## Abstract

Models for shape memory material behaviour can be posed in the framework of a structured continuum theory. We study such a framework in which a scalar phase fraction field and a tensor field of martensite reorientation describe the material microstructure, in the context of finite strains. Gradients of the microstructural descriptors naturally enter the formulation and offer the possibility to describe and resolve phase transformation localizations. The constitutive theory is thoroughly described by a single free energy function in conjunction with a path-dependent dissipation function. Balance laws in the form of differential equations are obtained and contain both bulk and surface terms, the latter in terms of microstreses. A natural constraint on the tensor field for martensite reorientation gives rise to reactive fields in these balance laws. Conditions ensuring objectivity as well as the relation of this framework to that provided by currently used models for shape memory alloy behaviour are discussed.

## 1. Introduction

Shape memory materials, especially in the form of metal shape memory alloys (SMAs), are nowadays well-known materials with an increasing number of applications in numerous fields [1]. Several analytical models are available for predicting their thermomechanical behaviour, as described in many comprehensive reviews on the state of the art for SMA constitutive modelling (e.g. [2–5]). Because the complex response of SMAs has several facets (pseudoelasticity, martensite reorientation, one- and two-way shape memory effects and constrained recovery), only a few of the existing models seek to account for all of them at the same time [6–8]. In addition, there are some aspects that have yet to receive a sufficiently comprehensive modelling treatment. One of these is the fact that, in certain situations, the phase transformations are observed to occur in spatially localized domains [9]. Another of these is the relatively recent discovery that, in some groups of alloys, phase transformations can be induced in specialized ways by magnetic fields [10].

In this paper, we describe a generalized modelling framework that is capable of consistently dealing with both the above aspects, as well as the conventional ones involving thermal and load activation. This is based on the idea, first introduced in [11], to model bodies made of SMAs as structured continua characterized by a fully fledged set of balance equations for a suitably chosen set of microstructural descriptors. Specifically, as in [11], these descriptors are taken in the form of a scalar field and a second-order tensor field. Even though the basic idea of seeking an enriched set of kinematic descriptors in continuum mechanics is not new, with antecedents tracing back to the Cosserat brothers, and unified general treatments presented in [12,13] among many others, the application of such ideas to SMA modelling remains underexploited. A related application limited to pseudoelasticity was given in [14] albeit with a different choice of microstructural descriptors and an alternative method of derivation.

There are at least two advantages of the proposed approach with respect to the more standard ones. The first one is its inherent capability for capturing the structure of spatially localized phase transformations, such as those discussed in [9]. Here, this happens not only due to consideration of constitutive functions that depend upon the spatial gradients of the microstructural descriptors, as has already been considered in other models (e.g. [15–18), but also because of the natural occurrence of transformation criteria in the form of local balance equations with divergence and source terms. A second advantage is the possibility to model, by means of source terms in the microstructural balance equations, phase transformations induced by non-mechanical external agents, such as in magnetic SMAs.

Besides the above advantages, the proposed framework has certain distinguishing features.

One is the choice of the two microstructural descriptors. The first descriptor is the conventional total martensite fraction *ξ* that describes how the material element at the microscale is partitioned between austenite and martensite [19]. The other one is a second-order tensor **M** that provides information on the specific nature of the martensite, an effect which is ultimately due to the variant structure at an underlying finer level than that which is resolved by the continuum treatment. In addition to [11], other current SMA models make use of a scalar and a second-order tensor internal variable, e.g. [8], but these do so in a different way with respect to the present approach, as the variability of the tensor field is strongly constrained by the scalar one. More generally, allowing for a fully independent variability of the two fields, a treatment that was advocated already in [11] and used in a one-dimensional setting in [20], provides a framework that can naturally describe phase transformations and martensite reorientations (e.g. [6,21–23]). Here, we present such a unified overall description of the microstructure within a finite strain generalized framework of a structured continuum whose constitutive functions may depend also on the spatial gradients of *ξ* and **M**.

A second distinguishing feature of the proposed framework is the fact that the constitutive information is specified through a free energy and a dissipation function following the approach proposed in [24,25]. This approach was suggested for use in the context of SMA modelling in [26], and subsequently began to be developed in a one-dimensional setting in [27,20]. By focusing on a specific and experimentally motivated aspect of rate-independent response, the dissipation function turns out to be completely determined by quantities that can be interpreted as constitutive functions that describe the internal resistance encountered by the phase transformations. In [27,20], it was shown that considerable modelling flexibility can be obtained by the consideration of internal resistance functions depending on the actual value of the phase fraction and its past history evaluated over suitable switching instants. This development enabled earlier phenomenological models to be placed on a firm thermodynamic basis, such as [28,29], which had already shown a strong ability to replicate complicated stress–strain–temperature paths.

A third distinguishing feature is that all aspects are developed in the context of a fully large strain setting. Under such circumstances, a careful linearization about a suitably identified base state is then able to recover the conventional strain theories [11] where, for example, an overall strain tensor is additively decomposed into a stress-free transformation strain and a load-induced mechanical strain. Further, the large strain setting permits a unified treatment of reactive terms associated with constraints on the microstructural descriptor **M**. Finally, it is well known that constitutive functions cannot depend upon the deformation gradient in an arbitrary way if both objectivity and second law requirements are to be satisfied, and conventional continuum mechanics treatments indicate precisely how such restrictions can be met in the most general way. The treatment here shows in a similar fashion how constitutive functions can depend upon **M** and its gradient in order for objectivity and second law requirements to be satisfied automatically.

Regarding notation, second-, third- and fourth-order tensors all enter the treatment. For a second-order tensor **A**, the transpose, inverse and inverse transpose are denoted as usual: **A**^{T}, **A**^{−1}, **A**^{−T}. Let **a**,**b**,**c**,**d**,**e**,**f**,**g**,**h** be arbitrary vectors in . The usual scalar product of two vectors is denoted by **a**⋅**b**. The tensor product ⊗ permits the construction of second-order tensors, e.g. **a**⊗**b**. Its action as a linear operator is that (**a**⊗**b**)**c**=(**c**⋅**b**)**a**. Using such dyads in the standard way allows for the definition of third- and fourth-order tensors, e.g. **a**⊗**b**⊗**c**⊗**d**. Algebraic operations can be defined with respect to such simple dyad tensors, and then, by linearity, these operations extend to general second-, third- and fourth-order tensors. The scalar product between two second-order tensors and between two third-order tensors is denoted by the same ‘⋅’ symbol. Its operation is the usual one, which means that its action on simple dyad tensors is
We have need of contraction operations that apply between tensors of different order. For this purpose, we use contraction operators denoted by and whose action begins from the operator location and then move outwards [30], meaning for example:
It is then formally the case that and . Standard multiplication of second-order tensors (**a**⊗**b**) and (**c**⊗**d**) will be denoted in the usual way of (**a**⊗**b**)(**c**⊗**d**) but it is also formally . The operations are associative so there is no ambiguity in an expression like , which is the type of operation that appears later in (5.5). We shall also have need for a transpose operation on third-order tensors (first encountered in (3.3) and continuing intermittently through to (5.5)) which is defined via
Finally in the first equation of (4.6), we have need for a transposition operation on fourth-order tensors which is defined via

## 2. Configuration fields and choice of the microstructural descriptors

A continuous body made of SMA is understood as an assembly of material elements endowed with an underlying microstructure made of an arrangement of an austenite phase and a martensite phase in a proportion determined by the *volume fraction of martensite* *ξ*.

Experimental observations shows that the martensitic phase can appear in several crystallographic variants and these variants can arrange themselves into patterned structures at a scale finer than that which is to be resolved here by the material points [10,31]. This strongly influences the behaviour at the macroscopic scale and, when these variant structures are expressed coherently, they contribute to the overall deformation. Changes in the variant pattern are usually referred to as martensite reorientations and, depending on the material length scale of interest, can be modelled in a variety of ways [5,4]. Our motivation here arises from the need to model engineered devices containing shape memory materials that, even for microdevices, perform sensing and actuation tasks at length scales that are several orders of magnitude greater than the length scales which characterize the underlying atomic arrangements in individual crystallographic variants.

In this context, the fraction *ξ* is not by itself sufficient to provide an appropriate description of the underlying microstructure. As a remedy, the orientation state of the martensite itself will be described by a second-order tensor-valued quantity **M** that will be called the *type of martensite* [11]. This tensor can be given various types of specific interpretations depending on the context (e.g. along the lines of [32] or [33]). For example, it can be interpreted as a local deformation that would convert an entirely austenitic patch into another one composed completely of martensite. This patch would contain, at microstructural scales, a fine pattern of martensite variants. The aggregate effect of this fine pattern then characterizes the type of martensite under consideration and could be considered as a finite strain counterpart of what, in the more usual small strain setting, is often called the *macroscopic transformation strain* (e.g. [34,35]).

The macroscopic motion of a material body is described, as usual, by an invertible mapping **x**=*χ*(**X**,*t*), where *t* denotes time, **X** denotes particle location in a reference configuration and **x** denotes the location of the particle at time *t* (the current location). The reference position of the body is chosen so as to coincide with a state of undeformed pure austenite. In turn, the motion *χ* determines the velocity and the deformation gradient
2.1
Three more fields are then used to characterize the configuration of each material particle: the absolute temperature *ϑ*>0, and the two microstructural descriptors *ξ* and **M**.

As already discussed, the value of the field **M**(**X**,*t*) describes the type of martensite that may be present at the material particle **X** at time *t*. Specifically, **M**=**I** describes the homogeneous deformation from austenite to the special state of martensite that has a fine scale microstructure of randomly oriented martensite variants—the so-called self-accommodated martensite. Such randomly oriented variants could generally be achieved by a process of unconfined cooling. However, a process in which martensite is obtained from austenite by either a cooling process with directional confinement or else by the application of external loading would generally give rise to special patterns of martensite variants at the microscale. At the operational scale of the continuum theory, this renders **M**≠**I** whenever the fine scale pattern of variants gives rise to a net macroscopic deformation with respect to the parent austenite phase.

A change in the phase fraction *ξ* is referred to as a *phase transformation* since it is associated with the conversion from austenite to a corresponding amount of martensite or conversely. A change in **M** is referred to as a *martensite reorientation* since this models an adjustment to the type of martensite. Both changes can occur either separately or in a coupled way as a consequence of thermal, mechanical or other stimuli.

This choice of microstructural descriptors was introduced in our previous work [11] and used in a small strain one-dimensional setting in [20] but is otherwise not common in the literature. Similar choices, but again in a small strain non-structured setting, have been successfully pursued in other works [6,21–23,].

One can also on this basis define appropriate finite strain tensors associated exclusively with the austenite to martensite transformation. Recalling the classical Lagrangian strain tensor with right Cauchy–Green tensor **C**=**F**^{T}**F**, a measure of pure transformation strain could be taken, for example, as .

Microstructural descriptors can be subject to various types of physically motivated constraints, the most obvious one being the restriction 0≤*ξ*≤1 arising from the fact that *ξ* has to represent a phase fraction. Also, in many SMAs, the austenite to martensite phase transformation can be regarded as volume preserving and this motivates the constraint
2.2
Moreover, other important constraints arise from the fact that macroscopic transformation strain is physically limited by the overall variant structure. This can be stated in various ways, for example in the form
2.3
where *k*_{mt} is a constitutive quantity characteristic of the maximum transformation strain achievable by a fully oriented martensite. Even if in most models *k*_{mt} is considered as a given material constant, it can also be specified as a suitable function of the state variables (e.g. phase fraction or even stress, as suggested in [7] and [8]). Other constraints can also be considered; for example, if the overall deformation is volume preserving then this causes to be equal to one.

In summary, the configuration of the body is described in terms of four fields {**x**,*ϑ*,*ξ*,**M**} that are each a function of (**X**,*t*). The velocity and spatial gradient field for **x** was already presented in (2.1). Corresponding to the additional configurational fields there are generalized velocities
2.4
as well as associated gradient fields
2.5
where and are scalars, **g** and **F**^{ξ} are vectors, is a second-order tensor and **F**^{M} is a third-order tensor.

## 3. Balance equations, thermodynamic laws and observer invariance

We adopt the point of view espoused in works such as [12,13] that processes causing changes in the microstructural descriptors (in this case *ξ* and **M**) are to be governed by balance principles that can be framed in terms of field quantities having the same tensor order as the corresponding configuration field. This is of course in keeping with standard continuum mechanics practice with regard to the conventional configurational quantities **x** and *ϑ* which can be regarded as necessitating balance principles that are framed in terms of linear momentum and entropy, respectively. Here, it should be acknowledged that there are varying schools of thought as regards precisely how such balances are to be derived or motivated. Specifically, there are alternative possible routes to derive microstructural balances from more primitive specifications like, for example, observer invariance of some type of power or the use of some extended form of the virtual power principle (e.g. [36]). Leaving aside the intriguing enquiry into matters of such a foundational nature, we shall here pursue an operational approach based on the strategy, consolidated after several well-known applications [12], to assume balance principles guided by a principle of analogy. Accordingly, changes in the content of an appropriate extensive thermodynamic quantity are treated as due to combined and potentially competing effects of either *surface exchange* by a flux through the boundary, or direct supply from a bulk *source* in the external environment, or a bulk internal *production*.

Specifically, following [11] we adopt the following approach: (i) assume standard balance equations for linear momentum and entropy: (ii) postulate direct microstructural balances associated with the new fields *ξ* and **M**; (iii) consider a modified balance of energy that directly acknowledges the influence of the changes in the microstructural descriptors in the expression of the flux and in the source terms [12,36]; and (iv) require that processes respect an observer invariance principle as well as any other physically motivated constraints (such as (2.2)).

This methodology, with its focus on paralleling procedures (i) and (ii), is not meant to support the idea that the structure of the physical laws which govern the chemistry and physics at the fine scale should necessarily replicate those at the macroscopic scale. Rather, it is believed that such a procedure, which could be later streamlined or perfected by adopting a more refined approach, is able to provide a framework that, in conjunction with additional constitutive specifications that are tailored on the basis of the particular material system under consideration, allows for engineering scale analysis of SMA behaviour in a manner that has yet to be fully explored.

At this point, it is useful to remark that chemical conversion between species and diffusion of different species with respect to each other are regarded as negligible effects in the present treatment. It is for this reason that it is not necessary to introduce explicitly a specific balance principle for mass density *ρ*>0. Namely, under the above stipulations any local statement of mass balance reduces to the simple condition . This not only allows for the elimination of *ρ* but also permits any bulk field quantities to be specified equally easily in terms of either mass density or volume density. Of course, if in some particular application, one finds that either chemical conversion or diffusion are no longer negligible, then a non-trivial mass balance would need to be introduced for the determination of *ρ*.

Local forms of all of the balance laws originate from integral statements over arbitrary control volumes. We employ notation as summarized in table 1.

In the mechanical balance, **p** is the linear momentum density, **t** is the nominal mechanical traction and **b** is the body force per unit mass in the reference configuration. No internal production of linear momentum is allowed because that would lead to a violation of observer invariance [25].

In the thermal balance, *η* denotes *entropy density* and can suggestively be thought of as a kind of ‘thermal momentum’ [37]. The associated surface exchange and bulk source are expressed, according to common practice, in terms of the corresponding *heat quantities*: the *surface heat exchange* *h* and the *bulk heat source* *r* after multiplication by the coldness 1/*ϑ*.

Concerning the microstructural balances, *p*^{ξ} and **p**^{M} have an interpretation as transformation momentum and reorientation momentum. In contrast to the standard linear momentum balance, these momenta may generally have non-trivial internal productions *Π*^{ξ} and *Π*^{M}, which can be interpreted as internal *net driving forces* of the transformation and reorientation. The fields *t*^{ξ}, **t**^{M} and *b*^{ξ}, **b**^{M} can be interpreted as external driving forces acting, respectively, across surfaces, due to contact interactions with the neighbouring material elements, and directly on the bulk material. The surface terms are especially useful for an accurate resolution of spatially localized transformation and reorientation [9], whereas the bulk source terms, which are absent in conventional SMA models, may be useful to describe the effect of external actions that can directly drive changes in the microstructural descriptors. This may be especially relevant for the case of transformation and reorientation driven by magnetic fields [10].

By assuming a continuous dependence of all the contact interactions upon the surface normal **n**, the balanced nature of the interactions ensures that a generalized Cauchy theorem holds for all of the surface fields [38]. This provides the existence of referential traction and flux fields such that
Here, **S** is the first Piola–Kirchhoff stress tensor, **q** is the heat flux vector, while the vector **s**^{ξ} and the third-order tensor **S**^{M} are the generalized microstresses with respect to the reference configuration.

Our attention is henceforth restricted to a quasi-static treatment and therefore the rates of change of the contents of all momenta **p**, *p*^{ξ} and **p**^{M} (i.e. macro- and micro-inertia) are neglected. Assuming sufficient smoothness, the local forms that emerge from the integral balance statements for these fields are as follows [11]:
3.1
and
3.2
The two equations (3.1) are standard macroscopic thermomechanical balances. The remaining two equations (3.2) then express the vanishing of the total rate of change of microstructural momenta. These serve to provide the appropriate generalization of the customary transformation and reorientation criteria that make their appearance in phenomenological models for SMA behaviour.

The first of (3.2) is the *phase transformation balance* and contains the sum of internal and external bulk driving forces *Π*^{ξ}+*b*^{ξ} in addition to a divergence term which is essential for treating any localization of transformation. The second of (3.2) is the *reorientation balance* that has an analogous structure but in the form of a second-order tensor balance equation.

Besides basic momentum balances, the intrinsic thermomechanical nature of the SMA behaviour requires the consideration of the first and second laws of thermodynamics. In the present framework, these play the decisive role in determining how constitutive equations can be specified.

The first law is stated in the form of a balance equation for internal energy *e* where the usual expression is generalized so as to take into account the energetic contribution of the changes in the microstructural descriptors [12]. The energy flux and the external energy source are then each additively decomposed into parts such that there is a part for each of the following rate quantities: the velocity of the macroscopic motion **v**, the rates of change of the microstructural descriptors and a remaining thermal part that is independent of these resolved rates. The local statement of energy balance then becomes
3.3

The second law is expressed as a standard requirement that the rate of entropy production *Γ* cannot be negative
3.4

There is a further basic ingredient that plays an important role in the theory, namely the requirement of observer invariance of the balance equations. A detailed enquiry on this issue is given in [11]. There it is shown that the quasi-static set (3.1) and (3.2) is automatically invariant with respect to changes in observers that undergo rigid body motions with respect to each other. In contrast, the energy balance (3.3) generates a non-trivial requirement in order for observer invariance to be satisfied. In particular, while translation invariance applied to (3.3) is automatic, rotational invariance as applied to (3.3) gives rise to the following additional tensor requirement on the various field quantities that have already been introduced [11]: 3.5

As elaborated upon later after (5.5), this same result would follow if an angular momentum balance principle were to be introduced.

## 4. Constitutive assumptions

The source terms **b**, **b**^{M}, *b*^{ξ} and *r* in (3.1)–(3.3) characterize the interaction between the body and the external environment and so are regarded as specified. The remaining stress and internal production terms have to be connected to the configurational fields by means of appropriate constitutive relations that express the special nature of the particular material under consideration. These are developed in terms of two basic ingredients: a *free energy function* that specifies how the material stores energy and the *dissipation function* that describes how processes involving the material dissipate energy [24,26,27]. Then, taking into account thermomechanical restrictions, the other constitutive equations follow as a matter of course.

It is at this point convenient and standard to eliminate the internal energy density *e* from (3.3) in terms of the Helmholtz free energy *ψ*=*e*−*ϑη*. Then a constitutive function for *ψ* is specified in the form
4.1
It is also to be noted at this point that neither the deformation **x** (or the displacement **x**−**X**) nor the temperature gradient **g**=∂*ϑ*/∂**X** is taken to appear as an argument of . Any such possible dependence would eventually be ruled out by second law and observer invariance requirements upon invoking the same argument that forbids such dependence in the standard theory (i.e. continuum mechanics with no microstructural descriptors) [39].

The second basic constitutive ingredient is the dissipation function which describes how the specific processes under consideration produce entropy by a requirement that . Unlike , which is a state function, will depend on the rate at which processes occur. Whereas the idea to specify as an independent constitutive function is old, its systematic introduction in the context of continuum thermodynamics is due to [24]. One of the advantages of this type of constitutive choice is that the enforcement of the second law (3.4) for all admissible processes is met by making this a requirement upon the constitutive function .

In general, the form of must take into account all possible dissipative mechanisms. Here, in order to focus on the aspects associated with the microstructural configurational fields, the contribution to associated with heat conduction is neglected in what follows (although it can be implemented in a straightforward way if desired [11]). More specifically, since experiments confirm that phase transformation and martensite reorientation occur in an essentially rate-independent way [10,31], the corresponding dissipation function is taken to be a positively homogeneous function of the rates of change of the various configurational quantities and their gradients:
4.2
The coefficients of the rates of change of the configurational fields that enter the expression of the dissipation function, namely
4.3
are called *internal resistance functionals*. These functionals are then able to account for the threshold levels that driving forces need to achieve in order to activate the transformations and reorientations.

Each of the internal resistance functionals (4.3) is subject to a constitutive specification in terms of not only the current (**X**,*t*) value of the fields {**F**, *ϑ*, *ξ*, **F**^{ξ}, **M**, **F**^{M}} but also the past history *s*<*t* of these fields at the material particle **X** under consideration. In words, the present value of the rate of entropy production is dependent upon the past process history. While, in principle, this history dependence could be quite general, the assumption of rate independence naturally restricts the way in which entities like the resistance functionals (4.3) can be stipulated in a thermodynamic framework of the type under consideration here [40].

It may be important to point out that even though the dissipation function is itself rate-independent, the overall modelling framework naturally captures rate-dependent effects in SMAs that are a consequence of thermomechanical coupling. Such phenomena arise because of the coupling between the first law (3.3) and the microstructural balances (3.2) that are induced by the temperature dependence of the free energy function (4.1). For example, the theory predicts one type of loading response under isothermal conditions and a separate type of loading response under adiabatic conditions. Then, under conditions of convective heat transfer, the loading response is intermediate between these limiting types, tending towards the former when strain rates are low and to the latter when strain rates are high [29,27].

A constitutive specification that is general enough to faithfully capture this history dependence while providing a relatively easy-to-use framework is obtained by expressing the history dependence of the internal resistance functionals (4.3) completely in terms of the configurational field values at specific past *switching instants*. These are past values of time at which at least one of the field values, or an appropriate norm of a field variable, changed from increasing to decreasing or vice versa [11]. Such a stipulation subsumes many previous theories for shape memory material modelling that have been presented in the literature. For example, many past works, while not developed in the present unified framework, build algorithms for simulating shape memory behaviour that make use of the material state at those past times for which there was a transition from martensite increase to martensite decrease or vice versa [19,20,28]. This is a particularly simple prescription for the determination of switching instants and yet, as indicated by the vast technical literature on phase transition modelling (especially in the context of one-dimensional shape memory effects), it gives rise to modelling frameworks that are capable of replicating a range of observed loading behaviours that exhibit complicated internal sublooping [29].

With these ingredients in place, the general form for all of the constitutive equations follows by exploiting the first law of thermodynamics along the lines of [24] as explained in detail in [11]. In brief, the procedure involves writing the time derivative of the free energy from the energy balance (3.3) in terms of the time derivative of *ψ* using (4.1). Subsequent simplification using the thermal balance (3.1)_{2} results in an expression that can be manipulated so that *Γ* appears on one side of the equality. Referring to this as the dissipation requirement, the combined requirements of and (4.2) then allow one to extract the additional constitutive equations. This is accomplished by invoking the relatively general nature of in an admissible physical process.

The first group of derived constitutive relations that are obtained in this way are the expressions for the entropy density and the Piola–Kirchhoff stress in the form
4.4
It also emerges from this argument that **q** is constrained in its relation to **g** in order for the inequality relation (3.4) to hold for all admissible processes. The standard way of meeting this condition is to stipulate the Fourier law type relation **q**=−*ρ***K****g**, where **K** is a second-order positive definite heat conduction tensor that may itself depend upon the configurational fields. Other ways of meeting the condition can also be considered, leading to more generalized theories of heat conduction. This group of constitutive relations now permits the standard thermomechanical balances (3.1) to be expressed in terms of the configurational fields.

The remaining group of derived constitutive relations are those associated with the phase transformation and reorientation balances (3.2). These are obtained by considering variations in the rates and from this one obtains 4.5

Here, and are reactive fields associated with the constraint on **M** in (2.2). For the specific condition (2.2), these constraint fields express themselves in terms of a scalar reactive production *a* and a vector reactive microstress **c** such that
4.6
Thus *a* and **c** enter the treatment as field quantities in their own right. This aspect parallels the way in which a general hydrostatic pressure, say *p*, would enter the treatment if the condition of overall volume preservation det **F**=1 were to be imposed. Indeed, if the condition of volume preservation were to be imposed then the Piola–Kirchhoff stress **S** in (4.4) would be amended so as to include the standard constraint stress term **S**_{0}=−*p***F**^{−T} on the right-hand side.

Also, with these constitutive stipulations in place, the energy balance (3.3) is identically satisfied for any admissible process and therefore it is no longer an independent condition. In going forward, the thermal balance (3.1)_{2} is now typically referred to as the heat equation and plays a decisive role in the determination of the temperature evolution.

The treatment has yet to address the issue of how the constitutive functions may depend upon **F**, **M** and **F**^{M} so as to provide necessary and sufficient conditions for them to be observer invariant. For a standard material in which there is no dependence upon **M** or **F**^{M}, and hence only a dependence upon **F**, the well-known result is that the dependence of the Helmholtz energy upon **F** can only be through **C**=**F**^{T}**F**. In the present more generalized framework, the same type of argument [11] leads to the conclusion that the dependence of the Helmholtz energy upon second-order tensors **F**, **M** and third-order tensor **F**^{M} must now be through the second-order tensors **C**, **H**^{M} and third-order tensor **H**^{FM} defined as follows:
4.7
Let be used to denote the state function for *ψ* under this more restricted dependence, i.e.
4.8
This connection also presumes that is expressed as a symmetric function of **C**. Under this dependence, the various derivatives with respect to {**F**,**M**,**F**^{M}} in (4.5) can now be re-expressed in terms of derivatives with respect to {**C**, **H**^{M}, **H**^{FM}} using the connections
4.9
and
4.10
Parallel considerations hold relative to the dissipation function (4.2). Namely, objectivity now requires that the rate of entropy production can be specified in terms of a constitutive function taking the form
4.11
In addition, each resistance functional {*Λ*^{C},*Λ*^{ξ},*Λ*^{Fξ},*Λ*^{HM},*Λ*^{HFM}} must now be expressible in terms of the time history of the fields {**C**,*ϑ*,*ξ*,**F**^{ξ},**H**^{M},**H**^{FM}}. This history dependence is again taken to be through appropriately defined *switching instants*. Furthermore, if one seeks a model in which the dissipation is due exclusively to changes in phase fraction and reorientation, then one may take a vanishing *Λ*^{C} . If *Λ*^{C} is non-vanishing, then it is defined so as to be symmetric.

Expressing the time derivatives in terms of it follows that a comparison of (4.2) with (4.11) confirms that {*Λ*^{ξ},*Λ*^{Fξ}} preserve their original meaning, and at the same time gives the following connections:
4.12
and
4.13
Going forward, we invoke an assumption that all dissipation is associated with changes in the microstructural fields by taking *Λ*^{C}=**0**.

## 5. Final equations and concluding remarks

Summarizing, the generalized constitutive framework presented here, and following [11], is specified in terms of the following ingredients:

— A free energy function of the type .

— Four internal resistance functionals {

*Λ*^{ξ},*Λ*^{Fξ},*Λ*^{HM},*Λ*^{HFM}} that characterize the dissipation function via (4.11) with*Λ*^{C}=**0**. These may depend, in general, on the current values of the fields {**C**,*ϑ*,*ξ*,**F**^{ξ},**H**^{M},**H**^{FM}} along with the values of these fields at specially identified past switching instants. This gives the possibility of an independent constitutive modelling of the forward and reverse transformations and reorientations.

The objective fields **H**^{M} and **H**^{FM} which appear in these constitutive relations are related to the original configurational field **M** and its gradient **F**^{M} via (4.7).

It is also necessary to specify a condition of heat conduction that relates **g** to **q** in a manner that is also consistent with (3.4). A standard treatment is then posed in terms of a positive definite heat conduction tensor **K**=**K**(**C**,*ϑ*,*ξ*,**F**^{ξ},**H**^{M},**H**^{FM}). The entropy and heat flux are then given by
5.1
whereas the Piola–Kirchhoff stress is given by
5.2
These are the functions that participate in the standard thermal and linear momentum balance equation (3.1).

The constitutive relations for the microstructural fields (4.5) then take the form
5.3
and
5.4
where and are given by (4.6). In (5.3) and (5.4), the fields *Π*^{ξ} and *Π*^{M} characterize the *net driving force* associated with internal rearrangements of the microstructure. In fact and yield the free energy release in the bulk that is associated with changes in *ξ* and **M**, respectively. Conversely, *ϑΛ*^{ξ} and *ϑ**Λ*^{HM} yield the internal resistance (i.e. the driving force consumption) encountered during transformation and reorientation. The microstresses **s**^{ξ} and **S**^{M}, which characterize actions on surface elements, contain terms describing a similar partitioning into energy release and energy consumption.

The results (5.1)–(5.4) are expressed in terms of objective constitutive functions. This allows one to rewrite the base expression appearing in (3.5) as 5.5 Upon making use of (4.6), it follows from the above expression that the objectivity condition (3.5) is now automatically satisfied.

The constitutive relations as specified above can be used directly in the balance equations (3.1) and (3.2). These are typically thought of as acting with respect to the reference configuration. It is often useful, and standard practice, to push the usual macroscopic balance equations (3.1) forward to the current configuration. Then instead of dealing with Piola–Kirchhoff stress **S**, one works with Cauchy stress **T**=*J*^{−1}**S****F**^{T}, and the form of this Cauchy stress immediately follows from (5.2). Now, because of the presence of the type of martensite **M** and its gradient **F**^{M}, the Cauchy stress **T** is not generally symmetric. This is the usual state of affairs in theories of structured continua [12,13]. It is also to be remarked that the development has not specifically posited a principle of angular momentum balance. If one were to be posited, it would simply replicate the previously stated (3.5) which was obtained here by requiring observer invariance of the energy balance principle. Then, a separate requirement that the constitutive theory is to be objective restricted the dependence of the constitutive functions upon **F**, **M** and **F**^{M} to be in terms of **C**, **H**^{M} and **H**^{FM}. This gives (5.5) which in turn ensures that (3.5) is automatically satisfied [41].

Taking into account all of the above constitutive stipulations, the phase transformation and reorientation balances (3.2) finally become
5.6
and
5.7
These equations provide the transformation and reorientation criteria that are basic for the description of the response of SMAs and involve four types of terms. One type, depending on the free energy , is associated with the *internal energy release*; a second type, arising from the dissipation function *Γ* via the various ** Λ** terms, provides the

*internal*

*resistance*; a third type is associated with the

*reactions*of the internal constraints; and, finally, there is a source term in each balance that describes the

*external actions*which directly influence either the phase transformation or the reorientation.

The presence of the terms associated with the derivatives of the free energy with respect to **F**^{ξ} and **H**^{FM}, as well as the corresponding terms *Λ* arising from the dissipation function, renders the criteria (5.6) and (5.7) differential equations in the fields of microstructural descriptors *ξ* and **M**. This feature offers the possibility to describe the occurrence of localized phase fraction profiles. A special case of this type of modelling has been presented in [15] with reference to SMA bars where a satisfactory agreement with experimental results is also reported.

Accordingly, (5.6) and (5.7) generally require that boundary conditions be specified in order for the resulting mathematical problem to be well posed. In this context, and in the absence of any specifically identifiable external agency whose action is restricted to a boundary surface, the natural boundary conditions would be a requirement of **s**^{ξ}⋅**n**=0 for (5.6) and **S**^{M}**n**=**0** for (5.7). Note also that the above equations are written relative to the reference configuration. However, when it is convenient, each equation could be pushed forward to the current configuration. In this process appropriate current configuration microstresses will emerge in place of **s**^{ξ} and **S**^{M}.

The external actions that enter transformation and reorientation criteria can be used to model interactions with the environment that may directly influence the phase transformations. While the term *b*^{ξ} is likely to be zero in usual cases, the term **b**^{M} could be specified as a function of an externally applied magnetic field in the case of ferromagnetic SMAs where reorientation phenomena can be activated due to different preferred directions of magnetization of the various martensite variants [10].

If the microstructural gradient terms associated with **F**^{ξ} and **H**^{FM} are dropped from (5.6) and (5.7) then these equations become algebraic in nature. If in addition the source terms *b*^{ξ} and **b**^{M} are also dropped then the balance statements (5.6) and (5.7) simplify to algebraic criteria of the type *Π*^{ξ}=0 and *Π*^{M}=**0**. This means that the bulk energy release is completely accounted for by the need to overcome a corresponding internal resistance, both to initiate and to sustain the microstructural evolution. The very early models [19,42] can each be viewed as a manifestation of such an algebraic balance. Specifically, the model in [19] implements an algebraic transformation equation for describing the austenite to martensite phase transformation in one spatial dimension. The resulting model was among the first to describe transformation hysteresis in SMA wires. Also in a small strain setting, but now with respect to three-dimensional states of strain, the model described in [42] can be viewed as a specification of a martensite reorientation kinetic as could be extracted from a small strain version of (5.7) in the absence of the divergence terms. The tensor nature of such an algebraic reorientation kinetic serves to establish a yield surface type of description. In addition, the dependence on switching instants in the dissipation function now serves to establish the specific hardening rules associated with the yield surface evolution.

Limitations of space in this article preclude the possibility of developing a detailed example in the context of specific functional forms for the constitutive entities that define this modelling procedure. Examples of this type are currently under development and will be reported in future publications. A simple preliminary example of an explicit constitutive model developed within this framework can be found in [11], §9.10.

Since these early works, a great deal of additional modelling development has taken place, as described in the previously indicated reviews [2–5]. Here, we have described a further generalization based on treating SMA bodies as structured continua with an underlying microstructure described by the total martensite fraction *ξ* and the type of martensite tensor **M**. This framework naturally accounts for both gradient effects and external source agencies capable of direct action at the microstructural level. The former offers a natural means for a more careful resolution of spatial localization with regard to both phase transformation and specialized patterns of martensite reorientation. The latter provides a means for treating transformations driven by non-mechanical external agents including magnetic SMAs. Additional features inherent in this framework, including a specific means for modelling complex internal sublooping arising from a range of thermomechanical inputs, are a consequence of the modelling flexibility afforded by the consideration of internal resistance functions with a memory of the configuration fields evaluated at past switching instants. These generalized features make this a promising tool for the development of specific material models that can be used to analyse a variety of engineering structures and processes.

## Authors' contributions

Both authors contributed equally to this work.

## Competing interests

We have no competing interests.

## Funding

We received no funding for this study.

## Footnotes

One contribution of 11 to a theme issue ‘Trends and challenges in the mechanics of complex materials’.

- Accepted January 20, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.