Materials Science and Engineering A 378 (2004) 459–464

Mechanistic simulation of deformation-induced martensite stabilisation

Yinong Liu∗

School of Mechanical Engineering, The University of Western Australia, Crawley, WA 6009, Australia

Received 22 April 2003; received in revised form 26 September 2003

Abstract

It is known that deformation of shape memory alloys, via either stress-induced martensitic transformation or martensite reorientation, causesstabilisation to the deformed martensite, as manifested in the increase of the critical temperature for the reverse transformation. Whereasa number of hypotheses have been proposed in the literature to explain this phenomenon, an actual mechanism by which the effect can bedemonstrated is yet to be established. This paper presents a heuristic mechanistic spring–slide model based on the thermodynamic conceptof elastic and frictional energies for thermoelastic martensitic transformations and plasticity concept of grain interior and grain boundaryphases. The model is able to demonstrate the alterations to the elastic and irreversible energies caused by deformation and the effects of thesealterations in causing the stabilisation of martensite.© 2004 Elsevier B.V. All rights reserved.

Keywords: Martensitic transformation; Martensite stabilisation; Shape memory effect; Spring–dashpot model

1. Introduction

It is known that in thermoelastic martensitic transforma-tion systems deformation via either martensite reorientationor stress-induced martensitic transformation causes stabili-sation to the martensite, as manifested in the increase of thecritical temperature for the reverse transformation of the de-formed martensite[1–6]. Accompanying the increase of thereverse transformation temperature, other effects are alsoobserved, including the restoration of transformation tem-peratures and the occurrence of two-way memory effect insubsequent transformation cycles[3,4]. To explain these ef-fects, a number of hypotheses have been proposed in the lit-erature for the mechanisms responsible for this phenomenon,including pinning effect of deformation-induced defects[1], destruction of transformation interfaces[6], and releaseof internal elastic energies[2]. From a thermodynamicviewpoint, the former two explanations can be regardedincreases of irreversible energies for the transformation.Whereas it has been generally regarded that the stabilisationeffect is related to alterations to the elastic and irreversibleenergies caused by the deformation, an actual mechanismby which the effects of these energies on the transformation

∗ Tel.: +61-8-9380-3132; fax:+61-8-9380-1024.E-mail address: liu@mech.uwa.edu.au (Y. Liu).

can be demonstrated is yet to be established. This paper pro-poses a heuristic model based on the spring–slide conceptto describe this effect. This model considers a simplifiedmechanistic system comprising a two-variant pair withina grain constrained by the grain boundary. The model isfound to be able to demonstrate the increase of the criticaltemperature for the reverse transformation after deforma-tion, and other associated effects, including the restorationof transformation temperatures in subsequent cycles, thereduction of transformation interval for the first reversionafter deformation, the expansion of transformation intervalin the subsequent cycles, and the occurrence of two-waymemory effect.

2. Experimental observations

Fig. 1shows the thermal dilatation measurements (TMA)of two specimens of Ti–50.0 at.% Ni. Specimen (a) was inthe as-annealed state. Specimen (b) had been deformed intension via martensite reorientation at room temperature tojust beyond the end of the stress plateau. The stress–straincurve of the pre-deformation is shown in the inset. Thetransformation behaviour after the deformation was mea-sured in a thermal sequence of heating–cooling–heating. Itis seen that after unloading a residual strain of 5.6% was

0921-5093/$ – see front matter © 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.msea.2003.10.339

460 Y. Liu / Materials Science and Engineering A 378 (2004) 459–464

0

100

200

300

400

0 2 4 6 8 10

Str

ess

(M

Pa)

Strain (%)

-7

-6

-5

-4

-3

-2

-1

0

200 250 300 350 400 450 500

Str

ain

(%

)

Temperature (K)

Y scale enlarged 10 times

(a) undeformed

(b) 6.2% tensile strain

(b)

two-way

memory effect

first reverse

transformation

Fig. 1. Thermal dilatation measurement of transformation behaviour aftertensile deformation via martensite reorientation.

retained. Upon heating, the reoriented martensite revertedback to austenite at a higher temperature compared to thereverse transformation on the second heating, with a re-duced transformation temperature interval. Associated withthe first reverse transformation a strain recovery of 4.9%was recorded, leaving behind∼0.7% permanent deforma-tion. In the subsequent thermal transformation cycle thespecimen exhibited a two-way memory effect of 1.5%. Theforward and the reverse transformations occurred at similartemperatures to that of the undeformed specimen.

Fig. 2 shows DSC measurements of the transformationbehaviour of two specimens of Ti–50.15 at.% Ni. Both spec-imens were annealed at 978 K. Specimen (a) was in theas-annealed state. Specimen (b) had been deformed in shearvia martensite reorientation to 6% prior to the measure-ment. The curves shown are actually integrated differen-tial calorimetry curves, with the original differential curvesshown in the inset. It is seen that the first reverse transfor-

260 280 300 320 340 360 380 400

Temperature (K)

250 300 350 400

Temperature (K)

0%

6.2%

1st h

eatin

g

2n

d h

eatin

g

co

olin

g

co

olin

g

heatin

g

(b) 6.2% tensile strain

(a) as-annealed

heat

flo

w

Mart

ensi

te F

racti

on

0

1

Fig. 2. Thermal dilatation measurement of transformation behaviour aftertensile deformation via martensite reorientation.

mation of the deformed martensite occurred at a higher tem-perature, consistent with the TMA measurement shown inFig. 1. In addition, it is also apparent that the transforma-tion interval of the reverse transformation of the deformedmartensite on the first heating decreased whereas that in thesecond thermal transformation cycle increased relative tothat of the undeformed specimen[5,6].

3. The model

3.1. The physical model

During martensite reorientation deformation, martensitevariant structure changes from a self-accommodation con-figuration to an oriented state. It is generally perceived thatreorientation deformation by a uniaxial stress produces ul-timately a single variant that is most favourably orientedrelative to the uniaxial stress in each grain. However, sucha reorientation deformation is practically impossible in apolycrystalline matrix without internal plastic deformationto co-ordinate for the mismatch in orientation among thefavoured variants in neighbouring grains[3,5,7]. This im-plies that, in addition to frictional resistance to phase/variantboundary movement, which is universal for both thermallyinduced transformation and mechanically induced trans-formation (or reorientation), there also exists a mechanicalresistance to global deformation in a polycrystalline matrix[8]. The effect, or the existence, of this resistance can beexplained as following. In the absence of this resistance, itwould be expected that only an infinitesimal external forcebe required to trigger a stress-induced martensitic transfor-mation in a polycrystalline specimen at a temperature justaboveMs, when the chemical resistance to the transforma-tion is practically zero (taking into account of the frictionalresistance to phase boundary movement). This, obviously,has never been observed. In fact, experimental evidenceshave shown that a finite external stress is required to inducea martensitic transformation for polycrystalline NiTi atMs[9,10]. Based on this understanding, from a mechanicalviewpoint, a polycrystalline aggregate can be modelled asshown inFig. 3, in which a network of finite volume is cre-ated to represent the “grain boundaries” or “grain boundaryaffected regions”. This is similar to the simulation of “grainboundary phase” and “grain interior phase” in the discus-sion of polycrystalline plasticity[11,12]. The interior ofthe cells (grains) represents the transforming/reorientationbody, which is capable of multiple-variant shape distortion.The grain boundary regions impose resistance to globalshape change and thus experience plastic deformation dur-ing stress-induced martensitic transformation and martensitereorientation deformation processes. It needs to be clari-fied that the width of the grain boundary affected regions(GBAR) is dependent on the nature of the deformation. Forplastic deformation, the width is compatible to the need toaccommodate dislocation activities and thus is generally

Y. Liu / Materials Science and Engineering A 378 (2004) 459–464 461

Fig. 3. Schematic illustration of polycrystalline aggregate of grains ofpreferential variants of various orientations.

significant only for ultrafine structures and nanocrystallinematrices. For deformation via stress-induced martensitictransformation or martensite reorientation, where the lat-tice distortion is large (up to 9%) and the physical size ofmartensite variant domains is large, the width is also ex-pected to be large. This also naturally extends to that thewidth of the GBARs for an isotropic matrix, where deforma-tion mismatch between grains is large, is greater than that inpreferentially textured structures where grains deform co-ordinatively in the same direction. Thus, during a thermallyinduced transformation where no global shape change isinvolved the boundary regions are ineffective (virtually ofzero width) and the transformation operates within the in-terior of the cells (grains). During a stress-induced process,via either a phase transformation or variant reorientation,both the boundary regions and the interior are active. It mayalso be postulated that during mechanical cycling, via eitherstress-induced martensitic transformation (pseudoelastic cy-cling) or martensite reorientation (ferroelastic cycling), thecritical stresses decrease in subsequent cycles than in thefirst, due to a reduction of the effective width of the GBARs.

3.2. The mechanistic model

The above concept can be expressed, with respect to thelocal environment indicated by the rectangle inFig. 3, us-ing a twin-variant mechanistic model shown inFig. 4 (state(1)). The model is based on the slide–dashpot concept, re-placing the time-dependent dashpot with a time-independentslide to simulate the dissipative, or hysteretic, aspects of themartensitic transformation. Such concept has been deployedpreviously to simulate thermal and mechanical hysteretic be-

haviour of thermoelastic martensitic transformations[13,14]and magnetic hysteretic behaviour of ferromagnetic materi-als. This scheme is effective as a general tool in describingthe so-called “elastohysteretic” phenomena that can be ex-pressed in thermodynamic context using reversible and irre-versible energy contributions.

The model consists of six elements. Element A repre-sents a finite unit that transforms to one martensite vari-ant upon cooling. This unit is accompanied by a frictionalslide (element B) and by a spring (element C). The fric-tional slide accounts for the internal resistance to transfor-mation/reorientation boundary movement. The slide has alimited sliding distance determined by the lattice distortionof the transformation. The spring expresses the elastic aspectof the transformation. Assembly A–B–C expresses a trans-formation unit. In this model two identical transformationunits are connected in parallel to form the transformationsegment. Outside the transformation segment there is an-other slide (element D) and another spring (element E). Theexternal spring expresses the elasticity of the GBARs. Theexternal slide expresses the frictional resistance to globalshape change imposed by the boundary regions, or plasticityof the GBARs experienced during stress-induced marten-sitic transformation or variant reorientation. It ought to bepointed out that this plasticity is in addition to the plasticityof the interior of grains, as defined in the strict sense of dislo-cation movement and dislocation production of martensite.The plasticity of the martensite is expressed by frictionalslide F. It is deemed that the friction on slide F is greaterthan the friction on either slide B or slide D, to allow marten-site reorientation prior to plastic deformation. It is clear bythese definitions that slide B is operative for martensite re-orientation in single crystals, slide B and slide D are effec-tive for martensite reorientation in a polycrystalline matrix,and slides B, D and F are operative for plastic deformationof oriented martensite in both single crystal and polycrys-talline matrices. The characteristic frictions on the slides areindicated next to each slide. The frictions on all three slidesare subjected to strain hardening when plastic deformationoccurs. It needs to be pointed out that having surrenderedits elastic and frictional aspects to the internal spring andthe internal slide and its plasticity to slide F, element A be-comes symbolic of the transforming unit preserving only itsgeometric attributes.

4. Thermomechanical behaviour of the mechanisticmodel

State (1) expresses the condition of austenite. In austen-ite, all springs are relaxed and no frictional force is createdon the slides. Upon cooling, the two units transform to twovariants of martensite oriented in opposite directions to rep-resent the self-accommodation structure of the martensite,as shown in state (2). Each variant produces a local deforma-tion corresponding to the lattice distortion of the martensite.

462 Y. Liu / Materials Science and Engineering A 378 (2004) 459–464

State

Thermally formed

self-accommodating

martensite

(2)ff=

S

fe= S

fe=S

ff=

SM

M

Austenite

C1

hg

D(2Sd)

E

C

B1(Str,Stw)

(1)

(3)ff=2S

Reoriented

martensite

Austenite formed

from reoriented

martensite by

heating (4)

A1

A2

fe=S

fe=S

(5)

fe=S

fe=S

fe=St

fe=St

dres

dtw

ff=2S

ff=2S

M

M

fe=S

fe=SM

M

A1

A2

dre

F1(Sdm)

Martensite

formed in

subsequent cycle

showing two-way

memory effect

B2(Str,Stw)F2(Sdm)

Fig. 4. Twin-variant mechanistic model of thermoelastic martensite in polycrystalline shape memory alloys based on the spring–slide concept.

The local deformations of the two variants are cancelledin the self-accommodating structure, giving a net zero dis-placement at location g. Within the transformation segmenta frictional force,ff = Str, is experienced by each unit dur-ing the process of transformation, whereStr is the frictionalresistance to the displacive movement of transformationboundaries. At the same time, an elastic force,fe = Sv, isgradually accumulated during the formation of martensitevariants (the original length of the springs are marked atopthe springs as reference to mark the states of the elasticforces). At the end of the forward transformation when thechemical driving force from the variants have vanished, thefrictional forces on the slides become the reactions to the in-ternal elastic forces stored in the internal springs and, hence,limit the magnitudes of the elastic forces, i.e.fe = ff = Sv.Clearly, Sv is limited by the critical force required to form

a new variant or for the movement of martensite varianttwin boundaries. Upon heating, these internal elastic forcesact as assistance to the reverse movement of the slides.It is seen that during the forward transformation both theelastic forces and the frictional forces resist the formationof a variant whereas during the reverse transformation, theelastic forces serve as the driving force and the frictionalforce as the resistance. According to the above analysis,it is obvious that the critical driving forces for the ther-mally induced martensitic transformation can be expressedas

forward transformation atT = Ms : Fth(Ms) = Str (1)

forward transformation atT = Mf : Fth(Mf ) = Str + Sv

(2)

Y. Liu / Materials Science and Engineering A 378 (2004) 459–464 463

reverse transformation atT = As : Fth(As) = Str − Sv

(3)

reverse transformation atT = Af : Fth(Af ) = Str (4)

Applying a force at point h (to the left) to the thermallyformed martensite shown in state (2) leads to martensitereorientation. The applied force acts to elongate the systemin three stages. It initially creates an elastic force in externalspring E, prior to the activation of external slide D (state(1)). Following that the compressive elastic force in springC1 is relaxed whilst that in C2 increased, during which allthree springs are activated. This process continues till thatthe elastic force in C2 reaches the friction on internal slideB2,Stw, and the reorientation of M2 starts, holding the elasticfor in C2 at constant whilst continuing to increase the elasticforce in C1, i.e. two springs are in action. These descriptioncan be mathematically expressed as following:

Fre = EiX (i = 1, 2, 3) (5)

whereEi is the apparent value of dF/dX of stagei and

E1 = 1

2Ee, E2 = EeEc

Ee + 2Ecand

E3 = kEeEc

2(Ee + Ec)(6)

whereEe is the stiffness of external spring E,Ec is the stiff-ness of internal springs C1 and C2, andk is a constant lessthan unity. It is understood that in reality the majority of thetransformation lattice distortion is accommodated inelasti-cally in the self-accommodation structure of the variants andonly a small fraction of the distortion is accommodated aselastic strain that causes internal elastic stresses. To accountfor this,k is included inE3 to expresses the “effective” stiff-ness of the system during reorientation. The critical forces(per variant) between the stages of the martensite reorienta-tion deformation process are expressed as following:

start of external slide : Fre = Sd (7)

start of M2 reorientation : Fre = Sd + Stw − Sv (8)

end of M2 reorientation : Fre = Sd + Stw (9)

start of reverse of external slide :Fre = Stw − Sd (10)

The condition at the end of the reorientation process, whenthe applied force is removed, is expressed in state (3),in which the frictional force on slide D, 2Sd, is the re-action of the elastic forces in the internal springs. It isobvious that Sd expresses the mechanical resistance totransformation/reorientation induced deformation of thepolycrystalline matrix and thatSd > Sv, otherwise globaldeformation would occur in preference to the formation ofself-accommodating martensite during a thermally inducedtransformation.

Heating of the reoriented martensite induces the reversetransformation. It is seen that the reverse transformation of

the oriented martensite (expressed in state (3)) is subjectedto the resistance of the frictions on both the internal and theexternal slides. Comparing with state (2) for the thermallyformed martensite, it is clear that the reverse transforma-tion of the oriented martensite requires an increased drivingforce:

reversion of oriented martensite :F1stth = Str = Sd (11)

This implies a stabilisation effect. The condition at the endof the reverse transformation of the oriented martensite is ex-pressed in state (4). Comparing with state (1) of the originalaustenite, it is seen that this reformed austenite is differentin two aspects, (i) the occurrence of a residual deformation,dr, and the creation of the internal elastic stresses,fe = Sd,which are aligned in the same direction as the original de-formation.

These directional internal stresses guide the selection ofmartensite variants in the subsequent thermal transformationcycles, resulting in the following transformation sequence.Martensite variants form in the same direction; internalsprings are relaxed and then compressed; the internal elasticforces increase and reach the (reduced) friction on the exter-nal slide, causing external deformation (two-way memoryeffect); the friction on the external slide increases with de-gree of deformation toSv, the critical force required to formnew variants; variants other than the preferential orientationare formed, resulting in reduced strain for the TWME. Thesituation at the end of the process is expressed in state (5).Driving forces required for this process may be expressed as

start of forward transformation : Fth(M∗s ) = Str − Sd

(12)

end of forward transformation : Fth(M∗f ) = Str + Sv

(13)

start of reverse transformation :Fth(A∗s) = Str − Sv (14)

end of reverse transformation :Fth(A∗f ) = Str + Sd (15)

The thermomechanical behaviour of the model, as discussedabove, is shown inFig. 5, including the force–displacementbehaviour of deformation by martensite reorientation, ther-mal recovery of the reoriented martensite, thermal transfor-mations in the subsequent cycle and thermal transformationcycle of virgin specimen prior to the deformation (thedashed lines). The numbers in parentheses correspond tothe numbers of the equations in the text. The apparent mod-uli of different deformation stages, as defined inEq. (6)arealso indicated in the figure. It is seen that model resemblesthe experimental observations. The specific features may besummarised as following:

• A continuous increase of stress is required for martensitereorientation deformation in polycrystalline matrices.

• Reoriented martensite induced by deformation is morestable than the self-accommodating martensite formedthermally.

464 Y. Liu / Materials Science and Engineering A 378 (2004) 459–464

(8)

(9)F

X

(7)

fm

T

(11)

(10)

(4)

(1)(2)(13)

(3)(14)

(12)

(15)

E1

E2

E3

E2

E1

(2)

Fig. 5. Thermomechanical behaviour of the mechanistic model.

• The transformation interval is reduced for the first reversetransformation of the reoriented martensite and enlargedfor the thermally induced transformations in subsequentcycles relative to that of the undeformed specimen.

• Two-way memory effect is induced after the simple de-formation via martensite reorientation.

At the conclusion of this discussion, which has beenlimited to the minimum configuration of the twinnedspring–slider pairs as the basic unit permitting martensitevariant self-accommodation and reorientation, it is of fur-ther benefit to make the following comments. A real poly-crystalline body is regarded to contain an entire spectrumof such units of varying mechanistic characteristics, such asresolved shear stresses and specific stiffness with respect toan applied stress. This requires an array of such basic unitsbeing assembled together to simulate more realistically thebehaviour of real polycrystalline materials, as expected.In the literature, double twinned configurations have beenused[14]. As shown inFig. 3, the twinned unit used in thisdiscussion can be regarded a semi-infinite solid system. Thedouble twinned configuration simply includes the other halfof the system, stretching across the grain boundary, and thusdoes not lead to further improvement to the performance ofthe current configuration. Configurations of multiple unitsof varying mechanistic characteristics have also been used[13]. This allows the smoothening and “round-up” of the“straight line” behaviour of the current model into the morefamiliar shapes of the hysteresis loops of the transforma-tion. This further development, whereas improving plausi-bly the appearance of the thermomechanical behaviour of

the model, does not contribute to enhancing the workingprinciple of the model. Thus, such attempt is omitted in thisdiscussion.

5. Conclusions

(1) Based on the principle of elastic and irreversible ener-gies, the model is able to reproduce, in a qualitative man-ner, the experimentally observed thermal transformationbehaviour and the deformation behaviour via martensitereorientation of polycrystalline shape memory alloys.

(2) The model demonstrates the complex phenomenon ofdeformation-induced martensite stabilisation, including(i) the increase of the critical temperature for the reversetransformation of deformed martensite, (ii) the restora-tion of the transformation temperatures after the firstreversion, (iii) alterations to transformation intervals in-duced by the deformation, and (iv) the occurrence oftwo-way memory effect in subsequent thermal trans-formation cycles. The model identifies that changes ofboth elastic and irreversible energies contribute to thedeformation-induced stabilisation of martensite in poly-crystalline alloys.

(3) The model also demonstrates that a directional internalelastic stress field in the direction of the pre-deformationis developed in the austenite after the first reversetransformation of the deformed martensite, causing thetwo-way memory effect in subsequent thermal trans-formation cycles.

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