On the Thermomechanics of Shape Memory Wires K. R. Rajagopal and A. R. Srinivasa Department of Mechanical Engineering Texas A&M University

Transcription

1 On the Thermomechanics of Shape Memory Wires K. R. Rajagopal and A. R. Srinivasa Department of Mechanical Engineering Texas A&M University Abstract The thermomechanical behavior of a shape memory wire is modeled based on a theory that takes cognizance that the body can possess multiple natural configurations[1]. The constitutive equations are developed by first constructing the form of the Helmholtz potential based on different modes of energy storage, and dissipation mechanisms. The internal energy includes contributions from the strain energy, the latent energy, the interfacial energy and thermal energy. The entropy of the system includes the entropy jump" associated with the phase transition. The role of the rate of mechanical dissipation as a mechanism for entropy generation and its importance in describing the hysteretic behavior is brought out by considering the difference between hysteretic and non-hysteretic (dissipation-less) behavior. Finally, simple linear or quadratic forms are assumed for the various constitutive functions and the full shape memory response is modeled. A procedure for the determination of the constants is also indicated and the constants for two systems (CuZnAl and NiTi) are calculated from published experimental data (see [2, 3]). The predictions of the theory show remarkable agreement with the experimental data. However, some of the results predicted by the theory are different from the experimental results of Huo and Muller[2] We discuss some of the issues regarding this discrepancy and show that there appears to be some internal inconsitency between the experimental data reported in Figure 6 and Figure 9 of Huo and Muller[2](provided they represent the same sample). Key words: Natural configurations, shapememory, dissipation, hysteresis, helmholtz potential, entropy, phase transition, Austenite, Martensite, interfacial energy, evolution equation, latent energy, multiconfigurational.

2 1 Introduction The ability of certain intermetallic alloys, notably NiTi to remember" certain predefined shapes has been the focus of intense study in recent years. Essentially, the alloys can be permanently deformed to a particular shape at some temperature, but when heated they revert to their original shape (the so called shape memory effect"). The reason for this behavior can be traced to the fact that the material has the ability to change its lattice structure from a high symmetry phase (called austenite")to a lower symmetry phase (called martensite"). The behavior has been studied from several viewpoints. From a thermodynamical perspective, it can be regarded as a solid to solid phase transition, with accompanying changes in entropy, shape etc. From this viewpoint the main interest is in the identification of the driving force" and other principal thermodynamical parameters that influence the phase transition (see e.g.,[4]). Martensitic transformations, have also been studied from a crystallographic viewpoint ([5], [6]) with a view towards elucidating the shape and orientation of the martensitic plates, the relative orientations of the crystal axes, and the distortions that result from the transformations. This analysis is based solely on the shape and the structure of the lattices and is not concerned with the process of phase transition. The results enumerate some of the possible juxtapositions of the individual phases for a given volume fraction of the two phases. Finally from a continuum viewpoint, one can study the macroscopic response of the material without regard to the precise details of the microstructural changes that take place (see for example [7]and [8] and the references cited therein). A recent paper that combines the thermodynamical and continuum perspectives and provides both an experimental and theoretical treatment of the problem is that by Huo and Muller[2]. The present paper is concerned with the macroscopic response of rods of shape memory material under various conditions of loading. This is not to say that the microstructural processes are ignored, but the theory deals with these effects in an averaged or smeared" out sense, retaining only the salient effects. This is achieved by the use of a continuum theory based on the concept of multiple natural configurations" developed in [1]. In this theory, each of the possible lattice structures that the material is capable of exhibiting is associated with a natural configuration and the response of the material depends upon deformations from all these configurations (see equation (13) below). Moreover, in this approach, the actual process of phase transition is 1

3 modeled as a change in the natural configurations. A kinematical constitutive equation" ( see equation (3)) is used to link the natural configurations associated with mixtures of the austenitic phase and the martensitic phase (at the microscopic level). In the current theory this kinematical constitutive equation takes the place of the crystallographic analysis mentioned in the previous paragraph by simply averaging" out the results over a representative volume element. One of the fundamental features of the shape memory effect, apart from the importance of the latent energy and latent heat 1 and the shape change, is the fact the force-deformation response is hysteretic and dissipative. We utilize the tools of continuum thermodynamics to model the process of phase transition, with certain modifications. In the present approach, an energy flow diagram" is first developed for the process that shows the flow of the incoming energy into the various storage and dissipation mechanisms that are present. This is then used for the development of the constitutive equations in such a way that the balance of energy is automatically satisfied. The importance of modeling the dissipative behavior cannot be overemphasized. In the current procedure, an explicit constitutive equation is provided for the rate at which the power supplied is converted into heat by the dissipative mechanisms. Finally, in a departure from standard practice in continuum thermodynamics, the non-negativity of the dissipation is used to restrict processes and not constitutive equations. Thus for example, under certain conditions, the material behaves in a non-dissipative manner simply because, under these conditions, hysteretic processes would violate the condition that the rate of dissipation be non-negative. The displacement and temperature fields are calculated from the momentum balance and the entropy equation, respectively. The principal advantages of the current approach are the following: 1. The constitutive equations involve only macrosopically measurable variables, all but one of which (the specific heat) can be obtained from the force displacement curves at different temperatures. 2. The ability to model the complex responses with constitutive assumptions that are affine in their respective variables with easily measurable 1 The term latent energy" refers to the difference between the specific internal energy between the two phases while the term latent heat" refers to the difference between the specific entropies of the two phases multiplied by the absolute temperature at which the transformation takes place. 2

4 constants. There are a total of six constants that need to be measured in order to characterize the full thermomechanical response. We outline a procedure for measuring them and provide measured values for two alloy systems, CuZnAl and NiTi, that were obtained from the experiments by Huo and Muller[2] and Miyazaki et al[3]. Five of the constants were obtained from the force displacement curves at constant temperature while the sixth(the specific heat) requires a separate calorimetric measurement. All the predicted results and graphs in the paper are based on the constants for the CuZnAl system. 3. The theory is robust in the sense that a wide range of phenomena can be modeled by changing the form of certain functions so that the characteristics of specific alloy systems can be incorporated. For illustrative purposes, we have chosen to use the simplest possible assumptions at the cost of complete generality. However, if the need arises, more general forms could have been chosen and we indicate these at the appropriate places in the paper. 4. The governing differential equations are amenable to standard numerical methods being somewhat similar to the system for thermoplasticity so that techniques that are used there could be adapted for use here. 5. We are able to predict the temperatures at which the the martensitic phase and the austenitic phase begin to form spontaneously (under no loads) by using the force-deformation curves at a couple of different temperatures to measure the latent energy and the latent heat (see section 5). The results are in agreement with experimentally determined values. 6. The entire force-deformation-temperature relation is determined by the choice of two scalar functions, the Helmholtz potential and the rate of dissipation function, together with the rate of energy dissipation equation. All the conditions for switching between dissipative and nondissipative behavior is derived from the above considerations, without the need for complicated switching rules(see eg. [9]) 2 2 The conditions for the change from dissipative to non-dissipative behavior is based on energetic conditions in the current paper. Thus, we have no difficulty in extending the cri- 3

5 These advantages come at a cost, however. The primary disadvantage is that the effect of the two phases is smeared out so that the exact nature of the juxtaposition is lost. Moreover, since the theory is entirely macroscopic, it does not account for the influence of such important parameters as grain size, alloy composition etc. Finally, the theory is one-dimensional by design. After much deliberation, we have chosen to develop such a theory due to the paucity of either qualitative or quantitative experimental evidence regarding the multidimensional response of the materials in question. Although fully three-dimensional theories have been developed based on analogies with classical plasticity, it is unclear whether such analogies are indeed valid, especially since the one dimensional response is quite complicated with the internal loops etc. ([2]). Instead, the one-dimensional setting was chosen in order to highlight the many new features of the theory presented here in as direct a manner as possible. Of course, one of the key ingredients of the theory, namely the change in symmetry and the accompanying changes in the physical characteristics of the material is lost in the one dimensional setting. Although the theory as presented is one-dimensional, care has been taken to ensure that the fundamental concepts such as the energy flow diagram (see Figure 1), the loading criteria etc., are generalizable to three dimensions, so that it would be possible to develop a three dimensional theory in the future in such a way that the current one dimensional theory would be recovered under appropriate specialization. The outline of the paper is as follows: In Section 2, we introduce the notion of multiple natural configurations and the kinematical constitutive equations for the intermediate states. We then define the parametrized family of response functions for this material. Section 3 deals with the energy flow during phase transitions and develops the reduced energy equation. This is followed by the modeling of the response for the individual phases. We introduce the notion of dissipation and the need for a memory variable" in Section 4. We first demonstrate that the material has a certain history dependent" inelastic behavior manifest by the study of the inner loops in the experiments of Huo and Muller[2]. We go on to show that this "history dependence" does not affect the Helmholtz potential but only the teria to a fully three-dimensional theory. Furthermore no restrictions on the hardening behavior are made. In contrast the switching rules" advocated by Fedelich and Zonzotto[9] are predicated on the fact that the force-deformation curve is horizontal. Moreover, the procedure in the latter paper cannot be extended to a fully three-dimensional theory. 4

6 rate of dissipation, so that the rate of dissipation function is history dependent. We then go on to develop a simple form for the constitutive equation for the rate of dissipation function. We are now in a position to predict the complete force displacement response with the internal loops, and we do so in Section 4.3. In order to gain a clear insight into the role of the dissipation function, we discuss the constitutive equations for a non-dissipative phase transition, and obtain a characteristic trilinear" (or non-monotone) response as a special case of the theory. Thermal effects are considered in Section 5, where the consequences of interrupting the isothermal loading with a temperature rise is developed. Also the force temperature curve at fixed length is investigated. Here we find an interesting and significant discrepancy with the experimental plots of Huo and Muller[2]. Thus, if one compares the force temperature plot that is obtained form the theory presented here (see Figure 5) with those of Huo and Muller[2, Figure 6], we find that there the theory reports a prediction of the temperature change that is about three times as large as that of the experiments. To elaborate, we find that at constant deformation, the A M transition takes place over a temperature change of approximately 300K while the experiments report about 100K. We discuss some of the issues regarding this discrepancy and show that there appears to be some internal inconsitency between the experiments represented in of Huo and Muller[Figures 6 and 9][2]. Section 6 deals with the procedure for the calculation of the material constants that appear in the theory and the constants for two systems are displayed. Finally, in section 7, we recapitulate some of the salient features of the current continuum model, and end with some discussions of the many questions that are as yet unanswered. 2 Preliminaries Consider a rod of initial length L lying along the x-axis. We shall assume that the entire rod is in a single phase at a uniform temperature θ 0. This is one of the natural states of the material and will be referred to as κ 0. The position of a generic material point in this phase is denoted as X. When the rod is stretched or compressed, it occupies a new configuration κ t referred to as the current configuration; the position of the material point 5

7 in κ t, that was originally at X, is denoted by x. Then, the motion of the material points with respect to κ 0 is given by the function x = χ κ0 (X, t). (1) The gradient of the function χ κ0 and is defined by is referred to as the deformation gradient F κ0 = χ κ 0 X. (2) Experimental evidence on shape memory alloys has shown that the rod is capable of existing in two different microstructures depending on conditions of temperature and load. At low temperatures and zero loads, its microstructure is termed "martensitic" while at a high enough temperature its microstructure is "austenitic". The temperature at which the microstructure is converted from the martensitic phase to the austenitic phase at no load is called the "austenite start temperature" while the temperature at which the reverse transformation occurs is called the "martensite start temperature". Moreover, it has also been shown that that there are many martensitic variants whose lattices are related to each other by various rotoinversions. Under various conditions of temperature and load, the following phenomena have been observed: 1. Pseudoelasticity : If the material is at a temperature above the austenite start temperature, it can be induced to undergo a phase change to martensite by the application of loads. However, the material returns to its austenitic phase upon unloading, so that there is no permanent shape change. The response is hysteretic and has a characteristic load deformation curve illustrated in [2, Figure 3]. Under these conditions, only one variant of martensite is formed. 2. One way shape memory : If the temperature of the rod is below the austenite start temperature but above the martensite start temperature, then a permanent strain remains upon a loading unloading cycle. This permanent strain (known as the transformation strain) is of the order of 8% and can be recovered by heating the specimen above the austenite start temperature. 6

8 3. Deformation twinning : If the material possesses more than one variant of martensite, then one variant can be caused to grow preferentially at the expense of others by the application of loads. This causes a residual strain upon unloading. 4. Combined deformation twinning and shape memory: The residual strain that was caused by the growth of a single martensitic variant can be recovered by heating it at no load so that the martensite is converted to austenite. We shall be concerned with the first two processes in this paper. The deformation twinning has been addressed in a separate series of papers ([10, 11, 12]). 2.1 Multiple natural configurations We first observe that the rod in question has two fundamentally different natural states: the state corresponding to the austenitic phase and that corresponding to the martensitic phase. On the macrosopic scale, these two states differ in their natural configurations, the elastic response functions as well as the material symmetry associated with them. These changes are a manifestation of the differences in lattice structure between the two materials. Following [1] we model the macroscopic behavior of this material by assuming that the mechanical response of the material depends upon the deformation gradient from two different reference configurations, each configuration being associated with one phase of the material. The question arises as to how to choose these configurations. As discussed in [1], any conveniently known configuration can be chosen to represent each phase. Here, we pick the stress free configuration of the martensitic phase at the austenite start temperature and that of the austenitic phase at the martensite start temperature as the two representative configurations. Since we shall work at temperatures above the austenite start temperatures, we shall refer to the former configuration as κ 0 and the latter configuration as κ 1. The deformation gradient of the mapping that takes the material from κ 0 to κ 1, denoted by G κ0, is a given number that is determined by the microstructure of the rod and its parent and product phases. It can be determined by measuring the rod in its stress free martensitic and austenitic 7

9 phases and calculating the ratio of their lengths. Of course, a tacit assumption is made that the rod is homogeneous so that G κ0 does not vary from location to location in the rod. 2.2 Intermediate states: The transition from the parent to product phase does not occur instantaneously but gradually in many stages that (at a microscopic level) are composed of juxtaposed layers of the parent and product, with the volume fraction of the product gradually increasing as the phase transition proceeds to completion. We now examine one such intermediate state. Let α be the fraction of the product phase. If we now consider a configuration of the rod wherein the parent and product phases are in their respective natural configurations, then owing to the juxtaposition (at a microscopic level) of the two structures, we can define the natural configuration of the intermediate states κ α by prescribing the deformation gradient G κα as G κα := 1 + αh, H := G κ0 1. (3) The above formula (3) represents the smearing out of the two phases, with each of them in their respective natural configurations. The form (3) can be derived based on crystallographic grounds [5, 12] for single crystals, based on the juxtaposition of piecewise homogeneous deformation gradients corresponding to the natural configurations of the individual phases. The kinematical assumption (3) is in the spirit of a constitutive assumption. It states that the traction-free states of the parent and product phases are such that they can be smeared out in the mixed state in the manner described in (3) while satisfying the compatibility conditions at the interface 3. Moreover, this smearing out allows the mixed state to be treated as if it were a single continuum rather than having to resort to the use of mixture theory and defining a partial stress associated with each constituent of the mixture. Such an alternative approach leads to serious problems with regard to specification of the individual partial tractions at the boundary. For future reference, we shall denote the gradient of the mapping from κ α to κ t as F κα. A routine 3 It should be noted however that incoherent interfaces are also possible resulting in the presence of dislocations along the boundary between the two phases. 8

10 calculation using the chain rule of differentiation reveals that We also observe that F κα = F κ0 (G κα ) 1 = G α = αh, F κα = 2.3 Balance Laws F 1 + αh. (4) F κ0 (1 + αh) F κ0 H α. (5) (1 + αh) 2 Within the context of the one dimensional theory presented here, the response of the material is governed by the balance of mass, linear momentum and energy which, with reference to the configuration κ 0, take the following local forms ϱ 0 = ϱf κ0, (6) and P X = ϱ 0ẍ, (7) ϱ 0 ɛ = ϱ 0 r q X + P F κ0, (8) respectively. In the above equations, ϱ 0 and ϱ are respectively the mass per unit length of the rod in the reference configuration κ 0 and current configuration κ t, P is the axial load on the rod, ɛ is the internal energy per unit mass of the rod, r is the body heating per unit mass and q is the axial heat flux and the superposed dot indicates the material time derivative (with X fixed). Furthermore if θ is the absolute temperature, η is the entropy per unit mass and ξ is the rate of mechanical dissipation (conversion of work into heat), then the local form of the entropy production equation can be written 9

11 in the form 4 ξ := ϱ 0 r q X ϱ 0θ η 0. (9) By introducing the Helmholtz potential ψ := ɛ θη, and substituting (8) into (9) we can obtain the reduced energy-rate of dissipation relation 5 as 2.4 Constitutive Theory We use (7)-(10) as follows: ϱ 0 ( ψ + θη) + P F κ0 = ξ. (10) Constitutive assumptions are made for the functions P, ɛ, η, ψ, and q as well as the rate of mechanical dissipation ξ. The non-negativity of the entropy production is automatically ensured for all admissible processes by the choice of the functional form for ξ The reduced energy-rate of dissipation relation (10) is used to provide restrictions on the constitutive assumptions as well as an evolution equation for the change of phase. The equations (7) and (9) are used for calculation of the current position x and the temperature θ as functions of time. Here we depart from the usual procedure in the sense that we assume a constitutive equation for the rate of dissipation. In this, we follow the approach of Green and Naghdi [13] who advocate the use of such an independent constitutive assumption. The conditions for the commencement and cessation of phase transition are governed by the satisfaction of the reduced energy-rate of dissipation relation (10), the non-negativity of the rate of dissipation and a maximum rate of dissipation condition. Intuitively speaking, 4 Note that mechanical dissipation is just one mode of entropy production. Other modes include heat transfer across a temperature gradient, mixing etc. the equation (7) accounts for entropy production due to mechanical processes and that due to temperature gradient.while that latter is present in all materials, even when they undergo purely thermal processes, the former provides the link between mechanical process and thermal processes. 5 Here and henceforth, we use the word rate of dissipation" to mean rate of entropy production due to mechanical working". 10

12 the maximum rate of dissipation criterion states that if the material is capable of responding in many different modes with different rates of dissipation, (including a mode that is non-dissipative), then the actual mode chosen will be the one with the maximum rate of dissipation. Thus, according to this criterion, a dissipative mode is always chosen over a non-dissipative mode whenever the former is possible. In other words, reversible, or non-dissipative processes occur only when dissipative processes are impossible. 2.5 Families of response functions Considering the fact that the material has more that one natural state and, as a consequence of the change in the lattice structure, its physical properties change, it is natural to assume that the material has at least two independent response functions one for the austenitic state and the other for the martensitic state. Thus, if the material responds elastically from an austenitic state, then its Cauchy stress is given by T = T κ0 (F κ0 ), (11) while, for the elastic response from the martensitic state is given by the Cauchy stress is given by T = T κ1 (F κ1 ). (12) It is evident that the form of the two functions are, in general, different and they each depend upon the choice of their respective natural configurations. If we now consider the possible intermediate states described by (3) it is evident that there exist families of response functions for the basic dependent variables parameterized by the volume fraction of the individual phases, whose forms are dependent upon the choice of the two configurations κ 0 and κ 1. For example, the family of stress response functions would take the form T = T κ0,κ 1 (F κ0, F κ1, α). (13) with similar equations for the other variables. Of course the kinematical constitutive assumption (3) shows that the other intermediate states are not primitives but are prescribed so that the use of the parameter α implies implicit dependence on the other configurations also. The notion of the family of response functions elaborated in [1] is central to the theory presented here and serves as a unifying theme tying together 11

13 different inelastic and dissipative phenomena into a common framework. In the next section we discuss the fundamental constitutive assumptions and the restrictions imposed upon the response functions by the reduced energy-rate of dissipation equation. 3 Restrictions on the Constitutive Equations 3.1 The energy flow through the system From the perspective of the conservation of energy, the shape memory alloy may be viewed as an energy transducer", converting the incoming energy into other forms and either storing or releasing it. We can consider the different energy conversion mechanisms in a unified way by constructing an energy flow diagram" (see Figure 1) that details all the energy storage, conversion and dissipation mechanisms. This, in turn will help in the development of the constitutive equations for the materials. Figure 1 reflects all the energy storage mechanisms that have to be accounted for in the internal energy of the system. Specifically, the term latent energy" refers to the difference in the specific internal energy levels between the two phases at a given temperature. The interfacial energy refers to the energy trapped in the incoherent interfaces 6 between the two phases (at a microscopic level) and the words thermal energy" refers to the energy stored in the lattice vibrations (thermal energy). Also, the entropy change refers to the change in the entropy with changing temperature as well as that associated with the change in phase. In the latter case, the entropy change results in heat exchange with the surroundings, referred to as the latent heat of the material. For mathematical reasons, it is more convenient to work with the Helmholtz potential rather than with the energy and entropy. We begin by assuming that the variables P, ψ, and η depend upon the variables {F κα, α, θ}, while 6 It is common to refer to this as the coherency energy", however, if the interfaces are perfectly coherent, there is no difference between the region surrounding the interface and other parts of the lattice. However, if there is a mismatch of the lattice parameters or if the orientations are not precisely matched, one needs to force the two regions together, either through pure local lattice distortions or through the generation of dislocations. It is for this reason that we referto these interfaces as incoherent. 12

14 the rate of mechanical dissipation ξ depends upon 7 {θ, α, α}. The above constitutive assumptions are based on the notion that the former three variables depend upon the current microstructure and temperature (characterized by α and θ) and the deformation from it, while the mechanical dissipation occurs during the process of phase transition alone and hence depends upon α. When this constitutive assumption is substituted into the reduced energyrate of dissipation relation (10) we get ψ (P ϱ 0 )(F κ0 ) ϱ 0 ( ψ F κ0 θ + η) θ ψ ϱ 0 α = ξ. (14) α By means of standard arguments that utilize the fact that (14) has to be satisfied for all admissible processes and further that (14) is affine in F κ0 and θ, we arrive at the following conclusions: Thus,(14) reduces to ψ P = ϱ 0, F κ0 (15) η = ψ θ. (16) ψ ϱ 0 α = ξ. (18) α In view of the above results, the constitutive assumptions for the material reduce to the specification of two scalar valued functions: The Helmholtz potential ψ and the rate of mechanical dissipation ξ. Once the functional forms of these functions are chosen, the form of the axial force can be calculated from (15), the entropy from (16) and the internal energy from the relation ɛ = ψ + θη. Finally, the equation (18) provides the equation for the evolution of α. For future convenience we shall define the driving force for phase change " D as D := ϱ 0 ψ α (19) 7 ξ could have a much more complicated structure. However, this simple form suffices for our purposes. 13 (17)

15 Of course, the constitutive equation for the rate of dissipation must be such that it must be compatible with (18). Thus for example, if processes with α = 0 are admissible for a certain range of values of α and F κα, then for those range of values, the rate of dissipation must vanish. In the next section we shall discuss specific forms for the Helmholtz potential and the rate of dissipation function that is appropriate for the behavior of shape memory alloys. 3.2 The Helmholtz potential: The elastic energy of the material is a consequence of it being strained from its current natural configuration κ α. Consequently, we stipulate that the elastic strain energy W per unit length in the configuration κ α be of the form W = (1 α)w 1 (F κα ) + αw 2 (F κα ) (20) where the functions W 1 and W 2 are respectively the strain energy functions of the parent and product phases. Similarly, if W 3 (α) is the interfacial energy per unit reference length and ψ 4 (θ) and ψ 5 (θ) are respectively the thermal free energies of the parent and product phases per unit mass, then the Helmholtz potential per unit reference length is ϱ 0 ψ = G κα [ W ] + W 3 (α) + ϱ 0 [(1 α)ψ 4 (θ) + αψ 5 (θ)]. (21) The factor G κα in the first term of (21) is due to the fact that (20) is defined per unit length in the configuration κ α while ϱ 0 ψ is per unit length in the reference configuration. The form of the constitutive assumption for the Helmholtz potential is essentially a simple addition of the various effects. Specifically, the strain energy is assumed to be independent of the temperature while the latent energy is assumed to be unaffected by strain. Of course, once could construct a model that has coupled thermo-mechanical effects but even this simple model is sufficient to predict the main effects of the shape memory behavior. The difference between the thermal free energies of the two phases contains contributions from two sources: (1) the difference between the internal energies and (2) the difference between the configurational entropies of the two phases multiplied by the absolute temperature. 14

16 Upon substituting (21) into (15) and (16), and using the fact the resulting equation is linear in and θ, we arrive at F κ0 P = W F κα, (22) η = ((1 α) ψ 4 θ + α ψ 5 ), (23) θ while the rate of dissipation equation (18) becomes with driving force" D given by D = ϱ 0 ψ α D α = ξ, (24) = [H( W F κα W ) ( W 3 α + (W 2 W 1 ))] ϱ 0 [ψ 5 ψ 4 ]. (25) Thus referring back to the energy flow diagram (Figure 1) we see that by using the Helmholtz potential, we have succeeded in modeling the internal energy, the entropy and the mechanical working of the system. It only remains to model the rate of rate of dissipation function and we shall turn to this task next. 3.3 The rate of dissipation function: We now assume that the rate of mechanical dissipation is given by { A + α if α 0, ξ = A α if α 0, (26) with A + 0 and A 0 referred to as the forward and backward dissipative resistances, respectively and are not constants. The constitutive forms of these two functions will be discussed presently. Before, proceeding with the specific forms of these functions we shall pause to consider one important point: We have independently modeled each of the mechanisms identified in the energy flow diagram without regard to the satisfaction of the energy equation. In other words, the various mechanisms are coupled by the fact that the total energy is conserved. We now use this coupling to obtain conditions for the onset of phase transitions and the rate of increase of the volume fractions of the individual phases. 15

17 Since it is the easier of the two, we first consider the rate of change of the volume fraction of the product phase. Substituting (26) into (24) and using the fact that the resulting equation must be valid for all admissible processes, we arrive at the following possibilities A < D < A + α = 0, (27) { A + if α > 0, α 0 D = (28) A if α < 0 The condition (27) represents the fact that so long as D lies between A and A +, the rate of dissipation equation (24) can only be satisfied with α = 0 so that the response is non-dissipative, and the stress response is that of an elastic material. On the other hand, whenever α 0 the driving force D must be equal to the dissipative resistance. An important consequence of the assumptions on the rate of dissipation function ξ is that the response of the material is rate-independent. To put it differently, the force deformation curve is the same irrespective of the speed with which the test is conducted. 3.4 Loading criteria For this material, we shall assume that the phase-transition is propagation controlled so that the rate of dissipation function itself provides the activation criterion [1]. As discussed by Rajagopal and Srinivasa [11], the conditions under which the new phase is created are determined by two criteria, namely the overcoming of the energy barrier for the initiation of the phase transition (the initiation condition) and the availability of sufficient energy to supply the dissipation that occurs during the transition. In processes that are initiation controlled, once the energy barrier for the creation of the new phase is exceeded, the energy release rate (or the driving force for phase change) is sufficient for continued propagation. In such cases, it was demonstrated that the phase change proceeds in intermittent bursts separated by periods of elastic response. For such processes, an independent initiation criterion must be postulated for the initiation of the transition. This has been done in the case of deformation twinning at low temperatures[12]. On the other hand, if the energy barrier for initiation is low, then the newly initiated phase nucleus is incapable of growing until the energy supply is increased sufficiently so that enough is available for overcoming the 16

18 dissipation that occurs during propagation. Such processes are called propagation controlled. Here the quasistatic response is smooth. For this case, the initiation condition is irrelevant for the macroscopic response. Instead, the rate of dissipation function itself provides the condition for the onset of phase transition. In the present case, we define the activation function for the start of phase transition by g := (D A + )(D + A ), (29) with D defined by (25). In terms of g, the results (27) (28), reduce to the following form g < 0 α = 0. (30) α 0 g = 0. (31) It should be observed that the activation function is defined entirely in terms of the quantities that appear in the reduced rate of dissipation equation (24) with the barriers given by the form of the rate of dissipation function (26). The elastic domain of the material is defined by those values of F κ0 for which g < 0. The condition (31) is the counterpart of the so-called consistency condition of plasticity. Here, the consistency condition is a result of the satisfaction of the rate of dissipation equation for all admissible processes; in contrast, the condition is independently postulated in classical rate-independent plasticity. The above considerations leave out the conditions for the commencement of phase transition. As outlined by Rajagopal and Srinivasa[11], for the case of propagation controlled mechanisms, we adopt the strain-space loading criteria[14] and hence obtain the following conditions for the beginning of phase transitions: { A + and ( D/ F κ0 ) F D = κ0 + ( D/ θ) θ > 0, A and ( D/ F κ0 ) F κ0 + ( D/ θ) θ (32) < 0. There is however one pathological" case to be dealt with,i.e., the case when both A + and A are zero. Under these conditions we have g = D 2, so that 0 is a minimum for g. This case requires careful consideration. As discussed by Rajagopal and Srinivasa[10], the criteria (32) are only applicable to regular points of the equation g = 0 and are obtained by showing 17

19 that elastic response is impossible under the conditions of (32), since this would lead to g > 0 and hence result in a violation of the rate of dissipation equation. A similar argument with g = D 2 = 0 leads to the conclusion that elastic response is impossible for any value of F κ0, i.e., loading takes place for every direction of further deformation. This last result is similar to that in the case of rigid-plastic materials, where the elastic domain reduces to a single point in strain space. Here, the condition D 2 = 0 represents a single point in deformation space for a given value of α. 3.5 Further specializations and the shape memory effect Hitherto, the constitutive equations have been specialized only to the extent that some broad features of the constitutive equation have been specified. We now consider further specializations that help us model shape-memory behavior in detail and enable comparisons with experimental data. To this end we further simplify the constitutive structure of the previous section by setting W 1 = W 2 = (1/2)M(F κα 1) 2, (33) W 3 = Bα(α 1), (34) ψ 4 = Cθ[1 ln(θ)], (35) ψ 5 = ψ 4 + f(θ). (36) In the above equations, the forms for W 1 and W 2 are assumed based on simple quadratic forms for the strain energy function of the two phases and with the further simplification that the axial elastic constant M is the same for the two phases. The interfacial energy" W 3 is a simple convex function that vanishes for the two pure phases α = 0 and α = 1. An extensive discussion regarding the choice of this form for the function is given by Huo and Muller [2]. The constant B is the interfacial energy constant and we shall see that that it plays an important role in the non-dimensionalization procedure. The form for ψ 4 and ψ 5 are simple and based on the standard form for rigid conductors, with C being the specific heat and the function f(θ) representing the difference in the free energies of the two phases. The form of this function will be later motivated by the experiments of Huo and Muller [2] and we shall see that it plays an important role in determining the temperature dependence of the hysteresis. 18

20 We see that the above assumptions imply that the elastic response is essentially uncoupled from the thermal response. Now, substituting (33) (36) into (22) (24), we obtain P = M[F κα 1], (37) η = C ln(θ) αf (θ), (38) ϱ 0 ɛ = ϱ 0 (ψ + θη) = G κα [1/2M(F κα 1) 2 ] + Bα(α 1) + ϱ 0 [Cθ + α(f(θ) θf (θ))]. (39) Furthermore, defining (1/2)((F κα ) 2 1) = E κα and using (25) the equation (32) can be written in the form α 0 D = (HME κα ) ϱ 0 f(θ) B(1 2α) = sgn( α)a ±, (40) where the notation sgn(.) stands for the signum function. With these definitions, the governing differential equations are X ( W F κα ) + ϱ 0 b = ϱ 0 ẍ, (41) ϱ 0 (C αθf (θ)) θ ϱ 0 αθf (θ) = ϱ 0 r q + ξ, (42) X with ξ given by (26) and α being determined by (40), whenever the conditions for the α 0 (strain space loading conditions) are satisfied. These equations, together with appropriate boundary conditions provide us with the requisite system for the determination of the axial displacement of the rod, its temperature and the amount of phase transition that takes place at a point. Rather than considering general solution procedures for such behavior, we shall confine ourselves to homogeneous, slow deformations with no temperature gradients and consider only the effect of phase transition. In this way, we shall be able to make comparisons with experiments and determine the material constants. Under these circumstances, the central equation that governs the shape memory behavior is (40) and we shall consider it further. In order to simplify matters and present the equations in a unified form, we nondimensionalize (40) by dividing throughout by B and obtain α 0 E κ α E 0 L(θ) = sgn( α)ā± + (1 2α), (43) 19

21 where E 0 := B MH, L(θ) := ϱ 0f(θ) B, Ā± := A± B. (44) The equations (40) (44) are central to the modeling of the shape memory effect and, together with (32) form the core of our analysis. 4 Response to homogeneous deformations Consider a rod of initial length l that is homogeneous in its reference configuration κ 0 and initially entirely made up of the parent material. Let the entire specimen be at a uniform temperature θ 0. One end of the rod is fixed and the other end is moved (under deformation control) slowly (so that inertial effects may be ignored), so that, at any instant t, the position of the end of the rod is given by x(l, t) = l + u(t), u(0) = 0. (45) Since the rod is homogeneous and the end motion occurs slowly, it is reasonable to seek homogeneous solutions of the form x(x, t) = (1 + u(t) )X, l (46) θ(x, t) = θ 0, (47) α(x, t) = α(t). (48) One immediately comes to the conclusion, that, neglecting inertia and body forces, the balance of linear momentum is satisfied since P = P (t) alone. 4.1 Heat released and absorbed during isothermal transformations For the case of isothermal transformations, it is an easy matter to calculate the heat released or absorbed. In order to maintain the temperature of the specimen, heat must be supplied or removed from the lateral surface of the bar to balance the change in the latent energy and that produced by the (49) 20

22 latent heat of material. The entropy equation (42) provides us with the equation required for the calculation of the amount of heat that has to be transferred to the material during the transformation process. Using (42) and noting that θ = const., we find that the rate of heat supply is given by ϱ 0 r = ξ ϱ 0 αθf (θ). (50) We can now obtain the total heat supply during the forward transformation of an initially unconverted material to be Q f = ξdt ϱ 0 θf (θ), (51) where the integration is carried out from the beginning of transformation to its completion and the temperature is held constant. Now using the constitutive assumption (26) for ξ, the above equation can be integrated to give Q f = A + dα θl (θ)), (52) similarly, the total heat supply during the reverse transformation is given by Q r = A dα + θl (θ)). (53) The above equations (52) and (53) then determine the amount of heat that is supplied or withdrawn from the lateral surfaces of the bar in order to maintain the temperature at θ 0. It is interesting to note that the heat released is not just the latent heat" but also the rate of dissipation, so that the heat released during the forward transition is not equal to the heat absorbed during the reverse transition. In order to fully understand the nature of the force-displacement curve and the way in which the volume fraction of the product phase changes with time, we shall consider the following possibilities: 1. A dissipationless process, i.e., A ± = Isothermal, force induced phase change with dissipation, i.e., θ = θ 0 3. Temperature induced phase change. 21

23 4.2 Nondissipative phase change Although it is well known that the process of martensitic transformation is accompanied by considerable dissipation, it is instructive to study a dissipationless process in order to highlight the role of the rate of dissipation function in modeling the shape memory behavior. For the case of a non-dissipative process, the equation for the evolution of the phase (43) can be easily solved for α to give where α = 1 φ(e κ α, θ), (54) 2 φ(e κα, θ) := E κ α E 0 L(θ). (55) Thus, we see that (54) provides valid solutions for α (i.e., those that lie in the range [0, 1]) only for those values of E κα and θ for which φ is in the range [ 1, 1]. Before proceeding further, keep in mind that the above equation (54) is valid only if D = 0. If D 0 then we have α = 0. We use the equation as follows: we begin with a given value of α. We calculate the value of D and if it happens to be zero, we calculate the value of α by differentiation of (54). The resulting force deformation curve is shown in Figure 2. Recall that the bar is fully in the parent phase initially. Then F κα = F κ0, since α = 0. Let the temperature of the bar be such that L(θ) > 1. Thus, initially, the value of D is negative and so nothing happens. Now, as u(t) is gradually increased, the value of (E κα /E 0 ) L(θ) gradually increases and when it reaches the value 1, the driving force D vanishes. Phase transition now begins. Now differentiation of (54) using (55), the definition of E κα and (46) reveals that H(1 + u(t))2 ( l 2 (1 + αh) 3 (1 + u(t)) u(t) + 2) α = l(1 + αh), (56) 2 which furnishes an equation for α. The above first order ODE can be solved(by quadrature, if necessary) for given u(t) and is valid for α in the range [0, 1]. The resulting solution is shown in Figure (2). One of key features of the resulting solution is that the forward and reverse transformations proceed along the same path (the line BC in the Figure (2)). It is also possible to eliminate the variable α entirely from the equations and obtain a non-monotonic elastic force-deformation curve from this analysis.thus, the theory outlined above recovers a purely elastic response in the limit of vanishing rate of dissipation. 22

24 4.3 Dissipative phase change It is well known that (see e.g.,huo and Muller[2]) the actual process does not proceed along the diagonal line as suggested by the non-dissipative analysis (see figures, 2 and 3). However, any point on the diagonal line is accessible by going through paths such as BF G (see figure 2). Moreover the response of the material is complicated by the presence of internal loops" in the force -deformation curve. We consider these loops first. When we observe the internal loops closely, we see that the material has a kind of memory". We demonstrate this by considering the paths L := BF GG B F and D := BF F in Figure 3. It is clear the point F can be reached in two different ways. One path is the direct" path D and the other is the loop" path L. However, the subsequent response (from point F) of the material depends strongly on the path that it has taken. For example, if the material reached point F along the direct path D, then, upon increasing the boundary displacement, the subsequent response is along F F, and there is no further phase change. On the other hand, if the point F is reached via the loop path" L, then the subsequent response is along the line B F with continued phase change. Thus the material remembers" when the diagonal line BC was touched last. This is crucial to the response of the material in the so-called internal loops that have been observed. At present there seems to be no consensus as to the physical origin of this hysteresis, although there seems to be some indications that it may be related to the number of interfaces that are formed during the process(see the discussions by Huo and Muller[2, section(8.2)] as well as Wilmanski[16]). Interestingly, the subsequent response after either path is the same when the boundary displacement is reduced and is essentially along the elastic unloading line F G with no phase change. We can therefore conclude that the elastic response of the material is unaffected by the previous deformation history, but the phase change behavior is strongly influenced by it. From a macroscopic viewpoint, irrespective of the physical basis of the memory", its manifestation can be explained by the introduction of a single scalar variable 8 (whose macroscopic interpretation is the value of the volume 8 It should be pointed out that the use of a discrete memory parameter that tracks the most recent loading, unloading event was originally suggested in models of metal plasticity by Dafalias and Popov[15] 23

25 fraction of the product(martensitic phase) when the driving force is zero) in the rate of dissipation function. The Helmholtz potential is unaffected by the memory, fully in keeping with the observation that the elastic response is unaffected by the memory. In view of the above description of the shape memory behavior, we keep the form of the Helmholtz potential ψ as described in (21) but now introduce dissipative effects by assuming that the dissipative response functions A ± are each given by Ā + (t) = k α(t) α(s), (57) Ā = k α(t) α(s). (58) where x = { 0 if x < 0, x if x > 0, (59) and where α(t) is the current volume fraction of the martensitic phase and α(s) is the martensitic volume fraction when the diagonal was last encountered. In the above equations, k is a non-dimensional material constant and α(s) is the value of α when the strain path last satisfied the condition D = 0. For example, in figure 3, for the direct path D described in the previous paragraph, the value of α(s) is 0 since the last time that D = 0 was satisfied was at the point B. On the other hand, for the loop path L when the material point is at E, the value of α(s) is equal to α at the point B. Thus as the material deforms, the value of α(s) takes sudden jumps whenever the line BC is touched. This value is a measure of the memory" of the material alluded to in the previous paragraph. Substituting (57) and (58) into (43) and using the definition (59) we get α 0 E κ α E 0 (1 2α) L(θ 0 ) = k(α(t) α(s)). (60) Moreover, the elastic domain is determined by the following conditions k(α(t) α(s)) < D < 0 if 0 < α(t) α(s) < 1, (61) 0 < D < k(α(t) α(s)) if 1 > α(t) α(s) > 0, (62) D < 0 if α(t) = 0, (63) D > 0 if α(t) = 1. (64) 24