of strain with respect to q K K = 1 . . . 6) are denoted by e K :el = e11 e? = e22 e3 = e 3 3e4 = 2eZ3 e5 = 2e13 e6 = 2e12

describing the strain in Voigt notation. In strain space S spanned by q K , ymmetryoperations of th e grou p Oh are represented by the 6 X 6 matrices dKL ollowing from th erepresentation Dmn o that th e strain transforms according toe k = dKLe , . (6)

The representation d,, of grou p Oh is reducible into th e rR s rl, 12nd r25'.Accordinglythe strain space S decomp oses into the invariant subspaces SI one-dimensional), SI2(two-dimensional) and S 25 three-dimensional):

S = S1 CB Si2 CB S25 ( 7 )which ar e spanned by @ (K 1 . . 6):

S1s 2 @5 = 2 q 3 - 91 - v2 @Pj = 9 1- 92 (8)s25 C P , = 2 q 4 @ j= 2Q5 0; 2 9 6 .

@i= q i + ~ 1 2 q 3

If referred' to the symm etry-adapted base vectors @i, he components of strain aredenoted by e i .e i = ( e , + e 2 + e 3 ) / 3= tr(e,,)/3 (9 )


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Landau theory of shape-memory alloys 65in the linear approximation describes the volume change during deform atio n.

represent shear deformation s on the (110) planes in the ( i1 0) directions. Th e remainingcomponentse: = e& = e23 e ; = e 5 / 2 = e13 e; = e 6 / 2= e12 (11)

denote shear deformations on the (100) planes in th e (010)directions.it can be ex pan ded into a powe r series with respect to strain:As in Lan dau th eory it is assumed t hat th e free energy is an analytic function so that

F ( e , T ) = F o ( T )+ F'(e,T) F 2 ( e ,T )+ . . . (12)F n ( e , ) = C L l L n ( T ) e L , . . . e L n L , = 1 . . . 6 , n a l . (13)

with F" an expression of the nth deg ree:

H ere and in the following the E instein conve ntion is use d, i.e . over indices occurringtwice summation is implied. Each F" has to m eet the symmetry requirements (3) and(6) individually. The expressions (13) can be interpreted as scalar products of twovectors, namely C and e 8 . . 8 e, oth ele me nts of a symm etrised product spaceS!y,,, spa nne d by the sy mm etrical n-fold tensorial products( v L I 8 . .8 V L , ) ~ ~ , , , v , ~ , u .. u v L n 1G L1 C L2 C . . . C L , =Z 6. (14)

C L l L n e L I . , e L , = CL1 LndLIM,. . d L , M , e M I . . . e M , (15)The symmetry requirement on F "( e , T ) xplicitly reads

and is valid for ar bit rar y strain e and every symmetry operation d of groupOh. hereforeit follows tha tCLl L n = d L I M l .. dLnM,CM1 M n . (16)

The products d L I M l . . dLnM,ene rate a representation d&, of the grou p Oh acting onspace S&,, which, in gene ral, is reducible even if the representation gen erat ed by dLM nspace S is irreducible. Equation (16) says that the coefficients C in order to obeysymmetry have to be built up by invariant directions n, of space Sa,,, according toI"

where the sum goes over all the invariant directions. Fjn are independent materialparameters of nth order . As invariants, the nj transform according to the identicalrepresentation of oh. There are as many invariant directions as the identical rep-resentation occurs in the decomposition of the representation d& into I R S . This numberis given byl gj " = - t r (d&m(i) )g , = 1

wh ere d;y,(i) is th e ma trix of group element number i in the representation d:y,,,. i runsover all the grou p elements , that is from 1 o gro up or de rg which is 48 for group Oh.T he


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66 F Falk and P KonopkaTable 1. Strain invariants J: of first and up to fourth order of the cubic group Oh.Straincomponents in symm etry-adapted coordinates (equations (9)-(11)).Invari ant Strain space

invariant directions q, can be calculated by Wigner projection ope rator s according toK

q;~ . M n =z L I M l ( 4 . . . d L , M , ( i ) ( P L , U . . . V c p L , (19)r = lwhere d L I M , ( i )s the represen tation matrix of grou p element number i in strain space S.To get all the invariant directions one has to insert the different base vectorscp L I ~. . vqL, 1s L , s . . . s L, s 6) of space S& in turn into th e right-hand side ofequation (19). Some of them coincide, but the number of different ones is given byequation (18). Performing the calculations on e finds 1,3 , 6 and 11invariant directionsof first , second, third and f our th or der , respectively. To get the invariants of th e nthord er in strain one ha s to multiply q,l L n by a general eLI . . eL, . The result is theinvariant J; ( j = 1 . , . y )of nth or der corresponding to material pa rameter F/so thatthe nth deg ree in the free-energy expansion reads

inFn = 2 FjJj

j = 1Th e invariants up to fou rth ord er are completely listed in table 1.Fo r later convenience,the fifth- and sixth-order invariants which a re from th e subspace S1@ S12 re listed intable 2.


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Landa u t heory of shape-mem ory al loys 67Table 2. Fifth- and si xth-ord er strain invariants of group Oh ro m space SI63SI?,Invariant Strain space

4. Adaptation of the free-energy function to copper-based alloysU p to fourth ord er there are 21 mater ial parameters necessary to determine the free-energy function. If th e m enti one d fifth- and sixth-order term s are included this num berincreases to 32. In ord er to handle the free-energy expansion on e has to reduce thisnumber by assumptions motivated by experimental results. The first observation is thatstress-free austenite exists at high tem pe rat ure . This means tha t for e = 0 the free energymust have a m inimum at least at those te m pera ture s. This is achieved by vanishing ofthe first-order term, i.e. F' = 0. The next observation is that at low-temperature thefree energy must have 24 symm etry-related monoclinic minima. H enc e there must exista minimum in a monoclinic direction which is not invariant with respect to a subgroupof Oh icher than the monoclinic gro up C2 .Translated into group-theoretical languagethis requirement is called the subd uction an d chain condition (Birm an 1978). Fu rthe r-more , experim ents suggest an approximation for copper-based and for so me otheralloys. O ne observes that in CuAlNi, Cu ZnG a, Cu Zn , CuAlZ n, AgCd and NiA l, toeach martensitic minimu m of strain eM nother on e with nearly -e M is associated (Sa buriand Wayman 1979, Ok am oto et a1 1986). W e assume this relation to hold exactly. H enc eon e is led to th e restriction

F( e ,7') = F ( - e , 7') (21)which can be o bey ed only if invariants of od d degree do not occur in F. Looking at thevalues of spo ntan eou s strain of mar tens ite in those alloys, on e finds moreover that e l ,e2and e3are in the o rde r of mag nitude of 0.1 whereas e4, e 5and e6are roughly one ord erof magnitude s m aller . Th e volume change, in the linear approximation given by 3ei =e , + e 2 + e3, is even smaller. Thu s the main contribution to the spontan eous strain isfrom the irreducible space SI2 f orthorho mb ic symmetry, whereas th e contributionsfrom space S25 are sm aller by a factor of 10 and that from space S1 s even smaller.T o obtain a first-order phase transition from a n even free-energy function o ne hasto include sixth-order terms. From the observation just described it is suggested thatonly sixth-order contributions from space SI2 re tak en , that is the sixth-order invariants


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68 F Falk and P K o n o p k aJ f = (J:)3 and J$ = (1;) re taken into account. In fourth or der we take into accountevery invariant of space SI23 S25 , .e. only volume chang es (space SI ) re d isregarded.In second o rd er every inv arian t is inclu ded . This results in the free-energy function

3 5 2

F = C. F:J~+ C. F;J; + C. F P J ; ( 2 2 )1 = 1 r = l I = 1with 10material constants F; which may d epend o n tem perature . The contribut ionF o(T ) s suppressed since i t only dete rm ines the specific heat and do es not influence thethermomechanical prop erties of the system. In ord er to find the minima of F one has tosolve the necessary condition dF/ de , = 0. This eq uatio n has six types of solutions ofinvariant plane s train. T he first type is the trivial on e e = 0 corresponding to austenitewhich is stable only if the matrix J2F/deKdeL s positive definite at e = 0 o r F: > 0,F: > 0 and F: > 0. Of th e oth er five types only one type agrees with the subduction andchain co nditio n. Only so lution s of this type yield a monoclinic s pon taneo us strain whichhas the structure

e i = O e i = -a e j = a eg = -/3e j = 0 e ; = P e , = 2 a e2 = 0 ( 2 3 )e3 = - 2 a e4= - 2 p e5 = 0 e6 = 2 p .

a and p are th e solutions of4 8 d e + 8 a 2 F j + 2 P 2 ( F ;+ FZ) + F$ = 04 a 2 ( F : + F i ) + 2 P 2 ( F ; + 2 F i ) + FZ = 0.

Generally there are two solutions for a an d p2 . Only o ne of them has positive valuesand in addition yields a minimum of the f ree energy. This solution is identified as one ofthe martensitic minim a. Th e oth ers follow by applying the symmetry operation s of thecubic gro up Oh. rrespective of a and /3, i.e. of the material parameters F:, the straingiven by equatio n ( 2 3 ) always repr esen ts an invariant plane strain. Th e correspondinginvariant plane is identified with th e habit pla ne. S om e algebra yields for their normal nand for the shear vecto rs , i f referred to cubic axes,n = ( l / k ) ( a , , a>s = ( l / k ) [ 2 k 2+ a(7 - ) ,p(7 - l ) , - 2 k 2 + a(?- I ) ] ( 2 5 )( 2 6 )k = Vp2+ 2 a 2 y = V l - 8 k 2 = 1- / s 1 2 / 2 .Th e structure of the habit-plane normal (equation ( 2 5 ) ) s confirmed by experimentalobservations, for example in the alloys CuZnGa, CuAlZn, CuZn, NiAl (Saburi andWayman 1979), AgC d (K rishnan and Brown 1973) and CuAlNi (Okam oto et af 1986).If the values of a and j? given by these autho rs are inserted into the sh ear of equ ation( 2 6 ) hen again t he result describes the ex perim ental values ra the r well with deviationsin the shear direction of a few degrees and in the amo unt of 1 to 2% of the mea suredvalues.

5 . CuAlNi alloysIn this section we deter min e the 10 material param eters F ; of the free-energy function


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Lan dau theory of shape-memory alloys 69(equ ation (22)) for the alloy CuA lNi by comparison with experimental results. Becauseof a lack of appro pria te dat a this is not so easily do ne . O ne would like to know the elasticmoduli of a uste nite single crystals and of martensite single variant crystals togethe r withtheir tem perature depend ence. U nfortunately there has been, as far as we know , onlya single experiment on martensitic elastic moduli of shape-memo ry alloys, namely onthe y ; martensi teof Cu-14 w t%Al-3 wt % Ni by Yasunaga et a1 (1983). W e thu s decidedto ad apt the free energy to this material. How ever, the experiment was not d one on asingle variant crystal but on untwinned marten site. To recalculate the d ata we have touse some othe r theoretical idea described late r. Furtherm ore no temperatu re depen -dence was reporte d. Instead we use data on the heat of transformation.The first step is to determ ine F ! , F i and F: from th e elastic moduli Ctl , Cfi andC$ of cubic austenite which a re the only ind epende nt ones and are defined by

C i L = d 2 F / d e K a e Le = O . (27)From equ ation (22) toge ther with table 1on e finds

F2 -F2 - C A - CA2 - I 1 12

1 - Z(C;: + 2 C 5 )(28)

F: = 2C$.T he elastic mod uli of a Cu-14.1 wt %A1-3 wt % Ni alloy were m easured by Yasuna ga et a1(1982) and by S uezawa and Sum ino (1976) for an alloy of a slightly different compo sition.Yasunaga et a1 report the temperature dependence of Ctl - C$ and of C$ whereasfrom the da ta of Suezaw a and Su min o the bulk m odulus may be calculated: this turnsout to be nearly indep endent of tem pera ture. As a result we have

F: = 592 GN m - 'F ; = (14.1 + ( T - 300 K ) X 4.6 X K- ' )GN m-* (29)F: = (148 - T - 300 K) X 9.4 X K - 1 ) G N m - 2 .

One observes that austenite is highly anisotropic with a very soft shear modulusC f i - C$. Th e seven param eters FP and at 300 K follow from five of th e m onoclinicelastic mod uli of ma rten site toge ther with values of the sp onta neo us strain measured byOkamoto et a1 (1986). From their da ta on habit-plane orientation n and on spontaneousshear 1 s 1 the valuesCY = 0.023 /3 = 0.0068 (30)follow (equ ation s (25) and (2 6)). The re is som e scatter in the rep orted value of y1 whichpropagates to 8. I t has been mentioned that no data on the elastic moduli of internallymicrotwinned martensite exist. Yasunaga et a1 (1983) detw inned y ; martensite prior totheir experiments by applying an ap prop riate external stress. Their elastic moduli dat areferring to o rtho rhom bic single crystals are listed in table 3 . We deal here with aninternally microtw inned m onoclinic martensite variant which consists of layers of ortho-rhombic twins. Thei r amo unt and their o rient ation with respect to the austenitic pare ntcrystal has been repo rted by O kam oto et a1 (1986). We assume that th e elastic moduliof the microtwinned variant follow as an average over those of the single crystals. Tothis end the moduli of the o rthorhom bic twins are referred to a common system ofcoordinates, e .g . to the cubic axesof the paren t phas e. Th en they are averaged accordingto their relative a mo un t. Th e averaging has been don e on the elastic moduli themselves


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70 F Falk and P K o n o p k aTable 3. Elastic moduli ( in GPa ) of or thorhombic y j , martensite single crystals (untwin ned)d u e to Yasunaga et a1 (1983) in Voigt notation.

~

189 141 205 54.9 19.7 62 .6 124 45.5 115

(as proposed by Voigt ( 1 9 1 0 ) ) and on their inverses which a re the elastic compliances(Reuss 1929) . Fo r fur ther calculations we use t he m ean values of bo th th e results listedin tab le 4 if referred to base @,or to the symmetry-adapted base @i. hese are comparedwithtakena t the spontaneousstra in ( e x ) = (0 , -a , a , /3,0,/3) of martensite. From equa tion( 2 2 ) ogether with tables 1and 2 on e finds

C i , = aF/ae,aeL 1 ,M

CE = 2FT CiR = O K > 1

( 3 1 )M

C: = 6F: + 120aF: + 12/3*F; + 10pF: + 1 1 5 2 a 4 F f + 8a4@C: = 2F: + 24n2F f + 48 F i + 2/3 F i + 1 9 2 d Ft + 8a4F9

Mc-.4 - C# = 2F: + 1 2 p 2 F i + 16/3F; + 8aF; + 8 a 2 F :Cz = 2F: + 8P2F! + 8aFi

Table 4. Elastic moduli ( in GPa ) of monoclinic internally twinned y ; martensite (referredto sym metry-adapted base vectors). Mean value of Voigt and of R e u s a ve ra ging p r oc e dur eapplied to th e elast ic constants of the or thorhombic mar tens i te of table 3 referred to baseOK nd to symmetry-adapted base O K .

c,, c22 C33 c44 cis C66 ClZ c ,? c2 3172 162 172 44 31 44 103 92 103-2 .6 5 1.5 3.7 -20 - 2 -0.3 5 0.8 1 -15 -3

1102 44 7 127 177 123 177 -21 - 2 -33

1.5 -21 0.3 -3.6 51 4.8 -12 50 7.3 3.9 -61 -12


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Land au theory of shape-me mory al loys 71Table 5.Elast ic moduli (i n GP a) of monoclinic internally twinned y ; martensi te (referred tobase Q K ) as calculated from the free-energy funct ion.

181 170 181 44 31 44 112 101 112~~

ci4 CIT cl6 c24 c?S c2 6 c3 4 c 3 S c36 c4 T c46 c5 635 0 -26 -9 0 -9 -26 0 35 0 -15 0

There follows the linear dependence 3C j j - C ii - 2C5j = 0, which is obeyed by thevalues of table 4. Toge ther with equat ions (29) and (30) th e five values C j i , Cjj, C,,,C;, and C,, are used to calculate the seven param eters FP and FP at 300 K :F l = -1.182 x l o 4 G N m -* F! = 3.13 x l o 5 G N m - 2 F i = 1.64 x l o 5 G N m -2F: = -5.53 x l o 4 G N m -* F i = -4.27 X l o 4 G N m - 2 (33)Ff = 3.35 x lo6G N m - * 6 3.71 X lo7GN m - 2 .

In the last step their tem pera ture depend ence is investigated. T he only hint is theequilibrium phase transition tem pera ture an d the heat of transform ation. To ependsstrongly on composition an d, con sequently, there is a large scatter in the values reportedby different auth ors. We decided to use the heat of transform ation reported by Otsu kaet a1 (1976) t o be -48.3 MJ m-3 (or -86.2 cal mol-') at 300 K . Irrespective of the valuechosen here the temperature dependen ce of F: and F: alo ne never sufficiently influencesthe value of th e free energy at the m artensitic minimum to produce a reasonable heat oftransformation. At least one of the parameters F: and FF also has to depend ontemperature . Since the major part of spontaneo us strain belongs to the space SI, wechose the only fourth-order parameter of this space, F' ; , to depend linearly on thetemperature:

F l ( T ) = Flo + ( T - 3 0 0 K ) F j T . (34)Th e heat of transformation Q follows from the entropy S ( e , T ) defined by

S ( e , T ) = - 8 F / d Taccording to

Q ( T )= TASwhere Tis the temp erature of t ransformation and A S represents the difference in entropyof bo th th e phases. F or a temper ature-induced transform ation of a stress-free crystalon e findsQ ( T )= T ( S ( e A ( T ) ,T ) - ( e ' ( T ) , T ) )= -T (4a2F$ , + 2 P 2 F & 16Lu4Fj~) .

(35)

F t T = 3 5 . 5 G N m - 2 . ( 3 6 )Togeth er with equation (29) this yields

Equat ions (29), (33), (34) and (36) give the full set of pa ram eters F:' for the free-energyfunction F(e,T ) f equation (22).


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72 F Falk and P Konopka

e1..................................................................

0.04

6. Consequences of the free-energy function

"

Th e free-energy function with the par am eter s just determ ined completely characterisesthe equilibrium prop erties of th e system . In the following som e of thes e are discussed.A t any tempe rature the austenitic minimum is at e = 0. Th e martensitic minima, on eof them determined by equatio ns (23) and (24), the other 23 by symmetry operations,exist below T , = 303 K . The ir positions in strain space, i .e. the spontaneo us strain ofmartensite with respect to austenite, depend on temperature as shown in figure 1.

-0.04 ........ Figure 1. Temperature dependence of th espontaneous strain of martensite. In thedotted range martensite is meta stable....... ...............

-0.06

Between T , and To= 271 K the m artensitic minima have a higher free energy than th eaustenitic minimu m. H ence in this tem peratu re range m artensite is metastable. BelowTomartensite is stable, having a lower free energy than austenite. How ever, since theaustenitic minimum does not vanish the austenite remains metastable. In figure 2 thefree energy is plotted for different temperatures along a straight line in strain spacejoining the austenitic minimum with one of the martensitic on es. N ote that the magn itude

-20 1Figure 2. Free -energ y curves for different tem peratu res along a straight line in strain spacejoining the austenitic an d martensitic minima (see text for details). To avoid intersections,the different free- energy curves are shifted along th ey axis.


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Landau theory of shape-memo ry al loys 7 3of the s t ra in a t the aus ten i t i c min ima as wel l as th e d i rec tion of th e section plo tted slightlydepends o n tempera tu re . Th e lowest energy barr ie r does no t exac tly l ie on t he jo in ingstraight l ine. Inste ad, i t is given by the en ergy of a sadd le point which, how eve r, is a l i t t lebit lower tha n the maxim um e nergy a long the s t raight line.Th e s tress-s train law derives from t he free ene rgy according to

O K = dF/deK (37)wh ere oK n d eK re fer to Voigt not at ion . In f igure 3 the derivat ive of F i n t h e d i rec ti onof the line joining th e austenitic an d martensitic minima is plot ted. I t describes a stress-s t rain curve during a shear deformation leading to on e of the martensi te variants . Th epart showing a negat ive s lope is unstable s ince here an elas t ic modu lus is negat ive.

fa1N

fb l /

no , / Is1,-,IS1 0.05/ o.8-1-/0.1 0.15.15 0.1 e 0 0 .0 5 0.1 0.15 0.15/ -1: / - 1 Ii

ic l- 2 1 .;-* I : /

i - I !I I1I - 2 '

I I C- 2 1 'Ez/ id)

I-l l D/ \ - . . .0.05 IS10 0.05 0.1 0.15

-1I

Figure 3. Stress-strain curve in shear along the deformat ion mode f rom aus ten i te to mar-tensite for di fferen t t em peratures : ( a ) T = 34 0 K . ( b )T = 30 0 K . (c) T = 27 1 K an d ( d ) T =230 K .

The refo re, with increasing s t ress the austeni te become s unstable w hen the m aximu m ofthe s t ress-st rain curve is reache d where the s t ress-induced t ransfo rmation to marten si tea long the upper a r row se t s in . On reducing the s t ress a t the min imum, mar tens i tebecomes uns tab le and th e re t rans fo rmat ion to aus ten i te s ta r ts . In thi s way th e f ree-energy curves give rise to a stress-strain hysteresis which, how eve r, is red uce d bynuclea tion phenome na no t included in the Landau approa ch . A t t emper a tu res T > T ,the stress-strain curv es ar e of pseudo-elas tic type where as at lower tem pera ture ferro-elastic cur ves follow.Th e elast ic moduli in Voigt notat ion are defined by

CKL, = d 'F/deKde[- ( 3 8 )where on the r igh t -hand s ide the a ppropr ia te va lues of s train have to be in serted. Infigure 4 the moduli of monocl inic ma rtensi te a re plot te d as funct ions of t em p era tu re .Th e va lues a t 300 K ar e given in table 5 . T h e m o du li C, , C l 2 ,C 1 3 , zz,CZ3, ii. Cj4,Cj5 n d Chh lotte d in figure 4 ( a ) charac terise an orth orho mb ic crystal where as the res t


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74 F Falk and P Ko n o p k a(figure 4(b)) are due to the monoclinic distortion of martensite. We would like toemphasise that the or tho rho mb ic moduli calculated from the free energy agree ratherwell with experiments (compare tables 4 and 5 ) . The monoclinic moduli are muchsmaller, differing from the values of table 4 . This may be a consequence e ither of theaveraging procedure necessary to get the moduli of microtwinned martensite or ofneglecting thi rd- and fifth-order term s in the free-energy expansion. This can be resolvedonly when ex perim ental data o n the m oduli of m icrotwinned m artensite are available.Th e stability or m etastability range of each ph ase is th e region in strain space in whichevery eigenvalue of the matrix C K Ls positive. If one appro aches the boundary of stabilitythen the eigenvector c orrespo nding to the vanishing eigenvalue gives the direction instrain space into which th e system will evolve to a stable configuration.

Gl, 33 250

-Ez%2-

... ,' .. ......

, . . ....

. . ...

( b )

65 c25 c35 [45 [56c2 L c26c4 6__cc Figure 4. ( a ) ( b ) Elastic moduli of n i x -44 Cl, tensite in Voigt notation agdinst tem-pe rd ture In the do t ted r ange mar tens i te is

metdstahle 1he cur ie s end where mar -tensite become s unstdhle

In figure 5 the h eat of transform ation is plotted against temperatu re: i t decreasescontinuously with increasing temperature. The equilibrium phase transition tem-perature wh ere the f ree energ y of both the phases coincides can be calculated numericallyto yield the resultT o = 271 K . (39)

Salzbrenner an d C ohe n (1979) suggest that T,, s identified with (M,+ A f ) / 2 .Calculatedvalues over a wide range a re reported by different au thors (see e. g ., Otsu ka e t a l 1976,Oka m oto er a1 1986, Yasunag a et a1 1982).


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Landau theory of shape-memory alloys 75

7. DiscussionAs it stands the free energy proposed in the present paper describes homogeneousshap e-m em ory alloys, i.e. one -ph ase single crystals in which th e strain does not vary. Inorder to deal with mo re involved cases, for example w ith strain dep end ing on positionor even with multiphase crystals, one has to em bed the free energy as a constitutiveequ ation into the basic laws of con tinuu m mechanics, namely into the balance of mass,of mom entum and of energy. Restrict oneself to isothermal equilibrium, then only thebalance of mo me ntu m is relevant which then rea ds

V * a + f = O (40)

n . a = a (41)within the body an d

at the surface, where f and a denote external volume and surface force densities,respectively, and n is the o ute r no rm al of th e surface of the bo dy. Typically, f s due togravity and may be n eglec ted. Since the free energy contains term s higher than secondord er the stress is a non -linea r function of s trai n, which results in the non-linear balanceequat ions (40) and (41) . It is thus rather cumbersome to solve these equations forarbitr ary surface forces o r in the case of defects such as dislocations. In so me specialcases, however, the problem reduces to one dimension. Assume for example a par-allelepiped with edg es parallel to the h abit pl ane, th e orien tation of which follows fromthe free energy according to eq uations (25) and (24). Fur therm ore, consider constantexternal forces acting o n th e surface parallel to the habit plane only. The n the solutionof th e balance equatio ns is a shea r defo rm ation parallel to the habit plane varying onlyin a direction normal to i t . Th e shear is propo rtion al to that given by equ ation (2 3). Ifone den otes its amo unt by s the fre e energ y for this special defo rma tion m ode is givenby

F ( s , T ) = UhSh - u4s4 + a$ (42)with temperature-dependent positive coefficients a , following from the par am eters Fi' ,To ge the r with the stress-strain curves in sh ear the result is plotted in figures 2 and 3.This free-energy function is exactly of the type proposed by Falk (1980) in a purelyone-dimensional approach. A two-phase system consisting of layers of austenite and


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76 F Falk and P Konopkamartensite is repr esen ted by a one-dimensional deformation of the type just discussedin which the strain jum ps from t he value of auste nite to that of martensite across thehabit plane. T he dynamics of this prob lem was treate d by Falk and Seibel (1987).Since the free energy proposed in the present paper yields 24 martensitic variantsone can deal with self-accommodating groups of four variants in an austenitic matrix.For exam ple, four variants with

(eK)I= (0,2a, 2cu, 0, 28, -28)(eK)lll (0,2a,- 2a, -2a, 0 , -28, -28)( e K I I V = (0, 2a, 2a,0,28,28)n l = (8,a, 4a,,, = (-8, a, 4

yield a vanishing mean value of strain. They have habit planes with normalsall = (8,a, a)n I v = (-8, a, 4 .

M oreo ver, the relative strain from o ne variant to an oth er is an invariant plane strain forth e pa irs 1-111, I -IV, 11-111 and 11-IV with normals (0 ,1 , -1) , ( l , O ,O ) , (1,0,0) and( O , l , - l ) , espectively. Th erefo re in this gr oup every interface is stress-free.In furth er investigations heterogen eous nucleation can be trea ted. U p to now therehave been attempts in this direction by Clapp (1973), Guenin and Clapp (1986) andOlson and Co hen (1982). In these pa pers , however, an inappro priate one-dimensionalfree energy is used. Taki ng the three-dimensional free energy proposed for the first timein this pap er, o ne has to solve the balance eq uation (40) for d efects such as dislocationsof different types and different orientations. Because of the highly non-linear stress-strain relation this can be done only numerically. It is expected that at least for ametastable phase around a dislocation core an embryo of the new phase exists inequilibrium.

ReferencesBarsch G R an d Krumhans l J A 1984 Phys . Rev. Lett. 53 1069Birman J L 1978 Gr oup T heore ticalM ethods n Physics, 6th In / . Colloq. ed . P Kramer and A Rieckers (LectureClapp P C 1973 Phy s. Status Solidi b 57 561Delaey L. Krishnan R V . Tas H and Warlimont H 1974 J . Muter. Sci. 9 1521Falk F 1980 Ac ta Metall. 28 1773-Falk F and Seibel R 1987 Int. J . Eizgng Sci. 25 785Guin in G and Clapp P C 1986 Proc. Int. Conf. on Martensitic Transformation (Tokyo: Japan Institute ofKoyama Y and Nittono 0 1982 J . Physique Coll. 43 C4 145Krishnan R V and Brown L C 1973 Metall. Trans . 4 423Landau L D and Lifshitz E M 1980 Statistical Physi cs (Oxford: Pergamon) Ch . XIVLiakos J K and Saunders G A 1982 Phil. M ag. A 46 217Lyubarsky G Y 1960 Th e Application of Group Theory in Physics (Oxford: Pergamon)Nittono 0 and Koyama Y 1982 Japan J . A p p l . Phys . 21 680Okamoto K. Ichinose S. Morii K. Otsuka K and Shimizu K 1986Actu Meuzll. 342065Olson G B and Cohen M 1982 J . Physique C oll. 43 C4 75Otsuka K , Wayman C M. Nakai K , Sakamoto H and Shimizu K 1976 Acta Meta ll. 24 207

Notes in Physics vo l79 ) (Berlin: Springer) p 203

1989 Int. J . Engng Sci. 27 277

Metals) p 171


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Falk - Three Dimensional Landou Theory Describing the Martensitic Phase Transformation of Shape...

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