microstructure and mechanical properties of constrained shape-memory alloy nanograins and nanowires

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Acta Materialia 56 (2008) 3558–3567

Microstructure and mechanical properties of constrainedshape-memory alloy nanograins and nanowires

Mathieu Bouville *, Rajeev Ahluwalia

Institute of Materials Research and Engineering, Agency for Science, Technology and Research (A*STAR), Singapore 117602, Singapore

Materials Theory and Simulation, Institute of High Performance Computing, Agency for Science, Technology and Research (A*STAR),

Singapore 117528, Singapore

Received 25 January 2008; received in revised form 25 March 2008; accepted 27 March 2008Available online 30 April 2008

Abstract

We use the phase-field method to study the martensitic transformation at the nanoscale. For nanosystems such as nanowires andnanograins embedded in a stiff matrix, the geometric constraints and boundary conditions have an impact on martensite formation, lead-ing to new microstructures – such as dots aligned on a square lattice with axes along h01i – or preventing martensite formation alto-gether. We also perform tension tests on the nanowires. The stress–strain curves are very different from bulk results. Moreover, theyare weakly affected by microstructures – the mechanical response of nanowires with different microstructures may be similar, while nano-wires with the same microstructure may have a different mechanical behavior. We also observe that at the transition temperature, orslightly below it, the narrowest wires behave pseudoelastically whereas wider wires are in the memory-shape regime. Moreover, the yieldstress does not change monotonically with width: it has a minimum value at intermediate width.� 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Nanomaterials; Phase-field simulations; Martensite; Mechanical testing; Shape-memory

1. Introduction

Martensitic transformations are displacive (diffusionless)phase transformations from a high-temperature high-symmetry austenite phase (usually cubic) to a low-tempera-ture low-symmetry phase known as the martensite phase(typically tetragonal, orthorhombic or trigonal). Thus thetransformation is accompanied by strain. Minimization ofelastic energy in the martensite phase results in the formationof a complex microstructure of the crystallographic variants.

The phase transformation and the complex microstruc-ture are responsible for unusual mechanical properties,which make materials undergoing martensitic transforma-tions useful for many technological applications [1,2].Martensite can exhibit the shape-memory effect, i.e. the

1359-6454/$34.00 � 2008 Acta Materialia Inc. Published by Elsevier Ltd. All

doi:10.1016/j.actamat.2008.03.041

* Corresponding author.E-mail addresses: [email protected] (M. Bouville), rajeev

@ihpc.a-star.edu.sg (R. Ahluwalia).

existence of a residual strain upon unloading that can berecovered upon heating. It can also have a pseudoelasticbehavior – a macroscopic deformation which is completelyrecovered when the load is removed. In the shape-memoryregime, stress–strain curves are characterized by a residualstrain when the stress goes down to zero after unloading,whereas with pseudoelasticity there is no residual strainand the system reverts to its initial state.

While martensitic transformations and microstructuresin bulk are well understood, the behavior of these transfor-mations at the nanoscale still requires investigation. For thinfilms [3–6], nanowires [7,8] and nanocrystals [9,10], the geo-metric constraints will have an impact on martensite forma-tion. Recently, Frick et al. [11] experimentally studied thestress–strain behavior of nanopillars of nickel–titaniumshape-memory alloys (SMA) using nanoindentation tests.They found that stress-induced martensitic transformationinitiates at relatively low stresses. Wang and Vlassak [10]studied the phase transformation behavior of grains of

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M. Bouville, R. Ahluwalia / Acta Materialia 56 (2008) 3558–3567 3559

nickel titanium embedded in an amorphous matrix at differ-ent grain sizes. They found that the nature of the phase trans-formation changes, indicating that geometry and boundaryconditions play key roles. Similarly, some applications ofmartensites make use of an SMA element which is embeddedin a matrix of a non-transforming material [12]. Since thesenanosystems are embedded in a matrix, there are geometri-cal constraints on the phase transformation and microstruc-tures: the martensite variants should arrange so that there isno macroscopic shape change (in the absence of externalstrain). Geometry also affects mechanical behavior. Thisleads us to the particular question of the effect of the sizeof the nanograins and nanowires – the smallest systems areexpected to behave very differently from the bulk. An under-standing of the size effects in such systems may provideinsights for further research into nanoscale devices that useshape-memory materials.

The effects of clamping [13,14] and non-transforminglayers [15,16] have been studied theoretically. However,the influence of geometry and size has not been systemati-cally studied. We use the phase-field method to study theeffect of size, shape and transformation strains on the for-mation, microstructure and mechanical properties of mar-tensite in the case of nanowires and nanograins embeddedin a non-transforming matrix.

1 For interpretation of color in Figs. 1,2,4–6,8,11–14,16, and 20–25 thereader is referred to the web version of this article.

2. Phase-field model

The free energy of our two-dimensional (2D) system isbased on the usual non-linear elastic free energy densityfor a square-to-rectangle martensitic transition [17–21]

G ¼Z

A22

2

T � T m

T m

ðe2Þ2 �A24

4ðe2Þ4 þ

A26

6ðe2Þ6

�

þ A1

2½e1 � x12ðe2Þ2�2 þ

A3

2ðe3Þ2 þ

k2kre2k2

�dr ð1Þ

Here T is the temperature and T m is the austenite–martens-ite transition temperature. If the martensite unit cell is big-ger or smaller than the austenite unit cell, then martensiteformation ðe2 6¼ 0Þ generates hydrostatic strain ðe1 6¼ 0Þ.This is taken into account in our model by setting x12 6¼ 0(even though our model is 2D, we call x12 a ‘volumechange’ to follow common practice). e1 is the hydro-static strain, e2 is the deviatoric strain and e3 is the shearstrain

e1 ¼ ðexx þ eyyÞ=ffiffiffi2p

ð2aÞe2 ¼ ðexx � eyyÞ=

ffiffiffi2p

ð2bÞe3 ¼ exy ð2cÞ

The feijg are the linearized strain tensor components. Inthe case of a square lattice, exx ¼ eyy so that e2 ¼ 0. Thiscorresponds to austenite. In the case of martensite,exx 6¼ eyy and e2 6¼ 0. The deviatoric strain e2 is thus usedto track austenite and martensite.

To the lowest order in e2, the energy is quadratic with anextremum at e2 ¼ 0. For temperatures above T m, the firstterm in Eq. (1) is positive, so that e2 ¼ 0 is a minimum,i.e. austenite is (meta)stable. On the other hand, ifT < T m, the first term in Eq. (1) is negative and e2 ¼ 0 isa maximum; here the austenite is unstable.

The evolution of the displacements is described by[22,23]

qo

2uiðr; tÞot2

¼X

j

orijðr; tÞorj

þ gr2viðr; tÞ ð3Þ

where q is a density, v is the time derivative of the displace-ments u and the stresses are given by

rijðr; tÞ ¼dG

deijðr; tÞð4Þ

with d the functional derivative. The second term on theright-hand side in Eq. (3) is a viscous damping term; it isa simplification of the more general damping of Ref. [23].We choose values for the parameters corresponding toFePd [24]: A1 ¼ 140 GPa; A3 ¼ 280 GPa; A22 ¼ 212 GPa;A24 ¼ 17� 103 GPa; A26 ¼ 30� 106 GPa and T m ¼ 265 K.The coupling constant x12 is varied in different cases tounderstand the effect of volume changes, as in Refs.[19–21].

In each 2D simulation, single-crystal austenite isquenched to T ¼ 250 K < T m and the system is allowedto transform and equilibrate. The stiff matrix in whichthe nanosystem is embedded is simulated by fixing the dis-placements to zero at the boundary (for nanowires thereare periodic boundaries along the axis of the wire).

3. Microstructures

3.1. Nanowires

Figs. 1 and 2 show the microstructure of nanowires (forlow and high values of x12, respectively). The length of thewires is 2 lm and their width w is a parameter we vary. Thefigures show only one half of the wire, for clarity. Here andin all other figures, austenite is shown in green and the twomartensite variants are in red and blue, respectively.1 Thewidest wires show twins similar to those observed in bulkmartensite, Fig. 1a. One can see that the twins do notextend unhindered to the interface. As long as this zone,which differs from the bulk, is thinner than the wire, themicrostructure is that of the bulk. However, in the caseof narrower wires, finite size effects dominate. For very nar-row wires, the martensitic transformation is completelysuppressed (Fig. 1f). For intermediate widths, e.g.Fig. 1d, the system exhibits a microstructure that is notobserved in bulk systems [19,20]: dots aligned on a square

Fig. 1. Microstructure of nanowires for several values of their width w.x12 ¼ 0 (except in c, where x12 ¼ 2). (a) w ¼ 200 nm, (b) w ¼ 90 nm, (c)w ¼ 86 nm, (d) w ¼ 72 nm, (e) w ¼ 71 nm, and (f) w ¼ 68 nm. Green:austenite; red and blue: martensite.

Fig. 2. Microstructure of nanowires for several values of their width w

and of the volume change x12. (a) w ¼ 90 nm; x12 ¼ 10; (c) w ¼ 74 nm;x12 ¼ 8; (b) w ¼ 72 nm; x12 ¼ 9; and (d) w ¼ 70 nm; x12 ¼ 10. Green:austenite; red and blue: martensite.

Fig. 3. Microstructure of nanowires and nanograins as a function of thevolume change x12 and of the system size. (a) Nanowire, (b) squarenanograin with faces along h01i, (c) square nanograin with faces alongh11i, and (d) circular nanograin. The microstructures of the nanowires areshown in Figs. 1 and 2 and those of the nanograins in Figs. 4–6.

3560 M. Bouville, R. Ahluwalia / Acta Materialia 56 (2008) 3558–3567

lattice with axes along h01i. For these widths, the transfor-mation is not completely suppressed and thus we observeretained austenite along with the two martensite variants,especially in Fig. 1e.

In some materials systems, there is a significant volumechange associated with the transformation (the volume ofthe martensite unit cell is different from that of austenite),so that the phase transformation generates hydrostaticstress, thus affecting the microstructure [19–21]. Figs. 2aand c show that, in the presence of a volume change, micro-structures change somewhat: dots may become triangularand/or elongated, and twins are no longer parallel.

There is also a range of widths (especially at large x12)exhibiting a mixture of twins and dots, Fig. 1c. For inter-mediate widths, we see clustered variants (Figs. 2b andd). For w ¼ 71 nm, the system exhibits dots, but these donot reach the usual strain associated with martensite andthere are only two of them across the wire rather thanthree, as shown in Fig. 1e. We obtain a ‘microstructure dia-gram’ – akin to a phase diagram – showing the microstruc-ture (austenite, twins, dots) as a function of the width ofthe wire and of volume change (Fig. 3a). There are five

possible microstructures for decreasing sizes: twins, twinplus dots, dots, clustered variants and austenite. Thevolume change x12 plays a minor role in determiningmicrostructures.

3.2. Nanograins

Similar work is carried out for nanoscale grains embed-ded in a non-transforming matrix. We study their micro-

Fig. 6. Microstructures for a circular nanograin. (a) Diameter of grain ¼150 nm; x12 ¼ 5; (b) diameter ¼ 142 nm; x12 ¼ 10; and (c) diameter ¼150 nm; x12 ¼ 10.


structure as a function of shape, size and orientation, aswell as volume change. We consider the cases of circulargrains and of square grains (with sides oriented either alongthe h01i direction of the matrix or along the h11idirection).

Fig. 4 shows microstructures and Fig. 3b the microstruc-ture diagram for a square grain with faces along h0 1i (notethat Fig. 4a is similar to the results of Jacobs [13]). Thetrend is similar to nanowires, with twins, dots and austenitesucceeding as the grain shrinks. A noteworthy differencebetween Fig. 3b and a is that the dependence on x12 is veryweak in the case of nanowire but noticeable for squaregrains. Also, no coexistence of twins and dots can beobserved because grains are too small to have severaldomains (they are small along both directions, whereasnanowires are nano only along one direction).

This dependence on x12 is even stronger in the case of asquare grain with faces along h11i, Fig. 3c: there can be nodot microstructure if x12 ¼ 0. Moreover, there exists anasymmetric dot microstructure for high x12 and intermedi-ate size, as shown in Fig. 5c. One can also notice that forsquare grains with faces along h11i the transition occursfor larger systems than in the case of square grains withfaces along h01i. The microstructure shown in Fig. 5a issimilar to results of spinodal decomposition in nanograins(Justin Song, private communication).

For circular grains (Figs. 3d and 6), the dependence onx12 is even stronger, with no dots for x12 < 7. As in othercases, the martensite–austenite transition depends weaklyon the volume change x12.

The results in this section show the important role geo-metric constraints can play in determining the microstruc-ture of nanowires and nanograins. In particular, certain

Fig. 4. Microstructures for a square nanograin with faces along h01i. (a)Side of square ¼ 120 nm; x12 ¼ 0; (b) side ¼ 103 nm; x12 ¼ 4:5; and (c)side ¼ 120 nm; x12 ¼ 10.

Fig. 5. Microstructures for a square nanograin with faces along h11i. (a)Side of square ¼ 116:7 nm; x12 ¼ 0; (b) side ¼ 126:6 nm; x12 ¼ 10; and (c)side ¼ 118:1 nm; x12 ¼ 10.

microstructures cannot exist in bulk martensite and ariseonly at the nanoscale.

4. Mechanical testing

4.1. Dynamic loading simulations

In the previous section we have shown that nanowiresand nanograins exhibit microstructures that are not foundin the bulk. An obvious question is whether this will affectthe mechanical properties of nanoscale systems. In this sec-tion and the next, we will simulate tension tests on nano-wires to determine how the mechanical response ofnanowires differs from that of the bulk.

A wire of length 2 lm is annealed until the microstruc-ture no longer evolves. It is then loaded at a constant strainrate along its axis (in tension) from times 0 to 70 (arbitraryunits), then unloaded. At every point the strain exx is givenby

exx ¼oux

oxþ _et

In this section, the strain rate will be 0.05% per time unitand there will be no volume change ðx12 ¼ 0Þ. In the nextsection, these two parameters will be varied to study theirimpact on microstructures and mechanical properties.Except in Section 4.5 the temperature will be T = 250 K(which is lower that the transition temperature,Tm = 265 K).

4.2. Difference between nanowire and bulk

In the simulations, the nanowire is embedded in a stiffmatrix so that eyy ¼ 0. Consequently, e1 ¼ e2 ¼ exx=

ffiffiffi2p

.Assuming that e3 is negligible and using x12 ¼ 0, the energybecomes

g ¼ A22

4

T � T m

T m

þ A1

4

� �ðexxÞ2 �

A24

16ðexxÞ4 þ

A26

48ðexxÞ6

Since rxx ¼ dg=dexx, we have

rxx ¼A22

2

T � T m

T m

þ A1

2

� �exx �

A24

4ðexxÞ3 þ

A26

8ðexxÞ5 ð5Þ

We also performed a bulk simulation (with periodicboundary conditions) of the loading process, startingfrom a twinned martensitic state. Fig. 7 shows that the

Fig. 7. Stress–strain curves. Dotted line: analytical curve from Eq. (5) fora bulk system; dashed line: simulation for a bulk system, and solid line:simulation for a nanowire of width w ¼ 1000 nm. (The dots correspond tothe microstructures shown in Fig. 8.)

Fig. 8. Microstructure evolution of a nanowire of width w ¼ 1000 nmduring a tension test. (a) t ¼ 0; e ¼ 0%; (b) t ¼ 5; e ¼ 0:25%; (c) t ¼ 30; e ¼1:5%; (d) t ¼ 35; e ¼ 1:75%; (e) t ¼ 40; e ¼ 2%; (f) t ¼ 43; e ¼ 2:15%; and(g) t ¼ 140; e ¼ 0%. (For each time, the simulated system is shown twiceover.)


stress–strain curve for the bulk system is very close tothe analytical result of Eq. (5). The bulk system eventu-ally reaches a single variant; on unloading it follows thesame stress–strain curve, so that there is no residualstrain.

Fig. 7 shows that the stress–strain curve for a relativelywide wire is very different from both the analytical solutionand the bulk simulation. Notice that there is a plateau inthe stress–strain curve. The microstructural evolution ofthis nanowire is shown in Fig. 8. There exist two martensitevariants, with positive and negative values of e2 (corre-sponding to rectangular unit cells elongated along the x-axis and the y-axis, respectively). When exx increases dueto the loading, e2 ¼ ðexx � eyyÞ=

ffiffiffi2p

also increases. The var-iant corresponding to e2 > 0 (in red in Fig. 8) thus becomesincreasingly favored while the variant with e2 < 0 (blue)becomes higher and higher in energy. The former thusgrows and the latter shrinks, so that eventually a single var-iant remains.

In the linear part of the stress–strain curve for this nano-wire, there is no change of microstructure (compare Figs.8a and b), this part corresponds to the elastic response ofthe martensite. When the strain becomes too large to beaccommodated elastically, the elongation of the wirerequired by the external strain comes from the switchingof the unfavorable variant (blue) to the favorable one(red). A consequence is that the unfavorable twins shrink– compare Figs. 8b and c. Each reduction of the twin sizedecreases the strain, which compensates the increase ofthe external strain. The result is that the stress does notchange and a plateau can be observed in the stress–straincurve. However, the twin width cannot decrease indefi-nitely: there is a critical size below which the interfaceenergy dominates so that very thin twins are unstable.When the twins reach this size, strain is reduced by the dis-appearance of some twins of the unfavorable variant. Thesystem then loses its periodicity (Figs. 8d–f). Each time atwin disappears there is an abrupt drop of the stress, whichresults in oscillations in the stress–strain curve. Eventuallymultiple twins disappear in a short time interval, which

leads to a noticeable decrease of the stress around point fin Fig. 7. Fig. 8f corresponds to the point when the lasttwin of the unfavored variant is finishing to disappear com-pletely. After that, the stress–strain curve merely shows theelastic response of a single martensite variant. Similarstress–strain curves have been observed experimentallyand in simulations [23].

4.3. Effect of nanowire width

Fig. 7 shows that the behavior of a nanowire (even arather wide one) is different from the bulk. This naturallyleads to the question of how the mechanical behavior


changes with the width of the nanowires. In particular, onecan expect that narrower nanowires will be even more dif-ferent from the bulk than are the wider wires. Fig. 9 showsstress–strain curves for several nanowire widths [25]. Notethat as the width is decreased, the martensite yield stressdecreases, i.e. it becomes easier to move the twin bound-aries. It is instructive to compare this behavior to the tem-perature dependence of the stress–strain curves shown inFig. 10: the effect of decreasing the width shows similaritieswith a decrease of temperature. On the other hand, Fig. 9shows that the residual strain is very weakly dependenton the width. We should also point out that the stress–strain curves with negative stresses we observe are a conse-quence of the strain loading. In strain loading, we can placethe system in mechanically unstable regions of the stress–strain curves.

It is possible that the differences in the mechanical behav-ior spring from microstructural differences (we observed inthe previous section that the microstructure was stronglyaffected by the width of the nanowire). Fig. 11 shows the

Fig. 10. Analytical stress–strain curves for different temperatures.

Fig. 9. Stress–strain curves (loading only) for several nanowire widths.Microstructures are shown in Figs. 8 and 11–16.

microstructure evolution for a narrow wire ðw ¼ 50 nmÞ.In Fig. 11 (and in other figures, unless otherwise specified)only half of the length of the nanowire is shown; austeniteis in green and the two variants of martensite are in redand blue.1 Since such a wire does not form martensite inthe absence of strain (see Fig. 3a), at the beginning of thetension test the wire is completely austenitic. The systemis more or less uniform throughout the transformation –no twins or dots form. The contrast observed in Fig. 11ais rather weak (compare to Fig. 8) and is closer to thegrowth of an instability than to the nucleation ofmartensite.

Fig. 12 shows the time evolution of the microstructureof a wire of width w ¼ 66 nm, i.e. one of the widest wiresthat is austenitic at equilibrium. Figs. 12a and b show thatmartensite appears in the form of dots of the favored var-iant (shown in red1 in Fig. 12). Note that this is differentfrom the microstructure of Fig. 1d, as it is made of onemartensite variant and austenite, rather than of the twomartensite variants. The austenite then disappears com-pletely and the system in completely martensitic (Fig. 12c).

Fig. 13 shows the microstructure evolution of a wire (ofwidth w ¼ 80 nm) that is made of martensite dots at equi-librium (Fig. 13a). Fig. 13b shows that, upon loading, thedots of the favorable variant (in red1) expand while thedots of the other variant (blue) shrink or transform to aus-

Fig. 11. Microstructure evolution of a nanowire of width w ¼ 50 nmduring a tension test. (a) t ¼ 20; e ¼ 1%; (b) t ¼ 35; e ¼ 1:75%; and (c) t ¼140; e ¼ 0%.

Fig. 12. Microstructure evolution of a nanowire of width w ¼ 66 nmduring a tension test. (a) t ¼ 10; e ¼ 0:5%; (b) t ¼ 15; e ¼ 0:75%; (c) t ¼ 35;e ¼ 1:75%; and (d) t ¼ 140; e ¼ 0%.

Fig. 13. Microstructure evolution of a nanowire of width w ¼ 80 nmduring a tension test. (a) t ¼ 0; e ¼ 0%; (b) t ¼ 10; e ¼ 0:5%; and (c) t ¼140; e ¼ 0%.

Fig. 14. Microstructure evolution of a nanowire of width w ¼ 90 nmduring a tension test. (a) t ¼ 0; e ¼ 0%; (b) t ¼ 10; e ¼ 0:5%; (c) t ¼ 25;e ¼ 1:25%; and (d) t ¼ 140; e ¼ 0%.

Fig. 15. Stress–strain curves for several nanowire widths. For each width,the top curve corresponds to loading and the bottom one to unloading.

Fig. 16. Microstructure evolution of a nanowire of width w ¼ 200 nmduring a tension test (unloading). (a) t ¼ 113; e ¼ 1:35%; (b) t ¼ 113:2;e ¼ 1:34%; and (c) t ¼ 113:6; e ¼ 1:32%. (Unlike in other figures, the wholesystem, rather than only one half, is shown.)


tenite. Finally a single variant remains, similar to Fig. 12c.(Wires of all widths up to about 250 nm have the samemicrostructure at t ¼ 35, i.e. e ¼ 1:75%, as shown inFig. 12c.)

Fig. 14 shows the microstructure evolution of a wire ofwidth w ¼ 90 nm; this is the narrowest wire exhibitingtwins at equilibrium. As expected, the twins correspondingto the favored variant grow and the unfavorable variantshrinks. Fig. 14b also shows a change in microstructurein some parts of the wire: some of the twins turn intosquare dots with faces along h11i.

Even though the microstructure evolution is quite differ-ent between w ¼ 50 nm (initially austenitic, no pattern for-mation; Fig. 11), w ¼ 66 nm (initially austenitic, forms dotsupon loading; Fig. 12), w ¼ 80 nm (initially made of mar-tensite dots; Fig. 13) and w ¼ 90 nm (initially twinned;Fig. 14), Fig. 9 shows that there is barely any differencein the mechanical response. The mechanical behavior ofthese narrow wires is not directly influenced by whetherthere is martensite initially or the way martensite forms,i.e. the mechanical properties of these narrow wires areindependent of the microstructure. (Note that the stressalways increases initially, before decreasing; even thoughthis linear increase may be small it always exists.)

4.4. Microstructural evolution during unloading

So far we have focused on the loading of the nanowires.There is a microstructure difference upon unloading too:when the strain is returned to zero (at t ¼ 140), the wiresof widths w ¼ 50 and 66 nm are austenitic (Figs. 11c and12d), while wider wires exhibits martensite dots (Fig. 13c)or twins (Fig. 8g). Wires of width w ¼ 90 nm exhibit amicrostructure after unloading that is different from theinitial configuration (compare Fig. 14d to a).

The stress–strain curves for loading and unloading arealmost identical for narrow wires; the difference betweenthem increases as the nanowire width increases, as seen inFig. 15. Since one expects that wide enough wires shouldbehave like the bulk, i.e. pseudoelastically, there must bea transition from shape-memory to pseudoelastic at somesufficient width. (Since we did not observe it, we canassume that the necessary width is too great to be compu-tationally tractable.)

One can notice a sharp increase in stress arounde ¼ 1:32% in Fig. 15 during the unloading of the wire ofwidth w ¼ 200 nm. Fig. 16 shows the microstructural evo-lution of this nanowire between e ¼ 1:35% and e ¼ 1:32%.A new variant (in blue1) is visible in Figs. 16b and c but isabsent from Fig. 16a. The jump in the stress–strain curve istherefore due to the single domain martensite state becom-ing unstable, so that the system switches to multipledomains.

4.5. Martensite yield stress

Fig. 17 shows, as a function of the width w, the value ofthe yield stress (i.e. the maximum value of the stress afterthe initial linear part in Fig. 9). One can see in Fig. 17 thatthe maximum value reached by the stress upon loadingincreases with the wire width (closed symbols). There areseveral regimes: for widths lower than about 200 nm,ryield has a power law dependence on w with a power of1.25, whereas the power is 2 for wider wires. For very nar-row wires, however, the yield stress is even larger.

Fig. 18 shows stress–strain curves for T ¼ 265 K (i.e. thetransition temperature). At this temperature, curves areshifted toward lower stresses for increasing thickness untilabout 100 nm. For wider wires the stress increases withthickness. Very narrow wires ðw < 30 nmÞ behave pseudo-elastically, whereas wider nanowires are in the shape-memory regime. (At T ¼ 250 K, all nanowires are in theshape-memory regime.) Fig. 17 shows the yield stress as afunction of the nanowire width. For T ¼ 265 K (open

Fig. 18. Stress–strain curves for nanowires of various widths at T = 265 Kand x12 ¼ 0.

Fig. 17. The martensite yield stress as a function of the width w of thenanowire, for different temperatures.

Fig. 19. The minimum value of the stress rxx in the stress–strain curve ofFig. 9 as a function of the width w of the nanowire. Crosses are the strainsat which the stress is minimum and empty triangles correspond to rmin fora faster loading of _e ¼ 0:5% per unit time.

Fig. 20. Microstructure of a nanowire of width w ¼ 130 nm at t ¼ 25 of atension test (corresponding to e ¼ 1:25%).


symbols), the yield stress decreases then increases withnanowire width, with a minimum around w ¼ 100 nm.Wires narrower than about 18 nm do not exhibit a yieldstress: the stress increases monotonically with the strain(see e.g. w ¼ 10 nm in Fig. 18). The yield stress obeys apower law of exponent �1.5 for narrow wiresðw 6 50 nmÞ and another power law for wide wires (ifw P 200 nm, the exponent is 1.8). For higher tempera-tures, such as T = 260 and 263 K, there are also tworegimes, but the minimum is reached at lower and lowerwidths as the temperature decreases. On the other hand,for the wider wires the yield stress is essentially the sameat all five temperatures.

4.6. Going to a single variant

Fig. 9 shows that the minimum value reached by the stressrxx upon loading and the strain at which this minimum isreached vary little for narrow wires but increase for widerwires. Fig. 19 shows these as a function of the nanowire widthw. There are three regimes for the stress: for narrow wiresðw 6 50 nmÞ; rmin � 439:2 expð�w=13:4Þ � 367:5; for wires

wider than 300 nm, rmin increases with the width linearlyðrmin � 0:45w� 405Þ; the smallest value of rmin is obtainedin the transition regime in-between, for nanowires 130 nmwide. (The linear fit of Fig. 19 is based on data for nanowirewidths up to w ¼ 2000 nm.)

The strain at which the stress is minimum (shown ascrosses in Fig. 19) also exhibits several regimes: for nano-wires narrower than about 130 nm, this strain does notdepend on width, but it increases with w for wider wires(above 300 nm, the dependence on width is weaker thanbetween 130 and 300 nm). Fig. 20 shows that a nanowireof width w ¼ 130 nm exhibits clear twins of the unfavoredvariant (blue1), whereas a nanowire of width w ¼ 90 nmdoes not (Fig. 14c). The change of regime aroundw ¼ 130 nm seems to come from these unfavorable twins,which require higher and higher strains to disappear whenthe thickness of the wire increases.

5. Effects of volume change and strain rate

All tension tests presented in the previous section werecarried for a system that does not have a volume change,i.e. x12 ¼ 0 in Eq. (1), and for a strain rate of 0.05% pertime unit. In the present section, we will study the conse-quences for microstructures and mechanical properties ofnanowires of varying these parameters. All simulationsare at T ¼ 250 K.

5.1. Volume change

The presence of a volume change (we will focus on thecase of x12 ¼ 10) has little effect on the microstructure ofnarrow nanowires (w ¼ 50 nm) (compare Figs. 21 and11a). For nanowires of width w ¼ 66 nm, the volume

Fig. 21. Microstructure of a nanowire of width w ¼ 50 nm for a volumechange x12 ¼ 10 at t ¼ 20 of a tension test (corresponding to e ¼ 1%).

Fig. 22. Microstructure of a nanowire of width w ¼ 66 nm for x12 ¼ 10 att ¼ 10 of a tension test (corresponding to e ¼ 0:5%).

Fig. 23. Microstructure evolution of a nanowire of width w ¼ 80 nmduring a tension test (x12 ¼ 10). (a) t ¼ 0; e ¼ 0%; (b) t ¼ 10; e ¼ 0:5%; (c)t ¼ 20; e ¼ 1%; and (d) t ¼ 35; e ¼ 1:75%.

Fig. 24. Microstructure evolution of a nanowire of width w ¼ 90 nmduring a tension test ðx12 ¼ 10Þ. (a) t ¼ 0; e ¼ 0% and (b) t ¼ 10; e ¼ 0:5%.

Fig. 25. Microstructure evolution of a nanowire of width w ¼ 200 nmduring a tension test ðx12 ¼ 10Þ. (a) t ¼ 0; e ¼ 0% and (b) t ¼ 20; e ¼ 1%.(Unlike in other figures the whole system, rather than only one half, isshown.)

Fig. 26. Stress–strain curves for two nanowire widths and two values of x12.

Fig. 27. Stress–strain curves for nanowires of widths w ¼ 50 nm andw ¼ 200 nm at strain rates of 0.05%, 0.5%, and 5% per time unit.


change noticeably affects the microstructure, while the sys-tem forms martensite dots for x12 ¼ 0 (Fig. 12a) there arenone if x12 ¼ 10 (Fig. 22). For wider wires (w = 80 or90 nm), the microstructure evolution in the presence of avolume change is also different from what is observed whenx12 ¼ 0 (see Figs. 23 and 24). While in Figs. 13 and 14 thefavored martensite variant (in red1) grows at the expense ofthe other variant (in blue), for x12 ¼ 10 the unfavorablevariant disappears almost completely but the favorableone shrinks too. This is especially noticeable in Fig. 23b.In other words, the system evolves from both martensitevariants to a single variant by first turning into a mixtureof austenite and (the favored variant of) martensite. Foreven wider wires ðw ¼ 200 nmÞ, the microstructure evolu-tion is similar to what is observed for x12 ¼ 0 (compareFigs. 25 and 16).

Fig. 26 shows stress–strain curves with and without vol-ume change. When x12 ¼ 10, the minimum stress is lower inmagnitude and reached at a higher value of the strain. One

can note that for w ¼ 200 nm there is no sudden increase ofstress upon unloading.

5.2. Strain rate

Fig. 27 shows that the stress–strain curves are affectedby the strain rate. Higher strain rates shift the curvetowards higher stress for loading and lower stress forunloading; consequently, the hysteresis increases with thestrain rate. This behavior is due to a competition betweenthe loading rate and the time scales of domain formationand motion. For slow loading, there is sufficient time fordomains to move and the yield stress is low. Upon slowunloading, the multi-domain state has time to form andtherefore the instability occurs at higher strain (i.e. earlier).Since faster loading does not give twin boundaries time tomove, part of the strain is accommodated elastically ratherthan through phase transformation, which raises the stress.This effect is greater for wider wires.

Fig. 28 shows that for faster loading the value of ryield

increases for all widths, but this effect is more noticeablefor wider wires. On the other hand, in Fig. 19 rmin doesnot depend on the strain rate for narrow wires (up to aboutw ¼ 70 nm).

Fig. 28 also shows that for lower strain rates (0.005% pertime unit) ryield does not depend on the width of the nano-wire (full squares in the figure). Table 1 sums up the depen-dence of ryield upon the nanowire width and the strain rate.

Fig. 28. The maximum value of the stress rxx in the stress–strain curve ofFig. 9 as a function of the width w of the nanowire for several strain rates.

Table 1Dependence of ryield upon nanowire width w and strain rate _e

ryield Slow Fast

Narrow independent of w / w/ _e�1=4 / _e

Wide / wnðn < 2Þ / w2

/ _e / _e


6. Conclusion

6.1. Limitations

One limitation of the present work is that the model is2D. Our 2D model does not account for the martensite var-iant elongated along z. Another issue is that, in threedimensions, the system may deform along the z-axis toaccommodate strain in ways that are missing in our work.A further limitation is that we set displacements to zero atthe boundaries, which corresponds to a clamped system ora system embedded in an infinitely stiff matrix.

We nevertheless believe that many of our observationsare qualitatively valid and would still hold in three dimen-sions or a different matrix. For instance we expect martens-ite formation to disappear for small systems (even thoughthe critical size may be different in three dimensions or withdifferent boundary conditions). The stress–strain curves arealso expected to be similar for 3D systems.

6.2. Summary

We used the phase-field method to study the martensitictransformation in constrained systems at the nanoscale.For nanosystems such as nanowires and nanograins, thegeometric constraints can lead to new microstructures orprevent martensite formation altogether. For all geome-tries, we observed a new microstructure – made of dotsaligned on a square lattice with axes along h01i – of inter-mediate size. In the case of a square grain with faces alongh11i, we also observed an asymmetric microstructure,which is not found in other geometries.

We also performed uniaxial tension tests on the nano-wires. The stress–strain curves are very different frombulk results and a size dependence of mechanical proper-ties is observed. Interestingly, the martensite yield stressshows non-monotonic behavior with respect to the nano-wire width: as the width is decreased, the yield stressdecreases until a critical thickness below which it startsto increase again. At T ¼ 265 K, the wider wires exhibita residual strain and are in the shape-memory regimewhereas the narrower wires show pseudoelastic behavior.In contrast, at T ¼ 250 K a residual strain is observedfor all widths. Another interesting result is that thestress–strain curves are not affected by microstructures– the mechanical response of systems with differentmicrostructures may be similar, while systems with thesame microstructure may have a different mechanicalbehavior. Finally, we also noticed that strain rate andvolume change associated with the martensitic transfor-mation have an impact of the mechanical response andon microstructures.

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1995;52:803.[25] Nanowires of width 2000 nm are 1000 nm long instead of 2000 nm.

Given that these systems have very neatly periodic twins, such smallersystems are not expected to behave noticeably differently (there areperiodic boundaries along the nanowire axis in either case). Simula-tions for w ¼ 1400 nm show the difference between lengths of 1000and 2000 nm to be about 10 MPa.

http://arxiv.org/abs/0712.2098

microstructure and mechanical properties of constrained shape-memory alloy nanograins and nanowires

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