Javier J Gutiérrez
email: javierj@us.es

Clémentine Nebut
email: nebut@lirmm.fr

María J Escalona
email: mjescalona@us.es

Manuel Mejías

Isabel M Ramos
email: ramos@us.es

Visualization of Use Cases through Automatically Generated Activity Diagrams

Functional requirements are often written using use cases formatted by textual templates. This textual approach has the advantage to be easy to adopt, but the requirements can then hardly be processed for further purposes like test generation. In this paper, we propose to generate automatically through a model transformation an activity diagram modeling the use case scenario. Such an activity diagram allows us to guess in a glimpse the global behavior of a use case, and can easily be processed. The transformation is defined using the QVT-Relational language, and is illustrated on a case study using a supporting tool.

Introduction

Requirement engineering is a crucial activity for a successful project. Among the different tasks to achieve in this activity is the writing of the elicited requirements in a given language (formal, semi-formal, or natural).

Structured use cases is a classical standard used in industry and taken as entry point of several research work [START_REF] Cockburn | Writing Effective Use Cases, 1st edn[END_REF][4] [START_REF] Escalona | Models and Techniques for the Specification and Analysis of Navigation in Software Systems[END_REF] [START_REF] Ben Achour | Writing and Correcting Textual Scenarios for System Design[END_REF]. By structured use cases, we mean use cases as defined in UML [START_REF]The UML Superstructure[END_REF], enhanced with additional information like pre and post conditions, and scenarii. Such use cases usually follow templates like Cockburn one [START_REF] Cockburn | Writing Effective Use Cases, 1st edn[END_REF], defining in natural language usage scenarii, quality of service constraints, or graphical interface elements, like [START_REF] Dustin | Quality Web Systems. Performance, Security, and Usability[END_REF].

A UML use case model does not contain much information. It allows to define the frontier of the system, actors, name of use cases, and relations between the use cases, the actors, and linking actors and use cases. UML methodologies like RUP [START_REF] Jacobson | The Unified Software Development Process, USA. Object Technology[END_REF] or Catalysis [START_REF] Souza | Objects, Components, and Frameworks with UML: The Catalysis(SM) Approach[END_REF] recommend to enhance this model with other UML artifacts like constraints to define contracts, and sequence diagrams to define system scenarii in a graphical way.

Both use case templates and UML methodologies thus incite the requirement writer to define scenarii, in a graphical or textual notation. Graphical notations for scenarii (system sequence diagrams [START_REF] Labiche | A UML-Based Approach to System Testing[END_REF] or activity diagrams [START_REF] Hartmann | TDE/UML: A UML-based Test Generator to Support System Testing[END_REF]) have the advantage to give in a glimpse a good idea of the dynamic aspects of a use case (loops, alternatives, choices). Moreover, they can be easily used by a tool to be analyzed (for example for test generation purposes, also in [START_REF] Labiche | A UML-Based Approach to System Testing[END_REF]). On the other hand, they can be difficult to design, and domain experts cannot reasonably asked to draw them. On the opposite, textual scenarii are easy to write: the scenarii are composed of various steps, each one is described by natural language sentences, and linked with natural language connectors. Such textual scenarii do not give an immediate intuition of the behavior of the use case, and cannot easily be processed by a tool.

The original contribution of this paper is a process to generate in an automated way an activity diagram modeling use case dynamic from textual scenarii. The generation is performed by a model transformation, taking as input use case textual scenarii (conform to a metamodel defined in this paper), and producing an activity diagram (conform to a restricted part of the UML activity diagram metamodel). The transformation is defined using QVT-relational language and implemented in Java in a prototype tool.

This transformation allows to benefit both from graphical and textual scenarii advantages. Textual scenarii can be written. Then they can be transformed in a graphical view. This allows an easier integration to UML tools, and to use the activity diagram for further treatments like test generation for example.

The rest of this paper is organized as follows. Section 2 introduces the source and target metamodels, this means, the textual requirement metamodel and an excerpt of the UML activity metamodel. Section 3 describes the transformation to generate an activity diagram from a tabular requirement. Section 4 introduces a case study. Section 5 describes other related approaches and, finally, section 5 summarizes conclusions and ongoing works.

Functional Requirements and Activity Diagrams Metamodels

This section introduces the artifacts we deal with: functional requirements and activity diagrams. Both ones are described by metamodels: our own metamodel for functional requirements and a subset of the UML metamodel for the activity diagrams.

Functional Requirement Metamodel

There are many approaches to define textual requirements [3][4][5] [START_REF] Ben Achour | Writing and Correcting Textual Scenarios for System Design[END_REF]. However the majority of the existing approaches are based on a common core of elements. These elements include the participants, preconditions, post-conditions, a set of main steps (to achieve the goal of the use case) and a set of exceptional steps (to manage alternative and erroneous scenarios). These common elements have been extracted from the cited approaches and modeled in the functional requirement metamodel introduced in figure 1 and briefly described in the following.

A FRActor element models an external actor who participates in the functional requirement performing the steps. The FunctionalRequirement element defines an interaction among the system and a set of external actors. The behavior of Function-alRequirement is defined with three different sequences of steps. The main sequence defines the steps performed by FRActor elements and the system to achieve the goal of the functional requirement. The alternative steps indicate variants or additional behavior for the steps of the main sequence. The erroneous steps indicate erroneous scenarios after performing a step from the main sequence. Alternative and erroneous steps are modeled with ExceptionalStep element. In this paper, the only relevant information for a MainStep element is the action performed (attribute action) and who performs the action. In an ExceptionalStep element, the action is not mandatory and the performer is the same one than the main step. The exceptionCondition attribute indicates a boolean expression that must be true to perform the step and the result attribute indicates the action to perform at the end of the exceptional step. Three possible results have been defined as the enumerated ResultType element: end the execution, continue the execution and repeat/go-to to another step. A continue result indicates that the execution of the functional requirement continues with the next step after the end of the alternative or erroneous step. The repeat or go-to result indicates that the execution of the scenario follows the execution of the step indicated in the nextStep attribute, and the end result indicates than the use case scenario ends. The nextStep attribute is only mandatory when the result type is repeat / go-to. An end result indicates that the execution of the scenario is finished. We have defined a concrete syntax to define functional requirements conform to this metamodel (figure 2). The concrete syntax is XML-based to make an approach available. One tool, available in [START_REF]Supporting web site[END_REF], has been development to generate this XML for our modeling tool (described in section 4).

Activity Diagram Metamodel

The UML activities metamodel contains a vast amount of elements which improves the flexibility and semantic of activity diagrams. A subset of the activity diagrams metamodel (as defined in UML 2.0) has been selected (figure 3). This subset defines the elements automatically generated by the transformations introduced in next section. Indeed, after the automatic generation of the activity diagrams, additional elements, like data flow, may be added by hand. Next section, describes how these elements represents the concepts defined in the functional requirement metamodel.

Transformation from Textual Scenarios to Activity Diagrams

This section defines the transformation to generate an activity diagram (as an instance of the metamodel of figure 3) from a functional requirement (as an instance of the metamodel of figure 1). The objective of this transformation is to generate from a functional requirement an activity that models the whole functional requirement (with the same name, pre-conditions and post-conditions). In this activity, the activity partitions model the different actors implied in the functional requirement and the system under development, control flows model the execution paths or scenarios and actions and decision nodes model the steps and the exception conditions. Transformation performs next tasks:

1. generating an activity for each functional requirement, 2. generating an action for each main step and generate a decision node and additional actions for each exceptional step, 3. all elements in the activity diagram are connected using control flows.

All the tasks are defined using QVT-Relational [START_REF]Meta Object Facility (MOF) 2.0 Query/View/Transformation Specification[END_REF]. We here present only the main ones in QVT-Relational; the whole transformation can be found in [START_REF]Supporting web site[END_REF]. The steps are described in detail in next sections.

From Functional Requirements to Activity Diagrams

Initially, one activity is generated for each functional requirement. The related relation (called FR_2_AD), in QVT-Relational language, is showed in figure 4. The name, preconditions and post-conditions of the activity are the same than the ones in the functional requirement (lines 17, 18 and 19). As seen in the previous section, one activity partition is also generated for each actor participant in the functional requirement (relation Actor_2_ActivityPartition, called in line 22) plus one additional partition for the system (as showed in figure 5(a), lines from 42 to 49). The next task is to transform the behavior, defined with three sequences of steps, of the functional requirement.

From Main Sequence to Actions

One action is generated for each step in the main sequence. The related QVT relation is given in figure 5(b). The attribute effect of the action is the attribute action of the step (line 67). The actions are included in the corresponding activity partition which may be identified using the performedBy attribute of the main step (lines 69, 70 and 71). However, if a step is performed by the system, this attribute is empty, so the action is classified in the system partition. This behavior is codified in the auxiliary function APName (line 76, figure 5(b)). The next step is to transform alternative and erroneous sequences in decision nodes and actions.

From Alternative and Erroneous Sequences to Actions

Two elements are generated from an alternative or an erroneous step. The QVT relation goes in figure 6.

The first element is a decision node (class DesicionNode, line 101), which evaluates the exception condition to allow or not the execution of the alternative or erroneous behavior. This exception condition is stored in the attribute desicionInput, which is generated with the auxiliary function generatePredicate (used in line 103). Due the exception condition was defined as constraints in the functional requirement metamodel (figure 1), this auxiliary function builds a string from a set of Constraint elements.

The second element is an action describing the additional behavior if any (as it has been seen in the functional requirement metamodel, an action attribute for exceptional steps is not mandatory). This action is generated by the auxiliary function Gener-ateAction (line 113). This function has been omitted from figure 7, but its code is quite similar to the QVT transformation MainStep_2_Action (figure 5

(b)).

Then, every possible result has also a different representation in an activity diagram, so three different relations are called to generate the adequate elements for each type of results (only one, relation GenerateEnd, has been included in the where clause of figure 6, line 115).

Fig. 6. From exceptional step to decision node

Fig. 7. Exceptional step of type end

With an end use case result, a new activity end node (class ActivityFinalNode) is added and it is linked with the decision (or action if any) generated in previous relation using a new ControlFlow (from line 124 to 136, figure 7).

For a continue use case, the action corresponding to the next step (in the main sequence) is obtained and linked with the decision (or action if any) using a control flow element. Finally, for a repeat/ go-to result type, the action corresponding for the next step attribute (see metamodel in figure 1) is obtained and linked with the decision (or action if any) using a control flow.

The last task is to add control flow elements among the elements that are still unlinked.

Linking Element with Control Flows

At this time of the transformation process, all elements have been created in the activity diagram, but only a few of them have been linked with other elements (the elements created in section 3.3). The final task is to link all the other elements with ControlFlow instances. No control flow may be added until all the elements are generated because the control flow source and target is different depending if the actions have been generated from main steps with or without alternative and erroneous steps (a step-by-step example is provided in next section).

This task is, mainly, an algorithmic task. However, it has been possible to define a set of relations to achieve every specific situation at the time to link the elements. These relations are briefly described in next points.

Initial node with first element and activity final node with last element

If the first main step of the functional requirement has not an alternative step, then, the initial node is linked with the action generated from the first step. In other case, the initial node is linked with the decision node generated from the exception condition of the first alternative. This relation is showed in figure 8. The same process is applied in the relation to link the last element in the activity diagram with an activity final node.

Decision nodes among them

As seen in the functional requirement metamodel (figure 1) all exceptional steps (alternative and erroneous steps) are ordered. So, the next task is to link the decision node generated from the first alternative step of a main step with the decision node generated from the second alternative step of the same main step, the decision node generated from the second one with the decision node generated from the third one, etc. This process is also done with the decisions nodes generated from the erroneous steps.

Decision nodes with activities

Next, the decision node generated from the last alternative step of a main step (if any) must be linked with the action element generated from this main step. In the same way, this action element must be linked with the decision node generated.

Nodes from one step with nodes from other step

Finally, the last decision node generated from the last erroneous step of the first main step is linked with the first decision node generated from the first alternative step of the second main step, and so on.

At this time, the activity diagram is finished.

Case Study

This section describes a case study of the transformation defined in previous sections. The case study chosen is an incident report system. This system is running in a public organism in Spain. Use cases were written one year ago using NDT approach [5][6], which contains all the common elements of the metamodel in figure 1. No additional information was needed to apply our transformation. This system has 9 use cases, however, two of them have been discarded since they describe interactions among a concrete digital sign service that is not available for us. The total number of steps is 46. One activity diagram has been automatically generated from each use case. The total number of activity nodes is 41 and the total number of decision nodes is 15. The complete results of this case study can be found in [START_REF]Supporting web site[END_REF]. One use case has been chosen (Login, from figure 2) to illustrate the transformations described in previous section.

To be able to apply our approach, we have first rewritten the use cases using our XML syntax. The transformations in previous sections have been implemented in a Java open-source tool (also available in [START_REF]Supporting web site[END_REF]). The input is a XML file as the one showed in figure 2 (generated automatically from Sparx Enterprise Architect tool) and the output is a XMI activity diagram file. In following figures, the activity diagram has been loaded also with Sparx Enterprise Architect tool.

the system show s and error message.

[Not(the number of tries is greater than three) ]

[the number of tries is greater than three.]

[the name of the key size is less than four characters.]

[the name or the key are not registered.] Fig. 10. From exceptional steps to decision nodes and actions

All the elements generated from the login functional requirement are showed in figure 9(a). The result of the relations introduced in sections 3.1 and 3.2 are showed in the incomplete activity diagram of figure 9(b). As described before, one activity, activity partitions and actions generated from main sequence have been added. (from RF-01. Logging)

[Not(the number of tries is greater than three) ]

[the number of tries is greater than three.]

[the name of the key size is less than four characters.]

[the name or the key are not registered.]

[Not(the name of the key size is less than four characters)]

[Not(the name or the key are not registered)]

Fig. 11. Final activity diagram

The result of the relation introduced in section 3.3 is showed in the incomplete activity diagram in figure 10. As described before, a decision node and an action (if any) have been added for each alternative and erroneous step and some elements have been linked with control flows.

Finally, the result of relations described in section 3.4 is showed in the activity diagram in figure 11 and, now, this activity diagram is complete. As described before, all the elements of the diagram, plus an initial and final node, are linked using Control Flow element.

Related Works

At the time to write this paper, few works exists dealing with automatic processing of textual requirements from a model-based point of view. Reference [START_REF] Baudry | Model-driven Engineering for Requirement Analysis[END_REF] introduces a requirement metamodel and a set of transformations (in Kermeta language) to build an executable model from the requirements (defined as a labeled transition system). Functional requirements are treated as black boxes and only pre-conditions and postconditions are used to generate the executable model. On the opposite, our approach focuses in the scenarios of each use case and not in the ordering of use cases. In [START_REF] Koch | Model Transformations from Requirements to Web System Design[END_REF] a complete requirement model for web system, called WebRE, is introduced. This new model merges two previous approaches: NDT [START_REF] Escalona | Models and Techniques for the Specification and Analysis of Navigation in Software Systems[END_REF] and UWE [START_REF] Koch | Software Engineering for Adaptative Hypermedia Applications[END_REF]. However, this model introduces web specific concepts like the browser or the links and navigation, so it is not as generic as the approach introduced in this paper.

There are also approaches which extend activity diagrams to describe use case behavior more precisely. For example, in [START_REF] Kösters | Coupling Use Cases and Class Models as a Mean for Validation and Verification of Requirements Specifications[END_REF] actions are stereotyped to indicate if they are performed by an actor, by the system or reflect inclusions and extensions. In our approach this goal is achieved using activity partitions. Another related work may be found in [START_REF] Almendros-Jiménez | Describing Use Cases with Activity Charts[END_REF]. This paper presents a method for describing use cases with activity charts. Support for inclusion and extension is provided. The main difference with our approach is that [START_REF] Almendros-Jiménez | Describing Use Cases with Activity Charts[END_REF] works with UML Use Case, but not with textual templates. This fact imposes that activity diagrams describes a sequence of use cases, instead of the behavior of one use case. Other papers about modeling sequences of use cases are [START_REF] Labiche | A UML-Based Approach to System Testing[END_REF] and [START_REF] Nebut | Automatic Test Generation: A Use Case Driven Approach[END_REF], which are test-generated oriented.

Another example is [START_REF] Boddu | RETNA: From Requirements to Testing in Natural Way[END_REF], which uses language analysis techniques to generate statecharts diagrams from unformatted textual requirements. Reference [START_REF] Fröhlich | Automated Test Case Generation from Dynamic Models[END_REF] describes a systematic but manual process to build state-charts from textual requirements. Reference [START_REF] Ziadi | Revisiting Statechart Synthesis with an Algebraic Approach[END_REF] defines an algebraic framework for composing state charts and shows how to leverage the algebraic structure of UML sequence diagrams to get a direct algorithm for synthesizing a composition of state charts out of them. In this paper, scenario diagrams are used to define scenarios, as an artifact to model functional requirement. This process goes from specific (a concrete scenario) to general (a statechart to describe the whole behavior) while work starts from general (the behavior of a whole scenario). In our approach, scenarios may be defined as a path obtained from the generated activity diagram.

Conclusions

This paper introduced a metamodel for functional requirements and a set of QVT relations to combine functional requirements in the form of use cases with activity diagrams (as described in UML 2.0) in an automatic way. These transformations allow requirement engineers and system users to work with the most valuable representation at any time. However, some drawbacks appear at the time of combining textual requirements with activity diagrams. The most obvious one is the loss of flexibility, for example at the time of adding new kind of result types. The task of adding a new result type implies add a new set of transformation for implementing the right activity diagram fragment for that new result and modify existing transformations for detecting the new result.

The representation of the functional requirement is not limited to XML. This paper has introduced a fragment of XML due it is the input of our supporting tool, as mentioned above. Other implementations may use different representations and it is even possible to design adapters. For example, in the supporting web site [START_REF]Supporting web site[END_REF], a simple Java program to obtain XML from Enterprise Architect project file is provided.

As mentioned in previous sections, the functional requirement metamodel includes a common subset of the elements that may be found in existing functional requirements approaches. The main advantage is that the metamodel may be used with any of those approaches with very little or no changes. Furthermore this metamodel may be easily extended to include additional information like priority, stability of requirements, performing issues, traceability, etc.

Although the transformation process has been defined with QVT-Relational, it has been implemented in Java language. Java allowed us to implement a prototype (also available in [START_REF]Supporting web site[END_REF]) in a shorter time than existing implementations of QVT-Relational or other transformation languages like ATL. However, one of our main ongoing works is to develop a stable version of the tool using one of the cited transformations languages.

Actually, we use the transformations described in this paper as a first step to generate test cases from use cases automatically. Transforming the textual requirements into activity diagrams allows us to apply graph coverage criteria, like covering all nodes or covering all flows, to generate concrete usage scenarios easily. Some papers describing our preliminary work (which was not formalized with metamodels and transformation) are [START_REF] Gutiérrez | Derivation of test objectives automatically[END_REF][8] [START_REF] Gutiérrez | Generación automática de objetivos de prueba a partir de casos de uso mediante partición de categorías y variables operacionales[END_REF]. Two additional case studies with empirical results are introduced in [START_REF] Gutiérrez | A Case Study for Generating Test Cases from Use Cases[END_REF].

Fig. 1 .Fig. 2 .

 12 Fig. 1. Functional Requirement Metamodel

1 Fig. 3 .

 13 Fig. 3. Activity diagrams metamodel

Fig. 4 .

 4 Fig. 4. From functional requirement to activity

Fig. 5 .

 5 Fig. 5. (a) From actor to activity partition. (b) From main step to action.

Fig. 8 .

 8 Fig. 8. From initial node to first element

Fig. 9 .

 9 Fig. 9. (a) Elements generated from the functional requirement. (b) Actions from main step.

Acknowledgements

This work is partially supported by the Ministry of Science and Education (Spain) under the National Program for Researching, Development and Innovation, project QSimTec (TIN2007-67843-C06-03) and REPRIS (TIN2005-24792-E). Authors also wants to remark the help provided for the student who developed the system used in the case study.


Philippe Hannequart

Michaël Peigney
email: michael.peigney@polytechnique.org

Jean-François Caron

A Micromechanics-based Model for Polycrystalline Ni-Ti Wires

Keywords: Shape memory alloy, wire, polycrystal, micromechanics, modelling, analytical solution

This paper proposes a constitutive model for polycrystalline shape memory alloy (SMA) wires arising from micromechanical arguments. The texture of the polycrystal is captured through the volume fractions and the maximal transformation strain in each crystalline orientation. As a result, the model is able to reproduce texture effects such as nonlinear hardening during phase transformation. An attractive feature of the proposed model is that closed-form expressions of the material response can be obtained for typical thermomechanical loadings of interest in SMA, such as cyclic traction at high temperature or thermal cycling at a fixed stress. Those analytical solutions are notably useful for identifying the constitutive parameters of the model. A temperature-controlled testing apparatus for SMA wires was developed for performing a reliable characterization of Nickel-Titanium wires. All model parameters have been identified by means of three tests: differential scanning calorimetry, isothermal traction test and thermal cycling at constant stress.

Introduction

The specific properties of shape memory alloys (such as the superelastic behavior and the shape memory effect) are the result of a diffusionless solid/solid phase transformation between different crystallographic structures (austenite and martensite). From their discovery in the 1930s until today, shape memory alloys (SMA) have slowly become a widely used industrial material, with applications in various fields of engineering such as aeronautics, robotics, biomedicals or civil engineering. However, the great capabilities of this material are rarely fully exploited. For example, when the shape memory effect is used in actuators, there are often only two operating positions, a "cold" position and a "hot" position. The intermediary shapes of the SMA are rarely used. There are several possible reasons for this, such as the difficulty to manufacture the correct material, the nonlinear behaviour of SMAs (which makes it difficult to use simple control laws) and the lack of accurate thermomechanical models to correctly describe the phase change of SMAs.

A strong research effort has already been put into modeling of SMAs, both in the one-dimensional and three-dimensional settings. Roughly speaking, SMA models fall in two main categories: phenomenological macroscopic models and micromechanical models. Phenomenological macroscoping models draw inspiration from plasticity by using inelastic strains as internal variables [START_REF] Auricchio | Shape-memory alloys: macromodelling and numerical simulations of the superelastic behavior[END_REF][START_REF] Boyd | A thermodynamical constitutive model for shape memory materials. part i. the monolithic shape memory alloy[END_REF][START_REF] Leclercq | A general macroscopic description of the thermomechanical behavior of shape memory alloys[END_REF][START_REF] Chemisky | Constitutive model for shape memory alloys including phase transformation, martensitic reorientation and twins accommodation[END_REF][START_REF] Souza | Three-dimensional model for solids undergoing stress-induced phase transformations[END_REF][START_REF] Tanaka | Thermomechanics of transformation pseudoelasticity and shape memory effect in alloys[END_REF][START_REF] Zaki | A three-dimensional model of the thermomechanical behavior of shape memory alloys[END_REF]. In micromechanical models, the internal variables (usually the volume fractions of the different phases) are directly related to the austenite-martensite microstructures that develop in the material [START_REF] Peigney | Numerical simulation of shape memory alloys structures using interior-point methods[END_REF][START_REF] Sagar | Contributions on the theory and computation of mono-and polycrystalline cyclic martensitic phase transformations[END_REF][START_REF] Bernardini | Models for one-variant shape memory materials based on dissipation functions[END_REF][START_REF] Govindjee | A multi-variant martensitic phase transformation model: formulation and numerical implementation[END_REF][START_REF] Stupkiewicz | Modelling of laminated microstructures in stress-induced martensitic transformations[END_REF][START_REF] Nae | Micromechanical modeling of polycrystalline shapememory alloys including thermo-mechanical coupling[END_REF][START_REF] Kelly | A micromechanics-inspired constitutive model for shape-memory alloys that accounts for initiation and saturation of phase transformation[END_REF]. Compared to phenomenological models, micromechanical models provide more information on the material behavior: in addition to the macroscopic stress-strain-temperature relation, they also give some insight in the microstructures that appear at a microscopic scale. However, micromechanical models are more difficult to use and to implement in a finiteelement code, partly because the number of internal variables is larger (and grows proportionally with the number of crystalline orientations). A third category of models that needs to mentioned is Landau/Landau-Ginzburg type models (see e.g. [START_REF] Song | Effect of microstructure on the hardening and softening behavIors of polycrystalline shape Memory alloys Part I: Micromechanics constitutive modeling[END_REF] and references therein). Those models are phenomelogical in nature, but they are able to describe the movements of martensite-martensite and austenite-martensite domain boundaries.

In a wide range of applications (such as actuators), SMAs are used in the form of wires loaded in tension. For such a simple geometry, the most widely used phenomenological models predict a piecewise linear response. This is in contradiction with experiments: for instance, experimental stress-strain curves at low temperature show some nonlinear hardening. It is possible to introduce nonlinear hardening in phenomenological models (as done in plasticity for instance), but this does not explain the physical original of that phenomenon. Numerical simulations based on three-dimensional models [START_REF] Anand | Thermal effects in the superelasticity of crystalline shape-memory materials[END_REF] show that the origin of the nonlinear hardening lies in the polycrystalline nature of the material: Phase transformation initiates in grains which are the most favorably oriented with respect to the loading, and progressively extends to the least favorably oriented grains. The corresponding simulations are computationally intensive as they require to consider a large number of grains for capturing the nonlinear hardening in a convincing manner.

In this paper, our objective is to propose a micromechanical model of SMA wires that: (i) is able to capture the mechanical stress-induced nonlinear hardening in relation with the polycrystalline texture, (ii) remains of simple use, in the sense that closed-form solutions can be derived. To fulfill those objectives, we start from an already established three-dimensional micromechanical model and make use of the wire geometry to perform some simplification. This construction is detailed in Sect. 2. In the model obtained, the polycrystalline texture is taken into account via 2 scalar functions representing the volume fractions and the uniaxial transformation strain for each crystalline orientation. An attractive feature of the proposed model is that closed-form expressions of the material response can be obtained for typical thermomechanical loadings of interest in SMA, such as cyclic traction at high temperature (superelastic regime) or thermal cycling at a fixed stress. Those analytical solutions are presented in Sect. 3. In Sect. 4 are reported some experimental results from calorimetry tests, traction at fixed temperature, and thermal cycling at fixed stress. Those results are used to identify the material parameters and discuss the validity of the model.

Micromechanical modelling of polycrystalline SMA wires

Reversible model

The geometrically linear setting is adopted throughout the paper. Consider a reference single crystal of shape memory alloy and denote the three-dimensional transformation strains of the N martensitic variants by e tr 1 , • • • , e tr N . These transformation strains are symmetry-related and have therefore the same trace k. They also verify the relation

1 N N i=1 e tr i = kI ( 1
)
where I is the identity second-order tensor. The value of k is very small for common shape memory alloys, which justifies using the approximation k = 0.

A polycrystalline SMA consists of numerous monocrystalline grains with distinct orientations. To each orientation corresponds a rotation R such that the transformation strains in the considered orientation are given by R • e tr i • R T with i = 1, • • • , N . Since the number of orientations can be very large, it is convenient to track the orientations with a continuously varying parameter s (which we can assume to vary in [0,1] without loss of generality). We denote by R(s) the rotation corresponding to orientation s. The transformation strains in orientation s are denoted by e tr i (s), i.e e tr i (s) = R(s) • e tr i • R(s) T . We also introduce the volumic density c(s) of orientation s. The function c(s) is positive and satisfies

1 0 c(s)ds = 1.
The functions R(s) and c(s) capture some information on the texture of the polycrystal. In three-dimensional reversible models of polycrystalline shape memory alloys, a commonly used expression for the free energy is

W (e, T ) = inf Θ 1 ,••• ,Θ N W (e, T, Θ 1 , • • • , Θ N ) (2)
where (3) In ( 2) and ( 3), e is the strain tensor, L is the elasticity tensor and λ(T ) is usually taken in the form

W (e, T, Θ 1 , • • • , Θ N ) = 1 2 (e -
λ(T ) = T -T 0 T 0 λ 0 ( 4
)
where T 0 is the critical temperature of phase transition and λ 0 is the latent heat. A common expression of the term m(T ) in (3) is

m(T ) = c(1 -log T T 0 )
where c is the specific heat [START_REF] Abeyaratne | A one-dimensional continuum model for shapememory alloys[END_REF]. Expression (3) rests on some assumptions. In particular, the elastic interaction between the individual austenite/martensite phases as well as the interaction between the grains are ignored. Those interactions would typically result in a additional mixing energy term in [START_REF] Leclercq | A general macroscopic description of the thermomechanical behavior of shape memory alloys[END_REF] (see e.g. [START_REF] Hackl | A micromechanical model for pretextured polycrystalline shapememory alloys including elastic anisotropy[END_REF][START_REF] Peigney | A non-convex lower bound on the effective free energy of polycrystalline shape memory alloys[END_REF][START_REF] Peigney | On the energy-minimizing strains in martensitic microstructures-part 1:geometrically nonlinear theory[END_REF] for some results in that direction). Expression (3) also ignores thermal strains which are supposed to remain small compared to transformation strains.In Eq. ( 3) it is also assumed that the austenite and the martensite have the elastic moduli: Such an assumption considerably simplifies the mathematical analysis but its validity remains questionable.

The scalar Θ i (s) in ( 2) can be interpreted as the volume fraction of martensitic variant i in the grains with orientation s. Accordingly, the function 3) is referred to as the free energy at fixed volume fractions. In (2), the infimum is taken over the functions Θ i (s) such that

W (ε, T, Θ 1 , • • • , Θ N ) in (
Θ i (s) ≥ 0, N i=1 Θ i (s) ≤ c(s). (5)
The stress-strain relation reads as σ = ∂W (e, T )/∂e, i.e.

σ = L : e - 1 0 N i=1 Θ i (s)e tr i (s)ds (6)
where Θ i (s) is a solution to the minimization problem in [START_REF] Boyd | A thermodynamical constitutive model for shape memory materials. part i. the monolithic shape memory alloy[END_REF]. Neglecting bending stiffness, the stress tensor σ in a wire is expected to be uniaxial with the form σ = σn ⊗ n

where n is a unit vector along the axis of the wire. Consider a infinitesimal element of length in which the stress is of the form [START_REF] Zaki | A three-dimensional model of the thermomechanical behavior of shape memory alloys[END_REF]. The strain e in the infinitesimal element is such that σn ⊗ n = ∂W (e, T )/∂e and therefore is a solution to the minimization problem

inf e W (e, T ) -σ(n • e • n). (8)
Replacing W with its expression (2), the problem [START_REF] Peigney | Numerical simulation of shape memory alloys structures using interior-point methods[END_REF] becomes

inf e inf Θ 1 , • • • , Θ N W (e, T, Θ 1 , • • • , Θ N ) -σ(n • e • n). (9)
where we recall from that the infimum is taken over volume fractions Θ 1 , • • • , Θ N satisfying [START_REF] Souza | Three-dimensional model for solids undergoing stress-induced phase transformations[END_REF]. Swapping the two infima in [START_REF] Peigney | Numerical simulation of shape memory alloys structures using interior-point methods[END_REF] and calculating the infimum with respect to e yields

e = σL -1 : (n ⊗ n) + 1 0 N i=1 Θ i (s)e tr i (s)ds. (10)
Substituting [START_REF] Bernardini | Models for one-variant shape memory materials based on dissipation functions[END_REF] in [START_REF] Leclercq | A general macroscopic description of the thermomechanical behavior of shape memory alloys[END_REF] gives

W (e, T, Θ 1 , • • • , Θ N ) -σ(n • e • n) = 1 0 N i=1 Θ i (s)(λ(T ) -σ(n • e tr i (s) • n))ds + m(T ).
The minimization problem (8) thus reduces to

inf Θ 1 , • • • , Θ N 1 0 N i=1 Θ i (s)(λ(T ) -σ(n • e tr i (s) • n))ds (11)
where the constant m(T ) has been dropped as it is independent of

Θ 1 , • • • , Θ N .
In a way similar to (2) and ( 9), the infimum problem in ( 11) is restricted to volume fractions Θ 1 , • • • , Θ N satisfying the constraints [START_REF] Souza | Three-dimensional model for solids undergoing stress-induced phase transformations[END_REF].

In order to solve the linear program [START_REF] Govindjee | A multi-variant martensitic phase transformation model: formulation and numerical implementation[END_REF], it is convenient to relabel the transformation strains e tr i (s) of each orientation s in such a way that

n • e tr 1 (s) • n ≥ n • e tr 2 (s) • n ≥ • • • ≥ n • e tr N (s) • n (12)
In such an ordering, variant 1 can be interpreted as the most favorably oriented variant with respect to the loading direction. Except for very special values of the orientation matrix R(s), the inequalities in [START_REF] Stupkiewicz | Modelling of laminated microstructures in stress-induced martensitic transformations[END_REF] are strict (which we assume in the following to simplify the presentation). If σ > 0 (i.e. the wire is under tension), it can easily be verified that the volume fractions Θ i (s) solutions to [START_REF] Govindjee | A multi-variant martensitic phase transformation model: formulation and numerical implementation[END_REF] satisfy

Θ i (s) = 0 for i > 1 (13)
i.e. the only martensitic variant that may appear in orientation s is the most favorably oriented one. If σ = 0 and λ(T ) ≤ 0, the minimum in ( 11) is attained for any set of volumes fractions such that N i=1 Θ i (s) = c(s) for all s. Such volumes fractions correspond to purely martensitic states. In particular, in view of (1), a self-accommodated state with negligible strain is obtained for

Θ i (s) = c(s) N . (14)
Self-accommodated states are typically observed when cooling down a stress-free specimen from a sufficiently high temperature.

In the model presented in this paper, we consider that the volume fractions Θ i (s) are always linear combinations of states of the form ( 13) and ( 14), i.e. Θ i (s) is assumed to be such that

Θ 1 (s) = 1 N θ 0 (s) + θ 1 (s), Θ 2 (s) = • • • = Θ N (s) = 1 N θ 0 (s) (15)
for some θ 0 (s) that can be interpreted as the volume fraction of self-accommodated martensite. The scalars θ 0 (s) and θ 1 (s) are submitted to the constraints

θ 0 (s) ≥ 0, θ 1 (s) ≥ 0, θ 0 (s) + θ 1 (s) ≤ c(s). (16)
which are consistent with [START_REF] Souza | Three-dimensional model for solids undergoing stress-induced phase transformations[END_REF]. For Θ i (s) satisfying ( 15), the free energy in the infinitesimal element can be written as

W (e, T, Θ 1 , • • • , Θ N ) = 1 2 σ 2 (n ⊗ n) : L -1 : (n ⊗ n) + λ(T ) 1 0 (θ 0 (s) + θ 1 (s))ds + m(T ) (17)
where [START_REF] Bernardini | Models for one-variant shape memory materials based on dissipation functions[END_REF]

has been used. Let ε = n • e • n and E = ((n ⊗ n) : L -1 : (n ⊗ n)) -1
be respectively the strain and the elastic stiffness in the direction of the wire. Setting

ε tr (s) = n • e tr 1 (s) • n (18)
and using [START_REF] Bernardini | Models for one-variant shape memory materials based on dissipation functions[END_REF], the expression (17) can be rewritten in terms of the uniaxial strain ε as

w(ε, T, θ 0 , θ 1 ) = 1 2 E ε - 1 0 θ 1 (s)ε tr (s)ds 2 + λ(T ) 1 0 (θ 0 (s) + θ 1 (s))ds + m(T ). (19)
In the proposed model, we adopt Eq. ( 19) as the free energy at fixed volume fractions. Expression [START_REF] Peigney | A non-convex lower bound on the effective free energy of polycrystalline shape memory alloys[END_REF] is derived from the three-dimensional energy (3) by performing a model reduction based on the wire geometry. In particular, the number of internal variables has been reduced from N (equal to 12 for NiTi alloys) to 2. A list of the main symbols used in the model is supplied in Table 1. Dissipative parameter for the self-accommodated (thermal induced) martensite G 1

Dissipative parameter for the oriented (stress induced) martensite A reversible model of SMA wire is obtained by minimizing [START_REF] Peigney | A non-convex lower bound on the effective free energy of polycrystalline shape memory alloys[END_REF] with respect to θ 0 and θ 1 . The corresponding free energy reads as

w(ε, T ) = inf θ 0 ,θ 1 w(ε, T, θ 0 , θ 1 ). ( 20
)
For later reference, we note that the Gibbs energy w * (σ, T ) = inf ε w(ε, T ) -σε corresponding to [START_REF] Peigney | On the energy-minimizing strains in martensitic microstructures-part 1:geometrically nonlinear theory[END_REF] is given by

w * (σ, T ) = inf θ 0 (s),θ 1 (s) w * (σ, T, θ 0 , θ 1 ) (21)
with

w * (σ, T, θ 0 , θ 1 ) = - σ 2 2E -σ 1 0 ε tr (s)θ 1 (s) + λ(T ) 1 0 (θ 0 (s) + θ 1 (s))ds + m(T ). (22)

Dissipative model

Experiments show that irreversible effects (hysteresis) play an important role in SMAs. To capture such irreversible effects, we wish to appeal to the framework work of standard generalized materials [START_REF] Halphen | Sur les matériaux standard généralisés[END_REF], viewing (θ 0 , θ 1 ) as internal variables and specifying their evolution law using a dissipation potential. In the classical framework of generalized standard materials (as applied for instance in hardening plasticity), the constitutive laws for a material with an internal variable α indeed read as

σ = ∂w ∂ε , - ∂w ∂α ∈ ∂Φ( α) ( 23
)
where Φ is the dissipation potential and ∂ is the subdifferential operator. Compared to hardening plasticity, a distinctive feature of phase transformation is that the internal variable (i.e. (θ 0 , θ 1 ) in our case) is constrained. More specifically, (θ 0 , θ 1 ) need to satisfy the inequalities ( 16) which result from mass conservation in the phase transformation process. In the presence of such constraints, the general constitutive laws [START_REF] Duerig | The influence of the r-phase on the superelastic behavior of niti[END_REF] need to be amended as

σ = ∂w ∂ε , - ∂w ∂α ∈ ∂Φ( α) + ∂I T (α) (24)
where I T is the indicator function associated to the constraints on the internal variable [START_REF] Frémond | Physics and astronomy online library[END_REF]. To be consistent with the second principle of thermodynamics, the dissipation potential Φ in (24) should be convex, positive and null at the origin. For the internal variable α = (θ 0 , θ 1 ), the requirements on Φ are satisfied by the following function

Φ( θ0 , θ1 ) = 1 0 G 0 | θ0 (s)| + G 1 | θ1 (s)|ds (25)
where G 0 and G 1 are positive parameters. We note that a similar dissipation potential has been used in [START_REF] Anand | Thermal effects in the superelasticity of crystalline shape-memory materials[END_REF][START_REF] Peigney | Numerical simulation of shape memory alloys structures using interior-point methods[END_REF] in the 3D setting. As will be seen later in Sect. 3.2.1, consistency with the experimental observations on the phase transformation requires that

G 0 < G 1 .
Adopting the expressions [START_REF] Peigney | A non-convex lower bound on the effective free energy of polycrystalline shape memory alloys[END_REF] for w and (25) for Φ, Eq. ( 24) yields the stress-strain relation

σ = E ε - 1 0 θ 1 (s)ε tr (s)ds (26)
and the evolution law

-∂w ∈ ∂Φ( θi ) + ∂I T (θ 0 , θ 1 ) (27)
with ∂w = ( ∂w ∂θ 0 , ∂w ∂θ 1 ) = (λ(T ), λ(T ) -σε tr (s)). (28)
In ( 27), the subdifferential ∂Φ( θi ) is the multivalued operator given by

∂Φ( θi ) = (B d 0 (s), B d 1 (s)) (29)
with

B d i (s) satisfying        |B d i (s)| ≤ G i if θi (s) = 0 B d i (s) = G i if θi (s) > 0 B d i (s) = -G i if θi (s) < 0 (30)
The subdifferential ∂I T (θ 0 , θ 1 ) is given by

∂I T (θ 0 , θ 1 ) = (z(s) -a 0 (s), z(s) -a 1 (s)) (31)
with conditions

z(s) ≥ 0, a i (s) ≥ 0, a i (s)θ i (s) = 0, z(s)(c(s) -θ 0 (s) -θ 1 (s)) = 0. ( 32
)
Collecting the expressions above, the evolution law [START_REF] Peigney | An incremental variational approach to coupled thermomechanical probles in anelastic solids. application to shape memory alloys[END_REF] becomes

-λ(T ) = B d 0 (s) + z(s) -a 0 (s) σε tr (s) -λ(T ) = B d 1 (s) + z(s) -a 1 (s) (33)
with conditions ( 30) and (32).

Consider a given local strain and temperature history (ε(t), T (t)) and denote by θ i (s, t) the value of θ i (s) at time t. Starting from a given initial state θ i (s, t = 0), the history θ i (s, t) is found by solving the time evolution problem defined by Eqs ( 26), [START_REF] Bhattacharya | Microstructure of Martensite: Why it Forms and how it Gives Rise to the Shape-memory Effect[END_REF], (32) and (33). This task greatly simplifies in situations where θ i (s, t) evolves monotonically with time. In such case, the values of θ 0 (s, t) and θ 1 (s, t) at any given time t can indeed be obtained by solving the incremental energy minimization problem

inf θ 0 ,θ 1 w(ε(t), T (t), θ i (s)) + 1 0 G 0 0 (s)θ 0 (s) + G 1 1 (s)θ 1 (s)ds (34) with i (s) = 1 if θ i (s, t) is increasing with t and i (s) = -1 if θ i (s, t) is decreasing. A justification of this result is provided in Appendix A.
A similar result holds for stress controlled loadings: for a given local stress and temperature history (σ(t), T (t)) such that θ i (s, t) evolves monotonically with time, the value of θ 0 (s, t) and θ 1 (s, t) at any given time t are a solution to the minimization problem

inf θ 0 ,θ 1 w * (σ(t), T (t), θ i (s)) + 1 0 G 0 0 (s)θ 0 (s) + G 1 1 (s)θ 1 (s)ds (35)
where w * is defined as in [START_REF] Frémond | Physics and astronomy online library[END_REF]. We will make use of the incremental variational formulations (34) and (35) in the next section.

Analytical solutions

As detailed in the following, closed-form expressions of the response of the material can be obtained for typical loadings of interest for SMAs.

3.1. Traction at high temperature 3.1.1. Reversible case It is insightful to first consider the reversible model. For a prescribed strain ε, the state of the material is obtained by solving the minimization problem inf

θ 0 ,θ 1 w(ε, T, θ 0 , θ 1 ) (36)
where θ 0 and θ 1 are submitted to the constraints [START_REF] Anand | Thermal effects in the superelasticity of crystalline shape-memory materials[END_REF]. The stress σ is given by ( 26) where (θ 0 , θ 1 ) is the solution of (36).

For solving (36) in the high temperature regime (λ(T ) ≥ 0), we first observe that w(ε, T, θ 0 , θ 1 ) > w(ε, T, 0, θ 1 ) for any (θ 0 , θ 1 ) satisfying ( 16) and such that θ 0 > 0. It follows that the minimization problem (36) reduces to

inf 0≤θ 1 (s)≤c(s) w(ε, T, 0, θ 1 ) (37)
i.e. there is no self-accommodated martensite in the energy-minimizing state.

Writing down the local optimality conditions in (37) yields

-σε tr (s) + λ(T )      = 0 if θ 1 (s) ∈ (0, c(s)) ≥ 0 if θ 1 (s) = 0 ≤ 0 if θ 1 (s) = c(s) (38) If λ(T ) > σε tr (s) for all s ∈ [0, 1] (39)
then it follows from (38) that θ 1 (s) = 0 for all s, i.e. the material is in the purely austenitic phase. Similarly, if

λ(T ) < σε tr (s) for all s ∈ [0, 1] (40)
then θ 1 (s) = c(s) for all s, i.e. the material is purely in the oriented martensite phase. With proper labelling of the cristalline orientations, we can assume without loss of generality that ε tr (s) is a decreasing function of s. Setting ε tr max = ε tr (0) and ε tr min = ε tr (1), the conditions (39) and (40) thus translate as λ(T ) > σε tr max and λ(T ) < σε tr min , respectively. In the intermediary case where λ(T )/ε tr max ≤ σ ≤ λ(T )/ε tr min , the conditions (38) show that

θ 1 (s) = c(s) for s ≤ s * , 0 for s > s * (41)
where s * is given by the consistency condition -σε tr (s * ) + λ(T ) = 0, i.e.

λ(T ) = Eε tr (s * ) ε - s * 0 c(s)ε tr (s)ds (42)
Let us now detail the explicit expression of the stress-strain curve provided by the model. The most natural way to parameterize the stress-strain curve is to use ε as a parameter, solving (42) for each given ε and substituting the result in [START_REF] Kim | Experimental studies of phase transformation in shape memory alloys[END_REF]. This turns out to be a relatively complex calculation, even for simple functions ε tr (s) and c(s). Such difficulties can be avoided if ε tr is strictly decreasing, which we assume from now. In such case, the stress-strain curve is strictly increasing and can therefore by parameterized by σ. Denoting by f the inverse function of ε tr , Eq. ( 42) can be equivalently rewritten as

s * = f λ(T ) σ . (43)
From ( 26) and (41) we have

ε = σ E + s * 0 c(s)ε tr (s)ds.
Using the change of variable s = f (x) in the above integral yields

ε = σ E + λ(T )/σ ε tr max f (x)c(f (x))xdx (44)
where ( 43) has been used. The relation (44) holds for λ(T )/ε tr max ≤ σ ≤ λ(T )/ε tr min and gives the stress-strain response during phase transformation.

It is insightful to calculate the tangent stiffness dσ/dε during phase transformation. By differentiation of ( 42) and (44) we find

dσ dε = E 1 -E λ(T ) 2 σ 3 f ( λ(T ) σ )c(f ( λ(T ) σ )) (45)
Since ε tr (and consequently f ) is a decreasing function, it is clear from (45) that 0 ≤ dσ/dε ≤ E, i.e. the tangent stiffness is positive (there is no softening) and always remains smaller than the Young modulus of the pure phases. Eq. ( 45) shows that the tangent stiffness dσ/dε is a function of σ, meaning that the response of the material is nonlinear during phase transformation.

A (ε tr max -ε tr min )(1 -s) 3 + ε tr min 1 - x -ε tr min ε tr max -ε tr min 1 3 B (ε tr max -ε tr min ) cos πs 2 + ε tr min 2 π arccos x -ε tr min ε tr max -ε tr min
Details of the stress-strain curve (44) depend on the functions ε tr and c which capture some information on the texture of the polycrystalline material. As an illustration, some stress-strain curves are plotted in Fig. 1 for two different textures labelled as A and B. The corresponding ε tr (s) functions are listed in Table 2 and plotted in Fig. 2. The values ε tr min = 0.03, ε tr max = 0.11, λ(T ) = 12 MPa, E = 28 GPa, c(s) = 1 have been used in Figs. 1 and2.

In the case of textures A and B, the integral in (44) can be calculated in closedform, leading to fully explicit expressions of the stress-strain response during phase transformation. The obtained expressions read as

ε = σ E + 1 4   ε tr max + 3ε tr min - λ(T ) σ -ε tr min ε tr max -ε tr min 1 3 3ε tr min + λ(T ) σ  
for texture A, and

ε = σ E + ε tr min + 2 π (ε tr max -λ(T ) σ )(ε tr max -2ε tr min + λ(T ) σ ) -ε tr min arctan   λ(T ) σ -ε tr min (ε tr max -λ(T ) σ )(ε tr max -2ε tr min + λ(T ) σ )   for texture B.
As can be observed in Fig. 1, the first elastic part of the stress-strain curves is the same for textures A and B. Then, once the threshold stress λ(T )/ε tr max is exceeded, the curves present different evolutions. We note that depending on ε tr (s), the stressstrain curve can present sharp slope changes or smooth changes, at the beginning and at the end of phase transformation. The stresses at the beginning and at the end of phase transformation only depend on the function ε tr through its extreme values ε tr max and ε tr min . A similar remark holds for the strain at the beginning of phase transformation. The strain at the end of phase transformation depends on the integral of ε tr (s) and is equal to

λ(T ) Eε tr max + 1 0 ε tr (s)ds.
The term 1 0 ε tr (s)ds can be interpreted as the maximum recoverable strain of the wire. It is equal to (ε tr max + 3ε tr min )/4 for texture A and 2(ε tr max -ε tr min )/π + ε tr min for texture B. At the end of phase transformation, which occurs for σ = λ(T )/ε tr min , the evolution becomes linear elastic again, which is represented by dotted lines in Figure 1.

Dissipative case

Let us now study how the results are modified in the dissipative case. We assume that the material is fully austenitic in the initial state, i.e. θ 0 (s) = θ 1 (s) = 0 for all s. An increasing strain ε is applied at a fixed temperature T such that λ(T ) > G 1 . For any given s, it is expected that θ i (s) increases with the applied strain, as in the reversible case. The exact value of θ i (s) can be obtained by using the variational formulation of the incremental evolution problem (34), which in the present case specializes as An important observation is that (46) has the same structure as the minimization problem (36) considered in Sect. 3.1.1. It follows that the solution to (46) is obtained by replacing λ(T ) with λ(T )+G 1 in the expressions of the stress-strain response found previously in Sect. 3.1.1. We thus have

inf θ 0 ,θ 1 w(ε, T (t), θ 0 , θ 1 ) + 1 0 G 0 θ 0 (s) + G 1 θ 1 (s)ds. ( 46
               ε = σ E for 0 ≤ σ ≤ λ(T )+G 1 ε tr max ε = σ E + λ(T )+G 1 σ ε tr max f (x)c(f (x))xdx for λ(T )+G 1 ε tr max ≤ σ ≤ λ(T )+G 1 ε tr min ε = σ E + ε tr min ε tr max f (x)c(f (x))xdx for λ(T )+G 1 ε tr min ≤ σ (47)
Eq. ( 47) give the stress-strain relations during loading, provided that λ(T ) + G 1 > 0.

Let us now assume that the applied strain ε decreases after having reached a maximum value ε max . We denote by θ max 1 be the value of θ 1 when ε = ε max . When ε decreases, it can be verified from the optimality conditions (33) that θ i does not evolve as long as σε tr min ≥ λ(T ) -G 1 . This corresponds to elastic unloading. For smaller values of σ, reverse phase transformation takes place, i.e. the volume fractions of martensite decrease. The problem (34) thus specializes as

inf θ 0 ,θ 1 w(ε, T, θ 0 , θ 1 ) - 1 0 (G 0 θ 0 (s) + G 1 θ 1 (s))ds. ( 48
)
Provided λ(T ) -G 1 > 0, the minimization problem (48) has the same structure as (36) so that the solution to (48) is obtained by substituting λ(T ) with λ(T ) -G 1 in the expressions obtained in Sect. 3.1.1. In summary, we have

               ε = σ E for 0 ≤ σ ≤ λ(T )-G 1 ε tr max ε = σ E + λ(T )-G 1 σ ε tr max f (x)c(f (x))xdx for λ(T )-G 1 ε tr max ≤ σ ≤ λ(T )-G 1 ε tr min ε = σ E + 1 0 θ max 1 (s)ε tr (s)ds for λ(T )-G 1 ε tr min ≤ σ (49)
Eq. ( 49) give the stress-strain relations during unloading, provided that λ(T ) -G 1 > 0. Such values of the temperature correspond to the superelastic regime of the material. In Fig. 3 is presented an example of stress-strain cycle as provided by Eqs (47) and (49), using texture B with parameters ε tr max = 0.11, ε tr min = 0.03, λ(T ) = 12 MPa, E = 28 GPa, G 1 = 3 MPa.

Thermal cycling under constant stress

Let us now consider a cyclic thermal loading at a constant stress σ. We assume that the material is fully austenitic in the initial state. During cooling, the volume fractions θ i (s) increase so that the minimization problem (35) becomes

inf θ 0 ,θ 1 -σ 1 0 ε tr (s)θ 1 (s) + λ(T ) 1 0 (θ 0 (s) + θ 1 (s))ds + 1 0 G 0 θ 0 (s) + G 1 θ 1 (s). (50)
It can be checked from (50) that the evolution is elastic as long λ(T ) ≥ min(-G 0 , σε tr max -G 1 ). The subsequent evolution depends on the value of the imposed stress σ, as detailed next.

0 2 4 6 8 λ(T )-G 1 ε tr max λ(T )+G 1 ε tr max 200 λ(T )-G 1 ε tr min λ(T )+G 1 ε tr min ε (%) σ (MPa)
Elastic behavior of austenite Loading Unloading Elastic behavior of martensite During cooling, the evolution is found to be elastic for λ(T ) > -G 0 . For λ(T ) ≤ -G 0 , the solution to (50) is found to be given by θ 0 (s) = c(s) for all s.

Assume now that the material is cooled down to a temperature such that λ(T ) < -G 0 and subsequently heated. It can be verified that the solution to (50) is given by θ 0 (s) = c(s), θ 1 = 0 as long as λ(T ) < G 0 , and that θ 0 (s) = θ 1 = 0 for λ(T ) > G 0 , i.e. phase transformation in austenite occurs brutally at λ(T ) = G 0 .

This evolution scenario holds provided σ ≤ (G 1 -G 0 )/ε tr max , i.e. for sufficiently small values of the applied stress. Since G 1 > G 0 , this evolution holds in particular for σ = 0, i.e. under stress-free condition.

Case

(G 1 -G 0 )/ε tr max < σ < (G 1 -G 0 )/ε tr min
The cooling is found to be elastic as long as λ(T ) > σε tr max -G 1 . When λ(T ) reaches the critical value σε tr max -G 1 , it can be checked that the solution to (50) is given by

θ 0 (s) = 0, θ 1 (s) = c(s) for s ≤ f ( G 1 +λ(T ) σ ) θ 0 (s) = θ 1 (s) = 0 for s > f ( G 1 +λ(T ) σ ) (51)
Accordingly the strain ε is given by

ε = σ E + G 1 +λ(T ) σ ε tr max f (x)c(f (x))xdx. (52)
These results can be interpreted as phase transformation into oriented-martensite in the most favorable orientations. The solution (51) holds as long λ(T ) > -G 0 . For λ(T ) < -G 0 , the solution to (50) can be verified to be given by

θ 0 (s) = 0, θ 1 (s) = c(s) for s ≤ f ( G 1 -G 0 σ ) θ 0 (s) = c(s), θ 1 (s) = 0, for s > f ( G 1 -G 0 σ ) (53)
so that

ε = σ E + G 1 -G0 σ ε tr max f (x)c(f (x))xdx (54)
The results (53) mean the material state is a mixture of oriented martensite and self-accommodated martensite. Assume now that the material is cooled down to a temperature such that λ(T ) < -G 0 and subsequently heated. Omitting the detail of the calculation, the following evolution is obtained:

• for λ(T ) ≤ G 0 , the evolution is elastic, i.e. θ 0 , θ 1 and ε are given by ( 53) and (54).

• for G 0 < λ(T ) ≤ 2G 1 -G 0 , the strain is given by (54) and θ 0 , θ 1 are given by

θ 0 (s) = θ 1 (s) = 0 for s > f ( G 1 -G 0 σ ) θ 0 (s) = 0, θ 1 (s) = c(s) for s ≤ f ( G 1 -G 0 σ ) (55) • for 2G 1 -G 0 < λ(T ) ≤ G 1 + σε tr max , θ 0 , θ 1 and ε are given by θ 0 (s) = θ 1 (s) = 0 for s > f ( -G 1 +λ(T ) σ ) θ 0 (s) = 0, θ 1 (s) = c(s) for s ≤ f ( -G 1 +λ(T ) σ ) ε = σ E + -G 1 +λ(T ) σ ε tr max f (x)c(f (x))xdx (56)
• for G 1 +σε tr max < λ(T ), we have θ 0 (s) = θ 1 (s) = 0 and ε = σ/E, i.e. the material is fully in the austenitic phase.

Case

σ > (G 1 -G 0 )/ε tr min
As in the preceding case, the cooling is elastic as long as λ(T ) > σε tr max -G 1 . When -λ(T ) reaches the critical value G 1 -σε tr max , the solution to (50) is given by (51), i.e. phase transformation into oriented-martensite occurs in the most favorable orientations. In contrast with the preceding case, the solution (51) holds as long λ(T ) > σε tr min -G 1 . For λ(T ) < σε tr min -G 1 , the solution to (50) is given by θ 1 (s) = c(s) for all s, i.e. all the grains are transformed in oriented martensite. Accordingly the strain ε is equal to σ/E + ε tr max ε tr min c(s)ε tr (s)ds. Omitting the detail of the calculation, the following evolution is obtained during heating from a temperature such that λ(T ) < σε tr min -G 1 : • for λ(T ) ≤ G 1 + σε tr min , the evolution is elastic, i.e. θ 1 (s) = c(s) and

ε = σ/E + ε tr max ε tr min c(s)ε tr (s)ds
• for G 1 + σε tr min < λ(T ) ≤ G 1 + σε tr max , θ 0 , θ 1 and ε are given (56). • for G 1 +σε tr max < λ(T ), we have θ 0 (s) = θ 1 (s) = 0 and ε = σ/E, i.e. the material is fully in the austenitic phase.

In contrast with the case covered in Sect. 3.2.2, it can be observed that θ 0 = 0 during the whole cycle, i.e. self-accommodated martensite never appears.

Three thermal cycles at constant stress corresponding to the three cases 3.2.1,3.2.2,3.2.3 are represented in Figure 4 for the texture B. Values of the parameters are ε tr min = 0.01, ε tr max = 0.17, G 1 = 4 MPa, G 0 = 3.2 MPa, E = 28 GPa. The term λ(T ) is taken in the form (4) with λ 0 = 74.2 MPa and T 0 = 294.9 K.

The first case (dotted curve in Figure 4) corresponds to an imposed stress which is too low to trigger phase transformation into oriented martensite during cooling. As a result, there is no change in the macroscopic strain.

The second case (solid curve in Figure 4) corresponds to an intermediary stress : the wire stretches during cooling and shrinks during heating. Note the temperature T at the end of phase transformation during cooling verifies λ(T ) = G 0 and therefore is independent of the applied stress σ. A similar remarks holds for the temperature at the beginning of phase transformation during heating. The variation ∆ε of the strain between the cool state and the hot state is given by

G 1 -G 0 ε tr max ε tr max f (x)xdx (
λ(T ) = σε tr max -G 1 λ(T ) = -G 0 λ(T ) = σε tr max -G 1 λ(T ) = σε tr min -G 1 Initial state Temperature T (K) ε (%) σ ≤ (G 1 -G 0 )/ε tr max (G 1 -G 0 )/ε tr max < σ < (G 1 -G 0 )/ε tr min σ > (G 1 -G 0 )/ε tr min Figure 4
. Thermal cycling at constant stress and therefore increases with the applied constant stress σ. The value ∆ε in (57) is always smaller than the maximum recoverable strain 1 0 ε tr (s)ds. The third (dashed curve in Figure 4) corresponds to a high applied stress σ. Both during heating and cooling, the temperatures at beginning and end of phase transformation increase. The variation ∆ of the strain between the cool state and the hot state is equal to the maximum recoverable strain 1 0 ε tr (s)ds.

Shape memory cycle

In this section we provide closed-form expressions of the material response during the shape memory cycle, i.e. when the material is cooled down, deformed in a cooled state and subsequently heated in a stress-free state.

Stress-free cooling

We consider a stress-free initial state at high temperature, in which the material is fully austenitic. The stress-free material is then submitted to a decreasing temperature T . As a special case of the thermal loading studied in Sect.3.2.1, we have θ 0 = θ 1 = 0 as long as λ(T ) > -G 0 . When λ(T ) reaches the critical value -G 0 , the volume fraction θ 0 (s) starts to increase. For λ(T ) < -G 0 , we have θ 0 (s) = c(s) i.e. the material is fully transformed in self-accommodated martensite. Note by ( 1) that there is no macroscopic deformation during the phase transformation. The behavior provided by the model is thus consistent with the experimental observations on martensitic transformation during stress-free cooling. We note that taking G 0 < G 1 is necessary to ensure that this consistency is respected.

Traction in the cooled state

Consider now that the material is cooled down to a temperature T such that λ(T ) < -G 0 . From that point, the temperature is kept constant and an increasing strain ε is applied. It is expected that the volume fraction θ 0 (s) decreases and that θ 1 (s) increases with the applied strain, i.e. that reorientation of martensite occurs. The incremental problem (34) thus specializes as

inf θ 0 ,θ 1 w(ε, T, θ 0 , θ 1 ) + 1 0 -G 0 θ 0 (s) + G 1 θ 1 (s)ds (58)
Let F be the functional to be minimized in (58). Since λ(T ) < G 0 , it can easily be verified that F (ε, c -θ 1 , θ 1 ) < F (ε, θ 0 , θ 1 ) for any positive (θ 0 , θ 1 ) such that θ 0 -θ 1 < c.The problem (58) thus reduces to

inf 0≤θ 1 (s)≤c(s) w(ε, T, c -θ 1 , θ 1 ) + 1 0 -G 0 (c(s) -θ 1 (s)) + G 1 θ 1 (s)ds (59)
i.e. the minimization is restricted to fully martensitic states. The problem (59) is almost identical to the incremental problem (46) obtained for traction at high temperature, so that the solution to (59) is easily adapted from (47). We obtain

               ε = σ E for 0 ≤ σ ≤ G 0 +G 1 ε tr max ε = σ E + G 0 +G 1 σ ε tr max f (x)c(f (x))xdx for G 0 +G 1 ε tr max ≤ σ ≤ G 0 +G 1 ε tr min ε = σ E + ε tr min ε tr max f (x)c(f (x))xdx for G 0 +G 1 ε tr min ≤ σ (60)
In contrast with the case of a high temperature, the stress-strain curve at low temperature is the same for all values of T (provided that λ(T ) < -G 0 ). Eq. ( 60) gives the stress-strain curve during loading, i.e. for increasing values of the strain ε.

Let θ max 1 be the value of θ 1 at the end of the loading, i.e. when the applied strain ε reaches its maximum value ε max . If the applied strain is subsequently decreased while keeping the wire under traction (σ ≥ 0), it can be verified that the unloading is elastic. In particular, when the material reaches its stress-free state, there remains a residual strain given by 1 0 θ max 1 (s)ε tr (s)ds.

Heating in the stress-free state

Starting from the stress-free state given by (60), we now assume that the material is heated. The equations of evolutions are similar to those obtained in Sect. 3.3.1 and can be solved in a similar manner. We obtain that

     θ 0 = c -θ max 1 , θ 1 = θ max 1 for λ(T ) ≤ G 0 θ 0 = 0, θ 1 = θ max 1 for G 0 ≤ λ(T ) ≤ G 1 θ 0 = θ 1 = 0 for G 1 ≤ λ(T ) (61)
For high temperature (G 1 ≤ λ(T )), the material is fully transformed in austenite.

Experimental validation

In this Section are reported some results of three types of experimental tests: a differential scanning calorimetry (DSC) test, mechanical traction tests at fixed temperature, and thermal cyclings at fixed stress. All those tests have been performed on a straight annealed 0.5mm diameter Nickel-Titanium wire, supplied by Memry Corp. (USA). The investigated alloy is Ni 50.4 Ti 49.6 (at.%). We use the experimental results to discuss the identification of the model parameters as well as the validity of the model.

Calorimetry

A DSC test has first been carried out in order to identify the model parameters T 0 and λ 0 in (4). Phase transition temperatures and enthalpies of transition have been measured. The test equipment was a DSC 250 (TA Instruments). The DSC experiments were performed at a 10 K per minute temperature variation speed, under an inert Nitrogen environment (purge gas). A 7.9 mg sample of a 0.5mm diameter virgin Ni-Ti wire has been tested. The test has been performed as detailed in Table 3.

Table 3. Experimental protocol The DSC chart is represented in Figure 5. The measured values are listed in Table 4. Two peaks are observed during cooling. Peak 1 (around 313 K) corresponds to the Austenite to R-phase transformation. Peak 2 (around 268 K) corresponds to the R-phase to Martensite transformation [START_REF] Duerig | The influence of the r-phase on the superelastic behavior of niti[END_REF]. During heating, a single peak, representing the Martensite to Austenite transformation, is observed. We choose not to model the R-phase here, because it tends to disappear under stress [START_REF] Duerig | The measurement and interpretation of transformation temperatures in nitinol[END_REF]. In this study, mechanical tests were always be performed on samples which had previously been heated up to 373 K and then cooled down to 253 K, in order to ensure that the material was in a martensitic state without R-phase at the beginning of the mechanical loading.

Peak

Temperature Enthalpy Formation of martensite 268.6 K 11.5 J/g Formation of austenite 321.3 K 22.9 J/g

Using the results of Sect. 3.2.1 with the expression (4) of the latent heat, the phase transformation temperatures at zero stress are :

T A = T 0 - T 0 G 0 λ 0 (cooling)
T B = T 0 + T 0 G 0 λ 0 (heating) (62)
It follows that

T 0 = 1 2 (T A + T B ) (63)
Using Eq. ( 63) with the values of T A and T B reported in Table 4 yields T 0 = 294.9 K. In (4), the parameter λ 0 represents the volumic phase transformation enthalpy. We choose to measure it experimentally from the mass phase transformation enthalpy on the martensite formation peak at low temperature. Adopting a density of ρ = 6.45 mg/mm 3 for Nickel-Titanium [START_REF] Churchill | Tips and tricks for characterizing shape memory alloy wire: Part 4 -thermo-mechanical coupling[END_REF], we obtain λ 0 = 11.5 × 6.45 10 -3 J.mm -3 = 74.2 MPa .

Traction tests under controlled temperature

Performing traction tests on SMA wires requires a careful experimental design.

The phase transformation indeed induces some self-heating, so thermal conditions and loading rate strongly influence the results. For this reason, a specific testing chamber has been set up in order to perform the traction tests in water, with a strain measurement by Digital Image Correlation (DIC). The testing chamber is made of PMMA and contains a volume of water which is pumped into the chamber by a heating and cooling apparatus (see Figure 6) and goes out by overflow.

The tested part of the wire finds is immersed in water, and its upper end is clamped on a displacement controlled ADAMEL DY31 testing machine. Two type K thermocouples monitor the water temperature at two different locations inside the water volume. The force in the wire was recorded using a 100N load cell and the local deformation in the wire was measured by DIC on images acquired by a Pike F421B camera (Allied Vision Technologies). A Matlab routine has been set up to record simultaneously the deformation images and the force values. We set up the applied strain rate to remain reasonably low (10 -4 s -1 ) in order to minimize temperature changes in the wire. This value is often found in the literature on SMA [START_REF] Kim | Experimental studies of phase transformation in shape memory alloys[END_REF] and can be justified from thermomechanical simulations [START_REF] Peigney | An incremental variational approach to coupled thermomechanical probles in anelastic solids. application to shape memory alloys[END_REF]. The experimental protocol described allows us to approximate isothermal conditions: the temperature in the wire is considered as uniform and equal to the water temperature. The first traction tests performed on a virgin SMA wire usually do not exhibit a homogeneous strain: some localization is observed. In order to reach a stabilized thermomechanical behavior, the tested wire was submitted to 20 cycles of traction loading up to 30 N followed by unloading and heating. Before the first cycle and between each of those 20 cycles, the wire was unloaded and heated up to 373 K in a furnace with no imposed stress. It was then cooled down to 253 K, in order to reach self-accommodated martensite state with no R-phase at the beginning of the next cycle. After that 'training' procedure, the resulting stress-strain curve was found to be reproducible. Accordingly, a homogeneous strain was observed along the wire as illustrated in Figure 7 showing the average axial strain and the corresponding standard deviation of the axial strain field on the wire. This data has been obtained with the software GOM Correlate. As can be observed in Figure 7, the strain field is homogeneous during the whole experiment, including phase transformation.

An example of stabilized stress-strain curve is shown in Fig. 9 at 298 K. Assuming that the test temperature is sufficiently low for the condition λ(T ) < G 0 to be fulfilled, the stress-strain curve predicted by the model is given by Eq (60). It notably depends on the material parameters E, G 0 +G 1 , ε tr (s), c(s), the identification of which is now discussed.

Consistently with Eq. ( 60), we first observe that the modulus E can be identified from the slope at the beginning of the curve, yielding E = 28000 MPa. Regarding the density c(s) and the function ε tr (s), we make use of some background information on the texture of NiTi wires. Such wires indeed show an oriented fiber texture in which the 111 crystallographic direction in each grain is along the axis of the wire [START_REF] Thamburaja | Polycrystalline shape-memory materials: Effect of crystallographic texture[END_REF][START_REF] Laplanche | Effect of temperature and texture on the reorientation of martensite variants in niti shape memory alloys[END_REF][START_REF] Bhattacharya | Microstructure of Martensite: Why it Forms and how it Gives Rise to the Shape-memory Effect[END_REF]. A perfect fiber texture would correspond to a constant function ε tr (s). In a true specimen, the 111 direction in most grains is not perfectly aligned with the axis of the wire. Moreover, there can be a small fractions of grains whose 111 direction differ significant from the axis wire. Recalling that the function ε tr is decreasing, the function ε tr in a true specimen is expected to be qualitatively of the shape depicted in Fig. 8,i 64) with the parameters β and γ set to 0.02 and 1, respectively. The value c(s) = 1 is adopted.

Applying Eq (60) for calculating the stress-strain response of the model requires the expression of the inverse function of ε tr and its derivative. To that purpose, we note that the inverse function of ε tr (s) in ( 64) is :

f (x) = 1 2 + 1 π arctan 1 β arctanh 1 -2 x -ε tr
we have

f (x) = β 2γπ 1 2 + 1 π arctan H(x) β 1 γ -1 1 (ε tr min -x)(β 2 + H(x) 2 ) 1 1 -
x-ε tr min ε tr max -ε tr min (67) Adopting (64), the function ε tr is determined from 4 scalar parameters. In principle, ε tr min and ε tr max can be obtained from the crystallographic theory of martensite. This calculation, reported in Appendix B, gives ε tr max = 0.107 and ε tr min = 0.024. According to Eq. (60), the threshold stress for the beginning of phase transformation is (G 0 + G 1 )/ε tr max . By tracking the end of linearity at the beginning of the stress-strain curve, it is thus possible to identify G 0 + G 1 . For the isothermal traction shown in Figure 9, the linear elastic domain ends for the stress σ lin = 50 MPa, yielding G 0 + G 1 = σ lin ε tr max = 5.35 MPa. The texture parameters β and γ are identified by trial and error, seeking for the best fit between the experimental curve and the analytical solution. The obtained parameters for this NiTi wire are listed in Table 5. The corresponding texture is labelled as texture C, and the resulting curve is plotted in Figure 9. A good fit is obtained for strains up to ε = 5%. However, the predicted phase transformation ends for about ε = 5%: beyond that value, the model gives a linear elastic behavior. This is in contradiction with the experiments, which show that phase transformation still takes place for strains up to 6.5% (this is indicated by the fact that the experimental loading and unloading curves do not overlap in the high strain domain). This discrepancy between the model curve and the experimental curve partially stems from the elastic interaction between the grains, which has not been considered in the model. 3.5 MPa An attempt at obtaining a better fit between the analytical curve and the experimental curve has been made by relaxing the parameters ε tr min and ε tr max : instead of using the values provided by the crystallographic theory, we consider ε tr min and ε tr max as free parameters. This allows to get a better fit with the experimental curve, especially in the high strains domain as shown in Figure 10. The material parameters of the corresponding texture (labelled as texture C*) are listed in Table 6. In Figure 11 is plotted the function ε tr (s) for texture C*. With texture C*, most transformation strains are found to fall between 4 and 7 %. This can be interpreted as the result of a privileged grain orientation in the wire. It is insightful to interpret the model stress-strain response in Fig. 10 in terms of the corresponding distribution of transformation strains shown in Figure 11. Firstly, the rounded part at the end of the linear elastic portion corresponds to the transformation of the grains parametrized by s close to 0. They are very favorably oriented with respect to the loading axis and transform first. Secondly, the stress-strain curve features an almost flat plateau, which corresponds to the transformation of the predominant grains with transformation strains in the range 4-7%. Finally, the end of the stress-strain curve features a significant nonlinear hardening, corresponding to the phase transformation of the less favorably oriented grains in the polycrystal. These grains present a low ε tr (s) value. Compared to texture C, the function ε tr (s) corresponding to texture C* yields a better match with the experimental results, but it probably deviates more from the real texture of the material. This all falls down to the simplified form of the free energy (3): Allowing the function ε tr to deviate from the real texture partially makes up for the fact that modeled energy does not exactly correspond to the true energy of the material.

Thermal cycling under constant stress

At this point, all the material parameters have been identified, except for G 0 and G 1 . More precisely, the sum G 0 + G 1 is known but the value of G 0 (and consequently G 1 ) is not known. Its value can be obtained from thermal cyclic under constant stress, as is now explained. The SMA wire is mechanically loaded in the testing chamber at a water temperature of 358 K, up to a traction stress of 127 MPa (25N). This stress is then maintained constant in time with the help of a PID controller included in the TestWorks software, which is used to control the electromechanical machine. Thermal loadings were carried out by varying the water temperature in the testing chamber between 283 and 363 K. In Figure 12, the experimentally measured strain in the wire is plotted as a function of the water temperature. We can now calculate the parameter G 1 by measuring the temperature difference ∆T between the heating and the cooling curve at a fixed strain. Using equations ( 52) and (56) as well as the expression (4) of the latent heat, we have indeed

∆T = 2G 1 T 0 λ 0 .
For ε = 0.03, we measure ∆T = 27.8 K which leads to G 0 = 3.0 MPa and G 1 = 3.5 MPa. We can note that the condition G 0 < G 1 introduced in Sect. 2.2 is fulfilled. Moreover, with the values obtained in Sect. 4.1 for λ 0 and T 0 , the condition λ(T ) < G 0 is satisfied at 298 K. This justifies using Eq. ( 60) to plot the stressstrain curve. All model parameters are now identified, and the analytical solution for thermal cycling, established in Eqs. ( 51), ( 53), ( 55) and (56), is plotted in Figure 12. That curve is in relatively good agreement with the experimental results, which contributes to validate the model. In Fig. 13(solid line) is plotted the experimental stress-strain curve obtained at 353 K (which is about the maximum temperature that can applied in the designed testing chamber). At such temperature, the material enters the superelastic regime. The stress-strain curve for texture C* is shown in dashed line. There is clearly a discrepancy between the experimental curve and the curve corresponding to texture C*. This is not entirely surprising because the martensitic microstructures that appear in the superelastic regime are much different from those that appear in the low temperature regime. It could be possible to identify a function ε tr (s) that would yield a good fit at high temperature, but then the corresponding fit at low temperature would be poor. This is a limitation of the model: using a single function ε tr (s) is probably not sufficient to describe accurately the nonlinear phase-transformation hardening at all temperatures.

Conclusion

The proposed model is based on a three-dimensional micromechanical expression for the free energy, using the wire geometry to perform some simplifications. In particular, the internal variables are reduced to two scalars per orientation (corresponding to the volume fractions of self-accommodated martensite and oriented martensite). Such simplifications allows analytical solutions to be obtained for the material response under typical loadings of interest. The obtained solutions depend on the polycrystalline texture through the functions c and ε tr representing the volumic density of crystalline orientations and the distribution of transformation strain, respectively. The identification of the material parameters have been discussed in detail and can take advantage of the presented analytical solutions. The model could be used for designing SMA actuators. Although a satisfactory fit with the experiments is obtained for the loadings corresponding to the shape memory effect (traction at low temperature, thermal cycling a constant stress), it would be interesting to refine the model by taking the elastic interaction between the grains into account. This would typically result in an additional term in the free energy. It would also be interesting to improve the model by using different elastic moduli for the austenite and the martensite. We also note that the proposed model could be extended to SMA rods (loaded in traction and compression) by adding an additional internal variable representing the volume fractions of the most favorably martensitic variant in compression.

Acknowledgments

This work has been partially funded by Arcora (Ingérop Group), and by ANRT under CIFRE grant No 2015/0495. The authors wish to thank Karim Benzarti who performed the DSC experiment, and Christophe Bernard who manufactured the experimental traction chamber.

Θ

  i (s)ds + m(T ).

  (s) transformation strain for the crystalline orientation s f inverse function of ε tr ε tr min minimum value of the function ε tr (s) ε tr max minimum value of the function ε tr (s) θ 0 (s) volume fraction of self-accommodated martensite for the crystalline orientation s θ 1 (s) volume fraction of oriented martensite for the crystalline orientation s G 0

Figure 1 .

 1 Figure 1. Superelastic stress-strain curves for the textures A and B, in the reversible case (no dissipation).

Figure 2 .

 2 Figure 2. Examples of functions ε tr (s)

Figure 3 .

 3 Figure 3. Example of superelastic response in the dissipative case -Texture B

1

  Test start at 293 K ; Heating up to 523 K (austenitic initial state) 2 Cooling down to 183 K ; Re-heating up to 523 K (first DSC cycle) 3 Cooling down to 183 K ; Re-heating up to 523 K (second DSC cycle)

Figure 5 .

 5 Figure 5. DSC test

Figure 6 .

 6 Figure 6. Experimental setup for mechanical tests at a controlled temperature

Figure 7 .

 7 Figure 7. Homogeneous strain in the wire during phase transformation

1 -

 1 .e with an almost flat plateau in its central part (stemming from a privileged orientation in the polycrystal) and larger fluctuations near s = 0 and s = 1. To model such a texture, we adopt the parameterized expression ε tr (s) = (ε tr max -ε tr min ) tanh(β(tan(π(s γγ) ∈ R 2 + . The functions ε tr in (64) is monotonically decreasing and varies from ε tr max to ε tr min as s varies from 0 to 1. The parameters β et γ in (64) control the length and the slope of the central part. The function shown in (8) correspond to Eq. (

Figure 8 .

 8 Figure 8. Example of function ε tr corresponding to a non perfect fiber texture.

Figure 9 .

 9 Figure 9. Model parameter identification -Texture C -Isothermal traction test at 298 K

Figure 10 .

 10 Figure 10. Model parameter identification -Texture C* -Isothermal traction test at 298 K

Figure 11 .

 11 Figure 11. Function ε tr (s) for the texture C*.

Figure 12 .

 12 Figure 12. Thermal cycle at a constant stress σ = 127 MPa.

Figure 13 .

 13 Figure 13. Texture C* -Isothermal traction test at 353 K

3 n 2 )

 32 Adopting the notations of Sect. 2, the uniaxial transformation strain ε tr (s) in orientation s isε tr (s) = max 1≤i≤12 n • R(s)e tr i R(s) T • n (B.1)where R(s) is the rotation corresponding to orientation s. It follows from (B.1) thatε tr min ≤ ε tr (s) ≤ ε tr max Re tr i R T • n, ε tr max = max R∈SO 3 max 1≤i≤12 n • Re tr i R T • n (B.2)The calculation of ε tr max can be done in closed-form. Swapping the order of the two maxima indeed yieldsε tr max = max 1≤i≤12 max R∈SO • Re tr i R T • nAs R varies in SO 3 , the vector R•n varies on the unit sphere S 2 = {m ∈ R 3 : m = 1}. Hence ε tr max = max i max m∈S 2 m • e tr i • m. For a given i, max m∈S 2 m • e tr i • m is equal to the maximum eigenvalue of e tr i . The transformation strains listed in TableB1all have the same maximum eigenvalue which can be expressed in terms of the lattice parameters as (α + β + δ + (α -β + δ) 2 + 8 2 )/2. In conclusion, we have β + δ + (α -β + δ) 2 + 8For NiTi, we obtain ε tr max 0.107. The calculation of ε tr min in (B.2) is tricky to perform in closed-form. Using numerical optimization to solve the min-max problem defining ε tr min yields ε tr min 0.024.

Table 1 .

 1 List of symbols for the model proposed

	σ	uniaxial stress
	ε	uniaxial strain
	T	temperature
	E	elastic modulus
	λ(T ) latent heat at temperature T
	T 0	critical temperature of phase transition
	c	specific heat
	s	continuous parameter (in [0, 1]) tracking the crystalline orientations
	c(s)	volume density of the crystalline orientation s
	ε tr

Table 2 .

 2 Examples of function ε

tr (s) Texture function ε tr (s) Inverse function f (x)

Table 4 .

 4 DSC : Measured values

Table 5 .

 5 Model parameter identification -Texture C

	E	28000 MPa
	λ	74.2 MPa
	T 0	294.9 K
	ε tr min	0.024
	ε tr max	0.107
	β	0.3
	γ	0.32
	G 0	3 MPa
	G 1

Table 6 .

 6 Model parameter identification -Texture C*

	E	28000 MPa
	λ	74.2 MPa
	T 0	294.9 K
	ε tr min	0.014
	ε tr max	0.131
	β	0.25
	γ	0.35
	G 0	3 MPa
	G 1	3.5 MPa

Appendix A. Variational principle for monotonic evolution

We denote by v(f ) the sign of variation of a given monotone function f , i.e. v(f ) = 1 if f is increasing and v(f ) = -1 if f is decreasing. For a given history (ε(t), T (t)) on a time interval [0, τ ], we are interested in finding a solution to the evolution equation (33) such that θ i (s, t) is monotonic in time and v(θ i (s, .) = i (s) where i (s) ∈ {-1, 1} is given. To that purpose, for any t in [0, τ ], consider a solution θi (s, t) to the minimization problem

If θi (s, t) satisfies v( θi (s, .)) = i (s), we claim that θi (s, t) is a solution to the evolution problem defined by Eqs ( 30), ( 32) and (33). For any fixed t, the stationarity conditions in (A.1) indeed read as

where (z(s), a 0 (s), a 1 (s)) are submitted to conditions (32). Let B d i (s) = i (s)G i . Since v( θi (s, .)) = i (s), the value B d i (s) thus defined satisfies [START_REF] Bhattacharya | Microstructure of Martensite: Why it Forms and how it Gives Rise to the Shape-memory Effect[END_REF]. Hence (A.2) can be rewritten as

where (z(s), a 0 (s), a 1 (s)) and B d i (s) satisfies the conditions (32) and [START_REF] Bhattacharya | Microstructure of Martensite: Why it Forms and how it Gives Rise to the Shape-memory Effect[END_REF]. The evolution equation ( 33) is thus satisfied at each time, which proves the claim.

A similar result can be obtained for stress-driven loading by replacing the Helmoltz free energy w with the Gibbs free energy w * in the reasoning.

Appendix B. Transformation strains in NiTi

NiTi alloys obey a cubic to monoclinic-I phase transformation. In such case, there are 12 martensitic variants with transformation strains listed in Table B1. Values of the lattice parameters for NiTi 49.75 alloys are α = 0.0234, β = -0.0437, δ = 0.058, = 0.0426 [START_REF] Knowles | Crystallography of the martensitic transformation in equiatomic nickel-titanium[END_REF].