Keywords: Shape Memory Alloy, Constitutive model, Finite deformation, Thermomechanical coupling, Cyclic loading, Numerical simulation

 is extended within a finite-strain thermodynamic framework. Moreover, the transformation strain is decomposed into phase transformation and martensite reorientation components to capture multi-axial non-proportional response. Secondly, in addition to the finite deformation, thermomechanical coupling effect is taken into account by developing a new finite-strain thermomechanical constitutive model. A more straightforward approach is obtained by using the finite Hencky strain. This model incorporates three important thermomechanical characteristics, namely the coexistence effect between austenite and two distinct martensite variants, the variation with temperature of hysteresis size, and the smooth transition at initiation and completion of phase transformation. Finally, with a view to studying SMA behavior under cyclic loading, the model developed in the second step is generalized to describe cyclic pseudoelasticity of polycrystalline SMAs. The generalized model captures four fundamental characteristics related to the cyclic behavior of SMAs: large accumulated residual strain, degeneration of pseudoelasticity and hysteresis loop, rate dependence, and evolution of phase transformation from abrupt to smooth transition. Numerical implementation of these models are realized by introducing proper integration algorithms. Finite element simulations, including orthodontic archwire, helical and torsion spring actuators, are carried out using the proposed models. The future directions of this work mainly involve plasticity and fatigue analysis of SMA structures.

2.1 Cyclic pseudoelastic response of the NiTi wire at strain rate of 1 × 10 -3 s -1 [START_REF] Morin | Thermomechanical coupling in shape memory alloys under cyclic loadings: Experimental analysis and constitutive modeling[END_REF]: (a) stress-strain curves (black for first cycle and red for stabilized cycle), (b) evolutions of the dissipated energy and the reversible strain with the number of cycles. 2.2 Strain rate dependent stress-strain curves in the cyclic experiments [START_REF] Kan | Experimental observations on rate-dependent cyclic deformation of super-elastic NiTi shape memory alloy[END_REF]

Introduction and motivation

In the last few decades, smart materials have been one of the most important subjects in materials science and technology. These materials, due to their unique properties, attracted the attention towards potential innovative applications in various industrial fields. Shape memory alloys (SMAs) are a typical class of smart materials capable of undergoing severe but recoverable inelastic deformation under the influence of appropriate thermomechanical loadings. The most salient features in SMA behavior are undoubtedly their pseudoelasticity (PE) and shape memory effect (SME), which take place at the crystalline scale by means of a first-order diffusionless transformation between austenite and martensite solid phases. These unusual properties promoted the use of SMAs in various industrial applications including aerospace, biomedical, automotive and robotic domains.

In industrial applications, SMA structures are subjected to a variety of service conditions. For some complex loading cases, such as large deformation, thermomechanical coupling, and cyclic loading, complexities of the thermomechanical analysis of these SMA structures increase greatly. First, SMA wires, films and spring actuators typically undergo very large rotations and strains, which will bring about both geometric and material nonlinearities in SMA-based boundary value problems. Second, since the deformation and shape recovery processes of SMAs can be either thermal or stressed induced, SMA structures are typically subjected to thermomechanically coupled boundary and loading conditions. For example, the thermally-activated SMA actuator is subjected to repetitive cooling and heating cycles accompanied with a preloaded force. Third, some SMA structures are typically subjected to cyclic thermomechanical loadings that may induce fatigue failure of the structures after a certain number of cycles. Thermomechanical characteristics related to the cyclic behavior of SMAs, such as the accumulation of residual strain during cycling, as well as the degeneration of pseudoelasticity and hysteresis loop with the number of cycles, are worthy of study for fatigue analysis of the SMA structures. To be able to accurately predict the thermomechanical response of SMA structures under arbitrary complex loading conditions, an in-depth and comprehensive understanding of the constitutive behavior of SMAs is indispensable. Thereby, the development of a reliable constitutive model becomes one of the most challenging subjects for current state of the art on SMAs.

This thesis aims to develop a comprehensive constitutive modeling approach to describe the thermomechanical and cyclic behavior of SMAs in finite-strain framework. The aforementioned three characteristics, i.e., finite deformation, thermomechanical coupling, and cyclic effect, are integrally addressed in this work. Meanwhile, another objective of this work is to numerically implement the constitutive model into finite element software so that it can be used as a computational tool in engineering analysis, design and optimization. The whole work is accomplished by the following three steps.

The first step deals with finite deformations in SMAs. To this end, the small-strain ZM model [START_REF] Zaki | A three-dimensional model of the thermomechanical behavior of shape memory alloys[END_REF][START_REF] Moumni | Theoretical and numerical modeling of solid-solid phase change: Application to the description of the thermomechanical behavior of shape memory alloys[END_REF] is extended, within a finite-strain framework, to describe large strains and rotations in SMAs. Before the finite-strain formulation, a summary of the ZM model is first presented. The development of the finite-strain model consists of the description of kinematics, the establishment of a finite strain thermomechanical framework, and the derivation of constitutive equations. Numerical implementation of the finite-strain model includes the time-discrete formulation of the constitutive equations, the symmetrization of the reorientation evolution equation, and material and spatial tangent operators.

The second step takes into account thermomechanical coupling in addition to finite deformations. To this end, a new 3D thermomechanically coupled, finite-strain constitutive model is developed. Hencky strain is used in the formulation, due to its remarkable properties in finite deformations, leading to a more straightforward modeling approach. The proposed model incorporates three important SMA characteristics: the effect of coexistence between austenite, single-variant and multi-variant martensites; the variation with temperature of the hysteresis size; and the smooth transition at the initiation and completion of phase transformation.

The third step addresses cyclic behavior of SMAs, by generalizing the model developed in the second step. The generalized model captures several fundamental characteristics related to the cyclic behavior of SMA, i.e., large accumulated residual strain, degeneration of pseudoelasticity and hysteresis loop, rate dependence, and evolution of phase transformation from abrupt to smooth transition.

These proposed SMA models are well validated against experimental data under various thermomechanical loading cases. Finite element simulations of SMA structures, such as archwire, helical and torsion spring actuators, are also carried out. The future work mainly involves irreversible plasticity of SMAs undergoing severe mechanical loading, and fatigue analysis of SMA structures.

Overview of shape memory alloys

Shape memory alloys (SMAs) are a class of smart materials that can undergo severe but recoverable inelastic deformation under the influence of appropriate thermomechanical loadings. They have been the focus of intense research effort by the scientific community in the last decades [START_REF] Müller | A thermodynamic finite-strain model for pseudoelastic shape memory alloys[END_REF][START_REF] Moumni | Theoretical and numerical modeling of solid-solid phase change: Application to the description of the thermomechanical behavior of shape memory alloys[END_REF][START_REF] Christ | Finite-element modelling of shape memory alloys-A comparison between small-strain and large-strain formulations[END_REF][START_REF] Lagoudas | Constitutive model for the numerical analysis of phase transformation in polycrystalline shape memory alloys[END_REF][START_REF] Xiao | An explicit, straightforward approach to modeling SMA pseudoelastic hysteresis[END_REF][START_REF] Auricchio | Theoretical and numerical modeling of shape memory alloys accounting for multiple phase transformations and martensite reorientation[END_REF][START_REF] Saleeb | On the modeling of the thermo-mechanical responses of four different classes of NiTi-based shape memory materials using a general multi-mechanism framework[END_REF][START_REF] Machado | Superelastic cellular NiTi tube-based materials: Fabrication, experiments and modeling[END_REF][START_REF] Chatziathanasiou | Modeling of coupled phase transformation and reorientation in shape memory alloys under non-proportional thermomechanical loading[END_REF]Cisse et al., 2016a).

The first discovery of shape memory effect was reported by Ölander (1932) in Au-Cd alloy in 1932. This alloy could not survive large stresses or strains and therefore was not utilized as actuators. The formation and disappearance of the martensite was observed by [START_REF] Greninger | Orientation Habit of Martensite[END_REF] by decreasing and increasing the temperature of a Cu-Zn alloy. The crystallography and thermodynamics of martensite transformation were later investigated by [START_REF] Grange | The temperature range of martensite formation[END_REF], [START_REF] Greninger | The mechanism of martensite formation[END_REF] and [START_REF] Bowles | The crystallography of martensite transformations I[END_REF]. In 1963, the nickel-titanium (NiTi) alloys were first observed having shape memory effect by the United States Naval Ordnance Laboratory [START_REF] Kauffman | Memory metal[END_REF]. Afterwards, several other alloys, such as Ag-Cd, Au-Cd, Cu-Sn, In-Ti, Ni-Al, Ni-Ti, Mn-Cu, were found that exhibit shape memory tendencies. Now, there are three main types of SMAs: Fe-Mn-Si, Cu-Al-Ni and NiTi alloys. Fe-Mn-Si alloys are called as ferromagnetic shape memory alloy (FSMA), which change shape under magnetic fields. Although Fe-Mn-Si and Cu-Al-Ni alloys are commercially available and less expensive, NiTi alloys are regarded as the most promising SMAs for various applications due to their stability and practicability [START_REF] Saadat | An overview of vibration and seismic applications of NiTi shape memory alloy[END_REF][START_REF] Machado | Medical applications of shape memory alloys[END_REF][START_REF] Hartl | Aerospace applications of shape memory alloys[END_REF]. The most salient features in SMA behavior are undoubtedly their pseudoelasticity (PE) and shape memory effect (SME), which take place at the crystalline scale by means of a first-order diffusionless transformation between austenite (A) and martensite (M) solid phases. This unusual behavior attracted the attention towards potential innovative applications of SMAs in engineering, examples of which are orthodontic archwires, endovascular stents, smart circuit breakers, robotic muscles, shape morphing actuators, etc. [START_REF] Williams | An automotive SMA mirror actuator: modeling, design, and experimental evaluation[END_REF][START_REF] Auricchio | Shape-memory alloys: effective 3D modelling, computational aspects and design of devices[END_REF][START_REF] Gu | Time integration and assessment of a model for shape memory alloys considering multiaxial nonproportional loading cases[END_REF]. Recently, [START_REF] Lexcellent | Shape-memory alloys handbook[END_REF] gave a detailed analysis, both physical and mechanical, properties of SMA materials. [START_REF] Mohd | A review of shape memory alloy research, applications and opportunities[END_REF] provided a review of SMA research, applications and opportunities. [START_REF] Hartl | Standardization of shape memory alloy test methods toward certification of aerospace applications[END_REF] introduced a specification and standardization effort of SMAs as employed as actuators for commercial and military aviation applications.

Martensitic transformation

The martensitic transformation (MT) is a first-order diffusionless solid-solid phase transformation occurring at lattice scale from a parent austenite phase, stable at high temperature and low stress, to a product martensite phase, metastable at low temperature and high stress [START_REF] Otsuka | Shape memory materials[END_REF]Cisse et al., 2016a). The crystal structure of austenite is generally a perfect cube and thereby has the higher crystallography symmetry, as shown in Fig. 1.1. In contrast, martensite has tetragonal, orthorhombic or monoclinic crystal structure and thereby lower crystallography symmetry [START_REF] Arghavani | Thermo-mechanical behavior of shape memory alloys under multiaxial loading: constitutive modeling and numerical implementation at small and finite strains[END_REF]. The lower symmetry of this phase enables the existence of several martensite variants characterized by their orientation with respect to the interface with austenite called "habit plane". The assembly of martensite variants may exist in two forms: multi-variant (twinned) martensite, in which variants with different orientations are "self-accommodated", and single-variant (detwinned) martensite, in which a specific variant is dominant. More crystallographic theories for MT are available in the literature [START_REF] Bowles | The crystallography of martensite transformations I[END_REF][START_REF] Otsuka | Shape memory materials[END_REF][START_REF] Sadjadpour | A micromechanics inspired constitutive model for shape-memory alloys: the one-dimensional case[END_REF][START_REF] Lexcellent | Two ways for predicting the hysteresis minimisation for shape memory alloys[END_REF]. This inherent phase transformation (PT) can be either thermally-induced or stress-induced. Fig. 1.2 shows a phase diagram with control variables of stress and temperature, in which MT can occur under different thermomechanical loading paths. Upon cooling at zero stress (path 1), austenite transforms to multi-variant martensite, termed the forward transformation (A→M). This phase transformation is initiated at the "martensite start temperature" M 0 s , and completed at the "martensite finish temperature" M 0 f . During heating, the multi-variant martensite transforms back to the austenite, termed the reverse transformation (M→A), which is characterized by the "austenite start temperature" A 0 s and the "austenite finish temperature" A 0 f . If a mechanical load is applied to the multi-variant martensite material at low temperature (path 2), the martensite will be detwinned by reorienting the variants in a certain direction depending on the applied stress. The "detwinning start stress" σ s and the "detwinning finish stress" σ f correspond to the stress levels at the initiation and the completion of the detwinning process. This martensite reorientation process results in a macroscopic deformation. A large part of this deformation remains after unloading. When the material is cooled with a constant applied stress σ, the austenite phase will transform directly to the single-variant martensite, producing a macroscopic deformation. Reheating the material will result in reverse transformation and thereby shape recovery. The loading path is shown in Fig. 1.2 (path 3). The characteristic temperatures M σ f , M σ s , A σ s , and A σ f represent, respectively, martensite finish, martensite start, austenite start, and austenite finish temperatures, which strongly depend on the applied stress level σ. Phase transformation from austenite to single-variant martensite can be also stress-induced by applying a sufficiently high mechanical load on the material in austenite phase (path 4). This phase transformation is initiated at the "martensite start stress" σ M s and completed at the "martensite finish stress" σ M f . When the load is released, the single-variant martensite will transform back to austenite. σ A s and σ A f represent the "austenite start stress" and "austenite finish stress". These characteristic stresses significantly depend on the temperature.

Shape memory effect

Shape memory effect (SME) is probably the most well-known property of SMA. It consists in the ability of the SMA to recover its original shape upon heating. This phenomenon is illustrated in the stress-strain-temperature diagram, as shown in Fig. 1.3. Starting from the parent phase (austenite), the stress-free material is cooled below the forward transformation temperatures (M 0 s and M 0 f ), resulting in the formation of multi-variant martensite. When subjected to an applied stress, the product multivariant martensite is reoriented to single-variant martensite. The reorientation process is initiated at the detwinning start stress σ s , and completed at the detwinning finish stress σ f , which is characterized by a transformation plateau in the stress-strain curve. It should be noted that the stress level for reorientation of the martensite variants is far lower than the permanent plastic yield stress of martensite. The applied stress is then released and the single-variant martensite is retained. Upon heating in the absence of stress, the reverse transformation is initiated at A 0 s and completed at A 0 f . Finally, the single-variant martensite completely transforms to the parent austenite phase, and the material recover its original shape. The shape memory cycle can be repeated if the material is again cooled to multi-variant martensite.

In addition to the above one-way shape memory effect (OWSME), the two-way shape memory effect (TWSME) is the ability of the material to remember its shape at both low and high temperatures. It is not an intrinsic property in SMAs, but is rather acquired by "training". Early work on the TWSME was reported by [START_REF] Perkins | Residual stresses and the origin of reversible (two-way) shape memory effects[END_REF], [START_REF] Miyazaki | Transformation pseudoelasticity and deformation behavior in a Ti-50.6 at% Ni alloy[END_REF] and [START_REF] Liu | Factors influencing the development of two-way shape memory in NiTi[END_REF]. Generally, the TWSME can be induced by thermomechanical cyclic loadings. The physical mechanism for the [START_REF] Perkins | Stress-induced martensitic transformation cycling and two-way shape memory training in Cu-Zn-Al alloys[END_REF], [START_REF] Tadaki | Shape memory alloys[END_REF] and [START_REF] Lovey | On the origin of the two way shape memory effect in Cu-Zn-Al alloys[END_REF] as follows: unrecoverable strain at the end of each loading cycle accumulates during training until saturation. The resulting permanent dislocations, defects and the residual internal stress stabilize a fraction of the martensite plates which are then identically recreated during subsequent transformations, thus producing macroscopic strain. This allows a trained SMA to switch configurations between stable austenite and oriented martensite phases by heating or cooling in the absence of external mechanical load. Due to the training requirements and to the fact that it averagely produces only half of the recovery strain compared to the OWSMA for the same material and its strain tends to deteriorate quickly, TWSMA is less applied commercially [START_REF] Schroeder | The two-way shape memory effect and other "training" phenomena in Cu-Zn single crystals[END_REF][START_REF] Huang | Training two-way shape memory alloy by reheat treatment[END_REF][START_REF] Ma | High temperature shape memory alloys[END_REF].

Pseudoelasticity

Pseudoelasticity (PE), sometimes referred to as superelasticity, is associated with the stress-induced phase transformation between austenite and single-variant martensite in SMAs. Fig. 1.4 shows a typical pseudoelastic stress-strain response of polycrystalline SMAs. It is featured with large reversible strain and hysteresis loop. The thermomechanical loading path for such an example is shown as path 4 in Fig. 1.2. At temperature above A 0 f , the material in austenite phase undergoes elastic deformation when a mechanical load is applied. The phase transformation from austenite to single-variant martensite is initiated at stress level of σ M s and completed at σ M f . This stress-induced transformation is accompanied by the generation of large transformation strains, as shown in the stress-strain curve of Fig. 1.4. A subsequent increase in the stress produces no further transformation but only the elastic deformation of the product single-variant martensite. Upon unloading, the reverse transformation is initiated at stress level of σ A s and completed at σ A f , accompanied by the recovery of the transformation strain. SMA reverts to its original shape when the mechanical load is completely released. It is noteworthy that the transformation stresses and the size of the hysteresis loop are significantly influenced by the testing temperature, as shown by path 4 in Fig. 1.2.

Application of SMAs

Since the 1960s, SMAs attracted great attention of scientists and engineers worldwide. Many SMAbased structures and systems were designed for various industrial applications. According to the primary function of the shape memory element, these applications can be classified into the following five categories:

• Free recovery. The one way or two way shape memory effects can be used for free recovery applications. The sole function of the SMA element is to cause motion or shape change without any externally applied stress.

• Constrained recovery. The constrained SMA element is prevented from changing shape and therefore generates a stress or force on the applications. This recovery stress can be used for clamping and coupling devices.

• Actuation. Since the SMA element can generate motion against the externally applied stress, it exhibits work production and thermal energy is directly converted into mechanical energy.

• Pseudoelasticity. This characteristic is mostly used in biomedical applications wherein the SMA element undergoes large reversible strains with high stress plateaus.

• Damping. The high damping capacity of the stress-induced martensite transformation is interesting for passive damping. The pseudoelastic hysteresis can be used for energy absorption uses.

Introduction

In the last three decades, SMAs have found many applications in industrial activities. The relevant SMA applications are presented in this section, particularly from the aerospace, biomedical, automotive and robotic domains.

Aerospace applications

The first and most well-known SMA application in aerospace domain was the hydraulic tube coupling used on the F-14 fighter jets in 1970s [START_REF] Otsuka | Shape memory materials[END_REF]. Since then the unique characteristics of SMAs were exploited in aerospace applications, which are mostly subjected to high dynamic loads and geometric space constraints [START_REF] Mohd | A review of shape memory alloy research, applications and opportunities[END_REF]. These applications include actuators [START_REF] Giacomel | Shape memory alloy actuator[END_REF][START_REF] Hartl | Aerospace applications of shape memory alloys[END_REF], couplings and fasteners [START_REF] Otsuka | Shape memory materials[END_REF], release or deployment mechanisms [START_REF] Huett | Design and development of miniature mechanisms for small spacecraft[END_REF][START_REF] Cleveland | Apparatus and method for releaseably joining elements[END_REF], vibration dampers, and manipulators [START_REF] Birman | Review of mechanics of shape memory alloy structures[END_REF][START_REF] Prahlad | Design of a variable twist tilt-rotor blade using shape memory alloy (SMA) actuators[END_REF].

Since the 1990s, aerospace communities mainly focused on the active and adaptive structures toward morphing capability and system-level optimization under various flight conditions. The most well-known examples are the Smart Wings program from Defense Advanced Research Projects Agency (DARPA) [START_REF] Kudva | Overview of the DARPA smart wing project[END_REF], and the Smart Aircraft and Marine Propulsion System Demonstration (SAMPSON) program [START_REF] Pitt | SAMPSON smart inlet SMA powered adaptive lip design and static test[END_REF]. To reduce the engine noise during take off and landing, designers from Boeing developed an active serrated aerodynamic device, known as a Variable Geometry Chevron (VGC), and installed on a GE90-115B jet engine for the Boeing 777-300 ER commercial aircraft (Hartl et al., 2009b,a;[START_REF] Oehler | Design optimization and uncertainty analysis of SMA morphing structures[END_REF]. Afterwards, different VGC solutions were proposed by Boeing, DARPA, NASA and other relevant research agencies, which were reviewed by [START_REF] Calkins | Shape memory alloy based morphing aerostructures[END_REF]. [START_REF] Sofla | Shape morphing of aircraft wing: status and challenges[END_REF] also provided a review of aircraft morphing wing technologies, and proposed a shape morphing wing design for small aircraft. In rotorcraft applications, SMA-based technologies have been used in rotor blade twisting [START_REF] Prahlad | Design of a variable twist tilt-rotor blade using shape memory alloy (SMA) actuators[END_REF][START_REF] Caldwell | Heat transfer model for blade twist actuator system[END_REF], rotor blade tracking tab [START_REF] Singh | Design of an improved shape memory alloy actuator for rotor blade tracking[END_REF][START_REF] Kennedy | Development of an SMA actuator for in-flight rotor blade tracking[END_REF], rotor control [START_REF] Loewy | Recent developments in smart structures with aeronautical applications[END_REF], and rotor blade trip morphing [START_REF] Testa | Feasibility study on rotorcraft blade morphing in hovering[END_REF]. Meanwhile, SMAs have been used in spacecraft applications to address problems related to actuation and low-shock release mechanisms in zero atmosphere environment [START_REF] Johnson | Non-explosive separation device[END_REF][START_REF] Birman | Review of mechanics of shape memory alloy structures[END_REF][START_REF] Carpenter | EO-1 technology validation report: lightweight flexible solar array experiment[END_REF], as well as vibration dampers and isolators during spacecraft launch [START_REF] Saadat | An overview of vibration and seismic applications of NiTi shape memory alloy[END_REF]. In addition, considerable SMA applications for aerospace, such as telescopic wing system [START_REF] Knowles | Telescopic wing system[END_REF], retractable landing gear [START_REF] Kutlucinar | Aircraft with shape memory alloys for retractable landing gear[END_REF], jet engine components [START_REF] Song | Shape memory alloy actuated adaptive exhaust nozzle for jet engine[END_REF], and flap edge fence [START_REF] Larssen | Deployable flap edge fence[END_REF], have been developed to exploit SMA advantages.

Biomedical applications

The shape memory and pseudoelastic characteristics, coupled with the biocompatibility, make NiTi alloys the most promising candidates for biomedical applications, such as orthodontic wires and drills [START_REF] Auricchio | A robust integration-algorithm for a finite-strain shape-memory-alloy superelastic model[END_REF][START_REF] Lagoudas | Shape Memory Alloys: Modeling and Engineering Applications[END_REF], filters [START_REF] Poletti | Long-term results of the Simon nitinol inferior vena cava filter[END_REF], stents [START_REF] Dotter | Transluminal expandable nitinol coil stent grafting: preliminary report[END_REF][START_REF] Itzhak'vinograd | A new intratracheal stent made from nitinol, an alloy with" shape memory effect[END_REF], intervertebral spacer [START_REF] Duerig | Superelastic nitinol for medical devices[END_REF], orthopedic staples (Laster et al., 1.3 Application of SMAs 2001), and surgical instruments [START_REF] Sanders | Nitinol spinal instrumentation and method for surgically treating scoliosis[END_REF]. According to the different uses, they are classified into four categories: orthodontic, cardiovascular, orthopedic, and surgical instruments applications.

Since the 1970s, NiTi orthodontic archwires have been successfully used in a variety of dental applications. They were proven to be more effective than other alternative materials because of their pseudoelastic plateau, during which a nearly constant and moderate force was provided over a larger deformation and thereby a longer period of time compared with stainless steel [START_REF] Andreasen | An evaluation of 55 cobalt substituted Nitinol wire for use in orthodontics[END_REF]. Another key dental application involves the use of NiTi drills in root canal surgery [START_REF] Cheung | Fatigue testing of a NiTi rotary instrument. Part 1: strain-life relationship[END_REF][START_REF] Lagoudas | Shape Memory Alloys: Modeling and Engineering Applications[END_REF]. The advantage of the NiTi drills is that they can with stand the high cyclic rotations with large bending and strains. The most common cardiovascular application is the self-expanding NiTi stent [START_REF] Cwikiel | Self-expanding stent in the treatment of benign esophageal strictures: experimental study in pigs and presentation of clinical cases[END_REF][START_REF] Yanagihara | Tracheal stenosis treated with self-expanding nitinol stent[END_REF][START_REF] Pelton | Fatigue and durability of Nitinol stents[END_REF]. This device is used to support the inner circumference of tubular passages in the body such as blood vessels. Compared to conventional balloon inflated stainless steel stents, NiTi stents can gently expand the blood vessels to the appropriate diameter. Another cardiovascular SMA devices are the Simon filters for trapping clots traveling in the blood stream [START_REF] Engmann | Clinical experience with the antecubital Simon nitinol IVC filter[END_REF], and the atrial septal occlusion for sealing the atrial hole located on the surface dividing the upper heart chambers [START_REF] Thanopoulos | Closure of atrial septal defects with the Amplatzer occlusion device: preliminary results[END_REF][START_REF] Chan | Transcatheter closure of atrial septal defect and interatrial communications with a new self expanding nitinol double disc device (Amplatzer septal occluder): multicentre UK experience[END_REF]. For orthopedic applications, the SMA devices are mostly used to support injured, weakened or fractured bones. These applications include intervertebral spacer used to provide local reinforcement to the vertebra [START_REF] Duerig | Superelastic nitinol for medical devices[END_REF], porous SMAs used as artificial bone implants [START_REF] Lagoudas | Processing and characterization of NiTi porous SMA by elevated pressure sintering[END_REF], orthopedic staples used to provide a compressive force at the interface of the separated bones [START_REF] Krone | The Potential of Powder Metallurgy for the Fabrication of Biomaterials on the Basis of Nickel-Titanium: A Case Study with a Staple Showing Shape Memory Behaviour[END_REF], and shape memory plates to hold the fractured joints together [START_REF] Tawfik | Thermal post-buckling and aeroelastic behaviour of shape memory alloy reinforced plates[END_REF]. Moreover, many SMA-based surgical instruments, including SMA baskets and pseudoelastic guide wires, have been introduced in Minimally Invasive Surgery (MIS), in which the instruments are inserted through small openings and subsequently expanded to a desired size [START_REF] Lim | Future of active catheters[END_REF][START_REF] Duerig | Superelastic nitinol for medical devices[END_REF][START_REF] Song | History and current situation of shape memory alloys devices for minimally invasive surgery[END_REF].

Automotive applications

In modern automotive industry, SMAs have been extensively used for applications ranging from sensors and actuators to vibration dampers and impact absorption uses, due to the demand of safer, more comfortable vehicles with better performance. The drive-by-wire technology offers a wide range of opportunities for SMA actuators as an alternative to conventional electromagnetic actuators in automotive applications. The mechanical simplicity and compactness of SMA actuators significantly reduce the scale, weight and cost of automotive components and provide substantial performance benefits, as demonstrated by [START_REF] Neugebauer | Numerical simulation of the activation behavior of thermal shape memory alloys[END_REF]. General Motors (GM) claimed that their 2014 Chevrolet Corvette became the first vehicle to incorporate SMA actuators to actuate the hatch vent that releases air from the trunk for easier closing of the trunk lid [START_REF] Mohd | A review of shape memory alloy research, applications and opportunities[END_REF]. SMAs can also be used for sensor and actuator purposes simultaneously as the SMA spring for continuous variable transmission system, wherein the spring acts as a sensor that monitors the temperature and actuates a valve at a specific temperature to adjust the oil flow [START_REF] Bhattacharyya | On the role of thermoelectric heat transfer in the design of SMA actuators: theoretical modeling and experiment[END_REF]. The pseudoelastic hysteresis Introduction behavior provides an effective system to dissipate vibration and impact. Taken the advantage of this property, SMAs are used in vibration isolation and impact absorption applications [START_REF] Paine | High velocity impact response of composites with surface bonded nitinol-SMA hybrid layers[END_REF][START_REF] Barnes | Panel deployment using ultrafast SMA latches[END_REF]. Several other SMA applications in automotive industry are the automatic pedestrian protection system to minimize pedestrian injuries during impact collisions [START_REF] Strittmatter | Long-time stability of shape memory actuators for pedestrian safety system[END_REF], the micro-scanner system for optical sensing of objects distance and angle [START_REF] Brugger | Ferromagnetic shape memory microscanner system for automotive applications[END_REF], the SMA activated automotive tumble flaps [START_REF] Bellini | Mechatronic design of a shape memory alloy actuator for automotive tumble flaps: a case study[END_REF], and a cost effective side mirror actuator [START_REF] Williams | Control of an automotive shape memory alloy mirror actuator[END_REF].

Robotic applications

With the increasing trends towards miniaturization of robotic system, SMAs have been exploited as micro-actuators and artificial muscles in a diverse range of commercial robotic systems. [START_REF] Kheirikhah | A Review of Shape Memory Alloy Actuators in Robotics[END_REF] recently presented a review of SMA applications in robotics' different branches such as crawler robots, jumper robots, flower robots, fish robots, walker robots, medical robots, and biomimetic robotic hand. The ideology of these researches is to utilize SMA actuators to design and develop innovative, lightweight, powerful, compact, and dexterous robotic system. [START_REF] Nishida | Development and control of a micro biped walking robot using shape memory alloys[END_REF] proposed a lightweight micro biped walking robot incorporating multiple SMA-based Flexible Flat Plates (FFPs) that consist of polyethylene plates and SMAs. A new biomimetic tendon-driven actuation system for prosthetic and wearable robotic hand applications was presented by [START_REF] Bundhoo | A shape memory alloy-based tendon-driven actuation system for biomimetic artificial fingers, part I: design and evaluation[END_REF], based on the combination of compliant tendon cables and one-way SMA wires that form a set of artificial muscle pairs for the required flexion/extension or abduction/adduction of the finger joints. Later, a novel sensory system for robotics was developed by [START_REF] Tuna | Object shape recognition with artificial whiskers using tomographic reconstruction[END_REF], they employed the pseudoelastic characteristic of a NiTi SMA to create an artificial rate whisker by utilizing the rat's sensing capabilities. [START_REF] Ali | Frequency-controlled wireless shape-memory-alloy microactuators integrated using an electroplating bonding process[END_REF] also developed passive micro-grippers that can be actuated wirelessly with magnetic field. The opportunity to control multiple selections of micro-SMA actuators became possible by applying different resonant frequencies. In addition to these, several other SMA applications are in fish robots [START_REF] Tao | Bio-inspired actuating system for swimming using shape memory alloy composites[END_REF][START_REF] Cho | Design, fabrication and analysis of a body-caudal fin propulsion system for a microrobotic fish[END_REF]Wang et al., 2008b), walker robots (Carlo and[START_REF] Metin | A biomimetic climbing robot based on the gecko[END_REF][START_REF] Nishida | Development and control of a micro biped walking robot using shape memory alloys[END_REF][START_REF] Berry | Bio-inspired shape memory alloy actuated hexapod robot[END_REF], and flying robots [START_REF] Furst | Design and fabrication of a bat-inspired flapping-flight platform using shape memory alloy muscles and joints[END_REF][START_REF] Colorado | Biomechanics of smart wings in a bat robot: morphing wings using SMA actuators[END_REF].

Research objectives

In industrial applications, SMA structures are typically subjected to complex thermomechanical boundary and loading conditions. Moreover, some of these SMA structures, such as SMA wires, films and spring actuators, typically undergo very large strains and rotations. A good understanding of constitutive relation of SMAs, especially in finite deformation regime, is indispensable to study the thermomechanical behavior of these SMA structures, and further to improve the reliability and durability of the SMA systems. To this end, the objectives of this research are: (i) to develop a comprehensive constitutive modeling approach capable of describing diverse SMA behavior under complex thermomechanical 1.5 Outline of dissertation boundary and loading conditions as well as large strains and rotations; (ii) to numerically implement the SMA models into finite element software so that it can be used as a computational tool in engineering analysis, design and optimization. The objectives are accomplished by the following three aspects:

• Theoretical modeling. The small-strain ZM model [START_REF] Zaki | A three-dimensional model of the thermomechanical behavior of shape memory alloys[END_REF] is extended, within a finite-strain framework, to account for large strains and rotations. Afterwards, to account for thermomechanical coupling effect, a new 3D finite-strain thermomechanical constitutive model is developed. The Hencky strain is used in the formulation for the straightforwardness of modeling approach. This model is then generalized to describe cyclic behavior of SMAs.

• Numerical implementation. With the introduction of the proper numerical integration algorithms, the proposed models are implemented into finite element software by means of user-defined material subroutines.

• Finite element simulation. The proposed SMA models are first validated against the experimental data under various thermomechanical boundary and loading conditions. Then, finite element simulations of SMA structures undergoing complex thermomechanical loadings as well as large strains and rotations, such as orthodontic archwire, helical and torsion spring actuators, are carried out using the proposed models.

Outline of dissertation

In this thesis, a comprehensive constitutive modeling approach is developed to investigate the thermomechanical and cyclic behavior of SMAs in finite deformations. After the motivation and introduction in Chapter 1 , the rest of the thesis is organized as follows:

• In Chapter 2, a literature review of constitutive modeling of SMAs is presented. Depending on the scale of the continuum, the existing SMA models are classified into microscopic, micro-macro and macroscopic models. The macroscopic models, also referred to as phenomenological models, are discussed in detail, respectively, with respect to finite deformation, thermomechanical coupling and cyclic behavior.

• In Chapter 3, a summary of the ZM model proposed by [START_REF] Zaki | A three-dimensional model of the thermomechanical behavior of shape memory alloys[END_REF] is first presented, which is then generalized, within a finite-strain thermodynamical framework, to account for large strains and rotations. The kinematics of the model features a two-tier decomposition of the deformation gradient. Constitutive equations are derived from a Helmholtz free energy and their time-discrete formulation is presented. Numerical algorithm of the model involves a logarithmic map integration, a symmetrization of the reorientation evolution equation, and material and spatial tangent operators. The integration algorithm is then implemented into Abaqus/Standard by means of a implicit user-defined material subroutine (UMAT). Model predictions are validated against Introduction experimental data under various loading conditions. The model is also used to simulate a SMA archwire subjected to large strains and rotations.

• In Chapter 4, in order to take into account thermomechanical coupling, a new 3D thermomechanically coupled, finite-strain model is developed based on Hencky strain. The kinematics description of the model characterizes a decomposition of the Hencky strain into elastic, transformation and thermal parts, and a decomposition of the transformation stretching into phase transformation and martensite reorientation components. The model incorporates three important SMA characteristics: the effect of coexistence between austenite, single-variant and multi-variant martensites; the variation with temperature of the hysteresis size; and the smooth transition at the initiation and completion of phase transformation. Finite element simulation of a SMA helical spring actuator subjected to complex thermomechanical loading and boundary conditions is carried out using the proposed model.

• In Chapter 5, the model developed in Chapter 4 is generalized to describe cyclic behavior of polycrystalline SMAs in finite deformations. The model captures several fundamental characteristics related to the cyclic behavior of SMA, i.e., large accumulated residual strain, degeneration of pseudoelasticity and hysteresis loop, rate dependence, and evolution of phase transformation from abrupt to smooth transition. The 3D model is reduced to a 1D version, and used to simulate the cyclic tensile experiments on NiTi wire at different loading rates under the assumption of uniform temperature field. Moreover, finite element simulation of a torsion spring undergoing large cyclic deformations is carried out, which shows the potential of the model for fatigue analysis of SMA structures.

Finally, in Chapter 6 a conclusion of our research as well as several future directions of the work are given.

Chapter 2

Literature review of the constitutive modeling of SMAs

Constitutive modeling approaches

Over the last three decades, substantial research was dedicated to developing constitutive models that can accurately describe the thermomechanical behavior of SMAs. Depending on the scale of the continuum, the existing models can be classified into microscopic, micro-macro and macroscopic models. Reviews of these models can be found in the literature of [START_REF] Lagoudas | Shape memory alloys, Part II: Modeling of polycrystals[END_REF], [START_REF] Khandelwal | Models for Shape Memory Alloy Behavior: An overview of modeling approaches[END_REF], [START_REF] Lexcellent | Shape-memory alloys handbook[END_REF] and Cisse et al. (2016a,b).

Microscopic models

Microscopic models are developed at the lattice or grain-crystal levels to describe microstructural features in SMA behavior such as phase nucleation-completion, interface motion, martensite twin growth, etc. [START_REF] Arghavani | Thermo-mechanical behavior of shape memory alloys under multiaxial loading: constitutive modeling and numerical implementation at small and finite strains[END_REF][START_REF] Guthikonda | Modeling martensitic phase transformations in shape memory alloys with the self-consistent lattice dynamics approach[END_REF]. They are mostly constructed by means of the Ginzburg-Landau (GL) theory or molecular dynamics (MD) Cisse et al. (2016a).

The microscopic models based on the GL theory are derived from a polynomial energy expression, usually written in terms of observable variables such as strain and temperature. [START_REF] Falk | Model free energy, mechanics, and thermodynamics of shape memory alloys[END_REF][START_REF] Falk | Ginzburg-Landau theory of static domain walls in shape-memory alloys[END_REF] proposed the earliest polynomial potential models for SMAs. In his work, a Landau-Devonshire-like free energy based on similarity of the electromagnetic curves of ferromagnetic materials was used to describe phase transformation and shape memory effect. The GL theory was later used to explore a theoretical approach to fine phase mixtures [START_REF] Ball | Fine phase mixtures as minimizers of energy[END_REF], to describe the atomic lattice vibrations [START_REF] Levitas | Three-dimensional Landau theory for multivariant stress-induced martensitic phase transformations. I. Austenite-martensite[END_REF], to solve the kinetic equations with FEM [START_REF] Idesman | Finite element simulations of martensitic phase transitions and microstructures based on a strain softening model[END_REF][START_REF] Zhong | Phase-field modeling of martensitic microstructure in NiTi shape memory alloys[END_REF] and to describe the mesoscale phase field [START_REF] Wang | Thermo-mechanical wave propagations in shape memory alloy rod with phase transformations[END_REF].

Molecular dynamics (MD) is a computer simulation method for studying the physical movements of atoms and molecules. It originated within the field of theoretical physics in the late of 1950s but now is mostly used in chemical physics and materials science [START_REF] Alder | Studies in molecular dynamics. I. General method[END_REF][START_REF] Rahman | Correlations in the motion of atoms in liquid argon[END_REF] Literature review of the constitutive modeling of SMAs 1964). Newton's equations are numerically solved to determine the motions of particles for a system consisting of a finite number of interacting particles. Depending the potential energy of interaction between particles considered, the MD models can be divided into the the embedded-atom-method (EAM) [START_REF] Daw | Embedded-atom method: Derivation and application to impurities, surfaces, and other defects in metals[END_REF][START_REF] Lai | Lattice stability of some Ni-Ti alloy phases versus their chemical composition and disordering[END_REF][START_REF] Uehara | Molecular dynamics simulation of shape memory behaviour using a multi-grain model[END_REF][START_REF] Mutter | Simulation of the shape memory effect in a NiTi nano model system[END_REF] and Lennard-Jones (LJ) [START_REF] Suzuki | A simple model for martensitic transformation[END_REF][START_REF] Deng | Origin of ultrafast annihilation effect of martensite aging: Atomistic simulations[END_REF][START_REF] Zhong | Patterning of martensitic nanotwins[END_REF] potentials.

These microscopic models are powerful and promising to understand the fundamental phenomena at the microscopic level. However, the use of these models for engineering applications remains impractical to-date because of the inconvenient scale-transition and the high computation cost.

Micro-macro models

Micro-macro models rely on micromechanics to describe the microscopic or mesoscopic material behavior. Macroscopic constitutive equations are derived by the method of scale transition [START_REF] Sun | Micromechanics modelling for the constitutive behavior of polycrystalline shape memory alloys-I. Derivation of general relations[END_REF][START_REF] Blanc | Micromechanical modelling of a CuAlNi shape memory alloy behaviour[END_REF][START_REF] Patoor | Shape memory alloys, Part I: General properties and modeling of single crystals[END_REF][START_REF] Levitas | Micromechanical modeling of stress-induced phase transformations. Part 1. Thermodynamics and kinetics of coupled interface propagation and reorientation[END_REF][START_REF] Khater | Dislocation glide in Fe-carbon solid solution: From atomistic to continuum level description[END_REF]. The development of these models requires the use of suitable observable variables (such as strain ε, stress σ and temperature θ), and internal variables (such as the martensitic volume fraction χ and the mean transformation strain εt ). Generally, the micro-macro models are developed by means of the micro-mechanical and micro-plane/micro-sphere approaches.

The essence of micro-mechanical approach lies in the modeling of a single grain and averaging the behavior of the grains using scale transition techniques to describe the macroscopic material response of polycrystalline SMAs. The scale transition can be accomplished using the Mori-Tanaka scheme [START_REF] Mori | Average stress in matrix and average elastic energy of materials with misfitting inclusions[END_REF][START_REF] Sun | Micromechanics constitutive description of thermoelastic martensitic transformations[END_REF][START_REF] Brassart | An extended Mori-Tanaka homogenization scheme for finite strain modeling of debonding in particle-reinforced elastomers[END_REF], the self consistent scheme [START_REF] Patoor | Thermomechanical behavior of shape memory alloys[END_REF][START_REF] Gall | The role of texture in tension-compression asymmetry in polycrystalline NiTi[END_REF][START_REF] Guthikonda | Modeling martensitic phase transformations in shape memory alloys with the self-consistent lattice dynamics approach[END_REF], the uniform stress (Sachs) or uniform strain (Taylor) mix approaches [START_REF] Šittner | Anisotropy of martensitic transformations in modeling of shape memory alloy polycrystals[END_REF][START_REF] Thamburaja | Polycrystalline shape-memory materials: effect of crystallographic texture[END_REF][START_REF] Hackl | A micromechanical model for pretextured polycrystalline shape-memory alloys including elastic anisotropy[END_REF], and the finite element technique [START_REF] Lim | Cyclic thermomechanical behavior of a polycrystalline pseudoelastic shape memory alloy[END_REF][START_REF] Anand | Thermal effects in the superelasticity of crystalline shape-memory materials[END_REF][START_REF] Sengupta | Multiscale finite element modeling of superelasticity in Nitinol polycrystals[END_REF][START_REF] Junker | Finite element simulations of poly-crystalline shape memory alloys based on a micromechanical model[END_REF]. This approach was used to develop polycrystalline SMA models considering various features, such as plastic deformation [START_REF] Kudoh | Crystal structure of the martensite in Ti-49.2 at.% Ni alloy analyzed by the single crystal X-ray diffraction method[END_REF]Wang et al., 2008a;[START_REF] Yu | Micromechanical constitutive model considering plasticity for super-elastic NiTi shape memory alloy[END_REF], grain-size effect [START_REF] Stupkiewicz | Grain-size effect in micromechanical modelling of hysteresis in shape[END_REF], transformation degeneration and shakedown [START_REF] Yu | A micromechanical constitutive model based on crystal plasticity for thermo-mechanical cyclic deformation of NiTi shape memory alloys[END_REF](Yu et al., , 2015a) and so on.

The micro-plane theory was introduced by [START_REF] Taylor | The plastic distortion of metals[END_REF] to describe the multiaxial macroscopic behavior of polycrystalline SMAs as a superposition of uniaxial responses with several micro-planes of different orientations under the assumption of static constraints or kinematic constraints. Sometimes the static and kinematic constraints are combined to get a so-called double constrained model [START_REF] Bažant | Microplane model M4 for concrete. I: Formulation with work-conjugate deviatoric stress[END_REF]. Meanwhile, micro-sphere models are based on the projection of the macroscopic strain or stress onto directions that are normal to a micro-sphere (Ostwald et al., 2010b,a). To obtain the evolution of the volume fractions of the different phases, statistical physics and the 1D constitutive model proposed by [START_REF] Govindjee | A computational model for shape memory alloys[END_REF] were employed. This approach was subsequently used to simulate the pseudoelastic and pseudoplastic behavior [START_REF] Ostwald | A Gibbs-energy-barrier-based computational micro-sphere model for the simulation of martensitic phase-transformations[END_REF] as well as the cyclic effect on SMAs [START_REF] Ostwald | An energy-barrier-based computational micro-sphere model for phase-transformations interacting with plasticity[END_REF].

Micro-macro models appear to allow accurate simulations, and in some cases they can successfully predict the reorientation and detwinning of martensite variants. However, they employ a large number of state variables and model parameters, which makes computation expense challenging and results in difficulty in engineering application.

Macroscopic models

Macroscopic models are based on simplified micro-macro thermodynamics, phenomenological thermodynamics or direct experimental data fitting. Suitable assumptions and approximations are commonly made to aim at capturing macroscopic SMA behavior such as pseudoelasticity, one-way or two-way shape memory effect, martensitic reorientation, tension-compression asymmetry and accumulation of residual strain. Plasticity models, thermodynamic potential models and hysteresis models are three main macroscopic modeling approaches [START_REF] Khandelwal | Models for Shape Memory Alloy Behavior: An overview of modeling approaches[END_REF]Cisse et al., 2016a).

The plasticity models are developed by analogy with a well-established elastoplasticity approach. In these models, the thermodynamic processes such as phase transformation and detwinning of the martensite variants are governed by loading functions and flow rules. The evolution rules normally have a certain structure and obey consistency conditions with the loading functions. [START_REF] Souza | Three-dimensional model for solids undergoing stressinduced phase transformations[END_REF] proposed a 3D phenomenological model that can describe the mechanical behavior of polycrystalline CuZnAlMn alloys undergoing stress-induced phase transformation. The model was built on the assumptions of an admissibility condition for thermodynamic forces and a locking constraint for phase transformations. This work was later improved by [START_REF] Auricchio | Improvements and algorithmical considerations on a recent three-dimensional model describing stress-induced solid phase transformations[END_REF] who proposed a robust algorithm to simulate the SME and PE under non-proportional loading. [START_REF] Boyd | A thermodynamical constitutive model for shape memory materials. Part I. The monolithic shape memory alloy[END_REF] proposed the first model that accounts for martensite reorientation by introducing an inelastic strain tensor, which is experimentally validated by [START_REF] Lagoudas | A unified thermodynamic constitutive model for SMA and finite element analysis of active metal matrix composites[END_REF]. Recently, [START_REF] Lagoudas | Constitutive model for the numerical analysis of phase transformation in polycrystalline shape memory alloys[END_REF] proposed a model that accounts for smooth response of polycrystalline SMA using nonlinear hardening functions.

For the models derived from the thermodynamic potentials, a suitable form of free energy or potential that represents the state of the thermodynamic system is first constructed from either microscopic, phenomenological or physical considerations. Macroscopic constitutive equations are then derived in compliance with thermodynamic principles. Zaki and Moumni (2007b,a) and [START_REF] Moumni | Theoretical and numerical modeling of solid-solid phase change: Application to the description of the thermomechanical behavior of shape memory alloys[END_REF] developed a phenomenological model for NiTi SMAs within the framework of generalized standard materials with internal constraints [START_REF] Halphen | Plastic and visco-plastic materials with generalized potential[END_REF]. The state variables used in the ZM model include the temperature θ, the martensite volume fraction χ, the local transformation strain tensors for austenite ε A and martensite ε M , and the local martensite orientation strain tensor ε ori . The model was later extended to account for tension-compression asymmetry [START_REF] Zaki | An approach to modeling tensile-compressive asymmetry for martensitic shape memory alloys[END_REF], plastic deformation [START_REF] Zaki | An extension of the ZM model for shape memory alloys accounting for plastic deformation[END_REF], thermomechanical coupling (Zaki et al., 2010a;Morin et al., 2011a) and cyclic loading effect (Zaki and Moumni, 2007a;[START_REF] Moumni | Cyclic behavior and energy approach to the fatigue of shape memory alloys[END_REF][START_REF] Morin | Thermomechanical coupling in shape memory alloys under cyclic loadings: Experimental analysis and constitutive modeling[END_REF]. [START_REF] Leclercq | A general macroscopic description of the thermomechanical behavior of shape memory alloys[END_REF] generalized the R L model proposed by [START_REF] Raniecki | Thermodynamic models of pseudoelastic behaviour of shape memory alloys[END_REF] to develop a general macroscopic model that explicitly takes into account the reorientation of martensite variants. To this end, they separated the martensite volume fraction into oriented and self-accommodation parts. The authors later used the model to discuss the validity of the four criteria proposed by [START_REF] James | A way to search for multiferroic materials with "unlikely" combinations of physical properties[END_REF] for hysteresis minimization in alloys exhibiting first-order transition [START_REF] Lexcellent | Two ways for predicting the hysteresis minimisation for shape memory alloys[END_REF].

Hysteresis models describe the hysteresis thermodynamic response in SMAs without necessarily going into the underlying physics. In analogy to input-output models, a suitable transfer function is constructed to capture the observed hysteresis between the forcing function and the response variable. Such models are commonly used in modeling magnetic and ferroelectric materials [START_REF] Smith | Smart material systems: model development[END_REF]. Depending on the nature of the hysteresis operator used, the modeling approaches can be generally classified into Preisach type and Duhem-Madelung models. Preisach models follow an integration of the response of individual relays that switch between two transformation type [START_REF] Ortin | Preisach modeling of hysteresis for a pseudoelastic Cu-Zn-Al single crystal[END_REF][START_REF] Ortin | Hysteresis in shape-memory alloys[END_REF]. Unlike, Duhem-Madelung models capture the hysteresis using differential equations. Two different operators are used, respectively, for the loading segment and unloading segment of the load path to capture the incremental material response [START_REF] Ivshin | A constitutive model for hysteretic phase transition behavior[END_REF][START_REF] Bekker | Phase diagram based description of the hysteresis behavior of shape memory alloys[END_REF].

In the current state of art in constitutive modeling, the macroscopic models, of which the present work is an example, are widely considered the most convenient for structural analysis and engineering design applications because of their smaller number of parameters and less computation expense [START_REF] Souza | Three-dimensional model for solids undergoing stressinduced phase transformations[END_REF][START_REF] Peultier | Macroscopic constitutive law of shape memory alloy thermomechanical behaviour. Application to structure computation by FEM[END_REF][START_REF] Lexcellent | About modelling the shape memory alloy behaviour based on the phase transformation surface identification under proportional loading and anisothermal conditions[END_REF][START_REF] Moumni | Theoretical and numerical modeling of solid-solid phase change: Application to the description of the thermomechanical behavior of shape memory alloys[END_REF][START_REF] Arghavani | A 3-D phenomenological constitutive model for shape memory alloys under multiaxial loadings[END_REF][START_REF] Baldelli | A gradient approach for the macroscopic modeling of superelasticity in softening shape memory alloys[END_REF].

Phenomenological models of SMAs

Throughout the history of SMA modeling, a large number of macroscopic phenomenological models have been proposed in the literature. Recently, with the increasing sophistication, these phenomenological models were capable of accurately describing complex SMA material responses such as the tensioncompression-torsion asymmetry [START_REF] Zaki | An approach to modeling tensile-compressive asymmetry for martensitic shape memory alloys[END_REF][START_REF] Reedlunn | Tension, compression, and bending of superelastic shape memory alloy tubes[END_REF][START_REF] Mehrabi | Anisotropic behavior of superelastic NiTi shape memory alloys; An experimental investigation and constitutive modeling[END_REF][START_REF] Rizzoni | A thermodynamical formulation for the constitutive modeling of a shape memory alloy with two martensite phases[END_REF], the multi-variant phase transformation [START_REF] Levitas | Three-dimensional Landau theory for multivariant stress-induced martensitic phase transformations. I. Austenite-martensite[END_REF][START_REF] Idesman | Finite element simulations of martensitic phase transitions and microstructures based on a strain softening model[END_REF][START_REF] Auricchio | Theoretical and numerical modeling of shape memory alloys accounting for multiple phase transformations and martensite reorientation[END_REF], the strain localization and softening [START_REF] Idesman | Finite element simulations of martensitic phase transitions and microstructures based on a strain softening model[END_REF][START_REF] Baldelli | A gradient approach for the macroscopic modeling of superelasticity in softening shape memory alloys[END_REF][START_REF] Bechle | Evolution of localization in pseudoelastic NiTi tubes under biaxial stress states[END_REF], the variation of the transformation hysteresis [START_REF] Lagoudas | Constitutive model for the numerical analysis of phase transformation in polycrystalline shape memory alloys[END_REF][START_REF] Sedlák | Thermomechanical model for NiTi-based shape memory alloys including R-phase and material anisotropy under multi-axial loadings[END_REF], the smooth transition in trained polycrystalline SMA [START_REF] Sittner | Experimental study on the thermoelastic martensitic transformation in shape memory alloy polycrystal induced by combined external forces[END_REF][START_REF] Grabe | On the viscous and strain rate dependent behavior of polycrystalline NiTi[END_REF][START_REF] Lagoudas | Constitutive model for the numerical analysis of phase transformation in polycrystalline shape memory alloys[END_REF], the residual strain accumulation and the cyclic effect (Zaki and Moumni, 2007a;[START_REF] Song | Non-proportional multiaxial transformation ratchetting of super-elastic NiTi shape memory alloy: Experimental observations[END_REF][START_REF] Yu | Study on the rate-dependent cyclic deformation of super-elastic NiTi shape memory alloy based on a new crystal plasticity constitutive model[END_REF]Yu et al., , 2015a;;[START_REF] Kimiecik | The effect of microstructure on stress-induced martensitic transformation under cyclic loading in the SMA Nickel-Titanium[END_REF][START_REF] Kan | Experimental observations on rate-dependent cyclic deformation of super-elastic NiTi shape memory alloy[END_REF], the crack growth [START_REF] Baxevanis | On the driving force for crack growth during thermal actuation of shape memory alloys[END_REF][START_REF] Hazar | Modeling of steady-state crack growth in shape memory alloys using a stationary method[END_REF], the size-dependent pseudoelasticity [START_REF] Qiao | Computational modeling of size-dependent superelasticity of shape memory alloys[END_REF], the plastic deformation [START_REF] Zhou | A macroscopic constitutive model of shape memory alloy considering plasticity[END_REF][START_REF] Zhu | Logarithmic stress rate based constitutive model for cyclic loading in finite plasticity[END_REF], the rate dependence [START_REF] Hartl | Three-dimensional modeling and numerical analysis of rate-dependent irrecoverable deformation in shape memory alloys[END_REF][START_REF] Andani | A rate dependent tension-torsion constitutive model for superelastic nitinol under non-proportional loading; a departure from von Mises equivalency[END_REF][START_REF] Yu | Rate-dependent cyclic deformation of super-elastic NiTi shape memory alloy: Thermo-mechanical coupled and physical mechanism-based constitutive model[END_REF], and so on. The vast majority of these models were developed under the assumption of infinitesimal deformation, and disregarded thermomechanical coupling effect in SMAs, which greatly simplifies the modeling framework that can be effective in presence of small strain/rotation and simple thermomechanical loading conditions. However, these models lose accuracy with increasing strains and rotations or complex thermomechanical loading paths and heat transfer conditions [START_REF] Christ | Finite-element modelling of shape memory alloys-A comparison between small-strain and large-strain formulations[END_REF][START_REF] Andani | A rate dependent tension-torsion constitutive model for superelastic nitinol under non-proportional loading; a departure from von Mises equivalency[END_REF].

Finite deformation

SMA models with finite strain formulation (FSF) are indispensable to accurately describe large rotations [START_REF] Christ | Finite-element modelling of shape memory alloys-A comparison between small-strain and large-strain formulations[END_REF] and distortions [START_REF] Ziolkowski | Three-dimensional phenomenological thermodynamic model of pseudoelasticity of shape memory alloys at finite strains[END_REF]. Early finite strain models, such as the work of [START_REF] Lubliner | Generalized plasticity and shape-memory alloys[END_REF], [START_REF] Auricchio | Shape-memory alloys: modelling and numerical simulations of the finitestrain superelastic behavior[END_REF] and [START_REF] Masud | A finite-strain finite element model for the pseudoelastic behavior of shape memory alloys[END_REF], were mostly built by generalizing small strain formulations (SSF) of elastoplastic materials. Following these pioneering work, many FSF models were developed using either hypoelastic-based additive or hyperelastic-based multiplicative approaches.

Hypoelastic FSF models are based on the additive split of the stretching tensor D, wherein constitutive relations are expressed directly in rate form and the formulation is limited to small strains but finite rotations. In this approach, standard infinitesimal elastoplasticity models are extended to the finite strain range by recasting the original evolution equations in terms of suitably chosen objective stress rates. An example of an isotropic hypoelastic SMA model was proposed by [START_REF] Müller | A thermodynamic finite-strain model for pseudoelastic shape memory alloys[END_REF] based on a self-consistent Eulerian theory of finite deformation using a logarithmic rate. The model is shown to be consistent with the definition of a Helmholtz free energy by fulfilling appropriate integrability conditions. [START_REF] Zaki | An efficient implementation for a model of martensite reorientation in martensitic shape memory alloys under multiaxial nonproportional loading[END_REF] proposed an efficient implementation for the finite deformation extension of the work of [START_REF] Zaki | Modeling and simulation of the mechanical response of martensitic shape memory alloys[END_REF] considering a hypoelastic framework. [START_REF] Teeriaho | An extension of a shape memory alloy model for large deformations based on an exactly integrable Eulerian rate formulation with changing elastic properties[END_REF] proposed an alternative modeling approach for isotropic deformation involving logarithmic strain definitions and internal variables with a Eulerian rate formulation. This approach was then utilized to derive SMA constitutive relations. [START_REF] Xiao | An explicit, straightforward approach to modeling SMA pseudoelastic hysteresis[END_REF] proposed a finite J 2 -flow elastoplastic model with nonlinear combined hardening that allows changes in the size and center of the elastic domain as the material deforms. This additive approach would be consistent with the multiplicative approach if appropriate integrability conditions are satisfied.

Hyperelastic FSF models are based on the multiplicative decomposition of the deformation gradient F into elastic and inelastic parts. [START_REF] Masud | A finite-strain finite element model for the pseudoelastic behavior of shape memory alloys[END_REF] presented a finite-deformation finite element model for pseudoelastic SMAs under stress-controlled loading conditions at constant temperature. [START_REF] Auricchio | A robust integration-algorithm for a finite-strain shape-memory-alloy superelastic model[END_REF] derived a finite-strain SMA model and presented an efficient and robust integration algorithms. [START_REF] Anand | Thermal effects in the superelasticity of crystalline shape-memory materials[END_REF] used the multiplicative decomposition to develop a FSF SMA model that successfully describes isothermal pseudoelasticity of an initially textured NiTi. [START_REF] Stupkiewicz | Finite-strain micromechanical model of stress-induced martensitic transformations in shape memory alloys[END_REF] extended the SSF model of [START_REF] Stupkiewicz | Modelling of laminated microstructures in stress-induced martensitic transformations[END_REF] to develop a micromechanical FSF model for SMAs with elastic anisotropy. [START_REF] Reese | Finite deformation pseudo-elasticity of shape memory alloys -Constitutive modelling and finite element implementation[END_REF] proposed a finite deformation constitutive model to describe the pseudoelasticity of SMAs and developed a new integration scheme to preserve the incompressibility of material during the phase transformation. [START_REF] Arghavani | A finite strain kinematic hardening constitutive model based on Hencky strain: General framework, solution algorithm and application to shape memory alloys[END_REF] presented a phenomenological finite-Hencky-strain formulation featuring kinematic hardening and non-associative flow rules integrated using a solution algorithm involving a nucleation-completion step. [START_REF] Panoskaltsis | Large deformation constitutive theory for a two-phase shape memory alloy[END_REF] introduced a general finite-strain model accounting for diffusionless phase transformations and discussed possible extensions to account for tension-compression asymmetry and the influence of loading rate. [START_REF] Paranjape | A phase field -Finite element approach to model the interaction between phase transformations and plasticity in shape memory alloys[END_REF] developed a finite element framework in which the constitutive formulation simultaneously accounts for phase transformations at the martensite correspondence variant (CV) scale and rate-dependent crystal plasticity in austenite.

Literature review of the constitutive modeling of SMAs However, a common drawback of these models is that the inelastic deformation of SMAs is accounted for by means of a single tensorial internal variable representing transformation strain. Therefore, they do not always guarantee complete recovery of transformation strain when SMAs undergo multiaxial non-proportional loading, wherein phase transformation and martensite reorientation may take place simultaneously.

Thermomechanical coupling

Extensive experimental investigations have shown that phase transformation of SMAs is a thermomechanically coupled process, with heat production due to latent heat and intrinsic dissipation [START_REF] Shaw | Thermomechanical aspects of NiTi[END_REF][START_REF] Müller | A thermodynamic finite-strain model for pseudoelastic shape memory alloys[END_REF]. The resulting temperature variations can readily influence the mechanical behavior of SMAs. From the experimental point of view, [START_REF] Shaw | Thermomechanical aspects of NiTi[END_REF] performed a series of experiments at different strain rates, temperatures and loading conditions. The results show stronger strain rate dependence in air than in water, indicating that the dependence is due to thermomechanical coupling. [START_REF] Peyroux | Thermomechanical Couplings and Pseudoelasticity of Shape Memory Alloys[END_REF] used infrared thermography to investigate heat exchange during phase transformation in SMAs and concluded that intrinsic dissipation can be neglected before latent heat. [START_REF] Grabe | On the viscous and strain rate dependent behavior of polycrystalline NiTi[END_REF] conducted biaxial tension/torsion experiments under both isothermal and non-isothermal conditions and found that the stress-strain responses for various strain rates coincide under isothermal conditions, proving that the dependence is due to temperature effect rather than strain rate.

Based on these experimental findings, a few models that take into account thermomechanical coupling were proposed. [START_REF] Müller | A thermodynamic finite-strain model for pseudoelastic shape memory alloys[END_REF] proposed a thermodynamic finite-strain model describing the pseudoelastic response of SMAs based on a self-consistent Eulerian theory. The model was used to simulate the material response under isothermal and adiabatic conditions. [START_REF] Christ | A finite element model for shape memory alloys considering thermomechanical couplings at large strains[END_REF] proposed a thermomechanically coupled SMA model and carried out finite element simulation involving a medical foot staple which interacts with a bone segment. Morin et al. (2011a) presented a generalized ZM model [START_REF] Zaki | A three-dimensional model of the thermomechanical behavior of shape memory alloys[END_REF] for SMAs accounting for thermomechanical coupling. The model showed that the mechanical hysteresis is mainly due to the intrinsic dissipation, whereas the thermal response is governed by latent heat. [START_REF] Andani | A rate dependent tension-torsion constitutive model for superelastic nitinol under non-proportional loading; a departure from von Mises equivalency[END_REF] modified the model originally developed by [START_REF] Lagoudas | Constitutive model for the numerical analysis of phase transformation in polycrystalline shape memory alloys[END_REF] to capture the multi-axial pseudoelastic behavior of SMAs under quasi-static isothermal and dynamic loading conditions. [START_REF] Chatziathanasiou | Modeling of coupled phase transformation and reorientation in shape memory alloys under non-proportional thermomechanical loading[END_REF] proposed a new 3D thermodynamic coupled model for SMAs to describe martensitic reorientation, forward and reverse phase transformation, dissipation and latent heat, wherein the strain mechanisms are associated with proper internal variables.

However, these models are either limited to small deformations [START_REF] Andani | A rate dependent tension-torsion constitutive model for superelastic nitinol under non-proportional loading; a departure from von Mises equivalency[END_REF]Morin et al., 2011a) or developed without considering thermal deformation [START_REF] Christ | A finite element model for shape memory alloys considering thermomechanical couplings at large strains[END_REF][START_REF] Müller | A thermodynamic finite-strain model for pseudoelastic shape memory alloys[END_REF]. Indeed, at high temperature and large stress magnitude the martensite structures has a much lower Young's modulus and can be readily deformed by application of an external force or heat, indicating that thermoelastic deformations can not be neglected [START_REF] Mohd | A review of shape memory alloy research, applications and opportunities[END_REF][START_REF] Chemisky | A constitutive model for cyclic actuation of high-temperature shape memory alloys[END_REF].

Cyclic behavior

Over the last decades, extensive experiments were carried out to investigate thermomechanical behavior of SMAs undergoing cyclic deformations. The following four primary characteristics were deduced from the experimental observations:

1. Large accumulated residual strain during cycling. Fig. 2.1(a) represents a typical pseudoelastic response of SMA under the stress-controlled cyclic loading. It is seen that the starting point of the stress-strain curves deviates with the number of cycles. This residual strain grows speedy at the beginning and tends to saturate after a certain number of cycles. This accumulated residual strain can be physically attributed to residual martensite and dislocations slipping in austenite during the cyclic deformation [START_REF] Kockar | Thermomechanical cyclic response of an ultrafine-grained NiTi shape memory alloy[END_REF][START_REF] Yu | Rate-dependent cyclic deformation of super-elastic NiTi shape memory alloy: Thermo-mechanical coupled and physical mechanism-based constitutive model[END_REF]. According to the literature [START_REF] Kan | Constitutive model for uniaxial transformation ratchetting of super-elastic NiTi shape memory alloy at room temperature[END_REF], the maximum residual strain reaches more than 7%. Thus, the total strain during cycling commonly reaches 10%-15%, which enters the finite strain regime. 2. Degeneration of pseudoelasticity and dissipated energy with the number of cycles. During the cyclic deformation, the hysteresis loop evolves progressively with the number of cycles as shown in Fig. 2.1(a), wherein the black curve represents the stress-strain response in the first cycle and the red curve represents the stabilized one. Fig. 2.1(b) shows the evolutions of the reversible strain and the dissipated energy with the number of cycles. It is seen that the reversible strain and the dissipated energy decrease, respectively, around 30% and 80% throughout the cyclic process. From a physical standpoint, the residual martensite and the dislocations slipping induce internal stress that can assist the formation of stress-induced martensite (Zaki and Moumni, 2007a;[START_REF] Wang | Dislocation slip stress prediction in shape memory alloys[END_REF]. As a consequence, the transformation stresses decrease with the number of cycles. Moreover, the increase of dislocation density obstructs the growth of martensite in a way similar to strain hardening in plasticity. Thus, the transformation hardening modulus increases gradually during cycling. Finally, the hysteresis loop stabilizes in a way comparable to plastic shakedown when the residual strain saturates.

3. Rate dependence. Fig. 2.2(a) shows the pseudoelastic stress-strain curves at different strain rates in the first cycle, and Fig. 2.2(b) the stabilized cycle. It is seen that the size of hysteresis loop, the reversible strain, the residual strain, the transformation stresses and hardening modulus are significantly influenced by the strain rate. The underlying cause for this rate dependence lies in four aspects: (i) phase transformation takes place with internal heat production due to the intrinsic dissipation and the latent heat; (ii) external heat transfer depends on the testing duration and therefore the loading rate; (iii) competition of the internal heat production against the external heat transfer determines the temperature in SMAs; (iv) pseudoelastic behavior of SMAs is strongly influenced by temperature. These experimental evidences can be found in the literature [START_REF] Shaw | Thermomechanical aspects of NiTi[END_REF][START_REF] Grabe | On the viscous and strain rate dependent behavior of polycrystalline NiTi[END_REF][START_REF] Thamburaja | A finite-deformation-based phenomenological theory for shape-memory alloys[END_REF][START_REF] Hartl | Three-dimensional modeling and numerical analysis of rate-dependent irrecoverable deformation in shape memory alloys[END_REF][START_REF] Morin | Thermomechanical coupling in shape memory alloys under cyclic loadings: Experimental analysis and constitutive modeling[END_REF][START_REF] Kan | Experimental observations on rate-dependent cyclic deformation of super-elastic NiTi shape memory alloy[END_REF][START_REF] Zhang | Experimental and theoretical investigation of the frequency effect on low cycle fatigue of shape memory alloys[END_REF]. As a matter of fact, in the experiments of [START_REF] Shaw | Thermomechanical aspects of NiTi[END_REF] and [START_REF] Grabe | On the viscous and strain rate dependent behavior of polycrystalline NiTi[END_REF], the stress-strain curves for different strain rates coincide under isothermal condition, which further evince that this rate dependence is due to thermomechanical coupling effect. 4. Evolution of the phase transformation from abrupt to smooth transition. It is well known that the cyclic loading is usually used to "train" the polycrystalline SMA material. For "untrained" polycrystalline SMAs, a sharp transition between the elastic and the transformation regimes is observed on the pseudoelastic stress-strain curves, as shown in Fig. 2.1(a) (black curve) and Fig.

2.2(a)

. This phenomenon is also commonly observed in single-crystalline SMAs. However, the phase transformation initiates and completes in a smooth manner in "trained" polycrystalline SMAs, as shown in Fig. 2.1(a) (red curve) and Fig. 2.2(b). This gradual phase transformation is due to the different crystallographic orientations of the grains, the strong heterogeneity of internal stresses and the presence of non-transforming precipitates during the cyclic deformation [START_REF] Lagoudas | Constitutive model for the numerical analysis of phase transformation in polycrystalline shape memory alloys[END_REF]. As a consequence, the pseudoelastic stress-strain curves evolve progressively from abrupt to smooth at the initiation and the completion of the phase transformation.

Based on these experimental findings, a few constitutive models that aim to describe the cyclic behavior of SMAs were proposed. [START_REF] Lexcellent | Thermodynamical model of cyclic behaviour of Ti-Ni and Cu-Zn-Al shape memory alloys under isothermal undulated tensile tests[END_REF] introduced the concept of instantaneous residual martensite that evolves with the number of cycles to describe cyclic pseudoelasticity. [START_REF] Abeyaratne | Cyclic effects in shape-memory alloys: a one-dimensional continuum model[END_REF] provided a 1D continuum model to qualitatively predict the cyclic behavior by capturing the idea that defects are precipitated during cycling and these defects tend to assist the nucleation of martensite. Subsequently, [START_REF] Bo | Thermomechanical modeling of polycristaline SMAs under cyclic loading, Part I: theroretical derivations[END_REF] proposed a more complete model. Micromechanical analysis over a representative volume element (RVE) was employed to describe the evolution of internal state variables during cyclic phase transformation. Zaki and Moumni (2007a) proposed a 3D phenomenological model. The cyclic effects, i.e., training and two-way shape memory are accounted for by introducing internal state variables such as internal back stress, residual strain and cumulated martensite volume fraction. The model was later improved by [START_REF] Morin | Thermomechanical coupling in shape memory alloys under cyclic loadings: Experimental analysis and constitutive modeling[END_REF] by accounting for thermomechanical coupling. The improved model was used to determine the area of hysteresis loop in stabilized state and therefore the fatigue life of pseudoelastic SMAs. [START_REF] Auricchio | A three-dimensional model describing stress-induced solid phase transformation with permanent inelasticity[END_REF] proposed a 3D model including permanent inelastic deformations, which was used to describe both pseudoelasticity and shape memory behavior under cyclic loading. Saint-Sulpice et al. ( 2009) performed a series of cyclic tests and developed a 3D macroscopic model to describe the pseudoelasticity and the evolution of permanent inelastic strain during cycles. [START_REF] Kan | Constitutive model for uniaxial transformation ratchetting of super-elastic NiTi shape memory alloy at room temperature[END_REF] constructed a cyclic constitutive model in the framework of generalized plasticity. The dependence of transformation ratcheting on the applied stress level and the anisotropic transformation surface are considered in the model. [START_REF] Zhang | Modeling evolutions of plastic strain, maximum transformation strain and transformation temperatures in SMA under superelastic cycling[END_REF] split the total cumulated martensite volume fraction into the forward transformation and the reverse transformation parts to construct a cyclic constitutive model. [START_REF] Yu | Rate-dependent cyclic deformation of super-elastic NiTi shape memory alloy: Thermo-mechanical coupled and physical mechanism-based constitutive model[END_REF] proposed a 3D thermomechanically coupled and physical mechanism-based constitutive model to describe rate-dependent cyclic deformation of pseudoelastic NiTi SMA. The interactions between the martensite transformation and dislocations slipping in austenite phase are included in the model.

However, these models suffer from the following limitations: (i) all these models were developed under assumption of small strain, though most of the strain levels in cyclic experiments have entered finite strain regime; (ii) most of them are not able to capture the rate dependence; (iii) none of them considered the abrupt to smooth transition during training process.

Therefore, this work is dedicated to developing a comprehensive constitutive model that is able to capture the aforementioned thermomechanical and cyclic characteristics of SMAs in finite deformations.

Chapter 3

Finite-strain extension of the ZM model

Introduction

This chapter introduces the first step of the comprehensive constitutive modeling approach to investigate thermomechanical and cyclic behavior of SMAs in finite deformation. The chapter addresses the constitutive modeling of SMA using finite strain formulation. The model developed in this chapter is a finite-strain extension of the small-strain ZM model [START_REF] Zaki | A three-dimensional model of the thermomechanical behavior of shape memory alloys[END_REF][START_REF] Moumni | Theoretical and numerical modeling of solid-solid phase change: Application to the description of the thermomechanical behavior of shape memory alloys[END_REF].

Before the development of the model, a summary of the ZM model is presented. The formulation of the model is based on a two-tier, multiplicative decomposition of the deformation gradient into elastic and transformation parts, where the transformation deformation is further split into phase transformation and martensite reorientation components. The finite-strain model is derived, within a thermodynamically consistent framework, from a Helmholtz free energy function. Numerical integration algorithm of the proposed model is presented featuring time-discrete formulation of the constitutive equations, proper symmetrization of the tensor variables, and explicit formulation of the material and spatial tangent operators involved. The algorithm is implemented into the finite element analysis (FEA) software Abaqus/Standard by means of a UMAT subroutine, to carry out simulations of SMAs subjected to various thermomechanical loading conditions. Simulation results of the proposed finite-strain model are compared against those of the small-strain ZM model. Moreover, the model is applied to simulate a SMA archwire undergoing large strains and rotations.

Summary of the small-strain ZM model

The ZM model for SMAs was proposed by [START_REF] Zaki | A three-dimensional model of the thermomechanical behavior of shape memory alloys[END_REF] based on the framework of generalized standard materials with internal constraints [START_REF] Moumni | Sur la modélisation du changement de phase solide: application aux matériaux à mémoire de forme et à l'endommagement fragile partiel[END_REF]. In this model, austenite is assumed to be elastic while martensite is inelastic. The assumption is physically motivated by the fact that reversible macroscopic inelastic deformation of SMAs is due to orientation of martensite variants. The Helmholtz free energy density ψ is constructed as the sum of free energy contributions Finite-strain extension of the ZM model from austenite W A and martensite W M as well as an interaction energy term I:

ψ = (1 -χ) 1 2 ε A : K A : ε A W A + χ 1 2 (ε M -ε tr ) : K M : (ε M -ε tr ) + C(θ) W M + G χ 2 2 + χ 2 [α χ + β(1 -χ)] 2 3 ε tr : ε tr I , (3.1)
where W A and W M are, respectively, the local austenite and martensite strain energies. ε A and ε M are the local strain tensors in austenite and martensite. K A and K M are the elastic moduli of austenite and martensite. χ is the martensite volume fraction and ε tr is the martensite orientation strain tensor. The interaction energy I consists of three components: G χ 2 /2 quantifies orientation-independent interaction between martensite variants, β χ(1 -χ)/2 2 3 ε tr : ε tr represents interaction between austenite and martensite and α χ 2 /2 2 3 ε tr : ε tr accounts for interaction increase due to orientation of martensite plates. G, α and β are model parameters. The heat density C(θ) associated with the thermal-induced phase transformation is given by

C(θ) = ξ(θ -A 0 f ) + K, (3.2)
where ξ and K are model parameters, A 0 f is the reverse phase transformation finish temperature at zero stress.

The state variables used for modeling obey the following physical constraints:

• Since the Reuss rheological model is adopted as rule of mixtures, the macroscopic strain tensor ε and the local strain tensor in austenite ε A and martensite ε M are subjected to

(1 -χ)ε A + χε M -ε = 0; (3.3)
• The martensite volume fraction χ is restricted within the [0, 1] interval,

0 ≤ χ ≤ 1; (3.4)
• The magnitude of martensite orientation strain ε tr may not exceed a material-specific upper limit H , i.e.

H -

2 3 ε tr : ε tr ≥ 0. (3.5)
The above physical constraints derive from the following potential:

W c = -ζ p : [(1 -χ)ε A + χε M -ε] -ζ tr H - 2 3 ε tr : ε tr -ζ f (1 -χ) -ζ r χ, (3.6
             ζ f ≥ 0, 1 -χ ≥ 0, ζ f (1 -χ) = 0; ζ r ≥ 0, χ ≥ 0, ζ r χ = 0; ζ tr ≥ 0, H -2 3 ε tr : ε tr ≥ 0, ζ tr H -2 3 ε tr : ε tr = 0. (3.7)
The sum of the Helmholtz free energy ( 3.1) and the potential (3.6) gives the generalized Lagrangian free energy function,

L = ψ + W c , (3.8)
which is then used to derive the constitutive equations. With some algebra, the following elasticity constitutive equation is obtained:

σ = K : (ε -χε tr ), (3.9)
where

K = (1 -χ)K -1 A + χK -1 M -1
is the equivalent elastic moduli of austenite and martensite mixtures.

In equation (3.9), the product χε tr represents the inelastic part of the total strain. Therefore, ε is additively split into elastic and transformation parts, ε = ε e + ε t , where ε e and ε t are given by

ε e = (1 -χ)ε A + χ(ε M -ε tr ) and ε t = χε tr . (3.10)
The thermodynamic forces associated with χ and ε tr are taken as sub-gradients of a pseudo-potential of dissipation D defined as

D = [a(1 -χ) + b χ] | χ| + χ 2 Y 2 3 ε tr : ε tr , ( 3

Development of the finite-strain model

The derivation of the finite-strain constitutive model for SMAs is based on the following two-tier decomposition: (i) a multiplicative decomposition of the deformation gradient into elastic and transformation parts; (ii) an additive split of the transformation deformation into phase transformation and martensite reorientation parts. Then, a thermodynamic framework and a Helmholtz free energy are introduced and used to derive the constitutive equations satisfying the requirement of thermodynamic consistency.

Kinematics

Let p be an arbitrary material point of a homogeneous body B. The motion of B is described with a smooth one-to-one mapping x = ϕ(p, t). The deformation gradient F, the velocity v, and the velocity gradient L of the motion ϕ are defined by

F = ∇ϕ = J 1 3 F, v = ϕ and L = ∇ x v = FF -1 , (3.12)
where ∇ and ∇ x denote, respectively, the material and spatial gradients of a general field; the dot denotes the material time derivative of a general field; J = det F and F denote, respectively, volumetric and isochoric components of F.

Using polar decomposition, F is written as

F = RU = V R, (3.13)
where the symmetric positive definite tensors U and V are the right and left stretch tensors, the proper orthogonal tensor R is the local rotation tensor. Thereby, the right and left Cauchy-Green deformation tensors, C and b, are defined by

C = F T F = U 2 and b = FF T = V 2 . (3.14)
The stretching and spin tensors, D and W , are defined as the symmetric and the skew parts of L: .15) In this work, the finite-strain response of SMAs is modeled by analogy with a well-established finite-strain elastoplasticity approach [START_REF] Auricchio | A robust integration-algorithm for a finite-strain shape-memory-alloy superelastic model[END_REF][START_REF] Souza Neto | Computational methods for plasticity: theory and applications[END_REF][START_REF] Reese | Finite deformation pseudo-elasticity of shape memory alloys -Constitutive modelling and finite element implementation[END_REF][START_REF] Arghavani | A finite strain kinematic hardening constitutive model based on Hencky strain: General framework, solution algorithm and application to shape memory alloys[END_REF][START_REF] Dimitrienko | Nonlinear continuum mechanics and large inelastic deformations[END_REF], in which the deformation gradient F is multiplicatively decomposed into elastic and transformation parts, such that

D = sym(L) = 1 2 (L + L T ) and W = skew(L) = 1 2 (L -L T ). ( 3
F = F e F t , (3.16)
where F e is defined with respect to an local unstressed intermediate configuration, while F t is defined with respect to the reference configuration.

Using equation (3.13), F e and F t are decomposed as

F e = R e U e and F t = R t U t , (3.17)
where R e and R t are elastic and transformation rotation tensors, U e and U t are right elastic and transformation stretch tensors. The corresponding right Cauchy-Green tensors for elastic and transformation deformation are given by

C e = U 2 e = F T e F e and C t = U 2 t = F T t F t . (3.18)
From equation (3.10) 2 , it can be concluded that the transformation deformation of SMAs is the result of phase transformation and martensite reorientation. Thus, a deformation gradient F tr associated with martensite orientation is defined with respect to the reference configuration, which will enhance the ability of the model to properly describe the martensite reorientation and the phase transformation simultaneously. The polar decomposition of F tr is written as

F tr = R tr U tr , (3.19)
where R tr is rotation tensor and U tr is right stretch tensor associated with martensite reorientation. The corresponding right Cauchy-Green strain tensor is then given by

C tr = U 2 tr = F T tr F tr . (3.20)
From equation (3.10) 2 , the stretch tensors U t and U tr are coaxial, which is verified if the rotation tensor for martensite orientation equals to the rotation tensor for transformation deformation:

R t = R tr . (3.21)
In small-strain formulation, the strain tensor conventionally adopted is an approximation of the Green-Lagrange strain tensor,

ε = 1 2 ▽ p u + (▽ p u) T ≈ 1 2 (C -1), (3.22)
where u is the displacement, C is the right Cauchy-Green strain tensor and 1 is the second-order identity tensor. Using equation (3.10) 2 , C t is expressed in terms of χ and C tr :

C t = χC tr + (1 -χ)1. (3.23)
This additive split of the inelastic deformation will enhance the ability of the model to properly describe the martensite reorientation and the phase transformation simultaneously.

Thermodynamic framework

In order to satisfy the principle of material objectivity, the Helmholtz free energy of SMA is assumed to depend on the right Cauchy-Green deformation tensors C e and C tr , the martensite volume fraction χ and the temperature θ as

ψ = ψ(C e , C tr , χ, θ) = ψ e (C e , χ) + ψ t (C tr , χ, θ), (3.24)
Finite-strain extension of the ZM model

where ψ e and ψ t represent the hyperelastic strain energy and transformation hardening energy, respectively. The dependence of ψ e on χ allows the definition of an equivalent elastic stiffness for the SMA that depends on its phase composition. This dependence is often neglected in SMA models [START_REF] Auricchio | A robust integration-algorithm for a finite-strain shape-memory-alloy superelastic model[END_REF][START_REF] Helm | Shape memory behaviour: modelling within continuum thermomechanics[END_REF][START_REF] Bouvet | A phenomenological model for pseudoelasticity of shape memory alloys under multiaxial proportional and nonproportional loadings[END_REF][START_REF] Reese | Finite deformation pseudo-elasticity of shape memory alloys -Constitutive modelling and finite element implementation[END_REF][START_REF] Arghavani | A 3-D phenomenological constitutive model for shape memory alloys under multiaxial loadings[END_REF], in which austenite and martensite are considered to have the same elastic stiffness. Thermal expansion is disregarded, with θ appearing only in the expression of ψ t . Both ψ e and ψ t are considered isotropic functions of C e and C tr . Thus, the time derivative of ψ in equation (3.24) gives

ψ = ∂ψ ∂C e : C e + ∂ψ ∂C tr : C tr + ∂ψ ∂ χ χ + ∂ψ ∂θ θ. (3.25)
The second law of thermodynamics, stated in the form of the Clausius-Duhem inequality, is written as

S : 1 2 C -ψ + η θ ≥ 0, ( 3
D t = 1 2 F -T t C t F -1 t , gives 1 2 S -F -1 t ∂ψ ∂C e F -T t : C + F -1 t C e ∂ψ ∂C e F -T t : C t - ∂ψ ∂ χ • χ - ∂ψ ∂C tr : C tr -η + ∂ψ ∂θ • θ ≥ 0. (3.30)
Taking time derivative of equation (3.23) then gives (3.31) where the term χ(C tr -1) corresponds to phase transformation at fixed martensite orientation, while χ C tr corresponds to martensite reorientation.

C t = χ(C tr -1) + χ C tr ,
1 2 S -F -1 t ∂ψ ∂C e F -T t : C + F -1 t C e ∂ψ ∂C e F -T t : (C tr -1) - ∂ψ ∂ χ • χ+ χF -1 t C e ∂ψ ∂C e F -T t - ∂ψ ∂C tr : C tr -η + ∂ψ ∂θ • θ ≥ 0.
(3.32)

For arbitrary thermodynamic processes, inequality (3.32) is guaranteed by the following choice of the constitutive laws with respect to S and η:

S = 2F -1 t ∂ψ ∂C e F -T t and η = - ∂ψ ∂θ , (3.33)
and by the following inequalities, which ensure nonnegative intrinsic dissipation in arbitrary evolutions of the internal state variables χ and C tr , so that

[M t : (C tr -1) -Z] • χ ≥ 0 and ( χM t -X tr ) : C tr ≥ 0, (3.34) with M t = F -1 t C e ∂ψ ∂C e F -T t , Z = ∂ψ ∂ χ , X tr = ∂ψ ∂C tr , (3.35)
where M t and X tr are, respectively, the Mandel stress and the symmetric stress-like tensors defined with respect to the reference configuration, Z is a scalar state variable. The nonnegative dissipation during phase transformation, as expressed in inequality (3.34) 1 , is guaranteed by the following choice of the evolution equation for χ: χ = γ χ S(A χ ), (3.36) where γ χ is a nonnegative multiplier, the sign function S(•) is used to extract the sign of A χ . The thermodynamic force A χ associated with phase transformation is then given by

A χ = M t : (C tr -1) -Z. (3.37)
The associated loading function for phase transformation is given by

F χ = A χ -Y χ , (3.38)
where Y χ controls the phase transformation threshold. γ χ and F χ are subjected to the Kuhn-Tucker consistency conditions:

γ χ ≥ 0, F χ ≤ 0, γ χ F χ = 0. (3.39)
Finite-strain extension of the ZM model Moreover, using the relation C tr = 2F T tr D tr F tr , a push-forward operation is carried out to rewrite the dissipation due to martensite reorientation (3.34) 2 in the unstressed intermediate configuration as

2F tr ( χM t -X tr ) F T tr : D tr ≥ 0. (3.40)
Then, the following associative evolution equation for martensite reorientation is formulated by analogy with the principle of maximum dissipation in elastoplasticity [START_REF] Simo | Computational inelasticity[END_REF]:

D tr = 1 2 γ tr A D tr ∥ A D tr ∥ , (3.41)
where γ tr is a nonnegative multiplier, (•) D = (•) -1 3 tr(•) is used to extract the deviator of a tensor, the thermodynamic force A tr associated with martensite reorientation is given by

A tr = F tr ( χM t -X tr )F T tr . (3.42)
Furthermore, in order to express the constitutive equations in a Lagrangian description, the evolution equation (3.41) is mapped into the reference configuration by means of a pull-back operation to get The associated loading function for martensite reorientation is given by

C tr = γ tr F T tr A D tr ∥ A D tr ∥ F tr . ( 3
F tr = ∥B D tr ∥ -Y tr , (3.46)
where Y tr defines the martensite reorientation threshold. γ tr and F tr are subjected to the Kuhn-Tucker conditions: γ tr ≥ 0, F tr ≤ 0, γ tr F tr = 0.

(3.47)

Constitutive equations

The Helmholtz free energy function ψ in equation (3.24) is additively split in the following form:

ψ = ψ e + ψ int + ψ cst , (3.48)
where ψ e is the hyperelastic strain energy, ψ int is the interaction energy between martensite and austenite phases [START_REF] Moumni | Theoretical and numerical modeling of solid-solid phase change: Application to the description of the thermomechanical behavior of shape memory alloys[END_REF], ψ cst is the potential energy due to the internal physical constraints. As aforementioned, ψ e is assumed to be an isotropic function of C e , allowing it to be expressed in terms of the invariants of C e as

ψ e = µ 2 I C e 1 -3 -µlnJ + λ 2 (lnJ) 2 + χC(θ), (3.49)
where

µ = 1-χ µ A + χ µ M -1 and λ = 1-χ λ A + χ λ M -1
are equivalent Lamé constants that depend on χ in the austenite and martensite composition, I C e 1 is the first invariant of C e , J is the determinant of the deformation gradient F.

The interaction energy ψ int is often derived from micro-mechanical or crystallographic considerations and finally expressed as a function of the martensite volume fraction and the local martensite orientation strain [START_REF] Zaki | A three-dimensional model of the thermomechanical behavior of shape memory alloys[END_REF][START_REF] Auricchio | Theoretical and numerical modeling of shape memory alloys accounting for multiple phase transformations and martensite reorientation[END_REF]. Likewise, ψ int is assumed to be an isotropic function of C tr . The following interaction energy is introduced in terms of χ and the invariants of C tr as

ψ int = G χ 2 2 + χ 2 [α χ + β(1 -χ)] I C t r 1 -3 , (3.50)
where

I C t r 1
is the first invariant of C tr , G, α and β are model parameters. The dependence of ψ int on the volume fraction χ and the orientation strain C tr allows the model to simultaneously capture the transformation deformation due to either phase transformation or martensite reorientation.

To satisfy the physical constraints on internal state variables χ and C tr , the following Lagrangian potential is defined as the constraint contribution ψ cst to free energy in equation (3.48):

ψ cst = -ζ tr (C -∥C tr ∥) -ζ f (1 -χ) -ζ r χ, (3.51)
where C is the upper limit of martensite orientation deformation, ζ tr is the Lagrange multiplier associated with the unilateral constraint ∥C tr ∥ ≤ C, ζ f and ζ r are the Lagrange multipliers associated with the bilateral constraints 0 ≤ χ ≤ 1.

In conclusion, the Helmholtz free energy function ψ is given by

ψ = µ 2 I C e 1 -3 -µlnJ + λ 2 (lnJ) 2 + G χ 2 2 + χ 2 [α χ + β(1 -χ)] I C t r 1 -3 - ζ tr (C -∥C tr ∥) + ζ f (1 -χ) + ζ r χ .
(3.52)

The constitutive equations derived from the free energy function (3.52) are finally summarized in Table 3.1.

In addition, the physical meaning of model parameters used in the model are given as

• a and b, respectively, control the width of the hysteresis loop for χ = 0 and χ = 1;

Finite-strain extension of the ZM model Table 3.1 Summary of the finite-strain model.

Constitutive equations:

S = µC -1 t + (λlnJ -µ) C -1 C t = χC tr + (1 -χ)1 M t = µ 2 C -1 t CC -1 t + 1 2 (λlnJ -µ) C -1 t Z = G χ + (α -β) χ + β 2 I C t r 1 -3 + C(θ) + ζ f -ζ r X tr = χ 2 [α χ + β(1 -χ)] 1 + ζ tr C t r ∥C t r ∥ A χ = M t : (C tr -1) -Z B tr = C tr ( χM t -X tr ) Evolution equations: χ = γ χ S(A χ ) and C tr = γ tr B D t r ∥B D t r ∥ C tr Yield functions: F χ = A χ -[a(1 -χ) + b χ] and F tr = ∥B D tr ∥ -χ 2 Y Kuhn-Tucker conditions: γ χ ≥ 0, F χ ≤ 0, γ χ F χ = 0; γ tr ≥ 0, F tr ≤ 0, γ tr F tr = 0.
• α and β, respectively, govern the evolution of the orientation strain C tr with respect to stress for χ = 0 and χ = 1;

• G measures interaction between martensite plates, affects the stress-strain curve slope during phase transformation;

• Y represents the stress level associated with the beginning of martensite reorientation;

• C defines the fully oriented strain magnitude of martensite variants;

• ξ represents the slope of phase diagram boundaries for forward and reverse transformations, which quantifies the influence of temperature on transformation yield stress;

• K is the heat density at reverse phase transformation finish temperature.

Numerical implementation

In this section, numerical implementation of the proposed model in an finite element analysis (FEA) framework is carried out. An implicit time-discrete formulation of the model is presented and then numerically integrated using a form of the return mapping algorithm. The symmetrization of the timediscrete evolution equation as well as the determination of the material and spatial tangent operators are also discussed.

Time-discrete formulation

The analytical equations of the finite-strain model are rewritten as a set of time-discrete incremental equations to be solved over discrete time intervals [t n , t n+1 ] covering the complete loading history, where n indicates the n-th kinematic time instant. The initial boundary value problem is stated as follows: given the deformation gradient F n and the state variables C tr n , χ n and S n at time t n , for a given deformation gradient F at time t n+1 , determine the state variables at time t n+1 satisfying the time-discrete constitutive equations. For convenience, deformation is considered to be controlled by the right Cauchy-Green deformation tensor C instead of F. Implicit integration is used to ensure the stability and accuracy of the time-integration algorithm.

The evolution equation (3.36) of χ is written using backward Euler approximation as

χ = χ n + ∆γ χ S(A χ ), (3.53)
where ∆γ χ is the phase transformation multiplier at time t n+1 . Since backward Euler integration does not preserve the incompressibility of martensite orientation, expressed as det(C tr ) = 1, exponential map integration is used instead. This integration method is well-established in computational plasticity [START_REF] Reese | Finite deformation pseudo-elasticity of shape memory alloys -Constitutive modelling and finite element implementation[END_REF]. Using backward exponential map integration to approximate the evolution equation (3.45) gives

C tr = exp ∆γ tr N tr C tr n , (3.54)
where N tr = B D tr /∥B D tr ∥ is the flow vector of martensite orientation, ∆γ tr is the martensite orientation multiplier at time t n+1 . Since the tensor N tr is traceless, i.e., tr(∆γ tr N tr ) = 0, a property of the exponential tensor function gives in this case det[exp(∆γ tr N tr )] = exp[tr(∆γ tr N tr )] = 1. The incompressibility of C tr is thereby guaranteed when the exponential integration scheme in (3.54) is used.

Symmetrization of tensors

Even though the exponential integration scheme (3.54) satisfies the incompressibility of deformation due to martensite orientation, it presents some disadvantages in numerical analysis. The first disadvantage is that due to the non-coaxiality of C tr and M t , their product B tr is generally not symmetric. This asymmetry causes the loss of an important property of exponential functions whereby the exponential of a symmetric tensor can be calculated in closed form by means of spectral decomposition. The closed form expression of exp (A) is given by exp

(A) = 3 i=1 exp(A i )n i ⊗ n i , (3.55)
where A i and n i are the eigenvalues and corresponding eigenvectors of the symmetric tensor A.

Finite-strain extension of the ZM model

Similarly, since exp(∆γ tr N tr ) and C tr n are not coaxial, their product is not symmetric. This is expressed mathematically by the relation:

exp ∆γ tr N tr C tr n C tr n exp ∆γ tr N tr . (3.56)
Hence, the second disadvantage of directly using exponential numerical integration (3.54) is the possible loss of symmetry in numerical calculation, which may result in immediate breakdown of the symmetry balance. In order to avoid these disadvantages, (3.54) must be symmetrized.

In order to address this issue, [START_REF] Reese | Finite deformation pseudo-elasticity of shape memory alloys -Constitutive modelling and finite element implementation[END_REF] proposed a exponential time-discrete scheme with guaranteed symmetry. This integration scheme was successfully used in subsequent work by [START_REF] Evangelista | A 3D SMA constitutive model in the framework of finite strain[END_REF], [START_REF] Vladimirov | Anisotropic finite elastoplasticity with nonlinear kinematic and isotropic hardening and application to sheet metal forming[END_REF] and others. The same approach applied to the timediscrete evolution equation (3.54), taking into account the equality

C tr C tr n -1 C tr = exp(∆γ tr N tr )C tr , gives -C tr n -1 + U -1 tr exp ∆γ tr U -1 tr N tr U tr U -1 tr = 0. (3.57)
An alternative logarithmic time-integration procedure was proposed by [START_REF] Arghavani | A finite strain kinematic hardening constitutive model based on Hencky strain: General framework, solution algorithm and application to shape memory alloys[END_REF] and shown to provide improved robustness and efficiency compared to the work of [START_REF] Reese | Finite deformation pseudo-elasticity of shape memory alloys -Constitutive modelling and finite element implementation[END_REF]. Thus, the exponential scheme in (3.57), taking into account the relations [exp(∆γ tr U -1 tr N tr U tr )] -1 = exp(-∆γ tr U -1 tr N tr U tr ), gives the following logarithmic time-discrete scheme:

ln U -1 tr C tr n U -1 tr + ∆γ tr G = 0, (3.58)
where G = U -1 tr N tr U tr is symmetric. The time-discrete evolution equations (3.53) and (3.58), together with the loading functions (3.38) and (3.46), are used to construct the following system of nonlinear time-discrete constitutive equations

               r χ = χ -χ n -∆γ χ S(A χ ) = 0, r tr = ln U -1 tr C tr n U -1 tr + ∆γ tr G = 0, F χ = | A χ | -Y χ = 0, F tr = ∥B D tr ∥ -Y tr = 0, (3.59)
which is solved using a Newton-Raphson method, taking into account the discrete Kuhn-Tucker conditions:

∆γ χ ≥ 0, F χ ≤ 0, ∆γ χ F χ = 0; ∆γ tr ≥ 0, F tr ≤ 0, ∆γ tr F tr = 0. (3.60)
For comparison with the implicit time-discrete formulation of the small-strain ZM model, the reader is referred to [START_REF] Zaki | An efficient implementation for a model of martensite reorientation in martensitic shape memory alloys under multiaxial nonproportional loading[END_REF][START_REF] Gu | Time integration and assessment of a model for shape memory alloys considering multiaxial nonproportional loading cases[END_REF].

Material and spatial tangent operators

Tangent operators used in the integration of the time-discrete equations are determined in two steps: a material tangent operator is first obtained by taking the material derivative of the constitutive relations in Lagrangian configuration; a spatial tangent is then obtained by a push-forward operation on the material tangent with the objective rates of Kirchhoff stress.

The material tangent is determined by differentiating the second Piola-Kirchhoff stress tensor to get dS = D : dC.

(3.61)

Using the definition of S in Table 3.1, dS is then written in terms of C t , J and C as follows:

dS = ∂S ∂ J • dJ + ∂S ∂C t : dC t + ∂S ∂C : dC, (3.62) with ∂S ∂J = λ J C -1 , ∂S ∂C t = µS, ∂S ∂C = (λlnJ -µ) T. (3.63)
In the above equations, S = ∂C -1 t /∂C t and T = ∂C -1 /∂C are fourth-order tensors, the components of which are given by

S i jkl = -C t ik -1 C t l j -1 and T i jkl = -C -1 ik C -1 l j . (3.64)
Using the definition J = detF = √ detC as well as equations (3.31) and (3.45) leads to the following material derivatives of J, C t and C tr :

         dJ = 1 2 JC -1 : dC, dC t = (C tr -1) • d χ + χ • dC tr , dC tr = dγ tr N tr C tr . (3.65) Substituting (3.65) 3 into (3.65) 2 gives dC t = (C tr -1) • d χ + χN tr C tr • dγ tr .
(3.66)

The loading conditions for phase transformation and martensite orientation are differentiated as follows:

F χ , χ • d χ + F χ ,C t r : dC tr + F χ ,C : dC = 0, F tr , χ • d χ + F tr ,C t r : dC tr + F tr ,C : dC = 0. (3.67)
Finite-strain extension of the ZM model Substituting equation (3.65) 3 into the above equations gives

F χ , χ • d χ + F χ ,C t r : N tr C tr • dγ tr + F χ ,C : dC = 0, F tr , χ • d χ + F tr ,C t r : N tr C tr • dγ tr + F tr ,C : dC = 0.
(3.68)

The following expressions of d χ and dγ tr are then obtained: where P and Q are second-order tensors. With the help of these tensors, equation (3.66) can be expressed as

d χ = (F χ ,C t r : N tr C tr )(F tr ,C : dC) -(F tr ,C t r : N tr C tr )(F χ ,C : dC) F χ , χ (F tr ,C t r : N tr C tr ) -F tr , χ (F χ ,C t r : N tr C tr ) , (3.69) dγ tr = F tr , χ (F χ ,C : dC) -F χ , χ (F tr ,C : dC) F χ , χ (F tr ,C t r : N tr C tr ) -F tr , χ (F χ ,C t r : N tr C tr ) , ( 3
dC t = ((C tr -1) ⊗ P) : dC + χ (N tr C tr ⊗ Q) : dC. (3.72)
The material tangent D is then obtained by substituting dJ and dC t into equation (3.62) to get

dS = λ 2 C -1 ⊗ C -1 : dC -(λlnJ -µ) C -1 [dC]C -1 - µC -1 t {[(C tr -1) ⊗ P + χ (N tr C tr ⊗ Q)] : dC} C -1 t , (3.73)
which gives

D = λ 2 C -1 ⊗ C -1 + µS : [(C tr -1) ⊗ P + χ (N tr C tr ⊗ Q)] + (λlnJ -µ) T. (3.74)
The implicit integration of the finite-strain model in Abaqus/Standard requires the specification of a spatial tangent operator based on the objective Jaumann rate of the Kirchhoff stress:

τ = τ -W τ + τW, (3.75)
where τ = Jσ is the Kirchhoff stress tensor and W = 1 2 (L -L T ) is the spin tensor. In contrast, the following Green-Naghdi stress rate is used for structural elements:

▽ τ = τ -RR T τ + τ RR T , ( 3
F -1 ⋄ τF -T = D : 2F T DF. (3.79)
The spatial tangent C, such that

⋄ τ = C : D, (3.80)
is then given by the following expression:

C =2 (µ -λlnJ) I + λ1 ⊗ 1 -2µFC -1 t (C tr -1)C -1 t F T ⊗ F PF T - 2 χµFC -1 t N tr C tr C -1 t F T ⊗ F QF T . (3.81)
where I is the fourth-order identity tensor.

Numerical simulation

The numerical integration scheme presented in the previous section is implemented into Abaqus/Standard by means of a UMAT subroutine. Several numerical simulations are carried out and compared to experimental data in different loading cases. The following examples are considered: (i) a single mesh hexahedral element subjected to uniaxial and non-proportional loadings; (ii) comparisons between the finite-strain model and the original ZM model; (iii) shape memory effect of SMA subjected to a thermomechanical loading; (iv) simulation of a SMA orthodontic archwire undergoing extensive strains and rotations.

Uniaxial proportional loading

A single hexahedral element (1 mm×1 mm×1 mm) subjected to uniaxial tensile loading is simulated at temperatures 293 K and 333 K. The tensile loading consists of uniform negative pressure applied on the top surface of the element with symmetry conditions prescribed on the bottom surface as well as on the Finite-strain extension of the ZM model two adjacent side surfaces. The pressure is increased in magnitude from zero to the maximum value then removed. The maximum pressure is set to 600 MPa at 293 K and 1000 MPa at 333 K. The model parameters used for the simulation are determined from the experimental data reported in [START_REF] Moumni | Theoretical and numerical modeling of solid-solid phase change: Application to the description of the thermomechanical behavior of shape memory alloys[END_REF] and listed in Table 3.2. The simulated stress-strain curves are reported in Fig. 3.1(a) at temperature 293 K and in Fig. 3.1(b) at temperature 333 K. In the first case, because the prescribed temperature is well below the austenite finish temperature A 0 f , the transformation strains are not recovered upon unloading. In the second case, however, in which θ > A 0 f , the transformation strains are completely recovered by way of pseudoelasticity. In both cases, the simulation results are in good agreement with the experimental data.

Multiaxial non-proportional loading

Following [START_REF] Stebner | Explicit finite element implementation of an improved three dimensional constitutive model for shape memory alloys[END_REF] and [START_REF] Gu | Time integration and assessment of a model for shape memory alloys considering multiaxial nonproportional loading cases[END_REF], the single hexahedral element is simulated considering multiaxial non-proportional loading consisting of combined axial and shear loadings. The model parameters used for the simulation are taken from [START_REF] Gu | Time integration and assessment of a model for shape memory alloys considering multiaxial nonproportional loading cases[END_REF] and listed in Table 3.3. The loading is strain-controlled, corresponding to the path in Fig. 3.2(a) where the maximum strain of 4% is achieved in tension/compression and in shear. Starting from an unloaded state, the hexahedral element undergoes tension, shear, compression, then reverse shear until all strain is recovered. The material response in this case shows pronounced nonlinearity, commonly reported for SMAs subjected to non-proportional loading. The simulation results in terms of shear vs. normal stress are reported in figure 3.2(b) and compared to the results of [START_REF] Stebner | Explicit finite element implementation of an improved three dimensional constitutive model for shape memory alloys[END_REF] and [START_REF] Gu | Time integration and assessment of a model for shape memory alloys considering multiaxial nonproportional loading cases[END_REF]. Overall, the results are in agreement, more so with those of [START_REF] Gu | Time integration and assessment of a model for shape memory alloys considering multiaxial nonproportional loading cases[END_REF], which were obtained using the original small-strain ZM model. The most pronounced deviation is observed during shear strain reversal. This is mainly because the angle between the orientation direction of martensite variants and the driving force direction is neglected at high values of the volume fraction χ. Consequently, the orientation strain tensor becomes aligned with the thermodynamic driving force. The evolution of the von Mises equivalent stress and the martensite volume fraction with time is plotted in Fig. 3.3. The results correlate well with those of [START_REF] Gu | Time integration and assessment of a model for shape memory alloys considering multiaxial nonproportional loading cases[END_REF]. Compared to the work of [START_REF] Stebner | Explicit finite element implementation of an improved three dimensional constitutive model for shape memory alloys[END_REF], however, lower values of the von Mises stress are obtained at approximate time instants 2-3, 5, and 6-7, and higher values of the martensite volume fraction at approximate time instants 3 and 7. The deviation in this case is due to intrinsic differences in the formulation of the two models, including the expressions of the loading functions and interaction energies.

Comparison of the finite-strain model with the original ZM model

To illustrate the superiority of the present finite-strain model, simulation results involving large deformation and rotation are compared to those of the original ZM model. As a first example, the hexahedral element in section 3.5.1 is simulated using both models. The applied stress-displacement response in Fig. 3.4 shows increased deviation with increasing displacement as the validity of the small-strain assumption breaks down. The observed deviation in the above example is mitigated by the absence of rotation. To illustrate the improved accuracy of the present model with increasing rotation as well as strain, an example of a SMA beam subjected to bending is considered. The dimensions of the beam are 200 mm×20 mm×10 mm. The boundary conditions are such that one end of the beam is fixed while the second is subjected to a downward acting force magnitude of 8 KN. Three sets of simulations are carried out using the two models: (a) original ZM model with NLGEOM OFF, (b) original ZM model with NLGEOM ON, (c) present finite-strain model. NLGEOM option is usually used in Abaqus to consider small strains but large rotations. The deformed beam and the contours of the volume fraction χ within are shown in Fig. 3.5. From the figures, the result obtained using the original ZM model with NLGEOM OFF shows exaggerated increase in element size and abnormal deformation in the transverse direction in the beam. The loading end is deflected downward with no horizontal displacement, which is in strong contrast with the results obtained using the ZM model with NLGEOM ON and the present finite strain model. Further deviation is observed when comparing the extent of phase transformation predicted by the two models, for which the simulated propagation of martensite into the volume of the beam is strongly exaggerated by the original ZM model with NLGEOM OFF. A possible explanation is that the absence of finite rotation makes for a less compliant beam that tends to deform inelastically in a more pronounced way to Finite-strain extension of the ZM model accommodate the stress field generated by the applied loading. The result obtained using the original ZM model with NLGEOM ON shows more accuracy on both deformations and evolution of martensite volume fraction. However, it is still somewhat stiff compared to the result obtained by the present model.

Simulation of shape memory effect

In order to demonstrate the the model capability in predicting the shape memory effect of SMAs, the material subjected to a thermomechanical loading is simulated and the stress-strain-temperature response is shown in Fig. 3.6. At a temperature of 300 K, the material is loaded from zero to the maximum stress 360 MPa. The martensite variants are fully orientated in this magnitude of applied stress. A mechanically unrecoverable residual strain on the value of 4% is obtained by unloading the stress to zero and keeping the temperature constant. The unstressed material then is heated from the temperature of 300 K to 340 K. The strain starts recovery at the temperature of 314 K, which is the austenite start temperature A 0 s . Finally, the strain is fully recovered after heating the material above the austenite finish temperature A 0 f = 332 K.

Application to the simulation of an orthodontic archwire

The present model is used to simulate a SMA orthodontic archwire with the initial geometry shown in Fig. 3.7, with 1 mm×1 mm rectangular cross-section. The archwire is subjected to a force F that increases from zero to 45 N then decreases back to zero. Because of symmetry about the vertical centroidal axis, only one half of the structure is considered in the simulation. The results in Fig. 3.8(a) and (b) illustrate the contours of von Mises stress and martensite volume fraction in the structure at maximum deformation, at which large strain and rotation are observed. Maximum phase transformation is achieved, expectedly, at the location of the maximum von Mises stress. It is worth noting that values Additionally, the force-displacement curve at the point of application of the load is reported in Fig. 3.9. The curve shows complete shape recovery upon unloading, even though the shape of the curve itself is quite different from that of the uniaxial force-displacement response.

Conclusion

In this chapter, a finite-strain constitutive model for SMAs is developed by generalizing the small-strain ZM model. The derivation of the finite-strain model is based on a multiplicative decomposition of the deformation gradient into elastic and transformation parts as well as a further decomposition of transformation deformation into phase transformation and martensite reorientation parts. Constitutive equations are derived within a thermodynamically consistent framework involving the Clausius-Duhem entropy inequality, the Helmholtz free energy density and the pseudo-potential of dissipation. Numerical implementation is performed including the logarithmic map integration, the symmetrization of tensors as well as the proper derivation of the material and spatial tangent operators. The numerical integration scheme then is implemented into the commercial FEA software Abaqus/Standard by using a UMAT subroutine. Finally, finite element simulations are carried out and discussed in comparison with the experimental data under different loading cases. An application example involving a SMA archwire subjected to large strains and rotations is also presented. With this finite-strain extension of the ZM model, a more accurate prediction of the mechanical behavior of SMAs undergoing large strains and rotations is achieved. However, this prediction will be valid only for quasi-static isothermal loading cases. Indeed, experimental evidence suggests that phase transformation in SMAs is a thermomechanically coupled process, with internal heat production due to the latent heat and intrinsic dissipation. Meanwhile, the mechanical behavior of SMAs is strongly influenced by temperature. This thermomechanical coupling effect will be the topic of the next chapter.

Chapter 4

A thermomechanically coupled Hencky-strain model

Introduction

This chapter introduces the second step of the comprehensive constitutive modeling approach to investigate thermomechanical and cyclic behavior of SMAs in finite deformation. In order to take into account the strong thermomechanical coupling effect in SMAs, a new thermomechanically coupled, finite-strain model is developed based on Hencky strain in this chapter.

The Hencky strain measure, sometimes referred to as logarithmic or nature strain, has been used in constitutive modeling of solids due to its remarkable properties in large deformations [START_REF] Arghavani | A finite strain kinematic hardening constitutive model based on Hencky strain: General framework, solution algorithm and application to shape memory alloys[END_REF]. Among the usual finite strain measures, the Hencky strain has two important features: (i) only Hencky strain maps the volumetric and isochoric parts of the deformation gradient onto the pure spherical and deviatoric strain measures, which allows straightforward additive split of the total strain [START_REF] Xiao | Explicit dual stress-strain and strain-stress relations of incompressible isotropic hyperelastic solids via deviatoric Hencky strain and Cauchy stress[END_REF] (ii) the logarithmic corotational rate of the Eulerian Hencky strain is the stretching tensor D [START_REF] Reinhardt | Application of objective rates in mechanical modeling of solids[END_REF], which makes it conjugate to the Cauchy stress in finite strain formulations. Hence, the kinematics description of the model characterizes a decomposition of the Hencky strain into elastic, transformation and thermal parts, and a decomposition of the transformation stretching into phase transformation and martensite reorientation components. The model incorporates three important characteristics of SMA behavior that have not been concurrently addressed in previous work.

First, the model accounts for the effect of coexistence between austenite, single-variant and multivariant martensites, which is often neglected in the most SMA models [START_REF] Zaki | A three-dimensional model of the thermomechanical behavior of shape memory alloys[END_REF][START_REF] Lagoudas | Constitutive model for the numerical analysis of phase transformation in polycrystalline shape memory alloys[END_REF][START_REF] Paranjape | A phase field -Finite element approach to model the interaction between phase transformations and plasticity in shape memory alloys[END_REF]. As a result, these models do not account properly for thermomechanical SMA behavior in situations involving coexistence of the three phases. At high stress level, phase transformation takes place only between austenite and single-variant martensite as temperature varies, whereas at low or moderate stress levels austenite transforms into both singlevariant and multi-variant martensites and vice versa [START_REF] Wu | The nonlinear relationship between transformation strain and applied stress for nitinol[END_REF]. The coexistence of these three phases leads to the unsaturation of transformation strain, which could impact the utilization of SMA, e.g., as actuators. Efforts to model this effect consisted in considering separate volume fractions for self-accommodated and reoriented martensite [START_REF] Zaki | A three-dimensional model of the thermomechanical behavior of shape memory alloys[END_REF][START_REF] Auricchio | Theoretical and numerical modeling of shape memory alloys accounting for multiple phase transformations and martensite reorientation[END_REF], or relating the magnitude of the transformation strain to the applied stress level [START_REF] Lagoudas | Constitutive model for the numerical analysis of phase transformation in polycrystalline shape memory alloys[END_REF]. The modeling approach in this work consists in introducing two scalar variables to represent the volume fractions of multi-variant and single-variant martensites. The associated evolution rules and yield functions are derived separately and can be active simultaneously upon thermomechanical loading.

Secondly, the model considers the variation with temperature of the hysteresis size, which is also rarely addressed by current models despite experimental evidence [START_REF] Shaw | Thermomechanical aspects of NiTi[END_REF]. The hysteresis area can be physically interpreted as the amount of energy dissipated due to irreversible processes during phase transformation. In some models, the variation of this area with temperature is accounted for by introducing different temperature influence coefficients for the forward and reverse transformations [START_REF] Saleeb | A multi-axial, multimechanism based constitutive model for the comprehensive representation of the evolutionary response of SMAs under general thermomechanical loading conditions[END_REF][START_REF] Sedlák | Thermomechanical model for NiTi-based shape memory alloys including R-phase and material anisotropy under multi-axial loadings[END_REF][START_REF] Lagoudas | Constitutive model for the numerical analysis of phase transformation in polycrystalline shape memory alloys[END_REF]. However, the underlying thermodynamics of such variation have been largely left unaddressed. In this work, the hysteresis size is controlled by the entropy difference ∆η t between different phases, which is related to latent heat and transformation temperatures. The forward entropy difference ∆η f and the reverse entropy difference ∆η r are determined, respectively, in terms of the forward transformation temperature θ AM and reverse transformation temperature θ M A .

Finally, the model can capture the smooth transition at the initiation and completion of phase transformation in polycrystalline SMAs. This smooth and gradual transition, compared to the abrupt transition in single-crystalline and untrained polycrystalline SMAs [START_REF] Shaw | Thermomechanical aspects of NiTi[END_REF][START_REF] Patoor | Shape memory alloys, Part I: General properties and modeling of single crystals[END_REF], is experimentally observed in trained polycrystalline SMAs [START_REF] Sittner | Experimental study on the thermoelastic martensitic transformation in shape memory alloy polycrystal induced by combined external forces[END_REF][START_REF] Grabe | On the viscous and strain rate dependent behavior of polycrystalline NiTi[END_REF]. To account for the smooth transition, several transformation hardening functions were proposed in the literature, including exponential [START_REF] Tanaka | Thermomechanics of transformation pseudoelasticity and shape memory effect in alloys[END_REF], trigonometric [START_REF] Liang | One-dimensional thermomechanical constitutive relations for shape memory materials[END_REF], power law functions [START_REF] Lagoudas | Constitutive model for the numerical analysis of phase transformation in polycrystalline shape memory alloys[END_REF], etc. In many cases, however, the introduction of these functions result in significant increase in the number of model parameters without achieving the desired accuracy. In the present work, a unique tangential transformation hardening function g t featuring only 5 parameters is introduced to describe smooth transformation in polycrystalline SMAs to a good degree of accuracy.

Incorporating these three characteristics, a thermodynamically consistent framework and a Helmholtz free energy including elastic, thermal, interaction and constraint components, are constructed to derive constitutive equations. Time-discrete formulation of the model and a Hencky-strain return-mapping integration algorithm are proposed. The algorithm is then implemented in FEA software Abaqus/Explicit by means of a explicit user-defined material subroutine (VUMAT). Simulation results using this model are validated against experimental data obtained under various thermomechanical loading conditions. The robustness and efficiency of the proposed framework are illustrated by simulating a SMA helical spring actuator subjected to complex thermomechanical loading and boundary conditions.

A thermomechanically coupled model based on Hencky strain

In this section, on the basis of a multiplicative decomposition of deformation gradient F into elastic F e , transformation F t and thermal F θ parts, a thermodynamically consistent framework and a Helmholtz free energy function are introduced. Then, a thermomechanically coupled constitutive model in terms of Hencky strains is derived to describe SMA behavior in presence of complex thermomechanical loading and boundary conditions.

Kinematics

In most SMA models [START_REF] Reese | Finite deformation pseudo-elasticity of shape memory alloys -Constitutive modelling and finite element implementation[END_REF][START_REF] Arghavani | A finite strain kinematic hardening constitutive model based on Hencky strain: General framework, solution algorithm and application to shape memory alloys[END_REF][START_REF] Auricchio | Theoretical and numerical modeling of shape memory alloys accounting for multiple phase transformations and martensite reorientation[END_REF][START_REF] Xiao | An explicit, straightforward approach to modeling SMA pseudoelastic hysteresis[END_REF][START_REF] Chatziathanasiou | Modeling of coupled phase transformation and reorientation in shape memory alloys under non-proportional thermomechanical loading[END_REF], the deformation gradient F is simply decomposed into elastic and inelastic parts. However, extensive experimental investigations have shown that phase transformation in SMA is a thermomechanically coupled process, which involves heat production and transfer due to latent heat and intrinsic dissipation [START_REF] Shaw | Thermomechanical aspects of NiTi[END_REF][START_REF] Grabe | On the viscous and strain rate dependent behavior of polycrystalline NiTi[END_REF]. Therefore, an extended multiplicative decomposition of the deformation gradient F into elastic F e , transformation F t and thermal F θ parts is assumed here, such that

F = F e F t F θ , J = J e J t J θ , F = Fe Ft Fθ , (4.1)
where F e is defined with respect to a local unstressed intermediate configuration, F t is defined with respect to a thermally expanded intermediate configuration, and F θ is defined with respect to the reference configuration. J e = det F e , J t = det F t and J θ = det F θ denote the volume-changing deformations, Fe , Ft and Fθ denote the volume-preserving deformations and det Fe = det Ft = det Fθ = 1.

Substituting equation ( 4.1) into the definition of the velocity gradient L in equation (3.22) gives the following additive split:

L = L + 1 3 δ1 = Le + Fe Lt + Ft Lθ F-1 t F-1 e + 1 3 δ e + δ t + δ θ 1, (4.2) with Le = Fe F-1 e , Lt = Ft F-1 t , Lθ = Fθ F-1 θ , δ e = J e J e , δ t = J t J t , δ θ = J θ J θ , (4.3)
where Le , Lt and Lθ are isochoric elastic, transformation and thermal velocity gradients, δ e , δ t and δ θ are respective rates of volumetric deformations. Thus, the corresponding elastic, transformation and thermal stretching and spin tensors are given as

D e = sym (L e ) = De + 1 3 δ e 1 and W e = skew (L e ) , D t = sym (L t ) = Dt + 1 3 δ t 1 and W t = skew (L t ) , D θ = sym (L θ ) = Dθ + 1 3 δ θ 1 and W θ = skew (L θ ) . (4.4)
With regard to the thermal and the transformation deformations, the following three kinematic assumptions are made:

1. First, the thermal deformation is assumed to be an isotropic thermal expansion, so that

F θ = J 1 3 θ 1. (4.5)
Hence, using equations (4.4) 3 ,

D θ = 1 3 δ θ 1 and W θ = 0. (4.6)
2. Second, the transformation deformation is assumed incompressible, that is 3. Last, for isotropic materials, the assumption of the irrotationality of inelastic flow is widely used in modern elasto-viscoplastic constitutive theories [START_REF] Dafalias | The plastic spin concept and a simple illustration of its role in finite plastic transformations[END_REF][START_REF] Gurtin | The decomposition F= F e F p, material symmetry, and plastic irrotationality for solids that are isotropic-viscoplastic or amorphous[END_REF][START_REF] Anand | A thermo-mechanically coupled theory for large deformations of amorphous polymers. Part I: Formulation[END_REF]. Therefore, the transformation flow is assumed irrotational (zero transformation spin)

J t = det F t = 1 and F t = Ft . ( 4
W t = 0. (4.9)
Hence, the transformation stretching tensor e be ,

D t = L t = F t F -1 t and F t = D t F t . ( 4
C t = F T t F t , b t = F t F T t , C θ = b θ = J 2 3
θ 1, (4.12) where det Ce = det be = 1 and det C t = det b t = 1.

The Lagrangian and Eulerian Hencky strain tensors are respectively defined as

H = ln U = 1 2 ln C = H + 1 3 δ1 and h = ln V = 1 2 ln b = h + 1 3 δ1, (4.13)
where H = ln Ū and h = ln V are the deviatoric components and δ = ln J the volumetric component of the Hencky strain. Using equation ( 4.13), the elastic, transformation and thermal Lagrangian and Eulerian Hencky strain tensors are written as

H e = 1 2 ln C e = He + 1 3 δ e 1, h e = 1 2 ln b e = he + 1 3 δ e 1, H t = 1 2 ln C t , h t = 1 2 ln b t , H θ = h θ = 1 3 δ θ 1, (4.14)
where He = 1 2 ln Ce , he = 1 2 ln be , δ e = ln J e , δ θ = ln J θ . (4.15)

By making use of equation (A.11), the logarithmic tensor function maps the unimodular tensors Ce , be , C t and b t onto the traceless tensors He , he , H t and h t , i.e. tr( He ) = tr( he ) = tr(H t ) = tr(h t ) = 0.

Thermodynamic framework

The model is derived based on the principle of virtual power (PVP) in compliance with the first (energy balance) and second (entropy inequality) laws of thermodynamics. A reduced form of the energy balance between internal energy E, internal stress power P i and heat supply Q is written

∫ B e v dV E - ∫ B Jσ : D dV P i + ∫ ∂B q • n dS - ∫ B h v dV Q = 0, (4.16)
where e v and h v are, respectively, the internal energy and heat source per unit volume in spatial configuration, q is the heat flux vector per unit area and n is the outward unit vector normal to the boundary in the spatial configuration. Substituting the additive split of D (4.11) 1 into the internal stress power in equation (4.16) gives

Jσ : D = p δ e + p δ θ + s : De + FT e s F-T e : D t , (4.17)
where p and s are the hydrostatic and deviatoric components of the Kirchhoff stress tensor, i.e., τ = Jσ = p1 + s.

Substituting equation ( 4.17) into equation ( 4.16) and localizing the obtained integral equation gives the following local form of the energy balance:

e v -p δ e + p δ θ + s : De + FT e s F-T e : D t -h v + ∇ • q = 0. (4.18)
The local form of the Clausius-Duhem entropy inequality is expressed as

η v θ -h v + ∇ • q - 1 θ q • ∇θ ≥ 0, (4.19)
where η v is the entropy per unit volume in the spatial configuration.

The Helmholtz free energy ψ per unit volume and its time derivative ψ in the spatial configuration are given by

ψ = e v -η v θ and ψ = e v -η v θ -η v θ. (4.20)
Combining the local energy balance in equation ( 4.18), the local entropy inequality in equation ( 4.19) and the free energy in equation ( 4.20) 2 gives the following entropy inequality:

p δ e + δ θ + s : De + FT e s F-T e : D t -ψ + η v θ - 1 θ q • ∇θ ≥ 0. (4.21)
The Helmholtz free energy per unit volume is assumed here to depend on the elastic deformation gradient F e through the elastic Lagrangian Hencky strain H e , on the transformation deformation gradient F t through the transformation Lagrangian Hencky strain H t , on the thermal deformation gradient F θ through the thermal Lagrangian Hencky strain H θ , on scalar internal variables χ M , χ S , and on temperature θ as

ψ = ψ H e , H t , H θ , χ M , χ S , θ , (4.22)
where χ M and χ S are, respectively, volume fractions of multi-variant and single-variant martensites.

The free energy ψ is assumed to be an isotropic function of H e , H t and H θ , which allows ψ to be expressed as the function of their argument invariants. As the Lagrangian and Eulerian Hencky strains share the same invariants, the Helmholtz free energy ψ can be written in terms of Eulerian Hencky strains as

ψ H e , H t , H θ , χ M , χ S , θ = ψ δ e , he , h t , δ θ , χ M , χ S ,
: D t - ∂ψ ∂ χ M χ M - ∂ψ ∂ χ S χ S - ∂ψ ∂θ + η v θ - 1 θ q • ∇θ ≥ 0. (4.26)
For arbitrary thermodynamic processes, inequality (4.26) is guaranteed by the following choice of the constitutive equations with respect to p, s and η v :

p = ∂ψ ∂δ e = ∂ψ ∂δ θ , s = ∂ψ ∂ he , η v = - ∂ψ ∂θ , (4.27)
and by the following inequalities, which ensure nonnegative intrinsic dissipation in arbitrary evolutions of the internal state variables D t , χ M and χ S , so that

FT e s F-T e - ∂ψ ∂h t : D t - ∂ψ ∂ χ M χ M - ∂ψ ∂ χ S χ S ≥ 0, (4.28)
as well as nonnegative dissipation due to heat conduction, so that

- 1 θ q • ∇θ ≥ 0. ( 4
ψ + η v θ = p δ e + δ θ + s : he + X t : h t + Z M χ M + Z S χ S , (4.34)
with equation (4.20) 2 , which yields the following second Gibbs relation between the internal energy e v and the entropy η v :

e v -θ η v = p δ e + δ θ + s : he + X t : h t + Z M χ M + Z S χ S . (4.35)
Combining equations (4.35) and ( 4.18) gives the following entropy relation:

θ η v = h v -∇ • q + Me -X t : D t -Z M χ M -Z S χ S . (4.36)
The internal energy and entropy densities are assumed to depend on the same state variables as the Helmholtz free energy:

e v = e δ e , he , h t , δ θ , χ M , χ S , θ and η v = η δ e , he , h t , δ θ , χ M , χ S , θ . (4.37)
The specific heat capacity is defined as

c v def = ∂e δ e , he , h t , δ θ , χ M , χ S , θ ∂θ , (4.38)
with equations (4.20) and (4.27) 3 , which can be written as

c v = θ ∂η δ e , he , h t , δ θ , χ M , χ S , θ ∂θ = -θ ∂ 2 ψ δ e , he , h t , δ θ , χ M , χ S , θ ∂θ 2 . ( 4
δ θ + ∂η ∂ χ M χ M + ∂η ∂ χ S χ S + c v θ = c v θ -θ ∂ 2 ψ ∂θ∂δ e δ e + ∂ 2 ψ ∂θ∂ he : he + ∂ 2 ψ ∂θ∂h t : h t + ∂ 2 ψ ∂θ∂δ θ δ θ + ∂ 2 ψ ∂θ∂ χ M χ M + ∂ 2 ψ ∂θ∂ χ S χ S . ( 4
c v θ =h v -∇ • q + Me -X t : D t -Z M χ M -Z S χ S D=intrinsic dissipation + θ ∂ 2 ψ ∂θ∂δ e δ e + ∂ 2 ψ ∂θ∂ he : De + ∂ 2 ψ ∂θ∂δ θ δ θ X=thermoelastic coupling + θ ∂ 2 ψ ∂θ∂h t : D t + ∂ 2 ψ ∂θ∂ χ M χ M + ∂ 2 ψ ∂θ∂ χ S χ S
T=thermo-transformation coupling .

(4.41) At this stage of the development of the theory and the experimental investigation, the thermoelastic coupling terms in equation ( 4.41) that give rise to a temperature variation due to variations in the state variables δ e , δ θ and he are not well characterized [START_REF] Ames | A thermo-mechanically coupled theory for large deformations of amorphous polymers. Part II: Applications[END_REF][START_REF] Bouvard | An internal state variable material model for predicting the time, thermomechanical, and stress state dependence of amorphous glassy polymers under large deformation[END_REF]. Hence, as approximations, the thermoelastic coupling is assumed to be neglected and instead a factor ω = (1 + X/D) is introduced to represent the ratio of intrinsic dissipation converted into heat. Consequently, equation ( 4.41) reduces to

c v θ = h v -∇ • q + ωD + T, (4.42)
where the thermo-transformation coupling term T is associated with the latent heat of phase transformation. The heat flux q is taken to be governed by Fourier's law

q = -k∇θ, (4.43)
where k is the nonnegative thermal conductivity.

Helmholtz free energy

The Helmholtz free energy function ψ in equation ( 4.22) is additively split as

ψ = ψ e + ψ θ + ψ int + ψ cst , (4.44)
where ψ e is the elastic energy, ψ θ is the thermal contribution to free energy (Morin et al., 2011a;[START_REF] Lagoudas | Constitutive model for the numerical analysis of phase transformation in polycrystalline shape memory alloys[END_REF], ψ int is the interaction energy between single-variant martensite, multi-variant martensite and austenite phases [START_REF] Moumni | Theoretical and numerical modeling of solid-solid phase change: Application to the description of the thermomechanical behavior of shape memory alloys[END_REF], ψ cst is the potential energy due to the internal constraints.

By applying the rule of mixtures [START_REF] Lagoudas | Shape memory alloys, Part II: Modeling of polycrystals[END_REF][START_REF] Moumni | Theoretical and numerical modeling of solid-solid phase change: Application to the description of the thermomechanical behavior of shape memory alloys[END_REF][START_REF] Auricchio | Theoretical and numerical modeling of shape memory alloys accounting for multiple phase transformations and martensite reorientation[END_REF], the elastic energy ψ e and the thermal energy ψ θ in equation ( 4.44) can be expressed in the following form:

ψ e = 1 -χ M -χ S ψ e, A + χ M ψ e, M + χ S ψ e,S , ψ θ = 1 -χ M -χ S ψ θ, A + χ M ψ θ, M + χ S ψ θ,S , (4.45)
where ψ * , A , ψ * , M and ψ * ,S are the free energies of the austenite, multi-variant and single-variant martensites, respectively. Considering a Reuss rheological model [START_REF] Moumni | Theoretical and numerical modeling of solid-solid phase change: Application to the description of the thermomechanical behavior of shape memory alloys[END_REF][START_REF] Auricchio | Theoretical and numerical modeling of shape memory alloys accounting for multiple phase transformations and martensite reorientation[END_REF], the elastic free energy of the mixture in equation ( 4.45) 1 is given by

ψ e = 1 2 Kδ 2 + µ∥ he ∥ 2 -3α θ Kδ(θ -θ 0 ), (4.46)
where δ = δ e + δ θ is the volumetric part of the Hencky strain, θ 0 is the reference temperature, α θ is the thermal expansion coefficient. The bulk modulus K is assumed to be equal for all phases and shear modulus µ are determined from the Reuss model [START_REF] Wagner | Lattice stability, elastic constants and macroscopic moduli of NiTi martensites from first principles[END_REF][START_REF] Auricchio | Theoretical and numerical modeling of shape memory alloys accounting for multiple phase transformations and martensite reorientation[END_REF] using the relations:

K = K A = K M = K S and 1 µ = 1 -χ S -χ M 1 µ A + χ M 1 µ M + χ S 1 µ S . (4.47)
The thermal energy ψ θ in equation ( 4.44), sometimes referred to as chemical energy [START_REF] Sedlák | Thermomechanical model for NiTi-based shape memory alloys including R-phase and material anisotropy under multi-axial loadings[END_REF][START_REF] Auricchio | Theoretical and numerical modeling of shape memory alloys accounting for multiple phase transformations and martensite reorientation[END_REF], is related to changes in the internal energy and entropy. The thermal energies in equation ( 4.45) 2 for the different phases, ψ θ, * , are defined as

ψ θ, * = e * 0 -η * 0 θ + c * v (θ -θ 0 ) -θ ln θ θ 0 , (4.48)
where * denotes either of the phases A, M and S, e 0 and η 0 are reference internal energy and entropy at reference temperature θ 0 , c v is the specific heat capacity. Substituting equation ( 4.48) into equation (4.45) 2 gives the following thermal energy ψ θ :

ψ θ =e A 0 -η A 0 θ + c A v (θ -θ 0 ) -θ ln θ θ 0 -χ M ∆e AM + χ S ∆e AS + χ M ∆η AM + χ S ∆η AS θ -χ M ∆c AM v + χ S ∆c AS v (θ -θ 0 ) -θ ln θ θ 0 , (4.49)
where

∆e A * = e A 0 -e * 0 , ∆η A * = η A 0 -η * 0 and ∆c A * v = c A v -c *
v are the differences of internal energy, entropy and heat capacity between austenite and multi-/single-variant martensites.

Regarding the three different phases, the following assumptions are introduced to simplify the formulation of the model : (i) since many DSC experiments showed that the heat capacities of austenite and martensite are almost equal [START_REF] Qidwai | On thermomechanics and transformation surfaces of polycrystalline NiTi shape memory alloy material[END_REF],

c A v = c M v = c S v = c v and ∆c AM v = ∆c AS v = 0;
(ii) equal internal energy and entropy between multi-and single-variant martensites, i.e. ∆e AM = ∆e AS = L h and ∆η AM = ∆η AS = ∆η t = L h /θ t , where L h is the latent heat of phase transformation between austenite and martensite, the transformation temperature θ t and the entropy 4.2 A thermomechanically coupled model based on Hencky strain 55 difference ∆η t are given by

θ t = θ AM A → M, S, θ M A M, S → A, and ∆η t = ∆η f = L h /θ AM , ∆η r = L h /θ M A , (4.50)
where θ AM and θ M A are forward and reverse phase transformation temperatures, ∆η f and ∆η r are forward and reverse entropy differences. With these assumptions, the thermal energy in equation ( 4.49) reduces to

ψ θ = e A 0 -η A 0 θ + c v (θ -θ 0 ) -θ ln θ θ 0 + ∆η t χ M + χ S (θ -θ 0 ). (4.51)
The interaction energy ψ int in equation ( 4.44) represents the crystal boundary energy between austenite, multi-and single-martensite phases, which is often derived from micro-mechanical or crystallographic considerations and finally expressed as a function of internal state variables [START_REF] Zaki | A three-dimensional model of the thermomechanical behavior of shape memory alloys[END_REF][START_REF] Auricchio | Theoretical and numerical modeling of shape memory alloys accounting for multiple phase transformations and martensite reorientation[END_REF]. In this work, the interaction energy ψ int is assumed to depend on the multi-variant χ M , single-variant χ S martensite volume fractions and transformation strain h t as

ψ int = ψ int χ M , χ S , h t = G t ( χ) + 1 2 µ t ∥h t ∥ 2 , (4.52)
where χ = χ M + χ S is the volume fraction of total martensite. The first term G t on the right-hand side of equation ( 4.52) represents the energy of interaction between austenite and martensite, while the second term 1 2 µ t ∥h t ∥ 2 represents increase in interaction due to orientation/reorientation of martensite plates (multi-variant martensite to single-variant martensite). The difference between the two martensite variants is neglected with respect to interaction with austenite.

The internal state variables χ M , χ S and h t used for modeling obey the following physical constraints:

• The volume fractions of multi-variant χ M , single-variant χ S and total χ M + χ S martensites are restricted to the [0, 1] interval,

0 ≤ χ M , χ S , χ M + χ S ≤ 1, (4.53)
which can be rewritten as the following three active constraints:

χ M ≥ 0, χ S ≥ 0, 1 -χ M + χ S ≥ 0. (4.54)
• The magnitude of transformation strain due to orientation of martensite variants must remain lower than a material-specific limit H , i.e.

H -∥h t ∥ ≥ 0. (4.55)

A thermomechanically coupled Hencky-strain model

The above physical constraints on internal state variables are guaranteed by the following potential as the constraints contribution to free energy ψ cst in equation ( 4.44):

ψ cst = -ζ M χ M -ζ S χ S -ζ M S 1 -χ M -χ S -ζ t (H -∥h t ∥) , ( 4
               χ M ≥ 0, ζ M ≥ 0, ζ M χ M = 0; χ S ≥ 0, ζ S ≥ 0, ζ S χ S = 0; 1 -χ M + χ S ≥ 0, ζ M S ≥ 0, ζ M S 1 -χ M -χ S = 0; H -∥h t ∥ ≥ 0, ζ t ≥ 0, ζ t (H -∥h t ∥) = 0. (4.57)
In conclusion, the Helmholtz free energy ψ is given by

ψ = 1 2 Kδ 2 + µ∥ he ∥ 2 -3α θ Kδ(θ -θ 0 ) + e A 0 -η A 0 θ + c v (θ -θ 0 ) -θ ln θ θ 0 + ∆η t χ M + χ S (θ -θ 0 ) + g t + 1 2 µ t ∥h t ∥ 2 - ζ M χ M + ζ S χ S + ζ M S 1 -χ M -χ S + ζ t (H -∥h t ∥) .
(4.58)

Constitutive equations

Substituting the Helmholtz free energy function ψ (4.58) into the thermodynamic framework in section 4.2.2, the hydrostatic stress p, deviatoric stress s and entropy η v are derived as Moreover, the partial derivatives of ψ with respect to the internal state variables h t , χ M and χ S are given by where g t = ∂G t /∂ χ is the phase transformation hardening function. In order to capture the smooth phase transformation response, as experimentally demonstrated [START_REF] Lagoudas | Shape memory alloys, Part II: Modeling of polycrystals[END_REF][START_REF] Hartl | Three-dimensional modeling and numerical analysis of rate-dependent irrecoverable deformation in shape memory alloys[END_REF] and theoretically formulated [START_REF] Lagoudas | Constitutive model for the numerical analysis of phase transformation in polycrystalline shape memory alloys[END_REF][START_REF] Auricchio | Theoretical and numerical modeling of shape memory alloys accounting for multiple phase transformations and martensite reorientation[END_REF] in SMA literature, a tangential hardening function g t is considered in the following form:

           p = ∂ψ ∂δ = K [δ -3α θ (θ -θ 0 )] , s = ∂ψ ∂ he = 2µ he , η v = -∂ψ ∂θ = 3α θ Kδ + η A 0 + c v ln θ θ 0 -∆η t χ M + χ S . ( 4
           X t = ∂ψ ∂h t = µ t h t + ζ t h t ∥h t ∥ , Z M = ∂ψ ∂χ M = ∆η t (θ -θ 0 ) + g t -ζ M + ζ M S , Z S = ∂ψ ∂χ S = ∆η t (θ -θ 0 ) + g t -ζ S + ζ M S ,
g t = κtan m 1 2 (ξ a χ + ξ b )π + r, (4.62)
where κ, m, ξ a , ξ b and r are model parameters obeying the following constraints:

κ > 0, 0 < m < 1, ξ a > 0, ξ b > 0, ξ a + ξ b < 1. (4.63)
The single-variant martensite volume fraction χ S , according to the literature [START_REF] Reese | Finite deformation pseudo-elasticity of shape memory alloys -Constitutive modelling and finite element implementation[END_REF][START_REF] Moumni | Theoretical and numerical modeling of solid-solid phase change: Application to the description of the thermomechanical behavior of shape memory alloys[END_REF][START_REF] Lagoudas | Constitutive model for the numerical analysis of phase transformation in polycrystalline shape memory alloys[END_REF], can be expressed in terms of the transformation strain as

χ S = ∥h t ∥ H . (4.64)
In addition, the transformation strain h t can be equivalently expressed as

h t = ∥h t ∥N t , (4.65)
where N t is the direction of transformation strain. Using equations (4.64) and (4.65), the time derivative of h t is given by (4.66) where the first term represents the phase transformation between austenite and single-variant martensite, the second term represents the reorientation of single-variant martensite. Substituting equation ( 4.66) into equation (A.4) gives the following split of logarithmic corotational rate of transformation Hencky strain:

h t = H N t χ S + H χ S N t ,
hL t = H N t χ S + H χ S N t -Ω L t N t + N t Ω L t . (4.67)
Considering the coaxiality of h t and N t , the following corotational rate of N t is introduced as

D N = N L t = N t -Ω L t N t + N t Ω L t . (4.68)
Therefore, the transformation stretching tensor D t is rewritten as

D t = H χ S N t + H χ S D N . ( 4

.69)

A thermomechanically coupled Hencky-strain model Substituting equation ( 4.69) into inequality (4.32) gives (4.70) where the thermodynamic forces A N associated to D N , A M associated to χ M , and A S associated to χ S are given using equations (4.60) and ( 4.61) as

H χ S Me -X t A N : D N -Z M A M χ M + H Me -X t : N t -Z S A S χ S ≥ 0,
         A N = H χ S 2µ He -µ t h t -ζ t N t , A M = -∆η t (θ -θ 0 ) -g t + ζ M -ζ M S , A S = 2µH He : N t -µ t H 2 χ S -∆η t (θ -θ 0 ) -g t -H ζ t + ζ S -ζ M S .
(4.71)

The nonnegative dissipation of multi-variant and single-variant martensite phase transformations is guaranteed by the following choice of the evolution equations:

χ M = γ M S(A M ) and χ S = γ S S(A S ), (4.72)
where γ M and γ S are nonnegative multipliers. The associated yield functions for the phase transformations are given by

F M = | A M | -Y M and F S = | A S | -Y S , (4.73)
where Y M and Y S control the thresholds of the phase transformations. γ M , F M , γ S and F S are subjected to the following Kuhn-Tucker consistency conditions:

γ M ≥ 0, F M ≤ 0, γ M F M = 0; γ S ≥ 0, F S ≤ 0, γ S F S = 0. (4.74)
The nonnegative dissipation of martensite reorientation is guaranteed by the following associative evolution equation:

D N = γ tr A N ∥ A N ∥ , (4.75)
where γ tr is a nonnegative multiplier. The associated yield function for martensite reorientation is given by 4.76) where Y tr defines the martensite reorientation threshold. γ tr and F tr are subjected to the Kuhn-Tucker conditions:

F tr = ∥ A N ∥ -( χ S ) 2 Y tr , (
γ tr ≥ 0, F tr ≤ 0, γ tr F tr = 0. (4.77)
Moreover, from the time derivative of ∥ N t ∥ = 1 and the equation (A.8), one gets

N t : N t = N t : D N = 0. (4.78)
Table 4.1 Summary of the thermomechanically coupled, finite-Hencky-strain model.

State and internal variables: δ e , he , h t , δ θ , χ M , χ S , θ Thermoelasticity: p = K [δ e + δ θ -3α(θ -θ 0 )] and s = 2µ he Multi-variant martensite transformation:

χ M = γ M S(A M ), where A M = -∆η t (θ -θ 0 ) -g t + ζ M -ζ M S Single-variant martensite transformation: χ S = γ S S(A S ), where A S = 2µH He : N t -µ t H 2 χ S -∆η t (θ -θ 0 ) -g t -H ζ t + ζ S -ζ M S Martensite reorientation: D N = γ tr A N ∥ A N ∥ , where A N = H χ S (I -N t ⊗ N t ) Me Transformation strain: D t = H γ S S(A S )N t + γ tr χ S A N ∥ A N ∥ Temperature evolution: c v θ = h v -∇ • q + ω γ tr ∥ A N ∥ + γ M | A M | + γ S | A S | + θ∆η t γ M S(A M ) + γ S S(A S ) Yield functions: F M = | A M | -Y M , F S = | A S | -Y S , F tr = ∥ A N ∥ -( χ S ) 2 Y tr Kuhn-Tucker consistency conditions:        γ M ≥ 0, F M ≤ 0, γ M F M = 0; γ S ≥ 0, F S ≤ 0, γ S F S = 0; γ tr ≥ 0, F tr ≤ 0, γ tr F tr = 0.
The numerical approximation of the evolution equation ( 4.81) of transformation strain is based on an exponential integration scheme as

F t = exp (T ) F t n , (4.84) with T = H ∆γ S S(A S )N t + ∆γ tr χ S A N ∥ A N ∥ , (4.85)
where T is the flow vector of transformation strain, ∆γ S and ∆γ tr are the multipliers at time t n+1 .

Regarding the temperature change in equation ( 4.82), the following time-discrete temperature evolution equation is given as

θ = θ n + 1 c v (h v -∇ • q + ωD + T ) , (4.86)
where o he is a second-order term in the elastic Hencky strain. Therefore, equation ( 4.92) is written as

exp 2 he trial = exp 2R e T R eT + he exp 2R e T R eT + exp 2R e T R eT he + o he , (4.94)
with the following definition of Green-Lagrange-type tensor of incremental transformation strain h t ∆ :

h t ∆ = 1 2 exp 2R e T R eT -1 , (4.95)
which can be rewritten equivalently as

exp 2 he trial = 1 + 2 h t ∆ + he + he h t ∆ + h t ∆ he + o he . (4.96)
By making use of equation (A.10), the logarithm of both sides of equation ( 4.96) gives

2 he trial = ∞ k=0 (-1) k k + 1 2 h t ∆ + he + he h t ∆ + h t ∆ he + o he k+1 . (4.97)
According to the multinomial theorem, equation ( 4.97) can be approximately expressed as

2 he trial = ∞ k=0 (-1) k k + 1 2h t ∆ k+1 + ∞ k=0 (-1) k k + 1 2 he k+1 + o h t ∆ + o he , (4.98)
where o(h t ∆ ) is the second-order term in the incremental transformation strain. In view of equations (4.93) and (4.95), equation (4.98) then is written as

he trial = he + R e T R eT + o h t ∆ + o he . (4.99)
Since the exponential map integrator (4.91) is first-order accurate and the elastic strain is assumed to be infinitesimal, the second-order terms can be neglected in equation ( 4.99) leading to the following approximate formula:

R t = he trial - he -R e T R eT = 0, (4.100)
which is used as a substitute for equation (4.88) 2 to improve numerical efficiency. Finally, the system (4.88) is solved using a Newton-Raphson method.

Calibration of the model parameters

In this section, the proposed three-dimensional (3D) model is first reduced to a one-dimensional (1D) version. Then, the 1D model is implemented into MATLAB using an explicit integration scheme to calibrate the model parameters against experimental data.

For the case of uniaxial loading, the Cauchy stress tensor σ and the deformation gradient F are given as σ = σ e 1 ⊗ e 1 and F = λ e 1 ⊗ e 1 + λ l e 2 ⊗ e 2 + λ l e 3 ⊗ e 3 , (4.101)

where σ and λ denote the uniaxial stress and stretch, λ l denotes the lateral contractions. e 1 , e 2 , e 3 are the three orthonormal bases. Since the spherical stress-strain response (4.59) itself is a scalar equation, the reduced 1D equations will not capture it and therefore volume-preserving deformation,

i.e. J = det F = 1, is assumed. This assumption gives the relation: λ l = 1 λ . Moreover, the polar decomposition can be written as

F = V R = V since R = 1.
The velocity gradient L is then computed as (4.109) where the entropy difference ∆η t takes different quantities ∆η f for the forward and ∆η r the reverse phase transformation. The 1D reduced model is then implemented into MATLAB to calibrate the model parameters from the uniaxial experimental data. The physical interpretation and the graphical illustration of the model parameters are provided in the following. • The elastic bulk modulus K and the shear moduli µ A and µ M are determined from experimental pseudoelastic stress-strain curve as follows: • The forward and reverse entropy differences, ∆η f and ∆η r , govern the variation of transformation stresses with temperature. This influence is demonstrated in Fig 4.1(a) with two stress-strain curves obtained at temperature θ (black) and θ + ∆θ (blue), wherein the temperature increase of ∆θ leads to the increase of the forward transformation stress of ∆σ f and the reverse one of ∆σ r .

L = FF -1 = λ λ T and T = 1 e 1 ⊗ e 1 - 1 2 e 2 ⊗ e 2 - 1 2 e 3 ⊗ e 3 , ( 4
F t = λ t e 1 ⊗ e 1 + 1 λ t e 2 ⊗ e 2 + 1 λ t e 3 ⊗ e 3 , ( 4
     A M = -∆η t (θ -θ 0 ) -g t + ζ M -ζ M S , A S = 2 3 H σ -µ t H 2 χ S -∆η t (θ -θ 0 ) -g t -H ζ t + ζ S -ζ M S .
K = E A + E M 6(1 -2ν) , µ A = E A 2(1 + ν) and µ M = E M 2(1 + ν) , ( 4
These two parameters are calculated as follows: • Y S and Y M control the thresholds of the single-variant and the multi-variant martensite transformations. For the sake of calculus simplicity, it is assumed Y M = Y S . They determine the height of the hysteresis loop σ h , as shown in Fig 4.1(a). They are computed using the following expression:

∆η f = H ∆σ f ∆θ and ∆η r = H ∆σ r ∆θ . ( 4
Y S = 1 2 H σ h . (4.112)
• The martensite reorientation threshold Y tr , the reorientation hardening modulus µ t and the magnitude limit of transformation strain H are determined from the stress-strain curve of martensite orientation experiment, wherein multi-variant martensite is completely oriented to single-variant martensite under the uniaxial tensile loading. Y tr takes the initial yield stress, µ t the slope of orientation plateau and H the magnitude of the residual strain, as shown in Figure 4.1(b).

• The parameters κ, m, ξ a , ξ b and r in the tangential hardening function g t control the smooth transition at the initiation and completion of the phase transformation. Figure 4.2 shows how the parameters κ, m, ξ a , ξ b and r affect the pseudoelastic stress-strain behavior.

Numerical simulations and experimental validations

The proposed model and the corresponding numerical algorithm presented in the previous sections are implemented into the finite element analysis software Abaqus/Explicit by means of a VUMAT subroutine.

To demonstrate capabilities of the model, numerical simulations are carried out and compared to the experimental data under a variety of thermomechanical loading conditions. Finite element simulation of a SMA helical spring actuator undergoing large deformations and temperature variation is also performed.

Isothermal pseudoelasticity

The first set of simulations is dedicated to pseudoelasticity under isothermal uniaxial tensile loading at temperatures of 298 K, 303 K and 313 K. Experimental data for a polycrystalline NiTi wire (50.8 at.% Ni, provided by Memry Corporation) reported by [START_REF] Lagoudas | Constitutive model for the numerical analysis of phase transformation in polycrystalline shape memory alloys[END_REF] is utilized for validation. The model parameters used for this set of simulations are calibrated from the experimental data and listed in Table 4.2. In the simulations, the applied stress is increased from 0 to the maximum value of 700 MPa then removed, with the temperature maintained at a constant value. Figure 4.3 shows comparisons between the model predictions and the experimental data reported by [START_REF] Lagoudas | Constitutive model for the numerical analysis of phase transformation in polycrystalline shape memory alloys[END_REF]. The pseudoelastic stress-strain responses at constant temperatures of 298 K,303 K and 313 K are,respectively,plotted in Figure4.3(a), (b) and (c). In addition, the evolution of the single-variant martensite volume fraction χ S is presented in Figure 4.3(d). From the figures, it is seen that the proposed model captures the pseudoelastic behavior of the considered polycrystalline SMA with good accuracy. In particular, complete shape recovery is achieved upon unloading, a smooth response is

Thermally-induced phase transformation

This section is devoted to the investigation of SMA response to temperature variation under zero-stress condition. Extensive experimental investigations have shown that phase transformation of SMA is a thermomechanically coupled process. The latent heat and the intrinsic dissipation contribute to heat generation and absorption during phase transformation, which can be measured by Differential Scanning Calorimetry (DSC) thermoanalytical technique. The difference in the amount of heat required to increase the temperature of a SMA sample compared to a reference specimen is measured as a function of temperature in DSC analysis. The thermally-induced phase transformation and the corresponding heat generation/absorption are predicted by the proposed constitutive model. Experimental date reported by [START_REF] Popov | A 3-D constitutive model for shape memory alloys incorporating pseudoelasticity and detwinning of self-accommodated martensite[END_REF] is utilized for model validation. The model parameters used in the simulation are determined from the uniaxial stress-strain response and the DSC data, listed in Table 4.3.

In the simulation, an unconstrained SMA component is first uniformly heated from 273 K to 393 K and then uniformly cooled back to 273 K. 4.4(a) shows a comparison between the model predictions and the experimental DSC data reported by [START_REF] Popov | A 3-D constitutive model for shape memory alloys incorporating pseudoelasticity and detwinning of self-accommodated martensite[END_REF], wherein the relative heat flow is plotted as a function of temperature. Heat absorption from martensite to austenite phase transformation results in a ridge along the heating part of the curve, while heat generation from austenite to martensite phase transformation results in a valley along the cooling part. The bottom and peak points on the cooling and heating parts, corresponding to temperatures θ M A = 303 K and θ AM = 339, denote average temperatures for forward and reverse phase transformation, respectively. It is worth noting that the ridge is short and wide, while the valley is deep and narrow. This is because the area under the curve in each case, representing phase transformation latent heat, is assumed to be the same. In this particular case, the temperature range of reverse transformation is larger than that of forward transformation, resulting in a low and wide ridge versus a deep and narrow valley. Overall, the heat absorption/generation during thermally-induced phase transformation is well described by the proposed model despite a small deviation at completion of the reverse phase transformation. Thanks to the tangential hardening function g t defined in equation ( 4.62), the smooth experimental DSC curves are simulated with good accuracy. In addition to simulating the heat exchange during phase transformation, the evolution of multi-variant martensite volume fraction is simulated and shown in Figure 4.4(b). According to the simulation, the reverse phase transformation of martensite to austenite is found to initiate at temperature 305 K and complete at temperature 353 K, while the forward phase transformation of austenite to martensite initiates at temperature 313 K and completes at temperature 300 K.

Isobaric thermal cycling tests

In isobaric thermal cycling tests, SMA samples are preloaded uniaxially to different constant stress levels, which are maintained while the material undergoes thermally-induced transformation cycles (actuation cycles). The evolution of transformations train with temperature and its dependence on the level of applied stress are predicted by the proposed model. The experimental data reported by [START_REF] Wu | The nonlinear relationship between transformation strain and applied stress for nitinol[END_REF] for equiatomic NiTi (50.0 at.% Ni) is utilized to validate the model. The model parameters used in the simulations are calibrated from the experimental data and listed in Table 4.4. Two constant stress levels of 160 MPa and 360 MPa are considered in the simulations, wherein temperature is decreased uniformly from 393 K to 253 K and then increased uniformly back to 393 K.

Figure 4.5 shows comparisons between model predictions and the experimental data reported by [START_REF] Wu | The nonlinear relationship between transformation strain and applied stress for nitinol[END_REF] for stress levels of 160 MPa and 360 MPa. Since the elastic SMA response was not addressed in the source work, the comparisons shown in Figure 4.5 is based on the variation of the transformation strain with temperature. At 160 MPa, the yield functions associated with both single-variant martensite transformation and multi-variant martensite transformation are activated. Thus, austenite transforms simultaneously into these two martensite variants, leading to unsaturated transformation strain of 0.021. In contrast, the stress at 360 MPa is too high to activate the yield function associated with multi-variant martensite transformation. In this case, austenite transforms entirely into single-variant martensite and the maximum transformation strain reaches the saturated value H = 0.047, shown as Table 4.4. It is worth noting that the material response shows abrupt transitions at the completion of forward transformation and at the initiation of reverse transformation at 160 MPa. This phenomenon is due to an assumption in the proposed model that the activation of the yield function associated with multivariant martensite transformation will shut down the transformation to single-variant martensite. This deficiency can be addressed by considering the coupling effect between these two types of martensite transformation. Overall, the evolution of transformation strain with temperature and the variation of the maximum transformation strain with the applied stress are successfully predicted by the proposed model.

Multiaxial non-proportional loading tests

In this section, a biaxial strain-controlled butterfly-shaped loading test is investigated to demonstrate the reliability of the model in presence of multiaxial non-proportional loading conditions. Experimental data reported by [START_REF] Grabe | Path dependence and multiaxial behavior of a polycrystalline NiTi alloy within the pseudoelastic and pseudoplastic temperature regimes[END_REF] is utilized as references. The model parameters used in the simulation are calibrated from the experimental data and listed in Table 4.5. In the source work, γ ′ = γ/ √ 3 and τ ′ = √ 3τ are used as shear strain and stress measures for simplicity of a von Mises-type equivalence. The overall agreement with the experimental data is satisfactory despite small deviation in presence of dominant shear and compression loadings. This is because the yield criteria for phase transformation and martensite reorientation of the von Mises type, which dose not account for the experimentally observed asymmetry in NiTi behavior in tension, compression and shear. A better simultaneous fit to tension, compression and shear data is achievable by means of more sophisticated criteria.

Pseudoelasticity under non-isothermal conditions

To demonstrate the influence of thermomechanical coupling on SMA behavior, simulations of pseudoelastic uniaxial tensile tests under non-isothermal boundary conditions are carried out. Experimental data reported by [START_REF] Shaw | Thermomechanical aspects of NiTi[END_REF] is used to validate the model. In the referenced work, pseudoelastic uniaxial tensile tests were performed on polycrystalline NiTi specimens (50.1 at.% Ni) at three different strain rates of 0.0004s -1 , 0.004s -1 and 0.04s -1 . In each case, heat exchange with the surrounding takes place by means of natural air convection with the external temperature maintained at 343 K. The model parameters used in the simulations are calibrated from the experimental data and listed in Table 4.6. In the simulations, the applied stress is increased from zero to the maximum of 1000 MPa then decreased to zero for each strain rate. The finite element model is defined with a uniform initial temperature field of 343 K and natural convection at the boundary with a convection heat transfer coefficient h = 50 Wm -2 K -1 . actuators can be found in literature [START_REF] Toi | Finite element analysis of superelastic, large deformation behavior of shape memory alloy helical springs[END_REF][START_REF] Hartl | Three-dimensional modeling and numerical analysis of rate-dependent irrecoverable deformation in shape memory alloys[END_REF][START_REF] Lagoudas | Constitutive model for the numerical analysis of phase transformation in polycrystalline shape memory alloys[END_REF][START_REF] Saleeb | Large scale simulation of NiTi helical spring actuators under repeated thermomechanical cycles[END_REF][START_REF] Auricchio | Theoretical and numerical modeling of shape memory alloys accounting for multiple phase transformations and martensite reorientation[END_REF]. The SMA helical spring actuator adopted in the simulation has a coil diameter of 20 mm, a wire diameter of 2 mm and a spring pitch of 10 mm. Figure 4.8(a) shows the initial geometry, meshed using 3312 coupled temperature-displacement reduced integration hexahedral elements (Abaqus/Explicit C3D8RT). The element highlighted in red is chosen for studying the local material response, i.e. stressstrain behavior and phase transformation. The mesh of the cross-section adjacent to the red element is shown in the figure . Loading is applied at the nodes on end surfaces of the spring. The thermomechanical loading cases considered in the simulations are shown in Figure 4.8(b).

In the first simulation, the applied forces are increased from zero to the maximum value of 15 N at the time of 5 seconds and then removed at the time of 10 seconds, with a prescribed temperature 380 K (black lines in Figure 4.8(b)). It is worth noting that the force is applied in a smooth manner to eliminate dynamic effect for quasi-static analysis in Abaqus/Explicit. Figure 4.9 shows the contour plots of the von Mises stress and the martensite volume fraction χ in the spring at the maximum force magnitude of 15 N. The maximum von Mises stress is achieved in central part of the spring, shown as Figure 4.9(a). Regarding its distribution through the cross-section, the von Mises stress is greater towards the outside compared to the core. In Figure 4.9(b), the martensite volume fraction is highest on the inner surface of the central segment of the spring. The local stress response and the phase transformation in the chosen element (red in Figure 4.8(a)) are shown in Figure 4.10(a). With increasing applied force, the von Mises stress on the element increases as well. Phase transformation from austenite to martensite then takes place when the von Mises stress exceeds 670 MPa. When the load is removed, martensite transforms back to austenite. The global force-displacement curve at the point of application of the load on the end surface is plotted in Figure 4.10(b). The figure shows complete recovery of inelastic strain upon removal of the load. In the second simulation, the spring is first preloaded with a force of magnitude 4 N in a smooth manner over a time period of 5 seconds at a constant temperature of 380 K. It is then cooled uniformly to 280 K before being uniformly heated back to 380 K over another time period of 5 seconds (blue lines in Figure 4.8(b)). Figure 4.11 shows contour plots of the von Mises stress and the martensite volume fraction χ in the spring at 280 K. The von Mises stress is maximum near the inner surface of the central portion of the spring as shown in Figure 4.11(a). In the same portion, the martensite volume fraction reaches its maximum value of magnitude 1 along toward the outside of the section, indicating complete transformation of austenite to martensite. The local stress response and phase transformation in the highlighted element (red in Figure 4.8(a)) are shown in Figure 4.12(a). In the first half of the loading sequence, the von Mises stress increases with increased applied force to nearly 480 MPa. However, as the temperature is decreased from 380 K to 280 K at constant applied force, the von Mises stress decreases to 230 MPa and austenite completely transforms to martensite. When temperature is increased again to 380 K, the martensite finally transforms back to austenite. The displacement of the point of application of the load on the end surface is plotted versus temperature in Figure 4.12(b). At 380 K, the applied force of 4 N generates approximately 4 mm of elastic displacement. when temperature is decreased to 320 K, the formation of single-variant martensite results in significant transformation strain.

The spring retrieves its original shape as the temperature is increased back to 380 K. The complete actuation cycle of the SMA helical spring actuator is shown in the figure.

Conclusion

A thermomechanically coupled, Hencky-strain based constitutive model for SMAs has been proposed in this chapter. The formulation of the model is based on a multi-tier decomposition of deformation kinematics comprising: (i) a multiplicative decomposition of the deformation gradient into elastic, transformation and thermal parts, (ii) an additive decomposition of the Hencky strain into spherical and deviatoric parts, (iii) an additive decomposition of the transformation stretching tensor into phase transformation and martensite reorientation components. A thermodynamically consistent framework was developed, and a Helmholtz free energy function including elastic, thermal, interaction and constraint components was introduced. Constitutive and heat equations were derived from the established thermodynamic framework and Helmholtz free energy. Three important characteristics of SMA response are considered in formulating the model. First, the effect of phase coexistence between austenite and two martensite variants is accounted for by deriving separate evolution laws of volume fractions for single-variant χ S and multi-variant χ M martensites, which can be active simultaneously. Then, the variation of the hysteresis size with temperature is characterized by means of the forward entropy difference ∆η f and reverse difference ∆η r . Finally, the smooth transition at initiation and completion of phase transformation is accounted for using a unique tangential transformation hardening function g t .

Time-discrete formulation of the constitutive equations were derived and integrated using a Henckystrain return-mapping algorithm. Whilst the proposed three-dimensional model was reduced to a one-dimensional form to perform the calibration of the model parameters. The model was then implemented into Abaqus/Explicit using a VUMAT subroutine. Numerical simulations are carried out and validated against experimental data under various loading cases, including proportional, non-proportional, isothermal, non-isothermal conditions. Finally, a simulation example is proposed in which a SMA helical spring is first subjected to mechanical loading resulting in a pseudoelastic response then to thermal loading resulting in a temperature-induced actuation. The simulation results illustrate the ability of the proposed model to describe the behavior of SMA devices subjected to complex thermomechanical loadings.

The proposed thermomechanically coupled finite-strain model has capabilities of capturing the non-isothermal material behavior of SMAs under complex thermomechanical boundaries and loadings. However, it can not predict the characteristics related to the cyclic behavior of SMA, such as the accumulation of residual strain, the degeneration of pseudoelasticity and dissipated energy, and the Chapter 5

Finite-strain cyclic behavior of SMAs

Introduction

This chapter introduces the third step of the comprehensive constitutive modeling approach to investigate thermomechanical and cyclic behavior of SMAs in finite deformation. The chapter deals with the development of a thermomechanical finite-strain constitutive model capable of predicting cyclic behavior of polycrystalline SMAs. The model can be used for fatigue analysis of SMA structures if a proper fatigue criterion is introduced.

As presented in 2.2.3, in order to predict the cyclic behavior of SMAs, a number of constitutive models have been proposed with different degrees of sophistication. However, these models suffer from the following limitations: (i) all these models were developed under assumption of small strain, though most of the strain levels in cyclic experiments have entered finite strain regime; (ii) most of them are not able to capture the rate dependence; (iii) none of them considered the abrupt to smooth transition during training process.

To this end, this chapter is dedicated to developing a 3D thermomechanical finite-strain constitutive model for cyclic pseudoelasticity of polycrystalline SMAs. The model is able to describe the four fundamental characteristics related to the cyclic behavior of SMA that have not been integrally addressed within the finite-strain framework: (i) large accumulated residual strain that results from the residual martensite and the dislocations slipping during cycling; (ii) degeneration of pseudoelasticity and hysteresis loop due to the increase of dislocation density and internal stresses with the number of cycles; (iii) rate dependence that can be attributed to the thermomechanical coupling effect; (iv) evolution of the phase transformation from abrupt to smooth manner, as a consequence of the diversified crystallographic orientations of the grains, the heterogeneity of internal stresses, and the presence of non-transforming precipitates during cycling. Based on the decomposition of finite Hencky strain into elastic, transformation, residual and thermal parts, the model is constructed within a thermodynamically consistent framework. Evolution equations associated with the internal variables are derived from the reduced form of energy balance, the Clausius-Duhem form of entropy inequality, and a Helmholtz free energy function that includes elastic, thermal, interaction and constraint energies. The residual Finite-strain cyclic behavior of SMAs strain evolves with the cumulated martensite volume fraction and stabilizes with a saturated value. The evolutions of the internal variables and the model parameters control the degenerations of the pseudoelasticity and the hysteresis loop with the number of cycles. The rate dependence is physically interpreted as the consequence of the competition between internal heat production and external heat transfer, and thereby is formulated by means of the temperature evolution and heat transfer equations. In addition, a tangential transformation hardening function g t is introduced to describe the evolution of phase transformation from abrupt to smooth manner during cycling. Finally, the model is used to simulate the cyclic tensile experiments on NiTi wire at different loading rates. The good agreement of the model predictions against the experimental data demonstrates the capabilities of the proposed model to well describe cyclic pseudoelasticity of polycrystalline SMAs, and to capture the aforementioned characteristics.

Constitutive model

Kinematics

The present work aims to develop a comprehensive constitutive model that simultaneously accounts for cyclic effect, finite strain, thermomechanical coupling, martensitic transformation and reorientation. To this end, the deformation gradient F is multiplicatively decomposed into elastic F e , transformation F t , residual F r and thermal F θ parts, such that

F = F e F t F r F θ , (5.1)
where F e is defined with respect to a local unstressed intermediate configuration, F t is defined with respect to a zero-transformation intermediate configuration, F r is defined with respect to a thermally expanded intermediate configuration and F θ is defined with respect to the reference configuration. It should be noted that the irreversible deformation F r stems from two physical mechanisms: residual martensite and dislocations slipping. However, the present work takes no account of this distinction, and therefore F r is not further split.

To simplify the following formulation, the transformation and residual deformation gradients, F t and F r , are grouped into a new component as

F * = F t F r and F = F e F * F θ .
(5.2)

With regard to transformation, residual and thermal deformations, the following three kinematic assumptions are made:

1. First, the thermal deformation is assumed to be an isotropic thermal expansion, so that

F θ = J 1 3 θ 1.
(5.3)

5.2 Constitutive model 81 2. Second, transformation and residual deformations are assumed incompressible, that is

J t = det F t = 1, J r = det F r = 1 and J * = J t J r = 1.
(5.4)

3. Last, for isotropic materials, the assumption of the irrotationality of inelastic flow is widely used in modern elasto-viscoplastic constitutive theories [START_REF] Dafalias | The plastic spin concept and a simple illustration of its role in finite plastic transformations[END_REF][START_REF] Gurtin | The decomposition F= F e F p, material symmetry, and plastic irrotationality for solids that are isotropic-viscoplastic or amorphous[END_REF][START_REF] Anand | A thermo-mechanically coupled theory for large deformations of amorphous polymers. Part I: Formulation[END_REF]. Therefore, the transformation and residual flows are assumed irrotational (zero transformation spin) W t = 0, W r = 0 and W * = 0, (5.5) where W t , W r and W * denote transformation, residual and total inelastic spin tensors.

Using the above kinematic assumptions (5.3), (5.4) and (5.5) and equations (3.12) 1 and (4.13), the elastic, transformation, residual and thermal Lagrangian and Eulerian Hencky strain tensors are written as

H e = ln U e = He + 1 3 δ e 1, h e = ln V e = he + 1 3 δ e 1, H t = ln U t , h t = ln V t , H r = ln U r , h r = ln V r , H θ = h θ = 1 3 δ θ 1, (5.6)
where He = ln Ūe and he = ln V e are deviatoric components, δ e = ln J e and δ θ = ln J θ are spherical components.

The velocity gradient L in equation (3.12) 3 is additively decomposed as

L = Le + Fe L * F-1 e + 1 3 δ e + δ θ 1, (5.7)
where Le = Fe F-1 e and L * = F * F -1 * are the isochoric elastic and inelastic velocity gradients, δ e = J e /J e and δ θ = J θ /J θ are the volumetric elastic and thermal velocity gradients. Then the stretching and spin tensors, D and W , in equation (3.15) 1 is written as

D = De + sym( Fe D * F-1 e ) + 1 3 δ e + δ θ 1 and W = W e + skew Fe D * F-1 e , (5.8)
where De and W e are the stretching and spin tensors of isochoric elastic deformation, D * is the inelastic stretching tensor.

Finite-strain cyclic behavior of SMAs

Thermodynamic framework

The global reduced form of energy balance and Clausius-Duhem form of entropy inequality can be expressed as (5.10) where e v and η v are the internal energy and the entropy per unit volume, q is the heat flux vector per unit area and n is the outward unit vector normal to the boundary of the body ∂B, h v is the heat source per unit volume. These state variables are defined in the spatial configuration.

∫ B e v dV - ∫ B Jσ : D dV + ∫ ∂B q • n dS - ∫ B h v dV = 0, (5.9) ∫ B η v dV ≥ ∫ B 1 θ h v dV - ∫ ∂B 1 θ q • n dS,
Considering the irrotationality of the transformation and residual flows in equation (5.5), the inelastic stretching tensor is additively split into transformation and residual parts, i.e., D * = D t + D r . Subsequently substituting this relation and equation (5.8) into the internal stress power in equation (5.9) gives

Jσ : D = p δ e + p δ θ + s : De + FT e s F-T e : (D t + D r ) , (5.11)
where p and s are the spherical and deviatoric components of the Kirchhoff stress tensor, i.e., τ = Jσ = p1 + s. D t and D r are the transformation and residual stretching tensors.

Substituting equations (5.11) into equation (5.9) and localizing the integral equations (5.9) and (5.10) gives the following local forms of the energy balance and entropy inequality: e vp δ ep δ θs : De -FT e s F-T e : (D t + D r )h v + ∇ • q = 0, (5.12)

η v θ -h v + ∇ • q - 1 θ q • ∇θ ≥ 0.
(5.13)

The Helmholtz free energy ψ per unit volume in the spatial configuration and its time derivative ψ are given by

ψ = e v -η v θ and ψ = e v -η v θ -η v θ.
(5.14)

Combining equations (5.12), (5.13) and (5.14) 2 gives the following entropy inequality: .15) In order to satisfy the principle of material objectivity, the Helmholtz free energy density ψ in the present work depends on the elastic deformation gradient F e through the elastic Lagrangian Hencky strain H e , on the transformation deformation gradient F t through the transformation Lagrangian Hencky strain H t , on the residual deformation gradient F r through the residual Lagrangian Hencky strain H r , on the thermal deformation gradient F θ through the thermal Lagrangian Hencky strain H θ , on scalar internal variables χ, χ c , and on temperature θ as ψ = ψ (H e , H t , H r , H θ , χ, χ c , θ) , (5.16) where χ denotes the volume fraction of martensite, χ c = ∫ t 0 | χ|dτ denotes the cumulated volume fraction of martensite. The free energy density ψ is assumed to be an isotropic function of H e , H t , H r and H θ , which allows ψ to be expressed as the function of their argument invariants.

p δ e + δ θ + s : De + FT e s F-T e : (D t + D r ) -ψ + η v θ - 1 θ q • ∇θ ≥ 0. ( 5
: D r - ∂ψ ∂ χ χ - ∂ψ ∂ χ c χ c - ∂ψ ∂θ + η v θ - 1 θ q • ∇θ ≥ 0.
(5.20)

For arbitrary thermodynamic processes, inequality (5.20) (5.23) as well as nonnegative dissipation due to heat conduction, so that (5.24) where Me is a symmetric and traceless Mandel stress tensor by rotating the deviatoric Kirchhoff stress s back into the intermediate unstressed configuration, X t and X r are symmetric stress-like tensors.

- 1 θ q • ∇θ ≥ 0,
Combining equations (5.14) 2 , (5.18) and ( 5.21) gives the following Gibbs relation between internal energy and entropy:

e v = θ η v + p δ e + δ θ + s : he + X t : h t + X r : h r + ∂ψ ∂ χ χ + ∂ψ ∂ χ c χ c , (5.25)
which, with the energy balance equation ( 5.12), yields the following entropy evolution:

θ η v = h v -∇ • q + Me -X t : D t -Me -X r : D r - ∂ψ ∂ χ χ - ∂ψ ∂ χ c χ c .
(5.26)

The internal energy and the entropy densities are assumed to depend on the same state variables as the Helmholtz free energy: e v = e δ e , he , h t , h r , δ θ , χ, χ c , θ , η v = η δ e , he , h t , h r , δ θ , χ, χ c , θ .

(5.27)

The specific heat capacity is defined as

c v def = ∂e δ e , he , h t , h r , δ θ , χ, χ c , θ ∂θ , (5.28)
which, with equations (5.14) and ( 5.21) 3 , can be written as

c v = θ ∂η δ e , he , h t , h r , δ θ , χ, χ c , θ ∂θ = -θ ∂ 2 ψ δ e ,
δ θ + ∂η ∂ χ χ + ∂η ∂ χ c χ c + c v θ =c v θ -θ ∂ 2 ψ ∂θ∂δ e δ e + ∂ 2 ψ ∂θ∂ he : he + ∂ 2 ψ ∂θ∂δ θ δ θ - θ ∂ 2 ψ ∂θ∂h t : h t + ∂ 2 ψ ∂θ∂h r : h r + ∂ 2 ψ ∂θ∂ χ χ + ∂ 2 ψ ∂θ∂ χ c χ c .
(5.30)

Then, combining equations ( 5.26) and (5.30) gives the following partial differential equation for the temperature θ: 5.31) wherein, the term D represents the intrinsic dissipations associated to plastic deformation, martensitic transformation and reorientation; the term T represents the entropy change due to plastic and transformation deformations as well as the latent heat during phase transformation; the term X represents the entropy change due to thermoelastic deformation; the heat flux q is taken to be governed by Fourier's law q = -k∇θ, (5.32) where k is the nonnegative thermal conductivity.

c v θ =h v -∇ • q + Me -X t : D t + Me -X r : D r - ∂ψ ∂ χ χ - ∂ψ ∂ χ c χ c D=intrinsic dissipation + θ ∂ 2 ψ ∂θ∂h t : D t + ∂ 2 ψ ∂θ∂h r : D r + ∂ 2 ψ ∂θ∂ χ χ + ∂ 2 ψ ∂θ∂ χ c χ c T=thermoplastic + thermo-transformation coupling + θ ∂ 2 ψ ∂θ∂δ e δ e + ∂ 2 ψ ∂θ∂ he : De + ∂ 2 ψ ∂θ∂δ θ δ θ X=thermoelastic coupling , (

Helmholtz free energy function

The Helmholtz free energy function ψ in equation ( 5.17) is additively decomposed as follows: (5.33) where ψ e is the elastic energy, ψ θ is the thermal contribution to free energy (Morin et al., 2011a;[START_REF] Lagoudas | Constitutive model for the numerical analysis of phase transformation in polycrystalline shape memory alloys[END_REF], ψ int is the interaction energy between martensite and austenite phases [START_REF] Moumni | Theoretical and numerical modeling of solid-solid phase change: Application to the description of the thermomechanical behavior of shape memory alloys[END_REF], ψ cst is the potential energy due to the internal constraints. By employing the the rule of mixtures [START_REF] Lagoudas | Shape memory alloys, Part II: Modeling of polycrystals[END_REF][START_REF] Moumni | Theoretical and numerical modeling of solid-solid phase change: Application to the description of the thermomechanical behavior of shape memory alloys[END_REF][START_REF] Auricchio | Theoretical and numerical modeling of shape memory alloys accounting for multiple phase transformations and martensite reorientation[END_REF], the elastic energy ψ e and the thermal energy ψ θ in equation (5.33) can be expressed in the phases, the second term 1 2 µ t ∥h t ∥ 2 represents increase in interaction due to orientation/reorientation of martensite variants, and the last term 1 2 µ r ∥h r ∥ 2 represents increase in interaction due to dislocation slipping.

ψ = ψ e + ψ θ + ψ int + ψ cst ,
Finally, the following Lagrangian potential is defined as the constraint energy ψ cst to guarantee the physical constraints on χ and h t : (5.40) where H is the fully oriented strain magnitude of martensite variants, ζ t is a nonnegative Lagrange multiplier associated with the unilateral constraint H -∥h t ∥ ≥ 0; ζ f and ζ r are nonnegative Lagrange multipliers associated with the bilateral constraints 0 ≥ χ ≥ 1. The constraint energy (5.40) is complemented by the following classical Kuhn-Tucker conditions:

ψ cst = -ζ t (H -∥h t ∥) -ζ f (1 -χ) -ζ r χ,
         H -∥h t ∥ ≥ 0, ζ t ≥ 0, ζ t (H -∥h t ∥) = 0; 1 -χ ≥ 0, ζ f ≥ 0, ζ f (1 -χ) = 0; χ ≥ 0, ζ r ≥ 0, ζ r χ = 0.
(5.41)

In conclusion, the Helmholtz free energy function ψ is constructed as

ψ = 1 2 Kδ 2 + µ∥ he ∥ 2 -3αKδ(θ -θ 0 ) + e A 0 -η A 0 θ + c A v -χ∆c t v (θ -θ 0 ) -θ ln θ θ 0 + χ∆η t (θ -θ 0 ) + G t + 1 2 µ t ∥h t ∥ 2 + 1 2 µ r ∥h r ∥ 2 -ζ t (H -∥h t ∥) -ζ f (1 -χ) -ζ r χ.
(5.42)

Constitutive equations

Substituting the Helmholtz free energy function ψ in equation ( 5.42) into the established thermodynamic framework, specifically, equations (5.21) and (5.23), gives the volumetric and the deviatoric stresses, p and s, and the entropy η v as follows: 5.43) as well as the following Mandel stress Me and state quantities X t and X r : (5.44) where N t = h t ∥h t ∥ denotes the direction of the transformation strain. Moreover, the partial derivative of ψ with respect to χ is given by (5.45) where g t = ∂G t /∂ χ is the phase transformation hardening function. In order to capture the abrupt to smooth transition during cycling at the initiation and completion of phase transformation, as experimentally demonstrated in the literature for polycrystalline SMAs [START_REF] Lagoudas | Shape memory alloys, Part II: Modeling of polycrystals[END_REF][START_REF] Hartl | Three-dimensional modeling and numerical analysis of rate-dependent irrecoverable deformation in shape memory alloys[END_REF][START_REF] Morin | Thermomechanical coupling in shape memory alloys under cyclic loadings: Experimental analysis and constitutive modeling[END_REF], a tangential transformation hardening function g t is considered as (5.46) where π is the circular constant, κ, m, ξ a , ξ b and r are model parameters. To account for the degeneration of the pseudoelasticity and dissipated energy during the cyclic deformation, these model parameters are considered to be functions of the cumulated martensite volume fraction χ c : 5.47) where f = {κ, m, ξ a , ξ b , r }, f 0 and f sat correspond, respectively, to untrained and completely trained SMA, τ is a dimensionless time constant. Additionally, the model parameters ∆η t and H also depend on χ c according to equation (5.47).

           p = ∂Ψ ∂δ = K [δ -3α(θ -θ 0 )] , s = ∂Ψ ∂ he = 2µ he , η v = -∂Ψ ∂θ = 3αKδ + η A 0 + c A v -χ∆c t v ln θ θ 0 -∆η t χ, (
Me = 2µ He , X t = µ t h t + ζ t N t and X r = µ r h r ,
∂ψ ∂ χ = ∆η t (θ -θ 0 ) -∆c t v (θ -θ 0 ) -θ ln θ θ 0 + g t + ζ f -ζ r ,
g t = κtan m 1 2 (ξ a χ + ξ b )π + r,
f = f 0 + ( f sat -f 0 ) 1 -exp - χ c τ , (
The martensite volume fraction χ, according to the literature [START_REF] Reese | Finite deformation pseudo-elasticity of shape memory alloys -Constitutive modelling and finite element implementation[END_REF][START_REF] Moumni | Theoretical and numerical modeling of solid-solid phase change: Application to the description of the thermomechanical behavior of shape memory alloys[END_REF][START_REF] Lagoudas | Constitutive model for the numerical analysis of phase transformation in polycrystalline shape memory alloys[END_REF], can be expressed in terms of the transformation strain, i.e., χ = ∥h t ∥/H . Thus, using this relation, the time derivative of deformation strain is additively split as follows: (5.48) where the first term represents phase transformation between austenite and martensite, while the second term represents martensitic reorientation. Substituting equation ( 5.48) into equation (A.4) gives the following split of the transformation stretching D t :

h t = H N t χ + χ N t ,
D t = hL t = H ( χN t + χD N ) , (5.49)
where

D N = N L t = N t -Ω L t N t + N t Ω L t
is the stretching tensor associated with martensitic reorientation. Substituting equation ( 5.49) into inequality (5.22) gives

H χ Me -X t : D N + Me -X r : D r + H Me -X t : N t - ∂ψ ∂ χ χ - ∂ψ ∂ χ c χ c ≥ 0. (5.50)
The nonnegative dissipation of equation ( 5.50) is guaranteed by the following choice of the evolution equations: 5.51) where Y χ and Y N control the thresholds of the phase transformation and the martensitic reorientation. γ t , γ N , F χ and F N are subjected to the following Kuhn-Tucker consistency conditions:

D N = γ N A N ∥ A N ∥ , D r = γ r A r ∥ A r ∥ and χ = γ t S(A χ ), (
γ t ≥ 0, F χ ≤ 0, γ t F χ = 0; γ N ≥ 0, F N ≤ 0, γ N F N = 0.
(5.58)

Finally, using equations (5.43), (5.51) and (5.31), the temperature evolution equation is written as

c v θ = h v -∇ • q + γ N ∥ A N ∥ + γ r ∥ A r ∥ + γ t | A χ | + γ t S(A χ )θ ∆η t + ∆c t v ln θ θ 0 -3αθK δ. (5.59)
A review of the constitutive equations derived in this section is presented in Table 5.1.

Table 5.1 Summary of the model accounting for cyclic effect.

State and internal variables: δ e , he , h t , h r , δ θ , χ, χ c , θ Thermoelasticity: p = K [δ e + δ θ -3α(θ -θ 0 )] and s = 2µ he Phase transformation:

D t = hL t = H ( χN t + χD N ) , D N = γ N A N ∥ A N ∥ , χ = γ t S(A χ ), where A N = (I -N t ⊗ N t ) Me , A χ = H 2µ He : N t -µ t H χ -ζ t + ∆c t v -∆η t (θ -θ 0 ) -∆c t v θ ln θ θ 0 -g t -ζ f + ζ r Residual deformation: D r = hL r = γ r A r ∥ A r ∥ , where A r = 2µ He -µ t h r , γ r = R s at τ exp -χ c τ χ c , χ c = | χ| Temperature evolution: c v θ = h v -∇ • q + γ N ∥ A N ∥ + γ r ∥ A r ∥ + γ t | A χ | + γ t S(A χ )θ ∆η t + ∆c t v ln θ θ 0 -3αθK δ
Evolution of model parameters:

f = f 0 + ( f sat -f 0 ) 1 -exp -χ c τ , f ∈ {κ, m, ξ a , ξ b , r, Y χ , ∆η t , H } Yield functions: F χ = | A χ | -Y χ and F N = ∥ A N ∥ -χY N Kuhn-Tucker consistency conditions: γ t ≥ 0, F χ ≤ 0, γ t F χ = 0 γ N ≥ 0, F N ≤ 0, γ N F N = 0

Model parameters calibration

In this section, the 3D constitutive model presented in Section 5.2 is first reduced to a 1D version. Then, the heat transfer conditions in the real experiment is simplified. Finally, the reduced constitutive and heat equations are implemented into MATLAB to calibrate the model parameters against the experimental data. (5.60) where σ and λ denote the uniaxial stress and stretch, λ l denotes the lateral contractions. e 1 , e 2 , e 3 are the three orthonormal bases. Since the spherical stress-strain response (5.43) itself is a scalar equation, the reduced 1D equations will not capture it and therefore volume-preserving deformation,

i.e. J = det F = 1, is assumed. This assumption gives the relation: λ l = 1 λ . Moreover, the polar decomposition can be written as F = V R = V since R = I . The velocity gradient L is then computed as

L = FF -1 = λ λ T and T = 1 e 1 ⊗ e 1 - 1 2 e 2 ⊗ e 2 - 1 2 e 3 ⊗ e 3 , ( 5
F t = λ t e 1 ⊗ e 1 + 1 λ t e 2 ⊗ e 2 + 1 λ t e 3 ⊗ e 3 , (5.63)
F r = λ r e 1 ⊗ e 1 + 1 λ r e 2 ⊗ e 2 + 1 λ r e 3 ⊗ e 3 , (5.64) with λ = λ e λ t λ r and J = J e J t J r = 1.

(5.65) Thus, the elastic h e , transformation h t and residual h r Hencky strains in equation (5.6) are written as h e = ln λ e T, h t = ln λ t T and h r = ln λ r T .

(5.66)

The deviatoric stress s in equation (5.43) is then written as s = 2µ ln λ e T = 2 3 σT .

(5.67)

The Mandel stress Me and the direction of transformation strain N t in equation (5.44) are written as 5.68) which indicates that Me and N t are coaxial for the case of uniaxial loading. Hence, according to equation (5.56), the thermodynamic force of martensitic reorientation A N is null, i.e. A N = 0. Moreover, the Finite-strain cyclic behavior of SMAs thermodynamic forces of residual deformation A r and phase transformation A χ in equation ( 5.52), with the assumption of equal heat capacities of austenite and martensite, i.e. ∆c t v = 0, are written as

Me = 2 3 σT and N t = 2 3 T, (
A r = 2 3 σ -µ t ln λ r T, (5.69) A χ = 2 3 H σ -µ t H 2 χ -H ζ t -∆η t (θ -θ 0 ) -g t -ζ f + ζ r .
(5.70)

Combining equations (5.51), ( 5.53) and ( 5.69) gives the following residual evolution equation:

h r = λ r λ r = R sat τ exp - χ c τ χ c N t .
(5.71)

Simplification of the heat transfer condition

In the present work, the proposed constitutive model is validated against the experimental data reported by [START_REF] Morin | Thermomechanical coupling in shape memory alloys under cyclic loadings: Experimental analysis and constitutive modeling[END_REF] for polycrystalline NiTi (56.1 at.% Ni) wire. The heat transfer conditions in the source work are shown in Fig. 5.1. According to [START_REF] Yin | Effect of deformation frequency on temperature and stress oscillations in cyclic phase transition of NiTi shape memory alloy[END_REF] and [START_REF] Yu | Rate-dependent cyclic deformation of super-elastic NiTi shape memory alloy: Thermo-mechanical coupled and physical mechanism-based constitutive model[END_REF], the heat flow through the two grips is simplified as heat conduction through the two boundary cross sections of the gauge-length of the wire. A cylindrical coordinate is defined with radius ρ, azimuth φ and altitude z, which is utilized in the following to formulate integral equations of the heat transfer over the NiTi wire.

Model parameters calibration 93

According to the literature of [START_REF] Morin | Thermomechanical coupling in shape memory alloys under cyclic loadings: Experimental analysis and constitutive modeling[END_REF], [START_REF] Yin | Effect of deformation frequency on temperature and stress oscillations in cyclic phase transition of NiTi shape memory alloy[END_REF] and [START_REF] Yu | Rate-dependent cyclic deformation of super-elastic NiTi shape memory alloy: Thermo-mechanical coupled and physical mechanism-based constitutive model[END_REF], the initial and boundary conditions of the heat transfer in the experimental NiTi wire are described as: 5.72) where n c and n l are outward unit vectors, respectively, normal to the cross-section surface S c and the lateral surface S l , θ a is the ambient temperature. The first equation represents the heat convection of the NiTi wire lateral surface to ambient media, and h is the heat convection coefficient. While the second equation represents the heat conduction through the cross-section surface, where k is the thermal conductivity and β is a constant. Using stokes's theorem, the integral form of the heat transfer ∇• q in equation ( 5.31) can be expressed as: (5.73) where n S is the outward unit vector normal to the surface of the NiTi wire.

         q • n c = h (θ -θ a ) in S c , q • n l = βk L (θ -θ a ) in S l , θ(t = 0) = θ a in B, (
∭ B ∇ • q dV = ∯ ∂B q • n S dS = ∯ S c q • n c dρdφ + ∯ S l q • n l dφdz,
Using the aforementioned 1D constitutive equations, the temperature evolution equation ( 5.59) is written as

∭ B c v θ dρdφdz = ∭ B h v dρdφdz - ∯ S c q • n c dρdφ - ∯ S l q • n l dφdz + ∭ B I r + |I χ | + L dρdφdz, (5.74) where                  I r = R s at τ exp -χ c τ 2 3 σ -3 2 µ t ln λ r χ c , I χ = 2 3 H σ -µ t H 2 χ -H ζ t -∆η t (θ -θ 0 ) -g t -ζ f + ζ r χ, L = θ ∆η t + ∆c t v ln θ θ 0 χ.
(5.75)

In equation ( 5.75), I r denotes the intrinsic dissipation due to residual deformation, I χ denotes the intrinsic dissipation due to phase transformation and L denotes the latent heat during the phase transformation. Using equation ( 5.72), the second and third terms in the right side of equation ( 5.75) can be written as Then, substituting equations (5.76) and ( 5.77) into equation (5.74) gives

∯ S c q • n c dρdφ = ∯ S c h (θ -θ a ) dρdφ = hA c θ c m -θ a , (5.76) ∯ S l q • n l dφdz = ∯ S l βk L (θ -θ a ) dφdz = βk L A l θ l m -θ a , ( 5
∭ B c v θ dρdφdz = ∭ B h v dρdφdz -hA c θ c m -θ a - βk L A l θ l m -θ a + ∭ B I r + |I χ | + L dρdφdz.
(5.80)

Considering the NiTi wire used in the experiment is an axisymmetric structure, the temperature field in the azimuthal direction is uniform. Meanwhile, since the diameter of the wire is very small compared to the gauge length, the temperature field in the radial direction is also assumed to be uniform. Moreover, by neglecting the temperature gradient in the altitude direction the temperature field in the NiTi wire can be regarded as uniform. Under this assumption of uniform temperature field, the temperature in the cross-section and lateral surfaces are considered to be the same as the mean temperature in the NiTi wire, i.e.,

θ c m = θ l m = θ m = 4 πd 2 L ∭ B θ(ρ, φ, z) dρdφdz.
(5.81)

Besides the assumption of uniform temperature field, it should be noted that the NiTi wire subjected to also a uniformly uniaxial stress field. Thus, all the state quantities and internal variables can be regarded as uniform, and therefore equation (5.80) yields

c v θ = h v - hd + 2βk dL (θ m -θ a ) + I r + |I χ | + L.
(5.82)

Model parameters calibration

The 1D constitutive model and the simplified heat transfer condition are implemented into MATLAB to calibrate the model parameters. All the model parameters are classified into four categories: thermoelasticity, phase transformation, cyclic deformation and heat transfer, and listed in Tab. 5.2.

Table 5.2 Model parameters required to be calibrated.

Elasticity: K, µ A , µ M , α; Phase transformation: κ, m, ξ a , ξ b , r, Y χ , ∆η t , H , µ t , Y N ; Cyclic deformation: µ r , R c , R 0 , R s , τ; Heat transfer: c v , ∆c t v , h, k, β, θ 0 , θ a .
With regard to the model parameters, the following two assumptions are introduced to simplify the calibration: (i) according to the differential scanning calorimetry (DSC) experiments [START_REF] Qidwai | On thermomechanics and transformation surfaces of polycrystalline NiTi shape memory alloy material[END_REF], the heat capacities of austenite and martensite are almost equal, i.e., ∆c t v = 0; (ii) since the evolution of residual deformation is directly related to the phase transformation (equation (5.53)) and controlled by the loading function (5.57) 1 , the change of loading surface due to dislocation slipping is therefore neglected, i.e., µ r = 0. Moreover, it should be noted that the eight parameters {κ, m, ξ a , ξ b , r, Y χ , ∆η t , H } depend on the cumulated martensite volume fraction χ c according to equation (5.47). Thus, the parameters in both the first and the stabilized cycles, i.e., {κ, m, ξ a , ξ b , r, Y χ , ∆η t , H } 0 and {κ, m, ξ a , ξ b , r, Y χ , ∆η t , H } sat , are required to be calibrated.

The elastic bulk modulus K and the shear moduli µ A and µ M are determined from experimental pseudoelastic stress-strain curve as follows:

K = E A + E M 6(1 -2ν) , µ A = E A 2(1 + ν) and µ M = E M 2(1 + ν) , (5.83)
where E A and E M are Young's moduli for austenite and martensite as shown in Fig 5.2(a). The thermal expansion coefficient α is obtained from the literature of [START_REF] Yu | Rate-dependent cyclic deformation of super-elastic NiTi shape memory alloy: Thermo-mechanical coupled and physical mechanism-based constitutive model[END_REF].

The entropy difference ∆η t governs the variation of transformation stress with temperature. This influence is demonstrated in Fig 5.2(a) with two stress-strain curves obtained at temperature θ (black) and θ + ∆θ (blue), wherein the temperature increase of ∆θ results in transformation stress increase of ∆σ T . The yield threshold for phase transformation Y χ controls the height of the hysteresis loop σ h as shown in Fig 5.2(a). These two parameters are calculated as follows:

∆η t = H ∆σ T ∆θ and Y χ = 1 2 H σ h .
(5.84)

The martensitic reorientation threshold Y N , the reorientation hardening modulus µ t and the magnitude limit of transformation strain H are determined from the stress-strain curve of martensitic orientation experiment, wherein self-accommodation martensite is completely oriented under the uniaxial tensile loading. Y N takes the initial yield stress, µ t the slope of orientation plateau and H the magnitude of the residual strain, as shown in Figure 5.2(b). It is noteworthy that the parameters ∆η 0 t , Y 0 χ and H 0 are determined from the experimental curves of the first cycle, while ∆η sat t , Y sat χ and H sat from the stabilized cycle.

The parameters κ, m, ξ a , ξ b and r in the tangential hardening function g t control the abrupt to smooth transition during the cyclic deformation, especially at initiation and completion of phase transformation. Figure 4.2 shows how the parameters κ, m, ξ a , ξ b and r affect the pseudoelastic stress-strain behavior of SMAs. Likewise, κ 0 , m 0 , ξ 0 a , ξ 0 b and r 0 are determined from the experimental curves of the first cycle, while κ sat , m sat , ξ sat a , ξ sat b and r sat from the stabilized cycle. The parameters R c , R 0 and R s are determined from the experimental curve of the saturated residual strain R sat versus the logarithmic strain rate log h eq (referring to [START_REF] Kan | Experimental observations on rate-dependent cyclic deformation of super-elastic NiTi shape memory alloy[END_REF]) according to equation (5.54). The dimensionless time constant τ in function f (equation (5.47)) controls the evolution rate of the parameters, which can be obtained from the curve of the transformation strain magnitude H versus the cumulated martensite volume fraction χ c (referring to Fig. 2.1(b)). Here, data points in the 1st, 5th, 10th, 20th and last cycles are used to calibrate τ.

The heat capacity c v is obtained from the literature of [START_REF] Morin | Thermomechanical coupling in shape memory alloys under cyclic loadings: Experimental analysis and constitutive modeling[END_REF]. The reference temperature θ 0 takes the equilibrium temperature of the phase transformation, i.e., (5.86) Thus, equation (5.82), without the external heat source h v , is written as 5.87) which is then utilized to determine ω from the pseudoelastic stress-strain curves at different strain rates (referring to Fig. 2.2).

θ 0 = 1 4 A s + A f + M s + M f , ( 5
c v θ = I r + |I χ | + L -ω (θ m -θ a ) , (
5.4 Numerical simulations and model validations 97

Numerical simulations and model validations

In this section, the experiments performed by [START_REF] Morin | Thermomechanical coupling in shape memory alloys under cyclic loadings: Experimental analysis and constitutive modeling[END_REF] are simulated using the reduced constitutive model and the simplified heat transfer condition presented in Section 5.3. Then, the model predictions are validated against the experimental data. In the source work, the transformation temperatures of the NITi wire were measured using DSC, i.e., martensite start temperature M s = 23 • C, martensite finish temperature M f = 13 • C, austenite start temperature A s = 9 • C and austenite finish temperature A f = 24 • C. The experiments were carried out at constant ambient temperature of 50 ± 1 • C, which is higher than the austenite finish temperature A f = 24. Thus, the initial phase of the NiTi wire is austenite and exhibits pseudoelasticity under applied stress. The NiTi wire is subjected to 50 cycles of quasi-static tensile loading at six different loading rates of 1 × 10 -4 s -1 , 2.5 × 10 -4 s -1 , 5 × 10 -4 s -1 , 1 × 10 -3 s -1 , 2.5 × 10 -3 s -1 and 5 × 10 -3 s -1 . The maximum applied stress is about 800 MPa, which is higher than the martensite finish stress σ m f , allowing austenite to completely transform to martensite. The model parameters used in the simulation are determined following the calibration procedure described in Section 5.3.3 from the experimental data. They are listed in Tab. 5.3. Fig. 5.3 shows the cyclic stress-strain curves at strain rate of 1 × 10 -4 s -1 . The two black hysteresis loops denote the simulated stress-strain curves for the first and the stabilized cycles, while the gray curves demonstrate the evolution of stress-strain relation during cycling. As expected, the dissipated energy and the reversible transformation strain degenerate while the residual strain accumulates with the number of cycles. Meanwhile, the martensite start stress σ ms and the austenite finish stress σ a f decrease but the transformation hardening modulus increases with the number of cycles. With regard to the initiation and completion of phase transformation on the stress-strain curves, the abrupt transition for the first cycle finally evolves into a smooth manner for the stabilized cycle. Figs. 5.3(b-d) show the Finite-strain cyclic behavior of SMAs comparisons of the model predictions against the experimental data and numerical results reported by [START_REF] Morin | Thermomechanical coupling in shape memory alloys under cyclic loadings: Experimental analysis and constitutive modeling[END_REF] for the 1st, 10th and stabilized cycles, respectively. In the experimental curve for the 1st cycle, instabilities appear at the initiations of phase transformations due to the propagation of the phase transformation front [START_REF] Li | The initiation and growth of macroscopic martensite band in nano-grained NiTi microtube under tension[END_REF]. However, both Morin's and the present models do not capture this phenomenon. This localization phenomenon is specialized in some other literature [START_REF] Favier | Mechanical instability of NiTi in tension, compression and shear[END_REF][START_REF] Shaw | A thermochemical model for a 1-D shape memory alloy wire with propagating instabilities[END_REF]. The experimental stress-strain curves for the 10th and the stabilized cycles show a smooth manner compared to the 1st cycle one. The proposed model captures this characteristic with a high degree of accuracy thanks to the introduction of the tangential transformation hardening function g t . This is an important improvement compared to Morin's model, in which the initiation and completion of phase transformation is abrupt and the transformation hardening is linear, as shown in Figs. 5.3(c,d). present the cyclic stress-strain curves at strain rates of 2.5 × 10 -4 s -1 , 5 × 10 -4 s -1 , 1 × 10 -3 s -1 , 2.5 × 10 -3 s -1 and 5 × 10 -3 s -1 , respectively. From these figures, it can be seen that the cyclic deformation of SMA strongly depends on the strain rate. The transformation hardening modulus, the martensite finish stress σ m f , the austenite start stress σ as and the saturated residual strain increase, but the reversible transformation strain decreases with the increasing strain rate. The underlying cause for this rate dependence lies in four aspects: (i) phase transformation takes place with internal heat production due to the intrinsic dissipation and the latent heat; (ii) external heat transfer depends on the testing duration and therefore the loading rate; (iii) competition of the internal heat production against the external heat transfer determines the temperature on NiTi wire; (iv) the transformation stress of SMA is strongly influenced by temperature. The proposed model accounts for fully thermomechanical coupling and therefore describes the rate dependence reasonably well.

Figure 5.9 Evolutions of the dissipated energy and the residual strain with the number of cycles: (a) strain rate of 1 × 10 -4 s -1 , (b) strain rate of 5 × 10 -3 s -1 , (c) strain rate of 2.5 × 10 -3 s -1 . Fig. 5.9 shows the evolutions of the dissipated energy (DE) and the residual strain (RS) with the number of cycles at strain rates of 1 × 10 -4 s -1 , 5 × 10 -4 s -1 and 2.5 × 10 -3 s -1 . Model predictions are validated against the experimental data reported by [START_REF] Morin | Thermomechanical coupling in shape memory alloys under cyclic loadings: Experimental analysis and constitutive modeling[END_REF]. In the source work, the experimental values for the 2nd, 20th and 40th cycles of DE and the 1st, 10th and 40th of RS are presented. In Fig. 5.9(a), it can be quantitatively observed the decrease of DE from 16.4 M J/m 3 for the 2nd cycle to the stabilized value of 5.6 M J/m 3 for the 40th cycle, while the increase of RS from 0.19% for the 1st cycle to a saturated value of 1.8% for the 40th cycle. The proposed model predicts the evolutions of these two variables in good agreement. In Fig. 5.9(b,c), with the increasing strain rate the stabilized values of DE and RS increase simultaneously. This rate dependence is also reasonably predicted by the proposed model. Fig. 5.10 shows the temperature evolutions during cycling at strain rates of 2.5×10 -4 s -1 , 1×10 -3 s -1 and 5 × 10 -3 s -1 . The experimental curves during the first 20 cycles are presented for the three different strain rates. For the given cycle, the temperature increases during forward transformation (austenite to 5.5 Finite elements simulation of a SMA torsion spring 105 martensite) and decreases during reverse transformation (martensite to austenite). This phenomenon is due to the predominance of latent heat with respect to other heat sources during phase transformation. The temperature amplitude decreases with the number of cycles and stabilizes at a certain value. However, with the increasing strain rate it increases dramatically. As a matter of fact, the size of hysteresis loop on stress-strain curve is strongly influenced by the temperature amplitude, as shown in Figs. 5.4, 5.6 and 5.8. At high strain rate, the hysteresis loop is larger. Overall, the temperature evolutions during cycling are reasonably predicted by the proposed model.

Finite elements simulation of a SMA torsion spring

The proposed model is implemented into finite elements analysis software Abaqus/Explicit by means of a user-defined material subroutine VUMAT. To demonstrate the capabilities of the proposed model in Finite-strain cyclic behavior of SMAs analyzing complex multi-axial boundary value problems, a SMA torsion spring undergoing large cyclic deformations is simulated in this section. The SMA torsion spring used in the simulation is shown in Fig. 5.11, with 1 mm × 1 mm rectangle cross-section. Because of symmetry with respect to the horizontal axis, only half of the structure is considered in the simulation. The spring is meshed using 1740 coupled temperature-displacement reduced integration hexahedral elements (Abaqus/Explicit C3D8RT). In the simulation, the pressure is repeatedly increased from zero to the maximum value 5.5 × 10 5 Pa then removed for 20 cycles during the time period of 20 s.

Figure 5.11 Finite elements model of the SMA torsion spring: geometry, mesh and loadings.

Fig. 5.12 shows the martensite volume fraction contour plots at the maximum loading for cycles of 1st, 5th, 10th and 20th. In order to highlight the evolution of phase transformation zone, the simulation result around the corner is magnified. From the figures, it is noted, with the number of cycles, the maximum value of the martensite volume fraction increases and the transformation zone dramatically expands. These results can be explained by the fact that during the cyclic deformations transformation stress gradually decreases with the number of cycles, as shown in Figs. 5.3-5.8. Thereby, austenite transforms to martensite more easily under the same applied loadings. Fig. 5.13 shows the residual deformation on the torsion spring. Specifically, Fig. 5.13(a) shows the deformation of the inside edge on the torsion spring in absence of loading for cycles of 1st, 5th, 10th and 20th. It is clearly observed the residual deformation increases with the number of cycles. Thereby, the torsion spring deviates increasingly from its initial configuration (reference in the figure). Furthermore, Fig. 5.13(b) shows the residual strain contour plot on the deformed torsion spring after undergoing 20 loading-unloading cycles. The residual strain mainly distributes at the round corner on the torsion spring, and the maximum value is approximate 1.5%. The global residual deformation on the torsion spring is also well shown by comparing the current torsion spring with its initial mesh.

Conclusion

In this chapter, a 3D thermomechanically coupled, finite-strain constitutive model is developed for cyclic pseudoelasticity of polycrystalline SMAs. The proposed model captures several fundamental characteristics deduced from the experimental observations, such as accumulated residual strain, degeneration of Finite-strain cyclic behavior of SMAs pseudoelasticity and dissipated energy, rate dependence, and evolution of the phase transformation from abrupt to smooth transition. The formulation of the model is based on the decomposition of the finite Hencky strain into elastic, transformation, residual and thermal components. A thermodynamically consistent framework and a Helmholtz free energy function including elastic, thermal, interaction, and constraint energies are constructed to derive the constitutive equations, i.e., thermoelasticity, evolution of martensite volume fraction, flow rule of residual strain, martensite reorientation, heat production, and temperature evolution. Then, the 3D model is reduced to a 1D version, also the heat transfer condition of the cyclic tensile experiment on NiTi wire is simplified under the assumption of uniform temperature field. Model parameters are calibrated from the experimental data. Numerical simulations of the cyclic experiments at different strain rates are carried out to validate the model. The good agreement between the model predictions and the experimental data well demonstrates the model capabilities to capture the fundamental characteristics related to the cyclic pseudoelasticity of polycrystalline SMAs. Finally, an application example involving a SMA torsion spring undergoing large cyclic deformations is presented.

The future direction of this step is to predict fatigue life of SMA structures. Indeed, according to [START_REF] Moumni | Fatigue analysis of shape memory alloys: energy approach[END_REF], the number of cycles to failure depends on the hysteresis area at the stabilized cycle. Thus, if a proper fatigue criterion is introduced, the model will be useful to predict fatigue life of SMA structures subjected to finite cyclic deformations.

Chapter 6

Conclusion and future work

Conclusion

Shape Memory Alloys exhibit many interesting properties when undergoing large deformations, thermomechanical or cyclic loading conditions. With the increasing industrial applications of these SMAs properties, a comprehensive and in-depth understanding of constitutive behavior became indispensable. To this end, this work developed a comprehensive constitutive modeling approach to investigate thermomechanical and cyclic behavior of SMAs in finite deformation. Moreover, the proposed models are implemented into finite element software as computational tools for the analysis, design and optimization of SMA structures. The whole work is accomplished by the following three steps.

The first step, detailed in Chapter 3, is the finite-strain extension of the ZM model proposed by [START_REF] Zaki | A three-dimensional model of the thermomechanical behavior of shape memory alloys[END_REF]. This step aims at the constitutive modeling and numerical implementation of SMAs undergoing large strains and rotations. The finite-strain formulation is based on a two tier decomposition of the deformation gradient into elastic and transformation parts, where the transformation deformation is further decomposed into phase transformation and martensite reorientation components. Constitutive equations are derived within a finite-strain thermomechanical framework. Numerical implementation of the model features the symmetrization of evolution equation to improve efficiency and robustness, and the material and spatial tangent operators involving the objective stress rates. The numerical algorithm is programmed into Abaqus/Standard by means of UMAT subroutine. Simulation results using the proposed finite-strain model demonstrate superiorities of the model dealing with SMA problems with large strains and rotations.

The second step, detailed in Chapter 4, takes into account thermomechanical coupling effect in addition to finite deformation. The Hencky strain is used, due to its remarkable properties in large deformations, to provide a more straightforward finite-strain formulation. The model incorporates three important characteristics of SMA response. First, the effect of phase coexistence between austenite and two martensite variants is accounted for by introducing two distinct internal variable, representing the volume fractions for single-variant χ S and multi-variant χ M martensites. Then, the variation of the hysteresis size with temperature is characterized by means of the forward entropy difference ∆η f and the reverse difference ∆η r . Finally, the smooth material behavior at initiation and completion of phase transformation is accounted for using a unique tangential transformation hardening function g t . Time-discrete formulation of the constitutive equations were derived and integrated using a Henckystrain return-mapping algorithm. Whilst the proposed three-dimensional model was reduced to a one-dimensional version to perform the model parameters calibration. The model was then implemented into Abaqus/Explicit using VUMAT subroutine. Numerical simulations are carried out and validated against experimental data for a variety of loading cases, including proportional, non-proportional, isothermal, non-isothermal conditions. Finally, a simulation example is presented in which a SMA helical spring is first subjected to mechanical loading resulting in a pseudoelastic response then to thermal loading resulting in a temperature-induced actuation. The simulation results show the ability of the proposed model to describe the behavior of SMA structures subjected to complex thermomechanical loadings.

The last step, detailed in Chapter 5, deals with cyclic pseudoelasticity of polycrystalline SMAs by generalizing the second-step model. This model captures several fundamental characteristics related to cyclic behavior of SMAs, i.e., large accumulated residual strain, degeneration of superelasticity and hysteresis loop, rate dependence, and evolution of phase transformation from abrupt to smooth transition. The formulation of the model is based on the decomposition of finite Hencky strain into elastic, transformation, residual and thermal components. A thermodynamically consistent framework is established in compliance with the reduced form of energy balance and the Clausius-Duhem form of entropy inequality, while a Helmholtz free energy function including elastic, thermal, interaction and constraint parts is constructed. Constitutive equations, i.e., thermoelasticity, evolution of martensite volume fraction, flow rule of residual strain, martensite reorientation and heat production are then derived from them. The 3D model is reduced to a 1D version and the heat transfer condition of the cyclic tensile experiment on NiTi wire is simplified. Model parameters are calibrated from the experimental data. Finally, numerical simulations of the cyclic experiments at different strain rates are carried out to validate the model. The model is able to capture the rate dependence, the temperature variation, the abrupt to smooth transition, and the evolutions of the dissipated energy, the reversible transformation strain, the residual strain, the transformation stresses and the transformation hardening modulus during cyclic deformations. Finite element simulation of a torsion spring shows the potential of the proposed model for fatigue analysis of SMA structures.

Future work

Though the present work has developed a relatively comprehensive approach to investigate the thermomechanical and cyclic behavior of SMAs in finite deformations, there are several improvements that would be the made for future work. Examples of them are plasticity and fatigue analysis.

• In general, the strain in pseudoelasticity of SMA is within 10%. This strain level is insufficient to demonstrate the advantages of the finite-strain models. Hence, the models might be extended to 6.2 Future work 111 account for large irreversible plastic deformations that occur when SMA is subjected to severe mechanical loading. Moreover, plasticity, also as a failure mechanism, compromises the integrity of SMA devices under certain service loading. Material behavior after the plastic yield point could be described by introducing a plastic deformation F p and its corresponding evolution law and yield criterion.

• The fatigue analysis of SMA structures requires two fundamental aspects: a reliable constitutive model capable of capturing thermomechanical characteristics related to the cyclic behavior of SMAs, as well as a proper fatigue criterion predicting the number of cycles to failure of the structures. Since the present model has shown the capabilities of well describing degeneration and transformation shakedown state in SMAs, if proper fatigue criterion is introduced, the model can be utilized to carry out the fatigue analysis of complex SMA structures.

• The internal loops in partially loaded and unloaded condition is a complex and tricky issue but is central to applications of SMAs since in most cases, due to the considerations of fatigue, the strains are severely restricted. If the relevant internal variables are introduced, the model can be generalized to address this phenomenon.

• Nowadays, the 3D printing technology is widely used to manufacture complex SMA structures. Thus, the model can be used to predict thermomechanical behavior of 3D printed SMAs, and therefore to study its difference from the classical SMAs. Titre : Modélisation du comportement thermomécanique et cyclique des matériaux à mémoire de forme en transformations finies Résumé : Cette thèse présente une approche globale de la modélisation du comportement thermomécanique et cyclique des alliages à mémoire de forme (AMF) en grandes déformations. Cette approche s'articule en trois étapes : i) La généralisation du modèle ZM de comportement des AMF en grandes déformations dans le cadre de la thermodynamique des processus irréversibles. Pour ce faire, le gradient de la transformation totale est décomposé sous la forme du produit de trois gradients : le gradient de transformation lié à la déformation élastique, le gradient lié au changement de phase et le gradient de transformation lié à la réorientation de la martensite. Cette décomposition permet ainsi la modélisation de la réponse des structures en AMF dans le cas de chargements multiaxiaux non-proportionnels en transformations finies.

Sciences

ii) La prise en compte du couplage thermomécanique en transformations finies. Pour ce faire, la déformation de Hencky a été introduite. Le modèle obtenu intègre trois caractéristiques thermomécaniques importantes des AMF, à savoir l'effet de la coexistence de l'austénite et de deux variantes de martensites distinctes, la variation de la taille de la boucle d'hystérésis avec la température et la transition du processus de changement de phase, d'abrupt à doux. iii) Enfin, en vue de prédire la réponse des structures en AMF sous chargement thermomécanique cyclique, le modèle développé dans la deuxième étape est généralisé pour décrire la pseudoélasticité cyclique des AMF polycristallins. Le modèle obtenu permet la prise en compte de quatre caractéristiques fondamentales liées au comportement cyclique des AMF : la déformation résiduelle accumulée, la dégénérescence de la boucle d'hystérésis, l'évolution de la transformation de phase, d'abrupte à douce. La mise en oeuvre numérique de ces modèles s'appuie sur des algorithmes d'intégration appropriés. Des exemples numériques on été traités pour valider chaque étape.

Mots clés : les alliages à mémoire de forme, modèle constitutif, transformations finies, couplage thermomécanique, comportement cyclique, simulation numérique.

Title: Constitutive modeling of the thermomechanical and cyclic behavior of shape memory alloys in finite deformations Abstract: Shape Memory Alloys (SMAs) are a class of smart materials that possess two salient features known as pseudoelasticity (PE) and shape memory effect(SME). In industrial applications, SMA structures are typically subjected to complex service conditions, such as large deformations, thermomechanically coupled boundaries and loadings, and cyclic loadings. The reliability and durability analysis of these SMA structures requires a good understanding of constitutive behavior in SMAs. To this end, this work develops a comprehensive constitutive modeling approach to investigate thermomechanical and cyclic behavior of SMAs in finite deformations. The work is generally divided into three steps. First, to improve accuracy of SMA model in finite deformation regime, the ZM model proposed by [START_REF] Zaki | A three-dimensional model of the thermomechanical behavior of shape memory alloys[END_REF] is extended within a finite-strain thermodynamic framework. Moreover, the transformation strain is decomposed into phase transformation and martensite reorientation components to capture multi-axial non-proportional response. Secondly, in addition to the finite deformation, thermomechanical coupling effect is taken into account by developing a new finite-strain thermomechanical constitutive model. A more straightforward approach is obtained by using the finite Hencky strain. This model incorporates three important thermomechanical characteristics, namely the coexistence effect between austenite and two distinct martensite variants, the variation with temperature of hysteresis size, and the smooth transition at initiation and completion of phase transformation. Finally, with a view to studying SMA behavior under cyclic loading, the model developed in the second step is generalized to describe cyclic pseudoelasticity of polycrystalline SMAs. The generalized model captures four fundamental characteristics related to the cyclic behavior of SMAs: large accumulated residual strain, degeneration of pseudoelasticity and hysteresis loop, rate dependence, and evolution of phase transformation from abrupt to smooth transition. Numerical implementation of these models are realized by introducing proper integration algorithms. Finite element simulations, including orthodontic archwire, helical and torsion spring actuators, are carried out using the proposed models.

  crystal structures and phase transformations of austenite, single-variant and multivariant martensites. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 A phase diagram and the phase transformations between austenite, multi-variant and single-variant martensites. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Schematic of the shape memory effect for a typical NiTi SMA. . . . . . . . . . . . . . 1.4 Schematic of the pseudoelasticity for a polycrystalline NiTi SMA. . . . . . . . . . . .

f

  : (a) the first cycle, (b) the stabilized cycle. . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Simulated results vs. experimental data under uniaxial loading: (a) martensite orientation at temperature 293 K, (b) pseudoelasticity at temperature 333 K. . . . . . . . . . . . . 3.2 Biaxial strain-controlled box-shaped loading test: (a) axial-shear loading path, (b) comparison of the simulated stress response with the results of Stebner and Brinson (2013) and Gu et al. (2015). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Evolution of the von Mises equivalent stress and the martensite volume fraction with time step. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Simulation results using the present finite-strain formulation (FSF) model and the small-strain formulation (SSF) ZM model. . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Deformation of the SMA beam in bending: (a) original ZM model with NLGEOM OFF, (b) original ZM model with NLGEOM ON, (c) present finite-strain model. . . . . . . . 3.6 Model prediction of the shape memory effect. . . . . . . . . . . . . . . . . . . . . . . 3.7 Initial geometry and dimension of the SMA orthodontic archwire. . . . . . . . . . . . 3.8 Orthodontic archwire at maximum deformation: (a) von Mises stress contour plot, (b) martensite volume fraction χ contour plot. . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Pseudoelastic force vs. displacement response of the SMA archwire at the point of application of the load. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Austenite finish temperature at stress level of σ A i , n i Eigenvalues and eigenvectors of the symmetric tensor A Skew part of the tensor A sym(A) Symmetric part of the tensor A tr(A) Trace of the tensor A

  Figure 1.1 The crystal structures and phase transformations of austenite, single-variant and multi-variant martensites.

  Figure 1.2 A phase diagram and the phase transformations between austenite, multi-variant and singlevariant martensites.

Figure 1 . 4

 14 Figure 1.4 Schematic of the pseudoelasticity for a polycrystalline NiTi SMA.

Figure 2 . 1

 21 Figure 2.1 Cyclic pseudoelastic response of the NiTi wire at strain rate of 1 × 10 -3 s -1 (Morin et al., 2011b): (a) stress-strain curves (black for first cycle and red for stabilized cycle), (b) evolutions of the dissipated energy and the reversible strain with the number of cycles.

Figure 2 . 2

 22 Figure 2.2 Strain rate dependent stress-strain curves in the cyclic experiments (Kan et al., 2016): (a) the first cycle, (b) the stabilized cycle.

  ) where ζ p , ζ tr , ζ f and ζ r are nonnegative Lagrangian multipliers. The potential (3.6) is complemented by the following Kuhn-Tucker conditions:

  .11) where a, b and Y are positive model parameters. The evolution equations and yield functions associated with the forward phase transformation (A → M), the reverse phase transformation (M → A), and the martensite orientation are then derived from the Lagrangian L and the dissipation D in equations(3.8) and(3.11).

3. 3

 3 Development of the finite-strain model 29 Substituting equation (3.31) into inequality (3.30) then gives

  .43) With the help of the following three transformations: tr (A tr ) = tr [C tr ( χM t -X tr )] = tr (B tr ) , F T tr A tr F tr = B tr C tr , ∥ A D tr ∥ = ∥B D tr ∥, (3.44) equation (3.43) is rewritten as C tr = γ tr B D tr ∥B D tr ∥ C tr .(3.45)

  .70) which can be equivalently written as d χ = P : dC and dγ tr = Q : dC, (3.71)

  the rotation tensor. Both τ and ▽ τ are special cases of the Oldroyd stress rate ⋄ τ (Lie time derivative of the Kirchhoff stress), expressed as ⋄ τ = τ -Lτ -τL T , (3.77) where L = FF -1 is the velocity gradient. Indeed, τ corresponds to ⋄ τ with the rate of deformation tensor D set to be zero, while ▽ τ corresponds to ⋄ τ with deformation gradient F substituted with the rotation tensor R. Taking the time derivative of the relation τ = FSF T and substituting into the expression of the Oldroyd stress rate (3.77), gives ⋄ τ = F SF T . (3.78) Moreover, equations S = D : C and C = 2F T DF give

Figure 3 . 1

 31 Figure 3.1 Simulated results vs. experimental data under uniaxial loading: (a) martensite orientation at temperature 293 K, (b) pseudoelasticity at temperature 333 K.

Figure 3 . 2

 32 Figure 3.2 Biaxial strain-controlled box-shaped loading test: (a) axial-shear loading path, (b) comparison of the simulated stress response with the results of[START_REF] Stebner | Explicit finite element implementation of an improved three dimensional constitutive model for shape memory alloys[END_REF] and[START_REF] Gu | Time integration and assessment of a model for shape memory alloys considering multiaxial nonproportional loading cases[END_REF].

Figure 3 . 3

 33 Figure 3.3 Evolution of the von Mises equivalent stress and the martensite volume fraction with time step.

Figure 3 . 4 Figure 3 . 5

 3435 Figure 3.4 Simulation results using the present finite-strain formulation (FSF) model and the small-strain formulation (SSF) ZM model.

Figure 3 . 6

 36 Figure 3.6 Model prediction of the shape memory effect.

Figure 3 . 7 Figure 3 . 8

 3738 Figure 3.7 Initial geometry and dimension of the SMA orthodontic archwire.

FiniteFigure 3 . 9

 39 Figure 3.9 Pseudoelastic force vs. displacement response of the SMA archwire at the point of application of the load.

3 e

 3 .10) Using the above three kinematic assumptions, the stretching and spin tensors, D and W , are written as D = De + sym Fe D t F-1 e + 1 3 δ e + δ θ 1 and W = W e + skew Fe D t F-1 e . (4.11) Using equation (3.14), the elastic, transformation and thermal right and left Cauchy-Green deformation tensors are written as C e = F T e F e = J 2 Ce , b e = F e F T e = J 2 3

  thermomechanically coupled model based on Hencky strain 57

  .102) with the stretching tensor D = sym(L) = L and spin tensor W = skew(L) = 0.The elastic F e and the transformation F t deformation gradients are defined byF e = λe e 1 ⊗ e 1 + 1 λ e e 2 ⊗ e 2 + 1 λ e e 3 ⊗ e 3 , (4.103)

  .104) with λ = λ e λ t and J = J e J t = 1. (4.105) Thus, the elastic h e and transformation h t Hencky strains in equation (4.14) are written as h e = ln λ e T and h t = ln λ t T . (4.106) The deviatoric stress s in equation (4.59) is then written as s = 2µ ln λ e T = Me and the direction of transformation strain N t in equation (4.65) are written as Me and N t are coaxial for the case of uniaxial loading. Hence, according to equation (4.80), the thermodynamic force of martensitic reorientation A N is null, i.e. A N = 0. Moreover, the thermodynamic forces associated to the multi-variant χ M and single-variant χ S martensite phase A thermomechanically coupled Hencky-strain model transformations are given by

Figure 4 . 1

 41 Figure 4.1 Calibration of the model parameters from the experimental stress-strain curves: (a) identification of E A , E M , ∆η f , ∆η r and Y S from the pseudoelastic response, (b) identification of Y tr , µ t and H from the martensite orientation response.

  .110) where E A and E M are Young's moduli for austenite and martensite as shown in Fig 4.1(a).

Figure 4 . 2

 42 Figure 4.2 Effects of the model parameters κ, m, ξ b , ξ a and r in the tangential hardening function on the stress-strain behavior, arrows indicate the value increase of the respective parameters.

Figure 4 . 3

 43 Figure 4.3 Model prediction vs. experimental data under isothermal uniaxial tensile loading: (a) θ = 298 K, (b) θ = 303 K, (c) θ = 313 K, (d) evolution of the single-variant martensite volume fraction.

Figure 4 . 4

 44 Figure 4.4 Unstressed thermally-induced phase transformation: (a) heat generation and absorption (latent heat), (b) evolution of the multi-variant martensite volume fraction.

Figure

  Figure 4.4(a) shows a comparison between the model predictions and the experimental DSC data reported by[START_REF] Popov | A 3-D constitutive model for shape memory alloys incorporating pseudoelasticity and detwinning of self-accommodated martensite[END_REF], wherein the relative heat flow is plotted as a function of temperature. Heat absorption from martensite to austenite phase transformation results in a ridge along the heating part of the curve, while heat generation from austenite to martensite phase transformation results in a valley along the cooling part. The bottom and peak points on the cooling and heating parts, corresponding to temperatures θ M A = 303 K and θ AM = 339, denote average temperatures for forward and reverse phase transformation, respectively. It is worth noting that the ridge is short and wide, while the valley is deep and narrow. This is because the area under the curve in each case, representing phase transformation latent heat, is assumed to be the same. In this particular case, the temperature range of reverse transformation is larger than that of forward transformation, resulting in a low and wide ridge versus a deep and narrow valley. Overall, the heat absorption/generation during thermally-induced phase transformation is well described by the proposed model despite a small deviation at completion of the reverse phase transformation. Thanks to the tangential hardening function g t defined in equation (4.62), the smooth experimental DSC curves are simulated with good accuracy. In addition to simulating the heat exchange during phase transformation, the evolution of multi-variant martensite volume fraction is simulated and shown inFigure 4.4(b). According to the simulation, the reverse phase transformation of martensite to austenite is found to initiate at temperature 305 K and complete at temperature 353 K, while the forward phase transformation of austenite to martensite initiates at temperature 313 K and completes at temperature 300 K.

Figure 4 . 5

 45 Figure 4.5 Comparisons between the model predictions and the experimental data reported by Wu et al. (2003): transformation strain vs. temperature at constant stress levels 160 MPa and 360 MPa.

Figure 4 .

 4 Figure 4.6 shows the butterfly-shaped strain input and the stress output in both axial and shear directions. The maximum axial and shear strain magnitudes reached in each case are ε = γ/ √ 3 = 0.015, as shown in Figure 4.6(a). The results of numerical simulation are compared to the reference experimental data in Figure 4.6(b-d). Specifically, Figure 4.6(b) shows the scaled shear stress √ 3τ versus axial stress σ, Figure 4.6(c) shows axial stress σ versus axial strain ε and Figure 4.6(d) shows √ 3τ versus the scaled shear strain γ/ √ 3. The overall agreement with the experimental data is satisfactory despite small

Figure 4 . 6

 46 Figure 4.6 Comparisons between the model predictions and the experimental data reported by[START_REF] Grabe | Path dependence and multiaxial behavior of a polycrystalline NiTi alloy within the pseudoelastic and pseudoplastic temperature regimes[END_REF]: (a) butterfly-shaped axial-shear loading path, (b) shear stress vs. axial stress, (c) axial stress-strain response, (d) shear stress-strain response.

Figure 4 . 8

 48 Figure 4.8 Simulation of a SMA helical spring actuator: (a) geometry and mesh, (b) force and temperature loading paths.

Figure 4 .Figure 4 .

 44 Figure 4.9 SMA helical spring actuator at maximum loading: (a) von Mises stress contour plot, (b) martensite volume fraction contour plot.

Figure 4 .Figure 4 .

 44 Figure 4.11 SMA helical spring actuator at temperature 280 K: (a) von Mises stress contour plot, (b) martensite volume fraction contour plot.

  version of the constitutive equationsFor the case of uniaxial loading, the Cauchy stress tensor σ and the deformation gradient F are given as σ = σ e 1 ⊗ e 1 and F = λ e 1 ⊗ e 1 + λ l e 2 ⊗ e 2 + λ l e 3 ⊗ e 3 ,

  .61) with the stretching tensor D = sym(L) = L and spin tensor W = skew(L) = 0. The elastic F e , transformation F t and residual F r deformation gradients are defined by F e = λ e e 1 ⊗ e 1

Figure 5 . 1

 51 Figure 5.1 Heat transfer conditions in the experiment and simplification in the model.

  .77) Finite-strain cyclic behavior of SMAs where θ c m and A c are mean temperature and area of the cross-section surface, while θ l m and A l are mean temperature and area of the lateral surface, i.e.,

FiniteFigure 5 . 2

 52 Figure 5.2 Calibration of the model parameters from the experimental stress-strain curves: (a) identification of K, µ A , µ M , ∆η t and Y χ from the pseudoelastic stress-strain curve, (b) identification of Y N , µ t and H from the stress-strain curve of martensitic orientation.

  .85) where A s and A f are austenite start and finish temperatures, M s and M f are martensite start and finish temperatures. They are determined by DSC tests. The heat convection coefficient h, the thermal conductivity k and the constant β, together with geometrical parameters of the NiTi wire L and d, constitute a new model parameter ω as ω = hd + 2βk dL .

Figure 5 . 3

 53 Figure 5.3 Cyclic stress-strain behavior of NiTi SMA at strain rate of 1 × 10 -4 s -1 : (a) evolution of stress-strain curves with the number of cycles, (b) comparison for the 1st cycle, (c) comparison for the 10th cycle, (d) comparison for the stabilized cycle.

Figure 5 . 4 Finite

 54 Figure 5.4 Cyclic stress-strain behavior of NiTi SMA at strain rate of 2.5 × 10 -4 s -1 : (a) evolution of stress-strain curves with the number of cycles, (b) comparison for the 1st cycle, (c) comparison for the 10th cycle, (d) comparison for the stabilized cycle.

Figure 5 .

 5 Figure 5.10 Comparisons of the model predictions against the experimental data for the temperature evolution during cycling: (a) strain rate of 2.5 × 10 -4 s -1 , (b) strain rate of 1 × 10 -3 s -1 , (c) strain rate of 5 × 10 -3 s -1 .

Figure 5 .

 5 Figure 5.12 Evolution of the martensite volume fraction contour plot at the maximum loading with the number of cycles.

Figure 5 .

 5 Figure 5.13 Residual deformation on the torsion spring: (a) deformation of the inside edge in absence of loading for different cycles, (b)initial mesh and residual strain contour plot for the 20th cycle.

  Schematic of the shape memory effect for a typical NiTi SMA. development of TWSME in trained SMAs was explained by

											Introduction
		400					Martensite reorientation
	Stress [MPa]	300 200
		100
					Shape recovery					300
		0 0	0.01	0.02	0.03	0.04	0.05	0.06	340	330	Temperature [K] 310 320
					Strain [-]
	Figure 1.3

Table 3 .

 3 2 Model parameters used in the uniaxial proportional simulations.

	Parameter Value	Unit Parameter Value	Unit
	µ A	15923 MPa λ A	23885 MPa
	µ M	6500	MPa λ M	9750	MPa
	α	588.24 MPa β	3186.3 MPa
	Y	265	MPa C	1.75	-
	a	28.21	MPa b	27.62	MPa
	G	37.3	MPa ξ	0.4398 MPa/K
	A 0 f	313	K	K	12.63	MPa

Table 3 .

 3 3 Model parameters used in the multiaxial box-shaped simulation.

		Parameter Value Unit Parameter Value Unit
		µ A	31798 MPa λ A	47698 MPa
		µ M	13462 MPa λ M	20192 MPa
		α	789.5 MPa β	3421	MPa
		Y	265	MPa C	1.735 -
		a	4.79	MPa b	5.39	MPa
		G	4.48	MPa ξ	0.19	MPa/K
		A 0 f	313	K	K	2.32	MPa
		0.04 0.06		Theoretical Numerical
	γ	0.02
	Shear strain	-0.02 0.00
		-0.04
		-0.06 -0.04 -0.02 0.00 -0.06	0.02	0.04	0.06
		Axial strain ε

  are symmetric and coaxial with he and h t . Using the definition of the logarithmic corotational rate of the Eulerian Hencky strain in 4.2 A thermomechanically coupled model based on Hencky strain

	θ . ∂ χ S χ S + ∂ψ and ∂ψ ∂ χ M χ M + ∂ψ Because of isotropy, the derivatives of the isotropic functions ∂ψ Hence, the time derivative of ψ gives ψ = ∂ψ ∂δ e δ e + ∂ψ ∂ he : he + ∂ψ ∂h t : h t + ∂ψ ∂δ θ δ θ + ∂ he equation (A.4) and equation (A.8), ∂ψ ∂ he : he = ∂ψ ∂ he : hL e = ∂ψ ∂ he : De and ∂ψ ∂h t : h t = ∂ψ ∂h t : hL t = ∂ψ ∂ψ ∂θ ∂h t : D t , θ. where hL e and hL t are, respectively, the logarithmic corotational rates of the elastic and transformation (4.23) (4.24) (4.25) Eulerian Hencky strains. Substituting equations (4.24) and (4.25) into the entropy inequality (4.21) gives ∂h t 51 p -∂ψ ∂δ e δ e + p -∂ψ ∂δ θ δ θ + s -∂ψ ∂ he : De + FT e s F-T ∂ψ e -∂h t

  is a new second Piola-Kirchhoff stress tensor defined with respect to the intermediate unstressed configuration. As s and V e commute, Me can be written as

	with		X t =	∂ψ ∂h t	, Z M =	∂ψ ∂ χ M and Z S =	∂ψ ∂ χ S .	(4.33)
	Substituting equation (4.27) into equation (4.24) gives the following first Gibbs relation between the
	free energy ψ and the temperature θ as
								.29)
	The following deviatoric Mandel stress tensor is defined with respect to the intermediate unstressed
	configuration:
					Me =	FT e s	F-T e = Ce Se ,	(4.30)
	where Se =	F-1 e s	F-T
				Me = R T e	VT e s	V -1 e R -T e = R T e sR e ,	(4.31)
	which is a symmetric and traceless tensor by rotating the deviatoric Kirchhoff stress s back into the
	intermediate unstressed configuration. Substituting Me into inequality (4.28) gives
				Me -X (4.32)

e t : D t -Z M χ M -Z S χ S ≥ 0,

  .56) where ζ M , ζ S , ζ M S and ζ t are nonnegative Lagrangian multipliers. The constraint energy (4.56) is complemented by the following classical Kuhn-Tucker conditions:

Table 4 .

 4 2 Model parameters used in isothermal pseudoelastic uniaxial tensile simulations.

			Parameter Value Unit		Parameter Value Unit
			K			23400 MPa	µ A	12500 MPa
			µ M S		9038	MPa	µ t		500	MPa
			∆η f		0.147 MPa/K ∆η r	0.133 MPa/K
			κ			5.6	MPa	m		0.2		-
			ξ a			0.998 -		ξ b	0.001 -
			r			-6.65	MPa	H	0.035 -
			θ 0			240	K		Y M	2.1		MPa
			Y S			2.1	MPa	Y tr	110	MPa
	observed at the initiation and completion of phase transformation, and the elastic modulus is found to
	depend on phase composition.
										700
		Model 298 K								Model 303 K
		Experiment 298 K							600	Experiment 303 K
										500
	Stress [MPa]								Stress [MPa]	300 400
										200
										100
										0
	0	0.01	0.02	0.03	0.04	0.05	0.06	0.07		0	0.01	0.02	0.03	0.04	0.05	0.06	0.07
				Strain								Strain
				(a)									(b)
										1
		Model 313 K								Model 298 K
		Experiment 313 K								Model 303 K
	Stress [MPa]								Martensitic volume fraction	0.2 0.4 0.6 0.8	Model 313 K
										0
	0	0.01	0.02	0.03	0.04	0.05	0.06	0.07		0	0.01	0.02	0.03	0.04	0.05	0.06	0.07
				Strain								Strain
				(c)									(d)

Table 4 .

 4 3 Model parameters used in unstressed thermally-induced phase transformation simulation.

	Parameter Value Unit	Parameter Value Unit
	K	41667 MPa	µ A	26923 MPa
	µ M S	11538 MPa	µ t	300	MPa
	∆η f	0.588 MPa/K ∆η r	0.408 MPa/K
	κ	9.6	MPa	m	0.2	-
	ξ a	0.998 -	ξ b	0.001 -
	r	10.2	MPa	H	0.06	-
	θ 0	313	K	Y M	8.1	MPa
	Y S	8.1	MPa	Y tr	110	MPa

Table 4 .

 4 4 Model parameters used in the simulations of isobaric thermal cycling tests.

	Parameter Value	Unit	Parameter Value Unit
	K	50000 MPa	µ A	23077 MPa
	µ M S	23077 MPa	µ t	300	MPa
	∆η f	0.2585 MPa/K ∆η r	0.282 MPa/K
	κ	7.52	MPa	m	0.08	-
	ξ a	0.998	-	ξ b	0.001 -
	r	-2.585 MPa	H	0.047 -
	θ 0	297	K	Y M	5.64	MPa
	Y S	5.64	MPa	Y tr	110	MPa

Table 4 .

 4 5 Model parameters used in butterfly-shaped non-proportional simulation.

	Parameter Value Unit	Parameter Value Unit
	K	52083 MPa	µ A	24038 MPa
	µ M S	24038 MPa	µ t	100	MPa
	∆η f	0.198 MPa/K ∆η r	0.198 MPa/K
	κ	6	MPa	m	0.2	-
	ξ a	0.98	-	ξ b	0.01	-
	r	-4.8	MPa	H	0.03	-
	θ 0	288	K	Y M	2.4	MPa
	Y S	2.4	MPa	Y tr	110	MPa

Table 4 .

 4 6 Model parameters used in non-isothermal simulations at different loading rates.

	Parameter Value Unit	Parameter Value Unit
	K	52083 MPa	µ A	24038 MPa
	µ M S	24038 MPa	µ t	100	MPa
	∆η f	0.33	MPa/K ∆η r	0.33	MPa/K
	κ	3.92	MPa	m	0.2	-
	ξ a	0.98	-	ξ b	0.01	-
	r	1.96	MPa	H	0.049 -
	θ 0	288	K	Y M	7.84	MPa
	Y S	7.84	MPa	Y tr	110	MPa
	k	18	W/m/K c v	440	J/kg/K
	ω	0.86	-	h	50	W/m 2 /K

  As the Lagrangian and Eulerian Hencky strains share the same invariants, ψ can be written in terms of Eulerian Hencky strains as ψ (H e , H t , H r , H θ , χ, χ c , θ) = ψ δ e , he , h t , h r , δ θ , χ, χ c , θ . and ∂ψ ∂h r are symmetric and, respectively, coaxial with he , h t and h r . Using equations (A.4) and (A.8),

																			(5.17)
	Hence, the time derivative of ψ gives
	ψ =	∂ψ ∂δ e	δ e +	∂ψ ∂ he	: he +	∂ψ ∂h t	: h t +	∂ψ ∂h r	: h r +	∂ψ ∂δ θ	δ θ +	∂ψ ∂ χ	χ +	∂ψ ∂ χ c	χ c +	∂ψ ∂θ	θ.	(5.18)
	Because of isotropy, the derivatives of the isotropic functions ∂ψ ∂ he ∂h t ∂ψ , ∂ψ ∂ he : he = ∂ψ ∂ he : hL e = ∂ψ ∂ he : De ,
									∂ψ ∂h t		: h t =	∂ψ ∂h t		:	hL t =	∂ψ ∂h t	: D t ,	(5.19)
									∂ψ ∂h r		: h r =	∂ψ ∂h r	:	hL r =	∂ψ ∂h r	: D r ,
	where hL e ,	hL t and		hL r are, respectively, the logarithmic corotational rates of elastic, transformation and
	residual Eulerian Hencky strains.
	Substituting equations (5.18) and (5.19) into the entropy inequality (5.15) gives
		p -	∂ψ ∂δ e	δ e + p -	∂ψ ∂δ θ	δ θ + s -	∂ψ ∂ he	: De +	FT e s	F-T e -	∂ψ ∂h t	: D t +
		FT e s	F-T e -	∂ψ ∂h r

  is guaranteed by the following choice of the constitutive laws with respect to the hydrostatic stress p, the deviatoric stress s and the entropy η v : Finite-strain cyclic behavior of SMAs and by the following inequalities, which ensure nonnegative intrinsic dissipation in arbitrary evolutions of the internal state variables D t , D r , χ and χ c , so that Me -X t : D t + Me -X r : D r -

									∂ψ ∂ χ	χ -	∂ψ ∂ χ c	χ c ≥ 0,	(5.22)
	with	Me =	FT e s	F-T e , X t =	∂ψ ∂h t	and X r =	∂ψ ∂h r	,
		p =	∂ψ ∂δ e	=	∂ψ ∂δ θ	, s =	∂ψ ∂ he	and η v = -	∂ψ ∂θ	,	(5.21)

  he , h t , h r , δ θ , χ, χ

	Hence, from equations (5.21) 3 , (5.27) and (5.29),
	θ η v =θ	∂η ∂δ e	δ e +	∂η ∂ he	: he +	∂η ∂h t	: h t +	∂η ∂h r	: h r +	∂η ∂δ θ
										∂θ 2	c , θ	.	(5.29)

Table 5 .

 5 3 Model parameters used in the simulation of cyclic tests.

	Parameter Value	Unit	Parameter Value	Unit
	K	41557	Mpa	α	22 × 10 -6 K -1
	µ A	26811	Mpa	µ M	11550	Mpa
	κ 0	2	Mpa	κ sat	5	Mpa
	m 0	0.1	-	m sat	0.3	-
	ξ 0 a ξ 0 b r 0	0.899 0.1 -0.15	--Mpa	ξ sat a ξ sat b r sat	0.98 0.001 -4.3	--Mpa
	Y 0 χ	5.8	Mpa	Y sat χ	2.5	Mpa
	∆η 0 t H 0	0.5 0.044	MPa/K ∆η sat t -H sat	0.3 0.035	MPa/K -
	µ t	500	Mpa	Y N	110	Mpa
	R c	0.145	s	R 0	2.345	%
	R s	1.68	%	τ	11	-
	c v	2.86 × 10 6 J/m 3 /K ω	0.55 × 10 6 J/m 3 /K
	θ 0	290	K	θ a	323	K

mécaniques et énergétiques, matériaux et géosciences (SMEMAG)

	ÉCOLE DOCTORALE ÉCOLE DOCTORALE
	Sciences mécaniques et Sciences mécaniques et
	énergétiques, matériaux énergétiques, matériaux
	et géosciences (SMEMAG) et géosciences (SMEMAG)

ÉCOLE DOCTORALE

  The future directions of this work mainly involve plasticity and fatigue analysis of SMA structures.

Keywords: Shape Memory Alloy; Constitutive model; Finite deformation; Thermomechanical coupling; Cyclic loading; Numerical simulation. Université Paris-Saclay Espace Technologique / Immeuble Discovery Route de l'Orme aux Merisiers RD 128 / 91190 Saint-Aubin, France

List of Tables

Substituting equation (4.75) Then, the thermodynamic force A N is written

where I is the fourth-order identity tensor. (I -N t ⊗ N t ) Me represents the component normal to N t of the Mandel stress Me . Substituting equations (4.72) 2 and (4.75) into equation (4.69) gives the following evolution equation for the transformation strain:

Using equations (4.41), (4.42), (4.72) and (4.75), the temperature evolution equation is written as

A review of the constitutive equations derived in this section is presented in Table 4.1.

Numerical implementation

In this section, numerical implementation of the constitutive model in an finite element analysis (FEA) framework is carried out. Time-discrete formulation of the proposed model is presented by using an exponential integrator for the transformation evolution equation. Then, a Hencky-strain return-mapping algorithm is introduced, which greatly increases the numerical simplicity and efficiency.

A time-discrete framework

The deformation-driven initial boundary value problem of the proposed thermomechanically coupled finite-strain model is described as: within the time interval [t n , t n+1 ], given the deformation gradient F n , the internal state variables N t n , χ M n , χ S n , the temperature θ n at time t n and the deformation gradient F at time t n+1 , determine the state variables and temperature change at time t n+1 satisfying the time-discrete constitutive equations.

The evolution equation (4.72) 1 of the multi-variant martensite phase fraction χ M is written using a backward-Euler integration scheme as

where ∆γ M is the multiplier at time t n+1 , the sign function S(•) is used to extract the sign of A M .

with

The time-discrete evolution equations (4.83), (4.84) and (4.86), together with the yield functions (4.73) and (4.76), construct the following system of time-discrete constitutive equations:

which is complemented by the following time-discrete Kuhn-Tucker conditions: Figure 4.7 Comparisons between model predictions and the experimental data reported by [START_REF] Shaw | Thermomechanical aspects of NiTi[END_REF] at different loading rates: (a) stress-strain response, (b) temperature variation.

Figure 4.7 compares the model predictions with the experimental data reported by [START_REF] Shaw | Thermomechanical aspects of NiTi[END_REF] at three different strain rates of 0.0004s -1 , 0.004s -1 and 0.04s -1 . In particular, Figure 4.7(a) illustrates the pseudoelastic stress-strain response and Figure 4.7(b) the variation of temperature with the strain. The pseudoelastic stress-strain response significantly depends on temperature and therefore is conjointly determined by the energy released or absorbed due to latent heat and intrinsic dissipation as well as the heat transfer by convection into surrounding medium. At low strain rate (0.0004s -1 ), the rate of heat transfer by convection into the surrounding medium is sufficient to evacuate dissipative and latent phase transformation heat resulting in minor variation in temperature. This variation increases at moderate strain rate (0.004s -1 ) for which the released heat is not convected into the surrounding medium timely, resulting in increased hysteresis. In this particular case, the finish temperature for forward phase transformation increased by 7 K and for reverse phase transformation decreased by 12 K with respect to the initial temperature of 343 K. Finally, at high strain rate (0.04s -1 ), the material behavior approaches approximately adiabatic conditions, with temperature increasing to 368 K upon loading then decreasing to 340 K after unloading. The rate of the released energy becomes dominant in this case, resulting in increased temperature and therefore higher thermodynamic stability of the austenite phase. As a result, it becomes increasingly difficult for forward transformation to proceed whereas the reverse transformation becomes easier. A manifestation of this effect is the increased slope of the stress-strain plateaus corresponding to forward and reverse transformations.

Simulation of a SMA helical spring actuator

To demonstrate the usefulness of the proposed model in analyzing complex structural SMA components subjected to non-trivial thermomechanical loading conditions, a SMA helical spring actuator is simulated in this section. The simulation is carried out considering separate loading cases to illustrate stressinduced pseudoelasticity and temperature-induced actuation. Previous work on the simulation of SMA Finite-strain cyclic behavior of SMAs following form: (5.34) where * ∈ {e, θ}, ψ * , A and ψ * , M denote the free energies of austenite and martensite phases, respectively. By assuming isotropic and homogeneous elasticity in austenite and martensite, a Reuss rheological model [START_REF] Moumni | Theoretical and numerical modeling of solid-solid phase change: Application to the description of the thermomechanical behavior of shape memory alloys[END_REF][START_REF] Auricchio | Theoretical and numerical modeling of shape memory alloys accounting for multiple phase transformations and martensite reorientation[END_REF] is considered to formulate the elastic energy ψ e for the phase mixture:

where δ = tr(h) = δ e + δ θ is the volumetric component of h, θ 0 is the reference temperature, α is the thermal expansion coefficient. The bulk modulus K is assumed to be equal for the two phases and the shear modulus µ is determined from the Reuss model [START_REF] Wagner | Lattice stability, elastic constants and macroscopic moduli of NiTi martensites from first principles[END_REF][START_REF] Auricchio | Theoretical and numerical modeling of shape memory alloys accounting for multiple phase transformations and martensite reorientation[END_REF] as follows:

(5.36)

The thermal energy ψ θ is sometimes referred to as chemical energy [START_REF] Sedlák | Thermomechanical model for NiTi-based shape memory alloys including R-phase and material anisotropy under multi-axial loadings[END_REF][START_REF] Auricchio | Theoretical and numerical modeling of shape memory alloys accounting for multiple phase transformations and martensite reorientation[END_REF], which is related to the changes in internal energy and entropy. The thermal energies in equation ( 5.34) 2 for the two different phases, ψ θ, * , are defined as

where * denotes either of the phases A and M, e 0 and η 0 are reference internal energy and entropy at reference temperature θ 0 , c v is the specific heat capacity. Substituting equation (5.37) into equation (5.34) 2 gives the following thermal energy:

where

are the differences of entropy and heat capacity between austenite and martensite.

The interaction energy ψ int represents the crystal boundary energy between austenite and martensite phases, which is often derived from micro-mechanical or crystallographic considerations and finally expressed as a function of internal state variables [START_REF] Zaki | A three-dimensional model of the thermomechanical behavior of shape memory alloys[END_REF][START_REF] Auricchio | Theoretical and numerical modeling of shape memory alloys accounting for multiple phase transformations and martensite reorientation[END_REF]. In the present work, the interaction energy ψ int is assumed to depend on the volume fraction of martensite χ, transformation strain h t and residual strain h r as (5.39) where µ t and µ r are transformation and residual hardening moduli. The first term G t on the right-hand side of equation ( 5.39) represents the interaction energy due to the coexistence of austenite and martensite with

where γ N , γ r and γ t are nonnegative multipliers, the sign function S(•) is used to extract the sign of A χ .

The rate of residual deformation γ r is determined by the rate of cumulated martensite volume fraction χ c as

where R sat denotes the saturated residual strain in the stabilized hysteresis loop. According to the experimental data reported by [START_REF] Kan | Experimental observations on rate-dependent cyclic deformation of super-elastic NiTi shape memory alloy[END_REF], R sat has an approximately linear relation with logarithmic strain rate, and therefore is defined in present work as

where h eq = 2 3 h : h is the equivalent strain rate and log h eq denotes the logarithmic strain rate; R c , R 0 and R s are model parameters that control the change of R sat with strain rate when h eq is greater than the threshold value h th .

Moreover, using equations (5.51) 1 and (A.8), with the relation ∥N t ∥ = 0, allows writing the Lagrangian multiplier ζ t as ζ t = Me : N t -µ t H χ.

(5.55)

Substituting equation ( 5.55) into equation ( 5.52) 1 , A N is written as (5.56) where I is the fourth-order identity tensor, and A N represents the component normal to N t of the Mandel stress Me . Then, the associated yield functions to describe phase transformation and martensitic reorientation evolutions are given by Appendix A

Mathematical supplementation

Corotational rates

The corotational rate of a symmetric Eulerian tensor A is defined by

where Q is a time dependent orthogonal tensor denoting the rotation of the frame, A is the material time derivative and Ω = QQ T is an antisymmetric tensor denoting the frame-spin. [START_REF] Xiao | On objective corotational rates and their defining spin tensors[END_REF] showed that the spin tensor Ω in equation (A.1) can be written in the following general form:

where λ i and P i are the distinct eigenvalues and eigenprojections and I = trb the first invariant of the left Cauchy-Green deformation tensor b, I = trb, h is a continuous antisymmetric function denoting the particular corotational objective rate. Moreover, for the Eulerian Hencky strain measure h, a logarithmic spin tensor is considered that has the following expression:

which yields the following relation between the logarithmic corotational rate of Eulerian Hencky strain hL and the rate of deformation tensor D:

Mathematical supplementation [START_REF] Khan | Continuum theory of plasticity[END_REF] proposed that the corotational integration for any spatial tensor A can, in general, be written

where ∫ C and ∫ m denote, respectively, the corotational integration and the material integration. When the spatial tensor A itself is a corotational rate, i.e., the logarithmic corotational rate hL , substituting equation (A.4) into equation (A.5) gives

where ∫ LC denotes the logarithmic corotational integration. The integration scheme (A.6) indicates that the logarithmic corotational integration of the rate of deformation tensor D is the Eulerian Hencky strain h.

By using the property of the skew symmetric tensor Ω = -Ω T , the inner product of Å and an arbitrary second-order symmetric tensor B is written Å : B = A : B + 2Ω : AB.

(A.7)

If A and B are coaxial, AB is also symmetric, indicating that the inner product of the symmetric tensor AB and the skew tensor Ω becomes null. Therefore, the following identity is obtained:

Å : B = A : B, (A.8) when the symmetric tensors A and B are coaxial for any arbitrary choice of the spin tensor Ω.

Tensor exponential and logarithm

For a generic tensor X, the tensor exponential function can be explicitly expressed by means of its series expansion as exp (X) = (A.13)