models for one-variant shape memory materials based on dissipation functions

International Journal of Non-Linear Mechanics 37 (2002) 1299–1317

Models for one-variant shape memory materials based ondissipation functions

Davide Bernardinia, Thomas J. Penceb

aDipartimento di Ingegneria Strutturale e Geotecnica, Universit�a di Roma “La Sapienza”, Via Eudossiana 18, 00184 Roma, ItalybDepartment of Mechanical Engineering, Michigan State University, East Lansing, MI 48824-1226, USA

Abstract

Some simple models for the macroscopic behavior of shape memory materials whose microstructure can be described asa mixture of two phases are derived on the basis of a free energy and a dissipation function. Keeping a common expressionfor the free energy, each model is based on a di.erent expression for the dissipation function. Temperature-induced as wellas isothermal, adiabatic and convective stress-induced transformations are studied. Attention is paid to closed form solutions,comparison among the models and parameter identi0cation. ? 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Shape memory materials; Dissipation function; Thermomechanics

1. Introduction

The peculiar behavior exhibited, at a macroscopicscale, by shape memory materials (SMM), like e.g.NiTi and CuZnAl alloys, is due to the occurrenceof phase transformations (PT) between the di.erentsolid phases of Austenite (A) and Martensite (M),the latter of which involves multiple crystallographicvariants [1,2]. Several models have been proposed inthe literature for the modeling of the macroscopic be-havior of SMM employing rather di.erent procedures(see e.g. the reviews [3,4] and the references therein).Most of the models are based on the prescription ofconstitutive equations for strain, or stress, and entropy,complemented by kinetic equations for the phase

E-mail addresses: [email protected](D. Bernardini), [email protected] (T.J. Pence).

fractions.While all such equations are interrelated and,desirably, have to be consistent with the laws of ther-modynamics, these connections are not always evi-dent nor explicitly veri0ed, even in cases in which athermodynamic framework is invoked at the outset.In this work, we employ a thermomechanical frame-

work related to the Ziegler–Green–Naghdi (ZGN)approach [5,6] to derive and discuss the basic featuresof some elementary models for the pseudoelastic beha-vior of SMM. The key features of the approach are:(i) all the constitutive equations are completely deter-mined after the speci0cation of two constitutive func-tions, respectively, expressing the free energy and therate of energy dissipation; (ii) the enforcement of thebalance equations for any admissible process imposesa restriction on the possible states that, after applica-tion of a maximum dissipation argument, translatesinto the kinetic equations consistent with the selectedconstitutive functions. In this way, the constitutiveinformation is clearly identi0ed and integrated in thestructure of the theory that, inherently, o.ers complete

0020-7462/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved.PII: S0020 -7462(02)00020 -3

1300 D. Bernardini, T.J. Pence / International Journal of Non-Linear Mechanics 37 (2002) 1299–1317

and automatic consistency with the laws of thermo-dynamics [7].The appellation ZGN has been selected because,

on one hand, Ziegler seems to be 0rst to have pursuedthe use of two scalar functions as basic constitutiveingredients of continuum mechanics (then widelyadopted by the French school, see e.g. [8,9], butalso by Houlsby and Puzrin [10] among the others);while, Green and Naghdi seem to be the 0rst to haveopened the way to derive kinetic equations for theinternal variables as restrictions induced by the bal-ance equations. Further development of this approachis provided by Rajagopal and coworkers (see e.g.[11] for general aspects and [12] for application toshape memory behavior) who extended the frame-work to allow for the concept of multiple naturalcon0gurations.In this paper, the general structure of the ZGN the-

ory is only brieHy recalled while the main attentionis devoted to the study of the role of the dissipationfunction on the thermodynamical modeling of pseu-doelasticity, paying special attention to obtain explicitresults and closed form solutions. Under this perspec-tive and at the expense of the roughness of the 0nalresults, the simplest settings are chosen systematicallywhenever meaningful to highlight the basic aspects,leaving the introduction of more re0ned assumptionsto the possible future developments. In particular, ourfocus is on establishing qualitative e.ects that stemfrom the basic form on the dissipation function. Assuch, neither re0ned quantitative prediction nor de-tailed comparison with experiment are at issue in thispaper. As discussed, for example, in [12], these issuescan be systematically developed in a theory of the typeconsidered here.Any macroscopic SMM modeling is a, more or less

direct, attempt to predict the e.ects of the microstruc-tural evolution induced by the energetic inputs thatpromote the PTs. In general, the actual microstruc-ture evolution is very complex and involves severalphenomena taking place at di.erent scales. However,in some cases, as for single crystals subjected to uni-axial stress states,M forms as a single family of inter-nally twinned correspondence variant pairs [13] and atwo-phase underlying microstructure model thus be-comes appropriate, at least in the pseudoelastic range.In more complex cases this becomes a more crude ap-proximation but can still be useful, provided further

constitutive information is introduced (for a discus-sion see [14]).For this basic purpose, it is, therefore, useful to con-

sider the simple setting of a one-dimensional shapememory element that is capable of transforming be-tween Austenite and a single variant of Martensite.The volume fraction � of Martensite, with �∈ [0; 1],is used as an internal state variable keeping trackof the microstructural geometry. While this proce-dure retains the inHuence only of the amount of thephases and rules out many of the microstrucural as-pects (such as, e.g., phase arrangement and shapes),more re0ned descriptions can be given by incorporat-ing additional microstructural variables, e.g. multiplevariants of Martensite [15].As a basic strategy, we systematically employ a

free energy function that, for the sake of simplicity,neglects the thermal strains and also posits identicalelastic moduli and speci0c heats for the A and Mphases. We consider three di.erent assumptions forthe dissipation function. Beginning with a primitivedissipation-free model (termed model M0), followedby a model (M1) in which the dissipation function isindependent of phase fraction, and concluding with amodel (M2) in which the dissipation function varieslinearly with �.For these separate models we study PTs induced by:

(i) prescribed temperature under constant stress, (ii)prescribed stress under constant temperature, (iii) pre-scribed stress in non-isothermal conditions, includingadiabatic and convective environments. The discus-sion is restricted to complete PTs, i.e. those that beginwith the material wholly in one phase (�=0 or 1) andtransform it wholly into the other phase (�= 1 or 0).The consideration of A → M → A transformation inthis framework allows for a useful comparison of thevarious models, but does not allow for the consid-eration of internal hysteresis paths (sublooping) andthe associated rich behaviors. These will be discussedin a separate account as their treatment requires ageneralization of the dissipation functions consideredhere that does not e.ect the complete PT behavior.Here attention is paid to the comparison among thevarious models with respect to the determination ofexplicit expressions for transformation stresses andtemperatures, the stress–temperature phase diagram,transformational heating, adiabatic temperaturechange and material parameter identi0cation.

D. Bernardini, T.J. Pence / International Journal of Non-Linear Mechanics 37 (2002) 1299–1317 1301

The stress-free transformation temperatures atwhich A → M and M → A start and 0nish will bedenoted in the usual way by Ms; Mf ; As; Af . Similarnotation will be used as subscript to denote otherquantities evaluated at the start and the 0nish of thetransformations.

2. Modeling framework

Consider a one-dimensional shape memory mate-rial element B. At time t0 it occupies the referencecon0guration �0 = (0; L0) ⊂ R, where the positionof each material point is labeled by X . At subsequenttimes t the material point X occupies the position x=(X; t), is at the absolute temperature #(X; t) and hasthe mass per unit reference length �0(X; t). Heat con-duction within B is neglected and, since the descrip-tion is macroscopic, the function satis0es the usualrequirements of twice continuous di.erentiability. TheelementB can experience mechanical and thermal in-teractions with the external environment, both by con-tact and at distance represented by the external forceand heat supplies per unit volume b; r acting on allpoints and by suitable boundary conditions. By con-vention, we assume that r ¿ 0 if heat Hows into B.Each part of B also experiences contact interactionswith the surrounding material, described by the con-tact force �. Moreover,B is endowed with an internalenergy, entropy and rate of energy dissipation densi-ties e, � and �. The above quantities are called observ-able and are assumed to satisfy the following equationsexpressing, respectively, the balance of energy, mass,linear momentum, entropy and the Clausius–Duheminequality [12] together with suitable boundary andinitial conditions

e = ��+ r; �0 = 0; �0 Mx =@�@X

+ b;

�=�#+r#; �¿ 0: (1)

Here �:=@=@X and superposed dot denotes materialtime derivative with a 0xed X .Constitutive equations have to be speci0ed in or-

der to complete the theory. To this end, the observ-able quantities �; # and the internal variable � areselected as arguments of the constitutive functions andthe Gibbs free energy is introduced as

�:=e − #�− ��: (2)

According to [5–12], constitutive equations are as-sumed to be given in a form that relates deformationgradient, entropy, free energy and rate of energy dis-sipation to s and �

�= �(s); �= �(s); �= �(s); � = �(s; �); (3)

with s:={�; #; �}. As is appropriate for models involv-ing a single variant ofMartensite, attention is restrictedto values of contact force of a common sign. Here, wetake �¿ 0 and henceforth regard � as tensile force.As shown in [12,7], in order that balance equations

are always ful0lled, the functions in (3) cannot begiven arbitrarily but must obey

�=−@�@�; �=−@�

@#; � +

@�@��= 0: (4)

The 0rst two relations in (4) coincide with the stan-dard restrictions among the response functions that areusually derived on the basis of the Clausius–Duheminequality [16], while the third one, for given func-tions � and �, restricts the possible choices of s in turngiving rise, as explained later, to the transformationkinetic. The Clausius–Duhem inequality here simplyplaces a direct restriction on the dissipation function,namely that � must be non-negative for any choiceof its arguments. The thermomechanical response ofthe material is, therefore, completely described by thetwo constitutive functions � and �.In what follows, we restrict attention on a SMM

element that, at time t0, is entirely in austenitic phase(�=0) and at homogeneous temperature #0, focusingon its response due to mechanical loading in a con-vective environment or to prescribed temperature his-tories. Speci0cally, we consider, in absence of bodyforces, a prescribed boundary force such that inertiais negligible. Linear momentum balance is thus satis-0ed by homogeneous �while entropy balance togetherwith constitutive equations yield

−# @2�@#@� �︷︸︸︷

piezocaloric e.ect

−# @2�@#2 #︷︸︸︷

temperature variation

−# @2�@#@� �︷︸︸︷PT

−�︷︸︸︷energy dissipation

−r︷︸︸︷exchange with environment

= 0

(5)

that, therefore, establishes a balance among heatingrates associated to various e.ects as explained in thesubscripts [17]. When needed, this equation enables


one to determine the temperature change due to thevarious thermomechanical couplings. For a speci0edheat supply r; the material heating is constitutive in na-ture and completely determined by the functions � and�. Di.erent assumptions on r are employed to modelcommon heating environments. In particular, isother-mal conditions arise when r is such that #= 0, adia-batic conditionswhen r=0; and convective conditionswhen some speci0c constitutive equation is speci0edto describe the heat exchange process between theelement B and the environment. Henceforth, New-ton’s law of cooling will be adopted for the convectivecase

r = �(#E − #); (6)

where #E is the environmental temperature and � is apositive constant [18]. In the limiting case of � = 0,there is no heat exchange with the environment ren-dering irrelevant the speci0cation of #E and recover-ing the adiabatic conditions. In the other limit case,�=∞, it follows that #= #E whereupon if the envi-ronmental temperature #E is constant then isothermalconditions are recovered, while if #E is regarded asa prescribed function of time, then a direct speci0ca-tion of # is formally obtained (prescribed temperatureconditions).Even though not strictly meaningful in the present

setting, in order to have the possibility to use some ter-minology widely used in literature, the quantities � and� will be referred sometimes also as stress and strain.

3. Constitutive equations

First, we describe the speci0c expressions adoptedfor the basic constitutive functions � and �; then, wediscuss further constitutive aspects such as equationsfor strain, entropy and driving force, transformationkinetic and phase diagram. In this framework, the lat-ter relations are all derived from the basic functions.

3.1. Free energy function

The free energy function is assumed to have thefollowing structure

�(�; #; �) :=−12�2

PE(�)− ��∗(�) +W ∗(�)

+ PEch(#; �); (7)

where PE is an e.ective elastic modulus, �∗ the macro-scopic transformation strain,W ∗ the phase interactionenergy and PE

chthe e.ective chemical free energy. To

within the e.ect of microscopic thermal expansion thathere has been neglected, this expression is consistentwith the average free energy in a representative vol-ume of SMM composed by two linear thermoelasticphases subjected to a prescribed local transformationeigenstrain � under the action of uniform boundarytractions � and temperature # [14]. Di.erent assump-tions on the underlying microstructure then lead to dif-ferent expressions for PE, �∗, W ∗ and PE

ch[14]. For our

purposes, we henceforth specify additional simpli0ca-tions for � as follows: (i) the e.ective elastic modulusPE is a positive constant; (ii) the transformation strainis given by

�∗(�) = ��; (8)

where �¿ 0 is a constant; (iii) the interaction energyis in the form

W ∗(�):=− ��(1− �); (9)

where, in general, � could be a function of � withfeatures dependent on the underlying microstructure.For the normalization W ∗(0) = W ∗(1) = 0, the in-equalityW ∗(�)¡ 0 promotes phase mixing since thislowers the free energy. Conversely, W ∗(�)¿ 0 pro-motes phase consolidation. Since 0ne microstructuralmixtures of Austenite and Martensite are ubiquitousin SMM, one anticipates W ∗(�)¡ 0. This case isadopted here, and � is simply taken as a positive con-stant; (iv) the e.ective chemical free energy is givenby

Ech(#; �) := c(#− #R − # ln #

#R

)− �0A#

+��(#− #R)�+ e0A; (10)

where c, #R, �0A; ��; e0A are constants. Here c is thespeci0c heat at constant force, #R de0nes a particu-lar reference temperature, e0A and �0A are the internalenergy and entropy of pure A at the reference temper-ature #R when � = 0, and �� is such that �0A − ��is the entropy of pure M at the reference tempera-ture #R when � = 0. Since Austenite is the naturallyhigh-temperature state, ��¿ 0.


Γ

ξ

Λ

ΛR

F

Fig. 1. Rate-independent dissipation function.

3.2. Dissipation function

The PTs in SMM have been observed to take placewithout appreciable atomic di.usion and therefore themicrostructural changes responsible for the macro-scopic behavior here at issue do not exhibit signi0cantrate dependence [1,2]. A rate-independent theory re-quires that the state variable response is invariant un-der a universal change in process time-scale. Such anapparently simple assumption, endows the dissipationfunction with a strong structure. Accordingly, it fol-lows from the last of (4) that the dissipation function� is required to be homogeneous of degree 1 in � andtherefore � must be in the form (Fig. 1)

�(s; �):= F(s)max{�; 0}+ R(s)min{�; 0}: (11)

In order to be always consistent with Clausius–Duheminequality, the state-dependent constitutive functions F and R must be such that

F¿ 0; R6 0: (12)

It is seen therefore that, after the enforcement of theabove physical restrictions, the possible choices for thefunction � are strongly guided as they are completelycharacterized by F and R, subject to (12).The subscripts F, R in the above formulas are rem-

iniscent of the fact that the A → M transformation,which supports an increase of �, is typically referredto as forward transformation (often abbreviated byFwT), whereas the M → A transformation, support-ing �¡ 0, is referred to as reverse transformation(often abbreviated by RvT).Understanding the e.ect of di.ering speci0cations

for F, R on hysteresis behavior is the major aimof this paper. In what follows, we consider the ef-fect of three separate and increasingly more general

speci0cations of F and R, each of which is regardedas de0ning, together with the free energy (7), a dif-ferent constitutive model for SMM:

model M0 :

{ F = 0;

R = 0;

model M1 :

{ F = M;

R = A;

model M2 :

{ F = Ms + ( Mf − Ms )�; R = As + ( Af − As )(1− �);

(13)

where M, Ms , Mf , are positive constants and R, As , As are negative constants so that (12) is satis0edfor all �∈ [0; 1]. These di.erent constitutive modelsare appropriate for the consideration of complete PTbehavior. In the event of PT reversal before trans-formation is complete, it would be necessary togeneralize these speci0cations. This might involve theinclusion of additional parameter dependence in (13)so as to track the history of partial transformation[12,19].

3.3. Constitutive equations for strain, entropy anddriving force

Once the free energy function is known, the con-stitutive equations for strain and entropy thus followfrom (4). Here in particular, the free energy � givenin (7) and subject to (8)–(10) yields

�=1E� + ��; (14)

�= c ln##R

− ��+ �0A: (15)

Moreover, the derivative of �with respect to the phasefraction plays a central role in the theory since it canbe identi0ed as a transformational driving force; inthe present case

!:=− @�@�= ��− ��(#− #R) + �(1− 2�): (16)

As! depends on the state s, its total time derivative! depends on the rates of both the observable and theinternal variables. It is useful to identify the portion of! that is independent of the phase fraction rate, i.e. tointroduce a speci0c function " such that

" =@!@�

� +@!@#

#: (17)


A possible choice is to de0ne, as the controllabledriving force,

"(�; #):=��− ��(#− #R) (18)

so that ! = " + �(1 − 2�). In the context of SMMmodeling, such a controllable driving force provides auseful consolidation of the various individual thermaland mechanical driving forces [19].With the above choices for � and �, Eq. (5) takes

the form

c#−#��− �︸︷︷︸PT heat rate

= r; (19)

where = F or R as appropriate. It is seen there-fore that, with the selected constitutive functions, themodel does not predict any piezocaloric e.ect (heredue to the absence of thermal expansion). Balance(19) simply involves the heat rates associated with:the temperature variation within each phase in the ab-sence of PT (described here by a common speci0c heatc), the latent heat due to the PTs (essentially depen-dent on ��), the energy dissipation, and the exchangewith the environment.

3.4. Transformation kinetic

A special feature of the ZGN approach is that thetransformation kinetic is not assumed a priori as istypically done in SMM models [4]. On the contrary,the transformation kinetic follows from restrictions (4)induced by the balance equations [12,7]. Speci0cally,by substituting from (11) into the third equation of (4)and considering that the resulting equation must holdfor any �, it follows that

�¿ 0⇒ ! = F and �¡ 0⇒ ! = R : (20)

The above equalities, also called transformation cri-teria, express energetic conditions necessary for PT.They take the form of a balance between the trans-formational driving force ! and the functions F and R that, therefore, represent the level attained by !whenever a PT process takes place. Assuming more-over that the above transformation criteria have to beful0lled at any time during a PT process, it then fol-lows that

�¿ 0⇔ ! = F and ! = F (FwT); (21)

�6 0⇔ ! = R and ! = R (RvT); (22)

hence providing the following transformation kinetic:

�=

max{

"@( F−!)=@� ; 0} if ! = F;

min{ "@( R−!)=@� ; 0} if ! = R ;

(23)

where otherwise �=0. The actual forms of the trans-formation criteria and of the kinetic are thus deter-mined by the free energy, via ! and ", and by thedissipation function via F and R. The functionsprovide, therefore, a macroscopic measure of the re-sistance encountered by the phase transition fronts tonucleate and propagate. The so-obtained structure forthe transformation criteria turns out to be analogousto those proposed in other contexts on the basis of thedetailed analysis of the energy balances at the trans-formation fronts as discussed e.g. in Ref. [3].

3.5. Stress–temperature phase diagram

A useful representation of the basic features of thePTs can be given by the phase diagram, namely adiagram that subdivides the stress–temperature regionD:={(�; #) |�¿ 0; #¿ 0} into zones wherein thevarious phases can exist and in which the variousPTs can take place [4,20]. A phase diagram de0nedfor both positive and negative values of � gener-ally calls for some kind of two-variant martensitemodel [15].In the present one-variant setting, D is partitioned

into three separate phase existence zones: the austen-ite zone DA on which � must obey �=0, the marten-site zone DM on which � must obey � = 1, and themixture zone DA=M on which � is not subject toeither restriction. Further, there are two transforma-tion zones: the FwT zoneDFwT that support �¿ 0 andthe RvT zone DRvT that support �¡ 0. The variousregions of the phase diagram de0ne loci of pointscharacterized by the same value of the phase fraction� and thus describe isofractional curves.While the transformation zones are clearly included

in the mixture zone, i.e. DFwT ⊂ DA=M and DRvT ⊂DA=M, in the event thatDFwT∪DRvT is a proper subsetof DA=M then its set complement in DA=M provides adead zone Ddead for PT.For example, when � = 0, such zones reduce to

intervals of the temperature axis determined by thestress-free transformation temperatures. In particular,


if Mf ¡Ms¡As¡Af then A and M can exist, at� = 0, in the intervals

{(�; #) |� = 0; #¿Af} ⊂ DA;

{(�; #) |� = 0; #6Mf} ⊂ DM;

whereas phase mixtures can be found on

{(�; #) |� = 0; Mf ¡#¡Af} ⊂ DA=M:

Moreover, phase transformations can occur on

{(�; #) |� = 0; Mf ¡#¡Ms} ⊂ DFwT;

{(�; #) |� = 0; As¡#¡Af} ⊂ DRvT;

whereas

{(�; #) |� = 0; Ms6#6As} ⊂ Ddead

is a dead zone. The complete regions DA=M, DFwT,DRvT, Ddead are then typically given by strips that ex-tend into D beginning from the above intervals on thetemperature axis [20]. The ensuing development willclarify how the phase diagram is determined directlyfrom the transformation criteria. The various assump-tions on the basic constitutive functions yield di.erentphase diagram zonal geometries and hence providedi.ering levels of modeling sophistication.

4. Transformation energetics

During PTs, various kinds of energy conversionstake place [1,2] and are regulated by the internalenergy and entropy balance equations in (1). Whencombined together and complemented by constitu-tive equations, the heat balance (19) is obtained.The integration of (19) over complete PT processesenables one to de0ne some quantities that provide aglobal characterization of the transformation energet-ics. These quantities are also useful in view of theirpossibility for making contact with experiment.In particular, the integral of r is the total heat

exchanged with the environment and, by virtue of(19), admits the decomposition∫r dt = c

∫d#︸︷︷︸

Q0

+∫(−#�� − F;R) d�︸︷︷︸

QF;R

; (24)

where the integrals are extended to complete FwT orRvT processes and the subscripts F, R have to be se-lected accordingly. Speci0cally, Q0 is that portion of

the heating independent of PTs while QF;R is the heat-ing speci0cally associated with PT and energy dissi-pation and can be computed after the knowledge ofthe temperature in terms of �. The sign conventiongives that QF;R¡ 0 and QF;R¿ 0 are associated withexothermic and endothermic transformation, respec-tively. Recall for SMM, that FwT is exothermic whileRvT is endothermic. The latent heat of transformationH is identi0ed with |QF;R| for temperature-driven PTunder conditions of zero stress.For a full transformation cycle, i.e. involving con-

secutive complete FwT and RvT, in which � beginsand ends at the same value, the work

W :=∮� d�; (25)

is the area enclosed by the stress–strain hysteresisloop. It then follows from (14) that this area can becomputed, after the knowledge of the stress in termsof �, through

W = �∮� d�: (26)

This result is independent of temperature variation andholds even if the temperature at the end of the cycledoes not match the temperature at the beginning of thecycle.

5. Model M0

The simplest model that can be contemplated in thepresent framework is based on a dissipation functionequal to 0. Besides its evident roughness that precludeit to capture the hysteresis inherent in PTs, such amodel is helpful to understand some of the basic fea-tures of the material behavior. Transformation criteria(20) require that both PTs occur at !=0 (see Fig. 2)and is equivalent to minimizing the free energy withrespect to �,

" + �(1− 2�) = 0: (27)

From (27), it turns out that the isofractional curvescoincide with the curves of constant ". Taking intoaccount the de0nition of ", it follows that such curves,in D, are lines of slope

m:=��: (28)


ξ

Π

0 1

ϑ

σ

ϑA,0

ϑM,0

DA

DM

A/M

FwTRvT

D

D

D

Fig. 2. Model M0: !–� relation (left) and phase diagram (right).

The various zones of the phase diagram are thusde0ned by two iso-" lines (Fig. 2).Speci0cally, the existence zones for A and M are

delimited by "¡− � and "¿�, i.e.

DA = {(�; #) | 06 �6 �A(#); #¿#A;0}; (29)

DM = {(�; #) |�¿ 0; �¿ �M(#); #¿ 0}; (30)

where

�A(#):=m(#− #A;0); �M(#):=m(#− #M;0) (31)

and

#A;0:=#R +��; #M;0:=#R − �

��: (32)

Thus, the mixture zone DA=M is the strip bounded bythe lines " =±�, so thatDA=M = {(�; #) |�A(#)6 �6 �M(#);

#¿#M;0}: (33)

The model gives no hysteresis as the two transfor-mation zones both coincide with the mixture zone, i.e.DFwT =DRvT =DA=M. When �=0, the region DA=M

collapses to the single line " = 0 resulting in abrupttransformations A ↔ M. It is, therefore, seen that theinteraction energy can be viewed as responsible forphase coexistence.From (23), the transformation kinetic becomes

�=12�

" if ! = 0: (34)

Since �¿ 0, FwT is promoted by increasing " whileRvT by decreasing ".

5.1. Temperature-induced constant-stresstransformation

Under constant stress � and prescribed temperature,PT is activated by temperature changes in the interval

#M;0 +1m�︸︷︷︸

#M(�)

¡#¡#A;0 +1m�︸︷︷︸

#A(�)

(35)

with FwT occurring for cooling #¡ 0 (corre-sponding to "¿ 0) and RvT occurring on heating(corresponding to "¡ 0). The model predicts twostress-dependent (at a rate 1=m) transformation tem-peratures, as the start conditions of RvT coincidewith the 0nish conditions of FwT and vice versa. At� = 0, the temperatures #M;0 and #A;0 are identi0edrespectively with both As; Mf and Ms; Af , giving riseto the following restrictions

#M;0 = As =Mf and #A;0 =Ms = Af ; (36)

which is a reHection of the lack of hysteresis in thismodel.The transformational heatings and the latent heat

are calculated to be

QF =−#R�� − ��; QR =−QF; H = #R��: (37)

It is seen, therefore, that QF¡ 0 hence con0rming, asexpected, the exothermic nature of the FwT. The sameheat produced during FwT must then be absorbed dur-ing the RvT.

5.2. Stress-induced isothermal transformation

Under constant temperature P# and prescribed stress�, PTs are activated by stress changes in the interval

�A( P#)¡�¡�M( P#) (38)

with FwT occurring on tensile loading (�¿ 0) andRvT on unloading (�¡ 0). The model predicts onlytwo transformation stresses, as there are the restric-tions

�A = �Ms = �Af and �M = �Mf = �As : (39)

Such stresses increases with the temperature at a ratem, while the di.erence �M−�A=2�=� is independentof the temperature.


σ

adiabatic

isothermal

ε

ϑ

t

adiabatic

isothermal

ϑ

∆ϑadF

Fig. 3. Isothermal versus adiabatic behavior of Model M0: stress–strain (left), temperature–time (right).

In the �–� plane, both PTs follow the same straightline with slope EPT (Fig. 3), as showed by the incre-mental isothermal stress–strain relation

� =2�E

2� + E�2︸︷︷︸EPT

�: (40)

In particular, for �=0 the coalescence of �M with �Agives a horizontal plateau corresponding to abrupt PTat the transformation stress m( P# − #R) (often calledMaxwell stress). The absence of PT-hysteresis clearlygives W =0. In order to sustain the constant tempera-ture P# during FwT, the heat QF must be removed fromthe system (recall (24) and Q0 = 0). During RvT theheat QR must be supplied, with

QF =− P#�� =−QR (41)

5.3. Stress-induced adiabatic transformation

In order to discuss some aspects of the non-iso-thermal response of SMM under mechanical loading,it is useful to consider adiabatic conditions. Thus, thePTs occur under the conditions arising from (19) withr = 0, i.e.

c#− #��= 0 (42)

so that temperature changes only occur if PT is active(� �=0).Consider the material initially in an austenitic state

at temperature P#. The FwT is activated under increas-ing � at �M( P#), after which the temperature evolvesexponentially with � according to

#F(�) = P# ea�; (43)

where the quantity

a:=��c

(44)

is the basic parameter that a.ects the magnitude of thePT-induced temperature variations. This parameter isespecially important since it continues to play a simi-lar role also in the more articulated models that will bediscussed later. The FwT is concluded at a tempera-ture greater than the initial one yielding a temperatureincrease

R#(ad)F = P#(ea − 1): (45)

Subsequent unloading provides RvT that supportsa temperature decrease that, after complete RvT, re-stores the initial temperature P#, i.e. R#(ad)R =−R#(ad)F .The overall adiabatic temperature variation after thefull transformation cycle is zero R#(ad) = 0.The stress–strain relation during PT becomes

non-linear and can be expressed parametrically interms of � by

�=1E� + ��; with

� = m( P# ea� − #A;0) + 2�� : (46)

The 0nal stress after complete FwT is greater thanthe corresponding one in isothermal conditions sinceadditional stress is necessary to complete the PT due tothe retained heat in the system that works against theFwT (Le Chatelier’s e.ect). The resulting adiabaticstress–strain curve during PT is thus steeper than thecorresponding isothermal one as shown in Fig. 3. Bycomparing the stress after complete FwT predicted bythe model in isothermal and adiabatic conditions, it


follows that the di.erence between the two values isproportional to the adiabatic temperature increase bythe factor m, i.e.

�adMf − �isothMf = mR#(ad)F (47)

as required by the phase diagram.Unloading from �¿�adMf to � = 0 activates RvT

and, since there is no hysteresis, all graphs for RvTsimply return along their original graphs for FwT.By virtue of (27), (32), (28), (43) it is possible

to evaluate the transformation path in DA=M followedduring adiabatic PT

� = m(#− #A;0) + 2�c��ln#P#: (48)

The absence of PT-hysteresis also gives W = 0,while, since r=0 then from (24) it follows that Q0 +QF = 0. Therefore,

QF =−cR#(ad)F =−�� P#[1a(ea − 1)

];

QR =−QF: (49)

By comparison with the isothermal conditions, con-sidering that the parenthesized term is ¿ 1, it is seenthat adiabatic conditions involve greater transforma-tional heating governed by the magnitude of a.

5.4. Stress-induced convective transformation

In this simplest setting, closed form computationscan be made to show how the heat exchange withthe environment gives rise to intermediate responsebetween isothermal and adiabatic conditions.The coeScient of convective heat transfer �¿ 0 is

given and the temperature of the external environment#E is regarded as constant. Then (19) with the con-vective heat exchange condition (6) gives

c#− #��= �(#E − #): (50)

While the thermomechanical response of B to pre-scribed � and #, and also to prescribed � in adiabaticconditions, gave output response of the other quan-tities that reHect the time-scale changes in the inputvariables, in this generic convective case a commontime scale change in � and #E alone does not give thesame time-scale change in the dependent quantities.This is due to the fact that � also embodies a charac-teristic time scale. Rate independence can be achieved

in this setting by simultaneously rescaling � [19]. Assuch a rescaling of a material parameter is physicallyarti0cial, convective processes are naturally viewed asrate-dependent even in the context of the otherwiserate-independent theory presented here. Such a ratedependence is a secondary e.ect due to the inHuenceon the PTs of the heat remaining in the material as aconsequence of the partial exchange a.orded by theconvective term.In the absence of phase transformation (�=0), (50)

is immediately integrable, giving a standard result, in-dependent of variation in �,

#= #E + ( P#− #E)e−(�=c)(t− Pt); (51)

where P# and Pt are initial values. This implies that, inabsence of PT, the temperature, if not already equalto #E, tends to reach the environmental temperature ata rate depending on the ratio �=c. As discussed previ-ously, the presence of an inherent time scale within �gives a rate dependence to convective transformations,unlike the rate-independent response for the previousloading cases. When PT is active, one may eliminate �in terms of # and � in (50) with the aid of (27) giving

c#− ��2�

#(�� − ��#) = �(#E − #) (52)

which can be rewritten as

(2�c + �2�#)d#d�= ��#+

2��(#E − #): (53)

In this form explicit rate e.ects enter only via theconsolidated expression

�∗:=��: (54)

In any interval of time during which � is constant,then �∗ is constant and (53) separates(A1#+ A2A3#+ A4

)d#= d� (55)

with

A1 = �2� ¿ 0; A2 = 2�c¿ 0;

A3 = (�� − 2��∗); A4 = 2��∗#E¿ 0: (56)

For initial values P# and P� integration of (55) gives thefollowing path segment in DA=M;

� − P� = 'con(#; P#); (57)


σ

ε

isothermal

adiabatic

convective

convective

isothermal σ

ϑD

D

A

M

Fig. 4. Convective versus isothermal stress–strain curve for Model M0 (left), isothermal, adiabatic and convective paths in the phasediagram (right).

where

'con(#; P#):=

A1A3(#− P#)

+(A2A3 − A1A4)

A23

lnA3#+ A4A3 P#+ A4

if A3 �=0;

A12A4

(#2 − P#2)

+A2A4(#− P#) if A3 = 0:

(58)

Consider now an initial state consisting of stress-free Austenite at the ambient temperature: (�; #) =(0; #E), #E¿#A;0 ⇒ � = 0: This initial state is thensubject to a constant loading rate �¿ 0 under con-vective conditions as described above. The tempera-ture then remains at #E prior to the initiation of FwT.Activation of FwT occurs at stress �M(#E) as in theisothermal and adiabatic cases, while, during FwT, thetemperature evolves with stress as

� − m(#E − #A;0) = 'con(#; #E) (59)

thus de0ning a path in the mixture zone DA=M begin-ning at the point (�; #)= (m(#E−#A;0); #E) (Fig. 4).The limit �∗ → ∞ gives isothermal response # =

#E, as can be inferred from (52). The other extremeof �∗ = 0, gives with the aid of (56)–(58) that

'con =��(#− #E) + 2�c��

ln(##E

)(60)

thus retrieving the adiabatic result (48). For 0nite�∗¿ 0 the response is between that of the isother-mal and adiabatic limit. More generally, for di.ering�∗ values �∗1 , �

∗2 such that �

∗1¡�∗2 the FwT convec-

tive path in DA=M for �∗1 will involve higher tempera-tures than the corresponding path for �∗2 , all else beingequal. The conclusion of FwT takes place when theconvective path enters DM; whereupon for continuedloading the temperature decreases back toward #E isgoverned by (51) with appropriate initial conditions.Unloading at a constant loading rate �¡ 0 is treated

similarly. Now, however, the stress–temperature re-turn path in D does not retrace the original loadingpath. This is because the continual decrease in tem-perature between conclusion of FwT and activation ofRvTwill cause RvT to initiate at a lower stress than thestress at the conclusion of FwT. The stress–strain un-loading graph, therefore, di.ers from the stress–strainloading graph, thus providing an apparent hysteresise.ect (Fig. 4). This type of e.ect, entirely due to thetime-scale embedded in �∗, is not the main concernof the present study. Accordingly, convective loadingwill not be discussed for models M1 and M2. We will,however, investigate both isothermal loading and adi-abatic loading for these two models in order to showhow their hysteresis-free status in model M0 is lost inthe more general models M1 and M2.

6. Model M1

The next model that can be contemplated in thisframework is characterized by constant non-zero during both FwT and RvT, so that, to activate


Π

0ξ

1

ΛM

ΛA D

s

DA/M

ϑ DA

FwT

MD

σ

RvTD

deadD

A

A f

s

Mf

M


PTs, the driving force must now attain the thresh-olds A; M. Due to the restrictions induced by theClausius–Duhem inequality, the thresholds have tobe of di.erent sign, with M¿ 0 and A¡ 0. Therelation between ! and � is described in Fig. 5.The transformation criteria (20) provide di.erent

expressions for FwT and RvT

" + �(1− 2�)− M = 0 (FwT);" + �(1− 2�)− A = 0 (RvT): (61)

The phase diagram is now determined by four iso-"lines that corresponds again to lines of slope m in D(Fig. 5).With respect to Model M0, the existence zones for

A andM are shifted by the thresholds s as they aregiven, respectively, by "¡ A−� and "¿ M+�,giving rise to

DA = {(�; #) | 06 �6 �Af (#); #¿Af}; (62)

DM = {(�; #) |�¿ 0; �¿ �Mf (#); #¿ 0}; (63)

where

�Mf (#):=m(#−Mf ); �Af (#):=m(#− Af ) (64)

and

Mf :=#R − M + ��

; Af :=#R − A − ��

: (65)

The mixture zone DA=M is given by intermediatevalues of "

DA=M = {(�; #) |�¿ 0; �Af (#)6 �6 �Mf (#);

#¿Mf} (66)

and now supports separate transformation subzonesDFwT, DRvT. Speci0cally, FwT can occur if "∈

( M−�; M+�), whereas RvT can occur if "∈ ( A−�; A + �). This de0nes two strips in D

DFwT = {(�; #) |�¿ 0; �Ms (#)6 �6 �Mf (#);

#¿Mf}; (67)

DRvT = {(�; #) |�¿ 0; �Af (#)6 �6 �As (#);

#¿As}; (68)

with

�Ms (#):=m(#−Ms); �As (#):=m(#− As) (69)

and

Ms:=#R − M − ��

; As:=#R − A + ��

: (70)

It is remarked that PT now requires not only thatthe (�; #)-path is within a transformation zone of thephase diagram, but also requires appropriate path di-rectionality. In particular, from (23) it follows thatFwT requires both (�; #)∈DFwT and "¿ 0. Similarly,RvT requires both (�; #)∈DRvT and "¡ 0.For � = 0, the transformation zones collapse into

lines eliminating the possibility of phase coexistenceand involving abrupt transformations.Notice from (65) and (70) that the model requires

the width of regionDFwT to match the width of regionDRvT, such width depending on the value �=��. IfMs¿As there is a non-zero intersection betweenDFwT

and DRvT, while in the physically more typical caseMs¡As, there is a no intersection between DFwT andDRvT giving rise to the dead zone

Ddead = {(�; #) |�¿ 0; �As (#)6 �6 �Ms (#);

#¿Ms}: (71)

It, therefore, follows that the presence of the drivingforce thresholds is responsible for the implementationof hysteresis in the model, while in this setting thephase coexistence is still governed by the interactionenergy only.The transformation kinetic (23) becomes

�=

{max{ 1

2� "; 0} if ! = A;min{ 1

2� "; 0} if ! = M(72)


from which it is observed that the phase fraction evo-lution takes place at the same rate as in model M0.However, unlike model M0, FwT and RvT are acti-vated at di.erent driving force levels. Thus, the e.ectof the introduction of the A; M in model M0 is feltonly on the activation, and not in the evolution, of thePT.

6.1. Temperature-induced constant-stresstransformation

Under constant stress � and prescribed temperature,starting from pure A, FwT progresses only if #¡ 0.According to (61), the relation between temperatureand phase fraction is

#F(�):=Ms +1m� − 2�

��: (73)

Thus FwT is activated and completed at thestress-dependent temperatures

Ms +1m�︸︷︷︸

#Ms (�)

¡#¡Mf +1m�︸︷︷︸

#Mf (�)

: (74)

Similarly, starting from pure M, RvT takes place onheating

#R(�):=As +1m� +

2��(1− �): (75)

Thus, RvT is activated and completed at thestress-dependent temperatures

As +1m�︸︷︷︸

#As (�)

¡#¡Af +1m�︸︷︷︸

#Af (�)

: (76)

The model now predicts four di.erent transforma-tion temperatures but with the restriction

Af − As =Ms −Mf = 2�� : (77)

The transformational heatings and latent heat havethe same expressions (37) predicted by the model M0.


Under constant temperature P# and prescribed stress,starting from pure A, FwT progresses only if �¿ 0

(tensile loading) and is activated and completed at thetemperature-dependent stresses �Ms ( P#) and �Mf ( P#).RvT takes place on unloading with start and 0nishstress given by �As ( P#) and �Af ( P#). The model thuspredicts four di.erent transformation stresses but suf-fers the restriction

�Af − �As = �Ms − �Mf =2��

(78)

so that the stress increase during transformations isthe same as in Model M0. The peculiarity of M1 isthat now the RvT start does not coincide with the FwT0nish but

�Mf − �As = M − A

�¿ 0: (79)

In the stress–strain plane FwT and RvT follow lineswith the same constant slope EPT and the incrementalstress–strain relations for both PTs have the same ex-pression (40) given by model M0 (Fig. 6). For �=0,horizontal loading and unloading plateaus correspond-ing to abrupt transformations are obtained.It is remarked that the model correctly predicts that

�Ms¿�Af and �Mf ¿�As , due to the sign of A and M as dictated by (12). This implies that, as expected,the constitutive restrictions imposed by the second lawensure a clockwise hysteresis loop.According to (26), the work as given by the area of

the stress–strain hysteresis loop follows from (61) as

W = M − A (80)

while the transformational heatings are now given by

QF =− P#�� − M; QR = P#�� + A: (81)

In contrast to the dissipation-free model M0, the pres-ence of driving force thresholds for PT now requiresheat removal during isothermal FwT that is di.erentfrom the required heat supply for RvT at the sametemperature. This di.erence balances with the area ofthe hysteresis loop, i.e. W + QF + QR = 0.


Following a development parallel to that givenfor Model M0, we again consider an initial state ofstress-free A at temperature P# subject to adiabatic


σ

adiabatic

isothermal ε

EPT

EPT

ϑ

t

adiabatic

isothermal

∆ϑ adF

∆ϑ ad

R

∆ϑ adϑ

Fig. 6. Adiabatic versus isothermal response of model M1: stress–strain (left), temperature–time (right).

loading �¿ 0. Eq. (19) then yields

c#− (#�� + M)�= 0 (FwT);c#− (#�� + A)�= 0 (RvT); (82)

so that temperature changes only occur if a PT is ac-tive. In particular, during FwT the temperature evolvesfrom the initial value P# in terms of � as follows:

#F(�) =(P#+

M��

)ea� − M

��: (83)

FwT is complete at temperature #F(1)¿ P# corre-sponding to an adiabatic FwT temperature increase

R#(ad)F =(P#+

M��

)(ea − 1): (84)

Comparing with model M0 as given by (45), it is seenthat the driving force threshold M directly contributesso as to give higher temperature increases.Upon unloading, the temperature during RvT

decreases from #F(1) according to

#R(�) =(#F(1) +

A��

)e−a(1−�) − A

��; (85)

reaching the value #R(0) upon return to pure A. Con-trary to Model M0, the temperature #R(0) does notmatch the initial temperature P# and, therefore, aftera full transformation cycle, there is a net temperatureincrease

R#(ad) = M − A��

(1− e−a)¿ 0 (86)

that is entirely due to the energy dissipation associatedwith the PT hysteresis.

adiabatic

isothermal σ

ϑD

D

A

M

D DFwTRvT

Fig. 7. Isothermal and adiabatic paths in the phase diagram forModel M1.

By virtue of (61) and (83), the stress during PTs isgiven, in terms of �, as

� =

(m P#+ M

� )ea� − m#R − �(1−2�)

� (FwT);

(m P#+ M� )e

a� − m#R − �(1−2�)�

+ A− M� e−a(1−�) (RvT);

(87)

which, in conjunction with (14), permits a parametricplot of the adiabatic stress–strain FwT curve. Com-pared to isothermal PT, the FwT 0nish stress is greater,so that the adiabatic stress–strain curve is steeper thanthe isothermal one. As for model M0, the di.erencebetween the two values is proportional to the adiabatictemperature increase by the factor m, i.e. (47) holds.The combined loading–unloading stress–strain curveis depicted in Fig. 6 and the adiabatic path in the phasediagram is shown in Fig. 7.


Under adiabatic conditions, the transformationalheatings obey QF;R + Q0 = 0 so that

QF =−cR#(ad)F ; QR =−cR#(ad)R ; (88)

where R#(ad)F is given by (84) and R#(ad)R :=R#(ad)−R#(ad)F is calculated with the aid of (86).The area of the hysteresis loop can be computed

with the aid of (26) and (87), giving

W = cR#(ad) = ( M − A)1− e−a

a: (89)

It is interesting to note that, since M− A is the areaof the isothermal loop and the right-multiplicative fac-tor ¡ 1, the model predicts that the adiabatic pseu-doelastic loop is smaller than the isothermal one. Thisloop area reduction induced in non-isothermal condi-tions increases exponentially with a, again con0rm-ing a as a basic quantity a.ecting the non-isothermalbehavior.

7. Model M2

In the logical path that we are following, Model M1carried out the central task to model the PT-hysteresisin the simplest possible way. This was done by as-suming the driving force thresholds F;R to be givenconstants. However, from a physical viewpoint, thisimplies that the resistance encountered by the tran-sition fronts is constant during the whole PTs andthis is not realistic. The resistance encountered by thePT in SMM originates from many complex phenom-ena including the interactions among product phaseinclusions and defects (like e.g. dislocations, grainboundaries, precipitates) [1,13]. Reasonably, such in-teractions increase when the concentration of the prod-uct phase increases. It seems, therefore, that an appro-priate SMM model should account for the variationof the resistance encountered by the transition frontsduring PT.As a prototype of variable -models, in the follow-

ing we consider functions F;R linear with � as givenin (13) (Fig. 8).ModelM2 is a generalization of modelM1, and all of the M2 formulae collapse onto the M1formulae if both Ms = Mf = M and As = Af = A.The transformation criteria (20) now provide

" + �(1− 2�)− Ms − PF�= 0 (FwT);" + �(1− 2�)− As − PR(1− �) = 0 (RvT); (90)

Π

Λ0As

ΛMs

ξ

1AfΛ

ΛMf

fDdead

D RvT

M

Mf

Ms

A s

FwTD

σ

A

ϑ D

D

D

AA/M


where for ease of notation we have introduced

PF:= Mf − Ms ; PR:= Af − As ; (91)

so as to easily collapse model M2 onto model M1 byrequiring the vanishing of PF;R. The formulas for M2thus inherit a structure similar to the correspondingones in Model M1 to within the addition of terms inPF;R.It follows that (62)–(71) continue to supply a de-

scription of the phase diagram for M2, however, nowthe width of DFwT can be di.erent from that of DRvT

(Fig. 8). The transformation kinetic is now given by

�=

max{

"2�+PF

; 0} if ! = Ms + PF�;

min{ "2�−PR ; 0} if ! = As + PR(1− �):

(92)

Note that the dependence of on � now permits dif-ferent transformation rates in FwT and RvT. Sinceboth the above denominators are positive, the PTs arealways associated with the same directionality of " asfor the previous models.

7.1. Temperature-induced constant stresstransformation

In this case the stress-dependent transformationtemperatures are, once again, given by (74) and (76).Now, however,

Ms = #R − Ms − ��

; Mf = #R − Mf + ��

;

As = #R − As + ��

; Af = #R − Af − ��

: (93)

The model now predicts four transformation temper-atures that can have independent values without re-strictions. The transformational heatings and the latentheat are again given by (37).



Temperature-dependent transformation stresses areagain given by (64) and (69).In the stress–strain plane, the PTs follows again

straight lines that, however, now have di.erent slopesin FwT and RvT (Fig. 9)

EF =2� + PF

2� + PF + E�2E;

ER =2� − PR

2� − PR + E�2 E: (94)

The area of the hysteresis loop is given by

W = Ms − As +PF − PR2

(95)

giving the energy dissipated during the complete cy-cle. The transformational heating is given by

QF =− P#�� − Ms −PF2;

QR = P#�� + As +PR2: (96)

Note again that W + QF + QR = 0.


Adiabatic FwT and RvT proceed, respectively,under the conditions

c#− (#�� + Ms + PF�)�= 0 (FwT);

c#− (#�� + As + PR(1− �))�= 0 (RvT): (97)

Beginning from pure A, the associated temperatureevolution from an initial temperature P# during FwT isgiven in terms of � by

#F(�) :=(P#+

Ms��

+PF��

1a

)ea� − Ms

��

− PF��

(1a+ �

): (98)

With respect to Model M1, the variation of the sinduces an additional term which is linear in �. Theadiabatic FwT temperature increase is given by

R#(ad)F =(P#+

Ms��

+PF��

1a

)(ea − 1)− PF

��: (99)

Upon unloading, the temperature during RvT evolvesfrom #F(1) according to

#R(�) :=(#F(1) +

As��

− PR��

1a

)e−a(1−�) − Af

��

+PR��

(1a+ �

)(100)

reaching the value #R(0) upon return to pure A. Thecomplete loading=unloading cycle gives an adiabatictemperature change

R#(ad) :=( Mf − As

��+c(PF + PR)

�2�

)(1− e−a)

− PF + PR��

: (101)

The stress during PTs is given in terms of � as

� =

(m P#+ Ms� + PF

�a )ea� − m#R

−�(1−2�)� − PF

�a (FwT);

(m P#+ Ms� + PF

�a )ea� − m#R

− �(1−2�)� + PR

�a + ( As− Mf

�

−PF+PR)a� )ea(�−1) (RvT)

(102)

which permits a parametric plot of the stress–straincurve. The adiabatic curve is always steeper than theisothermal one (Fig. 9) and the increase in the 0nishstress is still proportional to the temperature increasewhere now R#(ad)F is given by (99). The area of thehysteresis loop is again given by

W = cR#(ad): (103)

8. Parameter identi%cation

We now turn to discuss how the material param-eters de0ning the various models M0–M2 can bedetermined on the basis of experimental data. Suchidenti0cation can certainly be accomplished in var-ious ways depending on the available information.Here, we propose a possible procedure based on theassumption that the following parameters are avail-able from standard experimental tests:

• the stress-free transformation temperatures:Ms; Mf ; As; Af , the speci0c heat c and the latent heat


σ

adiabatic

isothermal ε

ER

EF

ϑ

t

adiabatic

isothermal

∆ϑ adF

∆ϑ ad

R

∆ϑ ad

ϑ

Fig. 9. Adiabatic versus isothermal response for Model M2: stress–strain (left) and temperature–time (right).

of transformation H (e.g. from one calorimetrictest);

• the elastic modulus E and transformation strain �(e.g. from one isothermal tensile test);

• the rate m at which transformation stresses increasewith temperature (e.g. from two isothermal tensiletests at di.erent temperatures).

Since m and � are available, the de0nition of mgiven in (28) immediately delivers a value for �� thatapplies to all of the models. We discuss the remain-ing parameter identi0cation for the three models sep-arately.Model M0 is characterized by six parameters

E; �; c; ��; #R ; �:

The 0rst four follow from the above considerations,while the last two remain to be determined. The modelsu.ers the restrictions (36) on the transformation tem-peratures that for convenience is here recalled as

#M;0 = As =Mf ; #A;0 =Ms = Af : (104)

Due to the oversimpli0cation in the model there is notenough information to use all the available data andthe parameters #M;0 and #A;0 have to be determined bysome interpolation from the available transformationtemperatures. Under this restriction, it then followsthat

#R =#M;0 + #A;0

2; � =

#A;0 + #M;02

�� (105)

completing the parameter identi0cation. Also the la-tent heat cannot be used to introduce information inthe model, since H follows from the model as

H =m�2(#M;0 + #A;0) (106)

providing an additional restriction on model M0.Model M1 is characterized by eight parameters

E; �; c; ��; M; A; #R ; �:

With respect to Model M0, the addition of the twoadditional parameters M, A causes the three restric-tions on the transformation temperatures (104) andon the latent heat (106) to consolidate into the singlerestriction

As − Af =Mf −Ms: (107)

Accordingly, the latent heat can be used to identify #Rand then H together with the transformation temper-atures enable the determination of �; A; M as fol-lows:

#R =H��; � =

��2(Ms −Mf );

M = H − ��2(Ms +Mf );

A = H − ��2(As + Af ): (108)

The Clausius–Duhem restrictions (12) are realizedprovided that

12(Ms +Mf )6#R =

H��612(As + Af ): (109)

The adiabatic temperature changes (84), (86) arepredicted from the model.Model M2 is characterized by the ten material

parameters

E; �; c; ��; Mf ; Ms ; Af ; As ; #R ; �:


The reference temperature #R is again determinedvia the latent heat from (108) as for Model M1. Itremains to determine Mf ; Ms ; Af ; As ; � so as toensure the values Ms; Mf ; As; Af . Thus, there is onedegree of freedom yet open. The FwT adiabatic tem-perature increase R#(ad)F given by (99) can be used forthis purpose. Determination of R#(ad)F requires a sin-gle adiabatic loading test with temperature sensing ca-pability beginning with pure austenite. Alternatively,R#(ad)F follows from a comparison of the adiabatic 0n-ish stress with the isothermal 0nish stress using (47).Elementary manipulations using (99), (93) now give

�=��(R#

(ad)F +Ms −Mf )

(ea − 1)(1− 2=a) + 2

− (ea − 1)[c(Ms −Mf ) + ��( P#+ #R −Ms)]

(ea − 1)(1− 2=a) + 2 :

(110)

With � determined from (110), the identi0cationof Mf ; Ms ; Af ; As now follows immediately from(93). With the parameter identi0cation now complete,the complete loading=unloading temperature changeR#(ad) follows from (101). Model M2 can be viewedas a two parameter generalization of M1, where oneof the two additional degrees of freedom so obtainedis used to eliminate the restriction (107) and theother is used to match a single additional experimen-tal observation. In the above treatment, R#(ad)F wasused for this latter purpose. It is useful to remark thatR#(ad) as given in (101) does not supply the samecompletion. Namely, substituting from (93),(91)into (101) to eliminate Mf ; PF; As ; PR in terms ofMs; Mf ; As; Af ; � gives cancelation of � such that

R#(ad)

As − Af +Ms −Mf + 1

= (1− e−a)(

As −MfAs − Af +Ms −Mf +

1a

):

(111)

Consequently (111) can be viewed as a restriction onM2 in a similar way that (36), (106) provided restric-tions on M0 and (107) provided a restriction on M1.More re0ned descriptions for the dissipation functions F; R can be contemplated to eliminate such a re-striction.

9. Conclusions

Models for the macroscopic behavior of SMM canbe conveniently built within the ZGN framework. Allthe constitutive information is included in two consti-tutive functions: the free energy � and the dissipationfunction �. The transformation kinetic is not assumeda priori but follows from the restrictions induced bythe balance equations. Its 0nal expression depends on� through the transformational driving force ! via(16), and on � through the driving force thresholds F and R. The basic essence of the models can besummarized by the relation between ! and �.In this simplest setting, explicit computations have

been carried out to show how increasing detail in therepresentation of the driving force thresholds F; Rprovide for the incorporation of additional physicale.ects. Even so, the simple assumptions consideredhere can easily be generalized by more re0ned expres-sions. Such re0nement to the constitutive descriptioncan be made on a physical basis and the associatedmodi0cations communicate throughout the structureof the theory in a simple way. For example, the ad-dition of a thermal expansion e.ect, di.ering phasespeci0c heats or di.ering phase elastic moduli, as wellas driving force thresholds varying non-linearly with� comprise a simple and immediate extension. Here,we have focused instead on certain issues that havereceived relatively less attention, such as the role ofadiabatic and convective heating environments andthe issue of parameter identi0cation. Ongoing devel-opment of the present framework is related to the is-sue of incomplete transformations and the extensionto multiple variant SMM.

Acknowledgements

The support of the Center for Fundamental Materi-als Research of the Michigan State University is grate-fully acknowledged.

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models for one-variant shape memory materials based on dissipation functions

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