A general macroscopic description of the thermomechanical behavior of shape memory alloys

.I. Mech. Phys. Solids, Vol. 44, No. 6, pp. 953-980, 1996 Copyright c 1996 Elsevier Science Ltd

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A GENERAL MACROSCOPIC DESCRIPTION OF THE THERMOMECHANICAL BEHAVIOR OF SHAPE MEMORY ALLOYS S. LECLERCQ Laboratoire

and C. LEXCELLENT

de Mkcanique Appliqube R. Chaltat, Facultk des Science: et Techniques, 24, chemin de I’kpitaphe, 25030 Besanvon Cedex, France

(Received 26 June 1995 : in revisedform

20 November

1995)

ABSTRACT The paper presents a macroscopic description .:hat allows the simulation of the global thermomechanical behavior of shape memory alloys @MA). Use is made of the thermodynamics of irreversible processes framework. Two internal variables are taken into account : the volume fraction of self..accommodating (pure thermal effect) and oriented (stressinduced) product phase. A specific free energy, valid in the total range of phase transition, is defined with particular attention paid to the interaction term. A study of the thermodynamic absolute equilibrium during phase transition proves its instability, and hence explains the hysteretic behavior of SMA. The kinetic equations for the internal variables are written in such a general way that the model could comply with the second law of thermodynamics. The postulate of five yield functions (each of them being related to one process) permits the phase transition criteria to be defined and the kinetic equations related to each process through consistency equations to be derived. The parameters of the model have been identified for three particular SMA, and the simulated results show good agreement with experiments. Copyright c 1996 Elsevier Science Ltd

1.

INTRODUCTION

Among the numerous studies published in the field of shape memory alloys (SMA), a not negligible number of which ccmcern pseudoelasticity, very few deal with the total set of their uncommon properties. In other words, very interesting works have already been performed to describe pseudoelasticity (see for instance Huo and Miiller, 1993 ; Raniecki and Lexcellent, 1994 ; Patoor et al., 1994) or recovery stress (Leclercq et al., 1994) as subjects of study, but not so numerous are the models that are able to take into account one way shape memory effect as well as recovery stress or pseudoelasticity, i.e. the non-isothermal behavior of shape memory materials. Indeed, even if the isothermal properties are much easier to consider, it is evident that the possible future applications of SMA shall require complete thermomechanical loadings, including obviously non-isothermal ones. Thus is seems clear to the present authors that the development of new models or improvement of existing ones is necessary to insure a broader industrial development of these materials. Some attempts to this goal have already been performed, from both micromechanics 953

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and C. LEXCELLENT

and macroscopic point of view. Sun and Hwang (1993a, b) use a classical MoriTanaka method to derive the constitutive equations of a polycrystalline shape memory alloy. Both phase transition and reorientation are taken into account, and they obtain good qualitative results. Nevertheless, the lack of quantitative experiments prevents verification of the power of prediction of their model. Moreover, even if the method used is pleasant and elegant, it is not a real unified one since each particular effect of SMA requires one particular constitutive equation. Brinson (1993) has developed a macroscopic description of the non-isothermal behavior of SMA, based on a model previously proposed by Tanaka (1986). The main result of this paper concerns a new concept of internal variable (the volume fraction of martensite), which is split into a self-accommodating one (pure thermal effect) and an oriented one (thermomechanical effect). Unfortunately the effect of reorientation of the self-accommodating product phase has not adequately been taken into account (see Leclercq, 1995) and this prevents a good prediction of the material’s behavior. Fremond (1987) developed a very nice model describing the behavior of SMA for any thermomechanical loading. This model is based on convex analysis applied to the now classical concept of thermodynamics of irreversible processes and local state postulate. The qualitative results of this approach are good. Finally, there exist other approaches which do not consider the volume fraction of product phase as an LI priori internal variable, but as a consequence of evolving interfaces between parent and product phases (see for instance Ball and James, 1992 or Abeyaratne and Knowles, 1993). These descriptions, closer to reality, need much time to become totally applicable for practical applications. As a matter of fact, there exists in each of the previous approaches some crucial point that makes easier the description and comprehension of shape memory materials, but no model is completely useful for a real prediction of the behavior of real structures. The goal of the present paper is therefore to propose a model quantitatively consistent with any thermomechanical response of SMA, except for reorientation of oriented product phase (a discussion of this case will be presented in a following study). After a general presentation of the constitutive equations derived from the postulate of a specific free energy for the system under study, a comparison between experiments and calculation will be shown. Concluding remarks will deal with the different assumptions related to the model, and the possible improvements in future.

2.

SOME BASIC ASSUMPTIONS

The model presented here rests upon the classical local state postulate, which assumes that there exists a representative volume where one can define and measure some internal quantities, called internal variables. In the following, the elastic strain $, the temperature T, the volume fraction of self-accommodating martensite zT and the volume fraction of oriented martensite z, will be taken as internal variables. A brief discussion is necessary here to clarify the sense of z, and zT. The product phase (namely the martensitic phase) has two possible origins. The first one is the classical parent phase (austenite) o product phase (martensite) due to changes of temperature only. Thus, the 24 variants of product phase may be created from the parent phase,

Thermomechanics

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alloys

Austenite

T

0, T

/ Self-accommodating martensite 1. Interrelations

Fig.

\ d between parent

Oriented w

and product

martensite phase.

each variant having its own “complement”. From the mechanical point of view, the creation of martensite induced by pure thermal effect does not induce any macroscopic phase transition strain. This is the reason why this product phase is called selfaccommodating martensite. On the other hand, martensite may be produced by an external mechanical action (namely a stress tensor a). In this case a small number of variants are created (those whose orientation is the most consistent with the stress tensor) and this results in the appearance of a macroscopic phase transition strain. One qualifies this product phase as oriented martensite. One should also note that oriented martensite can be produced directly from the self-accommodating one by orientation under an external mechanical load. Figure 1 summarizes the possible interactions between parent phase and product phases. From the crystallographic point of view, it is not possible to differentiate the two types of martensite distinguished above. Indeed the free energy of each product phase is obviously the same. Nevertheless their macroscopic effect on the shape of a SMA sample is not the same: the self-accommodating martensite does not produce any macroscopic phase transition strain, on the contrary to the oriented one. Moreover the micrographs shown in Fig. 2 prove that both martensites can be present at the same time inside the same SMA sample. In Fig. 2(a), one can see a micrograph of a sample being stressed (1D) in the pseudoelastic temperature range. The oriented martensite that appeared is easily recognizable. Holding the sample stressed, the temperature is then decreased below lMr (the classical martensite finish temperature). Then, the uniaxial stress is released. The micrograph of Fig. 2(b) clearly shows that some oriented martensite remains in.side the material, and coexists with the selfaccommodating one. Let us come back to the internal variables z, and zT. The micrographs shown in Fig. 2 make evidence that their physical meaning does exist, even if the free energy associated with each of these variables is the same. Indeed, (i) in the case of a pseudoelastic mechanical test, the martensite that is created is only the oriented one, (ii) in the case of cooling of an austenitic sample (without any mechanical action), the martensite that is created is only the self-accommodating one, and (iii) in the case of a thermomechanical loading, the two types of martensite may be created, each with its own kinetics. Thus it is possible, from the macroscopic point of view, to describe the different behaviors of SMA with two (and only two) variables, one (z,) being

S. LECLERCQ

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and C. LEXCELLENT

related to the creation of macroscopic phase transition strain, while the other (zT) is related to no macroscopic shape change. Obviously, the internal variables chosen here are only “averaged” variables, and do not have pretention of describing the microscopic results of the phase transformation. The second basic assumption that is laid down concerns the partition of strains. The terms of second order in the calculation of strains are here neglected, so the total strain 5 is assumed to satisfy the following equation

where 5’ and ctr are, respectively, the elastic and transformation strains. Such an assumption can be discussed. Indeed, SMA as TiNi polycrystals are well known for their ability to undergo up to 8% transition strains in tension (and 4% for copper-based alloys), which are at the boundary between small and large strains. But one can say that in most mechanical loadings of SMA, the strains remain small (up to 4%), because the mechanical tests often induce large displacements, but small strains of the material. So in these conditions (I) holds. A macroscopic formulation using large deformations shall be written in the future. One has here to point out that the thermal dilation will not be taken into account, because it is negligible with respect to the phase transition strain. In order to finish with the assumptions stated in this section, let us specify a question or “forward phase transof vocabulary. In the following, “forward phase transition” formation” will be used for qualifying the creation of self-accommodating martensite from austenite, or the creation of oriented martensite from austenite or self-accommodating martensite (see Fig. 1). On the contrary the creation of austenite from one of the two martensites will be called “reverse phase transition” or “reverse phase transformation”.

3.

THERMODYNAMIC

3. I

PRINCIPLES

AND CONSTITUTIVE

EQUATIONS

Spec$c ,fiee energy

The Helmholtz free energy of the three-phases chosen in the following form

system considered

in this study is

Q, =(l-z)~‘+z,~,‘+z,cD’+A~, where z = 2d + zT is the total volume Z, and zT must comply with 0
fraction

and

(2)

of product

phase. The three variables

z,

O
The term W (c( = 1,2,3) from (2) is the free energy of the cc-phase. It is easy for the reader to guess that : CI= 1 corresponds x = 2 corresponds x = 3 corresponds

to parent phase (austenite) ; to self-accommodating martensite to oriented martensite.

;

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b) Fig. 2. Micrographs ( x 500) of a CuAlNiMn polycrystal. (a) Creation of oriented martensite from austenite by mechanical loading (tensile test) (T = 200°C M, = 60’ C, D = 400 MPa). (b) Creation of self-accommodating martensite variants after cooling, the mechanical loading being released after cooling (T = 30 C. J4, = 4l’C, 0 = 0).

Thermomechanics

Equation

(4) gives the expression

of shape memory

959

alloys

of this energy

ox =u”,-Ts;,+L”:L:$+Cv (T-r,)-TLog 2p-a

[

11 .

f (

0

Here, ui and s; are the specific energy and entropy of the a-phase, E; is the second order tensor of the intrinsic elastic strain of the a-phase, L corresponds to the fourthorder tensor of isotropic elasticity. Moreover the terms p, T, To and C, are, respectively, the mass density, the actual temperature, the equilibrium temperature and the specific heat at constant volume. Let us point out here that the specific free energies of each martensitic phase must be the same, in order to be consistent with the statement that these two product phases are not physically differentiable. Let us come back to the term A@ in (2). It has been called by Raniecki et al. (1992) “configurational energy”, and represents the interactions that appear between the phases, typically the incompatibilities between deformations. One of the characteristics of this energy is that it must disappear when only one phase is present inside the material. Moreover, in the case of three phases coexisting, this term must take into account interactions between one phase and the two remaining ones, separately (one interface separates two phases and not three). Thus one can assume the following expression for the configurational energy (this expression has the great advantage to become fairly simple) A@ = +(I

-~)a,

’ +z,(l

-z)@13 +z,z,Qz3,

(5)

where @I*, @I3 and (D23are, respectively, the interaction energies between phases 1 and 2, 1 and 3, and 2 and 3. Here the reader should note that when two phases exist inside the material, i.e. z, or zT is zero, the configurational energy becomes the one expressed by Raniecki et al. (1992) and Huo and Mtiller (1993). Some considerations can also simplify the expression of A@. Indeed it seems completely realistic to postulate that since the two martensites are not physically differentiable, it is logical that their respective interactions with the parent phase are the same. Thus one can write the following system @‘* =
a,, =

K,,

@23= Q; = K2 > 0, where K, and K2 are constants. As a consequence, the combination

of (5) and (6) leads to (7)

3.2.

Some constitutive

Let us first lay down average strain in phase

assumptions an assumption concerning the intrinsic strain cC((i.e. the LX,in a representative volume, obviously). This strain is

960

supposed equation

S. LECLERCQ

and C. LEXCELLENT

to follow the same rule as the total strain. and one shall write the following

p‘1 = I? rc + $1 /1 .

(8)

where each term has a clear meaning. As we consider here that both parent and product phases have the same elastic constants represented by L, it is not difficult to establish that the elastic parts of the intrinsic deformations are the same for each phase, so that “L; = >;; = [;“; III cc. _.._

(9)

Some assumption must also be made about the phase transition part of the intrinsic strains E,. The discussion that was developed in Section 2 of this paper clearly shows that the oriented product phase is responsible for the phase transition strain. Thus the following holds $r = c” = (j ‘I -1 $3,r = ,‘,.

(10)

The expression of 5” raises several questions. Vacher (1991) has established the proportionality between the uniaxial phase transition strain and the volume fraction of martensite. In their first paper dealing with materials undergoing phase transformations, Raniecki et al. (1992) proposed an extension of Vacher’s statement to comply with the three-dimensional situation EPc= Jiz with in this case of isothermal

pseudoelasticity ti+:

zT = 0 and z0 = z

dev G d

-

(11)

where 7 is the total pseudoelastic uniaxial strain (for completed phase transition, see Fig. 3), G the classical Von Mises equivalent stress and z the volume fraction of martensite. One can easily note that the expression of 5 complies with the normality rule to an ellipsoid in the case of proportional loadings. Here use will be made of the preceding relation, but writing it in rates, and assuming the rate of 5 to be zero. One obtains (the dot denotes the time derivative) :I, _ = I$. iiZ,,__: ~ 21

dev (T 6

.

This form of 6” is satisfactory for the case of proportional loadings. it seems not to be sufficient for taking account of the non-proportional point will be discussed in the last section of this paper.

(12)

Nevertheless ones. This

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:J

_________.______________

n; = 0

_____________________

--

r

)

E

Fig. 3. Equilibrium

3.3.

states (u,E) (Tconstant).

Second law of therrno&namics

Let us write the expression

of the Clausius-Duhem

inequality

as follows

Here, s is the specific entropy of the system and @the heat flux received by the system. Taking account of (1) one can write (13) as (14) The inequality (14) must hold at each instant and particularly ii‘ (they are supposed to be independent). Thus one obtains equations 0

and the Clausius-Duhem

=

p

-

a2

inequality

=:

r, I&e,

-

a@

s=-z

whatever are he and the following state

(15)

as follows

(16) As usually done, one chooses to work in such conditions that the thermal dissipation term is non-negative (in particular temperature is homogeneous), so that one can write

S. LECLERCQ

962

and C. LEXCELLENT

(17) Introducing

(12) and dividing

by p, (17) reads 7-r::, + 7r:tT 3 0,

7r)i= ‘u” -(1-22)~,,,-=,~,::+n~~(T), 7r: = -(1-2Z)~,,-I,@,::f7-&(T), where ni; (T) is defined as a chemical

potential

(18)

of phase transition.

rr),( T) = Au* - TAP, Au* = u:, -uf

= u:, -u,:.

As* = s,: -s;

= s;, -s:, .

(19)

Let us point out that rcL(T) has a sense only in the case of effective phase transition, i.e. austeniteomartensite, and not in the case of reorientation of the self-accommodating martensite into the oriented one. Two terms arise in (I Sa), which have been called rr], and &. These two terms will be referred to as thermodynamical forces associated to z, and zr, respectively. One has to note that rck does not explicitly depend on the applied stress. This remark is in complete agreement with the fact that the oriented product phase appears only under a mechanical action. In case of orientation of the self-accommodating martensite, obviously i, = -2r. Thus, (18a) reads

where

Here r&,, is referred of self-accommodating 3.4.

Equilibrium

to as the thermodynamical product phase.

force associated

with reorientation

conditions and stabi&>,

In their first paper about the R,_ model (devoted to pseudoelasticity), Raniecki et al. (1992) developed a systematic study of the equilibrium related to such solid-solid phase transition. Thus, they used the technique of Lagrange’s multipliers and came to the conclusion that the line of equation rr: = 0 was the location of the absolute equilibrium states during forward and reverse phase transition. This means, by the way, that no dissipation occurs. Nevertheless simple considerations have shown that this absolute equilibrium could not exist, which justified the occurrence of the hysteresis loops. In this paper we shall not restart the study performed by Raniecki et al. (1992) but,

Thermomechanics of shape memory alloys

963

following their reasoning, it is clear that the two equations 7~: = 0 and rck = 0 give the absolute equilibrium states of the system. The stability of this equilibrium is related to the sign of the second derivative of the free energy with respect to the internal variables z, and zT, which corresponds to the first derivative of -7~: and - & with respect to z, and zr. Hence one has to deal with the matrix

ST

whose determinant

A complies

-

(21)

with A = Q$‘(4Qi, --a:).

(22)

The matrix S is obviously symmetric and thus has real eigenvalues. The product of these eigenvalues is equal to A. If A is not positive, there exists at least one eigenvalue that is not positive, and this leads to the conclusion of the instability of the equilibrium. Experiments show, as far as we know, that either the term 4@,,,-@: is not positive and (Dz > 0, or 0,:: = 0 (see the next section for applications) at least for the alloys that have been studied. Thus one can conclude the instability of the equilibrium in the phase transformation domain. On the other hand one can establish this instability by using simple deductions. First, let us focus on the austenite o martensite phase transition (7~: = 0 or 71: = 0). This also does not concern the reorientation of selfaccommodating martensite. From the purely mechanical viewpoint it is clear that equilibrium means 7~; = 0, and Raniecki et al. (1992) have already shown that this equation corresponds to a line of negative slope in a stress-strain plane. Thus the mechanical instability is obvious. From the viewpoint of temperature dependence (austeniteoself-accommodating martensite) the equation I& = 0 leads to a line of positive slope As*/2@,,,) in a (z, r) plane. This means that z increases with respect to T. This statement is in complete contradiction with experimental observations. In the same way, from the viewpoint of temperature dependence (austenite o oriented martensite) one can easily show that the sign of da/aT is opposite to that of az,/dT (at constant zT). This means that the equivalent stress decreases while T is increasing. Also it is in contradiction with the e.xperimental results. Hence, one can conclude that the austenite o martensite phase transition leads to unstable equilibrium states, whatever the driving motion of the transformation is. The particular case of the reorientation of the self-accommodating martensite can be treated the same way as Raniecki et al. (1992) did. In their work, one can simply replace rrf by &,, and z by z,. It follows immediately that the equilibrium is represented by &, = 0. The instability of this equilibrium follows from the derivation of -n$, with respect to z,. As a conclusion to this study, it is clear that the states of absolute equilibrium are unstable, so that the material will not follow them during phase transition. Figures 3-5 show the equilibrium states for pseudoelastic isothermal paths, non-isothermal paths at stress-free state, and non-isothermal paths at constant strain, respectively.

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Fig. 4. Equilibrium

and C. LEXCELLENT

states (z, T) at stress free state

I

e

T

Fig. 5. Equilibrium

states (u, T).

The presence of hysteresis loops characteristic of the SMA behavior is therefore justified. They correspond to entropy production (see for instance Rice, 1971, about the concept of constrained equilibrium). One can schematically draw (with dashed lines) the paths of phase transformation on previous figures. 4.

SYSTEM

EVOLUTION

AND KINETICS

The instability of the equilibrium induces that there exists up to now no thermodynamic relation that could give the equation of the branches of the hysteresis

965

Thermomechanics of shape memory alloys

loop. Nevertheless one needs such relations in order to write explicitly the evolution kinetics of z, and zT. These equations have to be combined with the behavior equations (12) and (15a) in order to define completely the behavior of the system. One should note that the same type of approach has already been used by Boyd and Lagoudas (1994), in a model describing pseudoelasticity and shape memory effect using a free energy function and a dissipation potential. In the same spirit, Graesser and Cozzarelli (1994) developed a model for pseudoelasticity and shape memory effect, on the basis of one constitutive equation and the definition of one kinematic internal variable (back-stress). 4.1.

Yield~functions

Let us assume that there exist five functions II/:, II/z, I,&, +z and $Ta that are, respectively, linked to the forward phase transition (F : austenite =Smartensite), the reverse phase transition (R : martensite * austenite), and the reorientation process of self-accommodating martensite. The superscripts c and T refer to the type of martensite involved in the process. Thus two functions are available for each kind of martensite, and one is devoted to the reorientation. Taking a similar framework as classical plasticity, the functions defined below are assumed to be constant during the phase transition. $;(z,,g,

T) = rf,-k”F = Y;,

$g(z,, 0, T) = -n;+k”, $;(z,,

T) = n;-k;

$:(zT,

T)’ = -n:+k;

= Y;;, = Y;, = Y’,,

t+bTu(z,,O;I= n;, - kT” = Y*“.

(23)

The constants Y”,, P, and YTB(CI= C-J’, 2) are non-negative and kUF,kR and kTo are functions taking zero value at the beginning of forward or reverse phase transition. These functions are extended from the kinetic forms proposed by Koistinen and Marburger (1959) and Raniecki et al. (1992). One has k;r = 2@&,-z:)-

sLog(l -AS*

-zz,+z:)

(7-T*)+$(Exp(-b,(T-M:))-Exp(-b’,(T*-My))) Ill k; = 20i,(z,-z!;)-

k;; = 2~,(z,-z~

-l)+

, I

%Log(l

-z,+z;t),

zLog(z,-z,M) -(T-T*)-

$(Exp m

-b,(T-A;))-

.

Exp(-b,(T*-A:)) I

966

S. LECLERCQ

and C. LEXCELLENT

90

go-

80

80-

; 70 o_ 5 60 z a, 50 L z 10 30

-interior

20 a

0

0 Strain

2.0

on CuZnAl

kT, = 2q,(z,-z,M

single crystal

3.0 L.0 Strain

b

[%I

Fig. 6. Subloops

1.0

(Huo and Miiller,

5.0

Loops(Z) 6.0

7.0

8.0 9.0

[%I

1993).

AS” - 1) + y Log@, - ZTM), 4x

AS* /CT”= 2@,n?z,- ,Log(l-&). a In (24) the parameters u;, a:, ai, ui, uTn, b,, hR, h& and 6, are identified from experimental tests. The temperature r” is that for which a mechanical loading is ongoing, i.e. T* can be the temperature at the beginning of an isothermal mechanical loading or the actual temperature in case of non-isothermal mechanical loading. Let us focus on the significance of $‘, $‘, 2,” and z,M, which characterize the “memory” of the material. Thus z,” and zy are the minimum values of z, and zT just before the last time the material was submitted to an active loading (which corresponds to a forward phase transition). On the other hand, z,” and zy are the maximum values of z,, and z7.just before the last time the material was submitted to unloading (reverse phase transition). Hence the introduction of these memory parameters allow account to be taken for one not yet well known behavior of SMA : the internal loops (see Fig. 6). This uncommon aspect of SMA reveals that the material behaves as if it could remember its loading history. As the physical phenomenon responsible for internal loops is not yet well defined, the simplest way to model it is to use some “memory variables”. This way has already been followed by Sun et al. (1994) in the modeling of the pseudoelastic behavior of single crystals.

4.2.

Phase trunsformution

kinetics

Before developing the kinetic equations for the phase transition, functions which will be helpful in the following

We\-) =

[

1 0

if x30 else

9

1 if x>O o else

H*(x) = [

let us define some

3

Thermomechanics

1

6(x) =

if

[0

of shape memory

x = 0

'

else

(s)

=

961

alloys

x

if

x30

[ 0 else

’

(25) The system (26) will then define the kinetic equations and zT involved in the modeling 2, =

for the internal

variables

z,

{(~~(7t~l.;)_i~(~~l~))~(~~ - Y”,)H*(l -z)}H*(lal) - {(L;;(-?$li)-L;(

-7GJ;))H($;;-

+ (~TU(&,I;))H($Td

- YTm)H*(+)d(( ?I),

iT = ~(~La(~CIZ)-~~(7t~J,))H(~r- f(&-Glt)-&(

Y;)H*(Z)}H*(z,)

Y;)H*(l

-z)}H*(lii’l)

-7i;li))H($:-

- (P(7&l._))H($‘”

Y;)H*(z)}H*(l~l)

- YTn)H*(+)6(1

T\).

(26)

The positive coefficients &, Ai, lLT”,,uT_, ,u;(a= c,T)can be directly identified from the derivation of consistency equations I& = 0, I& = 0, I,&~~= 0. For example, one obtains the following equations for a system undergoing phase transition and reorientation, where 0: is assumed to be zero (it appears that such a case is possible for some materials, see next section) : forward

:

phase transition

2,

=(l

-z,)

~&h,o;

1,

Exp(-b:,(T-M,P))F

(27) reverse phase transition

zrr = z,

:

*(5-1’1~ufR PAS* L-T =

reorientation

(self-accommodating

Exp(-b,(T-A;))f

1 ,

-aT,z,i;

martensite

(28) into oriented

martensite)

:

968

S. LECLERCQ

fT

zxc

and C. LEXCELLENT

-f,.

(29)

The compliance of the model with the ClausiussDuhem inequality is obvious, because of the way the kinetics have been written. Indeed, from (26) one can directly deduce that 11% 3 0 if and only if 7~: 3 0, t, d 0 if and only if 7~; < 0, (c( = rr, r) and this proves Duhem inequality. 4.3.

that

the proposed

mode1 complies

with the Clausiuss

About the phase transjbrmation criterion

There exist two viewpoints about this problem, and one has to choose between the following two hypotheses. The first one lays down that the phase transition appears as soon as the associated thermodynamic force becomes zero and changes its sign, i.e. 7rz = 0 (or 71: = 0). The second one states that there exists a yield value for the associated thermodynamic force, below which no phase transition can occur. Experimental studies tend to show that the second hypothesis is true (at least for TiNi SMA). This is the reason why the parameters Y”, YT and YTc have been introduced in the model. Nevertheless the formulation of these yield values is still not well defined, even if some recent works have already been published on this subject (Lexcellent, 1994). In order to make the comprehension of the following easier, it is postulated here that parameters Y are equal to zero. This assumption is nevertheless not too severe, because many SMA follow this statement (for instance CuZnAl, CuAlNi and some TiNi). A remark about this problem is that hypothesis 1 corresponds to the occurrence of internal loops as shown in Fig. 6. One the contrary, almost no internal loop is going to appear if hypothesis 2 is followed. Thus. the present mode1 is able to describe the behavior determined by any of these hypotheses, but one must say that the question “what is the hypothesis corresponding to the real behavior of any material’?” remains totally open.

5.

PARAMETER

DETERMINATION

This section is devoted to the determination of the parameters involved in the present model. In order to simplify the presentation, one assumes that the isotropic elastic tensor, the mass density, the latent heat of phase transition and the phase transition temperatures are known a priori. In the same way, the maximum pseudoelastic strain (y) has been measured on the complete phase transition plateau of a uniaxial loading-unloading test.

Thermomechanics

The thermodynamic

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alloys

corutan ts

The parameters are Au*, As*, O,t and @F. One needs four equations for their determination. First, the pseudoelastic behavior of SMA is used. Let oAM be the yield stress of phase transition at a constant temperature in the pseudoelastic range, Tpe 2 A: (being the conventional austenite finish temperature). One can write (30) As (30) depends on T, one should note that so does (rAM. Use is next made of the austenite o self-accommodating martensite (stress free state and temperature varying). Thus forward

phase transition

reverse phase transition

phase transition

=> &(zT = 0) = 0 o rcf,(Mf) -@it = 0,

(31)

=> TC:(+ = 1) = 0 Q r& (A:) + a,,, = 0.

(32)

Here MP and Ai are the conventional martensite start and austenite start temperatures. The last equation is the result of the reorientation of self-accommodating martensite. Let gTo be the yield stress for this reorientation. Thus, at a temperature below ME, and applying a uniaxial mechanical stress, one has

7c:,
(30-33)

-0,::

= 0.

allow the values of the parameters D

=

It

p(M:--A:) 2y(M:-T,,,)

to be determined.

(33) They read

@M

’

AM As*

=

y”

(34)

P
5.2.

The kinetic constants

These constants are involved in the writing of the yield functions and thus are directly linked to the kinetic equations for z, and zr. At present time, all these constants are calculated by data fitting. They are chosen by the computer on the basis of minimization of the area between experimental and simulated results. Nevertheless, more accurate calculations like inverse methods are about to be used with the present model.

970

S. LECLERCQ

0

0.2

and C. LEXCELLENT

0.4

Fig. 7. Loading

0.6 strain (%)

-unloading

0.8

1

tensile test (T = 70 C)

The values a; and ai correspond to the shape of the hysteresis loop according to pseudoelasticity. Thus they can be obtained by writing the response of a loadingunloading tensile test in the pseudoelastic range (see for instance Fig. 7). The slope of the non-linear curves is governed by a; and 0;;. The value aT” corresponds to the shape of the hysteresis loop according to the orientation of self-accommodating martensite. One can obtain it by fitting the data of a loading-unloading tensile test in the temperature range below ME (see for instance Fig. 8). The slope of the non-linear curve is governed by aTa. Finally, the constants & and ai are calculated from the results derived by Koistinen and Marburger (1959), and one has 0:

zz

or

-

aR =

Log(O,Ol) M,O-Iv: ’ km0

A’:--A:

1)

(35)

Those constants having been determined, the remaining parameters, i.e. b,, bR, 6, and bd, are related to the shape of the hysteresis loop which occurs during nonisothermal tests. Thus they are easily obtained from the data-fitting of non-isothermal tests such as recovery stress or thermal cycling at constant stress state (see for instance Fig. 9 or 10). In this case too, the slope of the non-linear curves is governed by those four parameters.

Thermomechanics

0.2

0

of shape memory

0.8

0.6

0.4

971

alloys

1.2

1

Strain (%) Fig. 8. Loading-unloading

5.3.

tensile test (T = - 115 ‘C).

Some examples

Three sets of parameters, corresponding to three different alloys, are presented here. The first alloy, which will be referred to as CuZnAl R205, has been manufactured by Trtfimetaux Co. (see Table 1). The second one is also a Trefimetaux one, and is called CuZnAl501 (see Table 2). The third alloy is a TiNi manufactured by Furakawa Electric Co. (Table 3). One has to say that this SMA does not undergo fully austenite o martensite phase transition, but also austenite o R-phase phase transition. Let us recall that the R-phase is a Rhombohedral pre-martensitic phase characteristic to TiNi. For more details, the reader can refer to the following authors (Miyazaki and Otsuka, 1986; Funakubo, 1987; Lexcellent et al., 1994; Leclercq et al., 1994).

Table 1. Materiaiparameters for CuZnAl R205 E (MPa) 72,000 Au* (J/k) 4332,8 rj ii1045 hn 0,15

P (k&-d

8000 As* (J/kg K) 23,4 a”R 0,27 b:, 0,15

;,04 % (J/kg) 50 a: 0,36 b,

1

M,” W

4

175

179

W (J/k) 271,2 d I,54 b, 1

W

uTm 0,Ol

S. LECLERCQ

972

and C. LEXCELLENT

250

250

200 Temperature (K) Fig. 9. Recovery

stress (8” = 2%).

2.2

(49 MPa)

21.0 I.& 1.41.2I0.8 0.6 0.4 -

TF,

0.2 0

( 10

20

30

40

Fig. 10. Recovery

50

60

70

80

90

strain (a = 49 MPa).

Table 2. Material parameters for CuZnAl501 E (MPa) 68,400 Au* (J/W 6230 ; &5 b:, 0,017

P

(k&3

8000

;,03*

M: WI

A: WI

313

315

@,t (J/W 36,77

0

a;; 0,34

4

aX

aTn

0,46

0,46

0,Ol

b:, 0,017

b, 0,84

ba

As* (J/kg K) 19,84

Pi’ (J/k)

1

913

Thermomechanics of shape memory alloys

Table 3. Material parameters for TiNi E (MPa) 68,000 Au* (J/W 2143,9 a”R 0,017

P (k/m3)

6500 As* (J/kg K) 6,66 a’R 0,767

6.

:,007 @,t (J/k) - 20 bm 0,067

M,O WI

-4: WI

325

319

b,

1

RESULTS OF THE SIMULATION

In this section some comparisons between experiments and theory will be presented in the case of tensile tests. For each alloy presented in the previous section, the influence of non-isothermal mechanical loadings will be observed. 6.1.

CuZnAl R205

Figures 7-9 show the behavior of this alloy for pseudoelasticity, orientation of the self-accommodating martensite, and appearance of the recovery stress, respectively. The calculations are compared with experimental tests performed by Vacher (1991). Let us point out that these tests have been used to determine the values of the material parameters. Thus the results here do not constitute any behavior prediction. In the case of Fig. 9, one has to recall that s0 represents the strain applied in the martensitic state, and maintained during heating. 6.2.

CuZnAl501

(Bourbon, 1994)

Figures l&15 show the behavior of CuZnAl 501 for pseudoelasticity, recovery stress and non-isothermal cycling at constant stress state (recovery strain). Let us point out that Figs 10-12 have been used for the determination of the material parameters, while Figs 13-l 5 are predictions of the behavior. 6.3.

TiNi

Figures 1618 present the behavior of TiNi for austenite o R-phase phase transition. Figure 16 shows the pseudoelastic behavior of this alloy. Remark that no hysteresis is obvious here. This is the reason why the term (I+,is a negative constant in Table 3 (see Lexcellent et al., 1994). Moreover, it was not possible to determine the non-negative constant Qf. Anyway, the instability condition of the equilibrium is always verified for this alloy, because 4Qit - (DF < 0 whatever be the value of QT. Figures 17 and 18 show the appearance of recovery stress for two imposed strains.

7. 7.1.

DISCUSSIONS

Definition Of bit and @T

During the presentation of the free energy, the interaction energy Qlt was supposed to be constant. This means that the yield stress of phase transition (in the pseudoelastic

S. LECLERCQ

974

and C. LEXCELLENT

250

150 r

100 -

50 -

300

360

340

320

Temperature (K) Fig.

140

1I. Recovery stress (c:,)= 1.15%)

,

120 -

100 -

0,5

1.5

1 Strain (%)

Fig. 12. LoadingPunloading

tensile test (r = 56 C).

2

Thermomechanics

of shape memory

2.4 2.2 2

975

alloys

(65 MPa)

1.8 1.6 i

10

20

30

40

Fig. 13. Recovzy

10

20

30

40

Fig. 14. Recovery

50

60

70

80

90

strain ((3 = 65 MPa)

50

60

70

60

90

strain (0 = 80 MPa)

3.5 3 2.5 2

1.5 1 0.5 0 10

20

30

40

Fig. 15. Recowry

50

60

70

80

strain (CJ = 100 MPa).

90

976

S. LECLERCQ

and C. LEXCELLENT

3

2 1

300-

B m

200 -

0

0.2

0.4

0.6

0.8

1

strain (%) Fig. 16. Loading

unloading

tensile test (T = 65 C).

80-

$

60-

40-

20-

310

320

330

Tempcrahwe (K) Fig. 17. Recovery

stress (i:,, = O,48u/,).

340

3

Thermomechanics

of shape memory

977

alloys

50-

0 300

I=,.

a,,

,

310

,

,

,

,

,

,

Temperature Fig.

( , ,

,

330

320

,

, 340

,

, ( 3 0

(K)

18.Recovery stress (8” = 0,57%).

range) increases linearly with the temperature [see (34a)]. The experimental observations are consistent with this result. Nevertheless one can see in the literature (Raniecki et al., 1992) some models iassuming that QD,,explicitly depends on temperature. Moreover, another assumption was made, stating that the interaction energy in the martensitic state was constant. This means that the yield stress aT” of reorientation is constant whatever be the temperature. This assumption is usually made, see Tanaka (1986) or Brinson (1993). But some experimental results from Dye (1990) or Vacher (1991) show that the stress increases slowly when temperature decreases (see Fig. 19, with the dashed line). Nevertheless, the small amplitude of this decrease justifies the choice that has been made in this paper. 7.2.

Thefrer

energy A@

Some recent works from Raniecki and Lexcellent (1994) show that the free energy of interaction between the three phases can be derived from microscopic calculations based on the lattice structure and the energy of each phase. The term A@ has been referred to as unaccommodation energy by these authors, and it can be written as follows A@ = z(1 -z)[K&$-KN], or simpler,

(36)

S. LECLERCQ

97x

and C. LEXCELLENT

UA

-1_/_1 OAM

--------_________ csT=

thermal phase transition domain I

,T

Fig. 19. Yield stress for forward

phase transition

vs temperature.

Aa = Z( I -z)[~fMi$

(37)

where M and N_ are, respectively. fourth-order and second-order tensors depending on the micromechanics of the material. The two equations (36) and (37) show explicitly that the unaccommodation energy can be a function of the direction of the stress tensor (x). This remark may be of importance in case of introduction in the model of a kinematic internal variable associated to K. _~

8.

VALIDITY

OF THE MODEL

AND CONCLUDING

REMARKS

Graesser and Cozzarelli (1994) and Boyd and Lagoudas (1994) have already proved that the modeling of SMA on the total scale of the phase transition domain was possible, in a phenomenological framework, but sometimes with a lack of physical meaning in the choice of the internal variables (Graesser and Cozzarelli, 1994), or much complication in the building of the model without any comparison with experiments (Boyd and Lagoudas, 1994). Other approaches (Ball and James, 1992) show great potential for modeling the thermomechanical behavior of SMA, considering twinned and detwinned martensites, but these approaches need time to become completely useful for engineers. It was the wish of the present authors to build a model allowing a good description of a large number of the very uncommon behaviors of SMA with a relatively simple formalism. The results presented in Section 6 of this paper show that this goal has been reached. indeed the isothermal behavior as well as the non-isothermal one are relatively well predicted. Moreover, the present model is up to now one of the first approaches that predicts the behavior of SMA on the total scale of the phase transition domain, according to experimental data on several materials. Nevertheless it is clear that this description can be improved on some domains. For

979

Thermomechanics of shape memory alloys

example the non-proportional tests in two or three dimensions are not satisfactory, and that is the reason why they have not been presented here. This requires the addition of some terms in the writing of the rate of transition strain Yr. This work is in progress at the present time and will soon give interesting results. Anyway, the reader can refer to the work of Rogueda (1993) for more information about tractiontorsion tests in the pseudoelastic range. In this work, it is shown that the RL model (Raniecki et al., 1992) is able to simulate at least two-dimensional tests for proportional loadings. The present work being a generalization of the RL model, obviously the simulation of two- or three-dimensional proportional tests can be performed by the present model. On the other hand, the two way shape memory effect has not been introduced here, but it seems that the introduction of a “memory” state variable like a kinematic backstress could solve this problem. Indeed, the goal to reach is to be able to produce oriented martensite from a change of temperature only. This work is also in progress. One can moreover say that the expressions of the kinetics for z, and zT could be changed to account more closely for the smoothness of experimental results. Anyway, the present model can be considered as a basic work allowing the description on a large scale of the SMA behavior. The price to pay for this is the increase of the number of parameters to be determined. But it is the necessary way to obtain a description suitable for the design of structures made of SMA.

ACKNOWLEDGEMENTS The authors wish to thank Professor B. Raniecki (I.P.P.T. Warsaw, Poland) for the useful scientific discussions that made this work possible. The micrographs shown on Figs 2(a) and (b) have been kindly provided Neuchatel, Switzerland).

by Professor

M. A. Morris

(Institut

de metallurgie

structurale,

REFERENCES Abeyaratne, R. and Knowles, J. K. (1993) A continuum model of a thermoelastic solid capable of undergoing phase transitions. J. Mech. Phys. Solids 41, 541-571. Ball. J. M. and James, R. D. (1992) Theory for the microstructure of martensite and applications. Proc. Int. Conf. on Martensiti.? Transformations, Monterey (ed. C. M. Wayman and J. Perkings), pp. 65-76. Bourbon, G. (1994) Contribution ii I’ttude isotherme et anisotherme du comportement cyclique des alliages B m&moire de forme. PhD Thesis #410, Universitk de Franche ComtC, France. Boyd, J. G. and Lagoudas, D. C. (1994) A constitutive model for simultaneous transformation and reorientation in shape memory materials. AMD-Vol. 189/PVP-Vol. 292, Mechanics of Phase Transformation and Shape Memory Alloys (ed. L. C. Brinson and B. Moran), pp. 159177. ASME. Brinson, L. C. (1993) One dimensional constitutive behavior of shape memory alloys: thermomechanical derivation with non-constant functions and redefined martensite internal variable. J. Intelligent Mater. Syst. Structures 4, 229-242. Dye, T. E. (1990) An experimental investigation of the behavior of Nitinol. M.S. thesis, Virginia Tech.

980

S. LECLERCQ

and C. LEXCELLENT

Fremond, M. (1987) Materiaux a memoire de forme. C. R. Acad. Sci. Paris, t. 304, SCrie II, 2399245. Funakubo, H. (ed.) (1987) Shape Memory Alloys. Gordon and Breach Science Publishers. Graesser, E. J. and Cozzarelli, F. A. (1994) A proposed three-dimensional constitutive model for shape memory alloys. J. Intelligent Mater. Syst. Structures 5, 78-89. Huo, Y. and Mtiller, I. (1993) Nonequilibrium thermodynamics of pseudoelasticity. Continuum Mech. Thermodyn. 5, 163-204. Koistinen, D. P. and Marburger, R. E. (1959) A general equation prescribing the extent of the austenite-martensite transformation in pure iron-carbon alloys and plain carbon steels. Acta Met. 7, 59-60. Leclercq, S. (1995) De la modelisation thermomecanique et de l’utilisation des alliages a memoire de forme. PhD thesis #450, Universite de Franche Comte, France. Leclercq, S., Lexcellent, C., Tobushi, H. and Lin, P. H. (1994) Thermodynamical modelling of recovery stress associated with R-phase transformation in TiNi shape memory alloys. Muter. Trans. JIM 35, 325-33 I. Lexcellent, C. (1994) Thermodynamical modelling of the hysteresis loops in shape memory alloys. Euromech 321, 23-27 May 1994, Microstructures and phase transitions in solids. Lexcellent, C., Tobushi H., Ziolkowski A. and Tanaka, K. (1994) thermodynamical model of reversible R-phase transformation in TiNi shape memory alloys. ht. J. Pres. Ves. Piping 58, 5 I-57. Miyazaki, S. and Otsuka K. (1986) Deformation and transition behaviour associated with the R-Phase in TiNi alloys. Met. Trans. A 17. Patoor. E., Eberhardt, A. and Berveiller, M. (1994) Micromechanical modelling of superelasticity in shape memory alloys. Pitman Research Notes in Mathematics Series 296, 38-54. Raniecki, B. and Lexcellent, C. (1994) R,-models of pseudoelasticity and their specifications for some shape memory solids. Eur. J. Mech. A/Solids 13,21-50. Raniecki, B., Lexcellent, C. and Tanaka, K. (1992) Thermodynamic models of pseudoelastic behaviour of shape memory alloys. Arch. Mech. 44, 261-284. Rice. J. R. (1971) Inelastic constitutive relations for solids : an internal variable theory and its application to metal plasticity. J. Mech. Phys. Solids 19,433. Rogueda, C. (1993) Modelisation thermodynamique du comportement pseudoelastique des alliages a memoire de forme. PhD Thesis #336, Universite de Franche ComtC, France. Sun, Q. P. and Hwang, K. C. (1993a). Micromechanics modelling for the constitutive behavior of polycristalline shape memory alloy: I Derivation of general relations. J. Mech. Phys. Solids 41, l-18. Sun, Q. P. and Hwang K. C. (1993b) Micromechanics modelling for the constitutive behavior of polycristalline shape memory alloy : II Study of the individual phenomena. J. Mech. Phys. Solids 41, 19-33. Sun, Q, P., Leclercq, S. and Lexcellent C. (1994) A micromechanics constitutive model of shape memory alloy singlecrystal with internal hysteresis loops, submitted to Znt. J. Plasticity. Tanaka, K. (1986) A thermomechanical sketch of shape memory effect : one dimensional tensile behavior. Res. Mech. 18,251-263. Vacher, P. (1991) Etude du comportement pseudoelastique d’alliages a memoire de forme CuZnAl polycristallins. PhD Thesis #215, Universite de Franche ComtC, France.