ub

Lo

Ala

23e 1

andre bfoelln an

the inuence of the microstructure on the mechanical response and investigate how the stressstrain behavior changes as a function ofstrain rate.

transition is usually rst order and is accompanied by aspontaneous strain. The martensitic transition is also

temperature that is higher than the austenite nish temper-ature of the material. In pseudoelastic deformation, a hightemperature cubic austenite phase typically transforms to

The martensitic transformation results in the formationof a complex microstructure consisting of twin boundaries

ants [2,3]. The motion of the domain walls can inuence thestrain rate dependence of the mechanical response and thusit is important to incorporate this aspect in a theoreticalframework. In the present paper, we study theoreticallythe role of the microstructural evolution on the mechanicalresponse of shape memory materials.

* Corresponding author. Tel.: +1 505 665 0419.E-mail address: txl@lanl.gov (T. Lookman).

Acta Materialia 54 (2006) 210921responsible for the shape memory eect, which refers tothe recovery by heating of an apparently permanent defor-mation undergone below a critical temperature [1]. Thisproperty makes shape memory materials suitable for alarge number of technological applications [2]. Anotherimportant property of shape memory materials is the so-called pseudoelastic behavior which arises due to a revers-ible stress/strain-induced martensitic transformation at a

between the crystallographic variants of the transforma-tion. This microstructure inuences the eective mechani-cal properties of these materials. For example, if a stressis applied to a material in the martensitic phase, there ismotion of the twin boundaries as the favored variants growat the expense of the unfavored variants [1]. Even duringpseudoelastic deformation, the strain-induced transforma-tion involves nucleation and growth of the martensitic vari- 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Shape memory alloys; Martensitic phase transformation; Microstructure; Modelling; Nucleation

1. Introduction

Shape memory alloys exhibit interesting mechanicalproperties due to a diusionless structural phase transfor-mation from a high temperature austenite phase (e.g.,cubic) to a low temperature martensite phase that is oftentetragonal, orthorhombic or monoclinic in structure. This

the martensite under deformation so that there is a plateauin the stressstrain curves. On removing the deformation,the material transforms back to the cubic austenite andthe deformation is recovered upon unloading. In fact, someshape memory materials can recover strains of up to 10%under tension, making these materials suitable for actuatorapplications [2].Dynamic strain loading of c

Rajeev Ahluwalia, Turab

Theoretical Division and Center for Nonlinear Studies, Los

Received 24 June 2005; received in revised formAvailable onlin

Abstract

We present three-dimensional simulations of the microstructuretetragonal transitions, using FePd as an example. The simulations astrain elds. The dynamics is simulated by force balance equationssipation function. Stressstrain properties in the pseudoelastic as wWe also study the eects of defect-induced heterogeneous nucleatio1359-6454/$30.00 2006 Acta Materialia Inc. Published by Elsevier Ltd. Alldoi:10.1016/j.actamat.2005.12.040ic to tetragonal martensites

okman *, Avadh Saxena

mos National Laboratory, Los Alamos, NM 87545, USA

December 2005; accepted 29 December 20053 March 2006

mechanical response of shape memory alloys undergoing cubic toased on a nonlinear elastic free-energy in terms of the appropriater the displacement elds with a damping term derived from a dis-as the shape memory regime are investigated using strain loading.d motion of twin boundaries during deformation. Thus, we probe

www.actamat-journals.com

20rights reserved.

ateA number of approaches have been used to model shapememory materials. These include energy minimizationtechniques that are subject to certain constraints thatensure the integrity of the lattice [1] and micro-mechanicalmodels that require prior knowledge of the habit planesand their volume fractions [4,5]. Such techniques have beensuccessful in estimating the recoverable strains for manymartensitic transformations. Over the past few years, con-tinuum models based on the GinzburgLandau approachhave also been used to study the mechanical response ofshape memory alloys [6,7]. The eect of microstructureon the eective stressstrain response has also been simu-lated using time-dependent GinzburgLandau approacheswith dynamics that is purely dissipational [810]. However,these studies do not consider the strain rate dependence ofthe stressstrain behavior. In the present paper, we use athree-dimensional displacement eld based dynamic modelthat includes inertial forces to study the eect of defectsand microstructure on the stressstrain properties of shapememory materials. We also investigate how microstruc-tural evolution inuences the strain rate dependence ofthe mechanical response.

The paper is organized as follows. Section 2 describesthe GinzburgLandau free energy for a cubic to tetragonaltransformation and Section 3 analyzes the properties of thehomogeneous part of the free energy. In Section 4, weintroduce a dynamic model for the displacement elds. Sec-tion 5 describes the simulation of microstructure during atemperature-induced cubic to tetragonal transformationusing the dynamic model. In Section 6, we describe oursimulations of stressstrain response in the pseudoelasticregime and in Section 7 we discuss the mechanical responsein the shape memory regime. Finally, we end the paper inSection 8 with a summary and discussion of our results.

2. Free energy functional for cubic to tetragonal transition

A free energy functional for cubic to tetragonal transfor-mations was proposed by Barsch and Krumhansl [11]. Thetheory is formulated in terms of symmetry adapted combi-nations of the components of the strain tensor. For the lin-earized strain tensor ij 12 ouioxj

oujoxi, the tetragonal

distortions are described by the deviatoric strains givenby e2 12p xx yy and e3 16p xx yy 2zz. The bulkstrain is e1 13p xx yy zz and the three shear strainsare denoted by e4 = xy, e5 = xz and e6 = yz. The appro-priate free energy functional is then written as

F Z

d~rftetragonal fgradient fbulk-shear fload; 1

where ftetragonal represents the free energy cost for a cubic totetragonal distortion and is given by

ftetragonal Ae22 e23 Be3e23 3e22 Ce22 e232. 2We note that the above functional form preserves the cubic

2110 R. Ahluwalia et al. / Acta Msymmetry as the strains e2, e3 transform appropriately un-der the three generators 3[111], 4[001] and 2[110] of thehigh symmetry cubic group. The origin of the cubic termlies in the threefold rotation. If the coecient of the har-monic term changes sign, e2, e3 take on non-zero values,thus transforming the cubic phase to the tetragonal phase.The coecient A is the deviatoric modulus expressed interms of the elastic constants as A = (C11 C12)/2. Thismodulus is temperature dependent and controls the cubicto tetragonal transition. The quantities B and C are theappropriate nonlinear elastic constants. As we have dis-cussed earlier, more than one crystallographic variantsare formed as a result of the transition. Thus, there are do-main walls between these variants and the energy cost asso-ciated with these variants is assumed to be

fgradient G2re22 re32; 3

where the coecient G can be obtained from phonon dis-persion curves or from experimental microstructure. Thetransformation can also result in bulk and shear strainsand the free energy cost associated with such deformationsis given by

fbulk-shear Abulk2

e1 E0e22 e232 Ashear2

e24 e25 e26.4

The quantities Abulk and Ashear are the bulk and shear mod-uli, respectively, that can be expressed in terms of the linearelastic constants. In particular, Abulk C112C123 andAshear = 4C44. Notice that the above form ensures thatthe transformation described by the deviatoric strains re-sults in non-zero e1 in equilibrium. Thus, the free energyalso incorporates volume changes associated with thetransformation. The volume change depends on the param-eter E0 that can be determined from the transformationstrains obtained from the lattice parameter data.

We rst rescale the free energy by introducing strainvariables

i ei=e0; 5where the condition F = 0 at s = 1 gives e0 = |B|/2C. Sincethe transformation is determined by a softening of thedeviatoric elastic constant A, we assume A to be tempera-ture dependent, i.e. A = A0s where A0 = B

2/4C is the valueof this elastic constant at the transformation start tempera-ture and s is a dimensionless temperature dened by

s 4AC=B2 T T c=T 0 T c; 6where T0 is the temperature at which the transformationstarts and Tc is the temperature at which the austenitephase becomes unstable. The rescaled free energy can bewritten aseF F =F 0Z

d~~r~f tetragonal ~f gradient ~f bulk-shear ~f load; 7

where F0 is given by

rialia 54 (2006) 21092120F 0 d2B4=16C3 d2A0e20. 8

E0 eO 0:47, where e1 3 . The valuesA0 = 1.97 1010 N/m2, Abulk = 19.23 1010 N/m2, Ashear

external stress contribution, rxxxx is added to the energyand the free energy is minimized to calculate the strains

Fig. 1. Stressstrain curves in the pseudoelastic regime calculated from thehomogeneous free energy fhomo in Eq. (13). The thick lines are the result ofstress loading and the thin lines are obtained by strain loading. The black,red and blue curves correspond to T = 313, 305, 300 K, respectively. (Forinterpretation of the references to color in this gure legend, the reader is

Fig. 2. Stressstrain curves in the shape memory regime calculated fromthe homogeneous free energy fhomo in Eq. (13). The black, red and bluecurves correspond to T = 297, 295, 290 K, respectively. (For interpreta-

aterialia 54 (2006) 21092120 2111= 28 1010 N/m2, T0 = 295 K and Tc = 270 K are takenfrom temperature-dependent elastic constant data [13].We then calculate the stressstrain curves for a uniaxialtensile loading (xx = applied) by minimizing the energyfhomo with respect to yy and zz, where

fhomo s22 23 B323 322 C22 232

eAbulk2

1 fE022 232 13as the shear strains xy, xz, yz are zero. The strains

minyy and

minzz represent the stress-free transverse strains if an externalxx is applied. These stress-free strains are obtained bynumerically solving _yy dfhomodyy , _zz

dfhomodzz

so that thesystem may be driven to the free energy minima. At conver-gence, the stress rxx = ofhomo/oxx is calculated atxx = applied, which is varied from 0 to 0.03. Fig. 1 showsthe calculated stressstrain curves in the pseudoelasticregime for three dierent temperatures. It is clear that thestressstrain curves for T = 305 and 300 K exhibit regionsof negative slope corresponding to mechanically unstableHere d is the length rescaling factor dened by

~r d~~r. 9With these denitions, the contributions to the free energycan be expressed in terms of dimensionless variables. Thefree energy for tetragonal distortions is expressed as

~f tetragonal s22 23 2323 322 22 232 10and the contribution to the strain gradients is now

~f gradient eG2d2

~r22 ~r32; 11

where eG G=A0 is the rescaled gradient coecient. Simi-larly, the contributions to the bulk and shear deformationstake the form

~f bulk-shear eAbulk2

1 eE022 232 eAshear2 24 25 26;12

where eAbulk Abulk=A0, eE0 E0e20 and eAshear Ashear=A0.3. Analysis of the homogeneous free energy

It is instructive to analyze the homogeneous free energy(~f gradient 0) as it provides insight into the nature of thestressstrain properties. To specify the free energy for theFePd system, we need the parameters A0, Abulk, Ashear,E0, e0, T0 and Tc. The transformation strains are calculatedfrom the lattice parameters [12] ao = 3.756 A, ac = 3.725 Aand at = 3.795 A using xx yy ataoao and zz acaoao .These allow us to obtain e0

oxxoyy2ozz

6p 0:0152 andf e01 0 oxxoyyozzp

R. Ahluwalia et al. / Acta Mregions. It is also observed that the stress required for thetransition increases as the temperature increases. InFig. 2, we show the stressstrain curves in the shape mem-ory regime. These curves correspond to the situation whereat zero load the material is initially in the austenite stateand the martensite is either a local or global minima. Thus,upon unloading, these curves exhibit a residual strain.Notice that the residual strain increases as the temperatureis decreased. We have also calculated the stressstraincurves by applying an external stress. In this case, an

referred to the web version of this article.)tion of the references to color in this gure legend, the reader is referred tothe web version of this article.)

1 2 3

ity, we choose A01 A02 A03 g and the equations of

G3 2s3 623 22 4322 23

atemotion for the rescaled displacement variables ~ui e0dui i 1; 2; 3 becomeo2~uxo~t2

o~rxxo~x

o~rxyo~y

o~rxzo~z

;

o2~uyo~t2

o~rxyo~x

o~ryyo~y

o~ryzo~z

;

o2~uzo~t2

o~rxzo~x

o~ryzo~y

o~rzzo~z

;

18

where ~x; ~y; ~z are rescaled space variables given by x d~x,y d~y and z d~z. The rescaled time variable ~t is written ast ~t

qd2=A0

q. The total (elastic + dissipative) rescaledfor dierent values of the applied stress. Note that thestressstrain curves obtained by stress-loading and strainloading are identical in the stable regions, as seen inFigs. 1 and 2.

The analysis in this section has shown that the stressstrain curves obtained by minimizing the homogeneous freeenergy predict mechanically unstable regions. Clearly, thebehavior of the system in these unstable regions cannotbe described by this simple approach and a dynamic frame-work that can describe the inhomogeneous microstructureis essential. In Section 4, we describe this framework whichis used to simulate microstructural evolution in the unsta-ble regions in Sections 57.

4. Dynamics

Dynamic equations for the elastic degrees of freedomcan be written in terms of the displacement elds by appro-priate force balance equations [1416]. The advantage ofworking with the displacement eld approach is that theelastic compatibility constraints that inuence the micro-structure are automatically satised [17,18]. Explicitly,

q~u r r$r r0$. 14

Here, the elastic stresses are obtained from the free-energyin Eq. (1) as

rij Xk

dFdek

dekdij

15

and r0$

represents the dissipative part of the stress tensorthat is determined from a dissipation functional R by [16]

r0ij Xk

dRd _ek

d _ekd_ij

. 16

The dissipation functional R is considered to be of the form

R Z

d~rA012

_e21 A022 _e22 _e23

A032 _e24 _e25 _e26

; 17

where A0 ; A0 ; A0 are the damping coecients. For simplic-

2112 R. Ahluwalia et al. / Acta Mstress elds can be determined from the elastic free energyand the dissipation functional using 2eE03eAbulk1 eE022 23 eGd2 ~r23 cm o3o~t ;G4 eAshear4 cm o4o~t ;G5 eAshear5 cm o5o~t ;G6 eAshear6 cm o6o~t

20and the rescaled damping constant cm g=A0

A0=qd

2q

.The above set of equations of motion can be used to simu-late the microstructural evolution in shape memory alloys.In the following section, we simulate the microstructureassociated with the cubic to tetragonal transformation asa function of temperature using these equations of motion.

5. Temperature-induced transformation

We apply the present theory to obtain the microstruc-tures for temperature-induced cubic to tetragonal transfor-mation in FePd shape memory alloys which have attractedattention due to interesting magnetoelastic properties [19].To completely specify the FePd system, we need experi-mental values for e0, E0, A0, Abulk and Ashear used inSection 3. To specify the spatial length scales, we use thevalue G = 3.15 108 N from microstructural dataobtained by transmission electron microscopy for FePdusing 30 per cent Pd [20]. The value is chosen such thatthe length scales of the simulated microstucture matchexperiment. With this choice, the smallest length scalebeing simulated becomes d 4 nm based on a simulationbox size of 64 64 64. Phonon dispersion data providea value [13,21] of G = 3.38 1010 N, corresponding tod 0.3 nm. The system size would then be too small and~rxx G13

p G22

p G36

p ;

~ryy G13

p G22

p G36

p ;

~rzz G13

p 2G36

p ;

~ryz G4;~rxz G5;~rxy G6;

19

where

G1 eAbulk1 eE022 23 cm o1o~t ;G2 2s2 1223 4222 23

2eE02eAbulk1 eE022 23 eGd2 ~r22 cm o2o~t ;

rialia 54 (2006) 21092120therefore we choose G from microstructural data. To solvethe full dynamic equations we need an estimate of the

damping constant g that can be obtained from measure-ments of ultrasonic attenuation as a function of frequency[22,23], x. The attenuation varies as gx2/vs, where vs is thesound velocity and thus this phonon viscosity model pro-vides an estimate for g. In the absence of ultrasonic mea-surements for FePd, we use the estimate g 0.015 N s/m2for V3S as it also undergoes a very similar cubic to tetrag-onal transformation [22]. As the rescaled time~t is related to

the real time t by t ~tqd2=A0

q, we use the estimate for

density q 104 kg m3 to obtain t=~t 2:8 1012 s.The form of the damping used in the simulation is a gen-

eralization of the result for a body with nite velocityundergoing internal friction to continuous elastic bodies[24]. The damping coecient corresponds to a domain wallmobility of 104 m3/N s at the transformation temperature[22,23]. The damping is 103 104 larger than theoreticallycalculated room temperature values for several metals, sothat the attenuation measured is not directly due to merelyphonon viscosity. Dislocation drag, thermoelastic eect,phonon scattering, electronic damping would all makesome contribution. Ultrasonic measurements [23] indicate

temperature T from T = 270 to 230 K in steps ofDT = 5 K. Note that after each change the system isallowed to relax for t 0.67 ns. Fig. 3 shows the develop-ment of the microstructure at two dierent temperaturesduring cooling. The columns show the spatial distributionof the strains xx, yy, zz. The yellow/red regions in the rstcolumn correspond to tetragonal variants stretched alongthe x direction characterized by xx 6 0. Similarly, the redregions in the second column depict the variant distortedalong the y direction (yy 6 0), and in the third columnthese red shaded regions represent the tetragonal variantdistorted along z (zz 6 0). The blue regions in the rst, sec-ond and third columns represent areas where xx = 0,yy = 0 and zz = 0, respectively. A cubic austenite phaseexists at T = 270 K where all the strains are close to zero.The austenite phase becomes unstable below this tempera-ture and the three tetragonal variants are formed. We cansee the emergence of the three variants in Fig. 3 for thesnapshots corresponding to T = 260 K. However, the inter-faces are not yet well dened and there are still traces of theremnant austenite phase as the microstructure is still evolv-ing at this stage. By T = 230 K, the microstructure equili-

30 Kcol

R. Ahluwalia et al. / Acta Materialia 54 (2006) 21092120 2113rapid but continuous increase in attenuation below thestructural transformation. However, there are limitationsin the interpretation of attenuation data. As the dampingterm helps to drive the system to the free energy minima,we do not expect the damping to aect the transformationstrains. However, anharmonicities in the dissipationalfunction may inuence the dynamics and transients.

Eqs. 1820 are discretized using nite dierencesD~x 1; D~t 0:02 and periodic boundary conditions.Starting from small amplitude random initial conditions(corresponding to austenite) for the displacement elds, aquasi-static cooling process is simulated by varying the

Fig. 3. Simulated microstructures in the martensite phase at T = 260 and 2variant distorted along x is represented by red/yellow regions in the rst

regions in the second column (yy 6 0) and the variant distorted along z iinterpretation of the references to color in this gure legend, the reader is refebrates and the interfaces become well dened. Littleaustenite is left by this time. Notice that on average theinterfaces adopt specic orientations that depend on thevariants that they separate. This is seen in the secondrow of Fig. 3. For example, the interface between the var-iant stretched along y direction and that stretched along thez direction makes an angle of 45 in the yz plane. In thexy and xz planes, this interface is parallel to one of theedges of the simulation box. These orientations are consis-tent with the predictions of Sapriel [25] that are based onstrain matching. In our model, the interfaces spontane-

. The columns show the spatial distribution of strains xx, yy and zz. Theumn (xx 6 0), the variant distorted along y is represented by red/yellow

s represented by red/yellow regions in the third column (zz 6 0). (Forrred to the web version of this article.)

ously assume these specic orientations due to the tendencyof the system to maintain elastic compatibility or producecoherent interfaces.

Earlier studies [17,18] of the cubic to tetragonal transi-tion based on this free energy considered an ideal martens-itic transformation where the volume change associatedwith the transformation was zero. In the present work,the volume change of 0.7% leads to interfaces which areirregular, although the overall pattern is similar to the caseof ideal martensitic transformations.

6. Strain-induced transformation and pseudoelasticity

A martensitic transformation can also be induced by anexternal deformation which results in large recoverablestrains. This behavior is referred to as pseudoelasticity. Inthis section, we study strain-induced transformations usingthe displacement eld approach. A loading process is sim-ulated by dening the strain

ij 12

ouioxj

oujoxi

appliedij . 21

This corresponds to a homogeneous strain applied every-where in the system. In the present simulation, we apply

ingunloading simulations reported in this paper corre-spond to a system that is clamped transverse to theloading direction. A stressstrain curve can be calculatedfor this situation by computing the stress analytically fromthe free energy in Eq. (13) using rxx = dF/dxx, subject tothe constraint that yy = zz = 0. This analytical solutionis plotted in Fig. 4. The analytical solution shows thatfor strain values in the range 0.006 to 0.017, the austen-ite is mechanically unstable, as is clear from the negativeslope of the stressstrain curve in that region. The pointC corresponds to where the austenite and martensite havethe same energy and so between C and D, the austenite ismetastable. As the behavior in the metastable and unstableregions is inuenced by the dynamics associated withnucleation and growth, a framework that describes thedynamics is essential.

The strain loading simulations are performed using thedenitions in Eqs. (19) and (20) by solving Eq. (18) asdescribed in Section 5. We simulate a system in the cubicaustenite phase at T = 313 K. The maximum applied strainfor this case is appliedxx 0:025 and the loading is completedin t 2.8 ns corresponding to a strain rate of_c 8:92 106=s. Thereafter, the applied strain is decreasedat the same strain rate to simulate the unloading process.

2114 R. Ahluwalia et al. / Acta Materialia 54 (2006) 21092120a uniaxial tensile strain and so appliedyy appliedzz appliedxy appliedyz appliedxz 0. The uniaxial strain along x is given byappliedxx _ct 22where _c represents the strain rate.

Before we describe our results, we remark that becausewe do not use stress free boundary conditions, all load-Fig. 4. The homogeneous stressstrain curve calculated by plotting ofhomooxx vs. xxthe corresponding potential energy proles.The stressstrain curves are computed by plotting the aver-age stress h dFdxxi vs. the average strain xx. Fig. 5 depicts themicrostructural evolution during the loading process. Here,the red regions in the columns also represent strains withxx 6 0, yy 6 0 and zz 6 0. At t = 0, the system exists inthe austenite state given by xx = yy = zz = 0. On loading,the homogeneous austenite phase persists until at, subject to the constraint that yy = zz = 0. The plots A, B, C and D show

ateR. Ahluwalia et al. / Acta Mxx 0.006 (point D in Fig. 4), the austenite becomesunstable and a non-equilibrium multi-domain state isformed. This is clear by comparing the microstructures atpoints A and B on the loading part of the stressstraincurve in Fig. 5. Notice that in the multi-domain state allthree tetragonal distortions occur, although the variantstretched along the x direction appears to be the favoredvariant. It is instructive to compare the orientation of

Fig. 5. Microstructural evolution during uniaxial loading along the x directioncorrespond to average stresses and strains indicated on the stressstrain curve. Nlimit corresponding to a stress level equal to the stress at point D in Fig. 4. (Foreferred to the web version of this article.)rialia 54 (2006) 21092120 2115domain walls for this multi-domain state to the low tem-perature microstructures without an external load thatare depicted in Fig. 3. For example, the angle betweenthe variant stretched along x and the variant stretched(not as much) along y, as seen in the top xy plane, is dier-ent from 45, unlike the low temperature microstructureshown in Fig. 3. This may be due to the two variants beingunequally distorted as the variant distorted along x is the

at T = 313 K for a strain rate _c 8:92 106=s. The snapshots A, B and Cotice that a multi-domain state is formed beyond the mechanical stability

r interpretation of the references to color in this gure legend, the reader is

simulations of the loading process with an initialquenched seed of the transformed phase that is embed-ded in the initial austenite matrix. The total uniaxial straineld in this direction is given by

xx ouxox appliedxx seedxx 23

where seedxx is the strain due to the defect.

seedxx 0 jx x0j 6 L0; jy y0j 6 L0; jz z0j 6 L0 0 jx x0j > L0; jy y0j > L0; jz z0j > L0.

24where the midpoint of the simulation box of size64 64 64 is given by x0 = y0 = z0 = 64/2 = 32. This rep-resents an inclusion of a cube of the martensite phase oflength L0 in the austenite matrix. A loading process for thissystem with strain rate _c 2:23 106=s (L0 24 nm) isshown in Fig. 7. This gure shows the stressstrain curveand the spatial distribution of xx at points A, B, C, Dand E on the stressstrain curve. In the snapshots A, B,C, D and E, the red/yellow regions correspond to regionsdistorted along x (xx 6 0). To clearly show the evolutionof the microstructure around the defect region, we displaytwo mutually intersecting perpendicular planes that pass

atefavored variant. It is also interesting to note that the stressrequired to maintain this multi-domain state is lower thanthe stress at the instability at which the variants areexpected to start forming. The jump in the stress at theonset of the transformation is typically associated with thisstress drop. The favored variant in the multi-domain stategrows at the expense of the other variants, as can be seenfrom the microstructure corresponding to the point C onthe stressstrain curve. Thereafter, the system stays in a sin-gle variant state of the favored variant and deforms accord-ing to linear elasticity. During the unloading, the materialremains in the single variant state until that state becomesunstable. A non-equilibrium state is formed again in whichthe system starts transforming to the austenite and there isa jump in the stressstrain curve at the onset of the reversetransition. At the end of the unloading, the material iscubic and all strains become zero, resulting in completestrain recovery.

We should remark that the stressstrain curve of thetype obtained in the present simulations has indeed beenseen in experiments on NiTi and CuZnAl shape memoryalloys [26,27]. Our simulations show that if there are nonucleating defects present in the system, the stress requiredto cause the transition corresponds to the intrinsic limit ofstability of the system. However, as the austenite becomesunstable, a jump in the stressstrain curve is observed. Sim-ilar behavior is seen at the onset of the reverse transforma-tion upon unloading. The present simulations show thatthe jumps in stress at the onset of the transitions are a gen-eric feature of strain loading as it places the system in themechanically unstable regions that result in multi-variantstates. Note that no jumps were observed in our earlierstress loading calculations as the unstable regions couldnot be accessed using that method [8,9].

As the deformation involves the formation and motionof domain walls, the rate of loading may inuence thestressstrain response. To investigate this, we have per-formed dynamic strain loading simulations describedabove for three dierent strain rates corresponding to_c 8:92 106=s, _c 4:46 106=s and _c 2:23 106=s.Fig. 6 shows the simulated stressstrain curves for the threestrain rates. We nd that the stressstrain curves for allthree strain rates match the analytical solution in the stableregions. However, a strain rate dependence is observed inthe unstable region of the stressstrain curve. As observedin Fig. 5, the loading process in this regime is governed byvariant formation and domain wall motion. As this processis not instantaneous, a competition between the time scalesof the domain dynamics and the loading rate is responsiblefor this strain rate dependence. For example, if the strainrate is fast enough that domain formation is avoided, thestressstrain curve is identical to the analytical solution inFig. 4. For slower strain rates, twin boundaries appearand the stress starts to deviate from the analytical solution.In fact, as seen in Fig. 6, the slower is the strain rate, the

2116 R. Ahluwalia et al. / Acta Msmaller is the strain at which the stressstrain behaviorbegins to deviate from the analytical solution.The next question we want to address is the eect ofdefect-induced nucleation on the stressstrain curves. It isexpected that the presence of defects will nucleate the trans-formation even before the system is in the unstable region,thereby reducing the stress required to cause the transition.To study defect-induced transformation, we repeated our

Fig. 6. Simulated stressstrain curves at T = 313 K for three dierentstrain rates. The blue curve corresponds to _c 8:92 106=s, the red curvecorresponds to _c 4:46 106=s, and the green curve corresponds to_c 2:23 106=s. For comparison, the homogeneous stressstrain curve ofFig. 4 is also shown. We see that the deviation from the homogeneousresponse occurs at lower strain values for slower strain rates. (Forinterpretation of the references to color in this gure legend, the reader isreferred to the web version of this article.)

rialia 54 (2006) 21092120through the defect. There is a point (C in Fig. 4) on thestressstrain curve where austenite and martensite have

R. Ahluwalia et al. / Acta Materialia 54 (2006) 21092120 2117the same energy. For strains higher than this critical strain,the nucleation of martensite can take place as martensitehas lower energy than the austenite. This nucleationprocess can be observed by comparing the microstructureat points B and C. We can see the growth of the martensitedomains (regions shaded red) from a seed defect that isembedded in the austenite matrix (regions shaded blue).On further loading, the favored variant grows, as can beobserved in the snapshot corresponding to D. Eventually,a single variant state of the variant stretched along the xdirection is established as can be observed in the snapshot

Fig. 7. Microstructural evolution during uniaxial loading (_c 2:23 106=s) alcubical simulation box. The snapshots A, B, C, D and E show the distributionstresses and strains indicated on the stressstrain curve. To clearly show the deUnlike the defect free case, a multi-domain state is created before the limit ofregion. (For interpretation of the references to color in this gure legend, the

Fig. 8. Distribution of xx, yy and zz on the surfaces for snapshot C of Figdistortions along the x direction are larger than those along y and z directions. (is referred to the web version of this article.)E. Notice that the strains in the defect regions are higher(4%) than the bulk value (3%). Even during this loadingprocess, the unfavored variants are also formed. Fig. 8shows the full microstructures for xx, yy and zz corre-sponding to the point C of Fig. 7. Notice that the magni-tudes of the distortions along the y and z directions aresmaller than those along the favored x direction.

To test the strain rate dependence of the stressstrainresponse in the presence of the defect, Fig. 9 comparesthe behavior for strain rates _c 8:92 106=s, _c 4:46106=s and _c 2:23 106=s with the analytical solution

ong the x direction for the case with a defect embedded at the center of theof the strain xx (along the loading direction) corresponding to the averagefect region, two mutually intersecting planes through the defect are shown.mechanical instability due to defect-induced nucleation in the metastablereader is referred to the web version of this article.)

. 7. It is seen that although all three variants appear to form, tetragonalFor interpretation of the references to color in this gure legend, the reader

for the defect free case. It is clear that the slower the rate ofloading, the smaller is the strain at which the stress deviatesfrom the analytical curve. This is due to the fact that nucle-ation is not instantaneous and so the time dependence ofthe nucleation will inuence the stressstrain response.For example, if the loading rate is fast enough that thereis little time for the martensite to nucleate, the apparentstress required to cause the transition will be higher andthe jump at the onset of the transition will be larger. Thus,the jump in stress becomes less pronounced as the strainrate becomes smaller. These results show how nucleationand growth of variants crucially inuence the mechanicalresponse of shape memory alloys in the pseudoelasticregime.

To further illustrate the role of defects on the mechani-cal response, we have performed loading simulations withthree dierent sizes of defects. The stressstrain curvesfor defect sizes L0 = 16, 24, 40 nm with the defect free caseat the xed strain rate _c 2:23 106=s are shown inFig. 10. As expected, nucleation occurs at smaller strainsfor larger defect sizes. In fact, for the defect free case, themartensite is formed only after the applied strain is higherthan the intrinsic limit of mechanical stability. Thus, thepresence of defects inuences the magnitude of the jump

than twin boundaries. Our simulations therefore model fastprocesses at short length scales.

7. Deformation in the martensite phase and the shape

memory eect

Here we investigate the mechanical response at low tem-peratures when the material has already transformed tomartensite in the absence of external deformation. Our ini-tial condition is the microstructure in the second row ofFig. 3, corresponding to T = 230 K. The loading condi-tions are identical to those in the previous section, althoughwe load to higher values of the maximum strain to span theregimes of interest. Fig. 11 shows the simulated loadingand unloading process. The stressstrain curve is calculatedin the same manner as the previous section but the averagestress at appliedxx 0 is subtracted for all points so that thestress is zero for the rst point. The strain rate used inthe present simulations is _c 12:5 106=s. The congura-tion at the rst point on the stressstrain curve is the ascooled martensite state of the second row in Fig. 3. Asthe strain is applied, there is a short linear elastic regimeduring which the microstructure does not change signi-cantly. This is clear by comparing the microstructure forthe point A in Fig. 11 to that of the second row in

2118 R. Ahluwalia et al. / Acta Matein stress at the onset of the transformation.We note that the strain rates used in the simulations

(106/s) are relatively higher than those typically observedin practice. The reason for the high strain rates lies in thetime scales that we are simulating, which are relativelyshort (1012 s). The real time is related to the scaled timevia the smallest length (d 0.110 nm). This limits the size

Fig. 9. Simulated stressstrain curves for the system with a defect atT = 313 K for three dierent strain rates. The blue curve corresponds to_c 8:92 106=s, the red curve corresponds to _c 4:46 106=s and thegreen curve corresponds to _c 2:23 106=s. The black curve representsthe homogeneous curve in Fig. 4. We observe that the magnitude of thejump at the onset of transformation is higher for a higher strain rate. (For

interpretation of the references to color in this gure legend, the reader isreferred to the web version of this article.)of the system and the times that can be simulated. Acoarse-grained model which averages over several twinboundaries could describe slower strain rates. The physicsat the coarse-grained scale would then be governed bythe movement of austenitemartensite interfaces rather

Fig. 10. Simulated stressstrain curves for dierent sizes of the embeddeddefect at T = 313 K. The cyan, blue and red curves correspond to defectsizes L0 = 16, 24, 40 nm, respectively. The green curve is for the defect freecase. The size of the defect determines the magnitude of the jump as thelarger the defect, the easier is nucleation. (For interpretation of thereferences to color in this gure legend, the reader is referred to the webversion of this article.)

rialia 54 (2006) 21092120Fig. 3. Beyond the linear elastic region, the twin boundariesstart to move, as can be seen in the microstructure

ateR. Ahluwalia et al. / Acta Mcorresponding to point B. Notice that the variant stretchedalong the x direction grows at the expense of the other vari-ants. This variant continues to grow and we can observe inthe snapshot corresponding to C that most of the materialhas transformed into this variant. Eventually, all variantsswitch in favor of this variant and we obtain a singledomain state. The unloading process is continued untilthe stress is zero. We observe that the material remains in

Fig. 11. Microstructural evolution during uniaxial loading (_c 12:5 106=s) astate at T = 230 K depicted in Fig. 3. The snapshots A, B and C correspond toa transient linear elastic region, the favored variant starts to grow at the expensthis gure legend, the reader is referred to the web version of this article.)rialia 54 (2006) 21092120 2119a single domain state with a residual strain even when thestress is zero. This residual strain can be recovered on heat-ing giving rise to the shape memory eect.

We performed simulations of the loadingunloading(starting from the same initial state shown in the secondrow of Fig. 3), for three dierent stain rates given by_c 12:5 106=s, _c 6:25 106=s and _c 3:125 106=s.Fig. 12 shows the strain-rate dependence of the simulated

long x in the shape memory regime for T = 230 K, starting from a twinnedthe stresses and strains indicated on the stressstrain curve. Note that aftere of the unfavored variants. (For interpretation of the references to color in

atestressstrain curves. During the loading part, we can seethat the detwinning stress appears to be lower for lowerstrain rate. This is due to the fact that for slower strain rates,the twin boundaries have time to move by their own inertiaand the stress required to move the interfaces is lower.

8. Summary and conclusions

We have carried out a detailed study of the microstruc-

Fig. 12. Simulated stressstrain curves for three dierent strain rates atT = 230 K. The black, red and blue curves correspond to_c 12:5 106=s, _c 6:25 106=s and _c 3:125 106=s, respectively.(For interpretation of the references to color in this gure legend, thereader is referred to the web version of this article.)

2120 R. Ahluwalia et al. / Acta Mture of cubic to tetragonal martensites and the inuenceof microstructural evolution on mechanical properties.The simulations are based on a nonlinear elastic free energyin terms of the appropriate strain components that drive thecubic to tetragonal transition. The dynamics is simulated bysolving the force balance equations for the displacementelds with a dissipational function providing the necessarydamping. The advantage of this method is that the elasticcompatibility relations are naturally satised.

We have applied the framework to the case of FePdshape memory alloys by using estimates for the free energyparameters and damping constants that are obtained fromexisting experimental data. We have simulated the micro-structure development as the material is cooled from thehigh temperature cubic austenite phase. The model hasbeen applied to study strain-induced cubic to tetragonaltransformation and the associated pseudoelastic behavior.Finally, the loadingunloading process in the shape mem-ory regime, starting from a low temperature martensitestate has been studied. Although the stressstrain curvessimulated are in reasonable agreement with experiments,there are limitations on our simulations, such as spatialand time scales accessed as well as loading and boundaryconditions, that make a more quantitative comparison dif-cult. Nevertheless, our simulations capture important fea-tures of the deformation process that are not described byexisting theoretical models. For example, the experimen-tally observed jumps in stress at the onset of the transfor-mation during loading are explained by the currentmodel. The simulations show that these are due to domainformation that reduces the total stress in the system. Thus,the stress required to initiate the transformation is higherthan the stress required to maintain the multi-domain state.The simulations also show that due to time-dependent phe-nomena such as nucleation and domain wall motion, astrain rate dependence of the mechanical properties isobserved in these materials.

Our present simulations describe processes at relativelysmall length (250 nm) and time (25 ns) scales. As lar-ger sizes and longer times will not appreciably increasethe scales, a coarse-graining is required to allow access toengineering length scales. However, the fact that we areable to capture the salient aspects of experiments at macro-scopic length scales, indicates that the essential physics(that of reversible elastic processes) is also valid at thesmaller scales we have simulated. Renement of ourscheme to a more coarse grained model in terms of volumefractions of the transformed phase is a possible approachto describing larger length scales.

References

[1] Bhattacharya K. Microstructure of martensite. Why it forms and howit gives rise to the shape-memory eect? Oxford: Oxford UniversityPress; 2003.

[2] Otsuka K, Wayman CM, editors. Shape memory materials. Cam-bridge (UK): Cambridge University Press; 1998.

[3] Vaidyanathan R, Bourke MAM, Dunand DC. Acta Mater1999;47:3353.

[4] Huang M, Brinson LC. Int J Plast 2000;16:1371.[5] Levitas VI, Idesman AV, Preston DL. Phys Rev Lett 2004;93:105701.[6] Falk F. Acta Metall 1980;28:1773.[7] Levitas VI, Preston DL. Phys Rev B 2002;66:134206.[8] Ahluwalia R, Lookman T, Saxena A. Phys Rev Lett 2003;91:055501.[9] Ahluwalia R, Lookman T, Saxena A, Albers RC. Acta Mater

2004;52:209.[10] Artemev A, Wang Y, Khachaturyan AG. Acta Mater 2000;48:2503.[11] Barsch GR, Krumhansl JA. Phys Rev Lett 1984;53:1069.[12] Seto H, Noda Y, Yamada Y. J Phys Soc Jpn 1998;57:3668.[13] Sato M, Grier BH, Shapiro SM, Miyajima H. J Phys F 1982;12:2117.[14] Lookman T, Shenoy SR, Rasmussen KO, Saxena A, Bishop AR.

Phys Rev B 2003;67:024114.[15] Ahluwalia R, Ananthakrishna G. Phys Rev Lett 2001;86:4076.[16] Bales GS, Gooding RJ. Phys Rev Lett 1991;67:3412.[17] Jacobs AE, Curnoe SH, Desai RC. Phys Rev B 2003;68:224104.[18] Ahluwalia R, Lookman T, Saxena A, Shenoy SR. Phase Trans

2004;77:457.[19] Cui J, Shield TW, James RD. Acta Mater 2004;52:35.[20] Oshima R, Sugiyama M, Fujita FE. Metall Trans A 1988;19A:803.[21] Saxena A, Barsch GR. Physica D 1993;66:195.[22] Barsch GR. J de Physique IV (Colloque) 1995;5:119.[23] Testardi LR, Bateman TB. Phys Rev 1967;154:402.[24] Landau LD, Lifshitz EM. Theory of elasticity, 3rd ed., vol.

7. Oxford: Butterworth-Heinemann; 1998.[25] Sapriel J. Phys Rev B 1975;12:5128.[26] Iadicola MA, Shaw JA. Int J Plast 2004;20:577.

rialia 54 (2006) 21092120[27] Otsuka K, Wayman CM, Nakai K, Sakamoto H, Shimizu K. ActaMetall 1976;24:207.

Dynamic strain loading of cubic to tetragonal martensitesIntroductionFree energy functional for cubic to tetragonal transitionAnalysis of the homogeneous free energyDynamicsTemperature-induced transformationStrain-induced transformation and pseudoelasticityDeformation in the martensite phase and the shape memory effectSummary and conclusionsReferences