Narrowing of hysteresis of cubic-tetragonal martensitic transformation by weak axial stressing of ferromagnetic shape memory alloy

Narrowing of hysteresis of cubic-tetragonal martensitic transformation by weak axialstressing of ferromagnetic shape memory alloyAnna Kosogor Citation: Journal of Applied Physics 119, 224903 (2016); doi: 10.1063/1.4953868 View online: http://dx.doi.org/10.1063/1.4953868 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/119/22?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Effect of niobium addition on the martensitic transformation and magnetocaloric effect in low hysteresisNiCoMnSn magnetic shape memory alloys Appl. Phys. Lett. 105, 231910 (2014); 10.1063/1.4903494 Microstructures and magnetostrictive strains of Fe-Ga-Ni ferromagnetic shape memory alloys J. Appl. Phys. 113, 17A303 (2013); 10.1063/1.4793609 Martensitic and magnetic transformation in Mn50Ni50−xSnx ferromagnetic shape memory alloys J. Appl. Phys. 112, 083902 (2012); 10.1063/1.4758180 Martensitic transformation and magnetic field-induced strain in Fe–Mn–Ga shape memory alloy Appl. Phys. Lett. 95, 082508 (2009); 10.1063/1.3213353 Magnetization rotation and rearrangement of martensite variants in ferromagnetic shape memory alloys Appl. Phys. Lett. 90, 172504 (2007); 10.1063/1.2730752

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Narrowing of hysteresis of cubic-tetragonal martensitic transformationby weak axial stressing of ferromagnetic shape memory alloy

Anna Kosogora)

Institute of Magnetism, Kyiv 03142, Ukraine and National University of Science and Technology “MISiS,”Moscow 119049, Russia

(Received 24 December 2015; accepted 2 June 2016; published online 14 June 2016)

An influence of axial mechanical stress on the hysteresis of martensitic transformation and ordinary

magnetostriction of ferromagnetic shape memory alloy has been described in the framework of a

Landau-type theory of phase transitions. It has been shown that weak stress can noticeably reduce

the hysteresis of martensitic transformation. Moreover, the anhysteretic deformation can be observed

when the applied mechanical stress exceeds a critical stress value. The main theoretical results were

compared with recent experimental data. It is argued that shape memory alloys with extremely low

values of shear elastic modulus are the candidates for the experimental observation of large anhyste-

retic deformations. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4953868]

I. INTRODUCTION

Ferromagnetic shape memory alloys (FSMAs) undergo

both the ferromagnetic ordering and the martensitic transfor-

mation (MT) on their cooling.1–3 The FSMAs are character-

ized by a number of unique features. In particular, giant (up

to 10%) deformations under a very small (�2 MPa) mechani-

cal stress or a moderate (� 5 kOe) magnetic field attract

much attention of researchers and engineers.3–8 These effects

are known as the superelasticity or magnetic shape memory,

respectively.

It is of common knowledge that the hysteresis is one of

the basic features of MTs, which are the first-order phase

transitions. The manufacturing and the scientific study of the

alloys with small hysteresis of MTs are among the modern

trends in materials science.9–11 In particular, an unusual

anhysteretic stress–strain behavior characterized by a giant

nonlinear deformation (more than 12%) was observed exper-

imentally for the Ni-Fe(Co)-Ga alloy above certain critical

temperature T�, whereas stress–strain loops showing a pro-

nounced hysteresis were obtained below this temperature.12

It has been argued that the reduction of hysteresis can occur

when the temperature of the deformed alloy exceeds T�,because this temperature corresponds to the critical point in

the stress–temperature phase diagram.12 In Ref. 13, it has

been shown experimentally that the temperature hysteresis

of MT in the Fe-Pd alloy decreases by one order of magni-

tude (or even more) with an increase in the absolute stress

value from 5 MPa to 40 MPa. Moreover, vanishing of MT

strain, which is a jump of strain value, was observed when

stress applied to the cooled/heated specimen was larger than

the critical stress corresponding to the critical point of the

phase diagram.13 It should be noted that the existence of

the critical point of MT was predicted much earlier using

the Landau theory of phase transitions,14 but experimental

evidences of this fact were obtained only recently for

Ni-Fe(Co)-Ga and Fe-Pd alloys.12,13

In the present article, Landau theory is used to demon-

strate a possibility of the reduction and disappearance of

temperature hysteresis under a small axial stress applied to

the representative FSMA with physical properties inherent

to Ni-Mn-Ga shape memory alloys. The stress-induced

changes of the MT temperature, MT strain, Young’s modu-

lus, and ordinary magnetostriction of the FSMA, as well as

the coordinates of the critical point at the phase diagram, are

evaluated. The conditions providing the observation of

anhysteretic deformations of a shape memory alloy (SMA)

are formulated.

II. QUANTITATIVE THEORY OF MARTENSITICTRANSFORMATIONS

Landau theory of phase transitions provides a consistent

theoretical description of cubic-tetragonal and cubic-

rhombohedral MTs.14–16 The basic properties of the alloys

undergoing the cubic-orthorhombic MTs are satisfactorily

described by this theory if the shape of the orthorhombic unit

cell of the crystal lattice is rather close to tetragonal. The

influence of the stress on the basic properties of SMAs

exhibiting the cubic-tetragonal and cubic-rhombohedral MTs

is fairly similar, so the case of cubic-tetragonal MT will be

considered below, for the sake of definiteness.

Landau theory describing the cubic-tetragonal marten-

sitic transformations starts from the following expression for

Gibbs potential:

G ¼ c2ðu22 þ u2

3Þ=2þ a4u3ðu23 � 3u2

2Þ=3

þ b4ðu22 þ u2

3Þ2=4� ðrðeff Þ

2 u2 þ rðeff Þ3 u3Þ=6; (1)

where

u2 ¼ffiffiffi3pðexx � eyyÞ; u3 ¼ 2ezz � eyy � exx; (2)

are the order parameter components of the cubic–tetragonal

MT, eik are the strain tensor components, and the values of

the effective stress rðeff Þ2;3 are expressed through the stress

components in the same way as the values u2;3 are related toa)E-mail: [email protected]

0021-8979/2016/119(22)/224903/7/$30.00 Published by AIP Publishing.119, 224903-1

JOURNAL OF APPLIED PHYSICS 119, 224903 (2016)


16:14:28

the strain components. The effective stress is a sum of me-

chanical stress applied to the alloy and magnetoelastic stress

rðeff Þik ¼ rðmeÞ

ik þ rik: (3)

The magnetically induced (magnetoelastic) stress rðmeÞik is

strictly interrelated with ordinary magnetostriction and

appears during ferromagnetic ordering of alloy specimen

(see Refs. 8 and 17 and references therein). The coefficients

c2; a4; and b4 are linear combinations of second-, third-, and

fourth-order elastic modules introduced in Ref. 14.

The equilibrium states of the elastically deformed single

crystal can be determined from the minimum conditions for

Gibbs potential (Eq. (1)). These conditions were analyzed in

Ref. 14. The results obtained are briefly mentioned below

and used then for the illustration of basic properties of the

axially stressed shape memory alloys.

According to Landau theory, the coefficient c2 � c2ðTÞinvolved in Eq. (1) is an increasing linear function of temper-

ature. The equilibrium values of deformations are prescribed

by the conditions @G=@u2 ¼ @G=@u3 ¼ 0, which result in

the equation system

c2ðTÞu2 � 2a4u2u3 þ b4u2 u22 þ u2

3

� �� rðeff Þ

2 =6 ¼ 0; (4)

c2ðTÞu3 þ a4ðu23 � u2

2Þ þ b4u3 u22 þ u2

3


3 =6 ¼ 0: (5)

In the case of a zero effective stress, this system has solu-

tions that correspond to the cubic phase and three variants

of tetragonal phase with four-fold symmetry axes oriented in

x-, y-, and z-directions, respectively.

For the sake of certainty, the phase transition to z-variant

of tetragonal phase may be considered. When the alloy spec-

imen is stressed in [001] or [110] directions, the condition

exx ¼ eyy is fulfilled and the value rðeff Þ2 is equal to zero. In

this case, Eq. (4) has a solution u2 ¼ 0 and Equation (5) is

reduced to the form

u3 c2ðTÞ þ a4u3 þ b4u23


3 =6 ¼ 0; (6)

where the order parameter of cubic-tetragonal MT is related

to the lattice parameters of a tetragonal lattice, cðTÞ and

aðTÞ, and to the transformation strain, eM, as u3ðTÞ¼ 2½cðTÞ=aðTÞ � 1� � 3eM. If the effective stress is equal to

zero, the order parameter is equal to zero in the austenitic

(cubic) phase and is expressed as

uð0Þ3 ðTÞ ¼ �ða4=2b4Þð1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� c2ðTÞ=ct

pÞ (7)

in the martensitic phase (ct � a24=4b4). If the effective stress

is not equal to zero, Equation (6) does not have the solution

u3 ¼ 0, which corresponds to a cubic phase. It means that

the effective stress rðeff Þ3 produces the elastic strain that

reduces the cubic symmetry of the lattice to a tetragonal one.

The MT in an elastically strained lattice is not accompanied

by the symmetry reduction and is referred to as the isomor-

phic (tetragonal-tetragonal) first-order phase transition.

The study of the extremum of potential Eq. (1) enabled

the construction of stress–temperature phase diagrams for

the shape memory alloys undergoing temperature-induced or

stress-induced MTs. This diagram was constructed in Ref. 14.

It was shown that the axially compressed high-temperature

(austenitic) phase is stable if

rðeff Þzz > rLAðTÞ ¼ �16c2

t ½1� RðTÞ�2½1þ 2RðTÞ�=9a4; (8)

where RðTÞ ¼ ½1� 3c2ðTÞ=4ct�1=2and rðeff Þ

zz ¼ rzz þ rðmeÞzz .

The low-temperature (martensitic) phase is stable when

rðeff Þzz < rLMðTÞ ¼ �16c2

t ½1þ RðTÞ�2½1� 2RðTÞ�=9a4: (9)

(Note that the compressive stress is negative.) The inequal-

ities (8) and (9) mean that the plots of the functions rLAðTÞand rLMðTÞ represent the lability lines of austenitic and mar-

tensitic phases at the stress–temperature phase diagram. On

the other hand, the temperatures T2ðrðeff Þzz Þ and T1ðrðeff Þ

zz Þ are

the solutions of the equations rðeff Þzz ¼ rLAðTÞ and

rðeff Þzz ¼ rLMðTÞ, respectively. These temperatures bound the

stability ranges of two phases in the axially stressed alloy

specimen. The austenitic phase is stable if T > T2, while the

martensitic one is stable if T < T1. Equations (8) and (9)

show that the lability lines cross each other if the condition

RðTÞ ¼ 0 is fulfilled. The critical stress determined from this

condition is expressed through the shear modulus, C0ðT1Þ,and lattice parameters as

r� ¼ ð8=27ÞC0ðT1Þ½cðT2Þ=aðT2Þ � 1�: (10)

The critical stress value r� and corresponding to this stress

temperature T� are the coordinates of the critical point on the

stress–temperature phase diagram.

For the second-order phase transition, the lability tem-

peratures are equal to each other and to the phase transition

temperature. Due to this, the second-order phase transition is

characterized by the anhysteretic continuous change of the

order parameter below the phase transition temperature.

Therefore, the hysteresis of phase transition temperatures is

a distinguishing feature of the first-order phase transitions in

comparison with the second-order ones.

The single crystals of FSMAs usually exhibit a jump-

like transformational behavior: a thermally induced strain

arises in these alloys at certain temperature value (see, e.g.,

Ref. 18). In the case when the phase transition from the ther-

modynamically stable to metastable phase takes place, the

maximal jump of strain value can be observed if the whole

specimen transforms from austenitic to martensitic phase at

T ¼ T2 and from martensitic to austenitic phase at T ¼ T1.

In this case, the hysteresis width is maximal and equals to

DTmax � T1 � T2. The value of DTmax tends to zero on

approaching the critical point of phase diagram.

In many cases, the MT goes gradually, through the

mixed austenitic-martensitic state that is observable in the

temperature interval T2 < T < T1. It was recently argued

that in such cases, the main physical property, which prede-

termines the hysteresis width, is the interrelation between

crystal lattices of martensitic and austenitic phases.19,20 The

ideal compatibility of crystal lattices of parent and product

phases provides the conditions for the observation of low

temperature hysteresis. The hysteresis observed for the for-

ward and reverse MTs going through the mixed two-phase

224903-2 Anna Kosogor J. Appl. Phys. 119, 224903 (2016)


16:14:28

state is narrower than maximal hysteresis DTmax and, there-

fore, vanishes when the DTmax tends to zero. The suitability

of the given description of hysteresis was already approved

in a very recent experimental work.13

III. THERMODYNAMIC, ELASTIC, ANDMAGNETOELASTIC CHARACTERISTICS OF THEAXIALLY STRESSED FSMA

A. Phase diagram of martensitic transformation

Ni-Mn-Ga alloys are among the most widely studied

FSMAs, so a representative alloy with physical characteristics

close to those of Ni-Mn-Ga alloys is worth consideration.

Although there exists a wide spread of the reported experimen-

tal values of Young’s modulus E (E � 3C0, C0 is a shear modu-

lus)17 from few tens of gigapascals21 to much smaller values,

between 3 GPa and 15 GPa,22–26 generally speaking, these

alloys are very soft near MT, especially, in the carefully grown

single crystalline form. To express this feature, the value of

Young’s modulus equal to EðT1Þ ¼ 3 GPa is used below for

computations. Moreover, typical experimental values of reverse

MT temperature T1ð0Þ ¼ 296 K and tetragonal distortion of

the cubic lattice cðT2Þ=aðT2Þ � 1 ¼ �0:06 are accepted.5

For the chosen values, the energy parameters a4 ¼ 11 GPa,

b4 ¼ 92 GPa and forward MT temperature T2ð0Þ ¼ 278 K

were estimated as explained in Refs. 25 and 27.

Another important measurable characteristic of FSMA

is a magnetoelastic stress.8,17 This stress causes a spontane-

ous and field-induced magnetostrictive deformation of

FSMA. In the twinned martensitic state, the magnetoelastic

stress can trigger a twin structure rearrangement causing a

giant magnetic field-induced strain effect.5,28,29 In particular,

for the twinned Ni-Mn-Ga alloy exhibiting magnetic field-

induced strain studied in Ref. 5, a theoretically estimated

magnetoelastic stress is negative and approximately equals

�5 MPa.8,17

The lability lines T1ðrÞ and T2ðrÞ of martensitic and

austenitic phases at the stress–temperature phase diagram are

determined from Eqs. (8) and (9) and shown in Fig. 1 as a

functions of stress r � rðeff Þ � rðmeÞ < 0, which is induced

by mechanical load. The maximal hysteresis width is

DTmaxðrÞ ¼ T1ðrÞ � T2ðrÞ. The magnetoelastic stress rðmeÞ

is equal to zero in a paramagnetic phase and is approxi-

mately equal to �5 MPa in a ferromagnetic phase.8,17

Strictly speaking, magnetoelastic stress depends on the mag-

netization value, which, in turn, depends on the temperature.

However, the temperature dependence of magnetization is

disregarded here, because for the chosen values of physical

parameters the lability lines (Eqs. (8) and (9)) lie in the MT

temperature range (from 280 to 300 K), while the Curie tem-

peratures of quasistoichiometric Ni-Mn-Ga alloys are much

higher (of about 370 K). For this reason, the rational magnet-

ization change in the MT temperature range is less than or

equal to 0.1. Both T1 and T2 temperatures increase if the

effective stress is negative and decrease otherwise. The

increase/decrease in MT temperatures reflects the martensite

stabilization/destabilization under the axial load.

Figure 1(a) illustrates that both austenitic and marten-

sitic phases are stable in the wide temperature interval (of

about 20 K) if the effective stress equals zero, that is,

r ¼ rðmeÞ ¼ 0. This is the case of MT occurring in an

unstressed paramagnetic crystal. The positive stress destabil-

izes the martensitic phase and therefore retards the forward

(austenitic-martensitic) transformation of the alloy. A desta-

bilizing stress arises, in particular, during the thermomechan-

ical cycling25 and rotation/deflection of magnetic moments

from the equilibrium direction. It should be noted that the

FIG. 1. Stress–temperature phase diagram of the ultra-soft Ni-Mn-Ga shape

memory alloy exhibiting the martensitic transformation in the paramagnetic,

(a), and ferromagnetic, (b), phases. The domain of stability of austenite lies

above the dashed line, while the stability domain of martensitic phase is situ-

ated under the solid line. (c) The experimental13 (dashed lines) and theoreti-

cal (solid lines) phase diagram obtained for ferromagnetic Fe-Pd alloy.



16:14:28

lability line of the paramagnetic austenitic phase is tangential

to the vertical line r ¼ 0 shown in Fig. 1(a) (for more details,

see Ref. 14). It means that the forward temperature-induced

MT cannot be completed and the vertical line r ¼ const > 0

(which is also shown in Fig. 1(a)) does not cross the lability

line of the austenitic phase. Therefore, the volume fraction of

this phase is not equal to zero. It should be emphasized that

this feature of theoretical phase diagram creates favorable

thermodynamic conditions for the “kinetic arrest” of MT

observed in the Ni-Co-Mn-(Ga,Sn,In) alloys,30–32 because

even small local stresses arising during MT can stabilize the

austenitic phase.

The magnetoelastic stress reduces the hysteresis of MT

in Ni-Mn-Ga alloy to the value of 6 K (see Fig. 1(b)). The

hysteresis of MT temperature range becomes equal to zero

when an additional compressive stress r reaches the value of

�12.5 MPa that corresponds to the critical effective stress r�

(Eq. (10)). The physical meaning of this theoretical result is

explained in Section III B. The experimental phase diagram

measured for Fe-Pd alloy13 is given in Fig. 1(c), where Ms is

the forward transformation start temperature and Af is the

reverse transformation finish temperature. Unfortunately, a set

of parameters needed for the quantitative comparison

of theoretical and experimental results is not measured for Fe-

Pd alloy yet. However, a computation of the maximal hystere-

sis width and its comparison with experimental one can be

carried out using reasonable values of theoretical parameters

eM ¼ 3%, rðeff Þ ¼ �15 MPa, and EðT1Þ ¼ 12:2 GPa. (The

theoretical value eM used for computations is close to the

value resulting from experimental strain-temperature loops,

presented in Fig. 2(b).) The parameters were chosen so that

the theoretical critical stress is equal to the experimental one.

The theoretical and experimental dependences of Ms and Af

on the stress value are presented in Fig. 1(c). The figure shows

that the real hysteresis of MT is much lower than the theoreti-

cal one for the small stress values, but the difference between

the theoretical and experimental values of hysteresis

decreases when the absolute stress value increases from zero

to 20 MPa. It should be emphasized that the theoretical critical

temperature T� ¼ 275 K appeared to be rather close to experi-

mental value of 285 K.

B. Martensitic transformation strain

The spontaneous deformation of the crystal lattice (MT

strain) is one of the main characteristics of MT. The theo-

retical temperature dependence of MT strain of axially

stressed Ni-Mn-Ga alloy is shown in Fig. 2(a). This depend-

ence was computed using Eq. (6). The computations show

that the larger the compressive stress is, the smaller the

jumps of MT strain value are in the course of both forward

and reverse MTs. The jumps disappear at the stress value

r� ¼ r� rðmeÞ � �12:5 MPa and continuous anhysteretic

deformation takes place. It means that the first-order phase

transition does not occur if the stress exceeds the critical

value. This statement also follows from the phase diagram

shown in Fig. 1(b): the cooling process at the fixed stress

value r ¼ const (where r < r� � �12:5 MPa) can be pre-

sented in this phase diagram as the vertical line that does

not cross the lability lines. It means that this process does

not result in the phase transition. Instead, a nonlinear con-

tinuous deformation takes place.

Figure 2(a) shows that the application of mechanical

stress decreases temperature hysteresis and after critical

stress r� hysteresis disappears. Moreover, at temperature

T ¼ 290 K, the MT strain computed for the stresses r ¼0 MPa and r ¼ �5 MPa is approximately equal, whereas the

hysteresis width corresponding to r ¼ �5 MPa is smaller by

a factor 0.5. It is also seen that the mechanical stress

increases the transformation temperatures. The similar trans-

formational behavior was observed experimentally in Ref.

13 for the Fe-Pd alloy. The experimental strain-temperature

loops observed for Fe-Pd alloy are shown in Fig. 2(b) to

compare with the theoretical ones calculated for Ni-Mn-Ga

alloy (Fig. 2(a)). Figure 2 shows that the qualitative agree-

ment between the theoretically described and experimentally

observed transformational behavior of axially stressed SMAs

takes place.

The hysteretic phenomena are inherent not only to the

temperature-induced but also to the stress-induced MTs.

However, the anhysteretic stress-induced deformation of

FIG. 2. (a) Hysteresis loops calculated for the forward and reverse MTs in

Ni-Mn-Ga alloy; the hysteresis is absent if the stress exceeds the critical

value. (b) Experimental strain-temperature dependences reported by Xiao

et al.13 for Fe-Pd alloy are shown in comparison with theoretical ones.



16:14:28

FSMA is predicted by the present theory and observed

experimentally at temperatures T > T� for the Ni-Fe(Co)-Ga

alloy (see Ref. 12). The anhysteretic postcritical deformation

is promising for applications because of its giant (>12%)

value.12

The critical stress that provides the experimental obser-

vation of an anhysteretic deformation of an alloy is described

by Eq. (10). According to this equation, a critical stress is

equal to jr�j � 17:5 MPa for the ultra-soft alloy with

C0ðT1Þ ¼ 1 GPa. If the shear modulus C0ðT1Þ exceeds the

value of 20 GPa, which is rather typical for the ferromag-

netic and nonmagnetic SMAs, the critical stress is larger

than 350 MPa. In many cases, the axial stress above 300 MPa

breaks the testing sample. Therefore, the ultra-soft SMAs

deserve a special attention of researchers.

C. Shear elastic modulus of Ni-Mn-Ga single crystal

The shear elastic modulus of the deformable ferromag-

netic crystal is different from that of unstressed paramagnetic

crystal and can be defined as

C0ðT; rÞ ¼ �½@u3ðTÞ=@r��1: (11)

The temperature-dependent lattice distortion is involved in

Eq. (11) through the function u3ðTÞ ¼ 2½cðTÞ=aðTÞ � 1�,which characterizes the tetragonal lattice distortion and pre-

determines the difference in the values of shear modulus of

the austenitic and martensitic phases. Moreover, the values

of lattice parameters and MT strain of the axially stressed

alloy specimen are noticeably different from those observed

at zero stress (Fig. 2(a)). Thus, the shear elastic modulus

depends on the value of mechanical loading.

Figure 3(a) shows the temperature dependences of

Young’s modulus (E � 3C0) of ferromagnetic crystal, which

undergoes the forward and reverse MTs. According to the

assumption formulated previously, the cooled specimen is in

austenitic phase above the temperature T2ðrðmeÞÞ ¼ 291:5 K

and in the martensitic phase below it. In the austenitic phase,

the equality EðTÞ ¼ 9c2ðTÞ occurs.12 The coefficient c2ðTÞof Landau expansion for Gibbs potential (1) vanishes at

T ¼ T2, and the Young’s modulus of austenitic phase

becomes equal to zero at this temperature. The computed

Young’s modulus of the martensitic phase has finite value of

5 GPa at T ¼ T2ðrðmeÞÞ ¼ 292 K. This value is close to the

experimental values of Young’s modulus measured for ferro-

magnetic SMAs at MT temperature.22–26 The heated speci-

men is assumed to be in the martensitic phase at the

temperature below T1ðrðmeÞÞ ¼ 297 K and in the austenitic

phase above it. Therefore, the distance between the minimal

values of Young’s modulus of heated and cooled specimen is

equal to the maximal hysteresis value.

The dependences shown in Fig. 3(a) have unusual fea-

ture: they practically coincide everywhere except for the nar-

row temperature interval DT � 5 K, corresponding to the

temperature hysteresis of MT. This feature was experimen-

tally observed, in particular, for the cubic-rhombohedral

(B2! R) phase transformation of Ti-Ni alloy.33

The influence of axial compressive stress on Young’s

modulus of cooled and heated FSMA is illustrated in Fig.

3(b). It is seen that the application of the compressive stress

increases the value of shear modulus of martensitic phase and

shifts the transformation temperature to a high-temperature

range. Moreover, Figure 3 shows that the rational change of

Young’s modulus under compressive stress increases signifi-

cantly on approach to the MT temperature range: the stress

of �15 MPa results in the change of Young’s modulus

DEðT; rÞ=EðT; 0Þ � 1 at room temperature. Similar behavior

was experimentally observed in Ref. 21 for the Ni-Mn-Ga

alloy under compressive stress. In this work, the rational

change of shear modulus DC0ðrÞ=C0ð0Þ � 0:7 was observed

for the compressive stress of �10 MPa.

D. Magnetostriction of Ni-Mn-Ga alloy

Let the magnetic field be applied in [110]jjz direction to

z-variant of Ni-Mn-Ga martensite with c=a < 1. In this case,

the x- and y-directions are physically equivalent, and there-

fore, rðeff Þ2 ¼ 0. Due to this, Eq. (4) has solution of u2 ¼ 0

and Eqs. (6) and (11) hold.

The magnetoelastic stress value depends on the tempera-

ture as

FIG. 3. (a) Temperature variation of Young’s modulus during heating

(dashed line) and cooling (solid line) of the unloaded ferromagnetic Ni-Mn-

Ga single crystal. (b) Influence of axial load on the temperature variation of

Young’s modulus of a ferromagnetic Ni-Mn-Ga single crystal.



16:14:28

rðmeÞzz ðT;HSÞ ¼ rðmeÞ

3 ðT;HSÞ=2 ¼ �6dM2SðTÞ; (12)

where HS and MS are the saturating magnetic field and the

saturation magnetization, respectively, and d ¼ �23 is a

dimensionless magnetoelastic constant (for more details, see

Ref. 17).

In the case if the alloy specimen is axially stressed by the

axial load, the elastic modulus depends on the stress value r.

The transversal (with respect to the field) magnetostriction of

martensite can be evaluated from an obvious relationship

eðmsÞzz ðT;HS; rÞ ¼ rðmeÞ

zz ðT;HSÞ � rðmeÞzz ðT; 0Þ

h i=EðT; rÞ (13)

received from Eqs. (6), (11), and (12), where EðT; rÞdependences depicted in Fig. 3 can be used for computations.

The inequalities rðmeÞzz ðT; 0Þ < 0, rðmeÞ

zz ðT;HSÞ > 0 hold and

the strain (Eq. (13)) is therefore positive. The plots of satu-

rated magnetoelastic strains (magnetostriction) of FSMA are

shown in Fig. 4 for forward and reverse MTs. The theoretical

curves, presented in Fig. 4, were obtained using the constant

value of saturation magnetization MS ¼ 600 G, which is

close to the value inherent to the Ni-Mn-Ga alloys at room

temperature. The constant magnetization value can be

accepted because the computations were carried out for the

temperatures well below the Curie point of Ni-Mn-Ga,

TC � 375 K.17

Figure 4(a) shows the theoretical temperature dependen-

ces of magnetostriction of the Ni-Mn-Ga single crystal that

undergoes the forward (solid line) and reverse (dashed line)

phase transitions between the cubic phase and single-variant

tetragonal state in the saturating magnetic field. The solid

line shows that the magnetoelastic strain of about 1% as

large can be achieved in the austenitic phase on approach of

the alloy temperature to the forward MT temperature. The

forward MT is accompanied by the abrupt decrease in mag-

netostriction to the value of about 0.2%. This theoretical

result is in qualitative agreement with the experimental data

reported in Ref. 17 for the low-temperature Ni-Mn-Ga as

well as with the data of Ref. 34, where Ni-Mn-Ga with the

martensite start temperature of about 271 K was studied.

In Ref. 34, a pronounced increase in magnetostriction of

the crystal being in cubic phase from the moderate value of

� 0:01% to the large value of 0:1% was observed on

approaching to MT temperature. The computations show

that the magnetostriction of 0.1% corresponds to the rela-

tively large values of shear elastic modulus (�20 GPa)

obtained for the Ni-Mn-Ga alloy specimens in the early

experiments.21 The elevated magnetostriction values shown

in Fig. 4 were computed for the value C0ðT1Þ ¼ 1 GPa, which

is close to the values measured in the ultra-soft alloys.22–26

Figure 4(b) illustrates the reduction and disappearance

of hysteresis of magnetostriction under the applied compres-

sive stress. It is also seen that the stress shifts the temperature

interval where the elevated values of magnetostriction are

observable.

IV. SUMMARY

The low values of shear elastic modulus are inherent to

the single crystalline ferromagnetic shape memory

alloys.12,22–26 The computations carried out for the represen-

tative FSMA showed that the weak axial stress of about few

megapascals drastically reduces the hysteresis width of

temperature-induced MT. The anhysteretic deformational

behavior of FSMA under the stress exceeding the critical

value is described. The critical stress value was expressed

through the shear elastic modulus of austenitic phase, C0, and

lattice constants of the martensitic phase (Eq. (10)). The esti-

mations carried for the representative FSMA showed that the

critical stress is roughly proportional to C0ðT1Þ, where T1 is

the temperature of the reverse MT, and varies from 17.5 MPa

to 350 MPa if the C0ðT1Þ rises from 1 GPa to 20 GPa.

Therefore, SMAs with low values of soft elastic modulus are

the candidates for the observation of large anhysteretic defor-

mations and deserve a special attention as the objects for the

fundamental studies and practical applications.

The recent experimental results obtained in Ref. 13 for

the Fe-Pd alloy were analyzed. A good agreement between

theoretical and experimental dependences of hysteresis width

on the axial stress value was demonstrated.

The hysteresis of temperature dependence of shear mod-

ulus was studied to estimate the temperature dependences of

ordinary axial magnetostriction during the forward and

FIG. 4. (a) Temperature dependences of saturation magnetostriction com-

puted for cooling and heating of Ni-Mn-Ga alloy in the absence of applied

stress. (b) The behavior of magnetostriction under different values of axial

stress in the same alloy.



16:14:28

reverse MTs. A drastic increase in magnetostriction in the

temperature ranges of these MTs was described.

ACKNOWLEDGMENTS

The author is grateful to Victor A. L’vov for his fruitful

discussions. The financial supports from the Ministry of

Education and Science of the Russian Federation in the

framework of Increase Competitiveness Program of NUST

“MISiS” (No. R4-2014-034) and the National Academy of

Sciences of Ukraine (No. 0112U001009) are acknowledged.

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Documents

Narrowing of hysteresis of cubic-tetragonal martensitic transformation by weak axial stressing of ferromagnetic shape memory alloy