Finite Strain Shape Memory Alloys Modeling - Scuola di Dottorato in ...

University of CassinoFaculty of EngineeringGraduate School in Civil EngineeringXXI CycleCassino, November 2008Finite Strain Shape Memory Alloys ModelingVeronica EvangelistaSupervisor: Prof. Elio SaccoTutor: Prof. Sonia MarfiaCoordinator: Prof.ssa Maura Imbimbo

AcknowledgementsIt is really difficult to express in few words my deep gratitude to my Ph.D.supervisor, Professor Elio Sacco, for his many suggestions and constant supportduring this research. He completely put at my disposal his knowledge; thank of him,I have had, during the last three years, the opportunity to participate to severalconferences meeting interesting people. With his enthusiasm, his inspiration and hisbrilliant ideas, he let me know how beautiful is the world of the research, teachingme the way to approach any kind of problem. I would have been lost without hishelp.I wish to thank Professor Sonia Marfia for the fruitful and stimulatingdiscussions. She helped the progress of my work. She could not even realize howmuch I have learned from her.I am really indebted to Professor Maura Imbimbo. Besides of being an excellentcoordinator, she is a really helpful and reliable person. She addressed me toward thebeautiful field of the scientific research. I am really glad that I have come to getknow her in my life. I always will be grateful to her for the important advices,constant encouragement and beautiful affective words she told me during this period.I wish also to thank Professor Raimondo Luciano for his disposal and helpfulsuggestions.I would like to express my deep and sincere gratitude to my friends andcolleagues Ernesto Grande and Maria Ricamato, who share with me thisexperience.I would like to remember all my friends for their friendship whenever I needed it.Special gratitude is for my family for providing me a loving environment. Mymother Marisa, my father Aldo, my brother Alcide and my sister Paola alwaysencouraged and supported me in these years.Finally, I wish to deeply thank my dear husband Sandro. His love has been for me apowerful source of inspiration and energy.

Contents1. INTRODUCTION ..................................................................................................11.1. Motivations of the Research and Outline of the Thesis ....................................31.2. Organization of the Dissertation .......................................................................52. SHAPE MEMORY ALLOYS................................................................................72.1. Introduction.......................................................................................................72.2. SMA Characterization.....................................................................................142.3. Biocompatibility of Shape Memory Alloys Ni-Ti ..........................................172.4. Applications of Shape Memory Alloys...........................................................172.4.1. Cardiovascular Applications.................................................................182.4.2. Orthodontic Applications......................................................................192.4.3. Orthopedic Applications .......................................................................202.4.4. Applications to Surgical Instruments....................................................202.4.5. Mechanical and Structural Applications...............................................212.5. Constitutive Models ........................................................................................223. Review of Some Basic Results in Continuum Mechanics:FINITE DEFORMATION.......................................................................................263.1. Introduction.....................................................................................................263.2.Governing Equations........................................................................................273.2.1. Kinematics and Deformations...............................................................273.2.1.1. Deformation Gradient .................................................................293.2.1.2. Material Strain Tensors...............................................................313.2.1.3. Spatial Strain Tensors .................................................................323.2.2. Push-Forward, Pull-Back Operations ...................................................333.2.3. Polar Decomposition.............................................................................353.3. Velocity and Material Time Derivatives.........................................................393.3.1. Velocity.................................................................................................39

3.3.2. Material Time Derivatives ....................................................................403.3.3. Rate of Deformation Tensors................................................................413.3.4. Stress Measures for Reference and Deformed State.............................433.3.5. Equation of Motion ...............................................................................463.3.6. Objectivity.............................................................................................483.3.6.1. Objective Stress Rate ..................................................................493.4. Constitutive equations-hyperelastic materials ................................................523.4.1. Hyperelasticity ......................................................................................523.4.2. Isotropic Hyperelasticity-Material Description ....................................554. A FINITE STRAIN CONSTITUTIVE MODEL...............................................564.1. Introduction.....................................................................................................564.2. Constitutive Model..........................................................................................584.2.1. Thermodynamic Framework.................................................................604.2.1.1. Elastic Free Energy.....................................................................614.2.1.2. Transformation Free Energy.......................................................624.2.2. Clausius-Duhem Inequality...................................................................654.2.3. Constitutive Equations ..........................................................................664.2.4. Pseudo-Potential of Dissipation and Evolution Law ............................684.2.5. Final Format of the Constitutive Model................................................735. FINITE STRAIN CONSTITUTIVE MODEL: NumericalProcedure...................................................................................................................775.1. Introduction.....................................................................................................775.1.1.Exponential Map....................................................................................775.1.2. Time Integration of the Constitutive Model .........................................805.1.3. Solution Algorithm ...............................................................................815.1.4. Newton-Raphson Method .....................................................................825.2. Principle of Virtual Work ...............................................................................855.2.1.Finite Element Implementation..............................................................885.2.1.1. Reference Configuration Formulation ........................................88

5.2.1.2. Finite Element Approximation...........................................................906. LINEARIZATION OF THE FINITE DEFORMATIONSMA MODEL ...........................................................................................................946.1. Introduction.....................................................................................................946.1.1. Linearized Deformation Gradient .........................................................946.1.2. Linearized Strain Measures...................................................................956.1.3. Linearized Volume Change ..................................................................966.1.4. Linearized Elastic Free Energy.............................................................976.1.5. Linearized Transformation Free Energy...............................................986.2. 3D Constitutive Model for Stress-Temperature Inducedsolid Phase Transformation in a Small strain Regime...............................996.3. Numerical Procedure.....................................................................................1066.3.1. Introduction.........................................................................................1066.3.1.1. Time Integration of the 3D SMA ConstitutiveModel ............................................................................................................1076.3.1.2. Newton-Raphson Method .........................................................1106.3.1.3. Consistent Tangent Matrix........................................................1127. 3D-1D Constitutive Model with 1D Evolution .................................................1157.1. Introduction...................................................................................................1157.1.1. Constitutive Model..............................................................................1157.2. 1D Model and Analysis of Model Parameters ..............................................1197.3. Numerical Procedure.....................................................................................1257.3.1. Time Integration of the 3D-1D SMA ConstitutiveModel ............................................................................................................1257.3.2. Consistent Tangent Modulus ..............................................................1287.3.3. Beam Finite Element...........................................................................1308. Numerical Results...............................................................................................1348.1. Introduction...................................................................................................134

8.1.1. Comparison between Large and Small DeformationModels...........................................................................................................1348.1.1.1. Uniaxial Response-Superelastic Effect.....................................1348.1.1.2. Shear Response-Superelastic Effect .........................................1368.1.1.3. Superelastic and Shape-Memory Effects..................................1398.1.2. Comparison between 3D and 3D-1D ConstitutiveModels...........................................................................................................1438.1.2.1. Uniaxial Response ....................................................................1438.1.2.2. Pure Bending Response ............................................................1478.1.2.3. Tension-Compression Test .......................................................1508.1.3. Experimental Comparison...................................................................154CONCLUSIONS .....................................................................................................158REFERENCES........................................................................................................161

1 INTRODUCTIONScience and technology have made amazing developments in the design ofelectronics and machinery using standard materials, which do not have particularlyspecial properties, i.e. steel, aluminium and gold. However, during the last decade,there was a great interest in advanced materials called ‘smart materials’. They haveone or more properties that can be significantly changed in a controlled fashion byexternal stimuli, such as stress, temperature, electric or magnetic fields. Theseinclude piezoelectric materials, magneto-rheostatic materials, electro-rheostaticmaterials and shape-memory alloys. For example, when a piezoelectric material isdeformed, it gives off a small but measurable electrical discharge. Alternately, whenan electrical current is passed through a piezoelectric material it experiences asignificant increase in size, up to a 4% change in volume. Piezoelectric materials aremost widely used as sensors in different environments. They are often used tomeasure fluid compositions, fluid density, fluid viscosity, or the force of an impact(airbag sensor). Electro-rheostatic (ER) and magneto-rheostatic (MR) materials arefluids which can experience a dramatic change in their viscosity. These fluids canchange from a thick fluid, similar to motor oil, to nearly a solid substance within thespace of a millisecond when exposed to a magnetic or electric field. MR fluidsexperience a viscosity change when exposed to a magnetic field, while ER fluidsexperience similar changes in an electric field. MR are being developed for use in carshocks, damping machine vibration, etc., while ER fluids have mainly beendeveloped for use in clutches and valves, as well as engine mounts designed toreduce noise and vibration in vehicles.In a different way, in shape-memory alloys, an input of thermal energy, which canalso be produced through resistance to an electrical current, alters the microstructurethrough a crystalline phase change. This change enables multiple shapes inrelationship to the environmental stimulus.1

As shape-memory alloys show some surprising features not present in materialstraditionally used in engineering, they are the basis for innovative applications.Shape-memory technologies are nowadays exploited in a variety of differentapplicative context ranging from sensors and actuators, to robotics, to dampingdevices. Recent experimental and numerical investigations have also shown thepossibility of using such materials in the area of seismic resistant design. Inparticular, they seem to be effective in improving the response of buildings andbridges under earthquake-induced vibrations. However, the largest commercialsuccess of shape-memory materials is related to some applications in the biomedicalfield. In particular, they are successfully employed in orthodontics (archwires),orthopedics, medical instruments, minimal invasive surgery technology (grippers,cutters), drug delivery systems and vascular scaffolding (cardiovascular stents, aorticaneurysm).A review of the available literature and personal contacts in the industry shows adiffusion of computational tools to support the design process of shape-memoryalloydevices. This is also due to the fact that conventional inelastic models do notprovide an adequate framework able to represent the unusual macrobehavior ofshape-memory materials.The present dissertation focuses on a new family of inelastic models, based on aninternal-variable formalism and known as generalized plasticity (Lubliner et al.,1993; Lubliner and Auricchio, 1996). This theory can be used to describe solid-solidphase transformations and, therefore, it is adopted herein as the framework for thedevelopment of one- and three-dimensional thermomechanical constitutive modelsfor shape-memory materials.The proposed constitutive models reproduce the basic features of shape-memoryalloys, such as superelasticity, the shape-memory effect together with a differentmaterial behavior in tension and compression.2

The implementations of the model in a finite-element scheme and the form of thealgorithmically consistent tangent are discussed in detail. The cases of large- andsmall-deformation regimes are considered.Based on the overall results, it appears that the proposed approach is a viable basisfor the development of an effective computational tool to be used in the simulation ofshape-memory alloys devices behavior.1.1 Motivations of the Research and Outline of the ThesisThe first reported steps towards the discovery of the shape memory effect were takenin the 1930s. According to Otsuka and Wayman (1998), A. Olander discovered thepseudoelastic behavior of the Au-Cd alloy in 1932. Greninger and Mooradian in1938 observed the formation and disappearance of a martensitic phase by decreasingand increasing the temperature of a Cu-Zn alloy. The basic phenomena of the shapememoryeffect governed by the thermoelastic behavior of the martensite phase waswidely reported a decade later by Kurdjumov and Khandros in 1949 and also byChang and Read in 1951. In the early 1960s Buehler and Wiley at the U.S. NavalOrdinance Laboratory discovered the shape memory effect in an equiatomic alloy ofnichel and titanium, which can be considered a breakthrough in the field of shapememorymaterials. This alloy with a composition of 53 to 57 % nickel by weight wasnamed Nitinol (Nichel-Titanium Naval Ordnance Laboratory). It exhibited an usualeffect: severely deformed, specimens of the alloy, with residual strains of 8-15 %,regained their original shape after a thermal cycle. This effect became known as theshape-memory effect, and the alloys exhibiting it were named shape-memory alloys.It was later found that not only other materials show the shape-memory property, butthat at sufficiently high temperatures such materials also possess the property ofsuperelasticity, that is the recovery of large deformations during mechanical loadingunloadingcycles performed at constant temperature.3

As a consequence of superelastic and shape-memory behaviors, shape-memoryalloys lend themselves to innovative applications in different fields of science andengineering. Recently proposed designes based on such materials range from selfexpandingmicro-structures for the treatment of body vessel occlusions to devices forthe control of space structures such as antennas and satellites.The first efforts to exploit the potential of NiTi as an implant material were made byJohnson and Alicandri in 1968. The use of NiTi for medical applications was firstreported in the 1970s. In the early 1980s the idea attained more support, and someorthodontic and mainly experimental orthopaedic applications were released. It wasonly in the mid-1990s, however, that the first widespread commercial stentapplications made their breakthrough in medicine. The use of NiTi as biomaterial isfascinating because of its superelasticity and shape-memory effect, which arecompletely new properties compared to the usual metal alloys.The conventional models of inelastic behavior, such as classical plasticity, do notprovide an adequate framework for representing superelastic and shape-memorybehaviors.Based on some pioneering work by Philips and collaborators, over the past twodecades a new family of inelastic models has been developed, that allows thedescription of features that cannot be represented by classical plasticity, though itincludes classical plasticity as a special case. It has been called generalized plasticity.As has recently been discussed, the numerical implementation of models belongingto this family is straightforward. Consequently generalized plasticity appears to be aviable and flexible environment for the development of constitutive materials withcomplex behavior.The present PhD thesis is aimed to derive and to develop a computational tool for theanalysis of shape-memory alloys through an exploration of the applicability ofgeneralized plasticity to the representation of their behavior. Specifically, someconstitutive models that reproduce both superelasticity and shape-memory effect aredeveloped and numerically implemented into a finite-element setting. Some4

applications are presented in order to show the viability of the proposed approach asan effective tool for the design of devices based on shape-memory alloys.1.2 Organization of the dissertationThe dissertation is organized as follows:• Chapter 2 focuses on shape-memory alloys. In particular, it presents a surveyof SMA-based applications explaining the great diffusion and interest of suchmaterials.• Chapter 3 review the basic equations of continuum mechanics in a largedeformation regime.• Chapter 4 describes a finite strain constitutive model for shape-memoryalloys able to reproduce both the superelastic effect and the shape-memoryeffect. It is developed within the framework of Generalized StandardMaterials and adopting the approach of thermodynamics of irreversibleprocesses.• Chapter 5 describes the time-discrete counterpart of the finite strainconstitutive model and its algorithmic implementation. The weak formulationof momentum balance equations is presented paying particular attention in itsformulation in the reference configuration.• Chapter 6 discusses the linearization of the equations governing the finitestrain model in order to enable the development of a constitutive model in theframework of the infinitesimal theory.• Chapter 7 introduces some assumptions in order to simplify the full 3D modelin a three-dimensional model governed by 1D evolutionary equations.Furthermore, an analysis of the material parameters on which the SMAconstitutive model depends is carried out.5

• Chapter 8 presents some numerical tests to check the proposed algorithmsand verify the efficiency of the constitutive models to simulate the realbehavior of shape-memory alloys.6

2 SHAPE MEMORY ALLOYS2.1 IntroductionShape memory alloys are materials that ‘remember’ their geometry. After a sampleof SMA has been deformed from its original crystallographic configuration withresidual strains up to 15%, it regains its original geometry by itself during heating(one-way effect) or, at higher ambient temperature, simply during unloading(pseudo-elasticity or superelasticity). These extraordinary properties rely on a stresstemperatureinduced athermal diffusionless thermoelastic martensitic transformationon an atomic scale between two solid phases: the austenite (A), characterized by anhigh symmetric crystallographic configuration, and the martensite (M), characterizedby a low symmetric crystallographic configuration. The austenite crystal structure isa simple body-centered cubic structure, while martensite is characterized by a morecomplex trigonal structure. Recent studies have shown that, depending on specificconditions, some SMA can present another crystallographic phase known as R-phase. The R-phase transformation can appear before the martensitic transformationaccording to the following sequence: austenite→R-phase → martensite. The crystalstructure of the R-phase is rhombohedric.The thermoelastic martensitic transformation causing the shape recovery is a result ofthe need of the crystal lattice structure to accommodate to the minimum energy statefor a given temperature (Otsuka and Wayman, 1998). There are many differentconfigurations that a crystal lattice structure can assume in the martensitic phase, butthere is only one possible configuration or orientation in the austenite state. In otherwords, the martensite can be present in different but crystallographically equivalentforms (variants). In particular, if there is no preferred direction along which themartensite variants tend to align, then multiple variants are formed. The twinned7

martensite has 24 variants, i.e. 24 subtypes with different crystallographicorientations. Instead, if there is a preferred direction for the formation of themartensitic phase, just one (single) variant is formed (see Fig. 2.2 Fig. 2.). When thematerial is in its martensitic form, it is soft and ductile and can be easily deformed(somewhat like soft pewter). Stress-induced martensite NiTi, one of the most usedshape memory alloys, is highly elastic (rubber-like), while austenite NiTi is quitestrong and hard (similar to titanium). The NiTi material has all these properties andtheir specific expression depend on the temperature in which it is used.8

transformation (M→A), indicated respectively asAsandAf. Clearly, the martensiteis stable at temperatures belowMf, while the austenite is stable at temperaturesaboveAf, with MfAf< .Composition and metallurgic treatments can have dramatic impacts on the abovetransition temperatures. The values of these temperatures vary in a nearly linear waywith the tensional state and with the entity and the length of the thermal treatments.Furthermore, the condition of the activation of the transition phase depends on thetemperature (T ); experimentally, it has been shown that in an uniaxial stresstemperaturediagram (Fig. 2.1), within the range of many applications, the regions inwhich the phase transformations may occur are delimited with good approximationby straight lines. The regions in which fractions are stable are indicated by S forsingle-variant martensite, by M for multiple-variant martensite and by A foraustenite. Phase transition occurs within the zones of the phase diagram marked ingrey while, outside of the phase transformation zones, the volume fractions of thedifferent phases are constant and their values depend upon the previous loadinghistory of the material.At the macroscopic level, depending on the temperature, SMA show two differentbehaviors, the superelasticity and the shape-memory effect, briefly explained in thefollowing:Superelasticity (Fig. 2.2). Consider a specimen in the austenitic state and at atemperature greater thanAf; accordingly, at a zero stress level only the austenite isstable. If the speciemen is loaded, while keeping the temperature constant, thematerial presents a non-linear behavior (ABC) due to a stress-induced conversion ofaustenite into single variant martensite. Upon unloading, keeping the temperatureconstant, a reverse transformation from single-variant martensite to austenite occurs(CDA) as a result of the instability of the martensite at a zero stress. As aconsequence, at the end of the loading-unloading process no permanent strains arepresent and the stress-strain path is a closed hysteresis loop.10

Shape-memory effect or one way shape-memory effect (Fig. 2.3). Consider aspecimen in the multi-variant martensite state and a temperature lower thanMf;accordingly, at zero stress only the martensite is stable in a multi-variantcomposition. During the loading phase, the material shows a non-linear response(AB) due to a stress-induced conversion of the multi-variant martensite into a singlevariantmartensite. During the unloading phase (BC), residual deformations show up(AC). However, the residual, apparently inelastic, strain may be recovered (shaperecovery) by heating the material to a temperature aboveA f, thus inducing atemperature-driven conversion of martensite into austenite. Finally, upon cooling theaustenite is converted back into multi-variant martensite.SAσ t,fMSβ tσ tASβ tAMMAM fM s A s A fσ cβ cσ c,fβ cFig. 2.1: Phase transformation zones11

Fig. 2.2: SuperelasticityFig. 2.3: Shape memory effectThe zero stress transformation temperatures,12Mf,Ms,Asandtransformation lines βtand βcand the critical stresses for detwinningand σc,fare all material parameters to be determined experimentally.Af, the slopes of theσ , σcσt,t,f

Ni-Ti 49/51 at % Ni -50 to 110 30Fe-Pt approx. 25 at % approx. -130 4PtMn-Cu 5/35 at % Cu -250 to 180 25Fe-Mn-Si 32 wt % Mn, 6 -200 to 150 100wt % SiTab. 2.1: Shape-memory alloys propertiesFusion temperature, °C 1300Densità, g/cm 3 6.45Austenite approx. 100Resistivity,micro-ohms•cmMartensite approx. 70Austenite18Thermical conductivity,W•cm•°CMartensite8.5Fusion latent calor, KJ/Kg•atoms 167Austenite approx. 83Young modulus, GPaMartensiteapprox. 28Austenite 195 to 690Yield Strength, MPaMartensite70 to 140Ultimate Tensile Strength, MPa 895Corrosion resistenceSimilar to Ti based alloys orto inox steel series 300Temperature of transformation, °C -200 to 110Shape Memory Strain8.5% maxPoisson ratio 0.33Tab. 2.2: Properties of Ni-Ti based shape-memory alloys2.2 SMA CharacterizationIn general, in order to describe the SMA behavior in its transformation range, somemethods accounting for the variation of temperature can be used.14

The Differential Scanning Calorimetry (DSC) measures the energy quantity adsorbedor regained by a sample, not stressed, when it is heated or cooled in itstransformation range. The exothermic and endothermic higher and lower values,corresponding to the energy variations, can be easily evaluated and show a clearphysical meaning. A result of DSC test is shown in Fig. 2.4, with the indication ofthe transformation temperatures.Fig. 2.4: DSC curve of a Ni-Ti alloyAnother technique is based on resistivity measures on the sample heated and cooled;using this procedure, significant variations of resistivity up to 20% are noted near thetransformation range.In order to define the mechanical behavior of SMA, the most effective method isthrough the application of load and temperature cycles; in particular, the sample,subjected to a constant value of load and to a temperature cycle, describes atransformation cycle in which the strains in both the directions are measured. Thecurve illustrated in Fig. 2.5 is a direct result of this test.15

Fig. 2.5: Transformation diagram obtained at constant value of stress varying the temperatureAnother test, able to define the transformation temperatures, is the so-called “ActiveA f ” test (or “Water Bath” or “Alcohol Bath Test” ): In particular, a sample of thealloy is subjected to a bending test at a temperature belowMs; then, it is heated andthe shape recovery is measured. As a consequence, a diagram as the one reported inFig. 2.6 is obtained. It can be pointed out that the bend angle regains its zero valuewhen the temperature reaches the A f value.Fig. 2.6: ActiveAfdiagram16

2.3 Biocompatibility of Shape-Memory Alloys-Ni-TiBiocompatibility is a crucial factor for the use of SMA devices in the human body; itrepresents the ability of a material to remain biologically innocuous during itsfunctional period inside a living creature. A biocompatible material must not produceany allergic reactions inside the host and, also, it mustn’t release ions into thebloodstream (Shabalovskaya, 1995).Generally, the biocompatibility of a material is strongly related to allergic reactionsbetween the material surface and the inflammatory response of the host. Manydifferent aspects can contribute to these reactions such as patient’s characteristics(health, age immunological state, and so on), and material characteristics (rugosityand porosity of the surface and individual toxic effects of the elements present in thematerial).Several investigations have been conducted in order to establish the biocompatibilityof Ni-Ti-based alloys, and to exclude intrinsic hazards involved in their applications.The analysis of aspects related to the biocompatibility of these alloys is performed byassessing each of their elements, nickel and titanium, separately.In particular, nickel, although necessary to life, is a highly poisonous element. Unlikenickel, titanium and its compounds are highly biocompatible; moreover, due to theirmechanical properties, they are usually employed in orthodontic and orthopaedicimplants. The oxidation reaction of titanium produces an innocuous layer of TiO 2which sorrounds the sample. This layer is responsible for the high resistance tocorrosion of titanium alloys, and the fact that they are harmless to the human body.In general, the properties of titanium confer good biocompatibility of Ni-Ti alloys.2.4 Applications of Shape Memory AlloysShape-memory alloys have unique properties which are not present in the materialstraditionally used in engineering applications. Accordingly, their use introduces new17

design capabilities, which make it possible to improve performance as well as topropose innovative solutions in different fields of human knowledge. Although firstdiscovered in the 1960s, shape memory alloys have found applications only in thepast 15-20 years. The high cost, lack of clear understanding of the thermomechanicalprocessing, and the inability to reliably predict their behavior are thereason for the SMA slow introduction into applications. Higher quality andreliability, coupled with a significant reduction in price has recently led to numerousapplications of shape-memory alloys in the biomedical, commercial and aerospaceindustries.The present paragraph reviews applications exploiting either the superelasticbehavior or the shape-memory effect.Superelastic-based applications take advantage of one of the following features:• the possibility of recovering large deformations (up to strain of 8-15%);• the existence of a transformation stress plateau, which guarantees constantstress over non-negligible strain intervals.Driven by a search for devices that could result in less invasive medical procedures(Machado and Savi, 2003), researchers have found numerous applications for SMAin the medical field.2.4.1 Cardiovascular ApplicationsStent is an important cardiovascular application that is used to maintain the innerdiameter of a blood vessel (Duerig, Pelton and Stockel, 1999). Stent (Fig. 2.8) is thetechnical word indicating self-expanding micro-structures, which are currentlyinvestigated for the treatment of hollow-organ or duct-system occlusions. The stent isinitially stretched out to reach a small profile, which facilitates a safe and a-traumaticinsertion of the stent. After being released from the delivery system, the stent, due tothe body temperature, self-expands to over twice its compressed diameter and exertsa nearly constant, gentle, radial force on the vessel wall. This device can be used not18

only in the angioplasty procedure, in order to prevent another obstruction of a vessel,but also in the treatment of aneurysms for the support of a weakened vessel.Fig. 2.8: Shape memory self-expanding stents2.4.2 Orthodontic ApplicationsDuring orthodontic therapy tooth movement is obtained through a bone remodellingprocess, resulting from forces applied to the dentition. Such forces are usuallycreated by elastically deforming an orthodontic wire and allowing its stored energyto be released to the dentition over a period of time. The optimal tooth movement isachieved by applying forces that are low in magnitude and continuous in time. Incontrast, high magnitude forces may cause irreversible tissue damage, while forcesdiscontinuos in time may lead to an erratic tooth movement.19

2.4.3 Orthopedic ApplicationsAnother area for interesting and commercially viable applications is orthopedics. Thehealing of fractured bones proceeds more rapidly if the fractured faces are understeady compressive stress. During surgery to reset fractures, shape-memory alloyplates can be attached on both sides of the fracture, producing the necessarycompressive stress in the fracture gap by exploiting the shape-memory effect(Duerig, Pelton and Stockel, 1996).Orthopedic treatment also exploits the properties of SMA in the physiotherapy ofsemi-standstill muscles adopting some special gloves (Fig. 2.9) composed of shapememorywires on regions of the fingers. These wires reproduce the activity of handmuscles, promoting the original hand motion. In particular, when the glove is heated,the length of the wires is shortened. On the other hand, when the glove is cooled, thewires return to their former shape, opening the hand. As a result, semi-standstillmuscles are exercised.AFig. 2.9: Shape memory alloy glove. A, Low temperature position. B, High temperature positionB2.4.4 Applications to Surgical InstrumentsIn recent years, medicine and the medical industry have focused on the concept ofless invasive surgical procedures. Shape-memory effects have been exploited formedical applications, such as intracranial aneurysm clips, artificial hearts,20

micropumps used to unblock blood vessel during angioplasty. These devices have aSMA tube whose diameter is reduced compared to polymer materials due to theirpseudoelastic effect. Moreover, it also allows greater flexibility and torsionresistance when compared to the same tube made of stainless steel (Lagoudas,Rediniotis and Khan, 1999). The SMA basket is used to remove kidney, bladder andbile duct stones. Fig. 2.10 presents a sequence of pictures related to the basketopening as it is heated.In general, for medical applications, only Ni-Ti alloys are used, due to their goodcorrosion resistance and biocompatibility.Fig. 2.10: Sequence of opening of the shape memory basket2.4.5 Mechanical and Structural ApplicationsIn the aerospace industry, shape-memory alloys have been used in adaptive aircraftwings and smart helicopter blades for increased efficiency and reduced noise andvibration.Recent years have seen numerous commercial and consumer applications as for eyeglassframes or cellular telephone antennas. Recently, there has been increasinginterest in using superelastic shape-memory alloys for applications in seismicresistant design of structures. for example as seismic dampers.21

2.5 Constitutive ModelsDue to the special properties of these active materials, new constitutive relationshipswhich are able to describe the macroscopic behavior resulting from their internalconstitution are required. Accordingly, during the last decade the area of constitutivemodelling of polycrystalline shape memory alloys has been the topic of manyresearch publications. Due to the novelty as well as to the complexity of theoccurring phenomena, most of the initial experimental investigations were referred tosimple stress states as tension or torsion and, as a natural consequence, the initialmodelling efforts were mostly directed toward the development of simple onedimensionalconstitutive equations. More recently, the increasing sophisticationachieved in SMA production technologies, combined with a deeper knowledge of themicro and macro-scale phenomena, has opened the possibility of performingcomplex experimental investigations, such as combined tension-torsion or generalnon-proportional loading tests. The majority of the published material models can beclassified into three different groups: micromechanically based approaches, conceptsbased on statistical thermodynamics and phenomenological models.The micromechanically based models for polycrystalline shape memory alloys haveits origin in models for single-crystal shape memory alloys. These single-crystalmodels are mainly derived by investigating the kinematics of the martensite phasetransformation in crystal lattices. The mechanism for martensitic phasetransformation are divided into homogeneous plane invariant strain, macroscopicshape deformation produced by heterogeneous martensitic transformation and latticeinvariant plastic slip (Patoor et al, 2006). These models are based on the descriptionof effects occurring at the micro-scale level, such as nucleation or interface motion.In general, most of the models deal with a single interface even if real transitionprocesses occur in a very high number of nucleation sites. Therefore, a micro-levelapproach is in general more suited for the development of a qualitative than for aquantitative description of macroscopic effects. Most of the models are based on the22

introduction of a multi-well free-energy function, of a nucleation criterium based onan energy barrier concept and of an interface kinetic law, based on a thermalactivation theory. By means of homogenisation techniques which consider geometriccompatibility at the grain boundaries, the single-crystal modelling can be transferredto polycrystal shape memory alloys (Huang and Brinson, 1998). In this way,averaging the investigated phenomena over a Reference Volume Element (RVE) themacroscopic constitutive laws are obtained.The second approach relies on the fundamental concept of equilibriumthermodynamics. Hence, the influence of interfacial energy on the phase boundariesis investigated and the phase equilibrium is determined by the minimisation of thetotal free energy (Liang and Rogers, 1990; Liang and Rogers, 1992; Boyd andLagoudas; 1996, Brinson, 1993; Brinson et al. 1996; Raniecki and Lexcellent, 1998;Govindjee et al., 2003). The latter concepts have the advantage that the microstructureof the material, such as habit planes, martensitic variants, etc., isincorporated. However, the identification of micromechanical parameters cannot becompletely avoided which is a rather difficult and elaborate task. Further thecomputational cost is in general quite high. The models are based on the introductionof an Helmholtz free energy as the sum of a chemical energy variation associated tothe phase transition, a surface energy associated to the presence of an interfacebetween the martensite and the austenite and a mechanical energy.On the other hand, phenomenological or macro-level models take into considerationonly the macroscopic behavior of the material; they have the advantage that theirmaterial parameters can be usually identified by classical experimental tests and thestructure of their materials equations is mostly well suited to be implemented intocomputer programs for structural analyses as finite element programs (Auricchio andLubliner, 1997; Auricchio and Sacco; 1997, Souza et al., 1998; Auricchio and Sacco,1999; Qidwai and Lagoudas, 2000; Auricchio and Sacco 2001; Auricchio andPetrini, 2002, 2004; Auricchio et al., 2003; Auricchio et al. 2008).23

Probably the first example along these lines is the 1D model of Tanaka (Tanaka,1986) who introduces an explicit exponential relation between a single scalar internalvariable, i.e. the martensitic fraction, and the external control variables, i.e. stress andtemperature.Most of the models proposed in research publications are limited to small strainsregime (Liang and Rogers, 1986, 1990; Brandon and Rogers, 1992; Liang andRogers, 1992; Tanaka et al., 1992; Brinson, 1993; Barret 1995; Barret and Sullivan1995; Boyd and Lagoudas, 1996; Brinson et al., 1996; Auricchio and Lubliner, 1996;Auricchio and Sacco, 1997; Raniecki and Lexcellent, 1994, 1998; Souza et al., 1998;Brinson et al., 1998; Bo and Lagoudas, 1999a, 1999b, 1999c; Auricchio and Sacco,2001; Auricchio and Petrini, 2002, 2004; Brinson et al., 2007; Auricchio et al. 1997).Large strain models to describe shape memory alloys behaviors have been derived byAuricchio and Taylor (1997, 1999), Auricchio (2001), Thamburaja and Anand(2001), Reese and Christ (2008), Christ and Reese (2008). Auricchio and Taylor(1997) and Auricchio (2001) proposed a phenomelogical model including largedeformations. They introduced a macroscopic free energy function depending oninternal variables which describe the state of the phase transition. Thistransformation coincides with a martensitic volume fraction where only one singlephase variant is considered. The activation of the transition is ruled by a Druger-Prager flow criterion known from plasticity in soil and concrete materials. On theother hand, the micromechanically based model of Thamburaja and Anand is able todescribe the effect of superelasticity. It is based on crystal plasticity whereas thepolycrystalline structure of SMA is approximated by the Taylor model. Finally,Reese and Christ (2008) presented new concepts for the modeling of the superelasticeffect in the finite strains regime. They assume that the deformation gradient can besplit into three parts: an elastic deformation gradient, a second part accounting for thedeformation occurring during the phase transition and a third one correlating with theenergy dissipation. Moreover, they suggest a numerical algorithm which uses the24

spectral decomposition allowing the numerical computation of the model variables ina closed form.In the present contribution the phenomenological approach is preferred.Characteristic for such concepts is a macroscopic free energy function orthermodynamic potential that depends on internal variables which describe the stateof the phase transformation and can be related to the actual microscopic materialcharacteristics. These variables account for the material history which, in dissipativephenomena, influences the material current behaviour (Germain et al., 1983). Asuccessful phenomenological model is required to be thermodynamically consistentor, in other words, the entropy inequalities constitutes a constraint to the admissibleconstitutive laws for real materials (Trusdell and Toupin, 1960; Trusdell and Noll,1992; Suquet 1995; Lemaitre and Chaboche, 1994). The constitutive modeling ofshape memory alloys is treated in the present work within the framework of finiteand infinitesimal strain regimes.25

3 Review of Some Basic Results in ContinuumMechanics: FINITE DEFORMATION3.1 IntroductionConsider a continuum body B with particle P ∈ B which is embedded in the threedimensionalEuclidean space at a given instant of time t . As the continuum body Bmoves in space from one instant of time to another, it occupies a continuoussequence of geometrical regions or configurations. In particular, it is necessary todistinguish between the reference configuration Ω where the initial shape of thebody to be analyzed is known and the current or deformed configuration ω reachedafter loading is applied. Fig. 3.1 shows the two configurations and the coordinateframes which will be used to describe each one. If quantities used to characterize thebehavior of the body whose motion is under consideration are described in terms ofwhere the body was before deformation the description is called material orLagrangian; if the same quantities are described in terms of where the body is duringdeformation the description is called spatial or Eulerian (Bonet and Wood, 1997).The deformed configuration of the body is unknown at the start of the analysis and,therefore, must be determined as a part of the solution process that is inherently nonlinear.26

Fig. 3.1: Reference and deformed configuration for finite strain problemThe chapter starts by describing the basic kinematics relations used in finitedeformation solid mechanics. This is followed by a summary of different stressmeasures related to the reference and deformed configurations and a description ofthe constitutive equations of hyperelastic materials in the framework of the finitedeformation.3.2 Governing Equations3.2.1 Kinematics and DeformationThe basic equations for finite deformation solid mechanics may be found in standardreferences on the subject (Mandel, 1974; Simo and Hughes, 1998).A body B is constituted by material points whose positions are given by the vectorX in a fixed reference configuration. In Cartesian coordinates the position vector isdescribed in terms of its components asX= X E ; I = 1,2,3(3.1)II27

whereEIare unit orthogonal base vectors and summation convention is used forrepeated indices of the same type (e.g. I). The position vector x in the currentconfiguration ω is given in terms of its Cartesian components asx= xe ; i = 1,2,3(3.2)iiwhere eIare unit base vectors for the current time and again summation conventionis used.The position vector at the current time is related to the position vector in thereference configuration through the mappingx = φ ( X , t)(3.3)i i IWhen common origins and directions for the coordinate frames are used, adisplacement vector may be introduced as the change between the two frames.Accordingly,x = δ ( X + U )(3.4)i iI I IwhereδiIis a shifter between the two coordinate frames defined by a Kroneckerdelta quantity such thatδiI⎧1 if i = I= ⎨⎩0 if i ≠ I(3.5)The shifter satisfies the relations28

δ δ = δ and δ δ = δ(3.6)iI iJ IJ iI jI ijwhereδIJand δ ijare Kronecker delta quantities in the reference and currentconfiguration, respectively. Using the shifter, a displacement component may bewritten with respect to either the reference configuration or the current configurationand related throughu = δ U and U = δ u(3.7)i iI I I iI i3.2.1.1 Deformation GradientA fundamental measure of deformation is described by the deformation gradientrelative toXI(Holzapfel, 2000) that enables the relative spatial position of twoneighboring particles after deformation to be described in terms of their relativematerial position before deformation. It is given byFiI∂x∂φi∂X∂Xi= =II(3.8)The requirement that during deformation there is not material penetration isexpressed by the assumption that the mapping φ is a one-to-one function. Inparticular, det F represents, locally, the volume after deformation per unit volume inthe reference configuration; it is therefore reasonable to assume that det F ≠ 0 .Furthermore, a deformation with det F < 0 cannot be reached by a continuousprocess starting in the reference configuration; that is, by a continuous one-parameterfamilyφξ(0≤ξ≤ 1) of deformations with φ 0the identity, φ 1= φ , and det ∇ φξ29

never zero. Indeed, since det ∇ is strictly positive at ξ = 0 , it must be strictlypositive for all ξ . It follows that:φξJ = det F > 0(3.9)The deformation gradient is a direct measure which maps a differential line elementin the reference configuration into one in the current configuration as∂φdx = dX = F dXii I iI I∂XI(3.10)Thus, it may be used to determine the change in length and direction of a differentialline element. The determinant of the deformation gradient also maps a differentialvolume element in the reference configuration into one in the current configuration,that isdv= JdV(3.11)where dV is a differential volume element in the reference configuration and dv itscorresponding form in the current configuration.In general, F has nine components for all t .The deformation gradient may be expressed in terms of the displacement asF∂u= δ + = δ + uiiI iI iI i,I∂XI(3.12)30

and it is a two-point tensor since it is referred to both the reference and the currentconfigurations. Expanding the terms in (3.12), the deformation gradient componentsare given by( 1+)⎡F11 F12 F13⎤ ⎡ u1,1 u1,2 u1,3⎢FiI=⎢F21 F22 F⎥⎢23⎥= ⎢ u2,1 ( 1+u2,2 ) u2,3⎢F F F ⎥ ⎢⎣u u + u( 1 )⎣ 31 32 33 ⎦ ⎢ 3,1 3,2 3,3 ⎥⎤⎥⎥⎥⎦(3.13)The direct use of FiIcomplicates the development of the constitutive equations and,as a consequence, it is common to introduce deformation measures which arecompletely related to either the reference or the current configurations.3.2.1.2 Material Strain TensorsFor the reference configuration, the right Cauchy-Green deformation tensor, C , isintroduced asTC= F F or C = F F(3.14)IJ iI iJNote that the tensor C is symmetric and positive definite at eachX ∈Ω. Thus,T( ) and 0 for all 0T T TC= F F= F F = C u⋅ Cu> u ≠ (3.15)Consequently, given the nine components F iI, it is easy to compute the sixcomponents CIJ = CJIvia (3.14); otherwise, given CIJit is impossible to computethe nine components FiI. With definition (3.14) and equation (3.9), it follows that:31

( ) 2 2Jdet C= det F = > 0(3.16)Alternatively the symmetric Green-Lagrange strain tensor, E , given as1 1E= ( C− 1 ) or EIJ = ( CIJ −δIJ)(3.17)2 2may be used.The Green-Lagrange strain tensor components may be expressed in terms of thedisplacements gradient asE1 ⎡ ∂ui ∂ui ∂ui ∂u⎤i= ⎢δ+ δ + ⎥2 ⎣ ∂X J∂XI ∂XI ∂XJ⎦1= ⎡δiIuiJ ,δiJuiI ,uiI ,uiJ,2⎣ + + ⎤⎦1= ⎡UI, J+ UJ, I+UK, IUK,J⎤2⎣⎦IJ iI iJ(3.18)3.2.1.3 Spatial Strain TensorsIn the current configuration, a common deformation measure is the left Cauchy-Green deformation tensor or Finger deformation tensor, b , expressed asTb= FF or b = F F(3.19)ij iI jIThe tensor b is symmetric and definite positive at eachx ∈ω,T( ) and 0 for all 0T T Tb= FF = FF = b u⋅ bu> u ≠ (3.20)32

Furthermore,( ) 2 2Jdet b= det F = > 0(3.21)The Euler-Almansi strain tensor, e , is related to the inverse of b as1 −1 1−1e= ( 1− b ) or eij = ( δij −bij)(3.22)2 2or inverting by−1−( 2 ) or b ( δ 2e) 1b= 1− e = −(3.23)ij ij ij3.2.2 Push-Forward, Pull-Back OperationAs already seen, vector and tensor quantities may be resolved along triads of basisvectors belonging to either the reference or the current configurations. Additionally,there are two-point tensors which are associated with both configurations, oneexample being the deformation gradient. The transformations between material andspatial quantities are typically called a push-forward operation and a pull-backoperation and are denoted by χ* (•)and χ − 1( )*• , respectively.In particular, a push-forward is an operation which transforms a vector or tensorquantity based on the reference configuration to the current configuration. Since theEuler-Almansi strain tensor e is defined with respect to spatial coordinates, it can becomputed as a push-forward of the Green-Lagrange strain tensor E , which is givenin terms of material coordinates. From equation (3.22), it can be concluded that:33

1 ⎡1⎤e= ( 1− F F ) = F ( )2 ⎢F 1−F F F2⎥F⎣⎦−T ⎡1 (T ⎤ )−1 −T−1= F⎢F F− 1 =⎣2⎥F F EF⎦= χ*( E)−T −1 −T T −T−1 −1(3.24)A pull-back is an inverse operation, which transforms a vector or tensor quantitybased on the current configuration to the reference configuration. Similarly to theabove, the pull-back of e is1 ⎡1= ( − ) = ( )2 ⎢−⎣2T T −T T−1E F F 1 F F F F 1 F FT ⎡1 (−T −1⎤ )T= F⎢1− F F =⎣2⎥F F eF⎦= χ−1*( e)⎤⎥⎦(3.25)As can be seen from the previous equations, the transformations are based onmultiplications by the deformation gradient in one of the following forms: F ,−1F ,T −TF , F . The form of the deformation gradient necessary to take depends on thetensor to be transformed. In particular, the push-forward and pull-back operations oncovariant second-order tensors, such as E , C , e ,−1b , are according toχT( ) ( ) 1 1T• = − • − , χ−(•) = (•)F F F F (3.26)* *However, the push-forward and pull-back operations on controvariant second-ordertensors, such as−1C , b and most of the common stress tensors, are according to1 1(•) = (•) T , (•) = (•)χ χ − − −T*F F*F F (3.27)34

3.2.3 Polar DecompositionThe deformation gradient tensor F (Simo and Hughes, 1998) discussed in theprevious section transforms a material vector X into the corresponding spatial vectorx . The crucial role of F is further disclosed in terms of its decomposition intostretch and rotation components. The use of the physical terminology stretch androtation will become clearer later. For the moment, from a purely mathematical pointof view, the tensor F is expressed as the product of a rotation tensor R times astretch tensor U to define the polar decomposition (Fig. 3.2) asF = RU (3.28)For the purpose of evaluating these tensors, recall the definition of the right Cauchy-Green tensor C asT T TC= F F=U R RU (3.29)Given that R is an orthogonal rotation tensor that isTRR= 1, and choosing U to bea symmetric tensor, gives a unique definition of the material stretch tensor U interms of C asU 2 = UU=C (3.30)In order to actually obtain U from this equation, it is first necessary to evaluate theprincipal directions of C , denoted here by the eigenvector triad { N1 N2 N3}and2their corresponding eigenvalues λ1,2λ2andthe form of its spectral decomposition as:352λ3, which enable C to be expressed in

3∑C= λ N ⊗N (3.31)α = 12α α αwhere, because of the symmetry of C , the triad { N1 N2 N3}are orthogonal unitvectors. Combining (3.30) and (3.31), the spectral form of the material stretch tensorU can be easily obtained as3∑U= λ N ⊗N (3.32)α = 1α α αOnce the stretch tensor U is known, the rotation tensor R can be evaluated fromequation (3.28) as−1R = FU .In terms of this polar decomposition, typical material and spatial elementar vectorsare related asdx= FdX=R( UdX )(3.33)In the above equation, the material vector dX is first stretched to giveUdX andthen rotated to the spatial configuration by R . Note that U is a material tensorwhereas R transforms material vectors into spatial vectors and is therefore, like F , atwo point tensor.It is also possible to decompose F in terms of the same rotation tensor R followedby a stretch in the spatial configuration (Fig. 3.2) asF = VR (3.34)36

which can now be interpreted as first rotating the material vector dX to the spatialconfiguration, where it is then stretched to give dx asdx= FdX=V( RdX )(3.35)where the spatial stretch tensor V can be obtained in terms of U by combiningequations (3.28) and (3.34) to giveTV = RUR (3.36)The right (or material) and left (or spatial) stretch tensors U and V measure localstretching or contraction along their mutually orthogonal eigenvectors, that is achange of local shape. Additionally, recalling equation (3.19) for the left Cauchy-Green or Finger tensor b givesT2( )( )Tb= FF = VR R V = V (3.37)Consequently, if the principal directions of b are given by the orthogonal spatialvectors { n1 n2 n3}with associated eigenvalues2λ1,2λ2anddecomposition of the spatial stretch tensor can be expressed as2λ3, then the spectral3∑V = λ n ⊗n (3.38)α = 1α α αSubstituting equation (3.32) for U into expression (3.36) for V gives V in terms ofthe vector triad in the undeformed configuration as37

3∑α = 1( ) ( )V = λ RN ⊗ RN (3.39)α α αComparing this expression with equation (3.38) and noting that ( RNα ) remain unitvectors, it must follow thatλ = λ ; n = RN ; α = 1,2,3(3.40)α α α αThis equation implies that the two-point tensor R rotates the material vector triad{ N1 N2 N3}into the spatial triad { 1 2 3}n n n . The rotation tensor measures thelocal rotation, that is a change of local orientation. Furthermore, the uniqueeigenvalues2λ1,2λ2and λ 2 3are the squares of the stretches in the principaldirections, that is they express the ratio between current and initial lengths of vectors.It is convenient to express the deformation gradient tensor in terms of the principalstretches and principal directions. To this end, substitute equation (3.9) for U intoequation (3.28) for F and use (3.40) to give3∑F= λ n ⊗N (3.41)α = 1α α αThis expression clearly reveals the two-point nature of the deformation gradienttensor in that it involves both the eigenvectors in the initial and final configurations.38

Fig. 3.2: Representation of the polar decomposition of the deformation gradient3.3 Velocity and Material Time Derivatives3.3.1 VelocityObviously, many processes are time-dependant; consequently, it is necessary toconsider velocity and material time derivatives of various quantities. However, evenif the process is not rate-dependant, it is nevertheless convenient to establish theequilibrium equations in terms of virtual velocities and associated virtual timedependantquantities (Bonet and Wood, 1997). For this purpose, consider the usualmotion of the body given by equation (3.3), from which the material velocity of aparticle is defined as the time derivative of φ as39

∂φ( X, t)VX ( , t)=∂t(3.42)Observe that the velocity is a spatial vector despite the fact that the equation has beenexpressed in terms of the material coordinates of the particle X . In fact, by inverting(3.3) the velocity can be more consistently expressed as a function of the spatialposition x and time as1VX ( , t) = v( φ − ( x, t), t) = vx ( , t)(3.43)Relationship (3.43) can be written in compact form as:V= v v=V1φor φ −(3.44)where denotes composition.3.3.2 Material Time DerivativeGiven a general scalar or tensor quantity g , expressed in terms of the materialcoordinates X , the time derivative of g( X ,t)denoted by g( X,t) is defined as( ,t)∂g Xg =(3.45)∂tThis expression measures the change in g associated with a specific particle initiallylocated at X , and it is known as the material time derivative of g . The spatialquantities are expressed as functions of the spatial position x , in this case thematerial derivative is more complicated to establish. The complication arises40

ecause, as time progresses, the specific particle being considered changes its spatialposition. Consequently, the material time derivative in this case is obtained from acareful consideration of the motion of the particle as( φ( , t+Δ t), t+Δt) − ( φ( , t), t)g X g Xg ( x, t)= lim(3.46)Δ→ t 0ΔtThis equation clearly illustrates that g changes in time as a result of a change in timebut with the particle remaining in the same spatial position and because of the changein spatial position of the specific particle. Using the chain rule, (3.46) gives thematerial derivative of g ( x ,t)as( , t) ( , t) φ ( , t) ( , t)∂g x ∂g x ∂ X ∂g xg ( x,t)= + = + ( ∇g)v (3.47)∂t ∂x∂t ∂tThe second term, involving the particle velocity in equation (3.47) is often referred toas the connective derivative.3.3.3 Rate of Deformation TensorsVelocity has been expressed as a function of the spatial coordinates as v( x ,t). Thederivative of this expression with respect to the spatial coordinate defines thevelocity gradient tensor l as( )∂v x,tl = =∇v∂x(3.48)To obtain an alternative expression for l , it must be observed that41

d ⎛ ∂φ⎞ ∂ ⎛∂φ⎞F = ⎜ ⎟= ⎜ ⎟=∇0v(3.49)dt ⎝∂X⎠ ∂X⎝ ∂t⎠It follows that∂v∂v∂φF = = = lF∂X ∂x ∂X(3.50)from whichl =−1FF (3.51)The symmetric part of l , denoted by d , is called the spatial rate of deformationtensor, and its skew-symmetric part is called spin, or vorticity, tensor, denoted by w .Thusd 1T= ⎡∇ +∇ ⎤2 ⎣ v v ⎦(3.52)andw 1T= ⎡∇ −∇ ⎤2 ⎣ v v ⎦(3.53)By (3.51) and (3.14)C T T T T T= F F + F F = F ⎡⎣∇ v +∇ v ⎤⎦ F = 2 F dF(3.54)42

It can be introduced the rotated rate of deformation tensor by the expression:TD=R dR (3.55)Using the polar decomposition F=from (3.54) that:RU and the symmetryUT =U, it can be derivedTherefore,( T)C = 2U R dR U=2UDU(3.56)1 11 − −2 2D=C CC (3.57)2Finally, a skew-symmetric tensor that gives the rate of changes of the rotation tensorcan be introduced as:⎛ ⎞∂RTTΩ = ⎜⇒ + =⎜⎝ ∂t⎟R Ω Ω 0 (3.58)⎠3.3.4 Stress Measures for Reference and Deformed StatesConsider a deformable continuum body B occupying an arbitrary region ω of thephysical space with boundary surface ∂ ω at time t .Arbitrary forces (external forces) act on parts or the whole of the boundary surface,and on an (imaginary) surface within the interior of that body (internal forces) insome distributed manner (Holzapfel, 2000).Let the body now be cut by a plane surface which passes through any given pointx ∈ ω . The plane surface separates the deformable body into two portions. Attention43

is focused on that part of the body lying on the tail of a unit vector n directed alongthe outward normal to an infinitesimal spatial surface element ds ∈ dω . Since theinteraction of the two portions is considered, forces are transmitted across theinternal plane surface. An infinitesimal resultant (actual) force acting on a surfaceelement is denoted as df .Initially, before motion occurred, the continuum body was in the referenceconfiguration. The quantities x , ds and n which are associated with the currentconfiguration of the body are denoted by X , dS and N , respectively, when they arereferred to the reference configuration. According to Fig. 3.3, for every surfaceelement, the following relations are introduced:df = tds= TdS(3.59)( ,, t ), ( ,, t )t= t x n T= T X N(3.60)Fig. 3.3: Quantities used in the definition of the stress measuresThe quantity t represents the Cauchy or true traction vector acting on ds withoutward normal n . The vector T represents the first Piola-Kirchhoff or nominal44

traction vector. For the Cauchy’s stress theorem, there exist unique second-ordertensor fields σ and P so that:t( x,, t n) σ( x,t)n( ,, t ) ( , t)⎧⎪ =⎨⎪⎪⎩TX N=PX N(3.61)As well as the Cauchy stress σ , also the Kirchhoff stress τ is introduced. It is asymmetric measure of stress defined with respect to the current configuration. Theyare related through the determinant of the deformation gradient asτ = Jσ τ = Jσ(3.62)orij ijand, often, they are the stresses used to define general constitutive equations formaterials. Notationally, the first stress subscript defines the direction of the force andthe second the direction of a normal to the area on which the force acts. The secondPiola-Kirchhoff stress S is a symmetric stress measure with respect to the referenceconfiguration and is related to the Kirchhoff stress through the deformation gradientasTτ = FSF or τ = FS F(3.63)ij iI IJ jJFurthermore, the unsymmetric first Piola-Kirchhoff stress P is related to S throughP= FS =(3.64)or PiI FiJ SJIand to the Kirchhoff stress by45

Tτ = PF or τ = PF(3.65)ij iI jIRelative to the rotated configuration, the rotated stress tensor can be defined asTΣ = R τR or Σ = R τ R(3.66)IJ iI ij jJAll of the stress tensors introduced above are conjugate to associated rate ofdeformation tensors through the following important stress-power relationships:1τijdij = PiI F iI=ΣIJDIJ = SIJCIJ2(3.67)or, in direct notation,1τ : d= P: F = Σ : D=S:C (3.68)2The double contraction of a stress tensor and of the associated rate of deformationtensor describes the real physical power during a dynamical process, i.e. the rate ofinternal mechanical work per unit reference volume.3.3.5 Equations of motioni. Lagrangian description:In order to derive the differential static equilibrium equations, consider the spatialconfiguration of a general deformable body B with boundary ∂B . The body isassumed to be under the action of body forces B per unit volume and traction forces46

Nt per unit area acting on the boundary. The deformation is prescribed on∂ B ⊂∂B as:φφ= φ (prescribed)on ∂ B (3.69)φand the nominal traction vector t N is prescribed on the part of the boundary∂ B ⊂∂B , with unit normal ˆN as:tNˆˆˆ N(prescribed) on tt = PN= t ∂ B (3.70)Assuming that∂ B ∩∂ B=∅and∂ B ∪∂ B=∂B , the local equations of motiontφtφtake the form:DIV P+ ρ0B=ρ0 A, in B (3.71)where ρ : → 0B is the reference density and A the material acceleration definedas the time derivative of the material velocity. Here,∂PP = (3.72)a ∂ XaA( DIV )Aii. Eulerian description:The counterpart of equation (3.71) in the Eulerian description takes the followingform:47( )divσ + ρ b=ρ a, inφB (3.73)0 0

⎛ρ⎞0 1where ρ= ⎜φ⎜⎝ −J ⎟⎠ and 1b=B φ − represent the density and the body force perunit of volume in the current placement, respectively. In a Cartesian coordinatesystem,∂σσ = (3.74)a ∂ xab( div )b3.3.6 ObjectivityAn important concept in solid mechanics is the notion of objectivity. This conceptcan be explored by studying the effect of a rigid body motion superimposed on thedeformed configuration. From the point of view of an observer attached to androtating with the body, many quantities describing the behavior of the solid willremain unchanged. Such quantities, like for example the distance between any twoparticles and, among others, the state of stress in the body, are said to be objective.Although the intrinsic nature of these quantities remains unchanged, their spatialdescription may change. To express these concepts in a mathematical framework,consider an elemental vector dX in the initial configuration that deforms to dx andis subsequently rotated to dx . The relationship between these elemental vectors isgiven asdx = Qdx=QFdX(3.75)where Q ( t)is a proper orthogonal transformation depending only on time anddescribing the superimposed rigid body rotation. Although the vector dx is differentfrom dx , their magnitudes are obviously equal. In this sense it can be said that dx isobjective under rigid body motion. This definition is extended to any vector that48

transforms according to a= Qa. Velocity is an example of a non-objective vectorbecause differentiating the rotated mapping φ = Qφwith respect to time gives,v ∂ φ ∂φ = = + φ∂tQ ∂tQ (3.76)Obviously, the magnitudes of v and v are not equal as a result of the presence of theterm Q φ , which violates the objectivity criteria.3.3.6.1 Objective stress ratesAssuming that the Cauchy stress tensor is objective, its material time derivativedefined as⎡ ∂ ⎤1σ = ( σφ) φ −(3.77)⎢ ⎣∂t⎥ ⎦is not objective. In fact, applying the chain rule, it follows that:⎡∂σ⎛ ⎞ φ⎤φ ∂σ∂σ φ φt x ⎟⎢ ⎜⎣∂ ⎝∂ ⎠ ∂t⎦⎥⎡∂σ−1⎤= +∇σ( V φ)⎢⎣∂t⎥⎦−1= + =(3.78)It can be concluded that:⎡∂σ⎤σ = + v⋅∇σ(3.79)⎢⎣∂t⎥⎦49

or, in components:⎡⎢∂σ⎣∂σab abab= + vc(3.80)⎢ ∂t∂x⎥cσ⎤⎥⎦Next, assume that σ transforms objectively, that is:Tσ = Q() t σQ () t(3.81)From equations (3.79) and (3.81), it follows that:σ T T T= Q () t σQ () t + ⎡ () t () t ⎤ − ⎡ () t () t ⎤⎢⎣ Q Q ⎥⎦ σ σ ⎢⎣ Q Q ⎥(3.82)⎦that is the material time derivative of the Cauchy stress tensor is not objective.Objective rates are essentially modified time derivatives of the Cauchy stress tensorconstructed in order to preserve objectivity.It is possible to show that any possible objective stress rate is a particular case offundamental geometric object known as Lie derivative (Simo and Marsden, 1984).• The Lie derivative of the Kirchhoff stress tensor or Trusdell stress rate isdefined as:⎧∂⎫Lvτ = ⎨⎪F ⎡ ( φ)⎤ ⎪ φt⎢⎣F τ F ⎥⎦F ⎬⎪⎩∂⎪⎭⎧⎪ ∂ST⎫⎪ −1= ⎨FF ⎬φ⎪⎩∂t⎪⎭−1 −TT −1(3.83)Using the expression for the derivative of the inverse:50

∂∂t=−∂F∂t−1 −1 −1F F F (3.84)along which the definition of material time derivative and spatial velocity gradient:v( ) ( ) TL τ= τ − ∇v τ−τ ∇v(3.85)• The Jaumann-Zaremba stress rate of the Kirchhoff stress is essentially acorotated derivative relative to spatial axes with instantaneous velocity givenby the spin tensor:τ= τ − wτ+τw(3.86)• The Green-McInnis-Naghdi stress rate of the Kirchhoff stress is defined by anexpression similar to (3.83), but with F replaced by R : ⎧∂⎫τ = ⎨⎪R ⎡ ( φ)⎤ ⎪ φt⎢⎣R τ R ⎥⎦R ⎬⎪⎩∂⎪⎭⎪⎧∂T⎫−1= ⎨R [ Σ φ]R ⎪⎬φ⎪⎩∂t⎪⎭−1 −TT −1(3.87)R Ω R ,Recalling that = ( φ)τ= τ − Ωτ+τΩ(3.88)51

3.4 Constitutive Equations - Hyperelastic MaterialStresses result from the deformation of the material, and it is necessary to expressthem in terms of some measure of this deformation such as, for instance, the strain.These relationship, known as constitutive equations, obviously depend on the type ofmaterial under consideration and may be dependant or independent on time. Forexample, the behavior of viscous materials is clearly entirely dependent on strainrate. Generally, constitutive equations must satisfy certain physical principles. Theequations must obviously be objective, that is, frame invariant. The constitutiveequations herein described are established for hyperelastic material, whereby stressesare derived from a stored elastic energy function. Although there are a number ofalternative material descriptions that could be introduced, hyperelasticity is aparticularly convenient constitutive equation because of its simplicity and, as aconsequence, it constitutes the basis for more complex material models such aselastoplasticity.3.4.1 HyperelasticityMaterials for which the constitutive behavior is only a function of the current state ofdeformation are generally known as elastic. A so-called perfectly elastic material isby definition a material which produces locally no entropy (Trusdell and Noll, 1992).In other words, the term ‘perfectly’ is used for a certain class of materials which hasthe special merits that for every admissible process the internal dissipationDintiszero (naturally, damage, viscous mechanisms and plastic deformations are excluded).Using as strain measure the deformation gradient F associated with a particle X inthe current configuration and its conjugate first Piola-Kirchhoff stress measure P thebasic material elastic relationship is given byP=P( F( X), X )(3.89)52

where the direct dependency upon X allows the possible inhomogeneity of thematerial.In the special case when the work done by the stresses during a deformation processis dependent only on the initial state at time t 0and the final configuration at time t ,the behavior of the material is said to be path-independent and the material is termedhyperelastic or Green-elastic. As a consequence of the path-independent behavior aHelmholtz stored energy function or elastic potential W per unit undeformed volumecan be established from which the first Piola-Kirchhoff stress is computed using∂W ( FX ( ), X)PFX ( ( ), X)=∂F(3.90)Because of the restrictions imposed by the objectivity, the stored energy functionremain invariant when the current configuration undergoes a rigid body rotation.This implies that W depends on F only via the stretch component U and it isindependent of the rotation component R . For convenience, however, W is oftenexpressed as a function of2 TC= U = F F asW( FX ( ), X) = W ( CX ( ), X)(3.91)Observing that 1 C= E is work conjugate to the second Piola-Kirchhoff stress S2enables a totally Lagrangian constitutive equation to be constructed in the samemanner as equation (3.90) to give( ( ), ) 2 ∂W= =∂WSCX X∂C∂E(3.92)53

The strain-energy function vanishes in the reference configuration, i.e. where F=1.This assumption is expressed by the normalization condition:W = W() 1 = 0(3.93)From the physical observation that the strain-energy function must increase withdeformation, it therefore follows that:W = W( F ) ≥0(3.94)which restricts the ranges of admissible functions occurring in expressions for thestrain energy. The strain-energy function attains its global minimum for F=1 asthermodynamic equilibrium. Relations (3.93) and (3.94) ensure that the stress in thereference configuration, called residual stress, is zero. The reference configuration isstress-free. For the behavior at finite strains the scalar-valued function W mustadditionally satisfy so-called growth conditions. This implies that W tends to +∞ ifeitherJ = det F approaches +∞ or 0 + , i.e.⎧⎪W( F) →+∞ as detF→+∞⎨+⎪⎩W ( F) →+∞ as detF→0(3.95)Physically, that means that is required an infinite amount of strain energy in order toexpand a continuum body to the infinite range or to compress it to a point withvanishing volume.54

3.4.2 Isotropic Hyperelasticity – Material DescriptionThe hyperelastic constitutive equations discussed so far are unrestricted in theirapplication. In particular they are applied to the isotropic case. Isotropy is defined byrequiring the constitutive behavior to be identical in any material direction. Thisimplies that, for the representation theorem for invariants (Gurtin, 1981, Trusdell andNoll, 1992) the relationship between W and C must be independent of the materialaxes chosen and, consequently, W must only be a function of the invariants of C asW ( CX ( ), X) = W ( I , II , III , X)(3.96)C C Cwhere the invariants of C are defined here asIIICCIII= trC=C:1= trCC= C : CC= det C = J2(3.97)As a result of the isotropic restriction, the second Piola-Kirchhoff stress tensor can berewritten from equation (3.92) as∂W ∂W ∂I ∂W ∂II ∂W∂IIIS = 2 = 2 + 2 + 2∂C ∂I ∂C ∂II ∂C ∂III∂CC C CC C C(3.98)(3.99)Introducing the derivatives of the invariants with respect to C , enables the secondPiola-Kirchhoff stress to be evaluated as∂W ∂W ∂WS= 2 1+ 4 C+2JC∂I ∂II ∂III552 −1C C C(3.100)

4 A FINITE STRAIN CONSTITUTIVE MODEL4.1 IntroductionA new phenomenological model for shape memory alloys in the framework of finitestrain is proposed.The model formulation is motivated by the well-understood micromechanicalanalysis of single-crystal metal plasticity (Fig. 4.1).Fig. 4.1: Kinematic model of elastoplastic deformation of a single crystalMetals in their usual form are polycrystalline aggregates, that is, they are composedof large numbers of grains, each of which has the structure of a simple crystal(Lubliner, 1990; Lubliner, 1991). A crystal is a three-dimensional array of atomsforming a regular lattice. The atoms vibrate about fixed points in the lattice but donot move away from them, being held more or less in place by the forces exerted byneighboring atoms. Because of the random orientation of individual grains in atypical metallic body, the overall behavior of the aggregate is largely isotropic, but56

such phenomena as the Bauschinger effect and preferred orientation, which occur asa result of different plastic deformation of grains with different orientations,demonstrate the effect of crystal structure on plastic behavior. In single crystals forwhich crystallographic slip is assumed to be the only mechanism of plasticdeformation, the material flows through the lattice via dislocation motion, while thelattice itself, with the material embedded to it, undergoes elastic deformation androtation. If the discrete dislocation substructure is ignored, the plastic deformationcan be considered to occur in the form of smooth shearing on the slip planes and inthe slip directions. This continuum slip model from the pioneering work of Taylor(1938) was employed and further developed by Hill (1950), Hill and Rice (1972) andAsaro (1983). For a crystal with a single slip system denoted by { sm , } withms ⋅ =0 and m = s = 1, this micromechanical description can be illustrated as inFig. 4.1. The unit vectors { sm , } are attached to the lattice, and the plastic flow ischaracterized by the tensor Fpdefined as:F = 1+ γs⊗m (4.1)pwhere γ is the plastic shearing on the crystallographic slip system defined by{ sm , }. Furthermore, the total deformation of the crystal is decomposed as:F= FF(4.2)epwhereFpis the part due to the slip only that moves the material through the latticevia dislocation motion, while Ferotates and distorts the crystal lattice.57

4.2 Constitutive ModelThe micromechanical behavior of crystal metals is assumed as the basis for thederivation of a new constitutive model for shape memory alloys. Nevertheless, itmust be pointed out that the plastic mechanism of deformation simply based oncrystallographic slips of the atomic planes results really different from thediffusionless thermoelastic martensitic transformation governing the phase transitionin the shape memory alloys. However, as for the modeling of the crystal metalplasticity, the SMA constitutive model considers the assumption of the localmultiplicative decomposition of the deformation gradient in the form:FX ( , t) = F( X, t) F( X , t)(4.3)etfor each material pointX ∈Ω, the reference configuration of the body. Feis relatedto the deformation caused by stretching and rotation of the material, Ftaccounts forthe inelastic flow associated to the phase transition in the SMA and F is the totaldeformation gradient. The multiplicative split of the deformation gradient in the formpresented in expression (4.3) is consistent with a large part of the literature on finitedeformationinelastic models (Lee,1969; Mandel, 1974).This decomposition introduces the idea of an intermediate configuration which canbe considered as the one associated to the body when the loads are removed and thetemperature is reduced to the initial one, thus releasing the thermoelastic strains.As a consequence, the intermediate configuration is obtained from the currentconfiguration by elastic destressing to zero stress. It differs from the initialconfiguration by a residual or inelastic deformation and from the currentconfiguration by a reversible or elastic deformation. Accordingly, it is implicitlyassumed that the local intermediate configuration defined bydeformation gradients F eandF is stress-free. Theconfiguration is not unique; in fact, arbitrary local material rotations can be58−1eFpare not uniquely defined because the intermediate

superposed to intermediate configuration preserving it unstressed. However, thedecomposition can be made unique by additional specifications dictated by the natureof the considered material model.Fig. 4.2: Multiplicative decomposition of the deformation gradientThe model has been conceived within the framework of Generalized StandardMaterials (Halphen and Nguyen, 1975): internal variables are introduced in order todescribe the phase transition processes and two convex potentials are defined, fromwhich the constitutive relations and the evolution law for the transformation strainare derived.In general, the state of a continuum material depends on the whole history of itsmechanical variables, and the modeling of its behavior may be based on differentkinds of approaches. In order to obtain a formalism which is directly accessible to themethods of functional analysis, the approach of thermodynamics of irreversibleprocesses is adopted by introducing state variables (Coleman and Noll, 1963;Coleman and Gurtin, 1967). This approach was initiated first by chemists, and wasapplied to continuum mechanics by Eckart and Biot around 1950. Thethermodynamic potential allows to define associated variables chosen for the studyof the phenomenon. This naturally leads to the state laws. Furthermore, the59

thermodynamic state of each material medium at a given point and time instant iscompletely defined by the knowledge of the values of a certain number of variablesat that instant, which depend only upon the point under consideration. In particular,the model assumes the temperature T and the right Cauchy-Green tensor C ascontrol or observable variables (accessible to direct observation) and thetransformation right Cauchy-Green tensorCtdefined asC = F F (4.4)Tt t tas internal or history variable. The quantityCtplays the role of describing the strainassociated to the phase transformation and, in particular, to the conversion fromaustenite or multiple-variant martensite to single-variant martensite. Its value at eachinstant accounts for the past history of the material, from which depends its currentstate.4.2.1 Thermodynamical FrameworkBefore going any further, it is convenient to consider a mechanical system consistingof two isotropic elastic springs in series. The tensors F eand F tdenote thecorresponding deformation gradient tensors of the two springs, the deformationgradient tensor for the whole system being F .Once the state variables have been defined, the existence of a thermodynamicpotential from which the state laws can be derived is postulated. Since the specificfree energy of a spring is a function of its elongation, the Helmholtz free energyfunction Ψ for the polycrystalline SMA material can be introduced through aconvex potential as the sum of the energies of the two springs:( ) ( T)Ψ ( CC , , T) =Ψ C +Ψ C ,(4.5)t e e t t60

whereΨerepresents the elastic strain energy due to the thermo-elastic materialdeformations, whileΨtis the energy connected to the phase transformation.Certainly, the material behavior of shape memory alloys is strongly temperaturedependent so that in general the temperature T is considered as additionalindependent variable.4.2.1.1 Elastic Free EnergyShape memory alloys can be assumed as isotropic materials; it follows that therelationship between the elastic free energydefined as:Ψeand the elastic Cauchy-Green tensorC= F F(4.6)Te e emust be independent on the material axes chosen. In fact, the free energy mustremain constant under rigid body rotations and, as a consequence, it must only be afunction of the invariants ofThe elastic free energyCe.Ψeis assumed as:1⎡ ⎛λ + 2μ ⎞ 2 ⎛3λ+ 2μ ⎞ ⎛9λ+ 6μ⎞⎤Ψe= I1 I1 μI22⎢⎜ ⎟ − ⎜ ⎟ + + ⎜ ⎟4 2 4⎥⎣⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎦(4.7)where I1( )2 2= trC eand I = 1 tr 2( Ce) − I are the principal invariant of the elastic2 1ν EECauchy-Green tensor and λ =and μ = are the Lamè(1− 2 ν )(1 + ν )2(1 + ν )constants being E the Young modulus and ν the Poisson coefficient. Note that the61

last term of equation (4.7) ensures that the free energy is null in the undeformedstate, i.e. when C=1.4.2.1.2 Transformation Free EnergyAdopting the basic idea of Souza et al. (1998), in order to account for differentexperimental features of shape-memory alloys the transformation free energyevaluated as the sum of some contributes.In particular, denoting with E ( C 1)computed by means of the following relation:tΨtis= 1 2t− the Green-Lagrange strain, Ψ tisΨ ( E ) =Ψ ( E ) +Ψ ( E ) +Ψ ( E ) +I ( E )(4.8)t t ch t tr t id t ε L twhere:• Ψchis the chemical energy, due to the thermally-induced martensitictransformation. The phase transformation in the SMA depends on the temperatureand in particular, increasing the gap between the test temperature and thecharacteristic temperatureMf, the transition from austenite to martensite starts foran higher value of the stress. Furthermore, experimental evidences show that forT ≤ M fthe stress do not depend yet on the test temperature. For all these reasons,the chemical energy is assumed to be a positive and monotonically increasingfunction of the temperature. Introducing the quantity ω = tr ( E)and the function:h( ω)⎧1 ifω> 0= ⎨⎩0 ifω≤ 0(4.9)it can be expressed as:62

( )( ) E 1 ( )Ψ = h ω β T − M + −h ω β T − M E (4.10)ch t f t c f twhere the parameters βtandβc, linked to the dependence of the critical stress on thetemperature, are introduced as the thermomechanical behavior of a SMA materialresults strongly sensitive to the deformation mode. In particular, experimentalevidences show that the shape-memory materials, expecially NiTi and CuZnAlalloys, exhibit a tension-compression asymmetry in their phase transformation.In the chemical energy expression (4.11), ⋅ is the positive part of the argument,Mfis the finishing temperature of the austenite-martensite phase transformation andT is the room temperature; finally, ⋅ represents the norm of the argument;• Ψtrrepresents the transformation strain energy introduced in order to accountfor the transformation-induced hardening:1 2Ψtr= h Et(4.11)2where h is a material parameter defining the slope of the linear stress-transformationstrain relation in the uniaxial case and related to the hardening of the material duringthe phase transformation;▪Ψidis the free energy due to the change in temperature with respect to thereference state in an incompressible ideal solid:⎡T ⎤Ψid= ( uo − Tηo ) + c⎢( T −To) −Tln⎥⎣To⎦(4.12)63

it is a function of the heat capacity c , of the internal energy uoand of the entropy ηoat the reference state defined by the temperature T 0below which no twinnedmartensite is observed;• ( E )I represents an indicator function introduced in order to satisfy theεLtconstraint on the transformation Green-Lagrange strain tensor norm:⎧⎪0 if Et≤ εLIε ( E )L t= ⎨ (4.13)⎪⎩ +∞ if Et> εLwhere εLis linked to the maximum transformation strain reached at the end of thetransformation during an uniaxial test. In fact, the Green-Lagrange strain tensordescribes the strain associated to the phase transformation and, in particular, to theconversion from austenite or multiple-variant martensite to single-variant martensite.Accordingly, the norm of this quantity should be bounded between zero for the caseof a material without oriented martensite and a maximum value ε Lfor the case inwhich the material is fully transformed in single-variant oriented martensite.The subdifferential of the indicator function results:Et⎧0 if Et≤ εL⎪ +∂ Iε( E ) ifL t= ⎨R Et = εL(4.14)⎪⎩ ∅ if Et> εLTo sum up, the transformation energy can be expressed asΨ = −1+ + − +⎡− −T ⎤+⎣⎦*2tβ T MfEt h Et ( uo Tηo) c⎢( T To) Tln⎥ IεEL t2To( h( ) t ( h( ))c)*with 1β = ω β + − ω β64( )(4.15)

4.2.2 Clausius-Duhem InequalityIn order to obtain the state laws of the model, a new variable must be introduced: theentropy. This quantity, indicated as η , expresses a variation of energy associated to avariation in the temperature.At this point, the second principle of thermodynamics is taken into consideration. Itpostulates that the rate of entropy production is always greater than or equal to therate of heating divided by the temperature. Accordingly, the Clausius-Duhem lawtakes the following form:1 q−( Ψ+ ηT ) + S : C− ⋅gradT≥0(4.16)2 Twhere S denotes the second Piola-Kirchhoff stress tensor, the dot between tensors itsscalar product and q the heat flux vector. This inequality states that the totaldissipation, obtained as the sum of the mechanical dissipation, linked to the internalvariables evolution, and the thermal one, associated to the heat conduction, must begreater than zero in all real materials. The Clausius-Duhem law is widely used inmodern thermodynamic research initiated by Trusdell and Toupin (1960) and firstclearly studied in the work of Coleman and Noll (1963).Introducing equation (4.5) into equation (4.16) the following inequality is obtained∂Ψe∂Ψt∂Ψ1 q− : C e− : C t− T− ηT + S : C− ⋅gradT≥0(4.17)∂C∂C∂T2 TetThe deformation ratesC eandC tcan be rewritten in the format65

C =− l C + F CF −C l(4.18)T −T−1e t e t t e tC (4.19)2 Tt= Ft dtFtwherel= FF and dtis the symmetric deformation rate tensor defined as:−1t t t1Tdt = ⎡∇ vt +∇v⎤t2 ⎣ ⎦ (4.20)with the spatial velocity gradient∇vt:∇ v = FF −(4.21)1t t tAfter some algebric elaborations (kintzel and Basar, 2006, Rosati, 1999) andexploiting the symmetry of ∂Ψe ∂Ce, the Clausius-Duhem inequality (4.17) isfinally transformed into the relation∂Ψ ∂Ψ ∂Ψ ⎛ ∂Ψ ⎞ q( S − 2 F F ): C + (2C −2 F F ): d − η + T ⎜ ⎟ − ⋅gradT≥0(4.22)∂ ∂ ∂ ⎝ ∂T⎠ T−1 e −T1e t Tt t e t t tCe 2 Ce Ct4.2.3 Constitutive EquationsThe constitutive law is thermodynamically admissible if, during the phasetransformation evolution, the Clausius-Duhem inequality is guaranteed. In order toderive the state laws of the model independently, an elastic deformation taking placeat constant and uniform temperature, i.e. grad T = 0 and T = 0 , which does not alterthe transformation strain ( dt = 0) may be imagined to occur. Since the Clausius-Duhem inequality holds regardless of any particular C , it necessarily follows that:66

−1∂Ψe−TS− 2FtFt= 0∂Ce(4.23)Assuming this equality to hold, a thermal deformation in which d t= 0 andgrad T = 0 can be supposed. Then, since T is arbitrary, it follows that:η + ∂Ψ = 0∂T(4.24)The developed final form of the second principle (4.22) is sufficiently satisfied bythe following state relations:••−1∂ΨeS=2FtFt∂Ce−T(4.25)∂Ψη =− (4.26)∂ TThese expressions show that the second Piola-Kirchhoff stress tensor is the variableassociated to the elastic right Cauchy-Green tensor, while the entropy is the variableassociated to the temperature.In an analogous manner, the thermodynamic force associated with the internalvariable can be defined as:T et=2 C ∂Ψ ∂Ψe−2t t∂CF ∂CFetT(4.27)67

The tensor T represents the driving force for the phase transformation processes; thefirst term in equation (4.27) is the symmetric so-called Mandel stress tensor definedas∂ΨM = C∂ Cee2e(4.28)while the second term is the back-stress which defines the origin of the elasticdomain given by:∂Ψtα = 2FtFt∂CtT(4.29)The quantities S , η and T constitute the associated variables. The vector formed bythese variables is the gradient of the function Ψ in the space of the variablesandCt. This vector is normal to the surfaceΨ= constant .Ce, T4.2.4 Pseudo-Potential of Dissipation and Evolution LawThe thermodynamic potential allows to write relations between observable statevariables and associated variables (Suquet, 1995). However, it does not give anyinformation about the evolution of the internal variables. It is therefore necessary tointroduce kinetic laws in order to describe the dissipation process, mainly theevolution of the internal variables. A complementary formalism is needed. This isprecisely the objective of the dissipation potential.Taking into account the state laws, the Clausius-Duhem inequality can be reduced toexpress the fact that the dissipation is necessarily positive:68

qφ = T : dt− ⋅gradT≥0(4.30)TThe dissipation φ is the sum of the products of the force variables or dual variablesT andgrad T with the respective flux variables dtand− q . The first term:Tφ = T d(4.31)1:tis called the intrinsic or mechanical dissipation. It consists of the dissipationassociated with the evolution of the internal variables; it is generally dissipated bythe volume element in the form of heat. The second term:φ q grad2= − ⋅ T(4.32)Tis the thermal dissipation due to the conduction of heat.In order to define the complementary laws related to the dissipation process, theexistence of a dissipation potential expressed as a continuous and convex scalarvalued function of the flux variables is postulated:qϕ( dt, )(4.33)TThis potential is a positive convex function with a zero value at the origin of thespace of the flux variable. The complementary laws are then expressed by thenormality property:69

∂ϕ∂ϕT= gradT=−∂d∂t( q T )(4.34)The thermodynamic forces are the components of the vectorϕ = constant surface in the space of the flux variables.grad ϕ normal to theThe Legendre-Fenchel transformation enables to define the corresponding potential*ϕ , the dual of ϕ with respect to the variables dtand− q . By definition:T⎧*Tϕ ( , T ) sup ⎪⎛⎞ ⎫grad: qTgrad = ⎨ t− ⋅ −ϕ( t, ) ⎪T d q⎜⎬⎝T ⎟d (4.35)⎪⎩⎠ T⎪⎭⎛q ⎞⎜dt,⎜⎝ T ⎟⎠It can be shown that, if the function ϕ * is differentiable, the normality property ispreserved for the variables d tandcan then be written as:− q and the complementary laws of evolutionT* *∂ϕq ∂ϕdt= − =∂TT ∂gradT(4.36)The properties that the potentials ϕ and ϕ * must possess for the automaticsatisfaction of the second principle of thermodynamics are the following: they mustbe non negative, convex functions, zero at the origin. It should be noted that thenormality rule is sufficient to ensure the satisfaction of the second principle ofthermodynamics, but it is not a necessary condition.The whole problem of modeling a phenomenon lies in the determination of theanalytical expressions for the thermodynamic potential Ψ and for the dissipationpotential ϕ or its dual*ϕ , and their identification in characteristic experiments. In70

fact the values of ϕ andenergy usually dissipated as heat.*ϕ are almost impossible to measure as they represent anFurthermore, introducing a limit criterion f to describe the onset of inelasticdeformations, an indicator function can be defined as:⎧0 if f , where I ( Tgrad , T ) =+∞.It can be proved that equation (4.37) is equivalent to write:RR* ∂F⎧ f = 0dt ∈∂ ϕ ( T) =∂ IR( T)and dt= ζif ⎪⎨ (4.38)∂ T ⎪⎩ ⎪ f= 0where F is a potential function defining the direction of the inelastic flow equal tof in the case of associated theories and ζ is a multiplier determined by theconsistency condition f = 0 .Therefore, the associative normality rule for the variable d t(4.38) can be restatedequivalently as:fdt= ζ∂∂ Tf( T) = dev( T) −R≤0(4.39)71

with R a material parameter defining the radius of the elastic domain and( tr )dev( T)= T− 1 T 1 is the deviator of the tensor T .3The quantity T is the thermodynamic associated variable to the symmetrictransformation rate tensor dt.Here, the derivative of the limit function f with respect to the dual variable definesthe direction of the internal variable increment , while ζ is the plastic consistentparameter.The limit surface defines the limit of the elastic domain. Inside the surface, that iswhen f < 0 , the internal variables do not vary, so that the plastic multiplier ζ iszero. Along the surface, that is when f = 0 , there is a variation of the internalvariables and the plastic multiplier is different from zero. Points outside of the limityield are not admissible.The described cases are summarized introducing the Kuhn-Tucker conditions ζ ≥0 f ≤ 0 ζ f = 0(4.40)that complete the model. The fulfillment of these conditions allows to evaluate theplastic multiplier and, as a consequence, the internal variables evolution.The so defined model can be classified as an associated model because the limitsurface governing the onset of the plastic deformations is assumed to be coincidentwith the plastic potential defining the plastic flow direction.It is well-known from experimental evidences that shape-memory materials,expecially NiTi and CuZnAl alloys, exibith a tension-compression asymmetry intheir phase transformation behavior which is not displayed by the here chosen ‘vonMises’ criterion. However the framework of this model also allows different phasetransition criteria which account for SMA tension-compression asymmetry.72

4.2.5 Final Format of the Constitutive ModelAll the model constitutive equations can be represented in terms of C (controlvariable) andCt(internal variable), tensors that live in the reference configuration(undeformed configuration).To rewrite all the constitutive equations of the model in terms of the model variables,some considerations have to be done.The equation describing the associative normality rule (4.39) is expressed using apull-back operation that leads to the relation:T fC ∂t= 2 ζ Ft Ft∂T(4.41)In the special case of von-Mises limit function the equation (4.41) becomes:T dev( T)C t= 2 ζ Ft Ftdev( T)(4.42)Using the expressions (4.25) and (4.28) and (4.29), the following expressions can beobtained∂ΨFMF= 2FCFF F C=CSCT T −1e −Tt t t e t t t t t∂Ce(4.43)F αF = F FαF F= CαC(4.44)T T Tt t t t t t t twhere the quantity α defined byα = F αF(4.45)−1 −Tt t73

has been introduced in order to define the back-stress only in terms of the variableCt.In fact, it can be easily shown that the stresses S and X only depend on C andIn particular, using the expression (4.7), it follows that:Ct.∂Ψ ⎡λ1 ⎤ 1e= ( I1−3)− μ + μe∂C⎢e ⎣4 2 ⎥⎦ 1 2C(4.46)so that the expression (4.25) becomes⎡λ1 ⎤ −1 −1 −1S= 2⎢( I1−3)− μt+ μt t⎣4 2 ⎥C C CC (4.47)⎦The differentiation of the transition free energy (4.15) with respect to thetransformation right Cauchy-Green tensor gives:∂Ψ t = − + +∂C2t1 (*β T Mfh Etγ )EEtt(4.48)being⎧ 0 if Et< εL⎪ +γ ∈∂ Iε( E ) ifL t= ⎨ R Et = εL(4.49)⎪⎩ ∅ if Et> εLUsing the same procedure, the expression (4.45) becomes74

1 ( C )*1t−1α = ( β T − M ( ) ) 2f+ h Ct− 1 + γ2 1( Ct−1)2(4.50)Furthermore, if C andPiola-Kirchhoff and the back-stress.Ctare known, it is always possible to compute the secondTo conclude, all the evolution equations for the internal variablein terms of the symmetric tensor C andCt.Ctare representedIn addition, it can be pointed out that the evolutive equation (4.42) can be rewrittenas:dev( YC )tC t= 2 ζdev( Y)(4.51)where the stress quantity Y is yet to be derived in what follows.Using formulas (4.43) it can be shown that:tr( M) = M⋅ 1= ( F CSCF ) ⋅ 1= CSC ⋅ F F = CSC ⋅ ( F F)=−T −1 −1 −T T −1t t t t t t t t t= CSC ⋅ C = CS ⋅ C C = CS ⋅ C C = CS ⋅ 1 = tr CS−1 T −1 −1t(t) t(t) t( t) ( )(4.52)and analogously the relation (4.44) leads totr( α) = tr( C α)(4.53)tThe application of the latter two statements (4.52) and (4.53) together with (4.43)and (4.44) yields:75

( ) 1 1T T T TFt dev T Ft = Ft ( M−α − tr( M − α) 1F ) ( )3t= CSCt −FαF t t− F3ttr M− α Ft=T 1 T( ) 1 T( ) 1 T= CSCt −Ft αFt − F ( )3ttr M Ft − F3ttr α Ft = CSCt −CαC t t− F3ttr CS F−(4.54)1 T+ F ( ) 1 ( ) 1 ( ) ( )3ttr CαtFt = CSCt −CαC t t− tr CS C3t− tr Cα3tCt = dev Y CtwithY= CS− Cαt(4.55)The tensor Y represent the new thermodynamic force introduced with the aim towrite all the constitutive model equations in terms of the variables chosen to descriedthe phase transition phenomena.To conclude this theoretical part the material laws can be summarized in whatfollows:second Piola-Kirchhoff stress tensor−1∂ΨeS=2FtFt∂C∂Ψtback-stress α = 2 ∂Crelative stress tensorY= CS−Cαtdev( YC )tevolution equationC t= 2 ζdev( Y)Kuhn-Tucker conditions ζ ≥0 f ≤ 0 ζ f = 0limit function f ( Y) = dev( Y)− Ret−T76

5 FINITE STRAIN CONSTITUTIVE MODEL:Numerical Procedure5.1 IntroductionThe derived material model in the framework of finite strain is implemented into thefinite element program FEAP (Zienkiewicz and Taylor, 1991). A numerical-discretetime integration scheme is adopted because of the strongly non-linear form of theevolution equations. In particular, the time integration is performed adopting abackward-Euler implicit procedure (Simo, 1992; 1998a; 1998b) and the inelasticstrain tensorC trelated to the phase transformation occurring in the SMA isevaluated. The time interval of interest is subdivided in subincrements and theevolutive problem is solved over a generic interval [ , ]tntn + 1, being t > n+ 1t . The nquantities at time n are denoting by the pedex n while the index n + 1 of allquantities evaluated at the time tn + 1is omitted.Once the solution at the time t nis known as well as the strain tensor C at time tn + 1,the stress history is computed from the strain history by means of a procedure knownas return-map.5.1.1 Exponential MapIn order to obtain the time integration of the evolutive equation given by formulas(4.40), some considerations have to be done. An exponential map is adopted inwhich the spectral decomposition is used to compute an exponential function (Reeseand Christ, 2008).In particular, equation (4.40) can be written in the following form:77

C = ζf( C, C ) = ζg( CC , ) C(5.1)t t t twhere the tensor f = f( C, C ) = 2 dev( Y) C / dev( Y)represents a function of twottarguments C andCt. Usingf= fC C it can be obtained a representation written− 1ttin terms of the function g=g( CC , ) . It can be emphasized that the tensor f istsymmetric, while the same do not hold for g .If the function ζ g was a constant g the evolution equation (5.1) could beanalytically integrated by means of the exponential map:C = exp( g( t−t )) C (5.2)t n t,nThe direct transfer of this integration rule to an implicit numerical integration of theevolution equation C t= ζ gCtwhere the term ζ g is held constant within the timeincrement Δ t = tn+1− tndoes not lead to a suitable integration rule because thesymmetry requirements are not fulfilledexp( ζ g Δt) C ≠C exp ( ζ g Δt)(5.3)Tn+ 1 n+ 1 tn t, n n+ 1 n+1To guarantee symmetry it can be used an updating formula ζ = ζΔtCC, C = exp( ζ g)C (5.4)−1t t n t tThe latter integration formula has the disadvantage that the exponent of a nonsymmetrictensor has to be computed. This requires to work with a seriesrepresentation78

2 3ζ 2 ζ 3exp( ζ g) = 1+ ζ g+ g + g + …2 62 3−1 ζ −1 2 ζ −13= 1+ ζ fCt + ( fCt ) + ( fCt) + …2 6(5.5)2 3⎛ −1 −1 ζ −1 −1 2 ζ −1 −1 3⎞−1= Ut⎜1+ ζ Ut fUt + ( Ut fUt ) + ( Ut fUt ) ⎟Ut⎝2 6 ⎠It follows that:( ζ )exp( ζ g) = U exp U fU U(5.6)−1 −1 −1t t t tThe latter derivation shows that exp( ζ g ) can be alternatively represented by( ζ )− −U exp U fU U− , where in contrast to the former format, the exponent of a1 1 1t t t tsymmetric tensor is included. This fact has the important advantage that theexponential function can be computed in closed form by means of the spectraldecomposition:3 3∑A T= A → A = A N ⊗ N ⇒ exp( A ) = exp( A ) N ⊗ N (5.7)∑b b b b b bb= 1 b=1In the latter relation Ab, with b= 1,2,3 represent the eigenvalues of A andN , with b = 1,2,3 its eigenvectors.bUsing (5.5) the integration rule (5.4) is finally given by( ζ )CC C = U exp U fU U C ⇒ C = U exp( ζ U fU ) U (5.8)−1 −1 −1 −1 −1 −1 −1 −1 −1t t, n t t t t t t t,n t t t t79

5.1.2 Time Integration of the Constitutive ModelThe time-discrete counterpart of the constitutive model is given by⎧ ⎡λ1 ⎤ −1 −1 −1⎪ S= 2⎢( I1−3)− μ4 2 ⎥Ct + μCt CCt⎪ ⎣⎦⎪1⎪ ( C )*1t−1( ( ) ) 2⎪α = β T − Mf + h Ct− 1 + γ⎪2 1( Ct−1)⎪2⎪⎪Y= CS−Cαt⎨γ ≥ 0⎪−1 −1 −1 −1 −1⎪ Ctn ,= Ut exp( ΔζUt fUt ) Ut⎪⎪Ut= Ct⎪ f = 2 dev( Y) Ct/ dev( Y)⎪⎪Et≤ εL⎪ F( Y) = dev( Y) −R≤0⎪⎩ Δζ≥0 Δ ζF( Y) = 0(5.9)where Δ ζ = ( ζ − ζ n ) is the consistency parameter, time integrated over the interval[ t , nt ] .From a computational point of view, the time-discrete model as presented in equation(5.9) shows a problem due to the fact that the transformation stress α is proportionalto the inverse of the Green-Lagrange strain norm and to its derivative so that, whenthe strain is zero, this quantity remains undefined. To overcome this difficulty, theEuclidean normEtcan be substituted with a regularized norm Et, defined as( d+1)dd( ) ( d−1) dEt = Et − Et+ d(5.10)d −180

where d is a user-defined parameter which controls the smoothness of theregularized norm. For small values of the parameter d , the regularized norm of theGreen-Lagrange strain tensor tends to the Euclidean one so that d indirectlymeasures the difference betweenEtand Et. In this way, the quantity Etisalways differentiable, even in the case Et = 0 with d > 0 .5.1.3 Solution AlgorithmThe solution of the modified time-discrete counterpart of the model is approachedwith a predictor corrector procedure, typical used for the classical theory ofplasticity. The algorithm consists in evaluating an elastic trial state, in which theinternal variables remain constant and in verifying the admissibility of the trial limitTRfunction. Then, if the trial state is admissible ( f ≤ 0 ), the step is effectively elastic,i.e. there is no transformation strain development and it represents the solution;TRotherwise, if the trial state is not admissible ( f > 0), the step is inelastic and a newsolution has to be evaluated. The transformation strain tensorthrough integration of the evolutionary equations.The inelastic step is solved using the following procedure.Ethas to be updatedFirst of all, the assumption γ = 0 is done, i.e. it is assumed that there is an evolvingphase transformation state in which 0 ≤ Et< εL. The equation (5.9) is rewritten inresidual form asC −1 −1 −1 −1 −1⎧ ⎪R =− Ctn ,+ Ut exp( Δ ζ Ut fUt ) Ut= 0⎨Δζ⎪⎩ R = dev( Y) − R=0(5.11)A new value ofa Newton-Raphson method.Ctis evaluated solving these seven scalar non-linear equations with81

If the above solution is not admissible, i.e. Et≥ εL, equation (5.9) is rewritten inresidual form assuming γ > 0 as⎧ C −1 −1 −1 −1 −1R =− Ctn ,+ Ut exp( Δ ζ Ut fUt ) Ut= 0⎪Δζ⎨ R = dev( Y) − R=0⎪γ⎪⎩R = Et− εL= 0(5.12)The non-linear equation system of eight scalar equations is solved with a Newton-Raphson method to compute a new value ofCtand γ .5.1.4 Newton-Raphson MethodThe non-linear evolutionary problem is solved using an iterative method. Inparticular, the solution is herein given only for the case of saturated phase transition,i.e. Et= εL, because the case of evolving phase transition can be simply obtainedeliminating from the governing system the last row.The iterative Newton-Raphson method requires the linearization of equation (5.12)asC C C C⎧ d( R ) = R, C: dC t t+ R, ΔζdΔ ζ + R,γdγ⎪ Δζ Δζ Δζ⎨dR ( ) = R, C: dCt t+ R, ΔζdΔ ζ + R,γdγ⎪ γ γ γ γ⎪⎩dR ( ) = R, C: dCt t+ R, ΔζdΔ ζ + Rd, γγ(5.13)hence the computation of the matrix:82

⎡R R Rt⎢⎢RR Rt⎢⎣R R RtC C C, C , Δζ, γΔζ Δζ Δζ, C , Δζ, γγ γ γ, C , Δζ, γ⎤⎥⎥⎥⎦(5.14)where the subscript comma indicates derivation with respect to the quantityfollowing the comma, i.e.R means the derivation of the first six scalar equationsC, CtCR with respect toDenoting as1 3( 1 1)Ctand so on.devI the fourth-order deviatoric identity tensor defined asI dev = I − ⊗ with I = 1 1 the fourth-order identity tensor with the genericcomponent equal to I ijkl= δ ikδ jl; the derivatives appearing in equation (5.14) assumethe following form:C• ( ) ( ) ( )( G)( C )R = B G U + U⎛⎛∂⎜⎞( U ) + G B⎝⎝⎠−1 −1 lt −1, Ctilhk lt ttjt ijhkilttjlt tjhk⎜⎜∂⎟t hk⎞⎟⎠(5.15)withBrjhk =• ( C−, Δζ) = ( t )1−−1 ⎛ ⎞⎛2∂U⎞t ⎜∂Ct⎟⎜ ⎟ =∂C⎜ ∂C⎟⎝ t ⎠rjhk⎜ t ⎟⎝ ⎠rjhk−1 −1( exp( ζ ))lt t tlt(5.16)G = Δ U fU(5.17)∂ ( G)lt( ζ Q)−( ) ( t )R U Q U1 1ij il gb∂ Δtjgb(5.18)with:83

−1 −1( )Q = U fU(5.19)lt t tlt• ( γ ) =Δζ( )−1 −1( exp( ζt t))− −∂( ΔζUtfUt)∂ Δ U fU ∂fR U U U U( ) ( ) ( )C −1 kl −1 rt −1 −1,ijtik 1 1tprttqt∂γljpq(5.20)with:∂frt= 2 Hrl ( Ct ) + ( dev( Y)) zlt rl∂γ1 ⎛ ∂α1H = ⎜− C + Crbdev( Y ) ⎝ ∂γ3bl( ) δ ( )∂α⎞δ ⎟∂γ⎠grrl t pr t pgrldev( I )( ) ( )ijhkz = dev( Y)C3 ij tdev( Y )hb∂αbk∂γ(5.21)(5.22)(5.23)∂ α =∂γEEtt(5.24)ζ• ( R,)( dev Y )( )Δ⎛hk∂dev( Y)⎞=t⎜ ⎟dev( Y)⎝ ∂C⎠C ijt hkij(5.25)Δ• R ζ , Δ ζ= 0(5.26)dev( Y) ijαlj 1 αΔ⎛ ∂ ∂pk ⎞⎜ t ilhk tδhpij⎟dev( Y) ⎝ ∂γ3 ∂γ⎠ζ• R = − ( C ) + δ ( C ), γ(5.27)84

γ• R,Ct=12EEtt(5.28)• R γ , Δ ζ= 0(5.29)• R γ = , γ0(5.30)The three-dimensional model herein proposed has been implemented into a standard2D four-node quadrilateral finite element. In the following, the weak formulation ofmomentum balance equations is developed as the first step toward the numericalimplementation of the model in the framework of the finite-element method (Ciarlet,1978; Bathe, 1982; Crisfield, 1986, 1991, 1997; Reddy, 1993).5.2 Principle of Virtual Worki. Spatial description.Generally, the finite element formulation is established in terms of a weak form ofthe differential equation (Reddy, 1984; Bonet and Wood, 1997). In the context ofsolid mechanics this implies the use of the virtual work equation. For this purpose,let η=η( x ) denote an arbitrary vector-valued function defined on the currentposition of the body. Using equation (3.73) for the static problem( )divσ + ρ0 b=0, inφB (5.31)and integrating it over the volume of the body it can be derived a weak statement ofthe static equilibrium of the body as:85

∫( ) ( ρ )f η = divσ+ b ⋅ η dv=(5.32)v00Recalling that:( div )div( ση) = σ ⋅ η+ σ : ∇η (5.33)and using the Gauss theorem enables equation (5.32) to be rewritten as:∫ ∫ ∫n⋅σηds− σ : ∇ ηdv+ ρ b⋅ η dv=0(5.34)∂v v v0The test function η in equation (5.34) is arbitrary. therefore, it is possible to lookupon η as the virtual displacement field δu , defined on the current configuration.The formulation in the weak form of the initial boundary-value problem (5.34) leadsto the fundamental principle of virtual work.Recalling that the first variation δe of the Euler-Almansi strain tensor e may beexpressed as:1 Tδe= ( ∇ δu+∇ δu) = sym( ∇δu )(5.35)2and using tn ( ) = σn = t on ∂φ( B ) and the symmetry of the Cauchy stress tensor,equation (5.34) becomes:∫ : ∫ 0 ∫v v ∂vσ δedv = ρ b⋅ δudv + t⋅δu ds(5.36)86

The spatial virtual work equation states that the virtual stress workσ : δe at fixed σis equal to the work done by the body force b per unit current volume and thesurface traction t per unit current surface along δu .It is important to emphasize that to establish the principle of virtual work is notnecessary to postulate the existence of a potential. No statement in regard to aparticular material is invoked. Therefore, the principle of virtual work is general inthe sense that it is applicable to any material including inelastic materials.i. Material description.In order to derive the expression of the material virtual work equation it must be usedthe following relation between the material and spatial virtual measures ofdeformation:( ) Tδ = χ* δ = δ− −1e E F EF (5.37)according to which the variation of the spatial tensor e is the push-forward of thedirectional derivative of the associated Green-Lagrange strain tensor E .Using equation (5.37), the expression of the internal virtual work done by the stressesbecomes:int−1−T( δ )−T−1( )δW = σ : δedv= Jσ : F δEFdV =∫V∫vtr F JσF E dV = S:δEdV∫V∫V(5.38)which shows that the second Piola-Kirchhoff stress tensor is work conjugate to Eand enables the material virtual work equation to be alternatively written in terms ofS as:87

∫ ∫ ∫S:δEdv = ρB⋅ δudV + T⋅δu dS(5.39)V V ∂V5.2.1 Finite Element ImplementationThe simplest approach is to start from a reference configuration since here integralsare all defined over domains which do not change during the deformation processand thus are not affected by variation or linearization steps. Later the results can betransformed and written in terms of the deformed configuration.To develop a finite element solution in the framework of the finite deformationproblem, the case of an elastic material is considered. Other material behavior maybe considered later by substitution of appropriate constitutive expressions for stressand tangent moduli.5.2.1.1 Reference Configuration FormulationMatrix notation to represent the stress, strain and variation of strain can beintroduced. For three-dimensional problems the matrix for the second Piola-Kirchhoff stress and the Green strain are defined, respectively, as[ S S S S S S ]S =(5.40)11 22 33 12 23 13T[ E E E 2E 2E 2E]E =(5.41)11 22 33 12 23 13Twhere, similar to the small strain problem, the shearing component are doubled topermit the reduction to six components. The variation of the Green strain is similarlygiven by88

[ E E E 2 E 2 E 2 E ]δE = δ δ δ δ δ δ(5.42)11 22 33 12 23 13Twhich permits to write the following matrix relation:δE SIJIJδT= ES(5.43)The variation of the Green strain is deduced from equations (3.14), (3.17) and fromthe variation of the deformation gradient expressed directly in terms of the currentconfiguration displacement:δ FiI∂δu∂Xi= =Iδui,I(5.44)and it can be written as1⎛∂δui∂δu⎞i1δE = ⎜ F + F ⎟= δu F + δu F2⎝∂XI∂XJ⎠ 2( , , )IJ iJ iI i I iJ i J iI(5.45)Substituting equation (5.45) into equation (5.42) it follows that⎧ Fi1δui,1⎫⎪Fi2 δu⎪⎪i,2⎪⎪ Fi3 δui,3⎪δ E = ⎨ ⎬⎪Fi 1δui,2 + Fi2 δui,1⎪⎪Fi2 δui,3 + Fi3 δu⎪i,2⎪⎪⎩⎪Fi3 δui,1+Fi 1δui,3⎭⎪(5.46)that represents the matrix form of the variation of the Green strain.89

5.2.1.2 Finite Element ApproximationThe reference configuration coordinates can be represented asaX = N () ξ X (5.47)I a Iwhere ξ are the three dimensional natural coordinates ξ 1, ξ2and ξ 3,Naarestandard shape functions. Similarly, the displacement field in each element can beapproximated byu = N () ξ u a(5.48)i a iThe reference system derivatives are constructed as followsu = N ξ u (5.49), ,() aiI aI iwhere explicit writing of the sum is omitted and summation convention for a isagain invoked. The derivatives of the shape function can be established by usingstandard routines.The deformation gradient and Green strain may now be computed using equations(3.12) and (3.18), respectively. Finally, the variation of the Green strain is given inmatrix form asˆaδ E= B δu (5.50)awhere B ˆ acan be split into two parts as90

ˆNLB = B + B (5.51)a a ain whichBais identical to the small deformation strain-displacement matrix and theremaining non-linear part is give byBNLa⎡ u1,1 Na,1 u2,1 Na,1 u3,1 Na,1⎢⎢u1,2 Na,2 u2,2 Na,2 u3,2 Na,2⎢ u1,3 Na,3 u2,3 Na,3 u3,3 Na,3= ⎢⎢u N + u N u N + u N u N + u N⎢u N + u N u N + u N u N + u N⎢⎢⎣u1,3 Na,1 + u1,1 Na,3 u2,3N + u N u N + u N1,1 a,2 1,2 a,1 2,1 a,2 2,2 a,1 3,1 a,2 3,2 a,11,2 a,3 1,3 a,2 2,2 a,3 2,3 a,2 3,2 a,3 3,3 a,2a,1 2,1 a,3 3,3 a,1 3,1 a,3⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦(5.52)It is immediately evident thatB NLais zero in the reference configuration andtherefore that B ˆ a= Ba.The variational equation may now be written for the finite element problem as:⎛⎞a ⎜∫a a⎟(5.53)⎝V⎠T( ) ˆ Tδu B SdV− f = 0where the external forces are determined asf∫ ∫ (5.54)= N ρ BdV + N tdSa a 0aV∂Vwith B and t the matrix form of the body and traction force vectors, respectively.The tangent term is given by91

ˆ Tˆ T ˆ ˆ ∂B∂fK = ∫B D BdV+ ∫ SdV− = K + K + K∂u ∂uT T M G LVV(5.55)where the first term is the material tangentmoduli.The second term,KM, in which D ˆ Tis the matrix of tangentKG, defines a tangent term arising from the non-linear form of thestrain-displacement equations and is often called the geometric stiffness. Thederivation of this term is most easily constructed from the indicial form written as⎛ ∂δE ⎞ ⎛ ⎞IJb S dV du = δ u N δ N S dV du = δ u ( K ) du⎜∫⎟ ⎜ ⎟δu∫ ⎝V j ⎠ ⎝⎠b a b a ab bIJ⎟j i a, I ij b,J IJ j i ijGjV(5.56)Thus the geometric part of the tangent matrix is given byabKG= Gab1 (5.57)whereG = ∫ N δ N S dV(5.58)ab, ij a, I ij b,J IJVThe last term in equation (5.55) is the tangent relating to loading which changes withdeformation. It is assumed for the present that the derivative of the force term f iszero so thatKLvanishes.The form of the equations related to the reference configuration can be easilytransformed to the current configuration replacing the reference configuration termsby quantities related to the current configuration, using for example Cauchy or92

Kirchhoff stress and converting integrals over the undeformed body to ones in thecurrent configuration.93

6 LINEARIZATION OF THE FINITEDEFORMATION SMA MODEL6.1 IntroductionThe strain quantities defined in the Chapter 3 are non-linear expressions in terms ofthe motion φ and they lead to non-linear governing equations. These governingequations need to be linearized in order to enable the limitation of the SMAconstitutive model proposed within the framework of finite strain to the smalldeformation regime. In particular, small deformations are presumed, reference andcurrent configurations coincide and strains are small enough that a linearized strainmeasure accurately approximates the more rigorous nonlinear measures. It istherefore essential to derive equations for the linearization of the strain quantitieswith respect to small variations in the motion.6.1.1 Linearized Deformation GradientConsider a variation of configuration ux ( ) from the current configuration x= φ( X )[Bonet and Wood, 1997]. Recall that the deformation gradient can be expressed as:F∂φ= (6.1)∂ XDenoting with D the linearization operator, the deformation gradient F can belinearized in the direction of u at this position as94

DF( φ)[ u]d=dεε = 0( φ εu)∂ +∂Xd ⎛ ∂φ∂u⎞= ⎜ + ε ⎟dεε = 0 ⎝∂X∂X⎠∂u= = ( ∇u)F∂X(6.2)In particular, if the linearization of the deformation gradient is performed at theinitial configuration, i.e. when x=X and therefore F=1, it results:DF[ u]= ∇ u(6.3)6.1.2 Linearized Strain MeasuresUsing equation (3.17) and (6.1), the Lagrangian strain can be linearized at the currentconfiguration in the direction u as:T1⎡d ⎪⎧⎛∂ ( φ+ εu) ⎞ ⎛∂ ( φ+εu)⎞ ⎪⎫⎤DEu [ ] = ⎢ ⎨⎜ ⎟ ⎜ ⎟−1⎬⎥2 ⎢dεε = 0 ⎪⎝ ∂X⎠ ⎝ ∂X⎩⎠ ⎪⎭⎥⎣⎦1⎡d ⎧⎪ ⎛⎛⎞= ⎢ ⎨ + + + −2 ⎢dε⎜⎜∂X ⎢∂X⎥ ⎟ ⎢∂X⎥⎪ ⎝⎝⎣ ⎦ ⎠ ⎣ ⎦ ∂X⎣ ⎩TT∂u ⎡∂u⎤ 2 ⎡∂u⎤∂u1 εε1TT1 ⎡⎛∂u ⎡∂u⎤ ⎞ ⎡∂u⎤∂u⎤= ⎢+ 2ε2 X ⎢+⎥X⎥ ⎢ X⎥⎢⎜∂ ∂ ⎟⎣⎝⎣ ⎦ ⎠ ⎣∂ ⎦ ∂X⎥⎦1 ⎛ ∂u⎡∂u⎤=+2 ⎜∂X⎢⎣∂X⎥⎝ ⎦T⎞⎟⎠ε = 0⎞⎫⎪⎤⎬⎥⎟⎠⎭⎪⎥⎦ε = 0(6.4)It follows that:95

DE[ u]= ε(6.5)Similarly, the right and left Cauchy-Green deformation tensors defined respectivelyin equations (3.14) and (3.19) can be linearized to give[ ] D [ ] 2DCu = bu = ε + 1(6.6)6.1.3 Linearized Volume ChangeThe volume change is given by the JacobianJ = det F . Physically, it expresses theratio between the elementar volume element defined in the current configuration andthe one defined in the reference configuration, i.e.:dvJ = (6.7)dVThe directional derivative of J with respect to an increment u in the spatialconfiguration is:( )DJ[ u] = D det F DF[ u ](6.8)Recalling the expression of the directional derivative of the determinant of a tensorand applying equation (6.3) for the linearization of F , it results:DJ[ u]⎛Jtr⎜⎜⎝−1= F ⎟= Jtr∇u∂u⎞∂X⎟⎠(6.9)96

In terms of the linear strain tensor ε , the above equation can be expressed as:DJ[ u]= Jtrε (6.10)In the initial configuration, it follows that:DJ[ u]= trε (6.11)6.1.4 Linearized Elastic Free EnergyThe expression of the elastic free energy, adopted to describe the elastic SMAbehavior in a finite strain regime is the following:1⎡ ⎛λ + 2μ ⎞ 2 ⎛3λ+ 2μ ⎞ ⎛9λ+ 6μ⎞⎤Ψe= I1 I1 μI22⎢⎜ ⎟ − ⎜ ⎟ + + ⎜ ⎟4 2 4⎥⎣⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎦(6.12)where I1and I 2are the principal invariant of the elastic Cauchy-Green tensor. Inparticular, the linearized expression of these quantities are derived as:[ ]*I = trC = tr 2E + 1 = 2trE + 3= 2trε+ 3= 2I+ 3 (6.13)1 e e e e11 2 12( ( ) )⎡ ( )2 2 ⎣1 2( 42tr ( Ee)3 4 tr EeI )21( ( εe) ( εe)εe)( 4 4 ⎤)I = tr C − I = tr E + 1+ E −I⎦2 22 e 1 e e 1= + + −2 2 * *2 1= 2tr −3−2 tr − 4tr = 4I −4I−3(6.14)97

where*I1and*I2are the principal invariant of the elastic strain tensor.Substituting equations (6.13) and (6.14) into the expression of the elastic free energy(6.12) it follows that:1Ψ ( )( * ) 2 *e= ⎡ λ+ 2μ I1 + 4μI⎤22 ⎢⎣⎥⎦(6.15)that is the elastic energy in a small deformation regime.6.1.5 Linearized Transformation Free EnergyThe transformation strain energy is assumed for a finite strain regime as:* 1 2⎡T ⎤Ψt= β T − MfEt + h Et + ( uo − Tηo) + c⎢( T −To) − Tln⎥+Iε ( E )L t(6.16)2 ⎣To⎦as a function of the Green-Lagrange strainLinearizing the expression (6.16) it follows the expression of the transformationenergy in the small strain regime as:Et.* 1 2⎡T ⎤Ψt= β εt + h εt + ( uo − Tηo) + c⎢( T −To) − Tln⎥+Iε ( ε )L t(6.17)2 ⎣To⎦with εtthe second-order transformation strain tensor.98

6.2 3D Constitutive Model for Stress-Temperature InducedSolid Phase Transformation in a Small Strain RegimeAssuming a small strain regime, the free energy for a polycrystalline SMA materialis, therefore, defined through the following convex potential:( ) ( T)Ψ (, εε, T) =Ψ ε +Ψ ε,(6.18)t e t tThe strain ε and the absolute temperature T are assumed as model control variableswhile the second-order transformation strain tensor ε trepresents the internalvariable. In accordance with experimental evidences, no volume variations duringthe phase transition are observed. The inelastic strain tensor ε trepresents a measureof the strain associated to the phase transition in the SMA. The norm of this quantityshould be bounded between zero for the case of a material without orientedmartensite and a maximum value ε L, for the case in which the material is fullytransformed in single-variant oriented martensite. The scalar quantity εLis related tothe maximum transformation strain reached at the end of the transition during auniaxial test.The Clausius-Duhem form of the entropy inequality, i.e. the second law ofthermodynamics, takes the following form:gradTσ: ε − ( Ψ+ ηT)−q⋅ ≥0(6.19)Twhere η is the entropy, q is the heat flux vector and σ is the stress tensor.Introducing in equation (6.19) the dependence of the energy function from the modelvariables ε , T and εtthe following inequality is obtained:99

∂Ψ ∂Ψ ∂Ψ gradTσ: ε− : ε− : ε t− T−ηT−q⋅ ≥0(6.20)∂ε∂ε∂TTtthat can be rewritten in the form:⎛ ∂Ψ ⎞ ∂Ψ ⎛ ∂Ψ ⎞ gradT⎜σ− ⎟: ε− : ε t− η + T⎜ ⎟ −q⋅ ≥0(6.21)⎝ ∂ε⎠ ∂ε⎝ ∂T⎠ TtAs a consequence, the state laws are:∂Ψσ = ∂ε(6.22)∂Ψη =− (6.23)∂ Tdefining the thermoelastic laws for the stress tensor and the entropy, while thethermodynamic force associate to the transformation strain, i.e. the transformationstress, can be defined as:∂ΨX =− (6.24)∂ εtThe equations (6.22) and (6.24) state that σ and X are the quantitiesthermodynamically conjugate to the deformation-like variables ε andrespectively.Accordingly, following standard arguments, the state laws can be derived as:εt,[ ]σ = C ε−ε t(6.25)100

T − Mf Tη = η0− ⎡⎣h( ω) βt + ( 1− h( ω)) β ⎤c⎦εt+ clnT − M Tf0(6.26)while the transformation stress assumes the following form:∂Iε( ε ) ∂ εX = σ− [( h( ω) βt + ( 1 −h( ω)) βc)T − Mf+ h εt+ ] ∂ ε ∂ εL t ttt(6.27)where the subdifferential of the indicator function results as:⎧ 0 if εt< εL∂Iε ( ε )L t ⎪ += ⎨R if εt= εL(6.28)∂ εt⎪⎩ ∅ if εt> εLEquation (6.27) can be rewritten in the following form:X = σ− α(6.29)where:εα = [( h( ω) βt + ( 1 −h( ω)) βc)T − Mf+ h εt+ γ]∂∂εtt(6.30)with:⎧ 0 if εt< εL⎪ +γ ∈∂ Iε( ε ) ifL t= ⎨ R εt = εL⎪⎩ ∅ if εt> εL(6.31)101

Accordingly, the tensor α plays a role similar to the back-stress of the classicalplasticity and X is the relative stress.Introducing a yield function that is assumed to depend on the deviatoric part of thedevthermodynamic force f ( X ), the equation describing the associative normalityrule for the internal variable εtis given by:devdev∂f( X ) dev ∂f( X )ε t= ζ = ζI (6.32)dev∂X∂Xwith ζ the so-called plastic multiplier and ( dev = − 1)( : )X X X 1 1. From the3analysis of the flow rule form (6.28) it can be pointed out that the transformationstrain tensor ε tis a deviatoric tensor. This fact appears in a perfect agreement withthe experimental evidences indicating no volume variations during the phasetransition.Furhermore, between tension and compression pseudoelastic experiments, anasymmetric behavior is observed. In particular, at a given test temperature it is foundthat: (i) the stress level required to nucleate the martensitic phase from the austeniticphase is considerably higher in compression than in tension; (ii) the transformationstrain measured in compression is smaller than that in tension; (iii) the hysteresisloop generated in compression is wider (along the stress axis) than the hysteresisloop generated in tension. These major differences between the tension andcompression response of a Ni-Ti alloy in pseudoelastic experiments are shown inFig. 6.1.To catch the asymmetric behavior of SMA during tension-compression tests, theactivation of the phase transition, i.e. the evolution of the inelastic strain ε t, is102

assumed to be ruled by the limit function introduced by Auricchio and Petrini (2002;2004):J( X ) = 2 + −(6.33)3f J2m RJ2where J2and J3are the second and the third invariant of the deviatoric part of therelative stress X , defined, respectively, as:Jdevdev(( ) ) J ( )( )1 1X : 1 X : 1 (6.34)2 32 32=3=while R is the radius of the elastic domain in the deviatoric space and m is amaterial parameter, with m ≤ 0.46 to guarantee the yield surface convexity. Theparameters R and m are linked to the uniaxial critical transformation stresses intension σtand compression σc, respectively, through the following relations:2 σcσt 27 σc −σtR= 2 m=3 σ + σ 2 σ + σt c t c(6.35)103

Fig. 6.1: Experimental stress-strain curves for Ni-Ti at T = 298Kin simple tension and simplecompression. For compression the absolute values of stress and strain are plotted.The model is completed introducing the classical complementarity Kuhn-Tuckerconditions: ζ ≥0 f ≤0 ζ f ≤0(6.36)that reduce the problem to a constrained optimization problem.It is well known that the normality properties are sufficient to guarantee thesatisfaction of the Clausius-Duhem inequality (Lemaitre and Chaboche, 1985).Some comments on the model features can be pointed out:♦The tensor X is not a deviatoric tensor, as it is assumed in the works byAuricchio and Petrini (2002; 2004). In fact, even if the transformation strain ε tistraceless , the equation (6.24) do not necessarily state that the derivative of the freeenergy with respect to it results in a deviatoric tensor.104

♦The limit function is assumed function of the deviatoric part of the relativestress X in order to ensure that εtis traceless.♦ The equation (6.33), describing the normality rule, is expressed in anassociated form; it defines a relationship between the associated variables of themodel, i.e. X and εt. The direction of the transformation strain evolution is given bythe derivative of the yield function with respect to the thermodynamic force. On thecontrary, the model modified by Auricchio and Petrini (2002; 2004) proposes adifferent expression of the evolutionary law as:fε t= ζ ∂(6.37)∂ σin which the rate of the transformation strain takes place along the derivative of thelimit function with respect to the stress tensor. This definition does not seem tosatisfy the thermodynamic theory of the associated variables, which states that thetransformation strain must take place along the derivative of the yield function withrespect to its associated variable. This assumption do not necessarily ensure thesatisfaction of the second law of thermodynamics that must be verified a posteriori.On the other hand, for the case under examination, it can be pointed out that:∂f ∂f ∂ ∂ ∂f= X s=∂σ ∂X ∂s ∂σ ∂XdevI (6.38)being s the deviatoric part of the stress and ∂ X= I . It follows that the evolutionary∂slaw proposed by Auricchio and Petrini in equation (6.39) coincides with the formpresented in equation (6.33).105

♦The deviatoric form of the transformation strain tensor ε tis not imposed apriori as in the constitutive model by Auricchio and Petrini but instead it is simplythe result of a consistent model.♦ The chemical energy Ψchis modified with respect to the model proposed bySouza et al. (1998), in order to account, in an effective manner, for the possibledifferent behavior of SMA in tension and compression.♦ The limit function defined in equation (6.33) is able to reproduce only the factthat the stress level required to active the transition from austenite to martensite ishigher in compression than in tension; nevertheless, it seems to be inadequate tosimulate that the transformation strain measured in compression is smaller than thatin tension together with the fact that the hysteresis loop generated in compression iswider than the one generated in tension.♦ The proposed model appears to be consistent with the thermodynamicformulation, since it satisfies the second principle of thermodynamics in the form ofthe Clausius-Duhem inequality.6.3 Numerical Procedure6.3.1 IntroductionThe constitutive model described above is implemented in the finite element codeFEAP. Due to the strongly non-linear character of the evolution equations this can bedone only in the context of a numerical-discrete time integration scheme (Simo andHughes, 1998). In particular, the time integration is performed adopting a backward-Euler implicit procedure and the inelastic strain tensor ε trelated to the phasetransformation in the SMA is evaluated. Once the solution at the time t nis known aswell as the strain tensor ε at time t , the stress at time t is computed from the strain attime t by means of a procedure known as return-map.The three-dimensional model has been implemented into a standard four-node solid2D finite element.106

6.3.1.1 Time Integration of the 3D SMA Constitutive ModelIntroducing the consistency parameter Δ ζ = ( ζ − ζ n ) , time integrated over theinterval [ t ]n ,t , the time-discrete counterpart of the 3D constitutive model is given by:[ ]⎧ σ = C ε−εt⎪⎪ *∂ εtX = σ−[ β T − Mf+ h εt+ γ]⎪∂εt⎪⎪γ ≥ 0⎪dev∂f( X )⎨ εt= εt,n+Δζ⎪∂X⎪εt≤ εL⎪⎪devJ3f( X ) = 2J2+ m −R≤0⎪J2⎪dev⎪⎩ Δζ≥0 Δ ζ f ( X ) = 0(6.39)In the framework of a predictor-corrector algorithm, an auxiliary state obtainedfreezing the transformation phase flow, is introduced. In other words, assuming nophase transformation, a purely elastic trial step is considered as:εTRtt,nTRTRσ C ε εtχTR= ε= ⎡⎣ −TR*TR t[ β T Mfh εt ]TR∂εtX = σ −αTR TR TR⎤⎦= − +1= − ( : )3Xf = J + m −Rdev,TR TR TRX X X 1 1dev,TRTR dev, TR dev, TR J3( )( X ) 22( X )dev,TRJ2( X )107∂ε(6.40)

The trial state is determined solely in terms of the initial conditions { n,t,n}given incremental strainε ε and theΔεn. Besides, this state may not, in general, correspond toany actual, physically admissible state unless the incremental process is elastic.TRTherefore, if the trial step is admissible, i.e. f < 0 , the step is elastic; otherwise, ifTRf ≥ 0 , the step is inelastic and the transformation strain has to be updated throughintegration of the evolutionary equations.The inelastic step is solved by the return-mapping algorithm.First of all, the assumption γ = 0 is done, i.e. it is supposed that there is an evolvingphase transformation state in which 0 < εt ≤ εL. The equation (6.39) is rewritten inresidual form as⎧∂f∂ ε= −( − Δ ζ ) + ⎡β− + ⎤ =⎪⎣ ⎦⎨⎪ ΔζJR = J + m − R=⎪⎩XTR*tR X σ CT Mfh εt0∂X∂εt3220J2(6.41)A new value of εtis evaluated solving these seven scalar non-linear equations with aNewton-Raphson method.If the above solution is not admissible, i.e. εt > εL, equation (6.39) is rewritten inresidual form assuming γ > 0 as108

⎧∂f∂ ε⎪ C ⎡⎤⎣ ⎦⎪⎪ΔζJ⎨R = J + m − R=⎪⎪ R = − =⎪⎪⎩XTR*tR = X−( σ − Δ ζ ) + β T − Mf+ h εt+ γ = 0∂X∂εt3220J2γεtεL0(6.42)The non-linear equation system, consisting of eight scalar equations is solved with aNewton-Raphson method to compute a new value of εtand γ .From a computational point of view, the presented time-discrete model shows aproblem due to the fact that the transformation stress X depends on thetransformation strain norm, which can be zero, so that this quantity remainsundefined. To overcome this difficulty, the Euclidean normεtcan be substitutedwith a regularized normεt, defined as( d+1)ddεt = εt − εt+d −1( ) ( d−1d)d(6.43)where d is a user-defined parameter which controls the smoothness of theregularized norm. For large values of the transformation strain tensor, the regularizednorm tends to the Euclidean one; for small values of ε t, the parameter d measuresthe difference betweenε tand ε t. In this way, the quantity ε tis alwaysdifferentiable, even in the case εt= 0 with d > 0 .109

6.3.1.2 Newton-Raphson MethodAs stated, the non-linear evolutionary problem is solved using an iterative method. Inparticular, only the case of saturated phase transition, i.e. εt = εL, is treated, sincethe case of evolving phase transition can be simply obtained eliminating from this thelast row.The iterative Newton-Raphson method requires the linearization of equation (6.42)asX X X X⎧ d( R ) = R, X: dX+ R, ΔζdΔ ζ + R,γdγ⎪ Δζ Δζ Δζ⎨dR ( ) = R, X: dX+ R, ΔζdΔ ζ + R,γdγ⎪ γ γ γ γ⎩ dR ( ) = R, X: dX+ R, ΔζdΔ ζ + Rd, γγ(6.44)i.e. the computation of the matrix:⎡R R R⎢⎢R⎢⎣RX X X, X , Δζ, γΔζ Δζ Δζ, XR, ΔζR,γγ γ γ, XR, ΔζR,γ⎤⎥⎥⎥⎦(6.45)where the subscript comma indicates derivation with respect to the quantityXfollowing the comma, i.e. R, γmeans the derivation of the first six scalar equationsXR with respect to γ and so on.The derivatives appearing in equation (6.45) assume the following form:110

RRRRRRRR*( f )2 2 2Xd, X= I+ hΔ ζ +Δ ζC + β T − M + γ ΔζB2 2 2X, ΔζX, γΔζ, ΔζΔζ, γ2γ, Xζ2γ, Δζγ, γ∂ f ∂ f ∂ f∂X ∂X ∂X∂f ∂f *d ∂f= h + C: + ( β T − Mf+ γ)B :∂X ∂X ∂Xd= N= 0= 0∂ f=Δ ∂ X∂f= : N ∂ X= 0d: Nd(6.46)with:Ndtε= (6.47)tεNXdev= X devX(6.48)d 1d dB =t( I −N⊗N)d(6.49)X 1X XB =dev( I −N⊗N)X(6.50)⎛⎞∂fdev ∂fJ3X X( ) 2 2= I = ⎜1− 4m ⎟N + 2m N − m1devdev3∂X ∂X ⎜⎟3⎝X⎠(6.51)dev dev 2 dev( X ) f( X ) ( X )⎛∂f ⎞ ⎛∂f ⎞ ∂ f⎜I ⎜ ⎟⎟ I I I (6.52)∂X ∂X ∂X ∂X⎝∂X ⎠ ∂X⎜ ∂X ⎟ ∂X⎝ ⎠ ⎝ ⎠2∂ f ∂ ∂ ⎛ dev ∂ ⎞ dev ∂dev dev= ⎜ ⎟= = ⎜ ⎟=2 dev dev dev2111

( )( ) ( )2 dev X X2X2X∂ f X JN ⊗ N + N ⊗N3 X X= 12m24 ( N ⊗N) −4mdevdevdev∂XX⎛⎞⎜ J+ − ⎟ +⎜⎟⎝X⎠3 X1 4mB 2dev3mX X1N + N 1XdevX(6.53)with the square tensor product defined as ( A B) = Aik BjlA⊗ B = A B .tensor product ⊗ defined as ( ) ij klijkl and the dyadicijkl6.3.1.3 Consistent Tangent MatrixThe time-discrete model is completed by the calculus of the consistent tangent matrixwhich preserves the quadratic convergence of the Newton-Raphson method.The consistent tangent can be evaluated as the derivative of the stress σ :dσD =(6.54)dεFor brevity, only the saturated phase transformation problem is treated; in fact, theevolving phase transition can be obtained simply removing all terms related to theparameter γ .Recalling equation (6.39), the linearization of the elastic constitutive equation gives:⎛dεdεt⎞ ⎛ dεt⎞dσ= C ⎜ − ⎟: dε= ⎜ − ⎟:dε⎝dε dε ⎠ C ⎝ I dε⎠(6.55)112

If equation (6.42) is considered as a function of the quantities X ,corresponding linearization gives:Δ ζ , γ and ε , theX X X X⎧ d( R ) = R, X: dX+ R, ΔζdΔ ζ + R,ε: dε = 0⎪ Δζ Δζ Δζ⎨dR ( ) = R, X: dX+ R, ΔζdΔ ζ + R,ε: dε= 0⎪ γ γ γ γ⎩ dR ( ) = R, X: dX+ R, ΔζdΔ ζ + R,ε: dε= 0(6.56)where:RX, εR = 0Δζ, εR = 0γ, ε= −C(6.57)Accordingly, it can be written that:−1X X X⎡ dX ⎤ ⎡R, XR, R ⎤Δζ, γ ⎡−C⎤⎢ ζ ζ ζd ζ⎥ ⎢ Δ Δ Δ ⎥R, XR, ζR⎢, γ0⎥⎢Δ⎥=− ⎢ Δ ⎥ ⎢ ⎥: dεγ γ γ⎢⎣ dγ⎥⎦ ⎢R, XR, ΔζR ⎥⎣⎢, γ ⎦ ⎣ 0 ⎥⎦(6.58)The linearization of equation (6.39 4 ) gives:d( ε ) = ε : dX + ε dΔ ζ + ε dγ(6.59)t t, X t, Δζt,γin which:113

2∂ fε =Δζ∂ X∂fεt,Δζ= ∂ Xε = 0tX , 2t,γ(6.60)It follows that:d( ε ) = E : dε(6.61)twith:X X X⎡R, XR, R ⎤Δζ, γ ⎡−C⎤⎢ Δζ Δζ Δζ⎥E =−⎡εtX ,εt, ζ0 R, XR, ζR⎢, γ0⎥⎣⎤Δ ⎦⎢ Δ ⎥ ⎢ ⎥(6.62)⎢γ γ γR, XR, ΔζR ⎥⎣⎢, γ ⎦ ⎣ 0 ⎥⎦−1In conclusion, the consistent tangent matrix assumes the following form:D= C( I−E )(6.63)114

7 3D-1D Constitutive Model with 1D Evolution7.1 IntroductionThe full three-dimensional model described in chapter 6, characterized by 3D SMAconstitutive and evolutive laws, can be simplified in a 3D model governed by 1Devolutionary equations. This assumption arises from the observation that, in manycases, the SMA devices as wires or ribbons are subjected to a load condition that isprevalent along a particular direction. Moreover, shape memory alloys are often usedin layered composite plates, obtained as staking sequence of thin layers, some ofwhich contain SMA wires. The SMA wires or fibers can be suitably oriented in thelaminate in order to obtain the desired performance along certain directions of theplate. As a consequence, the highest value of the stress occurs just along thosedirections.7.1.1 Constitutive ModelIf the 3D model is specialized to the case of uniaxial phase transition in the SMAmaterial, it can assumed that the phase transformation tensor takes the followingform:εt⎡⎤⎢1 0 0 ⎥⎢⎥1= ϑ ⎢0 − 0 ⎥⎢ 2 ⎥⎢ 1 ⎥⎢0 0 − ⎥⎣ 2⎦(7.1)115

in which only one scalar variable ϑ is used to describe the phase transition occurringin the material.Because of the particular relationship between the components of the transformationstrain tensor,1εt.22 = εt.33 =− εt.112ε = ε = ε = 0t.12 t.13 t.23(7.2)the components of the transformation strain rate tensor satisfy the followingconstraints:⎧ ε⎨⎩ ε+ 2ε = 0+ 2ε = 0t.11 t.22t.11 t.33(7.3)Taking into account the evolutionary equation (6.32) for the internal variable ε t,relations (7.3) are not automatically satisfied; consequently, they can be solved infunction of the stress components. It can be proved that equations (7.3) are satisfiedwhen it is set:σ22= σ33 = σ12 = σ13 = σ23 = 0(7.4)into the derivative of the activation function f . In other words, the phase transitionresults simply ruled by the value of the stress component σ11= σl.Thereby, the derivatives of the yield function with respect to the thermodynamicforce are:116

*∂f6 3σ9 3 2 2 T Ml− hϑ+mz β −= −∂X 9z sgn( ϑ)z∂f∂X∂f∂X111 ∂f=−2 ∂X22 111 ∂f=−2 ∂X33 11f(7.5)with:( h( ) t ( 1 h( ))c)*β = ω β + − ω β( ( ) ( ))2* ⎛*h ⎞z = 2σ l− 3 β T − Mfsgn ϑ + 3ϑh⎜− 2σl + β T − Mf6 sgn( ϑ) + 3 ϑ⎟⎝2 ⎠(7.6)being sgn ( ϑ ) the logical function defined as:sgn( ϑ )⎧− 1 ifϑ< 0= ⎨⎩1 ifϑ≥ 0(7.7)As a consequence, the form of the thermodynamic force X components becomes:⎧ X11= σl−μ⎪1⎨X22 = X33= μ⎪2⎪ ⎩X12 = X13 = X23= 0(7.8)with α defined as:* 3 2μ = [ β T − Mf+ h ϑ + γ] sgn( ϑ)(7.9)2 3117

under the constrain:⎧ 30 ≤ ϑ < εL→ γ = 0⎪ 2⎨⎪ 3ϑ = εL→γ≥0⎪⎩ 2(7.10)It can be noted that the quantity εLdoesn’t directly represent the uniaxial maximumtransformation strain; instead it is equal to this deformation amplified of the quantity3 2.The components of the deviatoric part of the thermodynamic force tensor are:⎧dev 2⎪X11= σl−μ3⎪⎨ dev dev 1 dev 1 1X22 = X33 =− X11=− σl+ μ⎪2 3 2⎪ dev dev dev⎩X12 = X13 = X23= 0(7.11)so that the yield function depends only on the first component of the deviatoricthermodynamic force tensor as:dev dev 3 dev m devf ( X ) = f( X11 ) = X11 + X11−R(7.12)2 3Finally, the activation of the phase transformation, i.e. the evolution of the inelasticstrain (6.32) is ruled by the following associative law:118

⎡⎤⎢1 0 0 ⎥⎢⎥1ε t= ϑ ⎢0 − 0 ⎥(7.13)⎢ 2 ⎥⎢ 1 ⎥⎢0 0 − ⎥⎣ 2⎦with: fϑ = ζ ∂(7.14)∂ X1120 ≤ ϑ ≤ αL= εL(7.15)3The model is completed introducing the Kuhn-Tucker conditions: ζ ≥0 f ≤0 ζ f ≤0(7.16)7.2 1D Model and Analysis of Model ParametersFocusing the attention only on the SMA behavior along the direction of the stressand assuming the uniaxial evolutive law given by equation (7.14), the previousmodel can be classified as a one-dimensional constitutive model.With those assumptions, the influence of the parameters appearing in the expressionof the thermodynamic force on the phase transition is analyzed.In particular, with the proposed model, the straight lines that delimit the phasetransformation zones, i.e. the starting and finishing lines of the austenite-singleσl119

variant martensite transformation ( A S )reverse transformation ( S → A)→ and the starting and finishing lines of the, are identified.In a uniaxial stress -temperature diagram and for a tensile state of stress, the region inwhich the A→ S phase transition may occur is delimited by straight lines with thefollowing equations:AM,+ 3 3 1.5Rσfinal= βt( T − Mf) + ( hεL+ )2 2 1.5 + m 3(7.17)σAM,+start3 1.5R= βt( T − Mf) +2 1.5 + m 3(7.18)The reverse transformation, i.e. the Sdelimited by the straight lines with equations:→ A transition, may occur in the regionMA,+ 3 3 1.5Rσstart= βt( T − Mf) + ( hεL+ )2 2 m 3−1.5(7.19)σMA,+final3 1.5R= βt( T − Mf) +2 m 3−1.5(7.20)The special choice of the transformation surface form, together with the introductionofβtand βc, tensile and compressive material parameter, allows the modeling ofthe asymmetry of tensile and compressive behavior. In fact, the A → S and S → Aphase transformations in a compressive state of stress occur in the region delimitedby lines with different equations.In particular, for the A→ S transition, these lines take the following form:120

(7.21)σ3 1.5R=− βc( T − Mf)+2 m 3−1.5(7.22)while, for the S→ A transformation:(7.23)σAM,- 3 3 1.5Rσfinal=− βc( T −M f) −( hεL− )2 2 m 3−1.5AM,-startAM,- 3 3 1.5Rσstart=− βc( T −M f) −( hεL− )2 2 m 3+1.5AM,-final3 1.5R=− βc( T − Mf) +2 m 3+1.5(7.24)These relationships contain an explicit dependence on the model parameters.It is clear that the SMA model taken into consideration is able to represent, with areduced number of parameters, the asymmetric behavior typical of SMA materials.Moreover, the model parameters βt, βcand h show a useful and clear mechanicalmeaning:♦the parameters β tand β care essentially linked to the slope of the phasetransition lines in tension and compression, respectively;♦the parameter h essentially controls the translation of the same lines in thestress-temperature plane.Furthermore, simple mathematical relations that give the starting and finishingtemperature of the transition from single variant martensite to austenite at zero stresslevel, can be obtained.Starting from tensile tests it results:121

⎛⎞⎜⎟1 1.5RAs = Mf− ⎜ + hε⎟Lβ ⎜t⎛ m ⎞ ⎟⎜⎜−1.5⎟⎟⎝⎝6 ⎠ ⎠(7.25)Af= M −f1.5Rmβt( −1.5)6(7.26)and, in an equivalent way, starting from compressive tests it results:⎛⎞⎜⎟1 1.5RAs = Mf+ ⎜ −hε⎟Lβ ⎜c⎛ m ⎞ ⎟⎜⎜+ 1.5⎟⎟⎝⎝6 ⎠ ⎠(7.27)Af= M +f1.5Rmβc( + 1.5)6(7.28)From equations (7.25), (7.26), (7.27) and (7.28) it can be concluded that in order toguarantee the uniqueness of the starting and finishing temperature of the transitionfrom single variant martensite to austenite at zero stress level, i.e.A sandAf, intension and compression it is necessary that the parameters β tand β csatisfy aparticular ratio depending on m .It follows that, on the basis of experimental data, it is possible to identify the valuesto assign to the proposed constitutive model parameters in order to reproduce thebehavior of the material under consideration.122

It is interesting to evaluate how many experimental tests are necessary to completelyidentify the constitutive model parameters with a sufficient degree of accuracy:Two loading and unloading tensile and compressive uniaxial tests should beperformed at two different high enough temperature values in order to assess theposition of the phase transformation straight lines which define the starting and thefinishing of the austenite to single variant martensite and of the reverse transitions.Considering the results of these tests in terms of stress-strain, the pointscorresponding to the starting and the finishing of the phase transition at the twodifferent temperatures can be defined in the diagram σ − T . In this way the straightlines defining the forward and the reverse phase transformation can be obtained.These lines have a slope which is linked, in tension, to the model parameter βt. Inparticular, if p represents the slope of these lines it can be pointed out that:2βt= p(7.29)3The same result can be obtained in compression.The material constant ε Lcan be easily derived from a uniaxial loading-unloadingtest estimating the maximum inelastic strain attained in the case of elastic unloadingwhen the material is fully martensite.Furthermore, the lines corresponding to the start and the end of the A→ S and of theS→ A transitions are far from each other of a quantity equal to 3 hε 2L, so thatalso the parameter h can be evaluated.The lines corresponding to the single variant martensite to austenite transition crossthe temperature axis in two points indicating the two characteristic temperaturesAsandAf. For temperature aboveAfthe pseudo-elasticity range occurs: at the end ofeach loading-unloading cycle the material recovers all the deformation, and the123

components of the transformation strain tensor go back to zero. For the values oftemperature betweenAsandAfthe reverse transition is not complete.One uniaxial tensile and compressive test forT≤ M should be performed in orderfto obtain the stress-temperature relationship as straight lines which are parallel to thetemperature axis. In this way, the parametersσt,t,fσ , σcand σc,fof the σ − Tdiagram (see Fig. 7.1) corresponding to the starting and finishing stress thresholds ofthe multi-variant to single variant martensite transition in tension and compressionare obtained. It results in tension:σ =t1.5Rm+ 1.53(7.30)3σt,f= σt+ hεL(7.31)2and in compression:σ =c1.5Rm− 1.53(7.32)3σc,f= σc+ hεL(7.33)2Starting from the values of the uniaxial stress values at which the transformationfrom parent to product starts in tension and compression, respectively, the parameterR and m to introduce into the constitutive model can be identified.124

It can be noted that equations (7.30) and (7.32) are in perfect agreement with thedefinition given in equation (6.35) concerning the model parameters R and m .SAσ t,fMSβ tσ tASβ tAMMAM fM s A s A fσ cβ cσ c,fβ cFig. 7.1: Phase transition zones in tension and compression7.3 Numerical procedureThe derived material model is implemented into the finite element program FEAP(Zienkiewicz and Taylor, 1991). The time integration is performed adopting abackward-Euler implicit procedure. The algorithm presented in the following is asimplification of the one previously described for the three-dimensional case.7.3.1 Time Integration of the 3D-1D SMA Constitutive ModelThe time integration of the 3D-1D constitutive model gives:125

[ ]⎧ σl= E ε −ϑ⎪⎪ ⎛*3 ⎞ 2⎪X11= σl− ⎜β T − Mf+ h ϑ + γ sgn ( )2 ⎟ϑ3⎪ ⎝⎠⎪γ ≥ 0⎪⎨2⎪ϑ ≤ εL⎪3⎪dev 3 dev m dev⎪ f( X11 ) = X11 + X11−R≤0⎪⎪ 2 3dev⎩ Δ ζ ≥ 0 Δ ζ f( X11) = 0(7.34)where Δ ζ = ( ζ − ζ n ) is the plastic multiplier, time integrated over the interval [ , n ]t t .In the framework of a predictor-corrector algorithm, an auxiliary state obtainedfreezing the transformation phase flow, is introduced. In other words, assuming nophase transformation, a purely elastic trial step is considered as:ϑTRn⎛3 ⎞ 2μ =⎜β T − M + h ϑ + γ sgn ϑ2 ⎟⎝⎠ 3XX= ϑTRTRσl= E ⎡⎣ε −ϑ⎤⎦( )TR *TR TR⎜f⎟= σ −μTR TR TR11 l2= σ −μ3dev,TR TR TR11lTR dev, TR 3 dev, TR m dev,TRf ( X11 ) = X11 + X11−R2 3(7.35)The trial state is determined solely in terms of the initial conditions { , }εnϑnand thegiven incremental strain Δ ε . Besides, this state may not correspond to any actual,physically admissible state unless the incremental process is elastic. Therefore, if theTRTRtrial step is admissible, i.e. f < 0 , the step is elastic; otherwise, if f ≥ 0 , the stepis inelastic and the transformation strain has to be updated through the integration ofthe evolutionary equations.126

The inelastic step is solved by the return-mapping algorithm.First of all, the assumption γ = 0 is done, i.e. it is supposed that there is an evolvingphase transformation state in which 0 ≤ η < 2 ε . The equation (7.34) is rewritten3 Lin a residual form as:Δζ 3 dev m devR = X11 + X11− R= 0(7.36)2 3A new value of ϑ is evaluated solving this non-linear equation with a Newton-Raphson method.If the above solution is not admissible, i.e. ϑ > 2 ε 3L, equation (7.34) is rewrittenin a residual form assuming γ > 0 as:⎧Δζ3 dev m devR = X11 + X11− R=0⎪ 2 3⎨⎪γ 3R = ϑ − εL= 0⎪⎩2(7.37)The non-linear equation system of two scalar equations is solved with a Newton-Raphson method to compute a new value of ϑ and γ .As stated, the non-linear evolutionary problem is solved using an iterative method. Inparticular, only the case of the saturated phase transition, i.e. ϑ = 2 ε 3L, is treated,since the case of evolving phase transition can be simply obtained eliminating fromthe governing system of equations the last row.The iterative Newton-Raphson method requires the linearization of equation (7.37)as:Δζ Δζ Δζ⎪⎧ d( R ) = R, ηdη+ R,γdγ⎨ γ γ γ⎪⎩ dR ( ) = Rd, ηη + Rd, γγ(7.38)hence the computation of the matrix:127

⎡R⎢⎣ RRΔζΔζ, η , γγ γ, ηR,γ⎤⎥⎦(7.39)where the subscript comma indicates derivation with respect to the quantityΔfollowing the comma, i.e. R ζ, γmeans the derivation of the first scalar equation R Δζwith respect to γ and so on.The derivatives appearing in equation (7.39) assume the following form:RRRRΔζ, ϑΔζ, γγ, ϑγ, γ⎛1 d⎛3⎛3⎞⎞⎞⎜⎛⎞⎜ ϑ ⎜1− ⎜dϑ + d⎟⎟⎟⎟*⎛ 3 m ⎞⎜⎛βT − M2 2f+ γ ⎞⎜ ⎜ ⎝ ⎠ ⎟⎟⎟= h1⎝⎠⎜⎟⎜ ⎟⎟⎜+ 2 3 ⎟− − −⎝ ⎠⎜⎜ ε ⎟tεt⎟⎜⎝ ⎠ ⎜ ⎜⎟⎟⎟⎜⎜⎟⎟⎝⎝⎠⎠(7.40)= 0== 03 sgn2( ϑ )7.3.2 Consistent Tangent ModulusThe time-discrete model is completed by the calculus of the consistent tangentmodulus which preserves the quadratic convergence of the Newton-Raphson method.The consistent tangent modulus can be evaluated as a linearization of the stress σl:Edσdεlt= (7.41)128

For brevity, only the saturated phase transformation problem is treated; in fact, theevolving phase transition can be obtained simply removing all terms related to theparameter γ .Recalling equation (7.34) the linearization of the elastic constitutive equation gives:⎛dε dϑ⎞ ⎛ dϑ⎞dσ l= E⎜ − ⎟dε = E⎜1−⎟dε⎝dε dε ⎠ ⎝ dε⎠(7.42)If equation (7.37) is considered as a function of the quantities ϑ , γ and ε , thecorresponding linearization gives:Δζ Δζ Δζ Δζ⎧ ⎪d( R ) = R, ϑdΔ ζ + R, γdγ + R,εdε= 0⎨ γ γ γ γ⎪⎩ dR ( ) = Rd, ϑΔ ζ + Rd, γγ + Rd, εε = 0(7.43)where:2 ⎛ 3 m ⎞ΔζR,ε= E3 ⎜+2 3 ⎟⎝ ⎠Rγ, ε= 0(7.44)Accordingly, it can be written that:ζ ζ−1⎡2 ⎛ 3 m ⎞⎤Δ Δ⎡dϑ⎤ ⎡R, R ⎤ϑ , γ ⎢ E3 ⎜+2 3 ⎟⎥⎢ dεγ γdγ⎥ =−⎢R, ϑR⎢ ⎝ ⎠⎥⎣ ⎦ ⎢⎣ , γ ⎥⎦ ⎢0⎥⎣⎦(7.45)The linearization of the quantity ϑ gives:d( ϑ)= ϑ dϑ+ ϑ dγ = dϑ+ ϑ dγ(7.46), ϑ , γ , γ129

in which:ϑ, γ= 0(7.47)It follows that:d( ϑ)= Edε(7.48)with:ζ ζ−1⎡2 ⎛ 3 m ⎞⎤Δ Δ⎡R, R ⎤ϑ , γ ⎢ E+⎥E =−[ 1 0]γ γ3 ⎜ 2 3 ⎟R, ϑR⎢ ⎝ ⎠⎥⎢⎣ , γ ⎥⎦ ⎢0⎥⎣⎦(7.49)In conclusion, the consistent tangent modulus assumes the following form:t( 1 )E = E − E(7.50)7.3.3 Beam Finite ElementA finite-element formulation for a beam made of shape-memory material, whoseconstitutive behavior is described adopting the 3D-1D constitutive model, isdeveloped.A classical small-deformation Timoshenko beam theory is herein adopted, whosekinematics, schematically illustrated in Fig. 7.2, can be expressed as:u = 0u123= vu = w+yϕ(7.51)130

Fig. 7.2. Timoshenko’s beam theoryThe strain field is given by:ε0= w'χ = ϕ'γ = v ' + ϕ(7.52)The prime on a variable indicates its derivative with respect to z . The kinematicsstrain vector is introduced as:f⎛ d ⎞⎜ 0 00'dz⎟⎧ε⎫ ⎧ w ⎫ ⎜⎟⎧w⎫ ⎧w⎫⎪ ⎪ ⎪ ⎪ d ⎪ ⎪ ⎪ ⎪= χ ϕ' ⎜0 0⎟⎨ ⎬= ⎨ ⎬= v = L v⎜ dz ⎟⎨ ⎬ ⎨ ⎬⎪γ ⎪ ⎪v' ϕ⎪ ⎪ϕ⎪ ⎪ϕ⎪⎩ ⎭ ⎩ + ⎭ ⎜ ⎟d ⎩ ⎭ ⎩ ⎭0 1⎜⎟⎝ dz ⎠(7.53)Regarding the finite element implementation, an interpolation of the dependentvariables that yields a locking-free finite element element is considered. In particularthe axial displacement w , the transverse displacement v and the rotation ϕ areapproximated as:131

w= N w + N w + N ww w w1 1 2 2 3 3θv= N v + N v + N v + N θv v v1 1 2 2 3 3ϕ = N ϕ + N ϕ + N ϕϕ ϕ ϕ1 1 2 2 3 3(7.54)The quantities w iand v iwith i = 1, 2 ,3 represent respectively the axial and thetransversal displacements while ϕiare the rotations of the cross sections about the xaxis of the i -th node with i = 1, 2, 3 , as schematically represented in Fig. 7.3, whileN θis a cubic bubble function.In a compact form, the formulas (7.54) can be expressed as:⎧w⎫⎪ ⎪u= ⎨v⎬=NU (7.55)⎪ϕ⎪⎩ ⎭where:w w w⎛N1 0 0 0 N2 0 0 0 N30 0 0 ⎞⎜v v vθ ⎟N = ⎜ 0 N1 0 0 0 N2 0 0 0 N30 N ⎟⎜ϕ ϕ ϕ0 0 N1 0 0 0 N2 0 0 0 N30 ⎟⎝⎠U ={ w v ϕ 0 w v ϕ 0 w v ϕ θ}1 1 1 2 2 2 3 3 3T(7.56)with:NNNw1w2w31= ξξ ( − 1)21= ξξ ( + 1)22= 1−ξ(7.57)132

NNNv1v2v3N θ1= ξξ ( − 1)21= ξξ ( + 1)22= 1−ξ= ξξ −2( 1)(7.58)NNNϕ1ϕ2ϕ31= ξξ ( − 1)2= 1 ξξ ( + 1) 22= 1−ξ(7.59)⎛ L 2and z2 ⎟ ⎞ξ = ⎜ − , z∈ [0, L].⎝ ⎠ Lxwv11ϕ1ywv3ϕθ33wv2ϕ22zFig. 7.3. Beam finite element133

8 Numerical Results8.1 IntroductionIn Chapters 4, 6 and 7 several constitutive models in large and small deformationregime able to reproduce the SMA macro-behavior are presented, paying attention onthe time-discrete models and their algorithmic implementation within a finiteelementscheme in the code FEAP. Some numerical applications are developed inorder to assess the ability of the model to reproduce the shape memory alloysmechanical response and to verify the efficiency of the proposed procedures.8.1.1 Comparison between Large and Small DeformationModelsFirst of all, the following material properties for shape-memory alloys are assumed:E = 53000MPa ν = 0. 36 T = 245K ε = 0.04 h=1000MPa(8.1)β β σ σ0L-1 -1t= 2.1MPaKc= 2.1MPaKc= 72MPat= 56MPa Mf= 223K( ± )where E and ν are the SMA Young modulus and the Poisson coefficient,respectively. A 2D four node quadrilateral element is implemented.8.1.1.1 Uniaxial Response-Superelastic EffectA two-dimensional clamped element with dimensions L= 100mmand h=20mm(see Fig. 8.1) is subjected to a tensile loading-unloading history, inducing a fulltransformation of austenite into martensite. The temperature is kept constant at thevalue of T = 285Kand after the maximum value of the axial load 7 KN is reached,the load is removed allowing the element to recover the starting conditions. In order134

to analyze the two-dimensional element response a regular mesh characterized by 10elements in the axial direction and 1 element in the transversal one is considered.Fig. 8.2 shows the uniaxial force-displacement response of a point along the freeedge of the two-dimensional element for the case of small and finite strains. It isinteresting to observe: (1) the ability to completely recover the deformation duringthe loading-unloading cycle being at a temperature aboveAf, i.e. to reproduce thesuperelastic effect in a uniaxial state of stress; (2) the similar prediction of the smalland finite strains models except in the part of the curve corresponding to the end ofthe phase transition and the saturated phase transformation.hPLFig. 8.1: Geometry and boundary conditions of the cantilever135

700060005000Axial force [N]4000300020001000finite strainsmall strain00 0.5 1 1.5 2 2.5 3 3.5 4Axial displacement [mm]Fig. 8.2: Uniaxial test: superelastic behavior in tension. Axial load versus axial displacement8.1.1.2 Shear Response-Superelastic EffectThen, a pure shear analysis is developed.A two-dimensional clamped element with dimensions L and h= 20mm(see Fig.8.3) is subjected to a shear loading-unloading history, inducing a partialtransformation of austenite into martensite. In particular, the analysis is carried outfor two different values of the ratio h . The temperature is kept constant at the valueLof T = 285K. In the first case,hL= 15and after the maximum value of the shear load600 N is reached, the load is removed allowing the element to recover the initialconditions. In the second example,h 1L = 10and the maximum load value is equal to136

300 N . A regular mesh characterized by 40 elements in the axial direction and 10elements in the transversal one is considered.BThLFig. 8.3: Geometry and boundary conditions of the cantilever700600h/L=1/5500Shear force [N]400300200100small strainfinite strain00 2 4 6 8 10 12 14 16 18Transversal displacement [mm]137

350300h/L=1/10250Shear force [N]20015010050small strainfinite strain00 10 20 30 40 50 60Transversal displacement [mm]Fig. 8.4: Pure shear test: superelastic behavior. shear force versus transversal displacement forh 1L = 5(upper) and h 1L = 10(lower)Fig. 8.4 shows the shear force-displacement response of the two-dimensionalelement for the case of small and finite strains and for the two values of the ratio h L .The results regard the transversal displacement of the point B along the free edge ofthe element. It is interesting to observe: (1) the ability to completely recover thedeformation during the unloading cycle being at a temperature aboveA f, i.e. toreproduce the superelastic effect also in a shear state of stress; (2) the similarprediction of the small and finite strains models of the mechanical response for theh 1caseL = ; (3) the difference between the small and finite strain results for the case5h 1L = 10due to the fact that the bending effect are more significant.138

8.1.1.3 Superelastic and Shape-Memory EffectsThe last example is referred to a little two-dimensional SMA arch clamped along oneedge and free along the other. The geometry is given in Fig. 8.5 in which:R = 5mm R = 8mm(8.2)ieIn order to analyze the structural element behavior, a regular mesh characterized by10 elements in the radial direction and 40 element in the tangent ones is considered(Fig. 8.5).F AR iR eFig. 8.5: Geometry and boundary conditions of the clamped archThe arch is subjected to tension tests under stress control at three different values oftemperature, and in particular at T > Af, Ms< T < Asand T < Mf. Moreover, forthe case T< A also the shape memory effect is analyzed under temperature control.s139

In Fig. 8.6 the applied force is plotted versus the displacement of the point A alongthe free edge for the small and finite strains models.In the first case, i.e. T > A , the arch is subjected to an increasing value of the forcefuntil a maximum of 180N , maintaining constant the temperature at 285K ; then theload is removed allowing the element to recover the starting conditions. Fig. 8.5shows the undeformed and the deformed configurations of the analyzed structure.Therefore, if the material is loaded at a temperature aboveAfmacroscopic nonlinearlarge deformations occur and they are totally recovered during unloading,since the product phase, i.e. the single-variant martensite, is not stable at this value oftemperature. An hysteretic loop in terms of stress and strain (pseudolelasticity) isdescribed.In the second case, i.e. Ms< T < As, the arch is subjected to an increasing value ofthe force until a maximum of 120N , maintaining constant the temperature at 240K .Finally, in the third case, i.e.T≤ Mf, the maximum value of the axial force is 90Nand the temperature is maintained constant at 223K .For the cases in which T < A = 248Kalso the strain recovery induced by a heatingscycle is indicated. In fact, after the loading-unloading cycle, a residual strainremains. With a thermal cycle in which the temperature reaches the maximum valueof 400K and then reduces to the starting value of temperature, the shape memoryalloy is able to recover the original configuration going back to the point defined byzero stress and strain level.All the results underline the model capability to predict:♦ the superelastic effect;♦ the shape-memory effect.140

200180T=285 K160140120Force [N]10080604020small strainfinite strain00 0.2 0.4 0.6 0.8 1 1.2 1.4Displacement of point A [mm]140120T=240 K100Force [N]80604020small strainfinite strain00 0.2 0.4 0.6 0.8 1 1.2 1.4Displacement of point A [mm]141

140120T=223K100Force [N]80604020small strainfinite strain00 0.2 0.4 0.6 0.8 1 1.2 1.4Displacement of point A [mm]Fig. 8.6: Tension test: applied load versus displacement of point A for T = 285K(upper),T = 240K(center) and T = 223K(lower)Analyzing the load-displacement response computed using the small and largedeformation regimes it is possible to conclude that the large-deformation analysesgive very different results than the small-deformation one. In particular, during thelarge deformation analyses a progressive stiffening of the structure due to geometriceffects can be observed. Accordingly, for the particular structure under investigationthe geometric effects are non-negligible.142

8.1.2 Comparison between 3D and 3D-1D Constitutive ModelsThe 3D and 3D-1D constitutive laws presented for SMA are used to perform somenumerical applications and a displacement based formulation is developedconsidering a four-node solid 2D finite element and a three-node beam elementbased on the Timoshenko’s theory. In the first finite element the 3D full constitutivemodel is used to reproduce the shape-memory alloys behavior; in the beam finiteelement the 3D-1D constitutive model in which the SMA phase transition isgoverned by the axial stress in the material is adopted. The analysis is developedassuming a plane stress regime.The numerical integration along the beam length is performed using 3 Gauss pointsper element. The cross-section integrals are computed dividing each section in 100strips and using 3 Gauss points in each strip.The following material properties are assumed:E = 53000MPa ν = 0. 36 T = 245K ε = 0.04 h=1000MPa(8.3)β β σ σ0L-1 -1t= 2.1MPaKc= 1.8MPaKc= 72MPat= 56MPa Mf= 223K( ± )It can be noted that, on the basis of this set of material parameters, the followingSMA properties can be derived:σA = 230K A = 250Ks= 105MPaσ = 120MPat, fc,ff(8.4)8.1.2.1 Uniaxial responseFirst a uniaxial test is developed in order to show the ability of the constitutive modelto properly reproduce the superelastic and the shape-memory effect is considered.143

• Pseudo-Elastic EffectA clamped beam with dimensions L = 10mmand b= h= 1mm(see Fig. 8.7), with band h the dimensions of the rectangular cross section, is subjected to a simpletensile-compressive loading-unloading history, inducing a full transformation ofaustenite into martensite. In order to verify the robustness and the effectiveness ofthe proposed simplified 3D-1D constitutive SMA model, the analyzed element isdiscretized with both two-dimensional finite elements and three-node finite beamelements. In the first case, the SMA phase transformation is represented by the fullthree-dimensional constitutive laws and the mesh is characterized by 10 elements inthe axial direction and 1 element in the transversal one; in the second application thephase transition in the SMA is assumed just ruled by the stress along the applied loaddirection and the structure is discretized in 1 element.In particular, the temperature is kept constant at the value of T = 285Kand after themaximum value of the axial load 400 N is reached, the load is removed allowing theelement to recover the starting conditions.Fig. 8.8 shows the uniaxial force-displacement response for the two kind ofdiscretizations. In particular the results are in terms of the axial displacement of apoint along the free edge of the elements. It is interesting to observe: (1) the ability tocompletely recover the deformation during the loading-unloading cycle being at atemperature aboveAf, i.e. to reproduce the pseudo-elastic effect in a uniaxial stateof stress; (2) the ability of the proposed 3D-1D SMA model implemented in a threenodebeam element to reproduce the exact behavior of the material with goodconvergence properties and a reduced computational effort.144

PhLFig. 8.7: Geometry and boundary conditions of the cantilever400300200Axial force [N]1000-100-200-3002D solid elementbeam element-400-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5Axial displacement [mm]Fig. 8.8: Uniaxial test: superelastic behavior in tension. Axial load versus axial displacement• Shape-Memory EffectAim of the application is to demonstrate the possibility to significantly govern theaxial displacement of the considered elements performing temperature cycles in theSMA. Some tests using the two different constitutive SMA models, the 3D and the145

3D-1D simplified, are performed considering the same beam element of the previoussection. It is first subjected to an increasing value of the axial force at a constanttemperature equal toMf, until a maximum load value P is reached; then, atemperature cycle is prescribed, i.e. the following loading and temperature history isassumed:t[s] 0 1 2 3 4 5 6P[N] 0 250 0 -250 0 0 0T[K] 223 223 223 223 223 400 223In Fig. 8.9 the applied load axial is plotted versus the displacement of a pointbelonging to the free edge, computed using the two different constitutive models. Itis interesting to observe: (1) the presence of a residual strain at the end of theloading-unloading pattern being at temperature equal toMf; (2) the ability tocompletely recover the initial state (zero stress, zero strain) applying a temperaturecycle characterized by a heating process reaching a temperature aboveAfand then acooling process going back to the initial value of the temperature, i.e. to reproducethe shape-memory effect in a uniaxial state of stress.It can be noted that, in a uniaxial state of stress, the use of the simplified SMA modelin a beam finite element does not induce differences in the solution with respect tothe ones recovered adopting the full 3D SMA model.146

250200150100Axial force [N]500-50-100-150-2002D solid elementbeam element-250-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4Axial displacement [mm]Fig. 8.9: Uniaxial test: shape-memory behavior in tension. Axial load versus axial displacement8.1.2.2 Pure Bending Response• Pseudo-Elastic EffectThen, a pure bending analysis is developed.In particular, a clamped beam with dimensions L = 20mmand b= h= 1mm(see Fig.8.10) is subjected to a bending loading-unloading history, inducing a partialtransformation of austenite into martensite. The analyzed element is discretized withboth two-dimensional finite elements and three-node finite beam elements. In thefirst case, a regular mesh characterized by 40 elements in the axial direction and 10elements in the transversal one is considered; in the second one the element isdiscretized in 10 beam finite elements. In particular the temperature is kept constant147

at the value of T = 285Kand after the maximum value of the bending load 83 Nmmis reached, the load is removed allowing the element to recover the initial conditions.BhmLFig. 8.10: Geometry and boundary conditions of the cantileverFig. 8.11 shows the bending moment-displacement response of the beam discretizedwith two-dimensional finite elements together with the ones of the beam discretizedusing three-node beam finite elements. The results regard the transversaldisplacement of the point B along the free edge of the elements. It is interesting toobserve: (1) the ability to completely recover the deformation during the unloadingcycle being at a temperature aboveAf, i.e. the ability to reproduce the superelasticeffect also in a bending state; (2) the ability of the proposed 3D-1D SMA model toreproduce the bending behavior with good convergence properties and a reducedcomputational effort.148

908070Bending moment [Nmm]60504030201002D solid elementbeam element-100 2 4 6 8 10 12 14 16 18 20Transversal displacement [mm]Fig. 8.11: Pure bending test: superelastic behavior. Moment versus transversal displacement• Shape-memory effectIn this analysis, the beam element of the previous sections is first subjected to anincreasing value of the bending moment at a constant temperature equal to149Mf, untila maximum load value is reached. Then the following loading and temperaturehistory is assumed:t[s] 0 1 2 3 4m[Nmm] 0 30 0 0 0T[K] 223 223 223 400 0In Fig. 8.12 the applied bending load is plotted versus the transversal displacement ofthe point B belonging to the free edge, computed using the two different constitutivemodels. It is interesting to observe: (1) the presence of a residual strain at the end ofthe loading-unloading pattern being at temperature equal toMf; (2) the ability tocompletely recover the initial state (zero stress, zero strain) during the temperature

cycle in which the maximum value of temperature is greater thanA f, i.e. toreproduce the shape-memory effect also in a bending state of stress.It can be noted that even in a bending analysis the use of the simplified SMA modelin a beam finite element does not induce significant differences in the solution withrespect to the one recovered adopting the full 3D SMA model.35302D solid elementbeam elementBending moment [Nmm]25201510500 2 4 6 8 10 12 14 16Transversal displacement [mm]Fig. 8.12: Pure bending test:shape-memory behavior. Moment versus transversal displacement8.1.2.3 Tension-Compression TestThe last example is referred to the two-dimensional SMA arch clamped along oneedge and free along the other, whose geometry is given in Fig. 8.5.In order to analyze the mechanical response of the arch, it is discretized both withtwo-dimensional finite elements considering a regular mesh characterized by 10elements in the radial direction and 40 element in the tangent one and beam finiteelements considering a mesh of 10 elements.150

The arch is subjected to tension-compression tests under stress control at threedifferent values of temperature, and in particular at151T> Af, As T Af< < andT = M f. Moreover, for the case T < Asalso the shape memory effect is analyzedunder temperature control.In Fig. 8.13 the displacement of the point A along the free edge is plotted versus theapplied force for the two-dimensional and the beam models.In the first case, i.e.T > A , the arch is subjected to an increasing value of the axialfforce until a maximum of 18N , maintaining constant the temperature at 285K ; thenthe load is removed allowing the element to recover the starting conditions.Therefore, if the material is loaded at a temperature aboveAfmacroscopic nonlinearlarge deformations occur and they are totally recovered during unloading,since the product phase, i.e. the single variant martensite, is not stable at this value oftemperature. An hysteretic loop in terms of stress and strain (pseudolelasticity) isdescribed.In the second case, i.e. As < T < Af, the arch is subjected to an increasing value ofthe axial force until a maximum of 10N , maintaining constant the temperature at240K ; then the load is removed.Finally, in the third case, i.e.the temperature is maintained constant at 223K .For the cases in whichT ≤ M , the maximum value of the axial force is 7N ,sfT < A also the strain recovery induced by a heating cycle isindicated. In fact, after the loading-unloading cycle, a residual strain remains. With athermal cycle in which the temperature reaches the maximum value of 400K andthen it reduces to the starting value, the shape memory alloy is able to recover theoriginal configuration going back to the point defined by zero stress and strain level.The constitutive model is able to catch the asymmetric response of the materialduring tension-compression test in both the case of pseudoelasticity and shapememory effect. In particular, setting all the model parameters equal in tension and in

compression except from the stress starting threshold of the A→ S phasetransformation that is chosen higher in compression than in tension and theparameter β governing the chemical energy that is set higher in tension than incompression, the phase transition starts first in the compressive state.All the results underline the model capability to predict:♦ the pseudo-elastic effect;♦ the shape-memory effect;♦ the asymmetric material behavior in tension and compression.The simplified proposed SMA model is able to catch, as in the previous applications,the main features of SMA materials as the 3D one with good convergence propertiesand a reduced computational effort when the highest value of the stress occursmainly along one direction.201510Axial force [N]50-5-10-152D solid elementbeam element-20-4 -3 -2 -1 0 1 2 3 4Axial displacement [mm]152

10864Axial force [N]20-2-4-6-8beam element2D solid element-10-4 -3 -2 -1 0 1 2 3Axial displacement [mm]642Axial force [N]0-2-4-6beam element2D solid element-8-3 -2 -1 0 1 2 3Axial displacement [mm]Fig. 8.13: Tension-compression tests: axial load versus axial displacement of point A forT = 285K(upper), T = 240K(center) and T = 223K(lower)153

8.1.3 Experimental ComparisonThe ability of the model to simulate and reproduce experimental data is tested. Inparticular, the numerical response obtained with the proposed constitutive model iscompared with experimental data available in literature. The pseudo-elastic SMAwire behavior is studied. All the analyzed structures are assumed to be constituted ofa specific Ni-Ti wire produced by GAC International Inc.; they have a rectangularcross-section of dimension h= 0.64mmand b= 0.46mm. In particular, this specificalloy has been experimentally investigated by Airoldi and coworkers (Airoldi et al.,1995) in two different sets of experiments:♦ tensile conditions;♦ three-point bending conditions.Details regarding the testing equipment such as the characteristics of the testingapparatus can be found in the aforementioned reference.In particular, first, the mechanical SMA property E and ν together with themechanical parameters σcandandσtand the other material properties h , εL,Mfare obtained following the procedure described in the subsection 2.3 on thebasis of the experimental curves related to a series of tensile tests performed byAiroldi et al. 1995.Then, some numerical simulations are carried out using the same material parametersthroughout the analyses.Finally, the experimental data are compared with the numerical results obtainedadopting the constitutive model herein proposed.In particular, on the basis of the experimental results related to tensile tests, the SMAproperties in tension are set equal to:βt,βcE = 47000MPa ν = 0. 36 h= 10MPa ε = 0.08βL-1t= 4.8MPaK σt = 140MPa Mf= 278K As = Af= 303K( + )154

Some parameters characterizing the SMA compressive behavior are not available inthe reference Airoldi et al. 1995 as a consequence they are set on the basis of thevalues deduced by Auricchio and Sacco (Auricchio and Sacco, 1999). It results:c( )β = =-15.2MPaK ε −L0.06The numerical integration along the beam length is performed using three Gausspoint per element. The cross-section integrals are computed dividing each section inone hundred strips and using three Gauss point in each strip.A uniaxial test is first simulated to show the ability of the constitutive model toreproduce the pseudo-elastic effect. The reference temperature T 0is assumed equalto 323K . Keeping the temperature constant ( T = 323K), the material is subjected toa simple tensile loading-unloaging history, inducing only a partial conversion ofaustenite into martensite. Fig. 8.14 shows the uniaxial stress-strain response togetherwith the corresponding experimental data.Then, following Reference Airoldi et al. 1995, the behavior of a simply supportedbeam subjected to a pointwise central force is studied. The beam length isL= 14mm.A complete loading-unloading test is simulated at two different temperatures,T = T0 = 310Kand T = T0 = 328K. The comparison between the numerical solutionand the experimental results are presented in terms of the applied load versus themidspan inflection in Fig. 8.15 and 8.16, respectively.It is interesting to observe the ability of the numerical solution to reproduce the trendof the experimental response both for the tensile and the bending behavior. Inparticular, it can be pointed out that only in the unloading phase of the bending testwhen T = T0 = 310Ksome differences between the numerical and experimentalresults appear; in all the other applications the numerical results are in very goodaccordance with the experimental ones.155

450400350Axial stress [MPa]30025020015010050Numerical modelExperimental00 0.5 1 1.5 2 2.5 3Axial strain [%]Fig. 8.14. Uniaxial test: superelastic behavior. Axial stress versus axial strain. Comparisonbetween experimental data and numerical solution.654Applied load [N]321Numerical modelExperimental00 0.5 1 1.5 2 2.5 3 3.5 4 4.5Deflection [mm]Fig. 8.15 Three-point bending test at T = 310K: superelastic behavior. Applied force versusmidspan deflection. Comparison between experimental data and numerical solution.156

9876Applied load [N]54321Numerical modelExperimental00 0.5 1 1.5 2 2.5 3 3.5 4 4.5Deflection [mm]Fig. 8.16. Three-point bending test at T = 328K: superelastic behavior. Applied force versusmidspan deflection. Comparison between experimental data and numerical solution.157

CONCLUSIONSThe research on shape memory alloys results, nowadays, mainly devoted to thedevelopment and to the design of new advanced SMA-based devices.As a consequence, the research activity presented in this thesis work has beenfocused on the development of three-dimensional thermomechanical constitutivemodels for shape-memory alloys in finite and infinitesimal strain regimes.The models have been conceived within the framework of Generalized StandardMaterials: internal variables are introduced in order to describe the phase transitionprocesses occurring in SMA materials and two convex potentials are defined, fromwhich the constitutive relations and the evolution law for the transformation strainare derived. In order to obtain a formalism which is directly accessible to themethods of functional analysis, the approach of thermodynamics of irreversibleprocesses is adopted.In particular, the finite strain constitutive model assumes the temperature T and theright Cauchy-Green tensor C as control variables while the transformation rightCauchy-Green tensorCtis the internal variable. Then, the governing equations ofthe problem are linearized in order to obtain a constitutive formulation within theframework of small strain. The strain ε and the absolute temperature T are assumedas model control variables while the second-order transformation strain tensor ε trepresents the internal model variable. Next, the full three-dimensional model in thisway deduced, characterized by 3D SMA constitutive and evolutive laws, issimplified in a 3D model governed by 1D evolutionary equations. This assumptionarises from the observation that, in many cases, the SMA devices as wires or ribbonsare subjected to a load condition that is prevalent along a particular direction. Then,the 1D formulation consistent with the 3D model is used to clarify the physicalmeaning of the material parameters governing the constitutive model. Moreover, thetest to be performed to evaluate the material parameters are specified.158

The proposed constitutive models seem to be able to reproduce the basic features ofshape-memory alloys, such as superelaticity, shape-memory effect and differentmaterial behavior in tension and compression.Furthermore, the numerical implementations within a finite-element scheme arediscussed in detail for both the case of large and small deformation regimes. Adisplacement based formulation is developed considering a four-node solid 2D finiteelement and a three-node beam element based on the Timoshenko’s theory.Next, the ability of the model to simulate experimental data is assessed.In order to check the numerical algorithm capability and the viability of the proposedapproach, some numerical applications are developed. The numerical results showthat it is possible to use the proposed model for the simulation of SMA-baseddevices.In particular, from the simulations it is possible to remark that:♦ the proposed models can be used to design SMA devices as it is able todescribe the classical pseudo-elastic and the shape-memory effects;♦ the results obtained adopting small deformation hypothesis are significantlydifferent from the one determined by the finite strain analyses especially atthe end of the phase transition, i.e., for high value of the deformation.Therefore, it can be pointed out that the estimation of the state of stress incase of small deformation formulation can be very inaccurate for multi-axialloading problems;♦ the simplified proposed SMA model is able to catch the main features ofSMA materials as the 3D one with good convergence properties and areduced computational effort when the highest value of the stress occursmainly along one direction;♦ the constitutive model produces results which are in good agreement withexperimental data.159

The overall discussion leads to conclude that the proposed models represent aneffective approach for the simulation of the SMA behavior, since they are able tocatch most of the significant features of shape-memory alloys.160

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