State equation for shape-memory alloys : Application to Cu

We deal with the hysteretic behavior of partial cycles in the two-phase region associated with the martensitic transformation of shape-memory alloys. We consider the problem from a thermodynamic point of view and adopt a local equilibrium formalism, based on the idea of thermoelastic balance, from which a formal writing follows a state equation for the material in terms of its temperature T, external applied stress (Y, and transformed volume fraction x. To describe the striking memory properties exhibited by partial transformation cycles, state variables (x,(J',T) corresponding to the current state of the system have to be supplemented with variables (x,o-,11 corresponding to points where the transformation control parameter ( (J and/or n had reached a maximum or a minimum in the previous thermodynamic history of the system. We restrict our study to simple partial cycles resulting from a single maximum or minimum of the control parameter. Several common features displayed by such partial cycles and repeatedly observed in experiments lead to a set of analytic restrictions, listed explicitly in the paper, to be verified by the dissipative term of the state equation, responsible for hysteresis. Finally, using calorimetric data of thermally induced partial cycles through the martensitic transformation in a Cu-Zn-AI alloy, we have fitted a given functional form of the dissipative term consistent with the analytic restrictions mentioned above,


I. INTRODUCTION
Shape-memory alloys are able to recover important deformations up to 10%.If the deformation process takes place at high temperatures, the alloys recover their original shape by simply eliminating the applied external stress (pseudoelastic behavior).Deformations produced at lower temperatures are recovered on heating the alloys (shapememory effect).I Among the materials exhibiting this unusual thermomechanical behavior, the more studied are Ni-Ti and some copper-based .B-brass alloys such as Cu-Zn, Cu-Zn-Al, and Cu-AI-Ni.
The pseudoelastic and shape-memory effects are intimately related to the thermoelastic martensitic transformation taking place in aU the alloy systems mentioned, The martensitic transformation is a solid-state transition with the following essential characteristics 2 ,:l; (i) it is a first-order transition, (ii) it is displacive, i.e" it takes place without atomic diffusion, (iii) it is accompanied by a distortion of the crystalline lattice with a dominant deviatoric component resulting in macroscopic shape change, and (iv) the transformation kinetics and the morphology of the resulting product, caned martensite, are dominated by elastic strain energy.
Martensitic transformations can be induced either by changing the specimen temperature T or by applying an external stress (T.A martensitic transformation is acknowledged to be thermoelastic when at each temperature andlor applied stress inside the transformation range a thermoelastic equilibrium is achieved.The equilibrium condition is de-fined by a local balance at the transforming interfaces between chemical forces associated with the structural change, and nonchemical forces such as elastic strain and dissipative energies. 1Depending on the crystal symmetry of the high-temperature phase, different crystaUographically equivalent martensitic variants may appear.In the absence of an external applied stress they form self-accommodating groups that minimize the total shape change.To the contrary, application of an external uniaxial stress results in the formation of the martensitic variant (or variants} best suited to accommodate the imposed strain.i The transformation exhibits hysteresis: On reverting the sense of variation of the external control parameter (temperatnre or stress), the system follows a reverse path different than the forward one. If the transformation is thermally induced, the conjugate variable of temperature is the entropy difference between the two phases, while if the control parameter is an externally applied stress, the conjugate variable is the transformation strain. I can be shown that in both cases the conjugate variable is proportional to x, the volume fraction of martensite in the specimen. 7For this reason, in what follows, the transformation path will be described by means of the variable x.
In a partially or fully transformed multi variant martensitic specimen, application of an external uniaxial stress (tension or compression) produces reorientation of martensitic variants and causes hysteretic behavior in the stressstrain space.In this study we are not dealing with hysteresis associated with martensitic reorientation, but with hysteresis produced by the transformation itself.For example, we explicitly exclude situations such as (i) application of an external stress once part of the material has been thermally transformed into martensite, and (ii) changes from an extensive to a compressive applied stress at constant temperature.
The transformation path depends not only on the instantaneous value of the control parameter but also on previous extreme values, corresponding to points where the sense of variation of the control parameter has been reverted.Associated with its hysteretic behavior, therefore, the system shows memory features.
Notwithstanding the fact of showing hysteretic and memory properties, it is a good approximation, at the usual experimental time scales, to assume that no relaxational processes occur during the transformation.Thus, the whole set of variables describing the transformation path may be considered time independent In particular, the transfonnation path is independent of the rate of change of the control parameter, at least within certain limits.Possible diffusional processes, foreign to the martensitic transformation, might take place simultaneously.They can actually be avoided by previous well-defined heat treatments. 8To conclude, if no diffusional processes take place, the system shows a stationary behavior in any state within the two-phase regionsometimes caned a static hysteretic behavior.Some of the above features of the transformation hysteresis in both temperatureand stress-induced transformations, including partial cycling, have been studied experimentally by several authors. 4 ,5,9-11It is worth noting that similar features are observed in the hysteretic behavior offerroelectric and ferromagnetic 12 materials.The phenomenological characteristics 13,14 of the hysteretic behavior displayed by shape-memory materials in the martensitic transformation region can be summarized by the fonowing items, schematically drawn in to complete transformation cycles, with x varying from 0 (parent phase P) to 1 (martensite it!)or from 1 to 0, are located on the boundary of the region.
(ii) Two branches can be defined for each cycle, the first one ( + ) corresponding to paths of increasing x and the second one ( -) corresponding to paths of decreasing x.
(iii) All paths tend tangentially to the boundaries of the two-phase region at x = 0 or at x = 1.Moreover, an internal loops performed between two fixed extreme values of the control parameter are congruent.
(iv) Given an original path in the two-phase region, it can be reproduced if none of the states corresponding to extreme values of the control parameter is modified.The system, therefore, keeps memory of the extreme values reached by the control parameter.Each time that the transformation path passes through a previous local maximum (minimum) of the control parameter, the memory of all the lower (higher) extreme points is erased.Hence, all the memory is erased on reaching the extreme values x = 0 or x = 1.
(v) Irreversible effects giving rise to hysteresis occur when a change in the control parameter results in a change of x.If a variation ofthe control parameter does not result in a modification of the transformed volume fraction, on reverting the control parameter the system fonows a path without hysteresis until the starting initial state.
In this work, a general expression of the state equation for shape-memory materials is obtained starting from a thermodynamic fundamental equation describing thermoelastic equilibrium. 7The hysteretic features appear through a dissipative term, which depends on the value of the state variables at the extreme points of the control parameter in the transformation path.The dependence determines the memory properties ofthe system.The formalism is applied to the Cu-Zn-Al alloy system, using experimental data partially obtained from the literature.

II. THERMOELASTIC BALANCE AND STATE EQUATION
The fundamental equation describing the thermodynamics of a thermoelastic transformation reads 7 : where dH and dS are differential changes in chemical enthalpy and entropy of the system, -dE el is the reversible internal work stored in the system as elastic strain energy and interfacial energy, (l:ij VOO"ij d€(i) is the llonhydrostatic part of the work performed by external forces to produce the macroscopic deformation, (p dV) is the hydrostatic part of this same work, liS; is the entropy production associated with dissipation of irreversible heat, and ltw, is an internal work irreversibly dissipated in forms other than heat and not giving rise to entropy production.10,15 A detailed description of the contributions included in the term -dE el is given in Ref. 7, In particular, the elastic strain energy originates from a total deformation consisting oflocal stress-free strain (including transformation strain and thermal expansion) and local elastic strain.Thermal expansion could be relevant since the transformation extends in a temperature range.
The free energy G * is introduced through the following Legendre transformation: Equation ( 1) can be rewritten in tenns of this potential and reads:  where G ~ and G it stand for the free energies of mutually noninteracting parent and martensitic phases, it follows that: Hence, from Eq. ( 5) we finally obtain: ) is the driving force for the transformation and dE diss = T dS, + aWi;;;;O no matter the sign of dx.It should be noted that dx > 0 in the forward (P -M) transformation, and dx < 0 in the reverse (1!1" -P) transformation.Equation ( 9) expresses the condition of thermoelastic balance in a continuous quasistatic formalism: Any differential variation of the control parameters Tor (T will result in a differential variation dx of the transformed volume fraction. However, highly sensitive experimental techniques, including calorimetry16 and resistivity, 17 make evident that x does not always evolve continuously.Actually, the system goes from x to x + ox through a relaxationaI process in which energy is dissipated.Part is released in the form of elastic waves, detected as acoustic emission, and part gives rise to entropy production.In some instances the relaxational effects become important, and experimentally one observes marked discontinuities in the paths giving x as a function of the control parameter.18 This is the case ofthe.BCbcc) -2H transformation in Cu-Zn-A1 19 and Cu-AI-Ni. 20Nevertheless, the characteristic times associated with the relaxational effects are always much shorter than the characteristic times associated with the variation of the external control parameters 1'-1 or &-1).
Let us now obtain explicit expressions for the driving force in the usual experimental situations.
First, we consider temperature-induced transformations at zero external stress.It is assumed that AN and AS are not temperature dependent; this is equivalent to assuming a negligible difference between the heat capacities of the two phases, as is the case in most of the aHoy systems considered.21 From Eq. ( 2), at a given temperature Twe have ( 10)   where G is the Gibbs free energy.To obtain this equation we recall that AH = T;)(O)AS, where To(O) is the temperature ofthermodynamic equilibrium between the two ideal phases at f7 = O.Dividing Eq. ( 9) by AH, it takes the form: where Eel =.Ec// fiB and Ediss =.Edissl AH.
A second case to consider is the stress-induced transformation at constant temperature.We restrict ourselves to a uniaxial stress, denoted by cr, and assume again that fiC p = o.The driving force at Tand a is now given by If we consider the thermodynamic eqUilibrium between the two phases in the absence of mutual interactions, both subjected to the applied external stress, Eq. ( 1) leads to fiB = Tb.S + Vo (JofiE,   (13)   equivalent to the result by W oHants, de Bonte, and Roos.22Finally, let us consider a general situation in which temperature and uniaxial stress are changed simultaneously.Equation ( 13) is a general expression of the enthalpy difference b.H at any temperature T, taking into account that ao = ao( T).Using Eqs. ( 12) and ( 13) it is easy to see that: On the other hand, combining Eq. ( 13) and /lB = To(O)LlS one finds: This expression is a first integral of the Clausiu.s-Clapeyronequation.A general expression of the thermoelastic balance is finally obtained by inserting ( 16) into (I 5), putting the result into (9) and dividing by !:lH.The result reads: ( T) VoAe aEe! + (fE di " _ 0 (17', 1 ------0"+------.

!:JI ax dx
Straightforward calculations show that the above equation reduces to Eq. ( 11) when (7 = ° and to Eq, ( 14) when Tis constant.Equation ( 17) is a formal expression of the thermod~~ namic state equation of the system in the (x,O", n space, It 1S important to note that ttE dis.;>0, and as a consequence, the equation displays two kinds of branches: branch ( + ), asso~ dated with dx> 0 (forward transformation), and branch ( -), associated with dx < 0 (reverse transformation), An explicit expression of the state equation for a given system will be available only when explicit expressions for Eel and

III. CHARACTERISTIC FEATURES OF THE DISSIPATIVE FUNCTiON
From now on we restrict our selves to hysteresis loops having x = 0 or x = 1 as one oftheir two extreme points.The associated reversal curves will be caHed first-order reversal curves.In such a situation the dissipative function associated with any branch will depend only on the parameters of a single extreme point.Hence, the situation stands for a firstorder approach to the general problem; the latter might be treated along the same lines but would give rise to a considerably more involved formalism.Let us suppose that (a€e/lax) and (ttediss/dx) depend only on x.This is reasonable provided that the system transforms in a narrow range of temperature and stress.
Let w+ be the function (ffedisJdx) in the ( + ) branch, and w _ the same function in the ( -) branch.It is not diffi~ cult to see that for (7 = 0: (20)   where T -'and T _ are temperatures corresponding to the same tra~sformed fraction x in the branches (+) and ( -), respectively.On the other hand, if the transformation is stress induced, we obtain: In what follows, only temperature-induced transformations will be studied.From the thermodynamic equivalence between (J" and T it is clear that the corresponding equations for stress-induced transformations can be obtained from the equations below by simply replacing Tby -0". From the general properties of the hysteresis loops described in the previous section, the characteristic features to be displayed by the functions (j) + and (j) _ can be deduced.Comparison between a complete and partial cycle leads to Here x is the extreme point of the partial cycle considered, and M f and Af are the final temperatures of forward and reverse transformation, respectively.The equality applies for x = 1 in the first equation and for x = 0 in the second one.The equations follow considering that, for a given x, the external loop is wider than or equal to the partial loop.For this reason, w + must be a monotonical decreasing function of x, and w_ must be a monotonical increasing function ofx.
"On the other hand, the state equation must satisfy: . ( 23 = ('aw+(X;x=O,Af ») __ ' Taking for the dissipative functions the ansatz: the condition ( 22) forcesf+ andf __ to be positively defined, and to satisfy; In addition, Eqs. ( 24) have to be satisfied as well, leading to (aJ+(X,X))

IV. FITTING TO EXPERIMENTAL OBSERVATIONS
To fit the dissipative functions to experimental observations, in addition to the conditions deduced in Sec.III, we consider that!+-andf_ can be written in the form: Then, for conditions ( 26) and ( 27) to apply, g+ and g_ must satisfy: These conditions are already satisfied by choosing for g +and g _ a polynomial function of second or higher degree without an independent term, of the arguments (1x) and x, respectively, Considering also that w + must be a monotonical decreasing function of :x and w _ a monotonical increasing function ofi, it follows that h +-must increase monotonically and h_ must decrease monotonically with x, The simplest choice is where n is an exponent to be determined.
First, a (x) is obtai.neddirectly from a fit to the width of the hysteresis loop in a complete transformation cycle . The equation to be used reads: Second, once a(x) is known, g +-(x) and g _ (x) are obtained from a fit to the width of partial hysteresis loops.For a and for a cycle x = l->xi:0-x = 1 we have It is worth noting that all partial cycles, corresponding to different values of X, are simultaneously fitted by single functions i-+ (x) and g_ (x) for the (-) and (+) branches, respectively.
As an application, the functions a(x), g+(x), and g_ (x), given by Eqs. ( 31), (32), and (33), have been fitted to experimental results of the thermally induced transformation of a Cu; 14.1 Zn; 17.0 Al (at.%) alloy.The values for the ( -) branch have been taken from Ref. 10, while the values for the ( + ) branch, obtained in the same experimental conditions, had not been published previously.23 a(x),g+-(x), andg Jx) have been chosen to bepolynomial functions, the latter ofx and (1x), respectively, and without independent term.The value of n together with the order of the polynomials have been optimized to give the best fit using the lowest possible values.The results are summarized in Table I and shown in Fig. 2.

Vo COMMENTS AND CONCLUSIONS
We have formulated, in very general terms, the state equation of a shape-memory material, adopting a continuous quasistatic approach based on the thermoelastic behav-2346 J, Appl.Phys" Vol.66, No.6, 15 September i 989 ior of the material.The state equation is composed of three terms: (i) a term explicitly dependent on temperature and/ or external stress, representing the driving force for the transformation, (ii) a term giving the reversible variation in stored elastic energy as a function of the transformed fraction, and (iii) a dissipative term including all the irreversible effects manifested at a macroscopic level by the hysteretic behavior.In a quasistatic formalism this last term is assumed to include in an effective (averaged) way all the microscopic irreversibilities, such as the irreversibiIities associated with nucleation and elastic energy relaxation, and takes them as a global steady dissipation, This is a reliable approach provided the characteristic times of the relaxational processes involved at a microscopic level are much shorter than the times of appreciable variation of the control parameter, as the experimental observations suggest, so that the quasistatic picture is preserved, Plastic flow processes that would give rise to long-time relaxational processes are explicitly excluded from our treatment since they are acknowledged to be absent in this kind of transformation.
The experimental behavior displayed by partial hysteretic loops inside the two-phase region enables phenomenological modeling of the dissipative term as a function of the transformed volume fraction.This is interesting for two reasons: On the one hand, there are no microscopic theories presently able to give such a dependency.On the other hand, it provides a way to predict the macroscopic behavior of shape-memory materials in the two-phase region, which is of considerable technological interest.
The memory effects displayed by partial hysteretic loops have to be described by thermodynamic variables different than the usual ones: the state ofthe system is described not only by the instantaneous value of the control parameter (T or -(}') and the transformed volume fraction, but depends also on the previous history of the system through the values of both the control parameter and the transformed volume fraction at the previous extreme values of the control parameteL As a first approximation to the general problem we have been only considering first-order transformation paths, which depend exclusively on the transformed fraction at a single extreme value of the control parameter.This leads to considerable simplification in the formalism and enables writing a set of analytic conditions to be satisfied by the energy dissipation term in the state equation.
We have suggested a very simple expression of the dissipative term consistent with the analytic conditions mentioned above, and this has been fitted to experimental results obtained by calorimetry of a Cu~Zn-AI alloy.A single polynomial function has been fitted to all partial cycles simultaneously; the order of the polynomial has been selected to be the minimum resulting in a reasonable fit.The result is reo markably good for the forward transformation cycles (partial cycles ( + )} and not so accurate for the reverse transformation cycles [partial cycles ( -)], the misfit being more important at low transformed-volume fractions.Besides the experimental uncertainty, the reason for the different behavior between the two kinds of partial cycles is not completely clear.Speculating, one could associate the different behavior to di.ssipative effects not included in the model and associated with the sudden disappearance of the martensitic plates at low transformed fractions, as observed in the optical microscope.
FIG.!.Schematic behavior of the transformation paths projected on a COIlstant Tor <T plalle.The continuous line represents a complete cycle which follows the boundary of the two-phase region.The broken lines represent internal paths within the two-phase region.The black points correspond to extreme values of the contro!parameter.
each T, p, and a the change in G :I< compensates for the changes in reversible elastic and dissipative energies.When the transformed fraction x changes to a value x + dx we have dG'" = (8G*) dx.ax T,p,fJ (6) Taking into account that: G* = (1-x)G~ +xG:tt, (7)

(
70 is the stress required for the transformation to take place in thermodynamic equilibrium at a temperature T. Inserting (13) in (12), considering Eq. (9), and dividing the final result by fiH we are led to ( a) 8Eel tJ'€diss 1 --fie o + --+ --= 0, o = (VoaotJ.E)/ fiH.In the equations above it is important to note that fiH, AS, and AE have been defined as AH = H M -H p < 0, AS = SM -Sp < 0, and AE = EM -lOp> O. Therefore, a comparison of Eqs.(11) and (14) shows that T and a play equivalent roles with opposite sign in the thermoelastic balance.
knowledges financial support from the K.u,L.(Belgium) and the C.S.I.e. (Spain), in the framework of cooperation between the two institutions.L.D. acknowledges financial support from the Belgian National Science Foundation (FKFO-Contract No. 2.0086.87).