Download - Effect of variable material properties and environmental conditions on thermomechanical phase transformations in shape memory alloy wires

Effect of variable material properties andenvironmental conditions on thermomechanical

phase transformations in shape memory alloy wires

K. Sadek a, A. Bhattacharyya b,*, W. Moussa a

a Department of Mechanical Engineering, 4-9 Mechanical Engineering Building, University of Alberta, Edmonton,

AB, Canada T6G 2G8b Department of Applied Science, University of Arkansas at Little Rock, 2801 South University, ETAS 575, Little Rock,

AR 72204-1099, USA

Received 17 April 2002; received in revised form 8 October 2002; accepted 10 January 2003

Abstract

This paper reports a computational study of the impact of variable material properties and environmental conditions

(thermal boundary conditions and convection coefficients) on shape memory alloy wires undergoing (i) zero-stress,

thermally-induced phase transformations, and (ii) stress-induced phase transformations at constant stress rates. A finite

difference numerical approach has been employed, and has been validated by comparing with two analytical solutions.

The results have been all given in non-dimensional form, and within the context of the range of parameters that have

been studied, the following recommendations can be made for shape memory alloys (SMA) actuator design: (i) an

uncertainty in the thermal boundary condition is not as important as long as the design process allows for a full

transformation back to martensite at the end of a cycle of martensite–austenite–martensite thermal transformation, (ii)

uncertainties in the thermal boundary condition, convection coefficient and thermal material properties are not as

important when the phase transformation in a SMA is induced by stress.

� 2003 Elsevier Science B.V. All rights reserved.

Keywords: SMA; Variable properties; Finite difference; Thermomechanical transformations

1. Introduction

Actuator design using shape memory alloys

(SMA) not only involves using appropriate con-

stitutive models (for a review, see [1]) but also in-

volves, in principle, the proper characterization ofthe effect of environmental conditions (heat ex-

change effects with the surroundings and through

the structural supports) [2]. This is of course im-

portant because of the well-known strong ther-

momechanical coupling during the phase

transformation. Also of consequence is the signif-

icant change in the evolution of the material

properties (heat capacity, electrical resistivity,

Computational Materials Science 27 (2003) 493–506

www.elsevier.com/locate/commatsci

* Corresponding author. Tel.: +1-501-569-8027; fax: +1-501-

569-8020.

E-mail addresses: [email protected] (K. Sadek), ax-

[email protected] (A. Bhattacharyya), walied.moussa@ual-

berta.ca (W. Moussa).

0927-0256/03/$ - see front matter � 2003 Elsevier Science B.V. All rights reserved.

doi:10.1016/S0927-0256(03)00049-1

mail to: [email protected]

thermal conductivity [3] and Young�s modulus) asthe SMA undergoes a phase transformation. If

these factors (and the corresponding input pa-

rameters in the design process) are not properly

characterized, then an uncertainty in the inputparameters will introduce an uncertainty in the

response of the SMA-based structure [4,5]. In re-

ality, however, depending on the particular appli-

cation, it is quite possible that uncertainties, even

significant ones, of some input parameters may not

have a significant impact on the uncertainty in the

system response. In such cases, characterization of

those input parameters need not be done as accu-rately. This of course translates to a cheaper design

and fabrication process for the SMA actuator.

Characterization of the convection coefficient

[2] as well as evolution in the material properties

[3–5] was done for SMA wires undergoing purely

thermal transformations (i.e. phase transforma-

tions due to heating and cooling, all of it at con-

stant stress). For this special case, there is nocoupling between the evolving temperature field

with the stress field. Once the purely thermal

problem has been utilized to determine the con-

vection coefficient and the evolving material prop-

erties [2–5], it is desirable to study the impact of

these parameters during a thermomechanical phase

transformation where both the stress and temper-

ature may change. The simultaneous evolution ofstress and temperature is often the case when a

SMA is used as part of an active structure [6]. Of

course, evolution of both the stress and tempera-

ture fields occurs with a strong coupling in SMAs,

even when the mechanical loading is quasistatic in

nature (dynamic effects are minimal enough to be

neglected). Such an attempt was beyond the scope

and focus of our previous papers.In this paper, we report a computational study

of thermomechanical phase transformations in

SMA wires (i.e. the stress and temperature may

both change simultaneously). The impact of envi-

ronmental conditions (convection to the surround-

ings and the thermal end-boundary conditions) as

well as variable material properties (heat capacity,

electrical resistivity, thermal conductivity andYoung�s modulus) are studied for stress-induced

thermomechanical phase transformations at con-

stant stress rates. The numerical approach that we

have used is based on implicit finite differences,

and due to the strong thermomechanical nature of

SMA phase transformations, an iterative scheme is

implemented within every time increment to solve

for the field variables. The numerical approach hasbeen validated by comparing its results with ana-

lytical solutions for two special cases involving

prediction of (i) the temperature field in a wire

with a uniform, constant heat source, temperature-

dependent thermal conductivity and heat capacity

(but their ratio is temperature-independent), (ii)

stress-strain response during an isothermal, stress-

induced transformation. The numerical approachis then used to study the problem outlined at the

beginning of this paragraph, and the results are

used to make certain recommendations for SMA

actuator design.

The paper is organized into five sections. Fol-

lowing the current introductory section, Section 2

outlines the boundary value problem of the SMA

wire actuator. Section 3 is on the implicit finitedifference method and its validation, Section 4 is

on the numerical results and Section 5 outlines the

conclusion of the paper.

2. The boundary value problem of a SMA wire

actuator

The boundary value problem of a SMA wire

actuator subjected to a combined thermomechan-

ical loading is illustrated in Fig. 1. The wire, ini-

tially in an austenitic state, is subjected to a cycle

of tensile loading and unloading. The possibility of

(t) (t)L L

Fig. 1. Schematic representation of the 1D boundary value

problem.

494 K. Sadek et al. / Computational Materials Science 27 (2003) 493–506

thermal actuation by electrical heating of the wire

is also allowed. There are several phenomenolog-

ical models on SMA phase transformations avail-

able in the open literature; for a review, see

Lagoudas and Bhattacharyya [1]. In this work, weshall adopt the model proposed by Boyd and

Lagoudas [7].

2.1. The dimensional equations

We focus on quasistatic mechanical loading,

such that inertia can be neglected and the uniaxial

stress in the bar is spatially uniform, i.e. r � rðtÞ.The material state of the wire at a given location

will be defined by the martensitic volume fraction,

n � nðx; tÞ, where ‘‘x’’ is the spatial co-ordinate

defined with its origin at the center of the wire (see

Fig. 1) and ‘‘t’’ is the time. The rate at which nevolves will be represented by _nn and defined as_nn � on=ot; a similar definition will be used for ratesof other quantities. The conservation of energyresults in the following equation:

o

oxKðnÞoT

ox

� �þ r ¼ qrðn;T Þ _rrþ qT ðnÞ _TT þ qnðT ;rÞ _nn;

ð1Þwhere KðnÞ is the material state-dependent thermal

conductivity, ‘‘r’’ is the heat source/sink and the

parameters qrðn; T Þ, qT ðnÞ and qnðT ; rÞ are given inthe Appendix A. The strain rate is expressed as

_ee ¼ erðnÞ _rr þ eT ðnÞ _TT þ enðT ; rÞ _nn; ð2Þ

where the parameters erðn; tÞ, eT ðnÞ and enðT ; rÞ aregiven in the Appendix A. The onset of transfor-

mation as well as its progress is governed by the

transformation criterion

Pðn; T ; rÞ ¼ Y A ! M�Y M ! A

�Y > 0; ð3Þ

where the expression for Y may be found in the

Appendix A. The parameter Y is a material

property and is equivalent to yield stress in clas-

sical plasticity. While Eqs. (1)–(3) are sufficient to

numerically evaluate n, T and r when the trans-formation is in progress, it is often more practical

from a numerical standpoint to write Eq. (3) in its

rate form during the phase transformation, and

use the rate form in conjunction with Eqs. (1) and

(2) in a numerical scheme. The rate form of Eq. (3)

is written as

_nn ¼ nrðT ; rÞ _rr þ nT ðT ; rÞ _TT ; ð4Þ

where nrðT ; rÞ and nT ðT ; rÞ are listed in the Ap-

pendix A. Note that if the stress rate, _rr, is zero,then Eq. (4) can be combined with Eq. (1) to write

o

oxKðnÞ oT

ox

� �þ r ¼ qT ðnÞ½ þ qnðT ; rÞnT ðT ; rÞ _TT :

ð5Þ

Eq. (5) is a non-linear partial differential equation

that can be solved for the temperature field; this

was done in our earlier work [2–5]. For the current

problem ð _rr 6¼ 0Þ, Eq. (1) has to be solved along

with Eq. (4). This is a coupled problem involving

the stress and the temperature, and was beyond the

scope of our previous work.

The boundary and initial conditions are nowintroduced, and are taken in such a way that the

entire problem is rendered spatially symmetric

about x ¼ 0. In the following, we shall focus on a

half-length of the wire, namely 06 x6 L. The

thermal boundary conditions are taken as

oTox

ð0; tÞ ¼ 0; �KðnÞ oTox

ðL; tÞ ¼ hB½T ðL; tÞ � Tamb;

ð6Þ

where hB is an effective heat transfer coefficient at

x ¼ L. The mechanical boundary conditions are

uð0; tÞ ¼ 0; _rrðtÞ ¼ _rrL; ð7Þ

where u � uðx; tÞ is the displacement and _rrL is a

constant stress rate imposed at x ¼ L. The initial

conditions for the temperature, stress and mar-tensite volume fraction are taken as

T ðx; 0Þ ¼ Tamb; rð0Þ ¼ 0; nðx; 0Þ ¼ n0; ð8Þ

where n0 is a spatially uniform martensitic volume

fraction at t ¼ 0. Finally we give the expression for

the heat source/sink term ‘‘r’’ (Eq. (1)). Allowingfor the possibility that the SMA wire may beelectrically heated and including the free convec-

tion along the wire as a source/sink term, we have

K. Sadek et al. / Computational Materials Science 27 (2003) 493–506 495

r ¼ qEðnÞJ 2 �2hLR

ðT � TambÞ; ð9Þ

where qEðnÞ is the electrical resistivity, ‘‘J ’’ is theelectrical current density, hL is the convection co-

efficient due to exchange of heat between the en-

vironment and the wire along its length, and ‘‘R’’ isthe radius of the wire.

The thermal conductivity (K), specific heat (Cv),

electrical resistivity (qE), Young�s modulus (E) andcoefficient of thermal expansion (a) are assumed tovary linearly with the martensite volume fraction

during the phase transformation process. There-

fore

KðnÞ ¼ KA þ nðDKÞ;

CvðnÞ ¼ Cv;A þ nðDCvÞ;

qEðnÞ ¼ qE;A þ nðDqEÞ; ð10Þ

EðnÞ�1 ¼ E�1A þ nðDE�1Þ;

aðnÞ ¼ aA þ nðDaÞ; ð11Þ

where we define

DK ¼ KM � KA; DC ¼ Cv;M � Cv;A;

DqE ¼ qE;M � qE;A;

DE�1 ¼ E�1M � E�1

A and

Da ¼ aM � aA:

2.2. The non-dimensional boundary value problem

Eqs. (1)–(4) and (6)–(9) in Section 2.1. are now

non-dimensionalised. The dimensional quantitiesand their non-dimensional counterparts are given

in Table 1. In particular, note that the non-

dimensionlised form of a rate term, e.g. _rr, willbe indicated as _�rr�rr and defined as _�rr�rr ¼ o�rr=o�tt. Thenon-dimensional counterpart of Eqs. (1)–(4) and

(6)–(9) are listed below

o

o�xxKðnÞ oT

o�xx

� �þ �qqEðnÞJ

2 � 2�hhLLR

� �ðT � 1Þ

¼ �qq�rrðn; T Þ _�rr�rr þ �qqT ðnÞ _TT þ �qqnðT ; �rrÞ _�nn�nn; ð12Þ

_�ee�ee ¼ e�rrðnÞ _�rr�rr þ eT ðnÞ _TT þ enðT ; �rrÞ _�nn�nn; ð13Þ

�PPðn; T ; �rrÞ ¼ Y A ! M

�Y M ! A

(Y > 0; ð14Þ

_nn ¼ n�rrðT ; �rrÞ _�rr�rr þ nT ðT ; �rrÞ _TT ; ð15Þ

where �qq�rrðn; T Þ, �qqT ðnÞ, �qqnðT ; �rrÞ, e�rrðnÞ, enðT ; �rrÞ,_nn�rrðT ; �rrÞ and nT ðT ; �rrÞ are given in the Appendix A.

The thermal and mechanical boundary conditions

are

oTo�xx

ð0;�ttÞ ¼ 0; �KðnÞ oTo�xx

ð1;�ttÞ ¼ �hhB½T ð1;�ttÞ � 1;

�uuð0;�ttÞ ¼ 0; _�rr�rrð1;�ttÞ ¼ _�rr�rrL; ð16Þ

whereas the initial conditions are

T ð�xx; 0Þ ¼ 1; �rrð0Þ ¼ 0; nð�xx; 0Þ ¼ n0: ð17Þ

3. The implicit finite difference method and itsvalidation

3.1. The implicit finite difference method

In general, analytical solution of Eqs. (12)–(17)

is not possible. Several numerical approaches may

be employed. Thus for example, Faulkner et al. [3]

wrote their own finite element code to study ther-mal transformations in SMA wires with variable

material properties. Another approach is to use a

commercial finite element software, e.g. ABAQUS

with a user subroutine that defines the SMA con-

stitutive response. In our work here, we use an

implicit finite difference method. The conduction

term is discretized using the thermal resistance

method [8–10]. We have written our own code andthis gives us a measure of control over the details

of the code itself.

The thermomechanically coupled system of

Eqs. (12)–(17) is solved using an implicit finite

difference method. Time derivatives are discretized

using the forward differencing technique, i.e. for

example


_�rr�rr ¼ o�rro�tt

� �rrð�tt þ D�ttÞ � �rrð�ttÞD�tt

;

_TT ¼ oTo�tt

� T ð�tt þ D�ttÞ � T ð�ttÞD�tt

;

_�nn�nn ¼ ono�tt

� nð�tt þ D�ttÞ � nð�ttÞD�tt

:

ð18Þ

The SMA wire ð06�xx6 1Þ is subdivided into Nelements of non-uniform length, with an element

node present at the center of each element (Fig. 2).

The values of the field variables n, T and �rr at each

node are taken identical to those for the corre-

sponding element, and are denoted as ni, T i and �rrrespectively. A column matrix of the nodal tem-

peratures is denoted as T. Further, a typical ma-terial property for the ith element that depends on

n will be defined using the value of n at the ithnode. Thus, for example, Ki � KðniÞ is defined as

the thermal conductivity of the ith element. The

conservation of energy (Eq. (12)) is written for a

typical element based on Fig. 2. Assuming that

values for all field variables are known at a given

time ‘‘�tt’’, we have to find their new values at a new

xi

i

iq i

qi−1

q i

q i

X

cond.

elect.

cond.

conv.

Fig. 2. Energy balance on node i.

Table 1

A list of the dimensional and non-dimensional parameters

Parameter Unit Non-dimensional form

x mm �xx ¼ xL

KðnÞ Wmm�1 K�1 KðnÞ ¼ KðnÞKA

qEðnÞ Xmm�1 �qqEðnÞ ¼qEðnÞqE;A

T �C T ¼ TTamb

a �C�1 �aa ¼ aTamb

CvðnÞ Jmm�3 K�1 CvðnÞ ¼CvðnÞCv;A

t s �tt ¼ tKA

Cv;AL2

r Nmm�2 �rr ¼ rCv;ATamb

r Wmm�3 �rr ¼ rL2

TambKA

DS0 Jmm�3 K�1 DS0 ¼DS0Cv;A

P Nmm�2 P ¼ PCv;ATamb

EðnÞ Nmm�2 EðnÞ ¼ EðnÞCv;ATamb

h Wmm�2 K�1 �hh ¼ hLKATamb

a8 Nmm�2a8 ¼

a8Cv;ATamb


time, �tt þ D�tt, where ‘‘D�tt’’ is the time increment.

Therefore, the energy balance for the ith node at

the instant, �tt þ D�tt, is written as

Aiðn̂n; bTT ; �̂rr�rrÞT i�1ð�tt þ D�ttÞ þ Biðn̂n; bTT ; �̂rr�rrÞT ið�tt þ D�ttÞ

þ Ciðn̂n; bTT ; �̂rr�rrÞT iþ1ð�tt þ D�ttÞ

¼ Diðn̂n; bTT ; �̂rr�rrÞ; 16 i6N ð19Þ

where Ai, Bi, Ci and Di are given in the Appendix

A. Note that these are all dependent on the current

values of the field variables. For numerical im-

plementation, one replaces them by trial values at

time ‘‘�tt’’, i.e. bntnt ;c�TTt�TTt and �̂rrt�rrt, which are then updated

in an iterative procedure within a given time step.

The N equations represented by Eq. (18) are as-

sembled into

Tð�tt þ D�ttÞ ¼ F�1D; ð20Þ

where F � Fðn̂n; bTT; �̂rr�rrÞ is a N � N matrix whereas T

and D � Dðn̂n; bTT; �̂rr�rrÞ are N � 1 column matrices.

The matrix F can be assembled from Ai;Bi and Ci

(see Eq. (19)) and is not given here. The numerical

approach is summarized in Fig. 3.

3.2. Validation of the numerical approach

The implicit finite difference method outlined in

Section 3.1 has been partially validated by com-

Fig. 3. Flow chart for numerical model.


paring its predictions with the two following cases

for which analytical solutions exist: (i) isothermal

mechanical response, (ii) purely thermal response.

These are described next.

If the wire is loaded with a very small stress rate( _�rr�rr � 0), the temperature field in the wire will not

change significantly with time. In this context, an

excellent approximation is to assume that the

stress–strain response of the wire is isothermal, for

which an analytical solution can be easily derived

using Eqs. (12)–(17). For the numerical imple-

mentation, we have used a dimensional length

L ¼ 20 mm. The spatial discretization used is: 100elements (06�xx6 0:5), 100 elements (0:56�xx6 0:75)and 200 elements (0:756�xx6 1), whereas the time

step was taken as D�tt ¼ 8:332� 10�6. We have

checked that the predictions of the stress–strain

response of an initially austenitic wire by our code

approaches the analytical solution as the stress

rate is decreased; the agreement is especially good

when _�rr�rr � 0:22.The thermal part of the code is validated by the

analytical solution of the following boundary va-

lue problem:

o

o�xxKðT Þ oT

o�xx

� �þ �rr ¼ CvðT Þ

oTo�tt

� �; ð21Þ

where KðT Þ and CvðT Þ are temperature-dependentproperties and KðT Þ=CvðT Þ ¼ constant, �rr is con-

stant, adiabatic and isothermal boundary condi-

tions are taken at �xx ¼ 0 and �xx ¼ 1 respectively. The

analytical solution for �rr ¼ 0 was given by Carslaw

and Jaeger [11], whereas the solution with �rr 6¼ 0

(but constant) was given by Amalraj et al. [4].

Here, we have reported the specific case studydone by Amalraj et al. [4] (see the discussion after

their Eq. (36)). The specific functions chosen for

KðT Þ and CvðT Þ are only temperature-dependent

as stated above and are given by, KðT Þ ¼ ðKM=KAÞ½1þ dðT � 1Þ and CvðT Þ ¼ ðCv;M=Cv;AÞ½1þdðT � 1Þ, where ‘‘d’’ is a constant parameter. Forthe numerical implementation, we have used a

dimensional length L ¼ 40 mm. The spatial dis-cretization used is identical to the isothermal

stress-induced case (see previous paragraph),

whereas the time step was taken as 2.1� 10�5. The

results have been reported in dimensional form in

Tables 2 and 3. The comparison between the an-

alytical solution and our numerical solution has

been given for T ð0; tÞ vs. t (Table 4) and T ðx; 10Þvs. x (Table 5). Our numerical approach shows an

excellent agreement, and is even better than that of

Amalraj et al. [4].

4. Numerical results

The numerical results will seek to shed some

light on the effect of variable material properties,

boundary conditions and the convection coeffi-

cient on (i) average strain vs. time response during

a zero stress thermal transformation, and (ii) stress

vs. average strain response during a stress-inducedtransformation at a constant stress rate. Note that,

Table 2

Time-dependent temperature evolution at x ¼ 0

t (s) T ð0; tÞ (K)

Analytical Numerical

0 300 300

1.0 303.475 303.6989

2.0 306.738 306.905

3.0 309.824 309.9474

4.0 312.758 312.8485

5.0 315.561 315.6263

6.0 318.248 318.2950

7.0 320.834 320.8663

8.0 323.329 323.3503

9.0 325.741 325.755

10.0 328.079 328.0876

Table 3

The spatial temperature profile at t ¼ 10 s

X (mm) T ðx; 10Þ (k)

Analytical Numerical

0 328.0799 328.0876

10 328.0799 328.0876

20 328.0799 328.0876

30 328.0799 328.0876

35 327.6542 327.5563

37 325.372 325.1422

38 321.7296 321.6672

39 312.541 312.5767

40 300 300


in reality, the phase transformation strain is non-zero only when stress is applied. Therefore, the

results on average strain vs. time response during a

zero-stress thermal transformation have to be in-

terpreted in the context of a constant applied

stress. Of course, if a constant stress had been

present, the transformation temperatures would

have simply shifted higher. Also of interest is how

much of the wire undergoes a phase transforma-tion during a cycle of heating/cooling, or loading/

unloading. Therefore the average martensitic vol-

ume fraction is of interest. The average strain and

martensitic volume fraction are defined as

eavgð�ttÞ ¼Z 1

0

eð�xx;�ttÞd�xx;

navgð�ttÞ ¼Z 1

0

nð�xx;�ttÞd�xx: ð22Þ

For all results in this section, the spatial and time

discretizations are identical to that of the isother-

mal stress-induced case (see second paragraph of

Section 3.2). Necessary dimensional parametersare given in Table 4 and the material properties of

the alloy are given in Table 5. We would like to

point out that while we have studied the purely

thermal transformation previously [3–5], recent

experiments [2] have underscored the importance

of the end boundary conditions. Therefore, the

results we report here will deal with the effect of

these boundary conditions. For completeness, wehave also included the effect of the convection

coefficient and the variable material properties on

the thermal field. This allows us to make certain

overall conclusions about SMA actuator design at

the end of this section.

4.1. The thermally induced transformation

A purely thermal transformation is studied for

a stress-free initially martensitic wire. Thus �rrL ¼ 0,

n0 ¼ 1 (see Eqs. (16) and (17)). A list of the

dimensional parameters used in generating the

results is given in Table 2. In particular, the non-

dimensional current density, J ¼ 1:8. The electriccurrent is switched on and the heating is continued

until the average martensitic volume fraction doesnot decrease any further (the code determines this

cutoff point if navg has reached a value of 0.04).

Table 4

A list of dimensional parameters used in the model

Parameter Description Value

R Wire diameter 0.18 mm

L Half wire length 20 mm

I Current 1.2 A

J Current density 11.8 Amm�2

hL Heat transfer

coefficient during

(a) Heating 7.6812E)5Wmm�2 K�1

(b) Cooling 6.6405E)5Wmm�2 K�1

Table 5

Properties of Ni–Ti alloy

Parameter Description Value Reference

EA Young�s modulus of austenite 30 Nmm�2 [6]

EM Young�s modulus of martensite 13 Nmm�2 [6]

KA Thermal conductivity of austenite 1.8E)2 Wmm�1 K�1 [4]

KM Thermal conductivity of martensite 1.8E)3 Wmm�1 K�1 [4]

Cv;A Specific heat of austenite 5.401E)3 Jmm�3 K�1 [4]

Cv;M Specific heat of martensite 4.861E)3 Jmm�3 K�1 [4]

qE;A Electrical resistivity of austenite 8.693E)4 Xmm�1 [4]

qE;M Electrical resistivity of martensite 6.993E)4 Xmm�1 [4]

Qlatent Latent heat 0.1596 Jmm�3 [5]

TM;s Martensite start temperature 20 �C [10]

TM;f Martensite finish temperature 0 �C [10]

TA;s Austenite start temperature 40 �C [10]

TA;f Austenite finish temperature 60 �C [10]


The electric current is then switched off, and free

convection is allowed to cool the wire back to its

ambient temperature.We begin the analysis by

giving the results for the evolution in the non-

dimensional temperature field, T ð0;�ttÞ, at the centerof the wire, �xx ¼ 0 (Fig. 4). For the adiabatic

boundary condition, T ð0;�ttÞ is also the temperatureat all other locations due to the absence of any

temperature gradients. Notice that, during the

heating, the curves corresponding to both bound-

ary conditions are positioned very close to each

other. However, in order to reach a targeted navgfor the entire wire (during the martensite to aust-enite transformation), the wire with an isothermal

boundary condition (and resulting temperature

gradients) takes a longer time to attain a temper-

ature high enough for transformation to occur.

This results in the overshoot of the dashed curve

(isothermal) as compared to the solid curve (adi-

abatic) in Fig. 4 during the heating process. The

overall response of the wire is dictated primarilyby navgð�ttÞ. Its evolution is shown in Fig. 5. Note

that the non-dimensional time taken by the wire

with the isothermal boundary condition to reach

the targeted value of navgð�ttÞ is about 0.035 greater

than the time taken due to the adiabatic boundary

condition. The effect of the isothermal boundary

condition is to act as a heat sink and delay theheating process. On the other hand, the ‘‘heat

sink’’ effect is beneficial during the cooling process;

compare the steeper gradient of the dashed curve

as compared to the solid curve as navgð�ttÞ increasesduring the martensitic transformation. In fact, we

estimate that at navgð�ttÞ ¼ 0:26, the previously

mentioned time difference of 0.035 has been re-

duced to about 0.02; a reduction of about 57%.The evolution of the average strain, navgð�ttÞ, is

shown in Fig. 6. Note that there will be a difference

between the two boundary conditions during

partial transformation, i.e. after partly completing

the heating process, the cooling process was to be

initiated. On the other hand, for a reasonably

complete M ! A and A ! M transformation as

shown, the average strain recovery occurs almostover a very similar average range of �tt for the twowidely differing boundary conditions.

Fig. 4. The evolution of the non-dimensional temperature, T ð0;�ttÞ, vs. non-dimensional time, �tt, during a zero stress, thermal phase

transformation to study the effect of thermal boundary condition.


The effect of the convective coefficient is shownin Fig. 7, for an isothermal boundary condition.

The solid curve in Fig. 7 corresponds to the ap-

propriate coefficients (see Table 2) taken during

the heating and cooling process. The dashed curve

sets the convective coefficient during cooling equal

to the convection coefficient prevailing during

heating. The outcome is that the curves are iden-

tical during the heating process whereas, thedashed curve shows an expectedly quicker trans-

formation back to martensite. The effect of the

variable material properties appears in Fig. 8.

Compare the dotted curve (K is fixed at its initial

martensitic value) and the dash-dot curve (�qqE is

fixed at its initial martensitic value) to the solid

curve (all varying properties).

Fig. 5. The evolution of the average martensite volume frac-

tion, navgð�ttÞ, vs. non-dimensional time, �tt, during a zero stress,

thermal phase transformation to study the effect of thermal

boundary condition.

Fig. 6. The evolution of the average strain, eavgð�ttÞ, vs. non-dimensional time, �tt, during a zero stress, thermal phase trans-

formation to study the effect of thermal boundary condition.

Fig. 7. The evolution of the average strain, eavgð�ttÞ, vs. non-dimensional time, �tt, during a zero stress, thermal phase trans-

formation to study the effect of convection coefficient with an

isothermal boundary condition.

Fig. 8. The evolution of the average strain, eavgð�ttÞ, vs. non-dimensional time, �tt, during a constant stress, thermal phase

transformation to study the effect of variable material proper-

ties with an isothermal boundary condition.


4.2. The stress-induced transformation

The stress-induced transformation is studied for

an initially austenite SMA wire subjected to a

constant stress rate, _�rr�rrL � 2:22 and n0 ¼ 0 (see Eqs.

(16) and (17)). Further, since no thermal actuationby electrical current is involved, we set J ¼ 0 in

Eq. (11). The navgð�ttÞ vs. �tt response for the stress-

induced transformation is given in Fig. 9 for an

isothermal boundary condition. The temperature

response at �xx ¼ 0 is shown in Fig. 10. Due to the

relatively low temperature levels (compare Fig. 10

with Fig. 4), the variable thermal properties as well

as the thermal boundary conditions do not have anoticeable impact on the SMA response. We have

numerically checked these effects at _�rr�rr ¼ 2:22. We

have repeated the checks at _�rr�rr ¼ 22:2 and the only

difference is that the time taken for the transfor-

mation is substantially lower. The stress–strain

response and the effect of the Young�s modulus areshown in Fig. 11.

4.3. Recommendations for SMA actuator design

The strong thermomechanical coupling during

the SMA response implies, in principle, that the

effect of the environment (heat exchange with the

surroundings as well as through the end grips of

the wires) needs to be characterized and includedin the design of actuators. Further the design

process needs to account for the significant vari-

ability in SMA material properties during design.

However, in reality, depending on the specific ap-

plication, some of the aforementioned factors may

not be dominant and therefore a complete and

accurate characterization of all the factors that

Fig. 9. The evolution of the average martensite volume frac-

tion, navgð�ttÞ vs. non-dimensional time, �tt, during a stress-inducedtransformation for an isothermal end boundary condition.

Fig. 10. The evolution of the non-dimensional temperature,

T ð0;�ttÞ, vs. non-dimensional time, �tt, during a stress-induced

transformation for an isothermal end boundary condition.

Fig. 11. The stress vs. average strain response during a stress-

induced transformation for an isothermal end boundary con-

dition.


can, in principle, influence the SMA actuator re-

sponse may not be necessary. This is where a

computational model can make valuable contri-

butions. Based on the results outlined in Figs. 4–

11, the following recommendations may be made(these are to be viewed within the range of pa-

rameters studied herein):

1. An uncertainty in the thermal boundary condi-

tion is not as important as long as the design

process allows for a full transformation back

to the martensite at the end of a cycle of mar-

tensite–austenite–martensite thermal transfor-mation.

2. Stress-induced transformations, while thermo-

mechanical in nature, do not cause thermal

fields significant enough for the variability in

material properties (thermal conductivity and

Young�s modulus) as well as the convection co-

efficient and thermal end boundary conditions

to matter significantly (Figs. 10, 11).

5. Conclusions

A computational study of the impact of

variable material properties and environmental

conditions (thermal boundary conditions and

convection coefficients) in SMA wires has been

reported in this paper. Two specific transforma-

tion processes have been studied: (i) zero-stress,

thermally-induced phase transformations, and (ii)

stress-induced phase transformations at constantstress rates. An implicit finite difference numeri-

cal approach has been implemented, and has

been validated by comparing with two analytical

solutions. The results have been all given in non-

dimensional form, and the following recommen-

dations have emerged for SMA actuator design: (i)

an uncertainty in the thermal boundary condition

is not as important as long as the design processallows for a full transformation back to martensite

at the end of a cycle of martensite–austenite–

martensite thermal transformation, (ii) uncertain-

ties in the thermal boundary condition, convection

coefficient and thermal material properties are not

as important as long as the design process for the

SMA actuator involves a stress-induced transfor-

mation. These conclusions need to be interpreted

within the range of the parameters studied.

Acknowledgements

The financial support of the Natural Sciences

and Engineering Research Council (NSERC) of

Canada through operating grants to A.B. and

W.M. is gratefully acknowledged.

Appendix A. List of parameters

The explicit form of the parameters used in Eqs.

(1)–(16) are given as

qrðn; T Þ ¼ aðnÞðT � TambÞ; qT ðnÞ ¼ CvðnÞ;

qnðT ; rÞ ¼ DaðT � TambÞr þ DCvðT � TambÞ lnT

Tambþ DS0T � P;

where DS0 is the difference in specific entropy of

both phases. Further we have

erðnÞ ¼1

EðnÞ ; eT ðnÞ ¼ aðnÞ;

enðT ; rÞ ¼1

DE

� �r þ DaðT � TambÞ þ H ;

where H is the uniaxial transformation strain in

a fully martensitic SMA wire with respect to itsfully austenitic state. The parameter Y (see Eq. (3))

is

Y ¼ 1

2HDðTA;s þ TA;fÞ � Qlatent:

The material property D represents the amount of

stress needed to cause a unit change in the trans-

formation temperatures of the wire (during a

constant uniaxial stress, thermal transformation).

The parameters TA;s, TA;f and Qlatent are the stress-

free austenite start and finish temperatures and themagnitude of latent heat evolved or absorbed per

unit volume of SMA during a purely thermal

transformation (i.e. no stress). Finally we have


nT ðT ; rÞ ¼Dar þ DC ln T

Tamb

� þ DS0

2a8;

nrðT ; rÞ ¼H þ 1

DE

�r2 þ DaðT � TambÞ

2a8;

where

a8 ¼

� 12HDðTM;f � TM;sÞðfor M ! A transformationÞ

� 12HDðTA;s � TA;fÞðfor A ! M transformationÞ

8>>><>>>: :

The non-dimensional parameters appearing in

Eqs. (11)–(16) are now given as

�qq�rrðn;T Þ¼ �aaðnÞðT �1Þ; �qqT ðnÞ¼CvðnÞ;

�qqnðT ; �rrÞ¼D�aaðT �1Þ�rrþDCvðT �1ÞlnT þDS0T � �PP;

Y ¼ YCv;ATamb

;

e�rrðnÞ¼1

EðnÞ; eT ðnÞ¼ �aaðnÞ;

enðT ; �rrÞ¼1

DE

� ��rrþD�aaðT �1ÞþH ;

n�rrðT ; �rrÞ¼Hþ 1

DE

� �rr2þD�aaðT �1Þ2�aa8

;

nT ðT ; �rrÞ¼D�aa�rrþDC lnT þDS0

2�aa8;

�aa8¼a8

Cv;ATamb:

The parameters needed in Eq. (18) are

Ai ¼�2D�xxi

D�xxiKi

þ D�xxi�1

Ki�1

!�1

;

Bi ¼2

D�xxiD�xxiKi

þ D�xxi�1Ki�1

� �1þ 2

D�xxiD�xxiKi

þ D�xxiþ1Kiþ1

� �1þ2�hhL þ Cv;i

D�tt þ �aa _�rr�rr þM1;iM3;i

0B@1CA;

Ci ¼�2D�xxi

D�xxiKi

þ D�xxiþ1

Kiþ1

!�1

;

Di ¼�rr �M1;iM2;i

�_rr_rr þ 2�hhL � Cv;i

D�tt

� T ið�ttÞ

� M1;iM3;i

D�tt

� T ið�ttÞ

0B@1CA;

where M1;i and M3;i have not been given here for

brevity.

The energy terms used in Fig. 2 are defined

as

�qqi�1��cond:

¼ 1

D�xxi2

D�xxiKi

þ D�xxi�1Ki�1

! ;

�qqiþ1��cond:

¼ 1

D�xxi2

D�xxiKi

þ D�xxiþ1Kiþ1

! ;

�qqi��conv:

¼ 2�hhLðT i � 1Þ; qijelect: ¼ �qqE;iJ2:

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