NiTiPillar Acta 2009

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Transformation-induced plasticity during pseudoelastic deformationin NiTi microcrystals

D.M. Norfleet a, P.M. Sarosi b, S. Manchiraju c, M.F.-X. Wagner d, M.D. Uchic e,P.M. Anderson c, M.J. Mills c,*

a Engineering Systems Inc., 3851 Exchange Ave., Aurora, IL 60504, USAb General Motors, R&D Tech Center, 30500 Mound Road, Warren, MI 48090, USA

c Department of Materials Science and Engineering, The Ohio State University, 2041 College Road, Columbus, OH 43201, USAd Lehrstuhl Werkstoffwissenschaft, Institut fur Werkstoffe, Ruhr-Universitat Bochum, Universitatsstr. 150, D-44801 Bochum, Germany

e Air Force Research Labs RXLM, Wright-Patterson AFB, OH 45433, USA

Received 20 February 2009; received in revised form 4 April 2009; accepted 7 April 2009Available online 10 May 2009

Abstract

[1 1 0]-oriented microcrystals of solutionized 50.7 at.% NiTi were prepared by focused ion beam machining and then tested in com-pression to investigate the stress-induced B2-to-B190 transformation in the pseudoelastic regime. The compression results indicate a sharponset of the transformation, consistent with little prior plasticity. Post-mortem scanning transmission electron microscopy reveals noapparent retained martensite but rather a macroscopic band of dislocation activity within which are planar arrays of $100 nm disloca-tion loops involving a single ah0 1 0i{1 0 1} slip system. Micromechanics analyses show that the angle of the band is consistent with acti-vation of a favored martensite plate. Further, the stress from the individual variants within the plate is shown to favor activation of theobserved slip system. The work done by the applied stress during the B2-to-B190 transformation is estimated to be $34 MJ m3 at ambi-ent temperature. 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Shape memory alloy; NiTi; Pseudoelastic response; Micromechanics; Stressstrain response

1. Introduction

Shape memory alloys (SMAs) have remarkable proper-ties that stem from a martensitic transformation that canbe induced by an appliedstress or cooling, and reversed upon

unloading or heating. Several well-known alloys exhibit thisresponse and they have enabled a variety of applicationsfrom valves and actuators to surgical devices[13].Thedom-inant transformation in NiTi SMAs is from the B2 crystalstructure to a martensite variant. In a solutionized form,the transformation is directly from the B2-to-B190 structures[4]. In the pseudoelastic mode, a NiTi SMA with a B2 struc-ture is stressed sufficiently to induce deformation via the

transformation to the B190 structure. Upon unloading,reverse deformation occurs via the reverse transformation.

The crystallographic aspects of the martensitic transfor-mation, including the orientation relationship with theordered B2 matrix, have been calculated and verified in sev-

eral systems [59]. However, some fundamental aspects ofshape memory and pseudoelastic behavior are not yetunderstood, notably how the matrix accommodates thelarge strain associated with the transformation. Theoreti-cally, accommodation may be achieved either by matrixplasticity or by inducing other transformation variants.As elaborated in Section 4.1, there are few direct experi-mental observations using transmission electron micros-copy (TEM) that directly explore these accommodationmechanisms [10,11], and no complementary modeling thatattempts to rationalize these mechanisms, even for the

1359-6454/$36.00 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

doi:10.1016/j.actamat.2009.04.009

* Corresponding author. Tel.: +1 614 292 1537.E-mail address: [email protected] (M.J. Mills).

www.elsevier.com/locate/actamat

Available online at www.sciencedirect.com

Acta Materialia 57 (2009) 35493561
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archetype SMA system, NiTi. This fundamental question istechnologically important since the useful service life ofcomponents under cyclic loading or loading/heating (i.e.functional and structural fatigue [12,13]) is determined bythe evolution of the microstructure and defect substructure.

Much attention has been focused on the effect of sample

size on dislocation plasticity, using compression sampleswith diameters less than 10 lm. A tremendous strengthincrease can be achieved in pure nickel and pure gold sam-ples when the diameter is decreased from 20 lm to submi-cron dimensions [1419]. The hypothesis is that smallsample dimensions can impede or interact with dislocationmotion, generation and plasticity. For NiTi, there is evi-dence that dislocation plasticity degrades the pseudoelasticeffect by causing irreversible or residual strain [20]. Thus,enhanced pseudoelastic effects might be achieved if smallsample sizes can inhibit plasticity in NiTi alloys, therebyraising the yield strength relative to the phase transforma-tion stress.

Similar to Frick et al. [21,22], the present investigationemploys focused ion beam machining to fabricate single-crystal cylinders that are subsequently compressed andunloaded. However, the sample diameters in this investiga-tion are larger (5 and 20 lm), and, in addition, significantimprovements are made in the machining and loadinggeometries to minimize artifacts associated with samplemisalignment and nonuniform sample cross-sectional area.Relative to the NiTi literature as a whole, a second signif-icant advance is to couple the microcrystal experimentswith post-mortem, ex situ, scanning transmission electronmicroscopy (STEM). An important finding is that such

microcrystalline tests can activate particular martensiteplates, enabling more precise measurements of the criticalstress for forward/backward transformation in the absenceof large-scale interaction between competing martensiteplates. Further, thin foils are prepared that encompassthe entire slip plane or entire gage length of the sample.The results reveal detailed dislocation configurations asso-ciated with the phase transformation. A final advance is to

interpret the stressstrain and STEM data via microme-chanics analyses of the most preferred martensitic platesto form, as well as the resolved shear stress on slip planesin the vicinity of martensitic plates.

2. Materials and methods

2.1. Experimental

Bulk polycrystalline NiTi bar stock containing50.7 at.% Ni was heat-treated in an argon atmosphereat 1000 C for 20 h, in order to increase the averagegrain size to >300 lm. The samples were then waterquenched to prevent the formation of metastable precip-itates (e.g. Ni4Ti3). They were cut to button-sized piecesand mechanically parallel-polished using an Allied HighTech MultiPrep System. Colloidal silica was used inthe final polishing step to achieve a surface finish of0.05 lm. Grain orientations were determined from elec-

tron backscatter diffraction (EBSD) on a FEI XL-30environmental scanning electron microscope. Fig. 1ashows a grain orientation image map (OIM). The largegrain indicated by the arrow in the OIM was selectedfor subsequent machining into cylindrical compressionsamples. Fig. 1b shows that the surface normal of thelarge grain is parallel to within 4 of the [1 1 0] direction,with a rotation toward [1 1 0]. Prior work by Gall et al.[20] has shown that, compared to other crystal orienta-tions, bulk [1 1 0] single-crystal compression samplesexhibit a moderate strain driving the B2-to-B190 transfor-mation during loading, and upon unloading, intermediate

values of remnant strain result. Micromachining was per-formed using a FEI DB 235 dual-beam focused ionbeam (FIB) and the procedure introduced by Uchicet al. [14]. Two 5 lm and one 20 lm diameter micropil-lars were fabricated from the large grain, each havingan aspect ratio of about 2.5:1. Fig. 2 shows an SEMmicrograph of a 5 lm pillar. The circular, fiducial markon the top surface aids the automated fabrication

Fig. 1. (a) OIM of polycrystalline 50.7 at.% NiTi sample. The large central grain (arrow) was chosen for pillar fabrication. (b) h1 1 0i pole figure showingthe near [1 1 0] orientation of this grain, together with the crystal indexing used in this paper. The grain normal (pillar compression axis) is at the center of

the pole figure, rotated by about 4 away from [1 1 0] toward [1 1 0], corresponding to an approximate orientation of [7 8 0]).

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process. A precise pillar shape results, with a constantcross-section and flat contact surface. This is in contrastto less-refined techniques that produce rounded cylindertops and significant tapers (e.g. $3) in the cylinderwalls, especially at submicron sample sizes [21,22].

The samples were compressed uniaxially at ambient lab-oratory temperature under constant displacement rate, giv-ing a nominal strain rate of 104 s1. A MTS NanoIndenter XP was used with a cleaved diamond indentertip, creating a flat platen to conform with the top cylindersurface [14]. An integrated goniometer enabled smalladjustments to the sample orientation during repeated

loading and unloading, so that the initial, nonlinear toe-in portion of the loaddisplacement trace was reducedto a minimum.

TEM samples were prepared with a foil plane that con-tains the [1 0 0] and [0 1 0] Burgers vectors (Fig. 2b). Theywere extracted with an in situ plucking device (OmniP-robe), and thinned by FIB to achieve an electron-transpar-ent thickness (


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possible plates. In particular, the martensitic transforma-tion induces an average deformation gradient

F I b m; or Fij dij bimj 1

within the plate. The identity matrix is I and the Kroneckerdelta function dij = 1 when iand jhave the same value (1, 2

or 3) and = 0 otherwise. The transformation causes a vec-tor v in the B2 phase to stretch and rotate to a vector v0 inthe martensite plate, where

v0 F v; or v0i X3j1

Fijvj 2

The resulting axial strain along the compression axis in-duced by the transformation is

e v0 v 1

1 vT F v 1 3

where v = ec is a unit vector along the compression axis.

The second goal is addressed by solving for the stressfield around an inclusion of volume X within which thereis a transformation strain

e 1

2FT F I 4

where F is defined in Eq. (1). The goal is to find a displace-ment field u(x) satisfying [23]

X3k1

X3l1

X3j1

Cijklo

2uk

oxloxjX3k1

X3l1

X3j1

Cijkloekloxj

5

The components Cijkl of the elastic moduli are approxi-mated to be the same for both the B2 and martensitephases. Recent first-principles calculations, althoughstrictly valid at 0 K, show the elastic moduli of B2 andB190 to be similar in magnitude [24,25]. Closed-form solu-tions to Eq. (5) are provided for particular inclusion shapes[23,26,27].

The stress field from transformation regions of arbitraryshape is obtained by introducing a scalar function g(x) that= 1 inside the transformed region and = 0 otherwise. Thisorder parameter can be expressed as a Fourier series

gx X

n1 ;n2 ;n3

gn1; n2; n3ein1x1 ;n2x2 ;n3x3 6

where the Fourier coefficients g(x1, x2, x3) are determined bythe shape of the transformation region and the dimensions(L1, L2, L3) of the periodic cell containing the transforma-tion region. Eq. (5) is solved for u(x) using the fast Fouriertransform (FFT) method with strain eij(x) = 0.5(oui/oxj + ouj/oxi) and stress rij(x) = Cijklekl(x), where ekl(x) isthe elastic strain field. Two approaches are taken to modelthe transformation. The plate approach takes X tobe a platewith a uniform, volumeaverage transformation strain (Eq.(4)). The variant approach uses several X transformationvolumes, each corresponding to an individual martensite

variant in the plate.

3. Results

3.1. NiTi microcrystal compression results

Fig. 4 displays the resulting uniaxial compression stressstrain curves from three NiTi microcrystals: two 5 lm

diameter samples and one 20 lm diameter sample. Thesesamples were strained to a total compressive value$2.53% and unloaded (Fig. 4a). Superimposed is theresponse of a bulk [1 1 0]-oriented single crystal from Gallet al. [20]. Fig. 4b details the response of the 5 lm samples,one of which has two cycles. Fig. 4c shows the one- andtwo-cycle response of the 20 lm sample, and Fig. 4d showsthe response for all four cycles of the 20 lm sample.

All of the micron-scale cases exhibit a flag-shaped hys-teresis loop, indicative of pseudoelastic behavior. As exem-plified in Fig. 4c, the typical loading response consists of (i)an initial linear-elastic portion, (ii) a small upper yieldpoint, (iii) a near-plateau region, and (iv) significant hard-

ening, especially in the 20 lm case. The unloading responseis (v) initially linear-elastic, giving way to (vi) a nonlinearregion where a large fraction of total strain is recovered,followed by (vii) linear-elastic unloading. Several compara-tive aspects are noted. The very small (e $ 0.1%) toe-inregion at the beginning of each test is evidence of the goodsample alignment that can be achieved using the goniome-ter stage. This is in contrast to the large (>1%) toe-inregions in Frick et al. [21,22]. Despite the reduced toe-inregion in the present tests, the elastic modulus is consis-tently smaller than Eh110i = 148 GPa based on publishedanisotropic elastic moduli [29]. The discrepancy is likely

due to error in the nominal strain as well as machine andsubstrate compliance.

The pillars show an abrupt transition to the plateau inregion (iii), occurring at a compressive stress of $610625 MPa. The abrupt nature of the transition suggestsnucleation of a single or limited number of dominant platesin a homogeneously stressed sample, with negligible priorplasticity. In contrast, the bulk single crystal has a smooth,gradual transition, beginning with a noticeable departurefrom linear-elastic behavior at $380 MPa (Fig. 4a) and ris-ing to a plateau at $600 MPa. The large ($75%) increase instress cannot be rationalized in terms of the activation of asingle plate. Rather, the internal stress state must be suffi-ciently inhomogeneous to nucleate plates over a large rangeof stress. The prospect of nucleating a single or limitednumber of plates under such conditions is small. Bulk sin-gle crystals with other orientations have similar gradualtransitions [20]. Thus, micron-scale tests are unique andoffer the capacity to precisely determine the onset oftransformation.

An additional consideration is that the bulk sample hasa concentration of 50.9 at.% Ni compared to 50.7 at.% Nifor the micron-scale samples. The forward transformationstress for the 50.9 at.% Ni bulk case is therefore decreasedby $60 MPa to replicate a 50.7 at.% Ni bulk case. This cor-

rection is based on a ClausiusClapeyron construction



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using data for NiTi [4] that Ms is about 20 C higher for50.7 at.% Ni. Thus, the plateau stress for the micron-scalepillars is $14% larger than that for a bulk sample of similarcomposition.

After the abrupt transition (iii), the hardening out to$3% strain is a function of specimen size. For example,the average hardening rate in this strain range varies from$1 to 3 GPa for the 5 lm case to $6 GPa for the 20 lmcase to $12 GPa for the bulk case. The bulk sample hasa maximal hardening rate at small strain with a monotonicdecay to nearly zero at 3% strain. In contrast, the 5 lmsamples have a negligible hardening rate initially but thisincreases with strain (Fig. 4b). Beyond 3% strain, the bulksample has negligible hardening but the 5 lm samples showcontinued hardening.

The micron-scale samples in the present work are consis-tently stronger in the forward (loading) direction than thebulk sample case. The difference is most pronounced early

in the nonlinear deformation process. At large strain, the

flow stress for the 20 lm case (Fig. 4d) far exceeds the bulkcase ($1.2 GPa vs. $600 MPa at 12% strain). Thus, theplastic flow stress at large strain appears to be enhancedin the micron-scale case.

The unloading behavior is also a function of sample sizeas well as cycle number. During the first cycle, both 5 lmcases display two transitionsone at $470 MPa (point B,Fig. 4b) that is consistent with the onset of the transforma-tion back to B2 and a very sharp transition at $360 MPa(point C, Fig. 4b) that is consistent with the completionof the back-transformation. These transitions are not assharp on the second cycle nor are they as sharp for the20 lm or bulk cases. These two transition points occur atsmaller stress as sample size increases, at least for the firstunloading cycle. The trend suggests that during the firstunloading cycle, smaller samples have less substructure toconstrain the back-transformation.

The limited data presented here does not show a strong

sample size effect on the magnitudes of remnant strain and

Fig. 4. Near [1 1 0] compressive stressstrain response of solutionized 50.7 at.% NiTi microcrystalline pillars with 5 or 20 lm diameters (present work)and a solutionized 50.9 at.% NiTi bulk crystalline sample [18] for (a) all samples; (b) 5 lm diameter samples; (c) 20 lm diameter sample; and (d) 20 lmdiameter and bulk samples.

D.M. Norfleet et al. / Acta Materialia 57 (2009) 35493561 3553


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reversible strain upon unloading. In particular, the 5 and20 lm cases have a comparable remnant strain ($0.2%)after a similar total imposed strain ($3%) on the first cycle.Second, the 20 lm (fourth cycle) and bulk (first cycle) casesshow a comparable reversible strain (4.5% vs. 5.0%) after alarge total imposed strain of $1012% (Fig. 4d).

The substructure generated by the first cycle producescharacteristic features in subsequent cycles. The secondcycle loading curves (5 and 20 lm cases) transitionabruptly when the first cycle plateau is reached. For the5 lm case, the stress drop is not present in the second cycle,suggesting that the substructure aids the forward transfor-mation upon reloading. The nature of the substructure isnot sensitively dependent on the total strain in the firstcycle. In Fig. 4b, the two first-cycle cases complete theback-transformation at $360 MPa and have $0.2% rem-nant strain, even though the total strain is different ($3%vs. $3.4%). In addition, the two unloading curves frompoint A nearly overlap initially, suggesting that the sub-

structure does not interfere with the first regions to back-transform, but interferes with the later regions to back-transform.

3.2. STEM results for the one- and two-cycle 5 lm pillars

The bright-field STEM images of Fig. 5a and b show aband of remnant dislocation structure in each of the one-

and two-cycle 5 lm samples, respectively. The bands aremost likely regions that transformed to martensite duringloading, and then transformed back to austenite uponunloading. Diffraction patterns from various locations ineach of the foils show no indication of residual martensite.The dislocation density is much greater in the top portion

of the band, suggesting that the transformation probablyinitiated at the upper surface. This is not surprising sinceeven a slight misalignment between the indenter and pillarproduces stress concentrations at the corners.

Fig. 6 shows images used for conventional dislocationanalysis. All dislocations inside and outside the transfor-mation zone satisfy an invisibility condition usingg = [1 0 0], indicating that they all have a Burgers vectorof a[0 1 0]. This is most clearly demonstrated in the lowerdislocation density, one-cycle sample shown in Fig. 6b.The relatively strong residual contrast is discussed in moredetail below.

The densely arrayed dislocation configurations inside

the former transformation zones exhibit similar attributesfor both the one- and two-cycle samples. At lower magni-fication (Fig. 6b and c), linear features are apparent thatrun nearly parallel to the [1 1 0] loading axis (rotated sev-eral degrees clockwise toward [1 1 0]). Fig. 7 shows thatthese linear features are arrays of elongated dislocationloops that appear to lie on parallel (1 0 1) planes. A seriesof images at different tilt angles shows that the dislocations

Fig. 5. Diffraction-contrast STEM images using g = [0 1 0] for (a) one-cycle and (b) two-cycle 5 lm pillars. The foil normal is [0 0 1].



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outside the apparent, former transformation zone are elon-gated along the nonorthogonal directions [1 4 1] and [1 4 1]in the parent phase, giving them a distinctive arrowheadshape. Thus, the small loops within the transformationzone and the arrowhead configurations at the peripheryof the transformation zone both lie on (1 0 1) planes.Fig. 6b and c have residual contrast since g (b u) isnon-zero. For instance, the arrowhead configurationshave |g (b u)| = 1 and generate a typical weak, double-peak contrast. Frequent intersections of the dislocationloop features with the foil surfaces may also contribute tothe strong residual contrast.

The result that much of the a[0 1 0] dislocation contentlies on (1 0 1) planes is rather surprising since the h1 0 0i/{1 1 0} family has a lower Schmid factor (0.35) compared

to the h1 0 0i/{1 0 0} family (0.50). Hence, the observeddislocations are not on a slip system with the highest Sch-mid factor, with respect to the macroscopic applied stress.The dramatic increase in dislocation content between theone- and two-cycle samples, both of which have beendeformed to the same maximum compressive strain, clearlysuggests a direct interplay between the transformation andthe observed dislocation structures.

3.3. Analytic modeling of the preferred martensite plate

The procedure outlined in Section 2.2 was used to iden-tify the most likely twinning modes (plates) to form, basedon: (i) the macroscopic forward transformation strain; (ii) aranking of the modes that generate the largest compressive

Fig. 6. Higher-magnification diffraction-contrast STEM images. Within the former transformation zone for the 5 lm one-cycle sample using (a) g[0 1 0]and (b) g[1 0 0]. At the periphery of the transformation zone for the two-cycle sample using (c) g[0 1 0] and (d) g[100]. The foil normal is [0 0 1].Arrows indicate the g-vectors for each image.



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strain; and (iii) the orientations of the band of residual dis-location content and planar arrays of dislocations. The first

aspect was addressed by noting that for the two-cycle sam-ple, the first forward cycle imparts a macroscopic compres-sive transformation strain of$1.65% (Fig. 4b). This straincould be achieved if the transformation strain in the bandwere $5%, since STEM images show the volume fractionof the band to be $1/3.

The second aspect is pursued by ranking the compres-sive axial strain produced by each of the 192 possible twin-ning modes, using Eq. (3). Table 1 lists the 32 twinningmodes that produce an axial [1 1 0] compressive strain of>5%. These are arranged into eight classes, T1T8, eachassociated with a mode (A, B or C) and type (I or II) fol-

lowing the convention in Table 7.3 of Bhattacharya [8].Each class is comprised of four crystallographic permua-tions of m(k,m) and b(k,m), with axial compressive transfor-mation strains ranging from 5.2% to 5.6%. Theconsistency with the macroscopic forward transformationstrain suggests activation of these highly favored twinningmodes.

The final aspect is pursued by superimposing the pre-dictions for the T4 class and a STEM image from the5 lm two-cycle sample, as shown in Fig. 8. The assump-tion here is that the observed dislocation content is inti-mately related to the activation of a particular martensiteplate. The projected value h0 = 31 fits remarkably well.

In principal, the h0 = 30 prediction for the T1 class alsofits well. However, the T4 class has a twin interface ori-

entation (k0 = 9) that aligns well with the linear, nearlyvertical lines in Fig. 8, which are traces of the dislocationloop arrays in Fig. 6. In contrast, the T1 class has pooralignment (k0 = 38).

Thus, the T4 class is viewed as most consistent. Theslight misorientation (see Section 2) puts the compressionaxis close to [7 8 0] rather than [1 1 0]. In this case, theT4 combinations (k, m) = (5, 8) and (8, 5) produce themaximum axial strain, relative to all 192 possible martens-ite plate types.

3.4. Analytic modeling of stressed slip systems near

a martensite plate

The stress state around the T4 class of plates mayexplain why dislocations are observed on a lower Schmidfactor (0.35) system, (1 0 1)/[0 1 0]. The FFT method (Sec-tion 2.2) is used with the package FFTW [28]. The periodiccell dimensions L1 = 4a = L2 = 4b and L3 = 4c isolate thestress fields of individual plates, and sufficient accuracy isobtained with a three-dimensional grid of 256 256 512 points. The thin plate geometry in Fig. 3 is used,with the invariant planes at x3 = c, where c = a/20 = b/20. The Cartesian coordinate system has x3 || m

(k,m) andx2 || b

(k,m)-(b(k,m) m(k,m))m(k,m).

Fig. 7. Tilt series of STEM images for the 5 lm one-cycle sample using g[0 1 0] for all images. The images are tilted about a [1 1 0] rotation axis. (a) 26tilt from [0 0 1] zone toward [1 1 0]. (b) Near the [0 0 1] zone. (c) 26 tilt from [0 0 1] zone toward [1 1 0]. (df) Corresponding schematicillustrations of the dislocation loop arrays observed in the former transformation zone. Arrows indicate the g-vectors for each image.



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Two representations are employed to study the stressfield. In the plate approach, the transformation strainin the plate has a uniform value 0.5(b m + m b). Iso-tropic elastic moduli (E= 148 GPa, m = 0.3) are alsoassumed. It was confirmed that the normalized stress dis-tribution r23(x1)/E from the FFT approach agrees wellwith the analytical solution by Chiu [27] for an isolatedcuboidal inclusion with a uniform transformation strain.This validates the FFT method, but the plate approachis unable to capture plasticity at the individual variantscale, as revealed by the STEM images.

In the variant approach, the stress field around a platecomposed of an arbitrary number of (k, m) = (4, 1) variantpairs is computed using the FFT approach. The B2-to-B190

transformation produces a strain 0.5(UTi UiI), where

U1

c e e

e a d

e d a

264

375 and U4

c e e

e a d

e d a

264

375 7

are referred to the austenite cubic basis, with a = 1.0243,c = 0.9563, d = 0.058 and e = 0.0427 [8]. The austenite elas-tic moduli are C11 = 130 GPa, C22 = 98 GPa andC44 = 34 GPa [29], with the same values assumed for mar-tensite, as discussed in Section 2.2. A macroscopic com-pressive transformation stress of 600 MPa issuperimposed as observed for the forward transformationstress in the experiments (Fig. 4a).

Fig. 9 shows the most stressed austenite slip systems justoutside the martensite plate (Fig. 3). Fig. 9ac show resultson the cuts x1 = 1.0125a, x2 = 1.05b and x3 = 1.05c,respectively. A type (i) plot indicates the favored austenite{1 0 0}/h0 0 1i or {1 1 0}/h0 0 1i slip system with the largestresolved shear stress. A type (ii) plot shows the correspond-ing resolved shear stress (in MPa) on the favored system.The stress is clearly localized along contours parallel tothe underlying twin interfaces.

Fig. 10 shows magnified views of the same three cuts inFig. 9, except that a filter is applied to show only slip sys-tems for which the resolved shear stress exceeds 1500 MPa.

A white background indicates regions where no slip system

Table 1Analysis of potential twinning modes and martensitic variants.a

Caseb eaxialc (%) (k, m)d m(k,m) b(k,m) h0e k0f Slip systemg

T1 (B,I) 5.2 (1,4), (4,1) (0.91, 0.25, 0.33) (0.05, 0.11, 0.05) 30 +38(5,8), (8,5) (0.25, 0.91, 0.33) (0.11, 0.05, 0.05)

T2 (B,I) 5.6 (1,4), (4,1) (0.35, 0.83, 0.43) (0.12, 0.03, 0.04) 68 38

(5,8), (8,5) (0.83, 0.35, 0.43) (0.03, 0.12, 0.04)T3 (B,II) 5.6 (1,4), (4,1) (0.38, 0.77, 0.51) (0.12, 0.02, 0.05) 71 9

(5,8), (8,5) (0.77 0.38, 0.51) (0.02, 0.12, 0.05)

T4 (B,II)h 5.3 (5,8) (0.22, 0.89, 0.40) (0.10, 0.06, 0.06) 31 +9 i

(8,5) (0.22, 0.89, 0.40) (0.10, 0.06, 0.06) 31 +9 j

(1,4) (0.89, 0.22, 0.40) (0.06, 0.10, 0.06) 31 +9 j

(4,1) (0.89, 0.22, 0.40) (0.06, 0.10, 0.06) 31 +9 i

T5 (C,I) 5.2 (1,5), (4,8) (0.0, 0.81, 0.58) (0.09, 0.04, 0.04) 45 0(5,1), (8,4) (0.81, 0.0, 0.58) (0.04, 0.09, 0.04)

T6 (C,I) 5.6 (1,5), (4,8) (0.86, 0.28, 0.43) (0.0, 0.09, 0.06) +63 0(5,1), (8,4) (0.28, 0.86, 0.43) (0.09, 0.0, 0.06)

T7 (C,II) 5.6 (1,5), (4,8) (0.91, 0.20, 0.36) (0.0, 0.1, 0.05) +57 90(5,1), (8,4) (0.20, 0.91, 0.36) (0.01, 0.0, 0.05)

T8 (C,II) 5.3 (1,5), (4,8) (0.04, 0.89, 0.46) (0.1, 0.03, 0.04) 42 +90(5,1), (8,4) (0.89, 0.04, 0.46) (0.03, 0.10, 0.04)

a See Fig. 3 for a description of m, b, n, and a.b (Mode, type) as indicated in the phenomenological theory of martensite, e.g. see Table 7.3 in Ref. [8].c As predicted by Eq. (3).d k and m refer to particular martensite variants as defined in Table 4.3 in Ref. [8].e The counterclockwise angle between the [1 1 0] axis and the invariant plane, projected on the [0 0 1] plane.f The counterclockwise angle between the [1 1 0] axis and the twin interface, projected on the [0 0 1] plane.g The max resolved shear stress, obtained from the micromechanics calculations of Section 3.4.h

a(5,8) = (0.02, 0.01, 0.29), a(1,4) = (0.01, 0.02, 0.29), and n(5,8) = (0.81, 0.59, 0), n(1,4) = (0.59, 0.81, 0). The a3 and n3 components are reversed

when (k, m) are reversed.i (1 0 1)/[0 1 0] has the maximum stress on x1 = 1.0125a and x2 = 1.0125b; (1 0 1)/[0 1 0] has the maximum stress on x3 = 1.05c.j (1 0 1)/[0 1 0] has the maximum stress on x1 = 1.0125a and x2 = 1.0125b; (1 0 1)/[0 1 0] has the maximum stress on x3 = 1.05c.

Table 2Legend for the austenite slip systems.

Label Austenite slip system

1 (0 1 0)/[1 0 0] and equivalent (1 0 0)/[0 1 0]2 (0 0 1)/[1 0 0] and equivalent (1 0 0)/[0 0 1]3 (0 0 1)/[0 1 0] and equivalent (0 1 0)/[0 0 1]4 (0 1 1)/[1 0 0] and equivalent (1 1 0)/[0 0 1]5 (0 1 1)/[1 0 0] and equivalent (110)/[0 0 1]6 (1 0 1)/[0 1 0]7 (1 0 1)/[0 1 0]



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meets this criterion. Although the 1500 MPa criterion isarbitrary, it serves as an effective filter to identify the mostlikely spatial regions for slip activity and the particular slipsystems involved. A value >1500 MPa would shrink thepredicted bands of slip activity and a smaller value wouldenlarge them, eventually reproducing the type (i) plots in

Fig. 10.Three significant observations are made from Fig. 10.First, the active slip systems in the vicinity of the platedo not include (1 0 0)/[0 1 0], even though it has the largestSchmid factor (0.50). Second, the experimentally observed(1 0 1)/[0 1 0] system is predicted to be dominant on twofaces of the plate. Third, the predicted regions for (1 0 1)/[0 1 0] and (1 0 1)/[0 1 0] activity are aligned to the twininterfaces in the martensite plates, correlating well withthe STEM images of planar loop arrays (Fig. 8).

Two additional comments pertain to the scope of thevariant approach. Figs. 9 and 10 show the results basedon the T4(4,1) plate mode in Table 1. However, the right-

most column in Table 1 shows that the other T4 modesalways favor slip systems (1 0 1)/[0 1 0] and (1 0 1)/[0 1 0]over the (1 0 0)/[0 1 0] system. Finally, these key resultsfrom the variant approach hold for other plate shapes, suchas an ellipsoid-shaped inclusion and a cuboid-shaped inclu-sion with hemispherical ends.

4. Discussion

4.1. Comparison with previous TEM studies

To our knowledge, the present results are the first

detailed analysis of the specific type and arrangement ofdislocations associated with pseudoelastic response in NiTialloys. This is significant since dislocation activity andretained martensite are associated with hysteresis loss, a

Fig. 9. Spatial distribution of (i) the most stressed slip system and (ii) corresponding resolved shear stress (in MPa) on three planes (a) x1 = 1.0125a; (b)

x2 = 1.0125b and (c) x3 = 1.05c that are just outside the faces of a martensite plate (T4, k= 8, m = 5 in Table 1) as shown in Fig. 10. The numbers in the

legend correspond to the slip systems in Table 2.

Fig. 8. Comparison of the crystallographic theory of martensitic trans-formations for NiTi with TEM observations from the preliminarymicropillar testing. Shown are the predictions for the T4 class in Table

1. The line labeled invariant plane shows the predicted intersection ofthe invariant plane with the [0 0 1] plane of the image. The line labeledtwin interface shows the intersection of the predicted martensitemartensite interface with the [0 0 1] plane of the image. The compressionaxis is approximately h= 4 from [1 1 0].



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decrease in critical transformation stress and an increase inremnant strain [3034]. Previous work [10,32,33] has iden-tified ah1 0 0i dislocations and dislocation activity [11], butwithout a detailed consideration of the stress generated bymartensitic variants and the applied stress. Chumlyakov

et al. [33] showed that dislocations within the B2 phaseof Ti50.8 at.% Ni0.3 at.% Mo are ah1 0 0i type on{0 0 1} and {0 1 1} planes. Although complete dislocationanalyses have not been performed, Hurley et al. [10] andHamilton et al. [11] have shown dislocation structures witha similar periodicity to the present work, in cyclicallydeformed NiTi single crystals and thermally cycled, solu-tionized Ti50.1 at.% Ni, respectively. Coupling betweenplasticity and transformations has been proposed byYawny et al. [34], but quantifying this in polycrystals,and even bulk single crystals, is difficult since multipleplates are usually activated.

The present work demonstrates that tests of micron-scale volumes can overcome these inherent difficulties.The plate level modeling identifies the likely transformationmode, taking into account the slight misorientation fromthe [1 1 0] compression axis. From this, the work to inducethe B2-to-B190 transformation in an isolated plate at roomtemperature is $34 MJ m3, based on the product re*,where r $ 600 MPa (Fig. 4) is the compressive transforma-tion stress along [7 8 0] and e* $ 5.7% is the axial strainalong [7 8 0] induced by either the (k, m) = (5, 8) or (8, 5)T4 class of plates in Table 1.

The variant level modeling (Figs. 9 and 10) identifies thatthe most stressed slip systems near favored martensite plates

are (1 0 1)/[0 1 0] and (1 0 1)/[0 1 0], consistent with post-

mortem STEM observations in one- and two-cycle pillars(Figs. 47). Thus, the variant modeling supports the hypoth-esis that the dislocation loop arrays observed in the STEMstudies are driven into the austenite by the local stress fieldof these transforming martensite variants. It is not clear what

stabilizes these loop structures in spite of the large line ten-sion forces that act to collapse them. The detailed formationof these loop arrays during/after the transformation is likelyto require in situ observation techniques.

4.2. Comparison with previous single-crystal and micropillar

testing

Frick et al. [21,22] reported a loss of pseudoelasticity asthe diameter of aged and solutionized single-crystal NiTi pil-lars is reduced from $1 lm; the loss is especially pronouncedin the diameter range $400200 nm. Further, the transitionsare not sharp as observed in the present work for [1 1 0]-ori-ented 5 and 20 lm diameter samples. There are possiblesources for these differences. First, the specimens in Fricket al.s work have rounded top surfaces, nonuniform cross-sectional areas due to tapering of sample walls, and are notaligned with a goniometer as in the present study. Theseissues become more pronounced for submicron samples,inducing localized deformation at the top, bending, devia-tions from uniaxial loading, and a large, nonlinear toe-inregion at the beginning of a test. Second, the compressionaxes in each study are different. However, the loss in pseudo-elastic behavior is observed for several orientations, rangingfrom [1 1 1] compression which favors plasticity, to [1 0 0]

which is intermediate, to [2 1 0] which favors transforma-

Fig. 10. Spatial distribution of slip systems with a resolved shear stress exceeding 1500 MPa for a stress axis of [7 8 0], on planes (a) x1 = 1.0125a, (b)x2 = 1.0125b and (c) x3 = 1.05c located just outside the faces of a martensite plate. These calculations assume aT4 type plate with ( k, m) = (8, 5), asidentified in Table 1. Slip system numbers correspond to those listed in Table 2.



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tion. The [1 1 0] compression axis in the present work isintermediate.

Overall, a comparison of the literature and present worksuggests that in solutionized form, the plateau value of for-ward transformation stress may not depend sensitively onsample diameter, at least down to $2 lm. In particular,

Frick et al. [22] reported $600 MPa for a $2 lm solution-ized sample. This is comparable to the $600 MPa value forthe 5 and 20 lm [1 1 0]-oriented pillars in the present work.The bulk [1 1 0] solutionized value is $60 MPa smaller,when adjusted for composition differences (Section 3.1).The martensite Schmid factor (MSF) % 0.43 in all of thesecases [20]. However, Frick et al. did not report a NiTi com-position, making exact comparison with their work diffi-cult. Despite the similar plateau values, the present workdisplays the sharpest transitions in stressstrain response,suggesting that micron-scale samples, when carefullymachined and aligned, can be used to study the abruptonset of the transformation.

5. Conclusions

Microcrystal compression specimens (5 and 20 lm) werefabricated from a large [1 1 0] grain in a solutionized50.7 at.%NiTi polycrystalline sample. Utilizing STEMtechniques, the substructure evolution was studied as afunction of loading cycle. The observations are:

At the sizes explored, there is no evidence to suggest acomplete loss in pseudoelastic behavior, as observed

by Frick et al. [21,22] for compression microcrystals inthe 200400 nm range. The compression results indicate a sharp elastic/trans-

formation transition, suggesting that little plasticityoccurs prior to the onset of the martensitictransformation.

Type a[0 1 0]/(1 0 1) dislocations are observed bothinside and at the periphery of the transformation zone.Those within the transformation zone are arranged asarrays of dislocation loops, while those at the peripheryhave a distinctive arrowhead morphology.

Four twinning modes that probably produced the rem-nant bands of dislocations are identified from 192 possi-

ble modes, based on a plate-level micromechanics modelthat employs the phenomenological theory of martensitetransformations.

The twin interface predicted for these four twinningmodes aligns with the observed planar arrays of disloca-tion loops, suggesting that loop formation may be dri-ven by local stress fields at the scale of individualvariants.

A variant-level micromechanics model reveals that theobserved (1 0 1)/[0 1 0] slip system is highly stressed bythe four likely twinning modes, in B2 regions that arejust outside the martensite plate and parallel to twin

interfaces. Further consideration of the small misorien-

tation (4) from the [1 1 0] deformation axis suggeststhat the most highly stressed martensitic plate systemis activated. An estimate of the stress work for activa-tion at room temperature is re* $ 34 MJ m3.

Application of the crystallographic theory of martensitetransformations to experimental STEM images offers a

powerful method to study the coupled interactionbetween transformation and plasticity. This preliminarytheoryexperiment approach suggests that fine-scaleplasticity is produced on a scale and orientation com-mensurate with individual martensite phases.

Acknowledgments

D.M.N., P.M.S. and M.J.M. acknowledge the supportof the AFOSR Science and Technology Workforce forthe 21st Century program. P.M.A. and S.M. acknowledgesupport from the Ohio Supercomputer Center (Grant

PAS0676-5) for computing resources in the micromechan-ics modeling.

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