difsudionless transformations

9 Diffusionless Transformations

Luc Delaey

Departement Metaalkunde en Toegepaste Materiaalkunde,Katholieke Universiteit Leuven, Heverlee-Leuven, Belgium

List of Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . 5859.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5879.2 Classification and Definitions . . . . . . . . . . . . . . . . . . . . . . . 5909.3 General Aspects of the Transformation . . . . . . . . . . . . . . . . . 5939.3.1 Structural Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5939.3.2 Pre-transformation State . . . . . . . . . . . . . . . . . . . . . . . . . . 5979.3.3 Transformation Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . 5999.3.4 Microstructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6009.3.5 Shape Changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6009.3.6 Transformation Thermodynamics and Kinetics . . . . . . . . . . . . . . . 6049.4 Shuffle Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 6079.4.1 Ferroic Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 6099.4.2 Omega Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 6109.5 Dilatation-Dominant Transformations . . . . . . . . . . . . . . . . . . 6109.6 Quasi-Martensitic Transformations . . . . . . . . . . . . . . . . . . . 6119.7 Shear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 6139.8 Martensitic Transformations . . . . . . . . . . . . . . . . . . . . . . . 6159.8.1 Crystallography of the Martensitic Transformation . . . . . . . . . . . . . 6159.8.1.1 Shape Deformation and Habit Plane . . . . . . . . . . . . . . . . . . . . 6159.8.1.2 Orientation Relationship . . . . . . . . . . . . . . . . . . . . . . . . . . 6169.8.1.3 Morphology, Microstructure and Substructure . . . . . . . . . . . . . . . 6189.8.1.4 Crystallographic Phenomenological Theory . . . . . . . . . . . . . . . . 6209.8.1.5 Structure of the Habit Plane . . . . . . . . . . . . . . . . . . . . . . . . . 6239.8.2 Thermodynamics and Kinetics of the Martensitic Transformation . . . . . 6249.8.2.1 Critical Driving Force and Transformation Temperatures . . . . . . . . . 6249.8.2.2 Nucleation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6309.8.2.3 Growth and Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6329.8.2.4 Transformation Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . 6349.9 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6349.9.1 Metallic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6349.9.1.1 Ferrous Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6359.9.1.2 Non-Ferrous Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6379.9.2 Non-Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6399.10 Special Properties and Applications . . . . . . . . . . . . . . . . . . . 6419.10.1 Hardening of Steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641

Phase Transformations in Materials. Edited by Gernot KostorzCopyright © 2001 WILEY-VCH Verlag GmbH, WeinheimISBN: 3-527-30256-5

9.10.2 The Shape-Memory Effect . . . . . . . . . . . . . . . . . . . . . . . . . 6419.10.3 High Damping Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . 6479.10.4 TRIP Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6479.11 Recent Progress in the Understanding of Martensitic Transformations 6499.12 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6519.13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652

584 9 Diffusionless Transformations

List of Symbols and Abbreviations 585

List of Symbols and Abbreviations

a lengthA factor (containing elastic terms)a, b, c constantsAd retransformation temperature (deformation induced)Aj amplitude of perturbation with polarization jAs starting temperature for austenite formationB pure strains associated with lattice correspondenceC lattice correspondenceC cubic sequenceC¢ elastic shear constantc* size of critical nucleusCij eigenvalues of elasticity tensorc/a axial ratioc/r thickness/radius ratio of nucleuse order parameterE electric fielde1, e2, e3 principal strainsEe strain energyF applied forceDG difference in chemical Gibbs energyDG* Gibbs energy of nucleationDgs surface Gibbs energy per unit volumeG* elastic state functionGa, Gb, Gg Gibbs energy of phases a, b, gGc, Gchem chemical Gibbs energyGelast elastic Gibbs energyGsurf surface Gibbs energyGtot total Gibbs energyGd defect Gibbs energyGi interaction Gibbs energyH magnetic fieldH* elastic state functionDH, DH* enthalpy changel molar lengthMd deformation-induced martensitic temperatureMf martensite finishing temperatureMs martensite starting temperatureP inhomogeneous lattice-invariant deformationq wave vectorr radius of a plater lattice vectorR rigid body rotationR rhombohedral sequence

r* size of critical nucleusS strain matrixDS entropy changeT temperatureT0 equilibrium temperatureTc critical transition temperatureTN Neel temperatureTTW temperature at which twins appearv volume of a plateVm molar volumex atomic fraction of elementsx, y lattice vectors

a name of phaseb name of phaseg name of phaseG interfacial energyd0 shear straine name of phasee0 strain associated with the transformationj surface to volume ratios stresssa applied stress

ASM American Society for Materialsb.c.c. body-centered cubicb.c.t. body-centered tetragonalf.c.c. face-centered cubicf.c.t. face-centered tetragonalG–T Greninger–Troianoh.c.p. hexagonal close packedHIDAMETS high-damping metalsHP habit planeHRTEM high-resolution transmission electron microscopyIPS invariant plane strainK–S Kurdjurnov–SachsLOM light optical microscopyN–W Nishiyama–WassermannPTFE polytetrafluoroethyleneSMA shape-memory alloysSME shape-memory effectTRIP transformation-induced plasticity


9.1 Introduction 587

9.1 Introduction

Diffusionless solid-state phase transfor-mations, as the name suggests, do not re-quire long-range diffusion during the phasechange; only small atomic movements overusually less than the interatomic distancesare needed. The atoms maintain their rela-tive relationships during the phase change.Diffusionless phase transformations there-fore show characteristics (such as crystal-lographic, thermodynamic, kinetic and mi-crostructural) very different from those ofdiffusive phase transformations.

Martensitic transformations, becausesome of the properties associated with themsometimes lead to specialized applications,are considered to be an extreme class ofdiffusionless phase transformation and wetherefore in this chapter concentrate onmartensite. Because of the similarity ofsome of the transformation characteristics,a number of other diffusionless solid-statephase transformations have sometimesbeen designated erroneously as marten-sitic. To avoid misinterpretations, Cohen et al. (1979) proposed a classificationscheme that identifies broad categoris ofdisplacive transformations showing fea-tures in common with martensitic transfor-mations, but distinct from them. Their clas-sification scheme, reproduced in Fig. 9-1,will largely be followed here. Martensitictransformations are here only a subclass ofthe broader class of displacive/diffusion-less phase transformations.

The classification proposed by Cohen etal. (1979) is discussed first, and subsequentsections deal with general aspects of thecrystallography, thermodynamics and ki-netics of the different displacive transfor-mations. Although it is not the purpose togive full details of all materials that exhibitthis type of transformation, the most im-portant material systems in which such

transformations have been observed arepresented.

A martensitic transformation can be de-tected by a number of techniques; some arein situ methods, whereas others are step-by-step measurements. The results are usu-ally plotted as a change in a physical prop-erty versus temperature (see the schematicrepresentation in Fig. 9-2), from which thetransformation temperatures can be deter-mined. Some of these plots can be trans-lated into the volume fraction of martensiteformed versus the temperature. Suchcurves allow us to determine the transfor-mation temperatures as indicated in thesefigures.

In situ detection becomes limited if thetransformation temperature is above roomtemperature, and dilatometry then seems tobe the most appropriate technique providedthat quenching – which is needed to avoidalterations in the sample due to diffusion –is possible inside the dilatometer. There arefar more ways of following the transforma-tion when the transformation temperatureis below room temperature – preparing thesample and carrying out the measurementscan take some time and at room tempera-ture diffusion is then almost negligible.During slow cooling after water quench-ing, the techniques frequently used includedilatometry, electrical resistivity and mag-netic measurements, calorimetry, in situmicroscopy, acoustic emission, elastic andinternal friction measurements, positronannihilation, and Mössbauer spectroscopy.Some of the less common techniques usedto study martensitic transformations werereviewed by Fujita (1982).

If crystallographic information is re-quired, X-ray and electron or neutron dif-fraction are used. X-ray diffraction meas-urements by Fink and Cambell in 1926 (lat-tice parameter of C-steel martensite), byKurdjurnov and Sachs in 1930, by Ni-

shiyama in 1934 and by Greninger andTroiano in 1949 (orientation relationshipbetween austenite – the parent phase – andmartensite) represent breakthroughs in thestudy of martensitic transformations.

The most frequently used techniqueswill now be briefly discussed and illus-trated.

If a sample is polished into the marten-site (= parent phase), a surface relief ap-pears. An edge-on section of such a sampleis shown in Fig. 9-2a (see Hsu, 1980). Theorigin of the surface relief is indicated bythe white arrows. Owing to the macro-scopic martensite shear (the two thinner ar-rows), a surface relief is obtained. This is


Figure 9-1. Classification scheme for the displacive/diffusionless phase transformations as proposed by Cohenet al. (1979).

9.1 Introduction 589

Figure 9-2. Some examples of how to see or measure the presence and growth of martensite (see text for de-tails).

explained further in Sec. 9.8.1.1 (Fig.9-19).

The electrical resistance shows at thetransformation temperatures a deviationfrom linearity versus temperature. This isshown in Fig. 9-2b for the martensitictransformation in an Fe–Ni alloy and aAu–Cd alloy. The difference in the resis-tance ratio for the two different materials isobvious and remarkable (see Otsuka andWayman, 1977). Measuring the electricalresistance while cooling or heating a sam-ple is a very convenient and relatively easyand accurate technique of determining thetransformation temperatures Ms , Mf , As

and Af .The heat exchanged with the surround-

ings is becoming a more popular method ofdetermining the transformation tempera-tures. An example is shown in Fig. 9-2c(Nakanishi et al., 1993). A DSC (differen-tial scanning calorimetry) curve allows anyparticular behavior of the martensitic sam-ple to be detected (for example, effects oc-curring during heat treatments and/or def-ormation steps).

A number of martensitic transformationsand materials are characterized by a so-called shape-memory effect (see Sec.9.11.2). Figure 9-2d (courtesy of MemoryEurope) shows the displacement of aspring made of a NiTi shape-memory alloy.The spring controls a small valve in a cof-fee-making machine. At the temperature As

the hot water starts to drop onto the coffeepowder. This valve is completely open assoon as the temperature Af is reached. Thetemperature range between As and Af seemsto be most suitable for making the best cupof coffee. On cooling, the valve closesagain. The “displacement– temperature”curve measured on cooling does not coin-cide with the heating curve.

During a martensitic transformation, notonly is the shape of the sample changed but

also the specific volume, which allows thetransformation temperatures to be deter-mined by dilatometry (Fig. 9-2e, fromYang and Wayman, 1993).

Changes in mechanical properties arealso measured while the sample is trans-forming. The Young’s modulus exhibits adip between the two transformation tem-peratures Ms and Mf , as clearly visible inFig. 9-2 f for four different alloys (see Su-gimoto and Nakaniwa, 2000).

9.2 Classification and Definitions

A structrual change in the solid state istermed “displacive” if it occurs by coordi-nated shifts of individual atoms or groupsof atoms in organized ways relative to theirneighbors. In general, this type of transfor-mation can be described as a combinationof “homogeneous lattice-distortive strain”and “shuffles”.

A lattice-distortive deformation is a ho-mogeneous strain that transforms one lat-tice into another; examples are shown inFig. 9-3. The homogeneous strain can berepresented by a matrix according to

y = S x (9-1)

where the strain S deforms the lattice vec-tor x into a lattice vector y. This strain is homogeneous because it transformsstraight lines into other straight lines. Aspherical body of the parent phase will thusbe transformed into another sphere or intoan ellipsoidal body. The actual shape of theellipsoid depends on the deformation in thethree principal directions. If a sphericalbody is completely embedded inside thematrix phase and is undergoing the strainS, the volume and shape change associatedwith this deformation will cause elasticand, sometimes, plastic strains in the parent


9.2 Classification and Definitions 591

and/or product phases. The lattice-distor-tive displacements therefore give rise toelastic strain energy. In addition, an inter-face separating the phases is created, gen-erating an interfacial energy. As is obviousfrom Fig. 9-1, the relative values of theseenergies play an important role in the clas-sification scheme.

A shuffle is a coordinated movement ofatoms that produces, in itself, no lattice-distortive deformations but alters only thesymmetry or structure of the crystal; asphere before the transformation remainsthe same sphere after the transformation.Shuffle deformations produce, in the idealcase, no strain energy and thus only interfa-cial energy. Two examples of the shuffledisplacement are given in Fig. 9-4. Shuffledeformations can be expressed by “lattice

Figure 9-3. Examples of the lattice-distortive defor-mations of a cubic lattice: (1) a dilatation in the threeprincipal directions transforms the lattice into an-other cubic lattice with larger lattice parameters; (2)a shear along the (001) plane leads to a monocliniclattice, and (3) an extension along the [001] axiscombined with a contraction along the [100] and[010] axis results in an orthorhombic lattice.

Figure 9-4. Examples of shuffle displacements in:(a1) strontium titanate; oxygen, strontium, ti-tanium; (a2) the displacement of some of the oxygenatoms can be represented by an alternating clockwiseand anti-clockwise rotation around the titanium at-oms; and (b) the (111) planes in a b (b.c.c.) latticeand the collapsed (0001) planes in the hexagonal wlattice (Sikka et al., 1982).

wave modulations” as

Dx = Aj (q) exp (i q · r) (9-2)

where r is a lattice vector, q is the wavevector giving the direction and inversewavelength of the modulation, A is the am-plitude of the perturbation, and j denotesthe polarization of the wave. An alternativedescription is represented by relative dis-placements of the various atomic sub-lat-tices that specify the structures of the twophases in terms of corresponding unit cells,which are not necessarily primitive.

Cohen et al. (1979) subdivided the dis-placive transformations into two maingroups, according to the relative contribu-tion of the two above-mentioned atom dis-placements and hence the ratio of the inter-facial/strain energy. In this context, theydistinguish between “shuffle transforma-tions” and “lattice-distortive transforma-tions”. Since the latter give rise to elasticstrain energy and the former only to inter-facial energy, major differences are foundin the kinetic and morphological aspects ofthe transformation, which justifies the dis-tinction. Shuffle transformations are notnecessarily pure; small distortive deforma-tions may additionally occur. They there-fore also include those transformations in-volving dilatational displacements, in addi-tion to the pure shuffle displacements, pro-vided that they are small enough not to al-ter significantly the kinetics and morphol-ogy of the transformation.

The lattice-distortive transformationsthemselves are subdivided according to therelative magnitudes of the two componentsof the homogeneous lattice deformation,i.e., the dilatational and the deviatoric (shear) components (see Fig. 9-3). The in-itial and the isotropically dilated sphereshave no intersection and it is therefore notpossible to find a vector whose length hasnot been changed by the transformation.

On the other hand, the ellipsoid obtainedafter a pure shear intersects the originalsphere; hence a set of vectors exist, whoselengths remain unchanged. Such a defor-mation is said to be characterized by an“undistorted line“. An undistorted line canonly result from a homogeneous latticedeformation if the deviatoric or shear com-ponent sufficiently exceeds the dilatationalcomponent.

Cohen et at. (1979) thus consider a trans-formation as deviatoric-dominant if an in-variant line exists. A further subdivisionwas made between phase transformationswith and without an invariant line, orbetween “dilatation-dominant” and “devi-atoric-dominant” transformations.

If the magnitude of the lattice-distortivedisplacements is large in relation to that ofthe lattice vibrational displacements, highelastic strain energies are involved. How-ever, if they are comparable, the strain energies will be small and hence will notdominate the kinetics and morphology ofthe transformation. In the latter case wedeal with “quasi-martensitic transforma-tions”. The former, i.e., the “martensitictransformations”, are therefore those dis-placive or diffusionless phase transforma-tions where the lattice-distortive displace-ments are large enough to dominate the ki-netics and the morphology of the transfor-mation. “Martensite” is the name nowgiven to the product phase resulting from amartensitic transformation. Because of thevolume change and the strain energy in-volved with the transformation, martensitictransformation requires heterogeneous nu-cleation and passes through a two-phasemixture of parent and product; it is a first-order diffusionless phase transformation.Consequently, the forward and reversetransformations are accompanied by an ex-othermic and endothermic heat effect, re-spectively, and forward and reverse trans-


9.3 General Aspects of the Transformation 593

formation paths are separated by a hystere-sis.

Among the various diffusionless phasetransformations which exist in solid-to-solid phase transformations, martensitictransformations have received much atten-tion in the past. Historically, the term “mar-tensite” was suggested by Osmond in 1895,in honor of the well-known German metal-lurgist Adolph Martens, as the name for thehard product obtained during the quench-ing of carbon steels. It was found that thetransformation to martensite in steel wasassociated with a number of distinctivecharacteristic structural and microstructu-ral features. During the last few decades itwas recognized that martensite also formsin numerous other materials, such as super-conductors, non-ferrous copper-based al-loys, zirconia, which recently became apopular research subject for ceramists,physicists, chemists and polymeric and bi-ological scientists.

Martensitic transformations have beenthe subject of a series of international con-ferences held in various places throughoutthe world. A list is given at the end of thischapter.

This increasing interest not only has aca-demic origins but can to a large extent alsobe attributed to a number of industrial ap-plications such as maraging, TRIP (trans-formation-induced plasticity) and dual-phase steels, applications involving theshape-memory effect, the high damping ca-pacity, and the achievement of transforma-tion toughening in ceramics.

It may also be of interest to draw atten-tion here to the “massive phase transforma-tions”. Although this type of phase trans-formation, which occurs upon fast cooling,is composition-invariant and the transfor-mation interface has a relatively rapidmovement, it does not fall into the presentcategory. Massalski (1984) defines mas-

sive transformation as a non-martensiticcomposition-invariant reaction involvingdiffusion at the interfaces (see also thechapter by Purdy and Bréchet (2001)).

“Bainite transformations” are also nottreated in this chapter, although they doshow some martensitic characteristics, butcombined with diffusional processes. Forfurther information, the reader is referredto Aaronson and Reynolds (1988) for an in-troductory review and to Krauss (1992).

9.3 General Aspectsof the Transformation

The various diffusionless phase transfor-mations have a number of features in com-mon, such as the crystallographic aspectsof the structural changes, the pretransfor-mation state, the transformation mecha-nisms, the microstructure and the shapechanges that result from the transforma-tion, and the thermodynamic and kineticaspects. The more general aspects aretreated in the following section before dis-cussing separately each subclass of trans-formation.

9.3.1 Structural Relations

This section is concerned with somecrystallographic aspects of the structuralchanges. It is always useful to first deter-mine a unique relationship – a lattice corre-spondence (C ) – between any vector in theinitial lattice and the vector that it becomesin the product lattice. A lattice correspon-dence thus defines a structural unit in theparent phase that, under the action of a ho-mogeneous deformation, is transformedinto a unit of the product phase. Such a cor-respondence therefore tells us which vec-tors, planes and unit cells of the productphase are derived from particular vectors,

planes and cells of the parent phase, with-out regard to their mutual orientation. Forevery structural change there exist manyways of producing a lattice correspon-dence; the one that involves minimalatomic displacements and which reflectsthe experimentally observed orientation re-lationships most closely should be se-lected.

The actual relationship between labelledvectors, planes, etc., before and after trans-formation (including their mutual orienta-tion) is given by the lattice deformation.Mathematically this lattice deformation isfactorized into a pure strain and a pure rotation, so the correspondence indicateswhat is the pure strain. Knowing the princi-pal axes of the strain ellipsoid, the direc-tions of undistorted lines, if any exist, canthen easily be found. In 1924, Bain pro-posed such a lattice correspondence for thef.c.c.-to-b.c.c. (or b.c.t.) transformation iniron alloys; it is referred to in the literatureas the prototype Bain correspondence and/or Bain strain. Since then, lattice corre-spondence values (C ) and their associatedpure strains (B) have been published for anumber of other structural changes; someexamples are given in Fig. 9-5.

In the original Bain strain, a tetragonalcell is delineated into two adjacent f.c.c.unit cells. Then, it is contracted along z byabout 20% and is expanded along x¢ and y¢by about 12%. In another example, thetransformation from a NaCl-type structureinto a CsCl-type structure, a contraction of40% along the [111] body diagonal and a19% isotropic expansion in the perpendicu-lar (111) plane is needed; the volumechange is about 17%.

Homogeneous strains alone, however, donot always describe the structural transfor-mation completely. Additional shufflesmay be needed to obtain the exact atom ar-rangements inside the deformed unit cell.

A special situation arises for some mate-rials, when the structural change can beachieved by a pure deformation that leavesone of the principal directions unaltered.Such a situation is found in some polymers.It occurs, for example, in polyethylene,which has an orthorhombic and a mono-clinic polymorph with chains parallel to thez-axis; these strong covalently bondedchains are unlikely to be distorted by thetransformation; consequently, the pure def-ormation along the z-axis (e3= 0) is zero.During transformation the chains are dis-placed transversely in such a way that one


Figure 9-5. Some examples of lattice correspon-dence and homogeneous deformation (expansionsand contractions) for (a) f.c.c. to b.c.c. or b.c.t. (afterBain, 1924) and (b) NaCl- to CsCl-type structures(the Shôji–Buerger lattice deformation) (Kriven,1982).


of the remaining principal strains becomespositive (e2 > 0) and the other negative(e1> 0) (Bevis and Allan, 1974).

The strain ellipsoid for the above exam-ple has a special shape. The cone of undis-torted vectors of the product phase degen-erates into a pair of planes, which rotate inopposite directions in the pure strain.Hence either of them may be invariant ifthe total deformation of the lattice includesa suitable rotation. Because all the vectorsin this plane are undistorted, the transfor-mation is said to be an “invariant planestrain (IPS)” type. The pure Bain strain isthen equivalent to a simple shear on that in-variant plane. Because this invariant planeis also a matching plane between the ma-trix and the product phase and both phaseshave to be present at the same time, a rigidbody rotation (R) over an angle q is re-quired in order to bring the product and theparent phases in contact along that plane,the habit plane.

A similar situation is found in structu-rally less sophisticated systems, namely thef.c.c. to h.c.p. transformation in cobalt.Both phases are close packed and a simpleshear on the basal plane transforms the cu-bic stacking into a hexagonal stacking. Be-cause the atomic distances in the basalplane do not change significantly duringthe transformation, the plane of simpleshear is the plane of contact or the habitplane (HP). This is the case, however, onlyif there is zero volume change in the trans-formation, i.e., in the case quoted above ifthe h.c.p. phase has an ideal axial ratio of1.633.

The situation becomes more complicatedif none of the principal strains is zero, butof mixed sign. To achieve matching alongthe plane of contact in cases where the twophases coexist, a deformation is needed ad-ditional to the pure Bain strain in order tohave an invariant plane. Because the final

lattice hase already been generated by theBain strain, this additional strain should bea “lattice-invariant strain”. Slip and twin-ning in the product phase or in both phasesare typical lattice-invariant strains; bothdeformation modes are shown schemati-cally in Fig. 9-6. The diffusionless phasetransformation can in this case be repre-sented by an analog consisting of a purelattice strain (B), a rigid lattice rotation (R)and an inhomogeneous lattice-invariantdeformation (P). The last factor is also as-sociated with a shape change, which mac-roscopically can be considered as homoge-neous. Such a combination is typical ofmartensite and is discussed in Sec. 9.8.1.4.

In cases where the lattice-invariant shearis twinning (as opposed to faulting or slip),type I twinning, where the twin plane orig-inates from a mirror plane in the parentphase, has been assumed. Otsuka (1986)carefully analyzed for a number of systemsthe possibility of a type II twinning as analternative inhomogeneous shear. In type IItwinning, the shear direction stems from atwo-fold symmetry direction of the parentphase. In a table, Otsuka (1986) compiledall the twinning modes observed in marten-

Figure 9-6. Schematic representation of (a) the ho-mogeneous lattice deformation, (b) the inhomogene-ous lattice-invariant deformation (slip and twinning),and (c) the lattice rotation.

site and found that most of them are type Ior compound but that type II twinning hadonly recently been observed. According toNishida and Li (2000), five different twin-ning modes exist in TiNi and other shapememory alloys such as Cu-Al-Ni, Cu-Snetc., namely the 111 type I, the 011type I, the ⟨011⟩ type II, the (100)-com-pound and the (001)-compound twins.Type II twinning has recently receivedmuch attention as a mechanism for latticeinvariant shear in some alloys. Sincetype II twins have irrational twin boundar-ies, the physical meaning of an irrationalboundary is still a controversial problem. Ithas proposed that an irrational boundaryconsists of rational ledges and steps, theaverage being irrational. Thereafter, Haraet al. (1998) carried out a careful study toobserve ⟨111⟩ type II twin boundaries in aCu-Al-Ni alloy by HRTEM, but they couldnot observe any ledges or steps. The boun-dary is always associated with dark straincontrast, and the lattice is continuousthrough the irrational boundary. Nishidaand Li (2000) also made extensive studieson ⟨011⟩ type II twin boundaries in TiNi byHRTEM, but they did not observe ledges orsteps either. Based on these experimentalresults, it is thus most likely that the type IIthin boundary is irrational even on a micro-scopic scale, and the strains at the boun-dary are elastically relaxed with wide twinwidth. To confirm this interpretation, Hara

et al. (1998) carried out computer simula-tions by using the molecular dynamicsmethod. The result showed that the irra-tional thin boundary did not show anysteps. Thus, the above interpretation for anirrational twin boundary is justified. Ot-suka and Ren (1999) have pointed outagain the importance of type II twinning inthe crystallographic aspects of martensite.They also stress the role that martensite ag-ing has on the rubber-like behaviour ofmartensite, a point that has been a long-standing unsolved problem. They showedthat the point defects play a primordialrole. The deformation mechanisms of thecold deformation of NiTi martensite havebeen thoroughly analyzed by Liu et al.(1999a, b). They also found an interplaybetween type I and type II twinning.

The cubic to tetragonal transformations,which occur in a number of metallic andnon-metallic systems, need some specialattention. The volume change with thesetransformations is sometimes very small oreven absent, and the c/a ratio does notchange abruptly but progressively (Fig.9-7); the transformation is then said to be“continuous”. The c/a ratio can be smalleror larger than unity, depending on compo-sition and temperature. The shape changeassociated with the transformation is smallenough in many systems, especially inthose belonging to the quasi-martensites,for elastic accommodation alone to be suf-


Figure 9-7. Temperaturedependence of c/a as mea-sured during the cubic totetragonal transformation;(a) second and (b) first-or-der phase transformation.


ficient for lattice matching. It is, however,possible for c and a to change abruptly withzero volume change.

Based on the crystallographic aspectsdiscussed above, a list of the most typicalcharacteristics of the diffusionless phasetransformations can be compiled (Table9-1).

9.3.2 Pre-transformation State

Diffusionless structural changes areachieved by atom displacements, such asshuffles and shears. The new atomic con-figuration is already prepared in some ma-terial systems at temperatures above thetransition temperature. Atoms in the parentphase then become displaced more easilytowards their positions in the new phasebecause the restoring force that is felt bythe displaced atoms diminishes on cooling.In certain cases the restoring force even vanishes at the phase transition tempera-ture.

Certain shuffle transformations resultfrom a vibrational instability of the parentphase and are therefore called “softmode”

phase transformations. A soft mode is, in simple terms, a vibrational mode, thesquare of whose frequency tends towardzero as the temperature approaches that ofthe phase transition. The average staticatom displacements resemble the frozen-inpattern of the vibrational displacements ofa certain vibrational mode. According toVallade (1982), “the crystal lattice vibra-tions can within the harmonic approxima-tion be separated into independent planewaves (phonons) characterized by a set of collective atomic displacements corre-sponding to a well defined frequency. Theenergy involved is a function of the squaresof the momentum and of the eigenfrequen-cies of the mode. The eigenfrequencies de-pend only on the mass of the atoms and onthe force constants. It is clear that the van-ishing of one eigenfrequency correspondsto the lack of restoring force for the mode:the amplitude can then grow without anylimit and the lattice is mechanically un-stable. Stability can be recovered only by changing atomic equilibrium positionswhich, in turn, changes the force con-stants“.

Table 9-1. A schematic overview of some characteristics typical of the various types of diffusionless phasetransformations.

Characteristics Structural change Pure lattice deformation

Type of Principal strains Volume changediffusionlesstransformation type* order ** Sign Value

Shuffle C or D F All zero Zero Zero up toS 10–5

Dilatational D F All positive or Large Large:All negative 10–1

Quasi-martensitic C or D F Mixed sign Small Small:S 10–4–10–3

Martensitic D F Mixed sign or Large Small or large:Zero and +, – 10–2–10–1

* C: continuous, D: discontinuous** F: first order, S: second order

As regards SrTiO3, the rotation-vibra-tional modes of the oxygen atoms are fro-zen into the low-temperature positions be-low 110 K. The temperature dependence ofthe softening, expressed by the square ofthe eigenfrequency of the mode, is repre-sented schematically in Fig. 9-8 for a sec-ond- and first-order phase transformation.Usually, the low-temperature phase alsoshows a soft mode as the temperature israised towards Tc.

Lattice softening can also be treated interms of a static approach in which thestability of the lattice is examined whensubmitted to small static or quasi-static ho-mogeneous strains. The free energy is thenexpressed as a function of the elastic con-stants; for a lattice to be stable when sub-mitted to small homogeneous strains, thefree elastic energy must increase for allpossible strains. For a cubic crystal this ismathematically expressed by saying thatall the eigenvalues of the elasticity tensormust be positive, in other words C44>0,(C11–C12) > 0, and (C11+ 2 C12) > 0.

The tendency toward mechanical in-stability can also be studied through exam-ination of the phonon dispersion curves,which gives the relationship between thewavevector q of the vibrational mode andits eigenfrequency. The lattice instabilitycan correspond to uniform (q = 0) or mod-ulated (q = non-zero) atom displacementsand the soft phonon may belong to an opticor an acoustic branch; an example of ameasured dispersion curve is given in Fig.9-9. The longitudinal acoustic branch inzirconium shows a dip at about 2/3 [111],which is the mode needed to transform thehigh-temperature b.c.c. structure of Zr intothe omega structure. The slope of the trans-verse acoustic branch of Nb3Sn is very flatat the origin on approaching the transitiontemperature of 46 K (Shapiro, 1981). Thiscorresponds fairly well with the experi-mental observation of a vanishing value ofthe elastic shear constant C ¢= (C11 – C12)/2.The atom displacements induced by thissoft mode coincide exactly with those as-sociated with the deformation from cubic


Figure 9-8. Temperaturedependence of the squaredfrequency of the softeningmode for (a) a second-orderand (b) a first-order transfor-mation, Tc and Tt being thecritical and the transforma-tion temperatures, respec-tively. (c) Phonon energy ofSrTiO3 measured below andabove Tc (after Rao and Rao,1978).


to tetragonal structure. For Cu–Zn–Al nosoft mode is present at 2/3 [111], althoughthe LA branch shows an anomalous dip;the branch measured perpendicular to itproves that the point in the reciprocal spaceis a saddle point and not a minimum. Thebranch TA2 [110] (polarization [11

–0]),

however, shows a small slope correspond-ing to a low value of C ¢ (Guénin 1982).Transformation models have been pro-posed for Cu–Zn–Al taking into accountboth the anomalous dip and the smallslope.

In a number of materials undergoing a cubic to tetragonal transformation, atweed-like pattern is observed in the parentphase by transmission electron micros-copy. This tweed contrast is characterizedby a ·100Ò direction of the modulation, atype of 110 ·11

–0]Ò shear strain and a

modulation which is incommensurate withthe parent phase. It is still debated whetherall the pre-transformational or precursoreffects are evidence of stable or metastablemodulated phases or whether they are well-defined artefacts determined by the kinet-ics of nucleation and the growth process.

A long-standing issue with b Cu-, Ag-and Au-base alloys that has now been re-solved is the appearance of extra maximain the electron diffraction patterns of theparent phase from quenched alloys. Overthe years these maxima have been givenvarious interpretations, often with an over-emphasis as possible pre-martensitic ef-fects. Systematic investigation, however,established that these effects are in fact ob-tained in martensitic structures located onthe surface of the thin foil and extendinginwards to a depth of 1 µm.

When considering martensitic transfor-mations, the role played by the combina-tion of lattice defects and of lattice instabil-ities is of particular interest; the large defor-mations present around the defects may in-

duce a localized lattice instability (Guéninand Gobin, 1982), which may trigger thenucleation of martensite on further coolingor stressing.

Pre-transformational lattice instabilitiesand soft modes and their relation to diffu-sionless phase transformations have beenreviewed by Delaey et al. (1979), Nakani-shi (1979), Vallade (1982), Nakanishi et al.(1982), and Barsch and Krumhansl (1988).The validity of the soft phonon or soft elas-tic stiffness approach to martensite is a dif-ficult and somewhat controversial subject.

9.3.3 Transformation Mechanisms

It should be emphasized that the pure lat-tice distortions considered above do notnecessarily imply the actual path the atomsfollow during the transformation. For sec-ond-order phase transformations, there is acontinuous change throughout the crystalwith decreasing temperature starting at Tc.Following the soft-mode concept, the ap-pearance of the new phase is considered asthe freezing of a particular wavelength vi-bration. The Bain-type strain for a second-order cubic to tetragonal transformation,for example, is equivalent to two 110·11

–0Ò shear strains whose corresponding C¢

Figure 9-9. Phonon dispersion curves for b.c.c. Zr.A pronounced dip occurs in the longitudinal (l) pho-non dispersion curve in the vicinity of q = 2/3 [111] (Sikka et al., 1982).

the fully formed product. The exact mecha-nisms for the various types of martensitictransformations are still under debate.

9.3.4 Microstructures

The microstructures that result from diffusionsless phase transformations showtypical features, which are related to thecrystallograph of the transformation. Thetransformation is associated with a reduc-tion in symmetry; consequently, differentequivalent orientational states of the prod-uct phase are formed. A single crystal ofthe parent phase thus transforms to a col-lection of the product-phase crystals, calledvariants, that are separated by interfaces.The higher the symmetry of the parentphase and the lower the symmetry of theproduct phase, the greater is the number ofequivalent paths of transformation. Thenumber of equivalent orientations or vari-ants is also determined by the symmetryelements that are maintained or broken dueto the Bain strain. The collection of vari-ants constitutes the microstructure.

The order of the transformation (whetherfirst or second order) also determines themicrostructure: in the former case parent/product or heterophase interfaces in addi-tion to product/product or homophase inter-faces are created, whereas in the latter onlyproduct/product interfaces are formed. Inthe former case the first plates formed cangrow to a larger extent than those formedlater, which can lead, for example, to mi-crostructures with fractal characteristics.

Fig. 9-10 shows a selection of character-istic microstructures obtained through dif-fusionless phase transformations.

9.3.5 Shape Changes

If we could transform a single crystal ofthe parent phase into a single crystal of the


shear constant vanishes at Tc. The trans-formation mechanism is therefore not acombination of expanding and contractingatom movements, but a lock-in of long-wavelength shear-type movements on110 planes in the ·11

–0Ò directions in this

scheme.As for the martensitic transformations,

the situation is not so straightforward. TheBain-type strains are concerned only withthe correspondence between initial and fi-nal lattices; they do not give the actual ob-served crystal orientation relationshipsbetween them. Based on the experimen-tally determined orientation relationships,different transformation mechanisms havebeen proposed, such as shears on the planesand along the directions involved in the orientation relationship. However, theseshear mechanisms have been found to betoo simple to be consistent with the experi-mental facts. More recently, a transforma-tion mechanism has been proposed formartensitic transformations of b.c.c. toclose-packed structures; a condensing stateof some soft phonon modes combined witha homogeneous shear explains the varietyof structures that are found. For the sametransformations, Ahlers (1974) proposed atwo-shear mechanism; the first shearcreates the close-packed planes, whereasthe close-packed structure is obtained bythe second shear.

Martensitic transformations are first-order phase transformations that occur bynucleation and growth. The growth stagegenerally takes place by the motion ofinterfaces converting the parent phase tothe fully formed product phase. Two typesof paths have to be considered for the caseof nucleation, the “classical” and the “non-classical” nucleation paths (Olson and Co-hen, 1982). The latter involves a continu-ous change in structure whereas the formerinvolves a nucleus of the same structure as


Figure 9-10. A selection ofmicrostructures obtained bydiffusionless phase transfor-mations: (i) schematic repre-sentation of the microstruc-ture of (a) martensite andaustenite, (b) the Dauphinétwins between the low tem-perature a- and the hightemperature b-phase ofquartz, and (c) the zig-zagdomain structure in neody-mium pentaphosphate whichundergoes an orthorhombicto monoclinic transforma-tion (after James, 1988); (ii) optical and transmissionelectron micrographs of (a, b)the twinned orthorhombicYBa2Cu3Ox high-tempera-ture superconductor (cour-tesy H. Warlimont, 1989),(c) the domain in SiO2 at thetransition temperature a to b(846 K) (courtesy Van Ten-deloo, 1989), and (d) thefractal nature of the marten-site microstructure in steel(courtesy Hornbogen, 1989).

product phase, the macroscopically visibleshape change would clearly reflect theBain-type strain; it is then the maximumtransformation-induced shape change thatcan be achieved. Depending on the symme-try relationship, this deformation can beobtained in as many orientations as productvariants exist.

In a martensitic transformation, the mac-roscopic shape change associated with theformation of a single martensite plate is notonly the result of the Bain strain but also ofa lattice-invariant deformation. The totalmacroscopic shape change is mainly ashear deformation along the habit plane ofthe martensite variant. The martensite platecontains either a large number of stackingfaults or has twins inside. It is therefore nota single crystal. If the single martensiteplate has twins inside and is subjected afterthe transformation to an externally appliedstress, an additional shape change is ob-tained by detwinning. Only then is the finalproduct a single crystal of the productphase. Fig. 9-11a shows the changes inshape after transforming a b-Cu–Zn–Alsingle crystal into a single martensite vari-ant and Fig. 9-11b shows the shape changeafter partially transforming an iron whisker.

The transformed sample usually containsa very large number of single-product do-mains arranged in a special configuration.In some systems the domains are arrangedsuch that the shape changes are mutuallyaccommodated. Because each product variant is associated with a differentlyoriented shape change, applying a mechan-ical stress during the transformation willpromote the formation of those variantsthat accommodate the applied stress. Thisprovides a resolved shape change in the di-rection of the applied stress. This is thefundamental concept of the shape-memoryeffect, as will be explained further in Sec.9.10.2.


Figure 9-11. Macroscopic shape change associatedwith martensite: (a) a Cu–Zn–Al single crystal beforeand after transforming to martensite and (b) a par-tially transformed Fe whisker (courtesy Wayman,1989).

a

b


If a single crystal of the product phase ismechanically strained it can either be trans-formed to another or be deformed to a dif-ferently oriented single crystal of the prod-uct phase (Fig. 9-12). Similar behaviour isalso typical of a number of ferroelastic materials; the reorientation is there referredto as “switching” (Wadhawan, 1982). Theswitching force in these materials is notonly a mechanical stress but can also be anelectric or magnetic field, the domains be-ing either electrically or magnetically po-larized.

In first-order phase transformations, asshown above, the full transformation shapechange is induced locally and is graduallyspread over the whole sample within asmall temperature interval, whereas in asecond-order phase transformation thesample changes its shape homogeneouslyand continuously as soon as the criticaltransition temperature Tc is reached.

Until now, shape changes have been dis-cussed that are induced by the forwardtransformation. It is evident that if a sam-ple of the low-temperature phase, a singlecrystal or a polyvariant, is heated to tem-peratures above the reverse transformation

temperature, similar shape changes are ob-served, provided that the reverse transfor-mation is also diffusionless. The situationfor second-order phase transformations isstraightforward; the sample whose shape ischanged during the forward transformationand possibly after deformation below Tc re-verts back to its original shape above Tc ina homogeneous and continuous way. Forfirst-order transformations, the reversetransformation is more complex and notyet well understood. Much depends onwhether the forward transformation iscompleted or not, and whether the growthof the martensite plate occurs by bursts orunder thermoelastic conditions (see be-low). Occurrence of the reverse transfor-mation does not necessarily imply that theoriginal shape is restored. Depending onthe crystal symmetry of the product phase,more than one crystallographically equiva-lent path can be followed for the reversetransformation. The shape changes that oc-cur during the reverse transformation are atthe origin of the shape-memory effect andare discussed in Sec. 9.10.2.

Figure 9-12. A series of macrographs representing the shape change while mechanically straining a Cu–Al–Nimartensite single crystal; (a) to (e) increasing with time (Ichinose et al., 1985).

9.3.6 Transformation Thermodynamicsand Kinetics

A diffusionsless phase transformationmay be of second or first order. The formeris generally dealt with in the phenomeno-logical Landau theory, while the latter is treated by classical thermodynamics(Kaufman and Cohen, 1958). The Landautheory has, however, been extended to alsocover first-order phase transformations(the Devonshire–Ginzburg–Landau theory)and has been applied by Falk (1982) tomartensitic transformations. The reader isreferred to the chapter by Binder (2001) foran introduction to those theories.

The chemical driving force occupies akey position in the classical thermodynam-ics of first-order diffusionless phase trans-formations, a subject that is introduced in thefirst chapter of this volume (Pelton, 2001). Inthe following section, therefore, only thoseaspects directly relevant to diffusionlessphase transformations will be dealt with.

Because no chemical compositionchange is associated with a diffusionlessphase transformation, the parent and prod-uct phases have the same homogeneouschemical composition and hence they aretreated as a single-component system. Forthose phase transformations whose structu-ral change is easily described by a dis-placement parameter, a phenomenologicaldescription of the free energy as a functionof the order parameter in terms of the Lan-dau theory leads to some interesting con-clusions. In the following, the free energyis discussed as a function of temperatureand composition. Other possible intensivethermodynamic state variables include ex-ternal pressure, mechanical stress, andmagnetic and electrical field strength.

The changes in chemical Gibbs energy,DG, as a function of temperature and com-position are shown schematically in Fig.

9-13 for first-order diffusionless phasetransformations between a parent phase,denoted P, and a product phase, denoted M.The product phase may be one of the low-temperature equilibrium phases or a meta-stable phase.

Taking again the martensitic transforma-tion as an example, the transformationstarts at Ms , which is lower than T0, andfinishes at Mf . This means that a higherdriving force is needed to complete thetransformation. On heating a fully marten-sitic stress-free single crystal, the reversetransformation sets in at a temperature As ,


Figure 9-13. Schematic representation of the molarGibbs energy (a) as a function of temperature butconstant composition and (b) as a function of compo-sition for an Fe–Ni alloy with T4 > T3 > T2 > T1 and T2 = T0 for XNi = X (after Mukherjee, 1982).


which is higher than T0. The differencebetween the forward and the reverse trans-formation temperatures is the transforma-tion hysteresis. The true first-order equilib-rium temperature, T0, which is calculatedfrom DG = 0, can thus only be bracketedfrom experimental data for the forward andthe reverse transformation temperatures,and is not necessarily halfway between Ms

and As.Strain energy resulting from the transfor-

mational shape change and interfacial en-ergy have been omitted from the free-en-ergy curves in Fig. 9-13. These two non-chemical-energy terms have to be consid-ered, however, in the overall free-energybalance. The strain energy associated withthe formation of a single domain of theproduct phase is proportional to the volumeof that domain. The interfacial energy isnot directly related to the volume of thetransformed domain but merely to its sur-face-to-volume ratio, and, in the case of ananisotropic interfacial energy, also to theorientation of the interface. Both quantitiesare positive and thus consume part of thechemical driving force for a forward trans-formation. Both terms will, however, in-crease the driving force for the reversetransformation, provided that the inter-facial coherence is not lost. The reversetransformation might start below T0 if anegligible net driving force is required fornucleation.

Considering the Gibbs energy per unitmolar volume, the total Gibbs energychange per unit molar volume for the for-mation of a single domain of the productphase embedded in the matrix phase isgiven by

DGtot = DGchem + (DGelast + j DGsurf) (9-3)

where j is the surface-to-volume ratio of the single domain. The two terms in parentheses are then the non-chemical con-

tributions to the Gibbs energy change,DGnon-chem, and Eq. (9-3) then becomes

DGtot = DGchem + DGnon-chem (9-4)

The transformation then proceeds untilDGtot becomes minimum or, if the phaseboundary is mobile, until the total force atthe parent-to-product interface is zero. Ifthe advancing direction of the interface isx, we can write

[∂ (DGtot)/∂x] dx = 0 (9-5)

or

[∂DGchem/∂x]dx + [∂DGnon-chem/∂x]dx = 0

The sum of the non-chemical restoringforces is then identical with the chemicaldriving forces. The difficult task now is to find expressions representing the non-chemical terms. Three thermodynamic ap-proaches have been worked out, dealing es-sentially with the influence of the two non-chemical contributions on the transforma-tion behavior (Roitburd, 1988; Ball andJames, 1988; Shibata and Ono, 1975, 1977).

According to Roitburd (1988), the strainenergy, which arises in crystals owing to adiffusionless phase transformation, can de-crease if the crystals are subdivided intodomains arranged such that a maximumcompensation of the individual strain fieldsis achieved. In order to determine which ar-rangements are energetically most favor-able, Roitburd calculates the strain energyfor arbitrary domain arrangements, andthen minimizes this energy. The formula-tion of this problem is complex and canhardly be solved in general, but he suc-ceeded for some specific cases.

Ball and James (1988) do not assumeany geometric restrictions on the shape orarrangements of the domains; they foundthis necessary to determine microstructuresoccurring in complex stress fields, or to explore new and unusual domain arrange-

ments. The general aim of their work wasto develop mathematical models, using cal-culus of variations, capable of predictingthe microstructure, especially the micro-structural details at the interface betweenthe parent and the product phases. At-tempts have been made to predict the pos-sible interfaces between austenite and mar-tensite from a minimization of a Gibbs en-ergy function, which depends on the defor-mation gradients of all possible domainvariants and on temperature. A deforma-tion or domain is then termed stable if it minimizes the total energy. They show,among others, that a martensite–austeniteinterface can exist as an energy-minimiz-ing sequence of very fine twins. A furtherexample of an intriguing application is the formation of triangular Dauphiné twinsin quartz, which become finer and finer inthe direction of increasing temperature. AGibbs energy function accounting for thisbehaviour could be constructed.

Shibata and One (1975, 1977) use theEshelby theory; the principle of their calcu-lation is in a corrected version (Christian,1976) illustrated schematically in Fig.9-14. An embedded part of the parentphase is cut out (step a) and is allowed totransform stress-free into the product phase(step b). A lattice-invariant deformation isapplied (step c) and the transformed crystalis subjected to forces along its surface suchthat it is deformed to the original shape (step d). The thus deformed part of theproduct phase is introduced in the emptyspace of the parent phase (step e), and theforces are removed, creating internalstresses in both the product and the parentphase. The total energy is then calculatedas a function of all possible lattice orienta-tions, taking into account the actual elasticconstants and the modes of lattice-invari-ant deformation, twinning or slip.

The total Gibbs energy of the system istherefore a function not only of the intrin-sic energies of the stress- and defect-freeparent and product phases, but also of thearrangement of the domains. The non-chemical component of the total Gibbs en-ergy of the transforming system is loweredby an appropriate rearrangement of the mi-crostructure and/or by irreversible plasticdeformation.

If the structural change can be repre-sented by an order parameter e, the Gibbsenergy of the system can then, according tothe theory of Landau–Devonshire, be rep-resented by

G = G0 + a (T – T1) e2 – B e4 + Ce6 (9-6)

where a, B and C are constants and T1 > 0.It can be shown that the high-temperaturephase becomes unstable with respect to any fluctuation of e, as soon as the temper-ature reaches T1 on cooling, and hence isthermodynamically metastable between T0

and T1. Accordingly, the low-temperature


Figure 9-14. The necessary steps in calculating theelastic stresses induced by a transforming ellipsoid(Christian, 1976). (For details see text.)

9.4 Shuffle Transformations 607

phase cannot exist at temperatures higherthan T2, which is the temperature abovewhich the low-temperature phase becomesunstable with respect to any fluctuation in e.

That additional undercooling is neededfor further transformation below Ms is due(in part) to the non-chemical contributions,which increase with increasing volumefraction of transformed product.

An interesting aspect of the diffusionlessphase transformations that are accompa-nied by a volume and shape change is therole played by external stresses, e.g., hy-drostatic or uniaxial. Both thermodynamicsand experiments show that the transforma-tion temperatures are affected by the appli-cation of stresses. According to Wollants etal. (1979), the relationship between a uni-axially applied stress s and the transfor-mation temperature T depends on thetransformation entropy and the transfor-mational strain in the direction of the ap-plied stress. This relationship, the Clau-sius–Clapeyron equation for uniaxiallystressed diffusionless first-order phasetransformations, is

ds /dT = – DS/e = – DH*/[T0 (s) e] (9-7)

where DH* = DH – FDl = DH – s e Vm =T0 (s) DS is itself a function of the appliedload, e = Dl/l, l is the total “molar length”of the sample, and F is the applied load(s = F/A). This equation is similar to thatrelating the equilibrium temperature to thehydrostatic pressure, except for the nega-tive sign on the right-hand side of Eq. (9-7).This relationship between ds and dT is ex-perimentally constant for most of the diffu-sionless transformations, which means thatthe thermodynamic quantity DS is, withinthe experimental scatter, independent oftemperature and stress. Knowing the trans-formation strain, uniaxial tensile tests arevery useful for determining the transforma-tion entropy.

The most relevant thermodynamic datafor the various diffusionless phase transfor-mations are presented in Table 9-2.

9.4 Shuffle Transformations

Shuffle transformations from a distinctclass of diffusionless phase transitions. Atthe unit-cell level the atom displacementsare intercellular with little or no pure strainof the lattice. The role of elastic strain en-ergy in shuffle-phase transformations issufficiently small that the transformationcan either occur continuously from the par-ent to the product phase or that it is com-pletely controlled by interfacial energy. Inthe former case the transformation is sec-ond order whereas in the latter it is a first-order phase transformation.

Cohen et al. (1979) gave three exampleswhich clearly illustrate the shuffle transfor-mations. The displacive transformation instrontium titanate is the prototype exampleof a pure shuffle transformation. The asso-ciated strain energy is so small that thetransformation occurs continuously. The b-to-w transformation in some Ti and Zr alloys shows, in addition to the shuffle displacements, small homogeneous latticedistortions. These distortions are smallenough for the transformation mechanismand the resulting microstructure to be dom-inated only by shuffling. In ferroelectrictransformations, which are accomplishedby shuffling, the interfacial energy is con-stituted largely by electrostatic interactionenergies and is therefore dependent on theorientation of the domain interfaces. Theinterfacial energy in those materials isstrongly anisotropic and controls the poly-domain structure.

Phase transformations that can be en-tirely described by shuffle diplacementsare often found where the change in crystal

structure is such that the point group towhich the crystal structure of the productphase belongs is a subgroup of that of theparent phase. In other words, some symme-try elements of the high-temperature phaseare lost on cooling below the transitiontemperature Tc. Because of this group/sub-group relationship, the product phase pos-sesses two or more equally stable orienta-tional states in the absence of any externalfield. The change in crystal structure caneasily be described by an order parameterwhich itself is related to the shuffle dis-

placement. For strontium titanate, the orderparameter would then simply be the rota-tion angle that describes the displacementof the oxygen atoms around the titaniumatoms (see Fig. 9-4). For convenience, theorder parameter is taken as zero for thehigh-temperature configuration and as non-zero for the low-temperature phase. Themajority of such transitions are found inchemical compounds (e.g., Rao and Rao,1978). As soon as the critical temperatureTc is reached on cooling, the order parame-ter changes continuously. The thermody-


Table 9-2. The elastic shear constant and some thermodynamic data characterizing the diffusionless phasetransformation (Delaey et al., 1982b).

Elastic shear constant C ¢ Trans- Thermodynamic quantitiesnear the Ms temperature formation

strain Heat of Change Chemical Transfor-C ¢ (1/C ¢) (dC ¢/dT ) transforma- in driving mation

(1010 Pa) (10–4 K–1) tion entropy force tempera-(J/mol) (J/(K · mol)) (J/mol) ture hys-

teresis (K)

Ferrous g Æ a¢ 2–3 negative ≈10–1 ≈2000–300 5.8 150–450 200–400g Æ e 3–10 (positive 600–1800

for Ni > 30%)

Co alloys negative ≈10–3 ≈400–500 ≈0.2 4–16 40–80rare-earthalloys

Ti and Zr 0.1 negative ≈2 ¥ 10–2 ≈4000 ≈1.0 ≈25 –alloys

b Cu–Ag–Au 0.5–1 4–20 ≈10–2 ≈160–800 0.2–3.0 ≈8–20 10–50alloys (positive)

In alloys 0.05–01 ≈1000 ≈10–3 ≈0 – ≈1.5 1–10(positive)

Mn alloys positive ≈10–3

(strongly)A 15 com- 0.5 ≈1000–3000 ≈10–4

pounds (positive)

Fe–Pt ≈320 – ≈16 ≈20–200(ordered) (ordered)

Fe–Pd ≈1 ≈100 ≈10–3 ≈1200 ≈1200alloys (positive) –10–2 (disordered) (disordered)

9.4 Shuffle Transformations 609

namics of such transformations are then inthe temperature range close to Tc, which isdealt with by a Landau approach.

9.4.1 Ferroic Transformations

Usually a phase transition that is domi-nated by shuffling is associated with achange in some physical properties, such asspontaneous electrical polarization, strainand magnetization. Because the crystalsymmetry of the parent phase decreasesduring the phase transition, two or moreequivalent configurations of the productphase are formed. In the absence of any ex-ternal field, the average polarization of theproduct phase is zero. However, under asuitably chosen driving force, which maybe an electrical field (E ), a mechanicalstress (s), or a magnetic field (H), the do-main walls of the product phase move,switching the crystal from one domain or-ientation to the other. Owing to the applica-tion of a uniaxial stress, for example, oneorientation state can be trensformed repro-ducibly into the other, and the crystal isthen said to be “ferroelastic”. The materialsexhibiting this property are called ferro-elastic materials. Similarly, we can defineferroelectric and ferromagnetic materials.According to Wadhawan (1982), “phasetransitions accompanied by a change of the

point-group symmetry are called ferroicphase transitions. We refer to a crystal asbeing in a ferroic phase if that phase resultsfrom a symmetry-lowering ferroic phasetransition”.

Not all ferroelastic phase transitions be-long to shuffle transformations as definedin Fig. 9-1. Indeed, in addition to shuffledisplacements, as for example those in-volved in the cubic to tetragonal transitionin barium titanate, the lattice may becomehomogeneously distorted. For the exampleconsidered here, the lattice distortion oc-curs discontinuously at the transition tem-perature; the lattice parameters changeabruptly (Fig. 9-15). Even below this tran-sition temperature, the lattice continues tobe homogeneously distorted. In caseswhere this lattice is tetragonal, the c/a ratiosteadily increases. For barium titanate thechange in c/a continues until the tempera-ture for another first-order phase transitionis reached. Many such phase transforma-tions are encountered in chemical com-pounds. In some cases the amount of spon-taneous strain is not large enough to con-trol the microstructure. In others, the strainenergy associated with the transformationwill be dominant. The transformation isthen, according to Fig. 9-1, quasi-marten-sitic or martensitic. In ferroelectric materi-als, the interfacial energy also has to be

Figure 9-15. Temperature de-pendences of the lattice parameters of the differentphases of BaTiO3.

taken into account and may even becomethe dominant parameter controlling the microstructure. A ferroic ferroelastic phasetransformation can thus be a shuffle, aquasi-martensitic, or a martensitic phasetransformation, a clear discrimination isonly possible by analyzing all the transfor-mation characteristics, and this not afterbut during the transformation.

9.4.2 Omega Transformations

The omega transformation is known tooccur as a metastable hexagonal or trigonalphase in certain Ti, Zr and Hf alloys oncooling from the high-temperature b.c.c b-phase solid solution or as a stable phaseunder the influence of high hydrostaticpressures or shock waves. The w-phasecannot be suppressed by quenching andforms as small cuboidal or ellipsoidal par-ticles with a diameter of 10–20 nm. Its lat-tice is obtained by collapse of one pair of(111) planes of the parent b.c.c. b-phase,

leaving the two adjacent planes unaltered(Fig. 9-4b). The collapse can be repre-sented as a short-wavelength displacementof atoms. The displacement of the atomsoccurs over a distance approximately equalto 2/3 ·111Ò. Each lattice site can thus beassociated with a forward, zero or back-ward displacement that can be representedby a sinusoidal wave dividing the repeatdistance along a [111] direction into sixparts. The collapse is not always complete,and then results in a “rumpled” plane. If thecollapse is incomplete the crystal structureof the w-phase is trigonal; if the collapse iscomplete, it is hexagonal. The figure alsoshows that reversing the direction of the dis-placement will not lead to a collapse of the111 planes. Moreover, it can be shownthat a 2/3 [111] displacement wave is equiv-alent to a 1/3 [1

–12] displacement wave.

If the displacement is taken as the orderparameter in a Landau-type approach, thetransformation is seen to be first order andthe Gibbs energy as a function of this orderparameter has an asymmetric shape. Con-sequently, no negative values of the orderparameter are then allowed (Fig. 9-16).The b-to-w transformation can also be par-aphrased in terms of a soft mode. The lat-tice tends to a mechanical instability for a2/3 ·111Ò longitudinal mode. This tendencycan be shown when measuring the phonondispersion curves by inelastic neutron scat-tering. Such curves are reproduced in Fig.9-9a for zirconium; a clear dip is visible atthe 2/3 [111] position.

The omega transformation has been re-viewed by Sikka et al. (1982).

9.5 Dilatation-DominantTransformations

A transformation is regarded by Cohenet al. (1979) as dilatation dominant if no


Figure 9-16. Gibbs energy change for the b.c.c. to wtransformation as a function of the order parameterfor various reduced temperatures (after de Fontaine,1973).

9.6 Quasi-Martensitic Transformations 611

undistorted line can be found in the lattice-distortive deformation. The f.c.c.-to-f.c.c.¢transformation in cerium is considered asthe prototype for a dilatational dominanttransformation. Below 100 K cerium un-dergoes a pure volume contraction of about16%; the ellipsoid of the f.c.c.¢ phase thusfalls completely inside that of the high-temperature f.c.c. phase. The low-tempera-ture cubic to tetragonal transformation intin also appears to be dominated by dilata-tion, although some deviatoric componentsare present; the volume expansion is about27%. The deviatoric component is notlarge enough to let the original sphereintersect with the dilated ellipsoid.

The name “dilatational diffusionlessphase transformation” has been used byBuerger (1951) but with a different mean-ing. In the systems he considers, for exam-ple the CsCl-to-NaCl transitions in manyalkali metal halides, he defines the termdilatational as follows: “the transformationcan be achieved by a differential dilatationin which the structure expands along thetrigonal axis and contracts at right-anglesto the axis”. Although the volume changein these and other related inorganic sys-tems may be very large (up to 17%), thetransformation is, in the context of Fig. 9-1,clearly not dilatation dominant but devi-atoric dominant. See Kriven (1982) for amore detailed review of these dilatationaldominant transformations.

9.6 Quasi-MartensiticTransformations

The quasi-martensitic and the marten-sitic transformation are both deviatoricdominant and are characterized by an un-distorted line. The morphologies of theproduct phases of the two transformations

are very similar (large plates, occurrence ofvariants and twins). A distinction betweenthe two transformations cannot be made bysimply judging only the product morphol-ogy, but rather a knowledge is required ofthe morphological relationships betweenparent and product phases during the trans-formation itself. It may be adequate to sayfirst what a quasi-martensitic transforma-tion is: a quasi-martensitic transformationis not a martensitic transformation, whichitself is “a first-order phase transformation,that undergoes nucleation, passes through atwo-phase mixture of the parent and prod-uct phases, and which product grows with atransformation front in a plate-like or lath-like shape being indicative of a tendencytoward an invariant-plane interface” (Co-hen et al., 1979). If a deviatoric dominanttransformation does not satisfy the abovecriterion, it should not be designated asmartensitic but as quasi-martensitic.

Three aspects are common to most of thematerials that transform quasi-martensiti-cally: (1) the lattice distortion is small anddeviatoric dominant and the change in lat-tice distortion is continuous or nearly con-tinuous; (2) a banded internally twinnedmicrostructure gradually builds up on cool-ing below Tc; and (3) a mechanical latticesoftening is expressed by elastic shear constants approaching zero as Tc is ap-proached. Because of the small lattice dis-tortion at the transformation, the ratio ofthe strain energy to the driving energy fortransformation is small; this ratio has beenused by Cohen et al. (1979) as an alterna-tive index to differentiate quasi-martensitictransformations from martensitic.

The three aspects are now illustrated bytaking the manganese-based magnetostric-tive antiferromagnetic alloys as an example(see Delaey et al., 1982a). One of the fourpolymorphic states of manganese is thegamma f.c.c. phase which is stable only at

high temperatures. Alloying with elementssuch as Cu, Ni, Fe, Ge, Pd and Au stabi-lizes the f.c.c phase and the latter can be re-tained by quenching. However, owing tothe antiferromagnetic ordering, the latticesbecome homogeneously distorted. This or-dering to the Mn atoms starts at a temper-ature TN, which is the Néel temperature for the paramagnetic to antiferromagnetictransition. The transformed product phasehas a banded microstructure containingfine twins. The temperature at which thisbanded microstructure is formed does notalways coincide with the transformationtemperature TN. Vintaikin et al. (1979) di-vide these antiferromagnetic alloys intothree classes according to the relative posi-tions of the temperature TN and the temper-ature TTW. The latter is the temperature atwhich the banded microstructure sets in.Depending on the type of lattice distortion,the alloys are grouped into three classes,each class being characterized by the rela-tive positions of the two temperatures. A

schematic representation of the phase dia-gram of the Mn-based alloys is given inFig. 9-17a, showing the temperature–com-position areas in which the various crystalstructures and microstructures are ob-served. The accompanying variation in thelattice parameters as a function of tempera-ture for the three classes of Mn-based al-loys is given in Fig. 9-17b.

The changes in lattice parameters showthat the transformation is almost second or-der, except for some alloys of class I and IIIwhere the transformation is weakly first order. A phase transformation is called “weakly first order” whenever the height ofthe discontinuous jump in the correspond-ing thermodynamic property is very small.The formation of the twinned banded mi-crostructure extends over the entire volumeof the sample quasi-instantaneously and isvisible in polarized light because of thenon-cubic structure of the product phase.Similar microstructures are observed inother quasi-martensitic product phasessuch as V54–xRu46Osx (Oota and Müller,1987). The microstructure, if properlyoriented with respect to the prepolishedsurface, exhibits a surface relief effect thatis enhanced as the temperature decreasesbelow TTW. This surface relief proves thatthe transformation is accompanied by ashape change associated with each domain.Because of the continuously changing lat-tice parameters, accommodation stressesare built up as the temperature decreases.An appropriate arrangement of these do-mains reduces the overall stored elastic en-ergy; further changes in microstructure aretherefore expected to occur even below thetransition temperature.

Class II alloys do not exhibit the twinnedbanded microstructure immediately belowTN. In the temperature region between TN

and TTW, broadening of some of the X-raydiffraction peaks is observed, which is


Figure 9-17. Schematic representation of (a) thephase diagram and (b) the variation of the lattice pa-rameters for the three classes of Mn-based alloys(Delaey et al., 1982a).

9.7 Shear Transformations 613

attributable to a chaotic distribution of thea and c axes with small undercooling. AtTTW the banded structure becomes visible(point A in Fig. 9-17) and a tetragonalstructure can now be clearly detected by X-ray diffraction. If the sample is now heated,only the banded microstructure disappears,not at A but at a temperature B that coin-cides with the Néel temperature. Thisproves that on cooling, very small, submi-croscopic tetragonal regions are firstformed as soon as TN is reached. Hocke andWarlimont (1977) have shown that whenthe distortion |c/a – 1| becomes greaterthan 0.005, a critical value is obtained atwhich the elastic strain is relaxed throughcoalescence of the small distorted regionsinto large banded twinned regions. Thus, atTTW there is not a phase transformation buta stress relaxation in the microstructure,which results in a twinned microstructure.The lattice distortive phase transformationitself occurs at TN, followed immediately (class I) or after some undercooling (classII) by a domain rearrangement and macro-scopic twinning.

Similar conclusions can be drawn forother quasi-martensitic transformations, asfor example in the iron–palladium alloys;the Pd-rich f.c.c. phase transforms on cool-ing first to an f.c.t phase and at lower tem-peratures to a b.c.t. phase. The f.c.c.-to-f.c.t. transformation, although sometimesregarded as martensitic, shows all the char-acteristics of a quasi-martensitic transfor-mation.

Because the formation of each single do-main is associated with a shape change andthus with accommodation stresses, the ap-plication of an external stress to the trans-formed product will result in a macro-scopic shape change. As the domain boun-daries, which for the Mn-based alloys coin-cide with the antiferromagnetic boundar-ies, are mobile, the banded structure will

gradually disappear and the product phasebecomes a single domain maximizing theshape change. The shape change thus ob-tained is gradually recovered on heatingthe sample and is completely recovered atTN and not (as in the case of Mn-based alloys of class II) at TTW, but at the point Bin Fig. 9-17. The quasi-martensitic alloysthus also exhibit the shape-memory effect.

Some of the materials characterized byshuffle displacements during the phasetransition may develop elastic strains astransformation proceeds. As in ferroelec-trics, for example, in addition to the elasticstrain energy, the dipole interaction energyalso contributes to the polydomain forma-tion. If the elastic strain energies are onlyslightly dominating, the transformation isquasi-martensitic; if, however, the elasticstrain energy is largely dominating, thetransformation can be martensitic.

Sometimes it becomes difficult to differ-entiate between martensite and quasi-mar-tensite, as for example in In-based alloys.In particular, if quasi-martensitic samplesare cooled in such a way that a temperaturegradient is created across the sample, theproduct phase and the parent phase thencoexist and apparently the transformationgoes through a two-phase region, the tworegions being separated by a blurred orplanar interface. Such observations do not,of course, facilitate the distinction betweenquasi-martensite and true martensite.

9.7 Shear Transformations

In this section we discuss a special groupof phase transformations, the so-calledshear transformations or polytypic transi-tions, which strictly belong to the marten-sitic transformations. According to Vermaand Krishna (1966), “polytypism may bedefined, in general, as the ability of a sub-

stance to crystallize into a number of dif-ferent modifications, in all of which twodimensions of the unit cell are the samewhile the third is a variable integral multi-ple of a common unit. The different poly-typic modifications can be regarded asbuilt-up of atom layers stacked parallel toeach other at constant intervals along thevariable dimension. The two unit-cell di-mensions parallel to these layers are thesame for all the modifications. The thirddimension depends on the stacking se-quence, but is always an integral multipleof the layer spacing. Different manners ofstacking these layers may result in struc-tures having not only different morpholo-gies but even different lattice types andspace groups”. Some random disorder oflayers (faulted sequences) is almost alwayspresent. Polytypic transitions are then tran-sitions among different polytypes; themovement of partial dislocations along thebasal plane constitutes the transition mech-anism, thereafter the name shear trans-formations. Polytypic transformations arefound in a variety of inorganic compoundsand also in metals and alloys.

Polytypic phases are constructed bystacking basic units in a cubic, hexago-nal or rhombohedral sequence. The stack-ing sequences are described by three keylayer positions, X, Y and Z; a cubic se-quence (C) is represented by the sequenceXYZXYZ …, a hexagonal (H) by, for example, XYXZ … or XYXZXYXZ …,and a rhombohedral (R) by, for example,XYZYZXZXYXYZYZXZXY … . Manyother stacking variants are possible andunit cells containing as many as 126 or 144layers have been reported. Each unit itselfcan contain a single layer, as in cobalt andits alloys, or two as in silicon carbide.

Transitions between different modifica-tions can be achieved either by a simpleshear, a shear combined with shuffle dis-

placements or the movement of partial dis-loctions along the basal plane. For exam-ple, the transition between a 2 H and a 3 Cstacking is easily performed by a shear,whereby the basic units are kept together inpairs (Fig. 9-18). This shear results in alarge deviatoric shape change. It should bekept in mind that the interlayer spacingneed not be constant, as is observed in thef.c.c.–h.c.p. changes in metals; the trans-formation may involve small changes andthus be IPS (invariant plane strain) ratherthan simple shear transitions.

The same transitions can be achieved bythe generation and movement of closelyspaced and repeatedly arranged partial dis-locations. The passage of a positive partialdislocation shifts the crystal in the direc-tion X Æ Y Æ Z Æ X and negative partialdislocations shifts the crystal in the direc-tion X Æ Z Æ Y Æ X. The following dis-tribution of partial dislocations on the unitlayers in the direction perpendicular to thelayers is proposed by Liao and Allen(1982) (a layer without a partial dislocationis denoted by a dot):

· – + · for 2 H Æ 4 H· – – – · · for 6 H Æ 3 C

The polytypic shear transformation fromone modification to the other is thus ac-complished by a coordinated propagationof groups of partials along the interfacebetween the two phases. The lateral dis-


Figure 9-18. Mechanism of the f.c.c.-to-h.c.p. trans-formation (Nishiyama, 1978).

9.8 Martensitic Transformations 615

placement of the interface, which is thethickening of the new phase generated bythe movement of those partials, is then dueto the formation of partial dislocations andtheir outward movement. During the tran-sition from a cubic to a hexagonal 2H se-quence, it has been implicitly assumedabove that the glide of the layers would al-ways occur in the same direction. There ex-ist, however, three different directions fortransforming an X stacked unit into a Zstacked unit. If the glide occurs alternatelyin these three directions, no shape changeresults from such a mechanism, as shownin Fig. 9-19 (Bidaux, 1988).

9.8 Martensitic Transformations

The characteristics necessary and suffi-cient for defining a martensitic transforma-tion are (a) displaciveness of the lattice-distortive type involving a shear-dominantshape change, (b) diffusion not required forthe transformation, and (c) sufficientlyhigh shear-strain energy in the process todominate the kinetics and morphology dur-ing the transformation (Cohen, 1982). Thedefinition is thus not based on the identityof the transformation product itself (its

structure, specific morphology or proper-ties), but rather on how it forms.

The crystallographic and thermody-namic aspects are fully discussed in the lit-erature and have already been introducedin a more general context in the above sec-tions; only a brief overview is given here.

9.8.1 Crystallography of the MartensiticTransformation

The relevant experimental observableparameters of the martensitic transforma-tion are the shape deformation, the habitplane, the crystallographic orientation rela-tionships, and the characteristic micro-structures.

9.8.1.1 Shape Deformationand Habit Plane

When a sample of the parent phase iscooled to below Ms , a relief gradually ap-pears on a prepolished surface of the par-ent-phase crystal. The surface relief disap-pears on heating to temperatures above As ,provided that no diffusion-controlled trans-formation interferes.

The martensite phase usually takes theform of plates; the plane of contact be-tween the parent and the martensite phasesis called the “habit plane”. A schematicrepresentation of such a martensite plateembedded in the matrix is shown in Fig. 9-20. During the formation of martensite,straight lines (for example, scratches on theprepolished surface) are transformed intoother straight lines and planes are trans-formed into other planes. No discontinu-ities are observed at the points of deflec-tion. This distortion can thus be repre-sented as a “linear homogeneous transfor-mation” of vectors and can be expressed bya matrix formulation. The macroscopicshape deformation can be decomposed into

Figure 9-19. Two different mechanisms, (a) and (b),to transform an h.c.p. to an f.c.c. structure (Bidaux,1988).

a component normal to the habit plane anda shear component parallel to a shear direc-tion located in this interface. The latter iscalled “macroscopic shear” and quantifiesthe shape deformation, whereas the formerrepresents the volume change associatedwith the transformation. A careful analysisof the surface relief reveals that the habitplane itself is unrotated and that any vectorin this interface is also left unrotated andundistorted by the shape change. The habitplane is thus essentially “undistorted” andthe macroscopic shape change associated

with the formation of martensite is thus an“invariant plane strain” deformation, ab-breviated to IPS. The most general invari-ant plane strain deformation, as observedin most martensitic transformations, can beachieved by combining an extension and asimple shear.

The habit plane and the direction of mac-roscopic shear are, with few exceptions,not simple low-indexed crystallographicplanes or directions of the parent or prod-uct phase. They are usually represented in a stereographic projection as shown sche-matically in Fig. 9-20.

9.8.1.2 Orientation Relationship

The next most important observable pa-rameter is the crystallographically well-de-fined “orientation relationship” that existsbetween the lattices of the parent and themartensite phases. It is described either bythe angles between certain crystallographicdirections in both phases or by specifyingthe parallelism between certain planes anddirections. This parallelism does not needto be rigorous, however, experimental re-sults usually deviate slightly. Nevertheless,the fact that such crystallographic parallel-ism is observed yields important informa-tion concerning the possible mechanismsexplaining the change in crystal structure.Some of those relationships observed insteels received great attention in the earlymartensite literature. Depending on the alloy composition, the f.c.c. austenite insteels transforms either to a b.c.c. or b.c.t.martensite or to an h.c.p. martensite, whichitself may further transform into b.c.c. mar-tensite. As regards the transformation off.c.c. austenite to b.c.c. or b.c.t. martensite,the orientation relationships are as follows:

– the Kurdjumov–Sachs (K–S) relations:

(111)P//(011)a¢ and [01–1]P//(11

–1)a¢


Figure 9-20. Schematic representation of (a) a sin-gle martensite plate embedded in a single crystal ofthe matrix phase, (b) a twinned plate and the positionof habit and twin plane, and (c) their stereographicrepresentation.


– the Nishiyama–Wassermann (N–W) re-lations:

(111)P//(101)a¢ and [12–1]P//(101

–)a¢

– the Greninger–Troiano (G–T) relations(here the planes and directions are nolonger exactly parallel):

(111)P ≈ (011)a¢ and [1–01]P ≈ [1

–1–1]a¢

Table 9-3. The crystallographic observables of the martensitic transformations in some metals and alloys(courtesy G. Guénin et al. 1979*).

Alloy system Structural change Composition wt.% Orientation relationship Habit plane

Fe–C f.c.c. 0–0.4% C (111)PΩΩ(101)M (111)P

Ø [110]PΩΩ[111]M

b.c.tetr. K–S relationship0.55–1.4% C K–S relationship (225)P

1.4–1.8% C Idem

Fe–Ni f.c.c. 27–34% Ni (111)PΩΩ[101]M

Ø [121]PΩΩ[101]M ≈ (259)P

b.c.c. N-relationship

Fe–C–Ni f.c.c. 0.8% C–22% Ni (111)P ≈1° of (101)M (3, 10, 15)P

Ø [121]P ≈2° of [101]M

b.c.tetr. G–T relationship

Fe–Mn f.c.c. 13 to 25% Mn (111)PΩΩ(0001)e (111)P

Ø [110]PΩΩ[1210]eh.c.p. (e-phase)

Fe-Cr-Ni f.c.c. 18% Cr, 8% Ni (111)PΩΩ(0001)eΩΩ(101)a¢ e (111)P

Ø [110]PΩΩ[1210]eΩΩ[111]a¢ a¢ (111)P

h.c.p. (e), b.c.c. (a¢)

Cu–Zn b b.c.c.Æ 9 R 40% Zn (011)PΩΩ?(11––

4)M ≈ (2, 11, 12)P

Cu–Sn idem 25.6% SN [111]PΩΩ[110]M ≈ (133)P

Cu–Al b.c.c. 11.0 to 13.1% Al (101)P at 4° of (0001)M 2° of (133)P

Ø [111]PΩΩ[1010]M

h.c.p. distorted 12.9 to 14.7% Al (101)PΩΩ(1011)M 3° of (122)P

[111]PΩΩ[1010]M

Pure Co f.c.c. (111)PΩΩ(0001)M (111)P

Ø ·110ÒPΩΩ[1120]M

h.c.p.

Pure Zr b.c.c. (101)PΩΩ(0001)M (596)P

Ø [111]PΩΩ[1120]M (8, 12, 9)P

Pure Ti h.c.p. (334)P

(441)P

Pure Li Burgers relations

* Gobin, P. F., Guénin, G., Morin, M., Robin, M. (1979), in Transformations de Phases à l’État Solide-Trans-formations Martensitiques. Lyon: Dep. Gènie Phys. Mat., INSA

Concerning the transformation of f.c.c. toh.c.p. austenite and that of h.c.p. to b.c.c.martensite (the b.c.c. to h.c.p. relation isknown as the Burgers relation), the follow-ing relations apply:

(111)P//(0001)e//(101)a¢

and

[11–0]P//[12

–10]e//[111

–]a¢

Taking the N–W relations as an exam-ple, any one of the four crystallographi-cally equivalent 111 austenite planes,(111), (1

–11), (11

–1) and (111

–), can be the

plane of parallelism. In each such planeany one of the three ·12

–1Ò directions, which

happen to be directions of the Burgers vec-tors, can be chosen. This therefore resultsin 12 different orientations of an a¢-crystalin one austenite crystal. These differentlyoriented martensite crystals are called“variants”. It can easily be shown that theK–S relations lead to 24 variants.

Orientation relationships and the orien-tation of the habit plane change from onealloy system to another, and within a givenalloy system from one composition to an-other. The observable crystallographic pa-rameters are summarized in Table 9-3 for a large number of alloy systems; a morecomplete list of these and other crystallo-graphic characteristics of various marten-sites is given by Nishiyama (1978).

9.8.1.3 Morphology, Microstructureand Substructure

Because the martensitic transformationis a first-order phase transformation, bothphases, the parent and the martensitephase, coexist on cooling in a temperaturerange between Ms and Mf and on heatingbetween As and Af . Martensite thus occursin physically isolated regions, the morphol-ogy of which is typical of the transforma-

tion. This morphology is easily observedby light optical microscopy (LOM) and themutual arrangement of these regions con-stitutes the microstructure at the LOMlevel. Electron microscopic analysis re-veals that also at the submicroscopic levelmartensite is characterized by a typicalsubstructure. The morphological, micro-structural and substructural aspects of mar-tensite are briefly discussed below.

The martensite regions are generallyplate-shaped, i.e. one lateral dimension ismuch smaller than the other two. If the twolarger dimensions are nearly equal they arecalled “plates”, and if they are very un-equal “laths”. A typical lath in low-carbonsteel (with a carbon content less than 0.4%)has dimensions 0.3 ¥ 4 ¥ 200 µm3. How-ever, martensite formed into the parentphase does not always appear as a geomet-rically well-shaped plate. Martensite platesthat form near a free surface or in a singlecrystal as a result of a single-interfacetransformation may show the idealizedplate-like shape. In such a single-interfacetransformation the habit plane extendsfrom one side of the crystal to the other(see Fig. 9-11). Because of the shapechange and the high elastic stresses that arecreated, a thick plate cannot terminate in-side a parent crystal. As is frequently ob-served, lenticular shapes or groupings ofdifferently oriented martensite plates willreduce these elastic stresses.

In the case of lenticular martensite, thehabit plane is no longer a plane but acurved surface and the normal average ofthe lenticular plate is then taken as the or-ientation of the habit plane. Sometimes thisorientation is visible as a “midrib” in somemartensites (Fig. 9-21). It is believed thatthe martensite could grow to a certain ex-tent as a plate, but that lenticular shapes areformed owing to the high elastic stressesthat are building up. The high stresses may



trigger other plates to form in the vicinityof the plate formed earlier, giving rise to an“autocatalytic growth” of martensite.

A multivariant martensite arrangementis the most commonly observed micro-structure. Often variants are arranged insome recognizable patterns and at timesnumerous variants present in a regular ar-ray give the impression of a martensite col-ony. The latter is typical of the “massive”microstructure, consisting of a packet ofparallel martensite “laths” separated bymore or less wavy interfaces. Each lath inthe packet maintains the same variant ororientation relationship with the parentcrystal. A single grain of the parent phasecan transform into one or more such pack-ets. The “plate” martensite arrangementwhich is observed in the same alloys differsfrom the lath configuration, because adja-cent martensite plates are generally notparallel to each other.

Diagrams have been constructed forcases where there is a variety in morphol-ogy, as for example for Fe–Ni–C alloys.Maki and Tamura (1987) showed that themorphology of the a¢-martensite in thesealloys is related to the transformation tem-perature and the carbon content (Fig. 9-22).

Distinct martensite plate arrangementscan also be recognized in alloys pertainingto the b-Hume–Rothery alloys. Schroederand Wayman (1977) classified these ar-rangements into spear, fork, wedge and di-amond forms. Each representation carrieswith it a definite crystallographic relation-ship between the variants constituting the arrangement. Grouping of martensiteplates in such arrangements will lead to aserious reduction in the elastic stresses. Byanalysis of the crystallography of the platesin a single group, it can be shown that therespective macroscopic shape changes an-nihilate each other (Tas et al., 1973). Such

group formation is then called “self-ac-commodation”.

In these b-Hume–Rothery alloys, threetypes of martensite form, the 3R-, 9R- and2H-types. A detailed analysis of the micro-structure reveals that the martensite vari-

Figure 9-21. Transmission electron micrograph of amartensite in steel showing the twinned midrib(courtesy C. M. Wayman, 1989, University of Illi-nois, Urbana (IL)).

ants form in six different groups, eachgroup consisting of four variants. The habitplanes of these four variants are locatedaround the same 110b pole, whereas eachbasal plane is located close to one of fourother 110b poles. The total macroscopicdeformation of such a group is to a first ap-proximation completely compensated. Amore complete reduction in the three-di-mensional strain is obtained if the totaltransformation strain is calculated for thesix groups as an entity. As the martensitetransformation involves a very small vol-ume change in all b-Hume–Rothery alloys,the strain accommodation is thus almostcomplete.

The shape change associated with a mar-tensite plate creates stresses in both theparent and the martensite phases. If thesestresses exceed the flow stress for plastic

deformation, strain accommodation is thenaccomplished not only by elastic but also byplastic deformation in one or both phases.

Partitioning of the parent crystal, withfiner plates forming subsequently in thepartitioned region, frequently occurs andillustrates the fractal nature of the transfor-mation (Fig. 9-10). It should be mentioned,however, that not all martensite micro-structures show fractal characteristics(Hornbogen, 1988).

Until now only the more macroscopicobservable features of the microstructureof martensite have been discussed. Trans-mission electron microscopy reveals thatthe substructure of martensite is also char-acteristic. It consists, depending on the al-loy system and alloy composition, of regu-larly spaced stacking faults (e.g. Cu-baseb¢-type martensite), twins with a constantthickness ratio (e.g. Fe–30% Ni), disloca-tions (e.g. Fe–20% Ni–5% Mn), stackingfaults and twins in the same martensiteplate (e.g. Cu–Ga), or twins in the midribregion surrounded by dislocations.

9.8.1.4 Crystallographic Phenomenological Theory

The formal phenomenological theoriesof martensite formation predict the crystal-lographic characteristics, such as the shapedeformation, the orientation of the habitplane, the orientation relationship betweenparent and product phase, and the ampli-tude of lattice invariant deformation. Thisprediction is obtained from the sole knowl-edge of the structures and lattice parame-ters of the two phases and with the basic as-sumption that the interface between parentphase and martensite is undistorted on amacroscopic scale.

The observation of the K–S and N–Worientation relationships led us to origi-nally believe that a martensite was formed


Figure 9-22. Relationship between a¢-martensitemorphology and Ms temperature as a function of car-bon content in Fe–Ni–C alloys (Maki and Tamura,1987).


by shear on those planes and directionsspecified in the orientation relationships.However, it was found that the shear mech-anisms proposed by the K–S and N–W relations are not consistent with these ex-perimental observations. The observationsmade by Greninger and Troiano (1949) onFe–22% – Ni–0.8% C martensite were thekey to the mathematical development ofthe crystallographic theory of martensite.They found that martensite plates exhibiteda surface relief that can be described by ahomogeneous shear along the habit plane,but this homogeneous shear could nottransform the f.c.c. lattice of the parentphase into the b.c.t. lattice of the marten-site. If the f.c.c. lattice had undergone thesame homogeneous deformation, the struc-ture of the martensite would have beentrigonal. They therefore suggested that twotypes of shear are involved in the marten-sitic transformation: a “first” simple shearwhich is responsible for the macroscopicshape change, and a “second” shear whichneeds to be added to obtain the structuralchange but which should produce no ob-servable macroscopic change in shape.Two years later, Bowles (1951) showedthat the shape deformation may be any in-variant plane strain. This opened the way tothe formulation of the general theory of thecrystallography by Wechsler et al. (1953)and, independently, by Bowles and Mack-enzie (1954). Almost equivalent theorieswere later developed by Bullough andBilby (1956) and Bilby and Frank (1960).The reader may consult the following moreelaborate reviews of these theories: Way-man (1964), Christian (1965), Nishiyama(1978) and Ahlers (1982).

The basic assumption in the crystallo-graphic theories is that the interface be-tween the product and the parent phases isundistorted, which means that any vectorthat lies in this interface on the side of the

martensite would be a vector of the samesize and the same orientation in the parentphase before transformation. As indicatedin Sec. 9.3.1, the macroscopic shapechange of an invariant-plane transforma-tion can be represented by a combinationof a pure lattice deformation (B), the so-called Bain strain, a rigid lattice rotation(R), and an inhomogeneous lattice-invari-ant deformation (P). The pure lattice defor-mation either increases or decreases somevectors in length. According to Wayman(1964), “the essence of the crystallographictheory of martensitic transformations is tofind a simple shear (of a unique amount, ona certain plane, and in a certain direction)such that vectors which are increased inlength due to the lattice deformation arecorrespondingly decreased in length due tothe simple shear, and vice versa. Such vec-tors which remain invariant in length tothese operations define the potential habitplane. Physically speaking, the ellipsoidgenerated from the initial sphere by the lat-tice deformation is distorted by the simpleshear into another ellipsoid which becomestangential to the initial sphere, the points oftangency being related along a diameter.”This is clearly illustrated in Fig. 9-23,where the problem becomes two-dimen-sional, because one of the principal axes ofthe lattice deformation is taken as normalto the plane of shear.

Some complementary remarks concern-ing the crystallographic theory should bemade. The input data for the calculationsare (i) the lattice parameters of the parentand martensite phases, (ii) the lattice corre-spondence, and (iii) the lattice-invariantshear. The output of the calculations is thenthe amount of inhomogeneous shear re-quired to obtain the invariant plane condi-tion, the macroscopic shape change, andthe orientation relationship. Because of thelattice symmetries, differently oriented

Brain relations and inhomogeneous shearsystems lead to a number of crystallo-graphically equivalent solutions. Becauseof the observed orientation relationships,the Bain relationship is fixed for mostcases. However, larger unit cells are some-times chosen, especially for those marten-sites where the crystal structure has a largeunit cell compared with that of the parentphase. The only variable in these calcula-tions is the choice of the inhomogeneousshear system. The orientation of the habitplane is found to be very sensitive to thechoice that is made. For the f.c.c. to b.c.c.or b.c.t. transformation, the twin shear(112)M[111

–]M gives a (3 15 10)P habit

plane, whereas a (011)M[1–1–1]M shear re-

sults in a (111)P habit plane.The phenomenological theory as ex-

plained above is based on one active shearsystem. However, for some alloy systemsthis is not adequate. For example, a single(112)M twinning system is not able to ex-plain the 225P habit plane in some steels,and even two inhomogeneous shear sys-tems do not give agreement with the ex-perimental observations. Similar disagree-

ments have been observed in other alloysystems. To test critically the validity ofthe crystallographic theories, all crystallo-graphic parameters should be measuredand compared with the theoretical predic-tions. Agreement should be obtained forthe complete set of parameters. For themartensites that are twinned, this includesa careful determination of the normal to thetwinning plane K1 relative to the parent lat-tice. In Cu–Al–Ni, for example, inconsis-tencies up to 12.5° have been found.

In cases where the lattice-invariant shearis twinning (as opposed to faulting or slip),type I twinning, where the twin plane orig-inates from a mirror plane in the parentphase, has been assumed. Otsuka (1986)carefully analyzed for a number of systemsthe possibility of a type II twinning as analternative inhomogeneous shear. In type IItwinning, the shear direction stems from atwo-fold symmetry direction of the parentphase. Otsuka (1986) compiled all thetwinning modes observed in martensiteinto a table and found that most of them aretype I or compound but that type II twin-ning had only recently been observed. According to Nishida and Li (2000), fivedifferent twinning modes exist in TiNi and other shape-memory alloys such asCu–Al–Ni, Cu–Sn etc., namely the 111type I, the 011 type I, the ·011Ò type II,the (100)-compound and the (001)-com-pound twins. Type II twinning has recentlyreceived much attention as a mechanismfor lattice-invariant shear in some alloys.Because type II twins have irrational twinboundaries, the physical meaning of an ir-rational boundary is still a controversialproblem. It has been proposed that an irra-tional boundary consists of rational ledgesand steps, the average being irrational.Thereafter, Hara et al. (1998) carried out acareful study using HRTEM, to try and ob-serve ·111Ò type II twin boundaries in a


Figure 9-23. Production of an undistorted plane byshear such that the shape ellipsoid touches the unitsphere along one of its principal axes (Christian,1965).


Cu–Al–Ni alloy, but they were unable toobserve any ledges or steps. The boundaryis always associated with dark strain con-trast, and the lattice is continuous throughthe irrational boundary. Nishida and Li(2000) also carried out extensive studies on·011Ò type II twin boundaries in TiNi usingHRTEM, but they did not observe ledges orsteps either. Based on these experimentalresults, it is thus most likely that the type IIthin boundary is irrational even on a micro-scopic scale, and the strains at the boun-dary are elastically relaxed with wide twinwidth. To confirm this interpretation. Haraet al. (1998) carried out computer simula-tions by using the molecular dynamicsmethod. The result showed that the irra-tional thin boundary did not show anysteps. Thus, the above interpretation for anirrational twin boundary is justified. Ot-suka and Ren (1999) have pointed outagain the importance of type II twinning inthe crystallographic aspects of martensite.They also stress the role that martensite aging has on the rubber-like behavior ofmartensite, a point that has been a long-standing unsolved problem. They showedthat the point defects play a fundamentalrole. The deformation mechanisms of thecold deformation of NiTi martensite havebeen thoroughly analyzed by Liu et al.(1999a, b). They also found an interplaybetween type I and type II twinning.

As already mentioned in Sec. 9.3.6, abetter and more complete agreement can beachieved when the strain energy terms,both bulk and interfacial, are included inthe calculation.

9.8.1.5 Structure of the Habit Plane

In a number of alloys, especially those inwhich the so-called thermoelastic marten-sites are formed, the interface betweenmartensite and the parent phase is mobile,

even at very low temperatures. This obser-vation shows that the interface migrationmust be accomplished without appreciablethermal activation. The interface is thus“glissile”. In searching for models to ex-plain the structure and mobility of theinterface, we are concerned with the idealand the actual interface morphology. Acareful experimental analysis of the inter-face structure is therefore required if wewant to verify the various models that havebeen proposed. As the models treat theinterface on an atomistic scale, the sub-structure of the interface should be studiedby conventional and by high-resolutiontransmission electron microscopy. Thesame holds for martensite-to-martensiteinterfaces, which in some alloys are alsomobile. Recently, atomistic imaging of themartensite/austenite and martensite/mar-tensite interfaces have been obtained. It istherefore not surprising that both aspects,the observation of interface substructuresand the atomistic models, are treatedjointly in the literature. For further readingconcerning the interface structures and thegrowth mechanism of martensite we referto the review papers by Christian (1982),Christian and Knowles (1982), and Olsonand Cohen (1986). A summary of these pa-pers is given below.

Let us first introduce the kinds of mar-tensite interfaces concerned: glissile andnon-glissile martensitic interfaces, with thelatter subdivided into the coherent andsemi-coherent interfaces. The two struc-tures, martensite and the parent phase, aresaid to be “fully coherent” if both latticeshave a matching plane parallel to the inter-face. If a fully coherent interface is dis-played, the crystal undergoes a shape de-formation leaving all vectors in the inter-face invariant. In general, the two phasesdo not have a plane of atomic fit, so thatfully coherent martensite interfaces are ex-

ceptional. A fully coherent martensitic in-terface is, for example, that between f.c.c.and h.c.p. structures with lattice parame-ters such that a (f.c.c.) = ÷–

2 a (h.c.p.); theatomic arrangement in the basal planes,which constitute the interface between thetwo structures, are identical. Such transfor-mations are found in Co and its alloys andin some Fe-based alloys. The situation atsemi-coherent interfaces becomes morecomplex. The models predict the presenceof dislocations to correct the mismatchalong the interface. If this coherent inter-face moves, it is suggested that not all vec-tors are left invariant and that the move-ment of dislocations causes shear in theproduct phase. Fig. 9-24 shows the slip as-sociated with the interface dislocations.

Internally twinned martensite has beenreported to show a zig-zag parent–marten-site interface, as observed by conventionalelectron microscopy in, for example,Ti–Mn and Cu–Al–Ni. Fine parallel stria-tions have been observed in the interfacebetween austenite and both the b¢-type andthe g¢-type Cu–Al–Ni-martensite. Thesestriations have been accounted for in termsof interfacial dislocations resulting fromrandom faulting on the basal plane of theb¢-type martensite and the twinning planesof the g¢-type martensite. High-resolution

electron micrographs show that the inter-face between martensite and the parentphase and also the intervariant interfacesand the interfaces between the internaltwins in one martensite plate contain dis-continuities (“steps”) on an atomic scale,the nature of which has not yet been com-pletely unravelled. These steps can be con-sidered as resulting from a small deviationof the ideal habit plane, and would then becomparable to those observed along theinterfaces of tapered twins.

An exact understanding of the structureof the interfaces involved in the martensitictransformation (the parent–martensite, theintervariant, and the twin/twin interfaces)is therefore essential in determining themechanism of transformation and the mo-bility of the interfaces.

9.8.2 Thermodynamics and Kineticsof the Martensitic Transformation

9.8.2.1 Critical Driving Forceand Transformation Temperatures

A quantitative thermodynamic treatmentof the martensitic transformation requires aprecise knowledge of the thermodynamicequilibrium temperature T0 and of thechange in Gibbs energy at the transforma-


Figure 9-24. Three-dimen-sional representation of asemi-coherent martensiteinterface; the vectors OA aredistorted into O¢A¢ but thelarge vectors OZ = O¢Z¢ areinvariant (Christian, 1982).


tion temperature Ms . Both can be calculated and/or derived from measureddata, as is shown here for two examples:the martensitic transformation in Fe–X(X = Ni, Ru, …) and in Cu–Zn–Al alloys.In the former example, both the parent andthe martensite phases have the same struc-ture as the equilibrium phases and hencethe data for the equilibrium phases can betaken. In the latter example, both structuresdiffer from that of the equilibrium phases,which requires a more elaborate calculation.

For the Fe–X alloys, the Gibbs energyper mole of the parent austenite phase gand of the martensite phase a are Gg andGa, respectively. The change in Gibbs en-ergy per mole, DGgÆa, which for a mar-tensitic transformation g Æ a is availableto the system at any temperature T, is then

DGgÆa|T = Ga – Gg (9-8)

This quantity is negative for temperaturesat which the a-phase is the more stable andpositive for temperatures at which the g-phase is the more stable. There is a charac-teristic temperature T0 corresponding to thethermodynamic equilibrium between bothphases, such that

DGgÆa|T =T0= 0 (9-9)

Because the transformation creates interfa-cial and elastic energies, the martensitictransformation g Æ a or a Æ g does notstart at T0, but at a temperature below orabove T0, respectively. It is therefore nec-essary to undercool or overheat, respec-tively, until Ms or As is reached. At thesetemperatures the Gibbs energy changeDGgÆa is sufficiently large to induce theforward or reverse transformation, respec-tively. DGgÆa (at Ms) is then the criticalchemical driving force.

The martensite phase, represented by M,is to be regarded as the a-phase embedded

in the g-phase. Because of the shape andvolume changes associated with the trans-formation, elastic strain energy also has tobe considered. The Gibbs energy is thuscomposed of chemical Gibbs energy, Gc,and strain energy, Ee, so that the Gibbs en-ergy change accompanying the transforma-tion may be written as

GgÆM = DGcgÆa + DEe

aÆM (9-10)

At temperatures below Ms , where bothphases coexist and thus are in equilibrium,DGgÆM |T = 0. DGc

gÆa|T is then exactlyequal, but opposite in sign, to the sum of allnon-chemical energies DGnc

aÆM |T . If thesurface energies are neglected in compari-son with the high strain energies, the non-chemical energy equals DEe

aÆM |T , andapproaches zero at T = Ms . The strain en-ergy stored in the material is the sum ofthat produced by shearing and by volumechange. The former depends on thestrength of the parent phase and thus alsoon the grain size, hence Ms also depends onthe grain size, as shown by Hsu and Xiao-wang (1989).

The necessary undercooling (T0 – Ms)and superheating (As – T0) vary for differ-ent alloy systems, and for certain materialseven with composition. A precise thermo-dynamic definition of Ms and As cannot begiven, however, if the non-chemical ener-gies, DGnc

aÆM, are not known. We can thenonly say that Ms or As is the temperature atwhich the quantity DGgÆa (at T = Ms orT = As , respectively) is sufficiently nega-tive or positive, respectively, to have a rea-sonable chance of nucleation.

Two approaches are found in the litera-ture for calculating the critical chemicaldriving force. The first is based on the ex-perimentally determined Ms temperatures(Kaufman, 1965) and the other on a theo-retical model for the non-chemical energies(Hsu, 1985).

The first approach has been used for fer-rous alloys, which can be classified intotwo systems: those with g-loops and thosewith stabilized g-phases (see Fig. 9-25).Figure 9-25 also gives the Ms temperaturesfor the a- and e-martensite. If A is the al-loying element for iron, the molar chemicalGibbs energy for the austenite phase (Gg)can be written as

Gg = (1 – x) GgFe + x Gg

A + Ggm (9-11)

where x represents the atomic fraction ofthe element A in solid solution in the g-austenite, (1 – x) the atomic fraction ofiron, Gg

Fe the chemical Gibbs energy ofpure iron as f.c.c. g-phase, Gg

A the chemicalGibbs energy of element A as f.c.c. phase,and Gg

m the Gibbs energy of mixing of the g-phase. Similarly, the Gibbs energy of thea-phase can be given as

Ga = (1 – x) GaFe + x Ga

A + Gam (9-12)

where GaFe and Ga

A are the Gibbs energies ofpure iron and pure element A as a b.c.c. a-phase, respectively, and Ga

m is the Gibbsenergy of mixing of the martensite phase.The change in chemical Gibbs energy ac-companying the martensitic transformationgÆa then becomes

DGgÆa = (1 – x) DGFegÆa + x DGA

gÆa

+ DGmgÆa (9-13)

The quantity DGFegÆa represents the Gibbs

energy change for transformation g Æ a ofpure iron and can be assessed experimen-tally from the measured heat of transforma-tion and the specific heat of both phases.The quantity DGA

gÆa cannot usually be ob-tained from experiments because the ele-ment A does not always exist in the twomodifications g and a; it must therefore beestimated from thermodynamic models forsolid solutions. The quantity DGm

gÆa is the


Figure 9-25. Schematic dia-grams for ferrous alloys thatform a g-loop (Fe–Cu, Cr,Mo, Sn, V, W) and that g-loop forms a stabilized aus-tenite phase (Fe–C, Ir, Mn,N, Ni, Pt, Ru); (a) equilib-rium diagrams; (b) Ms tem-perature diagrams (Kraussand Marder, 1971).


difference in Gibbs energy of mixing andcan in principle be measured experimen-tally through activity measurements; if not,it must also be estimated.

The equilibrium temperature T0 and thecritical driving force and Ms and As can becalculated from Eq. (9-12). Such calcula-tions have been performed by Kaufman(1965) for the iron–ruthenium alloy sys-tem, which is of particular interest becausethe g-phase transforms martensitically intotwo phases, the a-b.c.c. and the e-hexago-nal phases. Both phases also occur as equi-librium phases, as shown in Fig. 9-26 to-gether with observed Ms and As tempera-tures and the calculated T0 temperatures.The undercooling for the a-martensite formation is strongly composition depen-dent, whereas it is independent of composi-tion for the hexagonal martensite. Thecomputed T0 curves are seen to lie betweenthe appropriate transformation temperaturecurves. The calculated driving forces,DGgÆa and DGgÆe, for both transforma-tions are plotted as a function of temperaturefor various compositions. The intersectionsof these curves with the temperature axiscorrespond to the theoretically deduced T0

temperatures for the appropriate composi-tions. When the appropriate experimentallyderived Ms and As are cross-plotted, the crit-ical driving forces for the g/a and g/e mar-tensitic transformations are obtained. Thelatter is seen to be smaller, which is consis-tent with the closer lattice correspondenceof the former transformation.

Hsu (1985) presented a model by whichmore accurate computations of the non-chemical part DGaÆM are possible forFe–C, Fe–X and Fe–C–X alloys. This en-abled him to obtain from Eq. (9-9) thetheoretical Ms temperatures, which are ingood agreement with the observed values.

In the second example, martensite for-mation in Cu–Zn alloys, the change in

Figure 9-26. (a) The iron–ruthenium phase diagramand (b) the T0 and Ms and As temperature diagrams(after Kaufman, 1965).

Gibbs energy can, according to Hsu andZhou (1989), be described as

DGb¢–M = DGb¢–b + DGb–a + DGa–a¢

+ DGa¢–M (9-14)

where b¢ – M represents the transformationfrom the the ordered b.c.c. phase to the or-dered 9R-type of martensite, b¢ – b the or-der–disorder transition, b – a the transfor-mation from the disordered b.c.c. to thedisordered f.c.c. phase having the samecomposition, a – a¢ the disorder–ordertransformation in the f.c.c. phase, anda¢ – M the transition from the ordered f.c.c.phase to the ordered martensite phase. As-suming a simplified relationship betweenthe degree of ordering and temperature,Hsu and Zhou found good agreementbetween the calculated and observed Ms .Their calculations show that ordering ofthe parent phase, which cannot be sup-pressed even by severe quenching, stronglyinfluences T0.

It is known that martensite may also beinduced by an external stress at tempera-tures above Ms . The problem now is to cal-culate the change in T0 due to changes instress. As a first approximation, it is as-sumed that the driving force DGm

PÆM |T =Ms

required for nucleation remains constantwith temperature and thus independent ofstress. Patel and Cohen (1953) calculatedthe work done on the stressed specimen;their treatment provides a good under-standing of how an applied stress that is de-composed into a shear stress along thehabit plane and a normal stress perpendicu-lar to it, affects the transformation temper-ature. At Ms

s, which is the martensite starttemperature when cooling under an appliedstress s, the chemical Gibbs energy changeequals the transformation work of the ex-ternal stress:

DGsPÆM = 1/2sa (9-15)

¥ [d0 sin 2q ± e0 (1 + cos 2q )] Vm

where d0 is the shear strain, sa the appliedstress, q the angle between the stress axisand the normal to the operative shear plane,e0 the corresponding strain associated withthe transformation, and Vm the molar vol-ume. The quantity DHPÆM can be meas-ured by calorimetry and DSPÆM can beevaluated from stress-induced transfor-mation experiments or calculated from Eq.(9-16). The temperature T0 can be calcu-lated thermodynamically or obtained moreor less accurately from the relationships

DGPÆM = DHPÆM – T DSPÆM

DHPÆM = T0 DSPÆM (9-16)

and

T0 = (As – Ms)/2 = (Af – Mf)/2 (9-17)

However, it should be noted that the deter-mination of T0 does not always obey thesesimple relationships and that the calorimet-


Figure 9-27. Gibbs energy G* versus temperatureand force for stressed samples: P and M representfree energy surfaces for parent and martensite, re-spectively (Wollants et al., 1979).


rically measured heats do not always re-flect the exact heats of transformation.

As stress itself is also a state variable in-dependent of temperature, it should be con-sidered in the thermodynamic treatment asexplained by Wollants et al. (1979). To de-scribe the thermodynamic state of a uniax-ially stressed crystal, they introduced the“elastic” state functions H* and G*, whichincorporate the effect of stress as follows:

H* = U + P V – F l = H – F l

= H – s e Vm (9-18)

G* = U + P V – T S – F l = G – F l

= G – s e Vm (9-19)

where F is the applied force and l the “mo-lar length” of the crystal. Fig. 9-27 illus-trates how the equilibrium temperature andforce change when one of the variables ischanged; P and M represent the Gibbs energysurfaces of the parent phase and of marten-site, respectively. At the two-phase equilib-rium G*P = G*M and if, at constant hydro-static pressure, the intensive variables F(or s) and T are changed in such a way that there is thermodynamic equilibriumbetween martensite and the parent phase,then dG*P = dG*M, or

– SP dT – l P dF = – SM dT l M dF

so that dF/dT = – [DS/Dl]PÆM, or, sincethe molar work FDlPÆM = s ePÆM Vm andDSPÆM = DH*(s)/T0 (s), it also followsthat

(9-20)ds /dT = – [DH*(s)]/[T0 (s ) ePÆM Vm]

where Vm is the molar volume (Vm = V P =V M). Eq. (9-20) is the Clausius–Clapeyronequation for a uniaxial stress, which is sim-ilar in form to that for hydrostatic pressure,except for the negative sign.

The change in critical stress necessary toinduce martensite can be obtained fromtensile tests carried out at different temper-

atures (see Fig. 9-28). The elongation re-sulting from the transformation is orienta-tion dependent (Fig. 9-29). From data suchas those shown in Figs. 9-28 and 9-29, wecan calculate DSP–M. It is evident that foreach crystal orientation the slope ds /dT isdifferent.

Figure 9-28. Results of tensile tests for inducingmartensite in a Cu–34.1 Zn–1.8 Sn (at.%) alloy (Pops, 1970).

Figure 9-29. Orientation dependence of stress–strain curves for martensite formation in aCu–Al–Ni alloy (Horikawa et al., 1988).

Similarly, martensite can be induced bymagnetic fields. By taking into account thecomposition, the influence of grain boun-daries, and crystal orientation, the mag-netic invar effect and the austenite magne-tism, Shimizu and Kakeshita (1989) pro-posed an equation that describes the shiftof Ms as a function of the magnetic field.

9.8.2.2 Nucleation

Martensitic transformations are first-or-der phase transformations and hence occurby nucleation and growth. In most in-stances, except in the case of thermoelas-ticity (see below), the growth of a marten-site plate proceeds so rapidly that the trans-formation kinetics are dominated by thenucleation event. Various mechanisms ofmartensite nucleation have in the past beenproposed and can be considered under twosubheadings. In the first group of models,the so-called localized nucleation models,concepts of diffusional nucleation kineticsare applied; the second group of models isbased on lattice instability considerations

concerning both static and dynamic latticeinstability. All nucleation models can fur-ther be divided into classical and non-clas-sical. The former model involves latticeperturbations of fixed amplitude and vary-ing size, whereas the latter considers per-turbations of varying size (Olson and Co-hen, 1982b).

In the classical nucleation theory; mar-tensite nuclei form along a path of constantcomposition and structure and the state ofthe nucleus is given by its size. Because themartensitic transformation involves shearstrains, it can be shown that the strain en-ergy is minimized for a disc-like nucleus,but then the surface energy becomes verylarge. The critical nucleus, assuming anoblate spheroidal shape (Fig. 9-30), willthen have an aspect ratio (c/r) such that forany change in shape, the decrease in strainenergy will be exactly balanced by an in-crease in interfacial energy. The interfacialGibbs energy per plate is

v Dgs = 2p r2 G (9-21)

where v is the volume of the plate, Dgs thesurface Gibbs energy per unit volume, andG the interfacial energy. The strain energyper plate is

v Dge = (4/3) p r2 c (Ac/r) (9-22)

where Dge (= Ac/r) is the strain energy perunit volume and A is a factor to be deducedfrom linear elasticity and thus a function ofthe elastic constants and of the shear anddilatational strains. The chemical Gibbsenergy change per plate is

v Dgc or (4/3) p r2 cDgc (9-23)

If the nucleation occurs at a lattice defect,we have to consider also the Gibbs energyGd due to the defect and the nucleus–de-fect interaction energy Gi . According toOlson and Cohen (1982), the total Gibbsenergy describing the formation of a classi-


Figure 9-30. Shape of a nucleus of a martensiteplate.


cal martensitic nucleus becomes(9-24)

G (r, c) = Gd + Gi + v (Dgc + Dge + Dgs)

and is given schematically in Fig. 9-31.Three cases are considered in calculatingthe critical free energy for nucleation DG*and the critical nucleus size r* and c*.

In the case of homogeneous nucleation,Gd and Gi are zero, and on inserting thenecessary quantities into Eq. (9-24) wefind that the barrier DG* is too high by sev-eral orders of magnitude. Even assuminglocal compositional fluctuations or the ex-istence of pre-existing embryos does notgive full satisfaction. It was therefore soon

recognized that homogeneous nucleationof martensite is impossible. Recently,much progress has been made in the under-standing of nucleus formation at lattice de-fects. DG* and therefore also the criticalsize of the nucleus can be reduced by as-suming the nucleation at a defect. Undercertain special conditions, this heterogene-ous nucleation may even be barrierless.Such a case applies to the f.c.c.-to-h.c.p.transformation, which may take place bydissociation of a number of properlyspaced total dislocations present in the ma-trix phase into partial dislocations separ-ated by stacking faults. The stacking-fault

Figure 9-31. Schematicnucleus Gibbs energy (G)curves for nucleation via aclassical path: (a) homoge-neous, (b) heterogeneous,and (c) barrierless nuclea-tion (Olson and Cohen,1982b).

Figure 9-32. Electron mi-crographs of the nucleationand early growth stagearound inclusion particles ina Ti–Ni–Cu alloy (Saburiand Nenno, 1987).

energy is temperature dependent and be-comes positive below T0, which results in abarrierless nucleation.

In a number of alloy systems, softeningof certain elastic constants is observed andit is then argued that, although the homoge-neous soft-mode concept is definitely notadequate to describe the nucleation of mar-tensite, stresses and strains present arounddefects of the lattice can induce a local me-chanical instability. Such a model is called“the localized soft-mode concept” (Guéninand Clapp, 1986). In this model the latticeGibbs energy is a function of pure strainsand therefore of second- and third-orderelastic constants. The third-order constants(which relate the strain energy to theamount of strain) introduce anharmonicterms into the strain energy and may lead tomechanical instability. The region of me-chanical instability, or the “strain spino-dal”, is so defined that any further increasein strain will make the lattice unstable withrespect to a decomposition into strained re-gions. In these zones a nucleus can developwithout generating any strain energy, andthe only resisting term remains the surfaceenergy. This results in a reduced criticalsize of the nucleus, which is further de-creased as the temperature is lowered ow-ing to the increase in chemical drivingforce.

In situ electron microscope observationshave been made on the nucleation and earlystages of growth of martensite, as shown inFig. 9-32. Martensite nucleates at stressconcentrations, the nucleation takes placerepeatedly at the same place, and the straincontrast disappears as nucleation andgrowth proceed and reappears when mar-tensite disappears.

At present, the nucleation models are be-ing further refined by molecular dynamiccalculations.

9.8.2.3 Growth and Kinetics

A distinction is made between the kinet-ics of a single martensite plate and the glo-bal kinetics, which expresses the volumefraction of the parent phase that is trans-formed. According to the observed kinet-ics, martensitic transformations can be di-vided into two distinct classes: athermaland isothermal martensite. In athermal mar-tensite, the transformation progresses withdecreasing temperature, whereas in iso-thermal martensite, the transformation pro-gresses with time at a constant temperature.

The growth may be “thermoelastic” or ofthe “burst” type. The latter is the morecommon mode. It consists of the formationof comparatively large amounts of marten-site (typically 10–30 vol.%) in “bursts”that are caused by autocatalytic nucleationand rapid growth of numerous plates. Eachindividual martensite plate is completelyformed with a speed higher than 105 cm/sand the transformation progresses by theformation of new plates. The global kinet-ics of the transformation are therefore es-sentially controlled by the nucleation fre-quency. The thermoelastic growth mode ischaracterized by the formation of thin, par-allel-sided plates or wedge-shaped pairs ofplates (Fig. 9-33), which form and growprogressively as the temperature is loweredbelow Ms and which shrink and disappearon reversing the temperature change. Thisbehavior arises because the matrix accom-modates the shape deformation of the mar-tensite plate elastically, so that at a speci-fied temperature the transformation frontof the plate and the matrix are in thermody-namic equilibrium. Any change in temper-ature displaces this equilibrium and, there-fore, the plate grows or shrinks. A com-plete mechanical analog of this thermoelas-tic behavior is the pseudoelastic behavior.The growth or shrinkage of individual mar-



Figure 9-33. Thermoelastic behavior in Ag–Cd alloys showing thegrowth of self-accommodating groups of martensite plates (Delaey et al., 1974).

He

ati

ng

Co

olin

g

Figure 9-34. Schematic representation of some relevant features (volume-transformed product or transforma-tion strain) experimentally observed in hysteresis curves corresponding to thermally induced and stress-inducedthermoelastic transformations: (a, e) single interface transformation in a single crystal; (b, f) multiple interfacetransformation; (c, g) discontinuous jumps (bursts), (d, h) partial cycling behavior.

tensite plates is then a direct function of theincrease or decrease in stress. More elab-orate thermodynamic treatments of ther-moelasticity can be found in the papers byDelaey et al. (1974), Salzbrenner and Co-hen (1979), Ling and Owen (1981) and Or-tin and Planes (1989).

The best quantitative understanding ofthe kinetics of the martensitic transforma-tion is obtained from isothermal transfor-mations, because they permit both the nu-cleation and the transformation rates to bedetermined. In those alloys exhibiting iso-thermal martensite (Thadhani and Meyers,1986), it is shown that at each temperaturethe transformation starts in the austeniteand proceeds as a function of time. Thetransformation exhibits a C-curve behav-ior. Isothermal martensitic transformationkinetics consist of two effects: an initial in-crease in the total volume fraction of mar-tensite, which is attributed to an autocata-lytic nucleation of new martensite plates,followed by a decrease due to the compart-mentalization of the austenite into smallerand smaller areas.

9.8.2.4 Transformation Hysteresis

Hysteresis behavior is one of the pecu-liar characteristics of both the thermal andstress-induced martensitic transformations.In several studies the origin of the fric-tional resistance opposing the interfacialmotion of martensite plates has been inves-tigated and described. From a practicalpoint of view, the hysteresis phenomenonis an important problem in the applicationof shape-memory alloys. In general hyster-esis appears when, on passing through a lo-cal extreme value (maximum or minimum)of any control parameter such as tempera-ture or stress, one or more state variablesdo not follow the original path in statespace. When all the state variables, includ-

ing the control parameter, return to theiroriginal values, a closed loop is formed(Fig. 9-34). The loop is always contoured insuch a sense that it encloses a positive area,representing the energy lost in the cyclicprocess. Therefore, hysteretic behavior isalways related to an energy-dissipative pro-cess. The dissipated energy is much smallerin thermoelastic martensitic transforma-tions than in burst-type transformations.

9.9 Materials

9.9.1 Metallic Materials

A classification of the diffusionless dis-placive transformations in metallic materi-als is given in Table 9-4, where the alloysystems are subdivided into three groups.The origin of the martensitic transforma-tion in the first group lies in the allotropictransformation of the pure solvent. Theparent phase of the alloys of this group thusdoes not show any remarkable mechanicalinstability. The second group consists ofthe b b.c.c. Hume–Rothery alloys, whichare characterized by a moderate lattice in-stability in the temperature range aboveMs . The third group is characterized by adrastic mechanical instability of the parentphase. Because the transformation is onlyweakly first order (by this we mean a dis-continuous jump in the corresponding ther-modynamic property whose height, how-ever, is very small) or even second order, itis in this group of alloy systems that wefind, in addition to the martensitic, thequasi-martensitic transformations.

Traditionally, the ferrous and non-fer-rous martensites have been treated separ-ately in the literature. Before going into de-tail it will be an advantage to first compareand contrast ferrous and non-ferrous mar-tensites and to do it in such a way that we


9.9 Materials 635

can establish criteria to differentiate thetwo groups. Although not every detail oftransformation behavior will be discussed,a set of criteria have been chosen as shownin Table 9-5. A more detailed review ofmartensite in metallic systems is given byNishiyama (1978).

9.9.1.1 Ferrous Alloys

Martensitic transformations in ferrousalloys have been studied extensively, espe-cially the crystallography and morphologywhich have been reviewed by Muddle

(1982). Depending on alloy composition, adistinction is made among the various mar-tensites, based either on the crystallogra-phy, morphology or growth characteristics.

Essentially, three different crystal struc-tures appear: the b.c.c. or b.c.t. a¢-marten-site, the h.c.p. e-martensite, and the long-range ordered f.c.t. martensite. In plain car-bon steels martensite is regarded as asupersaturated, interstitial solid solution ofcarbon in b.c.c. iron (ferrite), with a crystalstructure that is a tetragonally distortedversion of the ferrite structure. The tetrago-nality is linearly dependent on the carbon

Table 9-4. Classification of metallic alloy systems showing diffusionless displacive transformations (Delaey et al., (1982).

1. Martensite based on allotropic transformation of solvent atom

1. Iron and iron-based alloys

2. Shear transformation, close packed to close packed1. Cobalt and alloys f.c.c. Æ h.c.p., 126 R SF*2. Rare earth and alloys f.c.c., h.c.p., d.h.c.p., 9 R(3. MnSi, TiCr2 NaCl Æ NiAs, Laves)

3. Body centered cubic to close packed1. Titanium, zirconium and alloys b.c.c. Æ h.c.p., orth. f.c.c tw, d*2. Alkali and alloys (Li) b.c.c. Æ h.c.p.3. Thallium b.c.c. Æ h.c.p.

4. Others: plutonium, uranium, mercury, Complex structuresetc. and alloys

2. b-b.c.c. Hume–Rothery and Ni-based martensitic shape-memory alloys

1. Copper-, silver-, gold-, b-alloys(disord., ord.) b.c.c. AB, ABABCBCAC, ABAC

2. Ni–Ti–X b-alloys b.c.c. Æ 9 R, AB tw, SF*Nickel b-alloys (Ni–Al) b.c.c. Æ ABC tw, SF*Ni3–xMxSn (M = Cu, Mn) b.c.c. Æ AB tw*(Cobalt b-alloys, Ni–Co–X)

3. Cubic to tetragonal, stress-relaxation twinning or martensite

1. Indium-based alloys f.c.c. Æ f.c.t., orth. tw, tws*2. Manganese-based alloys f.c.c. Æ f.c.t., orth. tws*3. A 15 compounds, LaAgxIn1–x b-W. Æ tetr.4. Others: Ru–Ta, Ru–Nb, YCu, LaCd

* SF: stacking faults; tws: (stress relaxation) twins; d: dislocated

content. This a¢-martensite is also found ina number of substitutional ferrous alloys,the martensite being either b.c.c. or b.c.t. Incertain alloy systems with an austenitephase of low stacking fault energy, a mar-tensitic transformation to a fully coherenth.c.p. product (e-martensite) is observed.

In some long-range ordered alloys, as inFe–Pt and Fe–Pd, f.c.t. in addition to b.c.t.martensite is observed.

As far as the morphology is concerned,plate, lath, butterfly, lenticular, banded,thin-plate and needle-like martensite canbe distinguished.


Table 9-5. A qualitative comparison between ferrous and non-ferrous martensites (Delaey et al., 1982b).

Ferrous martensite Non-ferrous martensite

Interstitial and/or substitutional Nature of alloying Substitutional

Martensitic state in interstitial Hardness Martensitic state is not muchferrous alloys is much harder harder and may even be softer

than the austenite state than the austenite state

Large Transformation hysteresis Small to very small

Relatively large Transformation strain Relatively small

High values near the Ms Elastic constants of Low values near the Ms

the parent phase

Negative near the Ms Temperature coefficient Positive near the Ms

in most cases of elastic shear constant in many cases

Self-accommodation is not obvious Growth character Well developed self-accommodatingvariants

High rate, “burst”, athermal Kinetics Slower rate, no “burst”,and/or isothermal transformation no isothermal transformation,

thermoelastic balance

High Transformation enthalpy Low to very low

Large Transformation entropy Small

Large Chemical driving force Small

No single interface Growth front Single interface possibletransformation observed

Low and non-reversible Interface mobility High and reversible

Low Damping capacity Highof martensite

9.9 Materials 637

A distinct substructure, crystallographicorientation of the habit plane and austeniteto martensite orientation relationship areassociated with each morphology, as sum-marized in Table 9-6.

Because the carbon atom occupies octa-hedral interstices in the austenite f.c.c. lat-tice, special attention is drawn to theFe–X–C martensite. In the martensite lat-tice, those interstitial positions are definedby the Bain correspondence (Fig. 9-35).Only those at the midpoints of cell edgesparallel to [001]B and at the centers of thefaces normal to [001]B are permitted. Thispreferred occupancy affords an explanationof the observed tetragonality c/a, the de-gree of which is a function of the carboncontent:

c/a = 1 + 0.045 (wt.% C) (9-25)

Careful X-ray diffraction of martensite,freshly quenched and maintained at liquidnitrogen temperature, has shown signifi-

cant deviations from the above equation.The tetragonality is abnormally lower forX = Mn or Re and abnormally higher forX = Al or Ni. Heating to room temperatureof the latter martensite results in a loweringof the tetragonality. The formation of do-mains or microtwinning in the former al-loys and ordering of the Al atoms in the lat-ter have been put forward as the origin ofthe abnormal c/a ratio. This behavior hasmoreover been related to the martensiteplate morphology (Kajiwara et al., 1986,1991). Kajiwara and Kikuchi found that inFe–Ni–C alloys the tetragonality is abnor-mally large and depends on the microstruc-ture. It is very large for a plate martensite,while it is normal or not so large for a len-ticular martensite. They conclude that “themartensite tetragonality is dependent onthe mode of the lattice deformation in themartensitic transformation. If the latticedeformation is twinning, the resulting c : ais large, while in the case of slip it is small”(Kajiwara and Kikuchi, 1991).

9.9.1.2 Non-Ferrous Alloys

A classification base of the non-ferrousalloy systems exhibiting martensite is

Table 9-6. Summary of substructure, habit plane(H.P.) and orientation relationship (O.R.) for the fourtypes of a¢-martensite (Maki and Tamura, 1987).

Morphology Substructure H.P. O.R.* Ms

Lath (Tangled) (111)A K–S Highdislocations

Butterfly (Straight) (225)A K–Sdislocations

andtwins

Lenticular (Straight) (259)A Ndislocations

and or ortwins (3 10 15)A

(Mid-rib) G–T

Thin-plate Twins (2 10 15)A G–T Low

* K–S: Kurdjumow–Sachs relationship, N: Nishiyamarelationship, G–T: Greninger–Troiano relationship

Figure 9-35. Schematic representation of the Baincorrespondence for the f.c.c. to b.c.t. transformation.The square symbols represents the possible occupiedpositions of the interstitial carbon atoms (Muddle,1982).

given in Table 9-4, while the alloying ele-ments are given in Table 9-7. A review ofthe non-ferrous martensites was given byDelaey et al. (1982a).

Two typical examples of the first groupare the cobalt and titanium alloys. Thestructure of the cobalt-based martensites is in general hexagonal close-packed, butmore complex close-packed layered struc-tures have been reported, such as the 126R,84R and 48R structures observed inCo–Al alloys. Because the transformationis a result of an f.c.c. to h.c.p. transforma-tion, the basal planes of the martensitephase are parallel with the (111) planes ofthe parent phase and constitute the habitplane. The structure of the martensite in Ti-based alloys is also hexagonal but that ofthe high-temperature phase is b.c.c. Bothplate and lath morphologies are encoun-tered in titanium, and also in the similarzirconium-based alloys. Slip is suggestedas the lattice-invariant deformation modein lath martensite, whereas the twiningmode is observed in the plate martensities.

A typical example of the second groupare the copper-, silver- and gold-based alloys, which have been extensively re-viewed by Warlimont and Delaey (1974).Depending on the composition, three typesof close-packed martensite are formedfrom the disordered or ordered high-tem-perature b.c.c. phase, either by quenchingor by stressing. The factors determining theexact structure are the stacking sequence ofthe close-packed structure, the long-rangeorder of the martensite as derived from theparent b-phase ordering, and the deviationsfrom the regular hexagonal arrangementsof the martensite. The last factor is due todifferences in the sizes of the constituentatoms. The stacking sequence of the mainthree phases are ABC, ABCBCACAB andAB, respectively.

One of the interesting findings is the suc-cessive stress-induced martensitic transfor-mations in some of the b-phase alloys dis-cussed above, as shown clearly if we plotthe critical stresses needed for the transfor-mation (Fig. 9-36). Stressing a single crys-tal of Cu–Al–Ni at, for example, 320 Kwe find the parent b1-to-b¢1, the b¢1 – g¢1,


Figure 9-36. Critical stresses as a function of tem-perature for the various stress-induced martensitetransformations in a Cu–Al–Ni alloy (Otsuka andShimizu, 1986).

Table 9-7. Schematic representation of some non-ferrous martensitic alloy systems; the Co-, Ti- andZr-based terminal solid solutions, the intermetallicNi-, Cu-, Ag- and Alu-based alloys, the antiferro-magnetic Mn-based alloys and the In-based alloys(Delaey et al., 1982).

9.9 Materials 639

the g¢1 – b≤1, and finally the b≤1 – a¢1 marten-site.

The other typical example of the secondgroup is the Ni–Ti-based alloy system;both prototype alloy systems constitute theshape-memory alloys (SMA). The occur-rence of a so-called “pre-martensitic” R-shape has long obscured the observations.The review by Wayman (1987), illustratesthe complexity of the transformation be-havior. During cooling, the high-tempera-ture ordered b.c.c. phase (P) transformsfirst to an incommensurate phase (I) and onfurther cooling to a commensurate phase(C), and finally to martensite. The P-to-Itransformation is second order, whereas theI-to-C transformation is a first-order phasetransformation involving a cubic-to-rhom-

bohedral (the so-called R-phase) structuralchange. At still lower temperatures therhombohedral R-phase transforms into amonoclinically distorted martensite. TheR-phase also forms displacively and can bestress-induced, and shows all the character-istics of a reversible transformation.

Concerning the third group, in only afew cases, as in In–Tl, has definite proofbeen provided to justify the conclusion thatthe transformation is martensitic. Most ofthe transformations in these systems haveto be classified as quasi-martensitic.

9.9.2 Non-Metals

Inorganic compounds exhibit a varietyof crystal structures owing to their diverse

Table 9-8. Non-metals with lattice deformational transformations (Kriven, 1982).

Inorganic compoundsAlkali and ammonium halides MX, NH4X (NaCl-cubic ¤ CsCl-cubic)

Nitrates RbNO3 (NaCl-cubic ¤ rhombohedral ¤ CsCl-cubic)KNO3, TlNO3, AgNO3 (Orthorhombic ¤ rhombohedral)

Sulfides MnS (Zinc-blende-type ¤ NaCl-cubic)(Wurtzite-type ¤ NaCl-cubic)

ZnS (Zinc-blende-type ¤ wurtzite-type)BaS (NaCl-type ¤ CsCl-type)

MineralsPyroxene chain silicates Enstatite (MgSiO3) (Orthorhombic ¤ monoclinic)

Wollastonite (CaSiO3) (Monoclinic ¤ triclinic)Ferrosilite (FeSiO3) (Orthorhombic ¤ monoclinic)

Silica Quartz (Trigonal ¤ hexagonal)Tridymite (Hexagonal, wurtzite-related)

Cristobalite (Cubic ¤ tetragonal, zinc blende-related)

CeramicsBoron nitride BN (Wurtzite type ¤ graphite-type)

Carbon C (Wurtzite type ¤ graphite)Zirconia ZrO2 (Tetragonal ¤ monoclinic)

OrganicsChain polymers Polyethylene (CH2–CH2)n (Orthorhombic ¤ monoclinic)

CementBelite 2 CaO · SiO2 (Trigonal ¤ orthorhombic ¤ monoclinic)

chemistry and bonding. Compared withmetals, the relatively low-symmetry parentstructures have fewer degrees of freedomon transforming to even lower symmetryproduct structures, or vice versa. Many ofthese transformations involve changes inelectronic states with relatively small volumechanges. They tend to proceed by shuffle-dominated mechanisms. However, sheartransformations involving large structuralchanges in terms of coordination numberor volume changes have also been reportedin inorganic and organic compounds, min-erals, ceramics, organic compounds, andsome crystalline compounds of cement.Some of the most prominent examples aregiven in Table 9-8 (Kriven, 1982, 1988).

Because of its technological interest as atoughener for brittle ceramic materials, zir-conia is considered as the prototype ofmartensite in ceramic materials. On cool-ing, the high-temperature cubic phase ofzirconia transforms at 2370 °C to a tetrago-nal phase. On further cooling, bulk zirconiatransforms at 950 °C to a monoclinic phasewith a volume increase of 3%. The lattertransforms on heating at about 1170 °C.The monoclinic to tetragonal phase trans-formation is considered to be martensitic.The Ms temperature can be lowered sub-stantially even below room temperature byalloying or by reducing the powder size.Small particles of zirconia, embedded in asingle-crystal matrix of alumina, remainmetastable (= tetragonal) at room tempera-ture for particle diameters less than a criti-cal diameter (Rühle and Kriven, 1982).These metastable particles can transform tothe monoclinic phase under the action of anapplied stress, and it is this property that isexploited in toughening brittle ceramics(see Becher and Rose (1994)).

Polymorphism is known to occur in several crystalline polymeric materials. Inmost of these systems the transformation

depends strongly on thermal activation.However, in PTFE (polytetrafluoroethy-lene) the conditions for no or weak thermalactivation are fulfilled, and the transforma-tion can then be regarded as a diffusionless


Figure 9-37. (a) Helix structure of the a- and b-modification of PTFE, and (b) dilatometric deter-mination of relative volume changes and tempera-ture range of transformations of PTFE (Hornbogen,1978).

9.10 Special Properties and Applications 641

or martensitic transformation (Hornbogen,1978). This polymer crystallizes as parallelarrangements of molecular chains parallelto the c-axis. The atoms along the chainsare arranged as helices; the period alongthe c-axis in the a-modification is 13C2F4

units while it is 15 units in the b-modifica-tion. The transformation from the a- to theb-helix occurs at about 19 °C (Fig. 9-37).Relaxation of the helix during this transfor-mation does not lead to an extension of thespecific length of the molecules in the c-di-rection. The diameters of the molecule andthus the lattice parameters in the a-direc-tion increase, which leads to an increase of about 1% in the specific volume. Theobserved shape change can be increased ifthe molecules have been aligned by plasticdeformation. An analysis of the shapechanges leads to the conclusion that thePTFE transformation is diffusionless by afree volume shear, a type of transformationnot yet known in metallic and inorganicmaterials.

Biological materials consisting of crys-talline proteins also undergo martensitictransformations in performing their lifefunctions. In a review entitled “Martensiteand Life”, Olson and Hartman (1982) dis-cuss some examples. The tail-sheath con-traction in T4 bacteriophages can be de-scribed as an irreversible strain-inducedmartensitic transformation, while polymor-phic transformations in bacterial flagellaeappear to be stress-assisted and exhibit ashape-memory effect.

9.10 Special Propertiesand Applications

9.10.1 Hardening of Steel

Much of the technological interest con-cerns martensite in steels. In a review onstrengthening of metals and alloys, Wil-

liams and Thompson (1981) consider mar-tensite as one of the most complex cases ofcombined strengthening. The hardness ofmartensite in as-quenched carbon steel de-pends very much on the carbon content. Upto about 0.4 wt.% C the hardening is lin-ear; retained austenite is present in steelcontaining more carbon, which reduces therate of hardening. Solute solution harden-ing by the interstitial carbon atoms is verysubstantial, whereas substitutional solidsolution hardening is low. For example,Fe–30 wt.% Ni martensites, where the car-bon content is very low, are not very hard.The hardening of martensite is not due onlyto interstitial solute solution hardening,however. The martensite contains a largenumber of boundaries and dislocations,and the carbon atoms may rearrange duringthe quench forming clusters that cause ex-tra dislocation pinning. The various contri-butions to the strength of a typical C-con-taining martensite are given in Table 9-9,from which it becomes evident that inter-stitial solid solution hardening is not themost important cause. Because many mar-tensitic steels are used after tempering,lower strengths than those shown in Table9-9 are found.

9.10.2 The Shape-Memory Effect

A number of remarkable properties havetheir origin in a martensitic phase trans-

Table 9-9. Contribution to as-quenched martensitestrength in 0.4 wt.% C steel (Williams and Thomp-son, 1981).

Boundary strengthening 620 MPaDislocation density 270 MPa

Solid solution of carbon 400 MPaRearrangement of C in quench 750 MPa

Other effects 200 MPa

0.2% yield strength 2240 MPa

formation, such as the shape-memory ef-fect, superelasticity, rubber-like behaviorand pseudoelasticity. The most fascinatingproperty is undoubtedly the shape-memoryeffect. Review articles were published byDelaey et al. (1974), Otsuka and Shimizu(1986), Schetky (1979), and Junakubo(1987). Recently, Van Humbeeck (1997)prepared a review on Shape Memory Mate-rials: “State of the Art and Requirementsfor Future Applications”. His review con-tains 104 references to recent articles onthe topic.

A metallic sample made of a commonmaterial (low-carbon steel, 70/30 brass,aluminum, etc.) can be plastically de-formed at room temperature. The macro-scopic shape change resulting from thisdeformation will remain unchanged if thesample is heated to higher temperatures.The only observable change in propertymay be its hardness, provided that the tem-perature to which the sample has beenheated is above the recrystallization tem-perature. Its shape, however, remains as itwas after plastic deformation. If the sampleis made of a martensitic shape-memorymaterial and is plastically deformed (bent,twisted, etc.) at any temperature below Mf

and subsequently heated to temperaturesabove Af , we observe that the shape thatthe specimen had prior to the deformationstarts to recover as soon as the As tempera-ture is reached and that this restoration iscompleted at Af . This behavior is called the“shape-memory effect”, abbreviated to SME.

If the SME sample is subsequentlycooled to a temperature below Ms and itsshape remains unchanged on cooling, wetalk about the “one-way shape-memory ef-fect”. If it spontaneously deforms on cool-ing to temperatures below Ms into a shapeapproaching the shape that it had after theinitial plastic deformation, the effect iscalled the “two-way shape-memory ef-

fect”. A more visual description of thesetwo effects is given in Fig. 9-38 and a clar-ifying example is shown in Fig. 9-39,where the applicability of the one-wayshape-memory effect is given for a space-craft antenna.

The shape that has to be rememberedmust, first of course, be given to the speci-men. This is done by classical plastic de-formation by either cold or hot working.This process, however, may not involveany martensite formation. The materialmust therefore be in a special metallurgicalcondition, which may require additionalthermal treatments. In Fig. 9-40, for example, depicting a temperature-actuatedshape-memory switch, two different “re-membering” shapes are used. The initialshape may be obtained by hot extrusion orwire drawing and may or may not receivean additional cold or hot working in orderto obtain the required shape. The shapesformed must receive a heat treatment, con-sisting of a high-temperature annealing,followed by water quenching. The speci-men is now martensitic, provided that thecomposition is such that Mf is above roomtemperature. In order to induce the shapememory, the martensitic specimens arebent either to be curved or to be straightand are placed into the actuator at roomtemperature. If the temperature of the actu-ator exceeds the reverse transformation


Figure 9-38. Schematic illustration of the shape-memory effect: (a) and (e) parent phase; (b), (c) and(d) martensite phase (Otsuka and Shimizu, 1986).


temperature of the shape-memory material,the specimens recover towards their “re-membered” position. The electrical con-tacts are either closed or opened.

Special procedures for handling of theshape-memory device are needed if wewant to induce the two-way memory effect.This can be explained by again taking thetemperature-actuated switch as an exam-ple. If the specimen taken in its remem-bered position is cooled back to room tem-

perature, we do not expect further shapechanges to occur. In order to reuse thespecimens after having performed theshape-memory effect, they must be bent tobe either curved or straight again. Reheat-ing these deformed specimens for a secondtime to temperatures above Af will result inshape memory. If this cycle, bending–heat-ing–cooling, is repeated several times,gradually a two-way memory sets in. Dur-ing cooling the specimen reverts spontane-

Figure 9-39. Application ofNitinol for a shape-memoryspacecraft antenna. From “Shape Memory Alloys” byL. McDonald Shetky. Copy-right (1979) by ScientificAmerican, Inc. All rights re-served.

ously to its “deformed” positions, thusopening or closing the electrical contactson cooling. This repeated cycling, defor-mation in martensitic condition followedby a heating–cooling, is called “training”.We can thus induce two-way memory byusing a training procedure.

A further comment should be made hereconcerning the shapes that can be remem-bered. We have to distinguish three shapingprocedures: the fabrication step from rawmaterial towards, for example, a coiledwire such as for the antenna, the fabrica-tion of the “to be remembered position”,such as the additional shaping for the actu-ator, and the final deformation in the mar-tensitic condition, such as the bending ofthe actuator. The first two fabrication stepsinvolve only classical plastic deformationand, therefore, the type and degree of de-formation are in principle not limited, pro-vided that the material does not fail. Thedegree of deformation, however, is limited

in the third deformation step, because itmay not exceed the maximum strain thatcan be recovered by the phase transforma-tion itself. Because these strains are asso-ciated with the martensitic transformation,the maximum amount of recoverable strainis bound to the crystallography of the trans-formation. Exceeding this amount of de-formation in the third fabrication step willautomatically result in unrecoverable de-formation.

Many examples of shapes that can be re-membered are possible. A flat SME speci-men can elongate or shorten during heat-ing, can twist clockwise or counter-clock-wise, and can bend upwards or downwards.An SME spring can expand or contract dur-ing heating. All this depends on the secondand third fabrication steps.

What happens now if, for one reason oranother, the specimen is restrained to ex-hibit the shape-memory effect? For exam-ple, what happens if an expanded ring is


Figure 9-40. Temperature-actuated switch designed so that it opens or closes above a particular temperature.From “Shape Memory Alloys” by L. McDonald Shetky. Copyright (1979) by Scientific American, Inc. Allrights reserved.


fitted as a sleeve over a tube with an outerdiameter slightly smaller than the inner di-ameter of the expanded ring but larger thanthe inner diameter of the ring in the remem-bered position? On heating, the ring willstart to shrink as soon as the temperature As

is reached. While shrinking it will touchthe tube wall and further shrinking will behindered. From this moment, a compres-sive stress will be built up, clamping theshrinking ring around the tube. Obviously,the composition of the alloy should be suchthat the Ms temperature is below the valueat which clamping is required; in manycases, this is below room temperature. Forclamping rings it is therefore important thaton cooling back to room temperature theclamping stress is still present. This meansthat two-way memory must be avoided,which is easily achieved by choosing ashape-memory alloy that exhibits a largetemperature hysteresis.

On heating a shape-memory device,stresses can thus be built up and mechanicalwork can be done. The latter would be thecase if a compressed shape-memory springhas to lift a weight as in a shape-memoryactuated window opener (Fig. 9-41). Avery useful device is realized when ashape-memory device is used, as shown in Fig. 9-42, in combination with a biasspring made of a conventional linear elasticmaterial, both being clamped between twofixed walls and attached to each other witha plate. At temperatures below Mf , theshape-memory spring is closed and com-pressed by the bias spring. The SME springhad to be deformed, in this case com-pressed, in order to fit into the clampingunit. The clamping unit with the twosprings installed is now heated to tempera-tures above Af ; as soon as As is reached, theSME spring will start to expand and try topush back the bias spring. At Af , the shape-memory spring will not yet have regained

its original length, and further heating is re-quired to overcome the force exerted by thebias spring. At a certain temperature higherthan Af the shape-memory spring will befully recovered. This temperature will de-pend on the strength of the bias spring.During this temperature excursion, theplate that is fixed between the two springswill have moved and can, if an “engine” isattached to it, deliver work. If the clampingunit is now cooled, the bias spring will tryto compress the shape-memory spring into

Figure 9-41. A simple shape-memory windowopener made from a copper-based shape-memory al-loy. From “Shape Memory Alloys” by L. McDonaldShetky. Copyright (1979) by Scientific American,Inc. All rights reserved.

Figure 9-42. A mechanism in which a shape-mem-ory alloy (SMA) spring is used in conjunction with abias spring. From “Shape Memory Alloys” by L.McDonald Shetky. Copyright (1979) by ScientificAmerican, Inc. All rights reserved.

its deformed position. The elastic energythat has been stored in the bias spring dur-ing the heating cycle is now released, al-lowing the plate to perform work also dur-ing the cooling cycle. To describe fullysuch a working performing cycle, a ther-modynamic treatment is needed (Wollantset al., 1979). The working performing cycle can best be illustrated by taking ashape-memory spring that expands or con-tracts during heating or cooling and thatcarries a load. The working performing cy-cle can then be represented in a displace-ment– temperature, a stress– temperature,or an entropy–temperature diagram.

Although the shape-memory effect hasbeen observed in many alloy systems, onlythree systems are commercially available,

mainly because of economic factors andthe reliability of the material. The three al-loy systems are Ni–Ti, Cu–Zn–Al andCu–Al–Ni. Generally, other elements areadded in small amounts (of the order of afew weight %) in order to modify the trans-formation temperatures or to improve themechanical properties or the phase stabil-ity. In all three cases the martensite is thermoelastic. Maki and Tamura (1987) reviewed the shape-memory effect in fer-rous alloys, where a non-thermoelasticFe–Mn–Si alloy has also been found toshow a shape-memory effect, and commer-cialization is being considered. The mostimportant properties of shape-memory al-loys are summarized in Fig. 9-43, in whichthe working temperatures, the width of the


Figure 9-43. Schematic representation of the most relevant shape-memory properties (courtesy Van Hum-beeck, 1989).


hysteresis, and the maximum recoverablestrain are given. Because of the superiormechanical, chemical and shape-memoryproperties of Ni–Ti alloys, this alloy sys-tem has been applied most successfully;about 90% of the present applications usethese alloys. Owing to the continuous im-provement of the properties of Cu-basedalloys, together with their lower price, CuSME alloys have been successfully used inseveral applications.

The commercial applications of shape-memory devices can be divided into fourgroups:

1. motion: by free recovery during heat-ing and/or cooling;

2. stress: by constrained recovery duringheating and/or cooling;

3. work: by displacing a force, e.g., inactuators;

4. energy storage: by pseudoelastic load-ing of the specimen.

Shape-memory effects have also beenreported in non-martensitic system, e.g., in ferroelectric ceramics (Kimura et al.,1981), and have found applications as micro-positioning elements (Lemons andColdren, 1978). The shape change is attrib-

uted here to domain-wall motion, as shownin Fig. 9-44.

9.10.3 High Damping Capacity

The hysteresis exhibited during a pseu-doelastic loading and unloading cycle is ameasure of the damping capacity of a vi-brating device fabricated from a shape-memory material, which is cycling underextreme stress conditions exceeding thecritical stress needed to induce martensiteby stress. Vibrating fully martensitic sam-ples also exhibit high damping. A fullymartensitic sample consists of a large number of differently crystallographicallyoriented domains whose domain boundar-ies are mobile. Under the action of an ap-plied stress these boundaries move but, be-cause of friction, energy is lost during thismovement. If a cyclic stress is applied, thisforeward and backward boundary move-ment will lead to damping of the vibration.Comparing the amount of this dampingwith the damping that we observe in othernon-SME alloy systems, it is found that thedamping capacity of martensitic shape-memory alloys is one of the highest. Theshape-memory alloys are said to belong tothe high-damping materials, the so-calledHIDAMETS.

9.10.4 TRIP Effect

TRIP is the acronym for TRansforma-tion-Induced Plasticity and occurs in somehigh-strength metastable austenitic steelsexhibiting enhanced uniform ductilitywhen plastically deformed. This uniformmacroscopic strain, up to 100% elongation,accompanies the deformation-inducedmartensitic transformation and arises froma plastic accommodation process aroundthe martensite plates. This macroscopicstrain thus contrasts with that occurring in

Figure 9-44. Schematic illustration of the mecha-nism for an electronic micro-positioning (Lemonsand Coldren, 1978).

shape-memory alloys in being unrecover-able.

TRIP has been extensively studied byOlson and Cohen (1982a) and we will fol-low their approach here. They distinguishtwo modes of deformation-induced trans-formation, according to the origin of thenucleation sites for the martensite plates: “stress-assisted” and “strain-induced” trans-formation. The condition under which eachmode can operate is indicated in a tempera-ture–stress diagram as shown in Fig. 9-45.At temperatures slightly higher than Ms

s,the stress required for stress-assisted nucle-ation on the same nucleation sites followsthe line AB. At B, the yield point for slip in the parent phase is reached, defining the highest temperature Ms

s for which thetransformation can be induced solely byelastic stresses. Above this temperature,plastic flow occurs before martensite canbe induced by stress. New strain-inducednucleation sites are formed, contributing tothe kinetics of the transformation. The

stresses at which this strain-induced mar-tensite is first detected follows the curveBD. At point D, fracture occurs and thusdetermines the highest temperature Md atwhich martensite can be mechanically in-duced.

When the transformation occurs at tem-peratures below Ms

s, the plastic strain isdue entirely to transformation plasticity re-sulting from the formation of preferentialmartensite variants. The volume of the in-duced martensite is therefore linearly re-lated to the strain. The existing nucleationsites are aided mechanically by the thermo-dynamic contribution of the applied stress,reducing the chemical driving force for nucleation. Above Ms

s, the relationshipbetween strain and volume of martensitebecomes more complex, because strain is


Figure 9-46. Transformation-induced plasticity intensile tests at various temperatures (Fe–29 wt.%Ni–0.26 wt.% C) (Tamura et al., 1969).

Figure 9-45. Idealized stress-assisted and strain-in-duced regimes for mechanically-induced nucleation(Olson and Cohen, 1982a).

9.11 Recent Progress in the Understanding of Martensitic Transformations 649

then a result of plastic deformation of theparent phase and of transformation plastic-ity. Strain hardening and enhancement ofnucleation of martensite also play an essen-tial role. When martensite is formed duringtensile deformation, the strain hardeningbecomes large. Necking is then expected to be suppressed, explaining the enhanceduniform elongation. Fig. 9-46 shows, as anexample, the amount of martensite, the elongation and the ultimate strength meas-ured after tensile tests of a TRIP steel as afunction of temperature, clearly illustratingthe enhanced elongation, especially in thetemperature range between Ms

s and Md.Such a large elongation (sometimes over200%) can also be produced by subjectinga TRIP steel specimen under constant loadto thermal cycles through the transforma-tion temperature.

9.11 Recent Progress in theUnderstanding of MartensiticTransformations

We draw attention here to some recentpapers that illustrate recent progress in theunderstanding of martensite, and in which

some new approaches are also explained.Most of the information referred to in thissection was presented at the most recentICOMAT international conference on mar-tensitic transformations held in 1998 atBariloche (Ahlers et al., 1999).

New directions in martensite theory arepresented by Olson (1999). The nucleationof martensite, the growth of a single mar-tensite plate, the formation of, for example,self-accommodating groups of martensiteplates, and this within single crystals of theparent phase as well as in polycrystallinematerial, and the constraints dictated by thecomponents where martensitic materialsare only one (maybe the most important)functional element of the component, areall influenced by different interactive lev-els of structures (ranging from solute atomsto components). Nucleation is the first stepin martensite life, and a component whosefunctional properties are attributed to thoseof martensite, can be considered the finalstep. Olson (1999) constructed a flow-block diagram in which the martensitictransformation is situated in a multileveldynamic system. This new system, shownin Fig. 9-47, and the one given in Fig. 9-1,offer powerful tools for a better under-

Figure 9-47. The flow-block diagramof martensitic transformation as amultilevel dynamic system (Olson,1999).

standing of diffusionless phase transforma-tions. Its use should lead to a better designof martensitic and bainitic alloys meetingspecific requirements. Close analysis of thepapers presented at ICOMAT 98 shows thatsuch an approach can already be found inmany papers.

Special attention is given to the influ-ence of external constraints, such as hydro-static pressure, the application of a mag-netic field, and to martensite formed in thinfilms prepared by either sputtering or rapidsolidification. Kakeshita et al. (1999) stud-ied the influence of hydrostatic pressure in-stead of uniaxial stress, in order to formu-late a thermodynamic approach for a betterunderstanding of the nucleation of marten-site. The strengthening mechanisms insteel due to martensite are reviewed, thediffusion of carbon in the various states(according to the dynamic system of Fig. 9-47) of martensite is highlighted. In thiscontext, the fracture mechanism is relatedto the tempering temperature and the car-bon diffusion.

As is commonly known, the mechanismof bainite transformation is a subject withmany unresolved issues. Bhadeshia (1999)gives an overview of the transformationmechanisms proposed to explain “amongothers” the growth of bainite. The develop-ment of bainite at both high temperatures (upper bainite) and low temperatures (lower bainite) is discussed and is illustrat-ed in Fig. 9-48. According to Bhadeshia,the unresolved issues are:

the growth rate of an individual bainiteplatea theory explaining the kinetics to esti-mate the volume fraction of bainite inaustenite obtained during an isothermaltransformationthe modeling describing quantitativelythe formation of carbidesand a number of features associated withthe interaction between plastic deforma-tion and bainite formation.

The influence of carbon on the bainitictransformation is treated in great detail andis shown to be a controlling factor of themechanical properties of different multi-phase TRIP-assisted steels (Girault et al.,1999; Jacques et al., 1999).

The martensitic transformation inFe–Mn-based alloys is treated in variouspapers, showing the increasing interest indeveloping ferrous shape-memory alloys.In these alloys, austenite transforms eitherinto a h.c.p. e-phase (g Æ e) or/and into a¢-martensite (g Æ a¢).

New approaches and strategies are dis-cussed for the application of shape-mem-ory alloys in non-medical (Van Humbeeck,1999) as well as in medical applications(Duerig et al., 1999). Only two examplesare shown here. The first example (Fig. 9-49) shows that Ni–Ti superelastic alloysimprove significantly the cavitation ero-sion resistance if compared with marten-


Figure 9-48. A schematic representation of themechanism explaining the growth and developmentof bainite (Bhadeshia, 1999).

9.12 Acknowledgements 651

sitic Ni–Ti. But it should be remarked thatthis figure is only an enlargement of a fig-ure giving an overall view of the cavitationresistance of other common alloys. For ex-ample, the weight loss after 10 h is already20–30 mg for the “common” alloys incomparison with the negligible weight lossof both Ni–Ti shape-memory alloys after10 h tested under the same conditions. Avery impressive example of the applicationof Ni–Ti shape-memory alloys is given inFig. 9-50. This figure shows an atrial septalocclusion device with Nitinol (Ni–Tishape-memory alloy) wires incorporated ina sheet of polyurethane. This device allowsholes in the atrial wall of the heart to beclosed without surgery. The two umbrella-like devices are folded in two catheters,which are placed on either side of the hole. Once the two folded umbrellas arewithdrawn from their catheters, they arescrewed together in such a way that thehole is closed. Because of the flexibility ofboth materials, the heart can again beatnormally. This device illustrates the con-cept of the elastic development capacity of shape-memory alloys. Because Ni–Tishape-memory alloys have proposed to bebiocompatible (see Van Humbeeck, 1977),many applications of these Ni–Ti alloys

are currently being developed and mar-keted.

9.12 Acknowledgements

The author would like to thank M. Ah-lers, J. W. Christian, M. De Graef, R. Gott-hardt, P. Haasen, H. S. Hsu, J. Ortín, K. Ot-suka, J. Van Humbeeck and P. Wollants forsupport and advice while preparing themanuscript, and M. Van Eylen, M. Nol-mans, H. Schmidt and K. Delaey for theirassistance. The “Nationaal Fonds voor We-tenschappelijk Onderzoek” of Belgium isacknowledged for financial support (pro-ject No. 2.00.86.87). The author especiallyacknowledges the continuing interest andencouragement he received from A. De-ruyttere. For help in preparing the revisedversion, I would like to thank M. Chandra-sekaran.

Figure 9-49. The weight loss of a martensitic (NiTi–1) and a pseudoelastic (Ni Ti–2) Ni–Ti shape-memory alloy (Richman et al., 1994).

Figure 9-50. A shape-memory device for repairingdefects in the heart wall (Duerig et al., 1999).

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Proceedingsof International Conferenceson Martensitic Transformations

“Physical Properties of Martensite and Bainite”(1965), Special Report 93. London: The Iron andSteel Institute.

“Shape Memory Effects in Alloys” (1975), Perkins,J. (Ed.). New York: Plenum Press.

“New Aspects of Martensitic Transformations”(1976), Suppl. to Trans. Jap. Inst. Met. 17.

“Martensitic Transformations ICOMAT ’77” (1978),Kiev: Academy of Science.

“International Conference on Martensitic Transfor-mations ICOMAT ’79” (1979), Cambridge (MA):Dep. of Mat. Science and Techn., M.I.T.

“International Conference on Martensitic Transfor-mations ICOMAT ’82” (1982), Delaey, L., Chan-drasekaran, M. (Eds.), Journal de Physique 43,Conf. C-4.

“International Conference on Shape Memory Al-loys” (1986), Youyi Chu, Hsu, T. Y., Ko. T. (Eds.).Beijing: China Academic Publishers.

“International Conference on Martensitic Transfor-mations ICOMAT ’86” (1987), Tamura, I. (Ed.).Sendai: The Japan Inst. of Metals.

“The Martensitic Transformation in Science andTechnology” (1989), Hornbogen, E., Jost, N.(Eds.). Oberursel: DGM-InformationsgesellschaftVerlag.

“Shape Memory Materials” (1989), Otsuka, K., Shi-mizu, K. (Eds.), Proceedings of the MRS Interna-tional Meeting on Advanced Materials, vol. 9.Pittsburgh: Mat. Research Soc.

“International Conference on Martensitic Transfor-mations ICOMAT ’89” (1990), Morton, A. J.,Dunne, B., Kelly, P. M., Kennon, N. (Eds.) to bepublished.

“International Conference on Martensitic Transfor-mations ICOMAT ’92” (1993), Wayman, C. M.,Perkins, J. (Eds.). Carmel, CA, USA: Monterey In-stitute of Advanced Studies.

“International Symposium and Exhibition on ShapeMemory Materials (SME ’99)” (2000). Zurich:Trans. Tech. Publications.

“International Conference on Shape Memory andSuperelastic Technologies (SMST ’97)” (1997),Pelton, A., Hodgson, D., Russell, S., Duerig, T.(Eds.). Santa Clara, CA, USA: SMST Proceedings.

“Displacive Phase Transformations and Their Appli-cations in Materials Engineering” (1998), Inoue,K., Mukherjee, K., Otsuka, K., Chen, H. (Eds.).Warrendale, OH, USA: TMS Publications.

“European Symposium on Martensitic Transforma-tions ESOMAT ’97” (1997), Beyer, J., Böttger, A.,Mulder, J. H. (Eds.). EDP-Sciences Press.

“European Symposium on Martensitic Transforma-tions ESOMAT ’94” (1995), Planes, A., Ortin, J.,Lluis Manosa. Les Ulis, France: Les Editions dePhysique.

“International Conference on Martensitic Transfor-mations ICOMAT ’98” (1999), Ahlers, M., Kos-torz, G., Sade, M. (Eds.). Amsterdam: Elsevier.

“Martensitic Transformation and Martensite” (1980),Hsu, T. Y. (Xu Zuyao) (Ed.). Shanghai: SciencePress (in Chinese).

“Pseudoelasticity and Stress-Induced MartensiticTransformations” (1977), Otsuka, K., Wayman, C.M. (Eds.), in: “Reviews on the Deformation Be-havior of Materials”.


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