THERMOMECHANICAL BEHAVIOUR OF

FUNCTIONALLY GRADED NITI WITH

COMPLEX TRANSFORMATION FIELD

BASHIR SAMSAM SHARIAT

B. ENG., M. ENG.

SCHOOL OF MECHANICAL AND CHEMICAL ENGINEERING

THIS THESIS IS PRESENTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY OF

THE UNIVERSITY OF WESTERN AUSTRALIA

2013

i

Abstract

This PhD thesis aims to investigate the thermomechanical behaviour of functionally

graded NiTi shape memory alloys (SMAs). Due to microstructural, compositional or

geometrical gradient, a complex transformation field is created within the SMA

structure. This provides gradient stress over stress-induced martensitic transformation,

widened controlling window for stress- and thermally-induced martensitic

transformations and better controllability of SMA element in actuation application.

Analytical and numerical models are introduced to predict the deformation behaviour of

such components under mechanical loads that are validated with actual experiments.

The analytical models provide closed-form solutions for global stress-strain variation of

such functionally graded alloys and can be used as effective engineering tools for

mechanism design.

The current thesis is presented in the form of research papers that are published or under

consideration for publication in scholarly international journals. It is categorised in the

following sections:

(1) Microstructurally graded 1D and 2D SMA structures

By applying designed heat treatment gradient along the length of a shape memory alloy

wire, transformation stress and strain gradients are created. Thus, the material exhibits

distinctive inclined stress plateaus with positive slopes, corresponding to the property

gradient within the sample. General polynomials are used to describe the transformation

stress and strain variations with respect to the length variable. Closed-form solutions are

derived for nominal stress-strain variations that are closely validated by experimental

data for shape memory effect and pseudoelastic behaviour of NiTi wires. The average

slope of the stress plateau is found to increase with increasing temperature range of the

gradient heat treatment.

The property gradient of the SMA plate is achieved by either a compositional gradient

or microstructural gradient through the thickness. The alloy behaviour can alter through

the thickness from shape memory effect to pseudoelasticity depending on the

ii

composition range and the testing temperature. Analytical model is established to

describe the deformation behaviour of such plates under uniaxial loading and during

recovery period. It is found that the martensitic transformation occurs partially over

nominal stress gradient unlike typical NiTi shape memory alloys. The analytical

solutions are validated with relevant experimental results.

(2) Geometrically graded 1D and 2D SMA structures

Analytical model is developed for the case of a uniformly tapered NiTi bar (or conical

wire), which describes the load-displacement relation of the sample at different stages

of loading cycle. The stress gradient for stress-induced martensitic transformation can

be adjusted by varying the taper angle. Proper experiment is conducted to compare with

the analytical solution.

Three types of geometrically graded NiTi strips linearly and parabolicly tapered along

their length are considered as sample geometries. This geometric gradient leads to local

stress gradient within the structure, which results in heterogeneous transformation

initiations throughout the sample. The geometrically graded sample exhibits positive

stress gradient for the stress-induced martensitic transformation. Closed-form solutions

are obtained for stress-strain variation of such components under cyclic tensile loading

and the predictions are validated with experiments. Finite element method,

implementing elastohysteresis model, is also attempted to simulate the local and global

behaviours that are verified with experimental data.

(3) Perforated NiTi plates under tensile loading

Perforated NiTi plates can be considered as geometrically graded SMA structures which

impose transformation localisation. This provides global stress gradient over stress-

induced martensitic transformation. Also, analysis of such structures provides a study

case as a 2D model for porous structures.

A computational model for deformation behaviour of near-equiatomic NiTi perforated

plates using elastohysteresis constitutive model and finite element method is presented.

In this model, the transformation stress is decomposed into two components: the

hyperelastic stress, which describes the main reversible aspect of the deformation

iii

process, and the hysteretic stress, which describes the irreversible aspect of the process.

It is found that, with increasing the level of porosity (area fraction of holes), the

apparent elastic moduli and the nominal stresses for forward and reverse

transformations decrease and the strain increases. Also, local strain distribution of the

NiTi plate is compared with that of a steel plate with similar geometry. The effect of

introducing holes into NiTi plates is explained by mean of mathematical expressions.

The effects of hole size, shape and number along the loading direction on pseudoelastic

behaviour are investigated through tensile experiments and modelling.

iv

Statement of candidate contribution (%)

The thesis contains research papers that are published or submitted for publication

consideration. The bibliographical details and the relative contributions of the candidate

to the papers are presented below. Each author has given permission for the work to be

included in this thesis.

Paper 1: Bashir S. Shariat (80%), Yinong Liu and Gerard Rio, Thermomechanical

modelling of microstructurally graded shape memory alloys, Journal of Alloys and

Compounds, Vol. 541, No. 407-414, 2012. (Chapter 2)

Paper 2: Bashir S. Shariat (70%), Yinong Liu, Qinglin Meng and Gerard Rio, Analytical

modelling of functionally graded NiTi shape memory alloy plates under tensile loading

and recovery of deformation upon heating, Under Review in Acta Materialia.

(Chapter2)

Paper 3: Bashir S. Shariat (80%), Yinong Liu and Gerard Rio, Mathematical modelling

of pseudoelastic behaviour of tapered NiTi bars, Journal of Alloys and Compounds, In

Press, doi: 10.1016/j.jallcom.2011.12.151. (Chapter 3)

Paper 4: Bashir S. Shariat (80%), Yinong Liu and Gerard Rio, Modelling and

experimental investigation of geometrically graded NiTi shape memory alloys, Smart

Materials and Structures, In Press. (Chapter 3)

Paper 5: Bashir S. Shariat (80%), Yinong Liu and Gerard Rio, Pseudoelastic behaviour

of perforated NiTi shape memory plates under tension, Under Review in Smart

Materials and Structures. (Chapter 4)

Paper 6: Bashir S. Shariat (80%), Yinong Liu and Gerard Rio, Numerical modelling of

pseudoelastic behaviour of NiTi porous plates, Under Review in Journal of Intelligent

Material Systems and Structures. (Chapter 4)

Candidate: Coordinating Supervisor:

v

Acknowledgements

I would like to thank my supervisors: Winthrop Professor Yinong Liu, Professor Gerard

Rio and Professor Hong Yang for their diligent guidance and continuous support during

my PhD study.

I would like to thank The University of Western Australia, particularly the School of

Mechanical and Chemical Engineering for their financial, technical and administrative

support.

I would like to thank The University of Western Australia for providing me with

Postgraduate Scholarship during 2008-2012 and Travel Awards for attending ICOMAT

2011 in Japan and PRICM 7 in Australia.

I would like to thank Laboratoire Génie Mécanique et Matériaux at Université de

Bretagne Sud in France for providing me with research fellowship during April – June

2009 through “France-Australia Science and Technology Linkage” Program, Grant

number: FAST080008.

I would like to thank the financial supports from the DIISRTE of the Australian

Government in Grant ISL FAST080008, Korea Research Foundation Global Network

Program in Grant KRF-2008-220-D00061 and French National Research Agency

Program N.2010 BLAN 90201 to this research.

My sincere thanks and appreciations are extended to my family for their kind support

and encouragement.

vi

Table of contents

Abstract i

Statement of candidate contribution iv

Acknowledgements v

Table of contents vi

Chapter 1: Introduction 1

1. Fundamentals of NiTi shape memory alloys 1

1.1. Thermoelastic martensitic transformation 1

1.2. Thermodynamics of thermoelastic martensitic transformation 4

1.3. Stress-induced martensitic transformation 6

2. Applications of NiTi shape memory alloys 7

2.1.SMAs as actuators 7

3. Thermomechanical behaviour of NiTi shape memory alloys 9

3.1. Lüders-like deformation 9

3.2. NiTi shape memory alloy under uniaxial loading 10

3.3. NiTi shape memory alloy under complex loading 11

4. Modelling of the thermomechanical behaviour 19

4.1. Phenomenological models 19

4.2. Modelling of porous SMA structures 20

4.3. Elastohysteresis model 21

5. Thesis objectives 27

6. Thesis overview 28

7. References 30

Chapter 2: Microstructurally graded 1D and 2DSMA structures 43

Paper 1: Thermomechanical modelling of microstructurally graded shape memory

alloys 45

Paper 2: Analytical modelling of functionally graded NiTi shape memory alloy plates

under tensile loading 67

vii

Chapter 3: Geometrically graded 1D and 2D SMA structures 93

Paper 3: Mathematical modelling of pseudoelastic behaviour of tapered

NiTi bars 95

Paper 4: Modelling and experimental investigation of geometrically

graded NiTi shape memory alloys 113

Supplement 1 to Paper 4: Additional experimental results for geometrically graded

NiTi strips with a wide range of width ratio 143

Supplement 2 to Paper 4: Finite element simulation of geometrically graded NiTi

strips with experimental validation 149

Chapter 4: Perforated NiTi plates under tensile loading 151

Paper 5: Pseudoelastic behaviour of perforated NiTi shape memory plates under

tension 153

Supplement to Paper 5: Numerical modelling of perforated NiTi plates based on

elastohysteresis model and finite element method 165

Paper 6: Numerical modelling of pseudoelastic behaviour of NiTi

porous plates 167

Chapter 5: Closing Remarks 183

1

Chapter 1

Introduction

1. Fundamentals of NiTi shape memory alloys

The phenomenon of shape memory effect in near-equiatomic NiTi shape memory alloys

(SMAs) was discovered in 1960s [1]. In the following decades, its unique features

attracted many researchers to investigate various metallurgical and mechanical aspects

of NiTi and to develop new alloys that exhibit the same effect. This has led to the

development of many engineering and technological applications of this effect. The two

most remarkable, and novel, properties of near-equiatomic NiTi are the pseudoelasticity

and shape memory effect [2, 3], which can also be observed at micro- and nano-scales

[4-6]. Pseudoelasticity refers to the recovery of large transformation strains (~6%)

spontaneously upon unloading. The shape memory effect is the ability of the alloy to

recover large inelastic mechanical deformation by the change, usually increase, of

temperature. In these processes, NiTi undergoes a martensitic transformation between a

parent phase (austenite) and a product phase (martensite).

1.1. Thermoelastic martensitic transformation

Martensitic transformation refers to a type of first order solid phase transformations that

involve no diffusion [7, 8]. Under such condition, the lattice change of the first order

transformation can only be realised by either shear motion of atomic planes or lattice

dilatation, or both. For polycrystalline alloys, a martensitic transformation involving

large volume change will inevitably proceed with tremendous internal mechanical

resistance and structural damages, such as cracking (e.g. martensite in high carbon

steels) and plastic deformation. Such transformations are often thermodynamically and

structurally irreversible. In contrast, transformations involving negligible volume

changes, or by pure shear, are often found highly reversible both thermodynamically

and structurally. These transformations are referred to as thermoelastic martensitic

2

transformations. Shape memory effect, which requires full structure reversion, thus

shape recovery, is exhibited by thermoelastic martensitic transformations.

In the process of a thermoelastic martensitic transformation by lattice shear, the shear

plane remains unchanged during transformation and is called the invariant plane. The

choice of an invariant plane is highly restrictive, often unique. That means the

transformation in a given alloy system always occurs by shear along that plane. Because

of this, the invariant plane is also known as the habit plane. The product phase

maintains certain crystallographic orientation relationship with the parent phase, known

as lattice correspondence [9, 10].

The difussionlessness and the unique shear direction imply that the lattice distortion at

the unit cell level is accumulated and manifested as the shape change of a crystal, or a

grain in a polycrystalline matrix. This inevitably causes large mechanical discontinuity,

or huge internal elastic strains, if all the grains in a polycrystalline matrix are allowed to

change freely. In order to minimize the strain energy caused by the crystal lattice

distortion, martensite is formed by altering its shear directions along the invariant plane.

The altering martensite domains are called variants [11]. Variants are heavily twinned,

and self-organized to provide a total zero macroscopic strain. For near-equiatomic NiTi,

the parent phase (austenite, A) has a B2 structure and the product phase (martensite, M)

has a B19’ structure [12, 13].

The lattice distortion due to martensitic transformation can be accommodated by

twinning [14]. The twin boundaries are so well-organised to allow easy deformation of

the lattice with no damage, providing zero macroscopic deformation. This structure is

called “self-accommodated martensite”. If sufficient mechanical load is applied to this

structure, the twin boundaries can move, converting one variant to another and

producing macroscopic deformation corresponding to the external loading. The product

structure is called “reoriented martensite”.

Fig. 1 illustrates schematically the thermomechanical behaviour of thermoelastic

martensitic transformation, including the pseudoelasticity, shape memory effect and

deformation by martensite reorientation. Illustration (I) shows the lattice of the parent

phase. Illustration (II) shows the lattice of a self-accommodating martensite, which

consists of twin-related martensite variants. Illustration (III) shows a single variant

3

martensite. It can be formed by deformation via detwining of the self-accommodating

martensite, known as martensite reorientation. This martensite is theoretically the same

as that formed by stress-induced martensitic transformation.

Fig. 1. Pseudoelasticity, shape memory effect, martensite reorientation and thermal

transformation behaviour of NiTi alloy

Illustration (IV) shows the behaviour of a thermally-induced martensitic transformation,

as in the case of NiTi, measured by differential scanning calorimetry (DSC). Here, the

two thermal flow peaks represent the forward A→M transformation on cooling and

reverse M→A transformation on heating, and sM and sA are the starting temperatures

and fM and fA are the finishing temperatures of the two transformations, respectively.

At 1T ( 1 fT A> ), the austenite can be converted to martensite by stress loading and

returned to the parent phase upon unloading. This feature is called pseudoelasticity. At

2T ( 2s sM T A< < ), NiTi exhibits shape memory effect upon heating the deformed

sample. In this case, the transformation strain in the deformed sample is only recovered

upon heating to above fA . When the austenite is cooled to below fM , self-

Stress loading

Unloading

Pseudoelasticity (T1)

Heating

Shape memoryeffect (T2)

Cooling Stress loading

Martensite Reorientation (T3)

(I) Parent phase (III) Deformed

(II) Twinned

T1T2T3

Austenite

Martensite

Mf Ms

As Af

M→A

M←A

(IV) Thermally inducedTransformation

4

accommodated martensite is produced. If mechanical loading is applied to the self-

accommodated martensite at 3T ( 3 fT M< ), the large macroscopic deformation is

appeared during martensite reorientation.

1.2. Thermodynamics of thermoelastic martensitic transformation

The free energy balance (G∆ ) of the system under uniaxial loading has been proposed

as [15]:

tG H T Sσερ

∆ = ∆ − ∆ − (1)

where H∆ and T S∆ are the changes of enthalpy and entropy energies, respectively. ρ

denotes the density of the material. σ and tε are the applied stress and transformation

strain, respectively. At equilibrium condition where 0G∆ = , the above equation gives:

00

tH T Sσ ε

ρ∆ = ∆ + (2)

where 0T and 0σ are the equilibrium temperature and stress of the martensitic

transformation, respectively. The effect of this thermodynamic approach on the

transformation is illustrated in Fig. 2(a). In this figure, Mf is the volume fraction of the

martensite and T denotes temperature. It is seen that A↔M transformations occur at the

constant temperature corresponding to the applied stress. Differentiating Eq. (2) with

respect to 0T yields:

0

0 0(0)t t

d S H

dT T

σ ρ ρε ε∆ ∆= − = − (3)

where 0(0)T is the equilibrium transformation temperature at zero stress (no mechanical

load). As ρ , tε and S∆ are all material and transformation properties, 0

0

d

dT

σ is a

constant which shows a linear relation between the transformation stress and

temperature as the driving forces of the transformation [16]. Eq. (3) is known as

Clausius-Clapeyron relation for thermoelastic martensitic transformation [17]. A wide

range of experiments have been carried out to verify the linearity of this relation,

practically all by measuring the relationship between the plateau stress for inducing the

martensite and the sM temperature, i.e. effectively SIM

S

d

dM

σ. A range of different values

5

for the slope 0

0

d

dT

σ have been obtained, varying between 5 and 8 MPa/K [18], partly

because of the variation in the actual transformation being measured (e.g., A→M or

R→M) and partly because of the difference between SIMσ and 0σ , which is dependent

on the metallurgical conditions of the matrix. Pseudoelastic cycling was found to

increase this slope, as the transformation strain tε is generally decreased by mechanical

cycling [18]. Also, it is found that the Clausius-Clapeyron slope depends on annealing

temperature [19]. For Ti-50.2at%Ni annealed at 703 K, the slopes were determined to

be 5.25 and 6.53 MPa/K for forward and reverse transformations, respectively [20].

Another thermodynamic approach is the phenomenological one which is based on

actual observation of hystoelastic behaviour of thermoelastic martensitic transformation.

The existence of thermal hysteresis requires an irreversible part in the free energy

equation in addition to the reversible part which accounts for thermoelastic behaviour of

the material. Thus, the free energy balance of the system can be expressed as [21, 22]:

el irG H T S E E∆ = ∆ − ∆ + ∆ + ∆ (4)

where, elE∆ and irE∆ are the elastic (reversible) and irreversible energies of the

transformation. elE∆ and irE∆ depend on not only the martensite volume fraction, but

also the metallurgical conditions and the martensite variant configurations. Therefore,

those values for thermally-induced transformation are different from those for stress-

induced transformation [16]. Fig. 2(b) illustrates the transformation behaviour based on

the second thermodynamic approach. In this figure, elE∆ is responsible for the

transformation interval, while irE∆ is responsible for transformation hysteresis.

Fig. 2. Schematic illustration of thermal transformation behaviour of NiTi based on two

thermodynamic approaches

0

1Mf

T

(a)

0 0( )T σ

0

1

fM

Mf

T

(b)

sM sA fA

6

1.3. Stress-induced martensitic transformation

The Clausius-Clapeyron relation, expressed by Eq. (3), provides a relationship between

stress and temperature as the driving forces of the martensitic transformation. The

negative sign implies that increasing stress is equivalent to decreasing temperature for

inducing martensite. Therefore, the thermoelastic martensitic transformation can be

induced by application of mechanical loading, i.e. stress, instead of decrease of the

temperature. In this case, stress is the driving force for inducing the A→M

transformation. The role of stress in thermodynamic balance of the NiTi component can

be observed in Eq. (2). Depending on the direction of the applied stress, the martensite

is formed in variants directed along the loading direction [11]. The applied stress must

be sufficient to induce the phase transformation, while remaining below the yield stress

of the lattice to avoid plastic deformation [23]. As the loading level reaches to the

critical transformation stress, the martensitic transformation occurs over a relatively

constant value of stress, resulting in a large transformation strain in a Lüders-like

manner [24, 25].

Fig. 3 shows the stress-strain diagram of a narrow strip of Ti-50.8%Ni alloy under

tensile loading. The sample is tested at three different temperatures. When tested at 303

K, the sample is in the parent phase (austenite) until the loading level reaches to ~360

MPa, which corresponds to point I. At this point, the A→M transformation starts and

propagates throughout the structure over a stress plateau. At point II, all structure has

transformed to the product phase (martensite), resulted in a large transformation strain.

Upon unloading, the structure is elastically deformed in martensite phase just before

point III, where the reverse martensitic transformation is initiated. The M→A

transformation is completed over a nearly constant stress level of ~125 MPa. At point

IV, the specimen is returned to the austenite phase and the transformation strain is

recovered. This behaviour is called pseudoelasticity or superelasticity, in recognition of

its large and non-linear but recoverable strain. It is also evident that the recoverability is

associated with a wide stress hysteresis. For this reason the behaviour is also called,

from a mechanics view point, hystoelastic behaviour. This unique feature opens the

door for many engineering and industrial applications of NiTi alloy. As observed in Fig.

3, the forward and reverse stress plateau levels highly depend on testing temperature.

They increase by increase of testing temperature, as per the Clausius-Clapeyron relation

expressed by Eq. (3). At high temperatures, where the transformation stress is beyond

7

the yield stress of the austenite phase, the structure undergoes plasticity before

martensitic transformation, and pseudoelasticity is not observed.

Fig. 3. Deformation behaviour of Ti-50.8%Ni alloy over stress-induced martensitic

transformation at different temperatures

2. Applications of NiTi shape memory alloys

SMAs can be regarded an apparatus that converts thermal energy into mechanical

energy, thus are often used as active materials in sensor and actuator designs. They have

applications in a variety of industrial sectors such as biomedical, automotive, aerospace

and oil exploration [26]. Their biocompatibility has been studied over the past two

decades, driven by the interest for medical application. In vitro and in vivo

investigations of NiTi in animal and human model systems revealed that appropriately-

treated NiTi is biocompatible [27]. However, coating technology may need to be

developed to increase the safety of NiTi alloy to prevent Ni release into the living

system [28]. The biomedical applications of NiTi includes orthodontic wires,

cardiovascular devices such as stents and filters, orthopaedic devices such as spinal

vertebrae spacers and artificial bone implants, and surgical tools such as guide wires and

grippers [26, 29-31].

2.1. SMAs as actuators

As a recovery force is generated in the direction of the recoverable shape change upon

the reverse transformation in a shape memory alloy, this force can be used to perform

work. Thus, the SMA component can be used as an actuator [32, 33] or a sensor [32,

34]. SMAs are capable of performing actuation task at the micro-scale [34-36]. SMA

0

100

200

300

400

500

0 0.02 0.04 0.06 0.08

Str

ess

(MP

a)

Strain

F

F

318T K=

311T K=

303T K=

I II

IIIIV

8

actuation mechanisms have the advantages, compared to other actuation mechanisms

like piezoceramics, solenoids and pneumatic systems, of being simple in mechanical

design, light in weight and compact in size, and high energy output density. In such

applications, SMAs are often used in the forms of thin wires (springs), tubes and plates

(films). These applications often involve control of actuation of the shape memory

component, either in stress or temperature.

Fig. 4 shows an example of SMA-actuating mechanism, which can be used as

automotive external side mirror. In this mechanism, the two DC motors of the

conventional side mirror are replaced by two pairs of the SMA wire. The typical rack

and pinion gear is substituted by a spherical joint. The new mechanism has less moving

parts, and the production cost is considerably reduced. Each SMA wire contracts when

Joule heated causing the mirror to rotate about one of the perpendicular axes, providing

its desired orientation. The temperature within the wire is controlled by the electric

current. A controlling algorithm is required to manage the actuation of the wires,

resulting in standard operation of the system [33].

Fig. 4. The design concept of a prototype mirror actuator [33]

Fig. 5 shows another example of SMA actuator design. It is for finger rehabilitation

treatment. After surgery of an injured flexor tendon of a finger, a dynamic splint should

be used to prevent tendon adhesion and unwanted deformation. The rehabilitation

period can take up to 4 months. In conventional dynamic splints, traditional

components, such as DC motor and accessories, were used to provide sufficient

dynamic forces. This made the device heavy and inconvenient to be used by the patient

for a long period of time. The proposed SMA-actuated splint is lighter, frictionless and

quiet. It employs Ti50Ni45Cu5 wires as SMA elements. The dynamic force is controlled

9

by a microcontroller, which adjusts the altering current in the wires to maintain required

heating temperature [37].

Fig. 5. The proposed layout of SMA actuator for finger rehabilitation [37]

3. Thermomechanical behaviour of NiTi shape memory alloys

3.1. Lüders-like deformation

Owing to the participation of the martensitic transformation, the deformation behaviour

of shape memory alloys is very different from that of conventional metallic materials.

Fig. 6 shows schematically comparison between a conventional generic metal alloy and

NiTi shape memory alloy in tension. The common metal exhibits typically a smooth

stress-strain curve involving elastic and plastic deformations. The plastic deformation is

characterized by a finite strain hardening coefficient which renders the deformation

mechanical stability. In contrast, the NiTi exhibits a large stress plateau in a Lüders-like

manner [25, 38, 39] (Stage II) prior to proceeding to the more conventional elastic

(Stage III) and plastic (Stage IV) deformations similar to those of the common metal.

The stress plateau is unique to shape memory alloys. The deformation over the stress

plateau is associated with stress-induced martensitic transformation and is thus sensitive

to temperature. In addition, the stress plateau represents a case of mechanical instability

of deformation with a zero strain hardening coefficient. It provides a discontinuity in

stress-strain curve. This unique behaviour and the distinctive mechanism of deformation

render shape memory alloys to obey very different laws. This has attracted much

attention in the past few decades for researchers to attempt to determine and to develop

10

theoretical understanding of the thermomechanical behaviour of shape memory alloys,

including NiTi. Such knowledge is fundamental and critical for the success of

application of these alloys.

Fig. 6. Comparison of deformation behaviour of SMA and all conventional metals

3.2. NiTi shape memory alloy under uniaxial loading

The fundamental difference in mechanical behaviour of SMA from that of conventional

metallic materials has attracted many researchers to investigate various aspects of the

deformation behaviour of NiTi in the past few decades. Most characterization of the

thermomechanical behaviour of NiTi has been done under simple loading conditions.

These include uniaxial tension and compression [40-57], which provide pure normal

stresses, torsion and pure shear testing [43, 58-60]. It is understood that the

transformation deformation is asymmetric between tension and compression. This

asymmetric feature is mainly due to detwinning of the martensite and low

crystallographic symmetry of its structure [47].

Fig. 7 shows the stress-strain variation of Ti-50.8at%Ni alloy under partial deformation

cycles of uniaxial tensile loading. The material demonstrates good pseudoelastic

behaviour, with a full deformation recovery of up to 8%, which is ~2% beyond the end

of the stress-induced transformation. Flat stress plateaus are observed over A↔M

martensitic transformations. It is known that the near-equiatomic NiTi exhibits good

cyclic stability under mechanical loading [61].

11

Fig. 7. Stress-strain variation of pseudoelastic NiTi alloy under partial deformation

cycles of uniaxial tensile loading

3.3. NiTi shape memory alloy under complex loading

One issue that has not been fully explored in the literature is the behaviour of NiTi

under complex loading. Sun et al. [54, 59] conducted systematic experimental

investigations on the effect of combined tension – torsion state on the deformation

behaviour of NiTi micro-tubes, in particular the occurrence of Lüders-type deformation.

They found that during uniaxial tension of the micro-tube, the transformation initiates

and propagates as a macroscopic spiral martensite band. During pure torsion, the alloy

exhibits axially homogeneous transformation field with monotonic hardening stress-

strain variation, and the transformation strain is much smaller than that of uniaxial

tension. McNaney et al. [62] conducted tension – torsion experiments on polycrystalline

NiTi thin-walled tubes using various loading paths (different ratios of tension and

torsion). Their experimental results reveal that the equivalent transformation stress level

and the equivalent transformation strain vary significantly by changing the loading path.

Wang et al. [63] studied the superelastic behaviour of NiTi thin-walled tubes under

combined tension and torsion. They concluded that the martensite formation rate is

higher in the sample under tension than that under torsion. This work was followed by

experimental investigation of NiTi under biaxial proportional and non-proportional

cyclic loadings [64]. It was discovered that the equivalent stress-strain curve during

proportional loading path is qualitatively similar to that during uniaxial loading. In non-

proportional loading, the deformation behaviour is quite different. Grabe et al. [65]

designed a novel device to apply tension – torsion test with temperature controlling

device. They obtained stress-strain plots of NiTi samples at different loading states.

0

100

200

300

400

500

600

0 0.02 0.04 0.06 0.08

Str

ess

(MP

a)

Strain

303T K=

12

Favier et al. [66] conducted a unique bulging test on a NiTi diaphragm, by which an

equibiaxial loading condition can be achieved. By this means, they analysed the

asymmetric behaviour of thin pseudoelastic NiTi plates with special attention to the

occurrence of transformation localisation. In this experiment, hydrostatic pressure is

applied on a clamped NiTi diaphragm. As shown in Fig. 8, the material along the rim of

the diaphragm is under uniaxial tension in the radial direction, the centre is under

equibiaxial loading, while the other parts in between are under various ratios of biaxial

tension. It was observed that the stress-induced martensitic transformation occurs at a

higher stress level at the centre of a NiTi diaphragm under bulging test compared with

uniaxial tensile testing. Also, the apparent elastic modulus prior to the forward stress

plateau under uniaxial tension was found to be smaller than the equivalent value for

equibiaxial state in the course of bulging. We can infer that the stress-induced A→M

transformation is retarded under equibiaxial tensile stress state [66].

Fig. 8. The distribution of stresses in a bulged NiTi diaphragm

3.3.1. Complex stress field

Complex loading condition can also be defined as the existence of complex stress field

within the SMA components. This can be maintained by application of uniaxial loading

on a SMA component with a complex structure. Fig. 9(a) shows real-life examples of

NiTi component with complex structure. One example is a porous or perforated

structure. A perforated structure undergoes a complex deformation when subjected to

uniaxial tensile loading. Owing to variations in geometry, different locations in the

structure may experience tension, bending, and even compression from the external

13

loading. Fig. 9(b) shows schematically a perforated plate under uniaxial tensile loading.

In this case, the plate can be modelled as a solid frame of thin elements. Here, element

A is under tension whereas element B is under bending. It is known that, unlike

conventional metallic materials, NiTi exhibits different behaviours in tension,

compression and shear [47]. Thus, the deformation behaviour of such structure is

expected to be inhomogeneous and complex.

Fig. 9. Complex geometries of NiTi; (a): porous cylinders and stent, (b): schematic

stress variation of a perforated plate under uniaxial tension

The deformation behaviour of porous SMA structures has been mainly studied under

compression [67-72]. The effect of porosity on global stress-strain variation has been

discussed. Li et al. [67] studied stress-strain behaviour of porous NiTi synthesised by

powder sintering. They found that the increase of sintering temperature increases the

pseudoelastic behaviour with decreased hysteretic width. They concluded that the

stress-strain variation of porous NiTi is not similar to those of bulk NiTi and other

porous materials. Porous NiTi does not show stress plateau like bulk NiTi. Greiner et al.

[68] produced porous near-equiatomic NiTi by powder metallurgy. The alloy exhibited

pseudoelasticity with recoverable compressive strains up to 6% and a maximum

compressive stress of 1700 MPa. The apparent modulus of elasticity of 16% porous

NiTi was found to be close to that of human bone. Zhao et al. [69] fabricated porous

14

NiTi alloy by spark plasma sintering. They examined the compression behaviour of the

alloy with the aim of its application as a high energy absorbing material. The higher-

porosity specimen demonstrated higher ductility and lower stress flow compared with

the lower-porosity sample. Zhang et al. [70] made porous NiTi alloys with gradient

porosity and large pore size. The sample structure could be effectively tailored by

changing the amount of NH4HCO3 as the temporary space-holder. The fabricated

samples with radial gradient porosity exhibited more than 4% of superelasticity.

Barrabés et al. [71] synthesised porous NiTi by self-propagating high-temperature

synthesis to be used for ingrowth of living tissues. They reported the mechanical

properties of the NiTi foam to be highly compatible with those of bone. Guo et al. [72]

investigated the compressive behaviour of 64% porosity NiTi alloy. Metallographic

analysis technique was used to monitor the metallographic changes during loading

process. It was observed that as the compressive load increases, many micro-cracks

were generated at the edges of holes due to stress concentrations close to the holes. At

regions far from holes, there existed no cracks to induce structural damage or phase

transformation.

Another example of complex stress field is bending of NiTi beams. A few experimental

works have been conducted on bending of SMA wires and beams [73-78]. When

bending, NiTi undergoes tension and compression at upper and lower surfaces,

respectively. For conventional materials, in elastic state, stress varies linearly across the

thickness in proportion to the strain, obeying the Hooke’s law of elasticity. In NiTi, the

stress-strain behaviour deviates from the classical linear elasticity, exhibiting a stress

plateau as shown in Fig. 7. In this case, the magnitude of the stress is limited by the

level of the stress plateau. As forward transformation is progressively initiated at each

layer, the local stress stops increasing while the strain continues to grow with the

increase of bending moment. Furthermore, the stress-strain behaviour of stress-induced

martensitic transformation of NiTi is asymmetric between tension and compression [47,

59]. NiTi has higher critical transformation stress but lower plateau strain in

compression than in tension. Thus, A→M transformation starts later in compression

side as the loading level increases. This asymmetric feature causes the neutral axis to

move from the central axis toward the compression side. This process results in a

different and very unique stress distribution of the NiTi beam for the same strain

distribution, as shown in Fig. 10. The stress at each layer restarts growing, when

transformation is completed within that layer. The whole transformation process causes

15

the structure moment-curvature relation to deviate from classical linear relation M

EIκ =

as soon as the phase transformation begins at the outer surface layer of the beam under

maximum tension.

In SMA beam bending, all the structure cannot be transformed to martensite as the part

of beam around the neutral axis experience low level of stress, not sufficient to induce

martensitic transformation. In SMA tubes, the transformation can be spread out to the

whole structure. Also, the local buckling due to compressive stress can be eased due to

low stiffness and occurrence of compressive transformation strain. In common metallic

tubes, local buckling or wrinkling occurs when the local compressive stress goes

beyond the critical buckling value. In SMA, if the compressive transformation stress of

the material is less than the buckling load, the highly compressed layers firstly

transformed to martensite with exhibiting compressive transformation strain according

to the level of bending moment before reaching the buckling stress value. In other

words, the structure collapse due to buckling is delayed in SMA tubes compared with

tubes of conventional materials.

Fig. 10. Stress and strain variations in a NiTi beam under bending

3.3.2. Complex transformation field in functionally graded NiTi structures

The SMA actuators are actuated by application of either stress or temperature. For

stress-induced B2-B19’ martensitic transformation, NiTi often exhibits Lüders-type

deformation behaviour, which is characterized by a flat stress plateau over a large strain

span of 5~7% (see Figs. 3 and 7). This presents a typical condition of mechanical

instability, in other words, inability for displacement (strain) control by controlling the

load (stress). Also, for thermally-induced transformation, NiTi has a narrow

transformation temperature range, typically ~5 K. The narrow window of the

controlling parameters, i.e. stress and temperature, weakens the controllability of SMA

element and is a challenge for actuator design. One solution to this problem is to use

MM

σε

Central axis

Neutral axis

16

functionally graded SMA. Due to existing complex transformation field in such

structures, the transformation occurs over wider intervals of external mechanical load or

temperature. This improves the controllability of SMA over martensitic transformation.

Functional gradients, more specifically gradients of the critical load or the critical

temperature for the transformation, can be created by either microstructurally [79, 80] or

geometrically [81] grading the SMA component.

In geometrically graded SMA, the structure geometry is graded in the loading direction

to create variation of cross-sectional area along the loading axis. This causes the

structure to experience progressive values of normal stress along the loading direction

during tensile loading. This stress gradient induces martensitic transformation

progressively within the graded sample as the loading level increases. The forward

transformation initiates at the high-stressed region corresponding to the lowest cross-

sectional area along the loading direction, and gradually propagates to the higher cross

sections as the load increases. This provides nonuniform transformation initiation in the

structure, resulting in the global stress-strain variation with positive stress-strain slope

over transformation which is deviated from the original flat stress plateau. In 1D SMA

structures, such as wires and bars, the structure can be uniformly tapered along its

length, creating 3D geometrical gradient of its shape. In 2D SMA structures, such as

films and plates, the geometrical gradient is obtained in planar direction by linear or

higher order variation of width with respect to the axial direction.

In microstructurally graded SMA, the gradient may be created by either compositional

variation or metallurgical variation. It is known that the transformation properties, i.e.

stress, strain and temperature, are highly dependent on NiTi composition and heat

treatment process [19]. Fig. 11 shows a schematic of microstructurally graded NiTi wire

made by gradient anneal along the wire length. 0x = and x L= correspond to the high

and low annealing temperatures, respectively. The effective annealing range has to be

designed based on the testing temperature and the yield strength of the material and the

requirement to observe full pseudoelastic behaviour. Owing to variation in annealing

temperature, the transformation stress tσ and strain tε vary in the longitudinal direction

as plotted in this figure, providing complex transformation field. The transformation

starts at the left end of the wire and propagate toward the right as the loading level

increases. This provides nominal stress gradient over stress-induced transformation.

17

The microstructural gradient can be created within the structure either along the loading

direction or perpendicular to the loading direction, e.g. 2D NiTi plates with

microstructural variation through the thickness. These two conditions constitute

correspondingly the serial connection and the parallel connection systems, thus different

mechanical behaviours. It has been reported that NiTi thin plates with microstructural

gradient in the thickness direction exhibit complex and unique shape recovery motion

upon heating after tensile deformation [79], referred to as the “fish tail” motion.

Fig. 11. Microstructurally graded NiTi wire with variation of transformation properties

Most of the studies on microstructurally graded SMAs have been carried out on multi-

layer or functionally graded NiTi-based films (or thin plates) [82, 83]. For 2D SMA

structures, i.e. plates and films, two suitable gradient directions can be assumed. One is

along planar loading direction, which has the same feature as explained for the case of

NiTi wire. Another approach is to maintain gradient across the thickness. A typical

approach to fabricate such NiTi films is to create a compositional gradient through the

film thickness by sputtering [84, 85]. It is well known that the variation in material

constituents in a typical functionally graded plate results in the variation of

thermomechanical properties [86-90]. In SMAs, this particularly leads to the variation

of transformation properties in the thickness direction. The alloy behaviour can alter

through the thickness from shape memory effect to pseudoelasticity because of

composition variation. The behaviour variation also depends on testing temperature.

Recently, compositionally graded thin NiTi plates have been created by surface

diffusion of Ni through the thickness of equiatomic NiTi plates [91]. This provides

composition range of 50.07-50.8 at.% for Ni during one-hour diffusion time. The

specimens exhibit reversible one-way shape recovery behaviour in a “fishtail-like”

motion.

( )t xε

( )t xσ

0 L

x

( )annealT x

18

Choudhary et al. [92] fabricated NiTi thin films coupled to ferroelectric lead zirconate

titanate (PZT) using magnetron sputtering technique. The intelligent structure exhibited

capability of demonstrating both sensing and actuating functions. The samples could

perform A→M phase transformation and polarisation-electric field hysteresis behaviour.

The transformation behaviour of these heterostructures was observed to highly depend

on NiTi film thickness. The hardness of the top NiTi layer of a fabricated film with

lower thickness was found to be affected by underneath PZT layer.

Birnbaum et al. [93] applied laser irradiation to functionally grade the shape memory

response and transformation aspects of NiTi films. Fabrication of thin functionally

graded NiTi plates has been reported by means of surface laser annealing [79]. Variation

of heat penetration through the plate thickness provides a progressive degree of

annealing, which results in a microstructural gradient within the thickness of the plate.

The plates exhibit a complex mechanical behaviour in addition to enlarged temperature

interval for thermally-induced transformation.

In the recent years, a few studies have been reported on the microstructurally grading of

NiTi wires. Mahmud et al. [94, 95] designed functionally graded NiTi alloys by

application of annealing temperature gradient to cold worked Ti-50.5at%Ni wires. As

the transformation properties are highly dependent on heat treatment conditions [96,

97], the gradient anneal imposes transformation properties gradient along the wire

length. Yang et al. [98] generated a spatially varying temperature profile by Joule

heating over a Ti–45Ni–5Cu (at%) wire. The graded sample demonstrated a low shape

recovery rate and stress gradient over stress-induced transformation. Park et al. [99]

applied time gradient annealing treatment on cold-worked Ti-50.9at.%Ni, and

investigated the shape memory behaviour via differential scanning calorimetry and

thermal cycling experimentation under constant load. They reported a temperature

gradient of 34 K in the R-phase transformation along the length of 80 mm of the

specimen due to time gradient annealing from 3 min to 20 min at 773 K. Meng et al.

[100] developed superelastic NiTi wires with variable shape memory properties along

the length direction by means of spatial electrical resistance over-ageing. The wire

exhibited two discrete stress plateaus over stress-induced martensitic transformation

during tensile testing.

19

4. Modelling of the thermomechanical behaviour

Numerous modelling and simulation works have been reported in the last three decades

on thermomechanical behaviour of SMAs. Each of these studies proposes a constitutive

model or extends an existing model for one or more aspects of SMAs properties. In

some studies, the proposed model is implemented in numerical codes, developed by

computational methods such as finite elements method, and the thermomechanical

behaviour of the SMA component is simulated. The main difficulty in the modelling of

SMAs is the fact that the local displacements and curvatures are engaged with

crystallographic transformation between two or more structural phases, and cannot be

directly described by classical constitutive theories.

Different approaches can be used to classify the existing models. The more common

one is to place them into two main categories: phenomenological macro-scale models

and micro-mechanical models. The phenomenological models are based on

experimental observations and do not consider microstructure of SMAs. They aim to

describe the global mechanical behaviour. Although they cannot describe SMA

microstructure, they are suitable for structure analysis and behaviour prediction, which

is a great advantage. In contrast, the micro-scale models focus mainly on parameters

like phase change kinetics and habit plane movement that are hard to be identified

experimentally. In the last category, we can define micro–macro-scale models [101-

104], which combine micro-mechanical parameters and macro-scale thermodynamics.

The local stress state is defined at micro-scale. Homogenisation techniques are

employed to obtain the macro quantities from the micro quantities [105]. This approach

deals with a large number of variables, which increase the computation cost and make

the engineering application of the model more difficult. Brief literature reviews of the

SMA models developed in the past decades have been presented by Lagoudas et al.

[106], Zaki and Moumni [107], Reese and Christ [108], Peng and Fan [109] and

Chemisky et al. [110].

4.1. Phenomenological models

The first phenomenological models, described 1D shape memory behaviour of SMA,

date back to 80’s [111]. In 90’s, the models took into account the thermodynamic

framework and martensite volume fraction. These models could predict more complex

thermomechanical features of SMAs [110]. These include the early constitutive models

proposed by Liang and Rogers [112], Brinson [113], Boyd and Lagoudas [114],

20

Leclercq and Lexcellent [115], Lexcellent and Bourbon [116], Abeyaratne and Kim

[117], and Auricchio and Lubliner [118]. Later, 3D constitutive models were developed

[107, 108, 110, 119-125]. In the past decade, researchers have attempted to implement

the developed constitutive models into finite elements codes [126-135]. This allows

them to describe the behaviour of SMAs with complex geometries or simple structures

under complex thermomechanical conditions. Some modelling works consider inelastic

behaviour upon cyclic loading and internal loops [136-141]. It is assumed that a fraction

of martensite is not recovered after each cycle, resulting in a permanent strain, which is

accumulated over the number of cycles [106].

4.2. Modelling of porous SMA structures

Several studies have been reported specifically on the modelling of porous SMAs [142-

147]. In general, the models aim to predict the global response of a porous SMA

component under compressive loading as a function of pore volume fraction. They are

mainly based on either using micromechanical averaging techniques [142-145] or

assuming a periodic distribution of pores [142, 146, 147]. In micromechanical

averaging method, the porous SMA is considered as a composite medium consisting of

parent phase as the matrix and randomly distributed pores as the inclusions. In this

method, the local variation of stress and strain cannot be investigated, although the

irregular distribution of pores is a near-to-fact assumption. In contrast, the other method

assumes a regular and periodic distribution of pores through the porous structure. This

arrangement deviates considerably from real porous SMAs, but allows the numerical

analysis to be reduced to that of a unit cell with appropriate boundary conditions. The

unit cell approach provides the possibility of obtaining the approximate values of local

quantities in periodic unit cells as presented by Qidwai et al [142]. Recently, Olsen and

Zhang [147] have studied the influence of micro-voids on the fracture behaviour of

shape memory alloys with low void volume fractions (<10%).

In part of this thesis, we explore the pseudoelastic behaviour of perforated plates, which

provides a study case of a 2D model for porous structures [148, 149]. The local and

global deformation behaviours of such structures are investigated by finite element

method, which implements the “elastohysteresis model”, and by analytical modelling

that are validated by tensile experimentation. It should be noted that the existence of

hole(s) in a SMA plate is considered as a geometrical defect to provoke transformation

localisation [150]. Therefore, a perforated SMA plate can be categorised as a

21

geometrically graded SMA structure, which creates gradient stress plateaus over A↔M

stress-induced martensitic transformations.

4.3. Elastohysteresis model

The elastohysteresis model is a phenomenological macro-scale model. However, the

deformation behaviour can be explained and analysed from a physical point of view.

The unique feature of this model is that it decomposes the SMA mechanical response

into hysteretic and hyperelastic parts as illustrated in Fig. 12. The hysteresis describes

the irreversible aspect and the classical elasticity, and the hyperelastic part describes the

reversible aspect of the transformation-deformation process, i.e. the pseudoelasticity

[151]. This approach is applicable not only for solids, but also for fluids, where we can

consider viscous and non-viscous irreversible power as well as reversible power. The

elastohysteresis model has been initially proposed by Guélin in 1980 [152]. The initial

objective was the modelling of shape memory alloys. Then, it has been applied for some

other materials like stainless steel, granular materials and elastomers. The model has

been theoretically studied and validated in 80’s. The theoretical investigations led to

two major documents by Favier and Pégon in 1988 [153, 154]. In 90’s, the simplified

expression of the behaviour has been implemented in numerical softwares [155-158].

This simplified version could precisely simulate the deformation behaviour of SMAs

with simple geometries. Since 2002, a new version of finite element code [159] has

been built and developed to take into account all aspects of the initial elastohysteresis

model and later propositions [160], facilitating the simulation of more complex

geometries.

Fig. 12. Decomposition of the SMA mechanical response to hyperelastic and hysteretic

contributions

-500

-300

-100

100

300

500

-0.06 -0.04 -0.02 0 0.02 0.04 0.06

Sh

ear

Str

ess

(MP

a)

Shear Strain

ElastohysteresisHyperelasticityHysteresis

22

4.3.1. Constitutive relations

The main objective of the elastohysteresis model is to simulate the deformation

behaviour of the SMA component under cyclic loops. In the case of monotonic loading,

the model does not provide much improvement, comparing with the classical

elastoplasticity modelling. The novelty of the model comes principally from the

hysteresis part. It introduces the concept of discrete (distinct) memorising of certain

events that appear during the loading history. It means that the behaviour at time t t+ ∆

depends on information at not only time t , but also several other older times. Thus, the

model cannot be represented only by differential equations in time, and must include

additional equations to take into account this specific memorised information. This

concept is in contrast with the classical plasticity theory and has taken a long time to be

approved by the mechanics community as an alternative of the classical concept.

Two additional concepts are considered in the modelling of the global behaviour.

Firstly, the decomposition of stress rather than that of deformation or strain is

considered in order to present the hysteresis and hyperelastic contributions of

deformation behaviour. Secondly, the idea of plasticity surface which separates the pure

elastic state from the elastoplasticity state is not applied. It means that even for a small

amount of loading, the irreversible part of the behaviour exists, although it is negligible.

The theoretical formulation of the model is so established to consider large geometrical

deformations that can be created by martensitic transformation.

The Cauchy stress tensor is expressed as the superposition of two stress components:

e hσ σ σ= + (5)

Here, eσ is the hyperelastic or the main reversible part of stress, which is time and

loading path independent. hσ is the pure hysteretic and irreversible part of stress, which

is deviatoric and time independent but dependent on the loading path. Eq. (5) defines an

elastohysteresis tensorial scheme with hyperelasticity and hysteresis contributions,

which allows simultaneous description of the reversible and irreversible phenomena.

Three particular states are considered to describe the kinematics: the initial

configuration at 0t = , the current configuration t and the next configuration t t+ ∆

corresponding to the next loading step. Each material point M is linked to a set of

23

material (constant) coordinates iθ . For each configuration, a local frame { }, iM g�

is

defined as:

a

i ai

Xg I

θ∂=∂��

(6)

where aX and aI ( 1..3a = ) are the global coordinates and the referential frame,

respectively.

The components of the Almansi strain and the rate of deformation are written as:

( )0

1 1( ) (0) ,

2 2t t

ij ij ij ij ijg t t g D gε+∆∆ = + ∆ − = ɺ (7)

where the components of the metric tensor g follows the classical tensorial calculus as:

. , . , .i i ij i jj j ij i jg g g g g g g gδ= = =� � � � � �

(8)

4.3.1.1. Hyperelastic contribution

The partial hyperelastic stress tensor is calculated from the hyperelastic potential

proposed by Orgéas et al. [161]. The potential ω is expressed as a function of three

invariants of the Almansi strain tensor: the relative variation of volume V , the intensity

of the deviatoric deformation Qε and the Lode angle εϕ defining the direction of the

deformation tensor in the deviatoric plane. These invariants have been chosen for their

ability to describe the kinematic values. They can be obtained from classical invariants

of the tensor. It is considered that there is no coupling between V and the other two

invariants. Here, we avoid presenting the potential and its expression based on the three

desired invariants, since it can be exhaustive and out of scope of this introduction. The

reader can refer to Ref. [161] for details. This potential depends on the bulk modulus

(equivalent to the elastic modulus: 1 2

EK

ν=

−) and 7 other parameters. Only one test,

for example a shear test, is required to determine all the parameters that are defined on

the stress-strain curve of Fig. 13. The hyperelastic stress component is written as:

( )1ij

ij

g

g

ωσ

ε

∂=

∂ (9)

To take into account the tension-compression asymmetry and more generally the

dependence of the transformation stress on the 3D loading condition (shear, tension and

24

compression), 1 2 3, , , , s eQ Qµ µ µ are considered to be dependent on εϕ . To fully

determine the parameters describing this dependency, three separate experiments are

required in tension, compression and shear conditions.

Fig. 13: Definition of hyperelastic parameters

4.3.1.2. Hysteresis contribution

The material is assumed to be isotropic and the hysteresis contribution is only

deviatoric. The constitutive relation is obtained by time integration of the relation:

2 tRDσ µ β σ= + Φ∆ɺ (10)

Here, σ is the deviatoric part of the Cauchy stress tensor (hysteresis contribution) and

σɺ is the Jaumann derivative of σ . D and Φ denote the deviatoric deformation tensor

rate and the intrinsic dissipation rate, respectively. µ corresponds to the Lamé

coefficient and β is a function of the Masing parameter. tR∆ is the variation between

the last reference state R and the current time t. This constitutive relation leads to three

independent parameters 0, , Q npµ that are illustrated in Fig. 14.

The management of the reference states is based on the monitoring of two quantities:

the intensity of the hysteresis contribution of the Cauchy stress and the intrinsic

dissipation rate.

Sh

ear

Str

ess

Shear Strain

1 2µ µ+

2sQ

1α 2µ

2 3µ µ+

2α

2eQ

25

Fig. 14: Definition of hysteresis parameters

4.3.2. Implementation of the model in finite element software

The elastohysteresis model is implemented in the finite element software called

Herezh++ [159]. This software is written in C++ language. It is capable of analysing

large deformations in solids which can be occurred due to material phase

transformation. Its objective is to be sufficiently flexible to easily adopt new concepts.

The various interesting concepts of object-oriented language were applied in the

software such as encapsulated data, template, static and dynamic polymorphism.

Standard Template Library (STL) is extensively used for containers such as lists,

adaptable arrays, stacks, associative containers and the basic associated algorithms such

as sorting, merging and suppression of redundancy.

We consider a non-dynamic equilibrium. The local balance equation is expressed by the

virtual power principle:

: . 0D D

P D dv T V ds Vσ∗ ∗ ∗ ∗

∂= − + = ∀∫ ∫ (11)

Here, D and D∂ are the solid volume and its boundary, where surface loading T is

prescribed. σ and D∗

are the Cauchy stress tensor and the virtual strain rate associated

with the virtual velocity V∗

. Eq. (11) must be satisfied for any virtual velocities.

In finite element method, the position of each physical point M (with respect to the

origin O) is estimated with the shape functions ( )r iϕ θ relative to the finite elements

containing the point:

Sh

ear

Str

ess

Shear Strain

µ

0

2

Q

np

26

OM ( ) ( )r arr i r i aM X Iϕ θ ϕ θ= = (12)

where summation convention on repeated index is applied. rM ( 1..nr = ) are the nodes

of the finite element. arX ( 1..3a = ) are the global coordinates of the node r . aI (

1..3a = ) are the 3 vectors defining the global referential frame. iθ ( 1i = , 1..2i = or

1..3i = according to the finite element dimension) are the local coordinates of point M

in the local referential frame of the element. In order to simplify the calculus, all the

elements of a specific type (e.g. linear triangle or quadratic hexahedron) are linked to a

unique reference element with iθ local coordinates. In particular, shape functions are

identical for all the elements of a specific type.

Considering deformations with the complex behaviour, elastohysteresis, the final set of

equations for each virtual velocity arV∗

is strongly nonlinear:

( )bss i bV V Iϕ θ

∗∗= (13)

Globally, Eq. (13) can be presented like:

( ) 0bsbsV R

∗

< > = (14)

where bsR represents the residual internal and external force for the node s and

direction b . Eq. (14) must be satisfied for all virtual velocities. Thus, we obtain a

general vector equation:

( )( ) 0arbsR X = (15)

where the final position of the node arX is unknown (degrees of freedom).

The simplest way to solve this nonlinear set of equation is to use the Newton-Raphson

technique, which requires calculating the gradient of these equations: ( )

( )bsar

R

X

∂ ∂

. In

order to calculate this gradient, we need to know the tangent evolution of the behaviour

relative to the position:

ij ijkl

ar arklX X

εσ σε

∂∂ ∂=∂ ∂ ∂

(16)

27

Quadrature operations use the well-known Gauss points, i.e. quadrature is estimated by

a pondered summation:

( )1

( ) ( )n

j jDj

f M dv f M W=

≈∑∫ (17)

( )jM are the Gauss point and jW are the Gauss ponderation. In order to simplify the

calculus, the quadrature is made in the local referential frame:

( ) { }( ) ( ) ( )1

( ) ( ) ( ) ( )n

i i ij ref j j jD Dref

elements j

f M dv f M J dv f J Wθ θ θ=

= ≈∑ ∑∫ ∫ (18)

The choice of the right number of Gauss points is a tricky problem. Of course, an

increased number improves the quality of the quadrature, but the computational time

increases at the same proportion. Therefore, the first objective is to use the smallest

possible number of Gauss points. But, if this number is too small, the stiffness matrix

becomes singular and leads to the occurrence of the well-known hourglass modes. This

is due to the fact that there are not enough independent values to generate the

coordinates of the matrix. Different techniques can be used to reduce the hourglass

effects, but generally no perfect solution exists. In Herezh++, an hourglass reducing

technique is applied which uses a simplified behaviour to quickly calculate a full

matrix, and add a small proportion of this matrix to stabilise the under-integrated

matrix.

For our simulations, we use linear pentahedral elements with 2 Gauss points. The

minimum rank of the stiffness matrix must be: 3 DOF per node µ 6 nodes - 6 rigid

nodes = 12. With 2 Gauss points and 6 independent values (deformations or stress

components) per Gauss point, we have 12 independent values. Thus, theoretically, no

stabilisation is required.

5. Thesis objectives

This thesis aims to investigate the thermomechanical behaviour of functionally graded

NiTi shape memory alloys with complex transformation field. It provides effective

engineering tools to predict the stress-strain variation of such structures in actuating

applications. This particularly includes:

(1) Thermomechanical modelling of microstructurally graded 1D and 2D SMA

structures and experimental validation

28

(2) Modelling and experimentation of geometrically graded 1D and 2D NiTi structures

(3) Numerical and analytical modelling of perforated NiTi plates under tensile loading

and experimental investigation

6. Thesis overview

This thesis is presented in the form of research papers that are published or under

review for publication in scholarly journals. The papers are placed in three chapters

according to their topics. Also, an introductory chapter is presented to provide the

background of the thesis topic and the literature review of the research and development

in the relevant areas. The contents of the chapters are briefly described below:

Chapter 1. Introduction

The first chapter is an introductory chapter. First, it presents the fundamental knowledge

of shape memory alloys, including martensitic transformation principles and

thermodynamics of transformation. A brief overview of SMA applications is presented.

Then, it introduces the thermomechanical features of NiTi and provides the current state

of research in the areas of simple loading, complex loading and functionally graded

SMA structures. A literature review of thermomechanical modelling of SMAs with the

consideration of thesis topic is provided. Finally, thesis objectives and the arrangement

of the thesis content are presented in this chapter.

Chapter 2. Microstructurally graded 1D and 2D SMA structures

Paper 1: Thermomechanical modelling of microstructurally graded shape memory

alloys, Journal of Alloys and Compounds, Vol. 541, No. 407-414, 2012.

Paper 2: Analytical modelling of functionally graded NiTi shape memory alloy plates

under tensile loading and recovery of deformation upon heating, Under Review in Acta

Materialia.

These two papers present analytical models to describe the deformation behaviour of

functionally graded 1D and 2D SMA components that are validated with relevant

experimental data. For 1D structure, the SMA wire is supposed to be microstructurally

graded along its length due to annealing temperature gradient. The nominal stress-strain

relation is then obtained based on linear and nonlinear variations of transformation

stress and strain. For 2D structure, a NiTi plate is considered to be microstructurally or

compositionally graded through the thickness. Closed-form solutions are obtained for

29

nominal stress-strain variations of such plate under uniaxial loading at different

deformation stages. Also, the curvature-temperature relations are established for

complex shape memory effect behaviour of the plate during recovery period.

Chapter 3. Geometrically graded 1D and 2D SMA structures

Paper 3: Mathematical modelling of pseudoelastic behaviour of tapered NiTi bars,

Journal of Alloys and Compounds, In Press, doi: 10.1016/j.jallcom.2011.12.151.

Paper 4: Modelling and experimental investigation of geometrically graded NiTi shape

memory alloys, Smart Materials and Structures, In Press.

Supplement 1 to Paper 4: Additional experimental results for geometrically graded NiTi

strips with a wide range of width ratio.

Supplement 2 to Paper 4: Finite element simulation of geometrically graded NiTi strips

with experimental validation.

This chapter provides modelling and experimental investigation of pseudoelastic

behaviour of 1D and 2D NiTi structures that are geometrically graded along the loading

direction. The geometrical gradient provides stress gradient through the SMA structure

along its length upon uniaxial loading. This results in stress gradients over stress-

induced A↔M transformations, which improve the controllability of SMA components

in actuation application. Analytical solutions are derived for deformation behaviour of

all proposed geometries. Tensile experimentations are conducted on a tapered NiTi bar

and linearly and parabolicly tapered NiTi strips with different width ratios. The model

can suitably satisfy the experimental results. It provides an effective tool for designing

SMA components with desired stress-strain slope and stress controlling window.

Also, finite element method is applied to numerically model the geometrically graded

NiTi structures based on elastohysteresis concept. The numerical results are validated

with experimental data related to tensile testing of different types of sample geometry.

Chapter 4. Perforated NiTi plates under tensile loading

Paper 5: Pseudoelastic behaviour of perforated NiTi shape memory plates under

tension, Under Review in Smart Materials and Structures.

Supplement to Paper 5: Numerical modelling of perforated NiTi plates based on

elastohysteresis model and finite element method.

30

Paper 6: Numerical modelling of pseudoelastic behaviour of NiTi porous plates, Under

Review in Journal of Intelligent Material Systems and Structures.

Perforated NiTi plate can be considered as a geometrically graded SMA structure which

imposes transformation localisation. This provides global stress gradient over stress-

induced martensitic transformation. Firstly in this chapter, the effect of introducing a

hole into a NiTi plate is explained by means of mathematical expressions. The effects of

hole geometry and number along the loading direction on pseudoelastic behaviour are

investigated through tensile experiments. Secondly, it presents a computational model

for deformation behaviour of near-equiatomic NiTi holey plates using elastohysteresis

constitutive model and finite element method. In this model, the transformation stress is

decomposed into two components: the hyperelastic stress, which describes the main

reversible aspect of the deformation process, and the hysteretic stress, which describes

the irreversible aspect of the process. It is found that, with increasing the level of

porosity (area fraction of holes), the apparent elastic modulus before and after the stress

plateau decrease, the nominal stresses for the A↔M transformation decrease and the

strain increases. The effects of hole size and number on global behaviour are

numerically investigated. Also, local strain distribution of the NiTi plate is compared

with that of a steel plate with similar geometry.

Chapter 5

This chapter is the closing remarks which provide the overall conclusion and

proposition of future work in this area.

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43

Chapter 2

Microstructurally graded 1D and 2D SMA

structures

Paper 1: Thermomechanical modelling of microstructurally graded shape

memory alloys

Paper 2: Analytical modelling of functionally graded NiTi shape memory alloy plates under tensile loading and recovery of deformation upon heating

44

45

Paper 1: Thermomechanical modelling of microstructurally graded

shape memory alloys

Bashir S. Shariata, Yinong Liua* and Gerard Riob

aLaboratory for Functional Materials, School of Mechanical and Chemical Engineering,

The University of Western Australia, Crawley, WA 6009, Australia

Tel: +61 8 64883132, Fax: +61 8 64881024, email: yinong.liu@uwa.edu.au

bLaboratoire d’Ingénierie des Matériaux de Bretagne, Université de Bretagne Sud,

Université Européenne de Bretagne, BP 92116, 56321 Lorient cedex, France

Journal of Alloys and Compounds, Vol. 541, No. 407-414, 2012.

Abstract

For better controllability in actuation applications, it is desirable to create functionally

graded shape memory alloys in the actuation direction. This is achieved by applying

designed heat treatment gradient along the length of a shape memory alloy wire,

creating transformation stress and strain gradients. This study presents analytical

solutions to predict the deformation behaviour of such functionally graded shape

memory alloy wires. General polynomials are used to describe the transformation stress

and strain variations with respect to the length variable. Closed-form solutions are

derived for nominal stress-strain variations that are validated by experimental data for

shape memory effect and pseudoelastic behaviour of NiTi wires. These materials exhibit

distinctive inclined stress plateaus with positive slopes, corresponding to the property

gradient within the sample. The average slope of the stress plateau is found to increase

with increasing temperature range of the gradient heat treatment.

Keywords: Shape memory alloy; NiTi; Martensitic transformation; Pseudoelasticity;

Functionally graded material; Modelling

1. Introduction

Shape memory alloys (SMAs) have been used in a wide range of engineering

applications including pipe couplings, sensors, actuators and medical devices [1]. They

have unique properties including shape memory effect and pseudoelasticity [2], which

can also be observed at micro and nano scales [3-5]. The shape memory effect is the

46

ability of the alloys to recover large mechanical strains, up to 8%, by increase in

temperature. Pseudoelasticity refers to the recovery of large non-linear strains

spontaneously upon unloading. In these processes, a shape memory alloy undergoes a

thermoelastic martensitic transformation between a parent phase (the austenite) and a

product phase (the martensite). The transformation proceeds by shear motion of atomic

planes. Owing to the participation of the martensitic transformation, the deformation

behaviour of SMAs is very different from that of conventional metallic materials. The

SMA exhibits a large stress plateau (as in the case of NiTi) in a Lüders-like manner

prior to proceeding to the more conventional elastic and plastic deformations similar to

those of the common metal [6, 7].

In some applications of SMAs, it is necessary that the shape memory component acts in

a controllable range with respect to the controlling parameters, i.e., temperature for

thermally-induced actuation or stress for stress-induced actuation. However, in typical

SMAs, e.g. equiatomic NiTi, the actuation range is narrow, which results in poor

controllability of the system. In the case of stress-induced transformation, the large

deformation occurs over the stress plateau [8], creating a situation of mechanical

instability. This instability is undesirable in many actuation applications. One way to

widen the controlling interval of the shape memory element is to create transformation

gradient across the desired direction of a SMA structure. This can be achieved by either

geometrically [9] or microstructurally grading a SMA component [10].

Most of the investigations on microstructurally graded SMAs have been focused on

multi-layer or functionally graded NiTi-based films [11]. A typical approach to fabricate

functionally graded NiTi films is to create a composition gradient through the film

thickness by sputtering [12, 13]. The variation in material constituents in functionally

graded plates results in variation of thermomechanical properties [14-18]. In SMAs, this

particularly leads to variation of transformation properties, i.e. forward and reverse

stresses and strains, in the thickness direction. The alloy behaviour can alter through the

thickness from shape memory effect to pseudoelasticity depending on the composition

range and the testing temperature. Birnbaum et al. [19] applied laser irradiation to

functionally grade the shape memory response and transformation aspects of NiTi films.

Choudhary et al. [20] fabricated NiTi thin films coupled to ferroelectric lead zirconate

titanate using magnetron sputtering technique. The transformation behaviour of these

heterostructures was observed to highly depend on NiTi film thickness.

47

In the recent years, a few studies have been reported on the microstructurally grading of

NiTi wires. Mahmud et al. [21, 22] proposed a novel design of functionally graded NiTi

alloys by application of annealing temperature gradient to cold worked Ti-50.5at%Ni

wires. As the transformation properties are highly sensitive to heat treatment conditions

[23, 24], the gradient anneal imposes graded transformation properties along the wire

length. They investigated the deformation behaviour of graded NiTi wires at different

annealing temperature ranges to obtain the effective range for a desired testing

temperature. Yang et al. [25] generated a spatially varying temperature profile by Joule

heating over a Ti–45Ni–5Cu (at%) wire. The Joule heated sample demonstrated a low

shape recovery rate of 0.02%/K and a Lüders-like deformation with a stress level

gradient from 340 MPa to 380 MPa. Park et al. [26] applied time gradient annealing

treatment on 30% cold-worked Ti-50.9at.%Ni, and investigated the shape memory

behaviour via differential scanning calorimetry and thermal cycling experimentation

under constant load. They reported a temperature gradient of 34 K in the R-phase

transformation along the length of 80 mm of the specimen due to time gradient

annealing from 3 min to 20 min at 773 K. They concluded that d

dT

ε of the functionally

graded sample is smaller than that of the isochronously annealed sample. They proposed

the most effective annealing temperature for time gradient heat treatment. Meng et al.

[27] developed superelastic NiTi wires with variable shape memory properties along the

length direction by means of spatial electrical resistance over-ageing. The wire was

partially over-aged by electrical resistance heating, providing the longitudinal variation

of mechanical properties. Two discrete stress plateaus over stress-induced martensitic

transformation were observed during tensile testing.

Recently, fabrication of thin functionally graded NiTi plates has been reported by means

of surface laser annealing [28]. Variation of heat penetration through the plate thickness

provides a progressive degree of annealing, which results in a microstructural gradient

within the thickness of the plate. The plates exhibit a complex mechanical behaviour in

addition to enlarged temperature interval for thermally-induced transformation. Also,

compositionally graded thin NiTi plates have been created by surface diffusion of Ni

through the thickness of equiatomic NiTi plates [29]. The specimens exhibit reversible

one-way shape recovery behaviour in a “fishtail-like” motion.

48

In previous modelling work, we have reported the nominal stress-strain behaviour of

geometrically graded, but property-wise uniform, NiTi alloys [9]. However, to date, no

theoretical model has been established to describe the stress-strain behaviour of

microstructurally graded SMAs. This paper presents unique closed-form relations for

nominal stress-strain variation of 1D SMA structures (i.e. wires and bars) with

longitudinally graded properties. The analytical work is validated with experimental

results.

2. Martensitic transformation parameters

Fig. 1 shows an ideal stress-strain diagram associated with the stress-induced

martensitic transformation of a typical SMA component. Six distinctive stages of

deformation in addition to the intrinsic transformation parameters are marked in this

figure. EA and EM are the elastic moduli of the austenite and martensite phases,

respectively. The forward and reverse transformation stresses are notified as tσ and tσ ′ ,

respectively. The forward and reverse transformation strains are defined as tε and tε ′ ,

respectively.

Fig. 1. Deformation stages and parameters of the pseudoelasticity of shape memory

alloys

3. Analysis for general polynomial gradient of transformation stress and strain

We consider a microstructurally graded SMA wire of diameter d and length L with

longitudinal variation of transformation stress and strain as illustrated in Fig. 2. This

variation can be created by gradient anneal of the sample in a furnace with designed

temperature distribution profile [21]. We assume the forward and reverse transformation

stresses and the forward transformation strain to be general polynomials of length

variable x. The reverse transformation strain can be expressed as a polynomial in terms

of the other transformation parameters by the geometrical relations shown in Fig. 1.

tε

tσ ′

tσAE ME

tε ′

σ

ε

(1)

(2)

(6)

(4)

(3)

(5)

49

( )

0

0

0

( )

( )

( )

1 1( ) ( ) ( ) ( )

ni

t ii

mi

t ii

pi

t ii

t t t tM A

x a x

x b x

x c x

x x x xE E

σ

σ

ε

ε ε σ σ

=

=

=

=

′ =

=

′ ′= − − −

∑

∑

∑ (1)

Fig. 2. Variation of transformation stress and strain in a microstructurally graded SMA

wire

The microstructurally graded sample is subjected to the tensile load F along its axis. The

nominal stress σ is defined as the axial force divided by the wire initial cross-sectional

area A:

2

4F F

A dσ

π= = (2)

The nominal strain is defined as the total elongation of the wire TotL∆ divided by the

initial length L [30]:

TotL

Lε ∆= (3)

The initial phase of the SMA component is considered to be 100% austenite. To

establish the nominal stress-strain relation of the microstructurally graded SMA, we

need to consider separate stages of the loading cycle. Stages (1-3) correspond to

loading, while Stages (4-6) correspond to unloading.

( )t xσ ′

( )t xε

( )t xσ

0 L

x

σ ε

d

50

3.1. Stage (1): 00 aσ≤ ≤

At this stage, the wire is entirely in the austenite phase as the applied load is less than

the critical value to initiate martensitic transformation at the left end of the wire shown

in Fig. 2. The nominal stress-strain relation is:

AE

σε = (4)

3.2. Stage (2): 00

ni

ii

a a Lσ=

< ≤∑

At this stage, the austenite to martensite transformation starts from the left end of the

wire and progressively propagates toward the right end as the loading level increases, as

schematically shown in Fig. 3(a). The structure consists of both austenite and martensite

regions, denoted by A and M. The displacement of the moving A-M boundary is defined

by variable -A Mx -(0 )A Mx L≤ ≤ . This stage ends when the A-M boundary reaches to the

right end of the wire with the highest transformation stress ( -A Mx L= ). At an instance

when the A-M boundary is at -A Mx , the nominal stress is:

-0

ni

i A Mi

a xσ=

=∑ (5)

The displacement of the loading point related to this stage can be expressed as:

A ML L L∆ = ∆ + ∆ (6)

where AL∆ and ML∆ are the elongations of the austenite and martensite regions related to

Stage (2). AL∆ is written as:

( )0-A A M

A

aL L x

E

σ −∆ = − (7)

The elastic elongation of a differential element at x in martensite area related to this

stage can be expressed in two parts. One part is related to the austenite period of the

element from the start of this stage to the instant when the stress level reaches to the

critical transformation stress of the element 0

ni

ii

a x=∑ . The other part is related to the rest

of this stage where the element is in martensite phase. ML∆ includes the elastic

elongation and the displacement due to martensitic transformation and is written as:

51

( ) ( )

( )

- - -

- - -

0 0 0

00 0 00 0 0

1 10 -

1 0

1 1( ) (0) ( ) ( )

1 1

1 1 1

1 1

A M A M A M

A M A M A M

x x x

M t t t tA M

x x x pn ni i i

i i ii i iA M

pni ii iA M A M A M

i iA M M

L x dx x dx x dxE E

a x a dx a x dx c x dxE E

a cx a x x

E E i E i

σ σ σ σ ε

σ

σ

= = =

+ +− −

= =

∆ = − + − +

= − + − +

= − + − + + +

∫ ∫ ∫

∑ ∑ ∑∫ ∫ ∫

∑ ∑

(8)

The total elongation of the wire from the start of loading is found by adding the total

displacement at the end of Stage (1) ( 0aσ = ) to L∆ related to Stage (2) and expressed

by Eq. (6):

1 1

0 0

1 1 1 1

1 1

pni ii i

Tot A M A M A Mi iA M M A A

a cL x x x L

E E i E E i E

σσ+ +− − −

= =

∆ = − + − + + + +

∑ ∑ (9)

Using Eq. (3), the nominal strain is found:

1 1

0 0

1 1 1 1 1 1 1

1 1

pni ii iA M A M A M

i iA M M A A

a cx x x

L E E i L E E L i E

σε σ+ +− − −

= =

= − + − + + + +

∑ ∑ (10)

Eqs. (5) and (10) can be used for plotting the nominal stress-strain diagram by variation

of A Mx − from 0 to L. Although, an ideal stress-strain cycle such as that shown in Fig. 1

has been considered for material behaviour, which is applied to the differential portion

of the wire (dx), the resulted nominal stress-strain variation of the structure is non-linear

as obtained from Eqs. (5) and (10).

Fig. 3. Microstructurally graded SMA wire undergoing transformation under tensile

loading; (a) forward transformation, (b) reverse transformation

L

F

A-M BoundaryxA-M

M A

x dx

L

F

A-M BoundaryxA-M

M A

x dx

(a)

(b)

52

3.3. Stage (3): 0

ni

ii

a Lσ=

>∑

At this stage, all structure has transformed to the martensite and undergoes linearly

elastic deformation with martensite modulus of elasticity EM. The total elongation with

respect to the initial (unloaded) condition is obtained by adding the total displacement at

the end of Stage (2), found by substituting A Mx L− = and 0

ni

ii

a Lσ=

=∑ in Eq. (9), to the

elastic deformation of this stage: 0

1 ni

iiM

a L LE

σ=

−

∑ . The corresponding nominal strain

is then obtained as:

0 0

1 1

1 1

pni ii i

i iM A M

a cL L

E E E i i

σε= =

= + − + + +

∑ ∑ (11)

3.4. Stage (4): 0

mi

ii

b Lσ=

>∑

This stage is related to the unloading in fully martensite phase. The strain varies versus

stress according to Eq. (11) until the stress level decreases to that required for reverse

transformation of the right end of the wire.

3.5. Stage (5): 00

mi

ii

b b Lσ=

< ≤∑

In this period, the reverse M→A transformation begins at the right end and the A-M

boundary continuously moves to the left as the load decreases, as shown in Fig. 3(b). At

an instance when it is at -A Mx , the nominal stress is:

-0

mi

i A Mi

b xσ=

=∑ (12)

The displacement of the loading point during this stage can be expressed by Eq. (6),

where ML∆ is written as:

( )( )0

1 1 mi

M t A M i A MiM M

L L x b L xE E

σ σ σ− −=

′∆ = − = −

∑ (13)

and AL∆ is for the reversely transformed area and takes into account the transition from

martensite to austenite of each differential element and the overall reverse

transformation strain through following equation:

53

( ) ( )

0 0 0

0 0 0

1 1( ) ( ) ( ) ( )

1 1

1 1

1

A M A M A M

A M A M

A M

L L L

A t t t tM Ax x x

L Lm m mi i i

i i ii i iM Ax x

L p n mi i i

i i ii i iM Ax

M

L x L dx x dx x dxE E

b x b L dx b x dxE E

c x a x b x dxE E

bE

σ σ σ σ ε

σ

− − −

− −

−

= = =

= = =

′ ′ ′ ′∆ = − + − −

= − + −

− − − −

= −

∫ ∫ ∫

∑ ∑ ∑∫ ∫

∑ ∑ ∑∫

( ) ( ) ( )

( )

1 1

0 0

1 1

0

1

1 1

1

pmi i ii

i A M A M A Mi iA

ni ii

A MiM A

cL L x L x L x

E i

aL x

E E i

σ + +− − −

= =

+ +−

=

− + − − −+

+ − − +

∑ ∑

∑

(14)

Using Eqs. (6), (13) and (14) and taking into account the displacement at the end of the

previous stage (0

mi

ii

b Lσ=

=∑ ), the nominal strain similar to Eq. (10) is found, where σ is

defined by Eq. (12). Here, Eqs. (10) and (12) provide stress-strain relation in terms of

variable A Mx − varying from 0 to L.

3.6. Stage (6): 00 bσ≤ ≤

All structure has returned to the austenite and recovers elastically to the original shape

according to Eq. (4).

Eqs. (4), (5), (10), (11) and (12) describe the stress-strain behaviour of the

microstructurally graded SMA wire based on general polynomial variation of

transformation stress and strain.

4. Closed-form stress-strain relation for linear variation of transformation stress

and strain

The set of equations obtained in the preceding section can be reduced to those based on

linear variation of transformation stress and strain by setting m, n and p equal to 1. To

establish the linear functions of forward and reverse transformation stresses and forward

transformation strain, it suffices to know their values at both wire ends, as illustrated in

Fig. 4(a). The coefficients of Eqs. (1) are defined as:

54

2 10 1 1

2 10 1 1

2 10 1 1

,

,

,

a aL

b bL

c cL

σ σσ

σ σσ

ε εε

−= =

′ ′−′= =

−= =

(15)

In the case of linear variation of forward and reverse transformation stresses, the A-M

boundary variable -A Mx applied in Stages (2) and (5) can be expressed in terms of the

nominal stress σ from Eqs. (5) and (12), respectively, and substituted in Eq. (10) to

form explicit stress-strain relations for these stages, providing more convenient

engineering tool for stress-strain prediction. Using Eqs. (15), the general equations of

the previous section result in a set of descriptive stress-strain equations for different

stages of the loading cycle:

Stage (1): 10 σ σ≤ ≤ and Stage (6): 10 σ σ ′≤ ≤

Eq. (4) is applied.

Stage (2): 1 2σ σ σ< ≤

( ) ( )2

12 1 1 11

2 1 2 1 2 1

1 1 1 1

2 M A A AE E E E

σ σε ε ε σε σ σσ σ σ σ σ σ

− −= − + + + − + − − − (16)

Stage (3): 2σ σ> and Stage (4): 2σ σ ′>

( ) 1 21 2

1 1 1

2 2M M AE E E

ε εσε σ σ += − − + +

(17)

Stage (5): 1 2σ σ σ′ ′< ≤

( )

( )

2

12 1 1 2

2 1

11 1

2 1

1 1 1

2

1 1

M A

M A A

E E

E E E

σ σε σ σ ε εσ σ

σ σ σσ σ εσ σ

′−= − − − + − ′ ′−

′−+ − − + + ′ ′−

(18)

As observed, linear variation of transformation stresses and strains results in a quadratic

variation of nominal strain ε versus nominal stress σ in Stages (2) and (5).

55

Fig. 4. Microstructurally graded SMA wire: (a) linear variation of transformation stress

and strain along the length; (b) linear variation of transformation stress and quadratic

variation of transformation strain along the length

5. Closed-form stress-strain relation for linear variation of transformation stress

and quadratic variation of transformation strain

The annealing temperature profile along the length of the sample can be so designed to

achieve non-linear variation of transformation stress and/or strain in the length direction.

Assuming linear variation of transformation stress, the set of general polynomial

equations obtained in Sec. 3 yields explicit stress-strain relations for higher order

variations of transformation strain. In this section, we present the solution for the case of

linear transformation stress and quadratic transformation strain ( 1, 2m n p= = = ) as

shown in Fig. 4(b). Coefficients ai and bi are defined in terms of the end values as in

Eqs. (15).

Stage (1): 10 σ σ≤ ≤ and Stage (6): 10 σ σ ′≤ ≤

Eq. (4) is applied.

( )t xσ ′ ( )t xε

( )t xσ

0 L

x

σ ε

1σ ′

2σ ′1σ

2σ

2ε

1ε

( )t xσ ′2

0 1 2( )t x c c x c xε = + +

( )t xσ

0 L

x

σ ε

1σ ′

2σ ′1σ

2σ(b)

(a)

56

Stage (2): 1 2σ σ σ< ≤

( ) ( )3 22

1 02 1 1 11

2 1 2 1 2 1 2 1

1 1 1 1

3 2 M A A A

cc L c L

E E E E

σ σσ σ σε σ σσ σ σ σ σ σ σ σ

− −= + − + + + − + − − − −

(19)

Stage (3): 2σ σ> and Stage (4): 2σ σ ′>

( )2

2 11 2 0

1 1 1

2 3 2M M A

c L c Lc

E E E

σε σ σ

= − − + + + +

(20)

Stage (5): 1 2σ σ σ′ ′< ≤

( )

( )

3 222 1 1

2 1 12 1 2 1

11 0

2 1

1 1 1

3 2

1 1

M A

M A A

c Lc L

E E

cE E E

σ σ σ σε σ σσ σ σ σ

σ σ σσ σσ σ

′ ′− −= − − − − ′ ′ ′ ′− −

′−+ − − + + ′ ′−

(21)

As observed in Eqs. (19) and (21), the strain ε is expressed as a cubic function of stress

σ in Stages (2) and (5).

6. Validation of the analytical model with experimental data

6.1. Material properties of gradient annealed Ti-50.5at%Ni wire

The presented analytical solution for the stress-strain relation of the microstructurally

graded SMA is compared with the experimental work reported by Mahmud et al. [21,

22, 31]. They carried out tensile testing on gradient-annealed NiTi wires in addition to

isothermally-annealed samples at different temperatures. Ti-50.5at%Ni wires with

diameters of 1.5 mm were annealed in the furnace with a temperature distribution

profile ranging from 630 K to 810 K over the full gauge length of the samples (100 mm)

as shown in Fig. 5 [21]. The tensile testing machine was equipped with a liquid bath for

temperature control, and the strain rate of ~10-4/sec was applied.

Fig. 6 shows the effect of annealing temperature on the tensile deformation properties of

a Ti-50.5at%Ni wire isothermally-annealed after cold work, with (a) showing the

variation of the forward and reverse transformation plateau stresses and corresponding

trend-lines, and (b) showing the forward transformation strain over the stress plateau

[31]. The tensile testing was conducted at 313 K.

57

Fig. 5. Temperature distribution profile of the furnace over the full gauge length of NiTi

sample [21]

Fig. 6. Effect of annealing temperature on B2-B19’ martensitic transformation

deformation properties of Ti-50.5at%Ni: (a) effect on the forward and reverse

transformation stresses at 313 K; (b) effect on the forward transformation strain

600

650

700

750

800

850

0 20 40 60 80 100Te

mpe

ratu

re (

K)

Sample Length (mm)

0

100

200

300

400

500

600

700

550 600 650 700 750 800 850 900

Str

ess

(Mpa

)

Annealing Temperature (K)

Forward Transformation

Reverse Transformation

1.58 1552.71t Tσ = − +

2.97 2170.81t Tσ ′ = − +

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

550 600 650 700 750 800 850 900

For

war

d T

rans

form

atio

n S

trai

n

Annealing Temperature (K)

7 2 35.76 10 1.01 10 0.37t T Tε − −= − × + × −

(a)

(b)

58

Using the temperature distribution profile depicted in Fig. 5 and also the stress-

temperature and strain-temperature relations defined in Fig. 6, the variation of

transformation stresses and strain along the full sample length is obtained. Fig. 7 shows

the variations of the forward and reverse transformation stresses (Fig. 7(a)) and the

forward transformation plateau strain (Fig. 7(b)) fitted with linear and quadratic trend-

lines with respect to the length variable x. At higher annealing temperatures where

shape memory effect is observed instead of pseudoelasticity, we can mathematically

assume negative values of reverse transformation stress varying consistently with that in

the positive range. In the descriptive equations, stresses are in MPa while variable x is in

mm. As observed, 2nd-order polynomials perfectly describe the variation of

transformation stresses and strain across the wire length.

Fig. 7. Properties of gradient annealed Ti-50.5at%Ni allow wire: (a) variation of the

forward and reverse transformation stresses along the sample length; (b) variation of the

forward transformation strain along the sample length

-300

-200

-100

0

100

200

300

400

500

600

0 20 40 60 80 100

Str

ess

(MP

a)

Sample Length (mm)

Forward Transformation

Reverse Transformation

2.89 238.97t xσ = +

20.02 0.75 267.44t x xσ = + +

5.42 297.58t xσ ′ = −

20.04 1.41 244.09t x xσ ′ = + −

0.04

0.05

0.06

0.07

0.08

0 20 40 60 80 100

For

war

d T

rans

form

atio

n S

trai

n

Sample Length (mm)

6 2 5 24.04 10 8.98 10 7.57 10t x xε − − −= − × + × + ×

4 23.14 10 8.11 10t xε − −= − × + ×

(a)

(b)

59

6.2. Full length sample tested at 313 K

The NiTi wire at the full length of 100 mm, annealed at the temperature gradient of 630-

810 K, was tested at 313 K [31]. Fig. 8(a) shows the comparison of the experiment

result with the analytical solutions described by Eqs. (4), (16), (17) and (18) based on

linear transformation stress and strain variations, and Eqs. (4), (19), (20) and (21) based

on linear transformation stress and quadratic transformation strain variations. Fig. 8(b)

shows the comparison of the same experimental data with the analytical solution based

on quadratic transformation stress and strain variations described by Eqs. (4), (5), (10),

(11) and (12) and setting 2m n p= = = . For this case, the analytical stress-strain curve

at Stages (2) and (5) can be plotted by variation of -A Mx over the full gauge length of

the sample and obtaining corresponding stress values from Eqs. (5) and (12),

respectively, and related strain values from Eq. (10). The elastic moduli of austenite and

martensite are determined from experimental stress-strain diagrams of isothermally-

annealed samples [31] as 22AE GPa= and 28ME GPa= and assumed to be constant

for different annealing temperatures. The other parameters in the applied equations are

determined from trend-line equations in Fig. 7 for linear and quadratic stress and strain

variations:

1 2

1 2

1 2

20 1 2

20 1 2

1 20 1 2

239 , 528

298 , 245

0.081, 0.05

267.44 , 0.75 , 21.35

244.09 , 1.41 , 40.12

0.076, 0.090 , 4.038

MPa MPa

MPa MPa

a MPa a GPa m a GPa m

b MPa b GPa m b GPa m

c c m c m

σ σσ σε ε

− −

= =′ ′= − == =

= = =

= − = =

= = = −

(22)

As observed in Fig. 8(a), the analytical solution based on the quadratic variation of

transformation strain predicts the overall forward transformation strain more precisely

than the solution based on linear variation of transformation strain. The deviation of

both analytical solutions from experimental curve can be noticed in forward and reverse

gradient stress plateaus (Stages (2) and (5)) and particularly in the amount of residual

strain. This is due to the linear assumption of transformation stress variation which

cannot perfectly describe the variation of actual forward and reverse transformation

stresses throughout the sample length as shown in Fig. 7(a). This deviation is effectively

reduced by applying quadratic variation of transformation stress and strain as illustrated

in Fig. 8(b). In this figure, the analytical solution perfectly predicts the experimental

stress-strain curve and the non-recovered strain.

60

Fig. 8. Comparison of the analytical solution with the experimental data for the full

length sample tested at 313 K [31]: (a) linear transformation stress and linear and

quadratic transformation strain; (b) quadratic transformation stress and strain

6.3. Partial length sample tested at 313 K

A fresh NiTi wire was annealed in the full gauge length of the furnace with the

temperature gradient profile shown in Fig. 5. Then, 30 mm of its length was removed

from the high temperature end giving annealing range of 630-783 K over its remaining

length of 70 mm. Tensile testing was performed on this sample [31]. Fig. 9(a) presents

the comparison of our analytical model based on linear and quadratic variations of

transformation stress and strain with the experimental data obtained at 313 K. To

determine the transformation parameters, the actual stresses and strain variations with

respect to the wire length shown in Fig. 7 are used, considering the origin of variable x

at 30 mm from high-temperature end of the full length sample and corresponding linear

and quadratic trend-lines. Note that the descriptive trend-lines for this case are different

from what plotted in Fig. 7 for the full gauge length of the wire. The linear and

quadratic transformation parameters for the partial length sample are defined as:

0 0.02 0.04 0.06 0.08 0.10

100

200

300

400

500

600

Nominal Strain

Nom

inal

Str

ess

(MP

a)

Quadratic StrainLinear Strain

Experiment

(a)

Ti-50.5at%NiAnnealing range: 630-810 KTensile testing at 313 K

0 0.02 0.04 0.06 0.08 0.10

100

200

300

400

500

600

Nominal Strain

Nom

inal

Str

ess

(MP

a)

(b)

Ti-50.5at%NiAnnealing range: 630-810 KTensile testing at 313 K

61

1 2

1 2

1 2

20 1 2

20 1 2

1 20 1 2

298 , 544

186 , 274

0.076, 0.047

309.2 , 2.03 , 21.35

165.7 , 3.82 , 40.12

0.075, 0.152 , 4.038

MPa MPa

MPa MPa

a MPa a GPa m a GPa m

b MPa b GPa m b GPa m

c c m c m

σ σσ σε ε

− −

= =′ ′= − == =

= = =

= − = =

= = − = −

(23)

As observed in Fig. 9(a), the gradient plateau starts at higher stress level for this sample

comparing with that for full length sample (in Sec. 6.2); since the maximum value of the

annealing temperature is lower in this case. Also, the residual strain is smaller as higher

volume fraction of the wire is in pseudoelastic range compared with the full length

sample. The quadratic transformation stress and strain assumption describes the

deformation behaviour of the NiTi wire more accurately than the linear one.

Fig. 9. Comparison of the analytical solution with experimental data for the partial

length sample based on linear and quadratic variations of transformation stress and

strain: (a) tested at 313 K; (b) tested at 333 K [31]

0 0.02 0.04 0.06 0.08 0.10

100

200

300

400

500

600

700

Nominal Strain

Nom

inal

Str

ess

(MP

a)

Quadratic Stress and Strain

Linear Stress and Strain

Experiment

Ti-50.5at%NiAnnealing range: 630-783 KTensile testing at 313 K

0 0.02 0.04 0.06 0.08 0.10

100

200

300

400

500

600

700

Nominal Strain

Nom

inal

Str

ess

(MP

a)

Ti-50.5at%NiAnnealing range: 630-783 KTensile testing at 333 K

(1)

(2)

(6)

(4)

(3)

(5)

(b)

(a)

62

6.4. Partial length sample tested at 333 K: Full pseudoelastic behaviour

As seen in the section above, the gradient annealed Ti-50.5at%Ni retains unrecovered

strain after deformation at 313 K. This is because that the portion of the wire at the high

temperature end demonstrates shape memory effect instead of pseudoelasticity. By

increasing the testing temperature, forward and reverse transformation stress levels

increase in all parts of the graded wire, leading to full pseudoelasticity at 333 K [31].

Here, we apply our model based on quadratic stress and strain variations to describe this

experimental result for the partial length sample of the preceding section. Since,

individual experiments on isothermally annealed samples of the same wire are not

available at 333 K, we determine some parameters, such as moduli of elasticity and the

stress values at the start and the end of forward and reverse plateaus, from the actual

stress-strain diagram of the gradient-annealed sample tested at 333 K. These stress

values define the corresponding values of high and low annealing temperature ends.

Considering linear change of forward and reverse transformation stresses versus

annealing temperature and using nonlinear temperature distribution profile of Fig. 5, the

variation of transformation stresses with respect to the length variable is obtained which

can be fitted with a quadratic curve. As reported by Tan et al. [32], the transformation

strain increases averagely by about 0.04% per 1 K increase in testing temperature.

Using the actual strain variation of Fig. 7(b) for the relevant range and considering

0.008 of strain increase due to change of testing temperature from 313 K to 333 K, the

final transformation strain variation relative to the variable x is established. All the

parameters for analytical formulations are defined as:

20 1 2

20 1 2

1 20 1 2

28 , 25

397.97 , 1.69 , 21.40

36.92 , 2.503 , 31.64

0.083, 0.152 , 4.038

A ME GPa E GPa

a MPa a GPa m a GPa m

b MPa b GPa m b GPa m

c c m c m− −

= =

= = =

= = =

= = − = −

(24)

Fig. 9(b) compares the analytical stress-strain curve with the experimental data obtained

at 333 K. The six distinctive stages of the loading cycle are also marked in this figure. It

is seen that the analytical solution appropriately describes the deformation behaviour of

the pseudoelastic graded NiTi wire. The gradient stress plateaus are appeared in both

forward and reverse martensitic transformations (Stages (2) and (5)); however the

average stress-strain slope over reverse transformation is higher than that of forward

transformation. The gradient stress for stress-induced martensitic transformation

provides increased controllability of SMA component over stress plateau.

63

6.5. The effect of annealing temperature range on deformation behaviour

In this section, we analytically demonstrate the effect of annealing temperature range,

achieved by taking different lengths of the full length gradient–annealed sample (100

mm) from high and low annealing temperature ends, on the mechanical behaviour of

gradient annealed Ti-50.5at%Ni. The analytical solutions are presented in Fig. 10. In

addition to the each sample length, the corresponding annealing temperature ranges is

given in the figure for easier understanding of that effect. The full length (2 100L mm= )

sample specifications and deformation behaviour is the same as what discussed in Sec.

6.2. All the analytical curves are based on quadratic transformation stress and strain

variations. The quadratic transformation parameters are determined by best curve fitting

of the actual curves shown in Fig. 7 for the relevant gauge length of each sample. It is

understood that the average slope of the forward stress plateau increases by increase of

the wire length L1 from the high temperature end while the plateau strain decreases.

Also, by decreasing the wire length L2 from the low temperature end, the forward

transformation starts at higher stress level and the plateau average slope and strain

decrease. We see a progressive change in NiTi wire behaviour from full shape memory

effect to full pseudoelasticity as L1 increases from 60 mm to the full length and then the

full length decreases to 2 10L mm= .

Fig. 10. The effect of annealing temperature range on the deformation behaviour of

microstructurally graded NiTi wire

64

7. Conclusions

1. This study proposes an analytical model to describe the deformation behaviour of

functionally graded shape memory alloys. The model takes into account the general

polynomial variations of transformation stresses and strains along the wire length and

provides closed-form solutions for nominal stress-strain relations of the graded SMA

components. The analytical solutions satisfy well the available experimental

measurements of the shape memory effect and pseudoelastic behaviour of NiTi

wires.

2. Annealing of cold worked NiTi wire within a temperature gradient field creates

gradient transformation stress and strain along the length of the NiTi wire. Such

wires exhibit distinctive stress plateaus with positive slopes on strain. Increasing the

temperature range of the gradient anneal increases the slopes of the stress plateaus

associated with the stress-induced martensitic transformation.

3. The analytical solution provides an effective engineering tool to predict the

deformation response of such components under tensile loading and design

mechanisms, such as actuators, which require high controllability. This solution is

derived based on 1D property variations (critical stress and transformation strain of

martensitic transformation) along the length of slender materials (e.g. wires,

ribbons), thus is generic and applicable to all shape memory alloy wires and ribbons

with transformation property gradient along the length, however the gradient may be

achieved. In this work, the model is applied to NiTi wires microstructurally graded

through heat treatment gradient.

Acknowledgements

We wish to acknowledge the financial support to this work from the Korea Research

Foundation Global Network Program Grant KRF-2008-220-D00061 and the French

National Research Agency Program N.2010 BLAN 90201.

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67

Paper 2: Analytical modelling of functionally graded NiTi shape

memory alloy plates under tensile loading and recovery of deformation

upon heating

Bashir S. Shariata, Yinong Liua*, Qinglin Menga and Gerard Riob

aLaboratory for Functional Materials, School of Mechanical and Chemical Engineering,

The University of Western Australia, Crawley, WA 6009, Australia

*Tel: +61 8 64883132, Fax: +61 8 64881024, email: yinong.liu@uwa.edu.au

bLaboratoire d’Ingénierie des Matériaux de Bretagne, Université de Bretagne Sud,

Université Européenne de Bretagne, BP 92116, 56321 Lorient cedex, France

Acta Materialia, Under Review.

Abstract

This article presents an analytical model to describe the deformation behaviour of

functionally graded NiTi plates under tensile loading and their shape recovery during

heating. The property gradient of the plate is achieved by either a compositional

gradient or microstructural gradient through the thickness. Closed-form solutions are

obtained for nominal stress-strain variations of such plates under uniaxial loading at

different deformation stages. It is observed that the martensitic transformation occurs

partially over nominal stress gradient unlike typical NiTi shape memory alloys. The

curvature-temperature relations are established for complex shape memory effect

behaviour of such plates during recovery period. The analytical solutions are validated

with relevant experimental results.

Keywords: Shape memory alloy; NiTi; Martensitic transformation; Pseudoelasticity;

Functionally graded material

1. Introduction

Shape memory alloys (SMAs) are materials which return to their original shapes upon

heating after being highly strained. Among them NiTi is the most widely used in many

engineering and industrial applications. The functional properties of SMAs often

manifest in two distinctive behaviours, known as the pseudoelasticity and the shape

memory effect [1]. One important type of application of SMAs is mechanical actuation,

68

owing to their ability to output mechanical work [2, 3]. SMA actuation mechanisms

have the advantages, compared to other actuation mechanisms like piezoceramics,

solenoids and pneumatic systems, of being simple in mechanical design, light in weight

and compact in size, and high energy output density. In such applications, SMAs are

often used in the forms of thin wires, tubes and plates. These applications often involve

control of actuation of the shape memory component, either in temperature or stress.

However, typical equiatomic NiTi has narrow transformation temperature windows,

typically <10 K, with a bulk of thermally-induced transformation occurring over a range

of ~5 K [4, 5]. Thus, the displacement of actuation generated by the transformation

period is difficult to be precisely controlled by controlling the temperature. For stress-

induced B2-B19’ martensitic transformation, NiTi often exhibits Lüders-type

deformation behaviour [6, 7], which is characterized by a flat stress plateau over a large

strain span of 5~7%. This presents a typical condition of mechanical instability, in other

words, inability for displacement (strain) control by controlling the load (stress). The

poor controllability of shape memory element in both thermally-induced and stress-

induced transformations is a challenge for actuator design. One solution to this problem

is to create functionally graded NiTi components [8, 9], which actuate over a widened

window of the controlling parameter, either stress or temperature.

Functionally graded material (FGM) is a material in which the composition or structure

gradually vary (typically in one direction) resulting in a variation of material properties.

This concept has been used for various structural and functional applications [10]. In

typical FGM plates, the material composition changes within the thickness of the

structure providing desired gradient of material properties [11-13]. The elasto-plastic

analysis of plates with gradient properties has been studied for thermal loading [14-16],

thermomechanical loading [17] and low-velocity impact loading [18]. For SMA plates

and thin films, several functionally graded designs can be envisaged. One approach is to

gradually change the Ni-Ti composition ratio in the thickness direction [19]. This

composition gradient can be effectively achieved via diffusion anneal of Ni into NiTi

[20-22]. Some people have also applied other materials to NiTi [19]. To improve

biocompatibility and wear resistance, Tan and Crone [23] created a three-layered graded

surface on NiTi sheets with oxygen by means of plasma ion implantation. Fu et al. [24,

25] used TiN layer to improve tribological features and load bearing capacity of

pseudoelastic or shape memory NiTi films. Chu et al. [26] formed bioactive sodium

titanate film with a trait of Ni2O3 on the NiTi substrate by NaOH treatment creating a

69

graded surface structure. Sui et al. [27] created a diamond-like carbon coating on NiTi

plates by plasma immersion ion implantation and deposition to improve corrosion

resistance and blood compatibility of NiTi components. Burkes and Moore [28]

proposed the production of functionally graded NiTi–TiCx by use of combustion

synthesis to combine the pseudoelastic and shape memory features of NiTi with the

high hardness and corrosion resistance of TiCx.

Property gradient can also be created with microstructural gradient [29]. It is known that

the transformation properties of NiTi are highly sensitive to heat treatment conditions

[30]. Microstructurally graded SMA can be achieved by annealing in a temperature

gradient after cold working. A few studies have been carried out on such functionally

graded wires of NiTi [4], Ti–45at%Ni–5at%Cu [31] and Ti–6at%Mo–4at%Sn alloys for

medical guidewire application [32]. Also, time gradient annealing treatment has been

applied to achieve a transformation temperature gradient of 34 K over the length of Ti-

50.9at.%Ni wire [33]. Meng et al. [34] used spatial electrical resistance over-ageing to

develop superelastic NiTi wires with variable shape memory properties along the

length, exhibiting two discrete stress plateaus over stress-induced martensitic

transformation. Recently, fabrication of thin functionally graded NiTi plates has been

reported by means of surface laser annealing [35]. Variation of heat penetration through

the plate thickness provides a progressive degree of annealing which results in a

microstructural gradient within the thickness of the plate. The plates exhibit a complex

mechanical behaviour in addition to enlarged temperature interval for thermally-induced

transformation. Also, compositionally graded thin NiTi plates have been created by

surface diffusion of Ni through the thickness of equiatomic NiTi plates [36]. The

functionally graded samples show reversible one-way shape recovery behaviour.

Another approach is to create geometrically graded NiTi structures. By changing the

cross-sectional area along the loading direction, different cross sections of the

component experience varied values of stress, causing stress gradient along the loading

direction. Thus, transformation occurs at varied loading levels within the structure,

creating complex transformation field. In NiTi plates and strips, this can be achieved by

variation of width as the thickness remains constant [37]. In NiTi bars and wires, the

stress gradient can be generated by uniformly tapering the components [38]. This

approach has the advantage of using commercially available conventional NiTi material

products.

70

To date, no theoretical model has been proposed to describe the deformation behaviour

of compositionally or microstructurally graded 2D SMA structures (i.e. plates and

films). This paper provides an understanding of the transformation-deformation

behaviour of such structures under tensile loading and the recovery of deformation upon

heating. It presents unique closed-form solutions for nominal stress-strain variations of

such NiTi structures at different stages of tensile loading in addition to recovery process

during heating. The analytical solutions provide an effective engineering tool for load-

displacement calculations. The solutions can be applied to describe the combined

pseudoelastic and shape memory behaviour of these alloys.

2. Definition parameters

The transformation parameters are defined based on the ideal pseudoelastic behaviour of

NiTi alloy under stress-induced martensitic transformation as shown in Fig. 1(a). tσ and

tσ ′ are forward and reverse transformation stresses, respectively. tε and tε ′

are forward

and reverse transformation strains, respectively. The apparent elastic moduli of the

austenite (A) and martensite (M) are noted as AE and ME , respectively. The reverse

transformation strain tε ′ can be expressed in terms of the other independent parameters

from the geometrical relation found in Fig. 1(a), as:

( )1 1t t t t

M AE Eε ε σ σ

′ ′= − − −

(1)

Fig. 1(b) shows a NiTi plate of length L , width b and thickness h . The plate is

functionally graded through its thickness. The critical stress for inducing the martensitic

transformation ( tσ ) and for the pseudoelastic reverse transformation ( tσ ′ ), and the

transformation strain (tε ) are schematically indicated on the cross-section. Variations of

these three parameters are assumed linear across the thickness of the plate (x-direction)

from Side (1) to Side (2), as seen in Fig. 1(b). The assumption of linear variation of

transformation properties simplifies the analytical derivations and is an accurate

estimation particularly for thin NiTi plates that are commonly used in SMA

applications. This property gradient can be achieved by either a microstructural gradient

[35] or compositional gradient [8]. The elastic moduli of austenite and martensite are

assumed to be constant within the structure. In Fig. 1(b), subscript 1 refers to the

transformation parameters of Side (1) and subscript 2 refers to those of Side (2).

71

Fig. 1. Definition of parameters of the functionally graded NiTi plate; (a):

transformation properties, (b): geometrical dimensions and transformation properties

gradients through thickness

The property profile across the property gradient can be determined from separate

experiment. In the case of a microstructural gradient created by gradient anneal, the

properties of the NiTi annealed at different temperatures are determined separately

using individual samples of the same alloy but annealed each at a given temperature.

This allows the establishment of calibration curves of the critical parameters used in the

model, such as apparent elastic modulus, critical stresses and plateau strains of the

upper stress plateaus of pseudoelasticity, and apparent yield strength, as functions of the

anneal temperature [35, 36]. Then by knowing the gradient of the temperature field used

to create the microstructurally graded NiTi, the property profile across the

microstructure gradient can be determined. For the analysis above, with respect to a

microstructurally graded plate created by gradient anneal, Side (1) corresponds to the

tε

tσ ′

tσ

AEME

tε ′

σ

ε

(a)

( )t xε

( )t xσ ′

( )t xσ

F

F

h

b

x

y

L

(b)

Side (2)Side (1)

2σ1σ

2σ ′

1σ ′

2ε1ε

72

side of higher anneal temperature, and thus has lower transformation stresses and higher

transformation strain, whereas Side (2) corresponds to the side of lower anneal

temperature, and thus has higher transformation stresses and lower transformation

strain.

The plate is subjected to a tensile loading F along its length (y-direction). The

transformation properties variations across the plate thickness are defined as:

2 11

2 11

2 11

( )

( )

( )

t

t

t

x xh

x xh

x xh

σ σσ σ

σ σσ σ

ε εε ε

−= +

′ ′−′ ′= +

−= +

(2)

3. Analytical solution for deformation behaviour of functionally graded NiTi plate

under tensile loading

In this section, we establish the nominal stress-strain relation of the functionally graded

NiTi plate under uniaxial loading as shown in Fig. 1(b). The nominal stress (σ ) and

nominal strain (ε ) are defined as:

F

bhL

L

σ

ε

=

∆= (3)

where L is the initial length and L∆ is the total elongation of the plate in the loading

direction. Due to the involvement of the martensitic transformation, we need to consider

separate stages of deformation to obtain the stress-strain relation. It is assumed that the

sample remains straight during all stages of tensile loading. Thus, all layers across the

thickness have the same total displacement along the axial direction.

3.1. Stage (1): 10AE

σε≤ <

At this stage, the entire structure is in austenite phase since the applied load is less than

the required value to induce martensitic transformation at Side (1) with the minimum

transformation stress within the structure. The nominal stress-strain relation is:

AEσ ε= (4)

73

3.2. Stage (2): 1 2

A AE E

σ σε≤ ≤

Fig. 2(a) shows how transformation starts within the graded NiTi plate. At this stage,

the stress-induced martensitic transformation initiates at Side (1), where the critical

stress for inducing the transformation is the lowest throughout the structure. As the

loading level increases, the transformation progressively propagates from Side (1) to

Side (2). This stage ends when the transformation starts at Side (2). During this stage,

the locus of the start of the transformation propagates from Side (1) to Side (2), with the

structure behind being partially transformed (denoted A→M) and that in front still in

fully austenitic state (denoted A). It should be noted that the local stress is varying

across the thickness of region A→M as all layers are partially transformed at different

stress levels, while region A experiences constant level of stress during this stage. The

boundary displacement is defined by A Mx − . The load F can be divided into two parts

carried by these two regions:

A A MF F F →= + (5)

where AF is found using Fig. 2(a) and Eqs. (2) as:

2 11( ) ( ) ( )A t A M A M A M A MF x b h x x b h x

h

σ σσ σ− − − −− = − = + −

(6)

Each differential layer located at x in the partially transformed region is stretched by a

constant differential load corresponding to its transformation stress. A MF → is the

integration of all the axial differential forces within the partially transformed region, and

is written in the following form using Fig. 2(a) and Eqs. (2):

22 11

0

( )2

A Mx

A M t A M A MF x bdx x x bh

σ σσ σ−

→ − −− = = +

∫ (7)

When the boundary is at A Mx − , the nominal strain of the plate is written as:

2 11

1A M

A

xE h

σ σε σ −− = +

(8)

Using Eqs. (3), (5), (6), (7) and (8) and by eliminating A Mx − , the nominal stress-strain

relation of the current stage is established:

74

21

2 1

( )

2( )A

A

EE

ε σσ εσ σ

−= −−

(9)

As observed the nominal stress is expressed as a quadratic function of the nominal

strain. At the end of Stage (2), the nominal stress reaches to 1 2

2

σ σ+.

Fig. 2. Evolution of martensitic transformation within the functionally graded NiTi

plate; (a): initiation of transformation at Stage (2), (b): completion of transformation at

Stage (4)

3.3. Stage (3): 2 22

A AE E

σ σε ε< ≤ +

During this stage, all the layers of the graded NiTi plate are in the course of martensitic

transformation. As stress-induced transformation occurs over constant value of stress at

each layer, the overall load F remains unchanged with respect to the maximum loading

level of the previous stage while the plate is uniformly stretched along the y-direction.

The nominal stress during this stage is:

1 2

2

σ σσ += (10)

This stage ends when the transformation is completed at Side (2), which has the

minimum transformation strain among all layers of the property graded plate, while

other layers of the plate with larger transformation strain are still in the course of

transformation.

x

y

dx

xA-M

F

F

Side (1) Side (2)

h

A→

M

A

(a)

x

y

dx

xA-M

F

F

Side (1) Side (2)

h

A→

M

M

(b)

75

3.4. Stage (4): 2 12 1

A AE E

σ σε ε ε+ < ≤ +

This stage starts when Side (2) is fully transformed to martensite and ends when the

whole structure becomes martensite. During this stage, to enable further elongation

when the transformation in other layers than fully transformed layers continues, stresses

at fully transformed layers increase along the line after the stress plateau (Fig. 1(a)). As

the loading level increases, the locus of the end of the transformation (i.e. the boundary

between full martensite region (M) and partially transformed region (A→M)) moves

gradually from Side (2) toward Side (1) as shown in Fig. 2(b). The boundary

displacement is specified by A Mx − . The load F can be written as:

A M MF F F→= + (11)

where A MF → is defined as:

1 2

2A M A MF bxσ σ

→ −+= (12)

MF is the integration of axial loads carried by differential layers at fully transformed

region. The axial load in each differential element of this region stretched at nominal

strain ε can be expressed in two parts related to austenite and martensite periods of the

element considering the transformation strain. Thus, MF can be written, using Eqs. (2),

as:

1 2 1 2

2

1 2 1 2 1 2 11 2 1

( )( ) ( )

2 2

1

2 2 2

A M

ht

M M t A MAx

A M A MM

A A A

xF E x bdx b h x

E

x xE bh

E h E h E

σσ σ σ σε ε

σ σ ε ε σ σ σε ε ε ε ε

−

−

− −

+ += + − − = − +

+ + − − − − − − + + −

∫(13)

At an instance when the moving boundary is at A Mx − , we can write by use of Eqs. (2):

1 2 11 2 1

( )( )t A M A M

t A MA A A

x xx

E E h E

σ σ σ σε ε ε ε ε− −−

−= + = + + + −

(14)

Using Eqs. (3), (11), (12), (13) and (14) and by eliminating A Mx − , the nominal stress-

strain relation of Stage (4) is obtained in the form of a quadratic equation:

76

2

11

1 2 1 2 1 2

2 12 1

2 2 2 2AM

MA

A

EEE

EE

σε εσ σ σ σ ε εσ ε σ σ ε ε

− −

+ + + = + − − − − + − (15)

At the end of this stage ( 11

AE

σε ε= + ), the nominal stress is found from Eq. (15) as:

1 2 1 21 22 2

M

A

E

E

σ σ σ σσ ε ε + −= + + −

(16)

3.5. Stage (5): 11

AE

σε ε> +

At this stage, the entire plate has already transformed to martensite and is elastically

loaded in the martensite phase. Using Eq. (16) and considering the elastic deformation

of the plate in martensite phase during this stage, the nominal stress-strain relation of

Stage (5) is found:

1 2 1 2 1 2

2 2 2MA

EE

σ σ σ σ ε εσ ε + + += + − −

(17)

Up to now, we have derived the nominal stress-strain relations of the functionally

graded NiTi plate for the five stages of loading. The same number of deformation stages

can be defined for unloading. To define the strain interval of each reverse stage and the

corresponding nominal stress-strain relation, we can substitute 1σ , 2σ , 1ε and 2ε by 1σ ′

, 2σ ′ , 1ε ′ and 2ε ′ in the strain interval and nominal stress-strain relation of the similar

forward stage and use Eq. (1) to express the final relation in terms of independent

transformation parameters. Then, Stages (6) – (10) related to the unloading path are

characterised as following:

3.6. Stage (6): ( )11 1 1

1

A ME E

σε σ σ ε′≥ − − +

This stage is related to unloading in full martensite phase. The NiTi plate deforms

according to Eq. (17) until the loading level decreases to the required value to induce

reverse martensitic transformation at Side (1) of the plate.

77

3.7. Stage (7): 2 2 2 1 1 12 1

A M A ME E E E

σ σ σ σ σ σε ε ε′ ′− −− + ≤ < − +

At this stage, the M→A transformation initiates at Side (1). As the loading level further

decreases, the reverse transformation progressively starts at other layers of the plate.

The related nominal stress-strain relation is:

( )

( )

( ) ( )

1 2 1 2

2

11 1 1

2 1 2 1 2 1

1 1

2 2

1 1

2 1 1 1

MM A

A M AM

M M A

EE E

E E EE

E E E

σ σ ε εσ ε

σε σ σ ε

σ σ σ σ ε ε

+ += + − −

′ ′− + − − − − ′ ′− − − − + −

(18)

3.8. Stage (8): 2 2 2 22

A A ME E E

σ σ σ σε ε′ ′−≤ < − +

During this stage, the entire structure is in the course of M→A transformation. Thus, the

nominal stress σ remains at a constant value as:

1 2

2

σ σσ′ ′+= (19)

3.9. Stage (9): 1 2

A AE E

σ σε′ ′

≤ <

At this stage the plate layers, starting from Side (2), progressively enter the full

austenite phase as the loading level decreases. The nominal stress-strain relation of this

stage is obtained as:

( )( )

2

1

2 12A

A

EE

ε σσ ε

σ σ′−

= +′ ′−

(20)

3.10. Stage (10): 10AE

σε ′≤ <

This stage is related to unloading in full austenite phase. The same relation as Eq. (4) is

applied.

78

Eqs. (4), (9), (10), (15), (17), (18), (19) and (20) describe the nominal stress-strain

relation of the functionally graded NiTi plate at the deformation stages defined in

Sections 3.1 – 3.10.

3.11. Analytical illustration

Fig. 3(a) illustrates the nominal stress-strain diagram of a functionally graded NiTi with

fully pseudoelastic behaviour under uniaxial loading based on the analytical solution

found in the preceding section and specifications of Table 1.

Table 1: The transformation and material properties of the functionally graded NiTi

plate with fully pseudoelastic behaviour

1( )MPaσ 2( )MPaσ 1( )MPaσ ′ 2( )MPaσ ′ 1ε 2ε ( )AE GPa ( )ME GPa

300 550 50 380 0.06 0.03 90 30

The two stress-strain loops shown in dashed lines represent the deformation behaviours

of the alloy on Sides (1) and (2) of the plate (if they were to be separated from the

plate). The red solid curve shows the nominal stress-strain loop of the functionally

graded plate. The numbers on the curve correspond to the relevant deformation stages

defined in the aforementioned sections. As observed, the stress-strain diagram of the

graded plate locates between those of Sides (1) and (2). The overall transformation

strain is less than that of Side (1) and more than that of Side (2). The forward

transformation (A→M) occurs in Stages (2), (3) and (4) creating non-uniform stress

plateau compared with the original behaviour of each layer of the graded plate. The

reverse transformation (M→A) proceeds in Stages (7), (8) and (9) providing nonlinear

stress gradient at portions of the reverse plateau (Stages (7) and (9)).

Although the analytical equations are derived based on fully pseudoelastic behaviour of

the graded NiTi plate, the model can also predict the deformation behaviour of a

partially pseudoelastic NiTi plate by mathematically assuming a negative value for

reverse transformation stress of the side of the plate with shape memory effect property.

Fig. 3(b) depicts the uniaxial stress-strain behaviour of a NiTi plate graded from one

surface with pseudoelastic behaviour to the opposite surface with shape memory effect.

The dashed lines show the stress-strain variations of the two surfaces of the plate. The

red solid curve presents that of the functionally graded plate. The transformation and

material parameters are defined in Table 2. Stress gradients are observed in portions of

79

the forward stress plateau. Residual strain appears upon unloading due to the part of the

plate with shape memory effect feature.

Table 2: The transformation and material properties of the functionally graded NiTi

plate with partially pseudoelastic behaviour

1( )MPaσ

2( )MPaσ

1( )MPaσ ′

2( )MPaσ ′

1ε 2ε ( )AE GPa

( )ME GPa

180 350 -120 100 0.070 0.045 90 30

Fig. 3. The nominal stress-strain diagrams of functionally graded NiTi plates under

uniaxial tensile loading; (a): fully pseudoelastic behaviour, (b): partially pseudoelastic

behaviour

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

200

400

600

800

1000

Nominal Strain

No

min

al S

tres

s (M

Pa)

(1)

(2)

(9)(8)

(6)

(4)(3)

(5)

(7)

(10)

Side (2)

Side (1)

(a)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

100

200

300

400

500

600

700

800

Nominal Strain

No

min

al S

tres

s (M

Pa)

Side (2)

Side (1)

(b)

80

4. Modelling of deformation recovery of functionally graded NiTi plates with

shape memory effect property upon heating

For functionally graded NiTi plates in shape memory state, the transformation strain is

not recovered upon unloading and a residual strain is produced, as illustrated in Fig. 4.

As seen in the figure, when the functionally graded plate is loaded to point I, the stress

levels at Side (1) and Side (2) are different, corresponding to points a and b,

respectively. As the sample is unloaded to II (0σ = ), Sides (1) and (2) are stressed at a’

and b’ respectively, Side (1) is under compression and Side (2) is under tension. Thus,

when the sample is released from the testing machine, Sides (1) and (2) displace to a”

and b”, respectively, causing the plate to bend toward Side (2). a” and b” correspond to

residual transformation strains of Sides (1) and (2), respectively, denoted as 1r

ε and 2r

ε .

These strains can be recovered by heating. Upon heating, the reverse transformation

commences from one side and propagates progressively to the other side. This causes

the plate (or strip) to continue to bend due to asymmetric strain recovery across the

middle plane of the plate, as experimentally observed [35]. In this section, we present

the expression of the change of the curvature of the strip based on kinematic relations

during the reverse transformation upon heating after a tensile deformation.

Fig. 4. The stress-strain states of the two sides of the functionally graded plate during

and after tensile testing

Fig. 5(a) shows a schematic of a functionally graded strip sample prepared by laser scan

anneal after cold rolling. The start and finish temperatures of the M→A transformation,

denoted sT and fT , are indicated in the figure. The left side (Side (1)) has higher

transformation temperatures (1s

T ,1f

T ) and is the side facing the laser. The right side is

Str

ess

Side (2)

Side (1)

Graded Plate

a”

a’

a

b

b’

b”

1rε

2rε II

I

Strain

81

the back side (Side (2)) and has lower transformation temperatures. The variations are

assumed linear through the thickness of the strip for simplicity. Also indicated in Fig.

5(a) is the residual transformation strain, higher on the laser side and lower on the back

side, to conform to previous observations [35]. Fig. 5(b) shows a schematic of the

variation of volume fraction of martensite during the reverse transformation upon

heating for Side (1) and Side (2), reflecting the difference in reverse transformation

temperature between the two sides. Naturally martensite volume fraction evolution

profiles of all the middle layers fall in between these two curves. Corresponding to the

positions and temperature windows of the two curves, five temperature regimes are

identified, as shown in the figure. As observed, the M→A transformation starts at Side

(2) and gradually propagates toward Side (1) as the temperature increases. The structure

is fully transformed to austenite at above 1f

T .

Fig. 5. Thermal transformation properties of the functionally graded NiTi plate; (a):

variation of transformation temperatures and residual strain after tensile loading across

the thickness, (b): variation of martensite volume fraction versus temperature on Side

(1) and Side (2) upon heating

( )r xε

( )sT x

( )fT x

h

b

x

y

L

(a)

Side (2)Side (1)

2fT

1fT

2sT1s

T

2rε

1rε

0

1

2sT

Mf

( )T K

(b)

2fT

1sT

1fT

I II III VIV

Side (2) Side (1)

>

>

>

>

82

Fig. 6(a) shows how the M→A transformation progresses across the plate thickness

with increasing temperature. The starting structure of the plate is martensite at the end

of tensile deformation and prior to heating. The vertical axis indicates the martensite

volume fraction ( Mf ), and the horizontal axis expresses the thickness of the plate in the

direction from Side (1) to Side (2), consistent to Fig. 5(a). The blue lines represent the

distribution of the volume fraction of martensite across the thickness of the plate at

some representative temperatures. When 2sT T= (refer to Fig. 5(b)), the reverse

transformation is about to start at Side (2) and 1Mf = at Side (2), as marked by point

“o”. When the temperature is increased to 21 sT T> , , (2)M Sidef lowers to

1f while the

transformation front moves to position 1x (where 1, 1M xf = ). Similarly, at 2T T= ,

, (2) 2M Sidef f= and transformation front moves to 2x . Considering that the reverse

transformation is associated with a shape recovery (length contraction) corresponding to

the transformation strain, it is easy to see that the red line connecting , (2) 2M Sidef f= and

, (1) 1M Sidef = (as at the moment of 2T T= ) effectively expresses the “length end” of the

strip at each layer position.

Increasing the temperature to 2f

T T= , the transformation at Side (2) is complete (

, (2) 0M Sidef = ) and the transformation front moves to 3x . Until this moment, , (1) 1M Sidef =

(Side (1) maintains the same length) and , (2)M Sidef continues to slide down from 1 to 0

(Side (2) continues to shorten the length), thus the strip continues to bend with

increasing curvature towards Side (2), as schematically shown in Fig. 6(b).

Continuing to increase temperature at 2 1f sT T T< < , the transformation starting point

continues to slide on the top axis from 3x to 4x and the transformation completion point

continues to slide on the bottom from 3x′ to 4x′ . During this process, both the lengths of

Side (2) and that of Side (1) remain unchanged. Based on kinematic assumption, the

curvature of the strip remains unchanged at 2 1f sT T T< < . At a temperature within this

range, where the transformation starting and completion points are at x∗ and x ∗′

respectively, three regions across the thickness can be considered: the region fully

recovered to austenite region (A) (right to x ∗′ ), the region in transition from martensite

to austenite (A+M) (between x∗ and x ∗′ ) and the region in martensite (M) (left to x∗ ).

83

Increasing the temperature to 1s

T T≥ , the reverse transformation occurs on Side (1),

resulting in shortening of the length of Side (1). This leads to relaxation of the curvature

back towards Side (1), until 1f

T T= , when the reverse transformation on Side (1)

reaches completion ( , (1) 0M Sidef = ).

Considering the free lengths of Side (1) and Side (2) ( 1L and 2L , respectively as in Fig.

6(b)), curvature λ of the plate may be computed as following (with an approximation of

assuming linear distribution of length in between the two sides):

1 2

1 2

1 2 L L

R h L Lλ

−= = + (21)

Fig. 6. Recovery Process of the functionally graded NiTi plate after tensile testing; (a):

propagation of M→A transformation within the plate thickness, (b): the structure

bending

h

R

Side (1) Side (2)

L1

L2

(b)

84

To establish the curvature-temperature relations during deformation recovery upon

heating after the tensile testing, we need to consider the five separate temperature

intervals defined in Fig. 5(b).

4.1. Stage (I): 2sT T≤

The residual strains of Side (1) and (2) are considered to be 1r

ε and 2r

ε as defined in Fig.

5(a). Using Eq. (21) with 11 (1 )rL L ε= + and

22 (1 )rL L ε= + , the initial curvature of the

plate (after tensile testing) is obtained:

1 2

1 2

0

2

2r r

r rh

ε ελ λ

ε ε −

= = − + + (22)

Note that the positive direction of the curvature is assumed to be toward Side (1).

4.2. Stage (II): 2 2s fT T T< ≤

At this stage, transformation starts at Side (2) of the plate. This stage ends when Side

(2) is fully transformed to austenite. If the sample has been loaded to the level where the

whole graded structure is transformed to martensite (point I in Fig. 4 and beyond), Side

(2) is likely to have experienced plastic deformation in the course of tensile loading as it

has generally lower transformation strain compared to that of Side (1). Thus, the total

2rε may not be recovered, and the amount of

2pε related to the plastic deformation of

Side (2) is remained at the end of Stage (II). Assuming that the transformation strain is

recovering linearly with temperature rise according to Fig. 5(b), 2L can be written as:

2

2 2 2

2 2

2 1 ( ) fr p p

f s

T TL L

T Tε ε ε

−= + − + −

(23)

Using Eq. (21) with 11 (1 )rL L ε= + and 2L defined above, the curvature-temperature

relation of this stage is found:

2

1 2 2 2

2 2

2

1 2 2 2

2 2

( )2

2 ( )

fr r p p

f s

fr r p p

f s

T T

T T

T Th

T T

ε ε ε ελ

ε ε ε ε

− − − − − = −

−+ + − + −

(24)

85

4.3. Stage (III): 2 1f sT T T< ≤

At this stage, the temperature level is beyond the final transformation temperature of

Side (2) and less than the temperature to induce transformation of Side (1). Therefore,

the lengths of the both sides would be unchanged (see Fig. 6(a)), resulting in a nearly

constant value of curvature during this period. This value is the maximum curvature

found during recovery, since Side (1) begins shortening when the sample is heated to

beyond 1s

T . Using Eq. (21) with the consideration of 11 (1 )rL L ε= + and

22 (1 )pL L ε= +

gives:

1 2

1 2

max

2

2r p

r ph

ε ελ λ

ε ε −

= = − + + (25)

4.4. Stage (IV): 1 1s fT T T< ≤

This stage begins with the initiation of M→A transformation at Side (1). The

transformation is completed within the entire structure by the end of this stage. The

curvature-temperature related to this stage is obtained as:

1

1 2

1 1

1

1 2

1 1

2

2

fr p

f s

fr p

f s

T T

T T

T Th

T T

ε ελ

ε ε

− − − = −

−+ + −

(26)

4.5. Stage (V):

As explained in Sec. 4.2, the structure can be partially deformed beyond the plateau

level to achieve full transformation of the graded plate to martensite during tensile

loading. In this case, the graded sample will not fully return to a straight shape after the

completion of the reverse transformation at 1f

T T> , retaining a final curvature, found

from Eq. (26) by setting 1f

T T= , as:

2

2

2

(2 )p

fp h

ελ λ

ε= =

+ (27)

Eqs. (22), (24), (25), (26) and (27) describe the curvature-temperature relation of the

functionally graded NiTi plate with shape memory effect property in the course of

deformation recovery after tensile loading.

86

5. Experimental validation

To validate the analytical solutions found in Sec. 3 and Sec. 4, Ti-50.2at%Ni strips,

microstructurally graded by laser surface scanning anneal after cold rolling, are tested.

The original NiTi strip was 40 mm × 4 mm × 0.5 mm in dimension. It was fully

annealed at 923 K for 1.8 ks, and then cold rolled to 0.37h mm= in thickness,

corresponding to 26% thickness reduction strain. Then, laser scanning anneal was

performed on one side of the strip, using 2.5 ms square profile laser pulses. A

temperature gradient across the strip thickness, provided by natural degradation of heat

penetrating into the material, created gradient partial anneal. Side (1) of the schematic

sample of Fig. 5(a) is the side facing the laser scanning, with the temperature in the top

layer being above the melting temperature of NiTi (1583 K). Based on micro-hardness

test, the temperature at the surface of Side (2) is estimated to be 680 K [35]. Fig. 7

shows the effect of annealing temperature of the forward and reverse transformation

stresses and forward transformation strain of Ti-50.2at%Ni alloy tested at 301 K. The

samples were initially tested at 333 K and 343 K [39]. The results are modified for

testing temperature of 301 K using

5.25 / , 6.53 / , 0.04% /t t td d dMPa K MPa K K

dT dT dT

σ σ ε′= = = [39]. It is known [40] that at

TAnneal > 840 K, the transformation properties of Ti-50.2at%Ni alloy are independent of

annealing temperature. Using Fig. 7, the transformation stresses and strain of Sides (1)

and (2) of the graded sample are defined as presented in Table 3.

Fig. 7. Effect of annealing temperature on the forward and reverse stresses and strain

of the stress-induced B2-B19’ martensitic transformation in Ti-50.2at%Ni at 301K.

0.025

0.03

0.035

0.04

0.045

0.05

-350

-250

-150

-50

50

150

250

350

620 660 700 740 780 820

For

wa

rd T

ran

sfo

rma

tion

Str

ain

Tra

nsf

orm

atio

n S

tre

ss (

MP

a)

Annealing Temperature (K)

tεtσ ′

tσ

87

Table 3: The transformation properties of the functionally graded NiTi plate under

experiment

1( )MPaσ 2( )MPaσ 1( )MPaσ ′ 2( )MPaσ ′ 1ε 2ε

60 275 -340 -45 0.050 0.032

Introducing parameters defined in Table 3 and 12.5AE GPa= , 11ME GPa= [35] into

the set of stress-strain relations derived in Sec. 3, the nominal stress-strain curve of the

functionally graded NiTi strip is obtained, as shown in Fig. 8. Also shown in the figure

is the experimental curve for comparison. It is seen that the analytical model agrees well

the experimental observation.

Fig. 8. Comparison of the analytical model with tensile testing result at 301 K of the

laser annealed Ti-50.2at%Ni strip

As observed in Fig. 8, the functionally graded sample has residual strain upon

unloading. When the deformed sample is heated in a water bath to different

temperatures, a complex mechanical behaviour for shape recovery is observed, due to

the progressive reverse transformation through the thickness of the functionally graded

structure. Fig. 9(a) shows shape change of a strip sample during heating after a tensile

deformation to 5.7%. The curved shape shows the edge of the strip sample. The

illustration is a montage of several images recorded at different temperatures. The

temperature of each position is marked in the figure. Side (1) is the side facing the laser.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

50

100

150

200

250

300

350

Nominal Strain

No

min

al S

tre

ss (

MP

a)

Side (2)

Side (1)

Model

Experiment

88

The curvature of the strip sample is measured, and its evolution with temperature is

shown in Fig. 9(b).

Fig. 9. Shape recovery of the laser annealed Ti-50.2at%Ni strip upon heating after

tensile deformation to 5.7%; (a): images of the shape of the strip sample at various

temperatures, (b): curvature evolution of the strip sample during heating showing

comparison between the analytical model and the experimental observation

This actual curvature change behaviour corresponds to the five heating stages defined in

Sec. 4. The sample is initially curved toward Side (2) after the tensile deformation (thus

the negative curvature). This corresponds to Stage (I). Upon heating, it curves further in

this direction, corresponding to Stage (II). During Stage (III), it remains at a maximum

(negative) curvature within a small temperature window. Then, it curves back toward

Side (1) during Stage (IV) upon further heating. The sample exhibits a residual positive

curvature at the end of the heating. This implies partial plastic deformation of the strip

during tensile loading, which has occurred on Side (2). Also shown in Fig. 9(b) is the

curvature evolution curve calculated using the analytical model developed in Sec. 4.

300 305 310 315 320 325 330 335 340 345-0.1

-0.075

-0.05

-0.025

0

0.025

Temperature (K)

Cu

rvat

ure

(1

/mm

)

Model

Experiment

(b)

(I)

(II)(III)

(IV)

(V)

89

The residual strains of Side (1) and Side (2) (1r

ε and 2r

ε ) are determined from Fig. 8 as

0.048 and 0.028, respectively. We assume 2

0.01pε = . The thermal transformation

properties of these sides are defined in Table 4 [40].

Table 4: Thermal transformation properties of the functionally graded NiTi plate under

experiment

1( )sT K

1( )fT K

2( )sT K

2( )fT K

319 325 309 313

It is evident that the model can effectively describe the complex thermomechanical

behaviour of the microstructurally graded NiTi sample during recovery of

transformation deformation. It is seen that the curvature in Stage (I) predicted by the

model is greater than that measured in the experiment. The difference can be due to the

plastic deformation of Side (2), which has been loaded far beyond the plateau level.

This leads to a real residual strain of Side (2) higher than that predicated by the model,

resulting in smaller initial curvature as observed in the experimental result.

6. Conclusions

1. An analytical model is established to describe the deformation behaviour of

functionally graded NiTi plates during uniaxial tensile loading and the recovery of

deformation upon heating. Closed-form solutions are obtained for nominal stress-

strain variations during tensile loading and curvature-temperature relations. The

model agrees very well with the experimental observations for both processes.

2. Functionally graded Ti-50.2Ni% strips (plates) are created by surface laser scanning

anneal after cold rolling. The material thus treated exhibit a microstructural gradient

through the thickness, which in turn results in gradients of transformation and

mechanical properties through the thickness. The functionally graded Ti-50.2Ni%

strips exhibit a residual curvature after tensile deformation and complex shape

change behaviour during heating.

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92

93

Chapter 3

Geometrically graded 1D and 2D SMA

structures

Paper 3: Mathematical modelling of pseudoelastic behaviour of tapered NiTi bars Paper 4: Modelling and experimental investigation of geometrically graded NiTi shape memory alloys

94

95

Paper 3: Mathematical modelling of pseudoelastic behaviour of

tapered NiTi bars

Bashir S. Shariata, Yinong Liua, Gerard Riob

aLaboratory for Functional Materials, School of Mechanical and Chemical Engineering,

The University of Western Australia, Crawley, WA 6009, Australia

Tel: +61 8 64888151, Fax: +61 8 64881024, email: bashirs@mech.uwa.edu.au

bLaboratoire d’Ingénierie des Matériaux de Bretagne, Université de Bretagne Sud,

Université Européenne de Bretagne, BP 92116, 56321 Lorient cedex, France

Journal of Alloys and Compounds, In Press, doi: 10.1016/j.jallcom.2011.12.151.

Abstract

Owing to their ability to exhibit the shape memory effect and pseudoelasticity, NiTi

alloys have been used in many applications as actuators with temperature- or stress-

controlling mechanisms. In some applications, it is necessary that the shape memory

component act in a controllable range with respect to the controlling parameters, i.e.

temperature for thermally induced actuation or stress for stress-induced actuation.

However, in typical equiatomic NiTi the actuation range is narrow, which results in

poor controllability of the system. This study proposes improved controllability of a

typical NiTi component by tapering its structure and presents closed-form solution for

load-displacement relation at different stages of loading cycle. The analytical solution is

in good agreement with experimental data. The nominal stress-strain diagram shows

gradient stress for stress-induced martensitic transformation. The nonlinear gradient can

be controlled by geometrical designs and is influenced by elastic moduli of the austenite

and martensite phases.

Keywords: Shape memory alloy; NiTi; Martensitic transformation; Functionally graded

material; Pseudoelasticity

1. Introduction

NiTi shape memory alloys (SMAs) have been used as sensors and actuators in many

engineering applications [1]. This is due to their remarkable properties, most notably the

pseudoelasticity and shape memory effect. In both properties, NiTi exhibits large

96

Lüders-type deformation in tension due to martensitic transformation from the austenite

to the martensite phase [2-4]. Thermoelastic martensitic transformation in NiTi has a

small transformation temperature range, typically <10 K [5]. Also, in the case of stress-

induced martensitic transformation (SIMT), the large deformation occurs over a

constant value of stress due to Lüders-like deformation under tensile loading, creating a

situation of mechanical instability. However, in many actuating applications, this

sudden movement is not acceptable and it is necessary that the NiTi component acts in a

controllable range with respect to the controlling parameter, i.e., stress. Therefore, it is

essential to widen the controlling interval of the shape memory element. One way to

achieve this goal is to create transformation load gradient along the direction of

deformation. This may be achieved by using either microstructurally or geometrically

graded NiTi components.

Most of the studies on microstructurally graded NiTi alloys have been focused on multi-

layer or functionally graded NiTi-based thin plates (films). There are different

approaches to fabricating functionally graded NiTi films [6]. A typical approach is to

maintain composition gradient through the film thickness by sputtering [7]. The

variation in material constituents in functionally graded plates leads to variation of

thermomechanical properties [8, 9]. In SMAs, this particularly provides variation of

transformation properties, i.e., forward and reverse stresses and strains, in the thickness

direction. The alloy properties can change through the thickness from pseudoelasticity

to shape memory effect depending on the composition range and the testing

temperature. Recently, Birnbaum et al. [10] has proposed a new method to functionally

grade the shape memory response of NiTi films. By laser irradiation, they alter

thermomechanical and transformation aspects of NiTi thin films.

A few studies have been reported on microstructurally grading of NiTi wires. Mahmud

et al. [5, 11] reported a novel design of functionally graded NiTi alloys by utilizing

annealing temperature gradient along the length of NiTi wire. Due to the sensitivity of

the alloy’s thermomechanical properties with respect to heat treatment condition,

functionally graded properties were achieved in the length direction. This resulted in

longitudinal transformation stress gradient for both forward and reverse transformations.

They proposed the effective temperature ranges for gradient anneal depending on testing

temperature. Yang et al [12] investigated spatially varying temperature profile of a Ti–

45Ni–5Cu (at%) wire generated by Joule heating. The Joule heated sample

97

demonstrated a low shape recovery rate of 0.02%/K and a Lüders-like deformation with

a stress level gradient from 340 MPa to 380 MPa.

Another way to impose the transformation stress gradient is to geometrically grade the

SMA component. One design is tapering a NiTi bar to provide a progressive change in

cross-sectional area. Upon axial loading, different cross sections of the bar experience

varied values of normal stress, causing transformation stress gradient. Bars and columns

either uniform or tapered have been widely used as structural elements. Among them,

NiTi bars, columns and beams have been used in many engineering applications [1]

including seismic protection [13]. They can accommodate unexpected large structural

deformation due to heavy earthquakes. They remain functional after earthquakes thanks

to their excellent fatigue features. Also, the hysteretic pseudoelasticity of NiTi provides

additional damping to a structure, reducing plastic deformation of critical members [14].

In some applications of NiTi alloy [15], tapered bars are preferred because of

geometrical or functional considerations. Tapered members are especially used in

weight-sensitive structures and to minimise material use for construction cost reduction.

In the past decades, various mechanical aspects of tapered bars and columns made of

common materials have been studied, such as load-bearing capacity [16], bending [17],

buckling [18-20] and vibration [21]. To date no analytical or numerical model has been

reported for pseudoelastic behaviour of SMA structures with continuously varying

dimensions. This paper presents unique closed-form relations for nominal stress-strain

behaviour of tapered bars (wires) made of NiTi under axial loading in addition to

experimental investigation.

2. Stress-induced martensitic transformation parameters

To achieve an analytical solution of the stress-strain behaviour of a geometrically

graded NiTi alloy component, intrinsic parameters of the deformation behaviour are

defined using idealised stress-strain behaviour of pseudoelastic NiTi alloy, as illustrated

in Fig. 1. EA and EM are the elastic moduli of the austenite and martensite phases,

respectively. The forward and reverse transformation stresses are defined as tσ and tσ ′ ,

respectively. The forward transformation strain is tε . The reverse transformation strain

tε ′ can be expressed in terms of the other transformation parameters defined above by

the geometrical relation shown in Fig. 1, as:

98

( )1 1t t t t

M AE Eε ε σ σ

′ ′= − − −

(1)

Six distinctive stages of deformation are also marked in the figure.

Fig. 1. Definition of transformation parameters and deformation stages

3. Analytical solution

A tapered NiTi bar of solid circular cross section and length L is subjected to axial load

F as shown in Fig. 2. The cross-sectional radius r(x) changes linearly from one end to

the other with respect to length variable x, which is measured from point O where the

sides of the tapered bar meet. r1 and r2 are the cross-sectional radii at the top and bottom

ends of the bar, respectively (r1≠r2). The radius ratio α is defined as:

1

2

r

rα = (2)

We define the nominal stress σ as the axial force divided by the bottom end cross-

sectional area:

22

F

rσ

π= (3)

It is assumed that the angle of taper Ө is small. Because of the transformation involved,

we need to consider several stages of loading cycle to establish the load-displacement

relation.

tε

tσ ′

tσAE ME

tε ′

σ

ε

(1)

(2)

(6)

(4)

(3)

(5)

99

Fig. 2. Tapered NiTi bar under tensile loading

3.1. Stage (1): 20 tσ α σ≤ ≤

At this stage, the bar is entirely in the austenite phase as the applied load is less than the

critical value to initiate transformation at the top end of the bar which holds the highest

normal stress in the structure. The displacement of the loading point (elongation of the

entire bar) is determined by the following relation [22]:

2

1( )

L

AL

FdxL

E A x∆ = ∫ (4)

where L1 and L2 are distances from the origin point to the top and the bottom ends,

respectively. Considering Fig. 2, the cross-sectional area at x is:

( )2

2 1

1

( ) ( )r x

A x r xL

π π

= =

(5)

Using Eqs. (4) and (5), the total displacement is obtained as:

1 2Tot

A

FLL

E r rπ∆ = (6)

The nominal strain of the NiTi bar under tensile loading is found by dividing the total

elongation TotL∆ by initial length L:

TotL

Lε ∆= (7)

r1

r2

F

L

L1

L2

O

x

dx

r(x)

Ө

100

The nominal stress-strain relation of this stage is found using Eqs. (6) and (7):

AE

σεα

= (8)

3.2. Stage (2): 2t tα σ σ σ< ≤

At this stage, the austenite to martensite transformation starts at the top end of the bar

and progressively propagates downward as the loading level increases. The structure

consists of both austenite and martensite regions, notified by A and M, respectively in

Fig. 3. The displacement of the moving A-M boundary with respect to the origin O is

defined by variable A-Mx . This stage ends when the A-M boundary reaches to the

bottom end of the bar (A-M 2x L= ). At an instance when the A-M boundary is at A-Mx ,

we can write:

( ) ( ) 22 2 1 A-M

A-M( )t t

r r xF r x

Lσ π σ π

− = =

(9)

Fig. 3. Tapered NiTi bar under tensile loading along its axis during forward

transformation

The displacement of the loading point related to this stage can be written as:

A ML L L∆ = ∆ + ∆ (10)

r1

r2

F

L

L1

L2

O

x

dx

r(x)

A

M

xA-M

A-M Boundary

101

where AL∆ and ML∆ are the elongations of the austenite and martensite areas related to

this stage. AL∆ is:

( )( )

( )2

A-M

2 2 21 1 1

2 21 A-M 2

1 1

( )

Lt t

AAx A

F r dx F r LL

E r x LE r x

σ π σ πππ

− − ∆ = = −

∫ (11)

The elastic elongation of a differential element at x in martensite area M related to this

stage can be expressed in two parts. One part is related to the austenite period of the

element from the start of this stage to the instant when the loading level reaches to the

critical transformation load of the element ( )t A xσ . The other part is related to the rest

of this stage where the element is in martensite phase. ML∆ includes the elastic

elongation and the displacement due to martensitic transformation and is written as:

( )( )( )

( )( )( )

( )

( )

A-M A-M

1 1

2 221

A-M 12 2

2211

A-M 121 1 A-M

( ) ( )

( ) ( )

1 1 1 1

x xt t t

M t

L LA M

t tt

M A A M t

r x r dx F r x dxL x L

E r x E r x

LFLx L

E r E L x E E

σ π σ π σ πε

π π

σ εσπ σ

− −∆ = + + −

= − − + − − +

∫ ∫ (12)

Using Eq. (9), the total elongation of the bar from the start of loading is found by adding

the total displacement at the end of stage (1) ( 2tσ α σ= ) to L∆ related to stage (2) and

expressed by Eq. (10):

1/21/2 11 2

1/22 1 2 1 2 1

1 11 1 1 1

2( ) ( )

t t t tM ATot

A M t A M t

L r Lr E r EL FL F

r r r r E E r r E E

σ ε σ επ π σ σ

−

∆ = + − + − − + − − −

(13)

Using Eqs. (2), (3), (7) and (13), the nominal stress-strain equation for this stage is

obtained as:

1/21 2 3

1

1/2

2

3

1 1

1

1 12

1

1 1

1

M A

t t

A M t

t t

A M t

c c c

E Ec

cE E

cE E

ε σ σ

αα

σ εα σ

ασ εα σ

= + +

−=

−

= − + −

−= − + −

(14)

102

3.3. Stage (3): tσ σ>

At this stage, all structure has transformed to martensite and undergoes linearly elastic

deformation with martensite modulus of elasticity. L∆ of the bar related to this stage is

found by following relation:

( )2

1

2 22 2

21 21

1

Lt t

ML

M

F r dx F rL L

E r rr xE

L

σ π σ ππ

π

− −∆ = =

∫ (15)

The total elongation with respect to the initial (unloaded) condition is obtained by

adding L∆ at the end of stage (2) ( tσ σ= ) to L∆ expressed by Eq. (15):

1 2

1 1 tTot t

M A M t

FLL L

E r r E E

εσπ σ

∆ = + − +

(16)

Eqs. (2), (3), (7) and (16) give the nominal stress-strain relation as:

4

4

1 1

M

tt

A M t

cE

cE E

σεα

εσσ

= +

= − +

(17)

3.4. Stage (4): tσ σ ′≥

This stage is related to the unloading in fully martensite phase. The strain varies versus

stress according to Eqs. (17) until the stress in the bottom end of the bar reaches to the

reverse transformation stress.

3.5. Stage (5): 2t tα σ σ σ′ ′≤ <

In this period, the reverse M→A transformation begins at lower end and the A-M

boundary continuously moves upward as the load decreases (See Fig. 4). The

displacement of the loading point during this stage can be expressed by Eq. (10), where

ML∆ is written as:

( )( )

( )A-M

1

2 2 22 2 1

2 21 1 A-M

1 1

( )

xt t

MML M

F r dx F r LL

E r L xE r x

σ π σ πππ

′ ′− − ∆ = = −

∫ (18)

103

and AL∆ is for the reversely transformed area and takes into account the transition from

martensite to austenite of each differential element and the overall reverse

transformation strain through following equation:

( )( )( )

( )( )( )

( )2 2

A-M A-M

2 222

2 A-M2 2

( ) ( )

( ) ( )

L Lt t t

A t

x xM A

r x r dx F r x dxL L x

E r x E r x

σ π σ π σ πε

π π

′ ′ ′− −′∆ = + − −∫ ∫ (19)

Substituting Eq. (1) in Eq. (19) and applying integrations yield:

( )2 22

2 11A-M 22 2

1 1 A-M 2

1 1 1 1t tA t

A M A M t

r LFLL x L

E r E r x L E E

σ εσπ σ

′ ∆ = − − + − − +

(20)

The relation between A-Mx and F can be expressed as:

( ) ( ) 22 2 1 A-M

A-M( )t t

r r xF r x

Lσ π σ π

− ′ ′= =

(21)

The load-displacement equation is derived using Eqs. (10), (18), (20) and (21) and

considering the displacement at the end of previous stage (at tσ σ ′= ):

( )

1 2

2 1

1/2 11/2

2 12 1

1 1

( )

1 1 1 11

( )

M ATot

t t t t t

A M t t A M tt

r E r EL FL

r r

L r LF

E E r r E Er r

π

σ σ ε σ εσ σ σπσ

−∆ =

−

′ + − + + − − + −′ −

(22)

The corresponding nominal strain is obtained as:

1/21 2 3c c cε σ σ′= + + (23)

where c1 and c3 are defined in Eqs. (14) and 2c′ is:

2 1/2

1 11

(1 )t t t

t A M t t

cE E

σ σ εσ α σ σ

′ ′ = − + + ′ − (24)

104

Fig. 4. Tapered NiTi bar during reverse transformation

3.6. Stage (6): 20 tσ α σ ′≤ <

All structure has returned to parent phase, austenite, and recovers elastically to the

original shape according to Eq. (8).

The stress-strain relations derived for Stages (1)-(6) are summarised in Table 1.

Table 1. Nominal stress-strain relations of the six deformation stages

Stage Nominal stress interval Nominal stress-strain relation

(1) 20 tσ α σ≤ ≤

AE

σεα

=

(2) 2t tα σ σ σ< ≤ 1/2

1 2 3

1

1/2

2

3

1 1

1

1 12

1

1 1

1

M A

t t

A M t

t t

A M t

c c c

E Ec

cE E

cE E

ε σ σ

αα

σ εα σ

ασ εα σ

= + +

−=

−

= − + −

−= − + −

r1

r2

F

L

L1

L2

O

x

dx

r(x)A

M

xA-M

A-M Boundary

105

(3) tσ σ>

4

4

1 1

M

tt

A M t

cE

cE E

σεα

εσσ

= +

= − +

(4) tσ σ ′≥

4

4

1 1

M

tt

A M t

cE

cE E

σεα

εσσ

= +

= − +

(5) 2t tα σ σ σ′ ′≤ < 1/2

1 2 3

1

2 1/2

3

1 1

1

1 11

(1 )

1 1

1

M A

t t t

t A M t t

t t

A M t

c c c

E Ec

cE E

cE E

ε σ σ

αα

σ σ εσ α σ σ

ασ εα σ

′= + +

−=

− ′ ′ = − + + ′ −

−= − + −

(6) 20 tσ α σ ′≤ <

AE

σεα

=

3.7. Illustrations

As analysed above, with regard to the mechanical behaviour in the form of stress-strain

relations, the final solution is independent of the bar length. However, it should be noted

that equations in the form of Eq. (4) are written using the assumption that the stress is

uniformly distributed over each cross section of the bar. This assumption gives

satisfactory results for a tapered bar provided that the angle of taper is small. As

reported by Gere and Goodno [22], if this angle is 20o, the uniformly distributed stress is

3% less than the exact stress. For smaller angles, this error decreases.

To illustrate the analytical solutions, we assume a pseudoelastic NiTi bar with the

following specifications:

400 , 200 , 0.06, 90 , 30t t t A MMPa MPa E GPa E GPaσ σ ε′= = = = = (25)

The nominal stress-strain diagram of the tapered NiTi bar ( 0.83α = ) is depicted in Fig.

5 as the solid-line curve. The numbers on the curve correspond to the stages defined in

Sections 3.1 to 3.6 and summarised in Table 1. The dash-line curve is the stress-strain

diagram of the prismatic bar (with constant cross-sectional radius) of the same material.

106

Positive stress gradients are evident in stages (2) and (5), as described by nonlinear Eqs.

(14) and (23). The average stress-strain slope of Stage (2) is greater than that of Stage

(5), providing varying hysteresis width over pseudoelastic loop. As seen, the plateau

length in the tapered NiTi bar is slightly larger than that of the prismatic one.

Fig. 5. Nominal stress-strain diagram of the tapered NiTi bar

Fig. 6 shows the effects of α variation on stress-strain behaviour, while other

specifications are kept constant as defined in Eqs. (25). By decreasing α, the plateau

length and slope increase. The dashed lines are linear reference lines. It is seen that the

stress–strain curve (solid line) deviates from linearity over the stress plateau.

Fig. 6. The effect of α variation on stress-strain behaviour of the tapered NiTi bar

Fig. 7 demonstrates the effect of EM variation on the mechanical behaviour of a NiTi

tapered bar with 0.67α = , while other properties are defined as Eqs. (25). As EM

decreases, the curve slope in stage (2) gradually decreases, and the overall plateau strain

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

100

200

300

400

500

Nominal Strain

Nom

inal

Str

ess

(MP

a)

(1)

(2)

(6)

(4)

(3)

(5)

α = 0.83

0 0.02 0.04 0.06 0.080

100

200

300

400

500

Nominal Strain

Nom

inal

Str

ess

(MP

a)

α = 0.83

α = 0.67

α = 0.5

107

increases. It is noted that the stress-strain curve for each value of EM passes through an

intersection point I with coordination values defined in Fig. 7.

Fig. 7. The effect of EM variation on stress-strain behaviour of the tapered NiTi bar

4. Experimental Investigation

A pseudoelastic Ti-50.8at.%Ni bar of 3 mm in diameter is used for experiments. The

transformation behaviour, as measured by differential scanning calorimetry, of the alloy

is shown in Fig. 8. Cyclic tensile test has been carried out on the prismatic bar at 293 K

at a strain rate of 10-4/sec. Fanned air cooling was used to maintain the isothermal

condition. The stress-strain diagram is shown in Fig. 9. The stress-induced martensitic

transformation proceeded in a typical Lüders-type manner, with a clear upper-lower

yielding behaviour at the onset of the stress plateau. According to this figure, the

transformation parameters defined in Fig.1 are obtained as following:

390 , 110 , 0.067, 24 , 17t t t A MMPa MPa E GPa E GPaσ σ ε′= = = = = (26)

Fig. 8. Thermal transformation behaviour of the NiTi bar

0 0.02 0.04 0.06 0.08 0.10

100

200

300

400

500

Nominal Strain

Nom

inal

Str

ess

(MP

a)

EM=10GPa

EM=60GPa

EM=30GPa

,tt t

A

IE

σ ε ασ

+

α = 0.67

150 180 210 240 270 300 330 360

Hea

t Flo

w

Temperature (K)

108

Fig. 9. Pseudoelastic tensile stress-strain loops of a straight NiTi bar

A tapered element from that NiTi bar is fabricated with following geometrical

dimensions:

1 240 , 1.2, 1.5 ( 0.8)Gaugelength mm r r α= = = = (27)

Fig. 10 demonstrates the nominal stress-strain diagram of the tapered bar under cyclic

tensile loading. The gradient stress for SIMT is observed in the pseudoelastic cycles.

This provides controllability of load and displacement over the plateau. The internal

loops follow the same forward and reverse paths of the full cycle similar to deformation

behaviour of the uniform bar. Fig. 11 compares the experimental data (last cycle) with

the analytical solution using parameters defined in Eqs. (26) and (27). It is seen that the

analytical solution can reasonably predict the experimental behaviour of the sample.

Fig. 10. Pseudoelastic tensile stress-strain loops of a tapered NiTi bar

0

100

200

300

400

500

0 0.02 0.04 0.06 0.08 0.1

Nom

inal

Str

ess

(MP

a)

Nominal Strain

0

100

200

300

400

0 0.02 0.04 0.06 0.08 0.1

Nom

inal

Str

ess

(MP

a)

Nominal Strain

109

Fig. 11. Comparison of the analytical solution with experiment

5. Discussions

The analysis of this study, in particular the modelling, revealed some interesting aspects

in the mechanical behaviour of tapered NiTi bars. These features may be discussed as

following.

It is evident that the pseudoelastic stress-strain loops of the tapered bar have non-

constant stress hysteresis, despite the constant stress hysteresis of the ideal uniform NiTi

bar defined in figure 1. This is related to the generally higher d

d

σε

for the forward

transformation (Stage 2) relative to the lower value for the reverse transformation (Stage

5). The difference in the d

d

σε

value is mainly determined by the fact that tσ is higher

than tσ ′ . The ratio between the maximum stress and the minimum stress for inducing

martensitic transformation (for both the forward and the reverse processes) of a tapered

NiTi bar is determined by the ratio of the cross-sectional area of the big end and that of

the small end, which is the same for both the forward process and the reverse process.

Given t tσ σ ′> , naturally ( ) ( )A M M Aσ σ

→ →∆ > ∆ . It is easy to demonstrate that the actual

mathematical relationship between A M

d

d

σε →

and M A

d

d

σε →

is reflected by the values

of 1c , 2c and 2c′ , which are in turn affected by material and geometrical properties.

Another point of interest is point I identified in figure 7, which defines a common point

in Stage (4) for all pseudoelastic loops of different EM values. The same is also proven

0 0.02 0.04 0.06 0.08 0.10

100

200

300

400

Nominal Strain

Nom

inal

Str

ess

(MP

a)

α = 0.8

110

invariably for a range of other sample geometries (unpublished work by the same author

on 2-D geometrically graded NiTi components under tensile loading). The interpretation

of this point is unclear.

It is also worth mentioning that, due to the geometrical gradient, the stress-induced

martensitic transformation in the component always occurs in a localised manner,

nucleating always at the small end and propagating progressively towards the big end.

This implies that during actuation cycling, the small end always experience more

deformation that the big end, thus is the location of first failure, both in fatigue and over

loading.

Finally, we wish to point out that the derivation of the constitution equations presented

in Table 1 is purely mathematical and the method is equally applicable to pseudoelastic

stress-strain loops with positive stress-strain slopes instead of flat stress plateaus as

defied in Figure 1.

6. Conclusions

1. Gradient stress for stress induced martensitic transformation is achieved by tapering

a NiTi bar. The stress gradient, or the widened stress interval, for inducing the

martensitic transformation renders the component better controllability in stress-

induced martensitic transformation.

2. The stress gradient for stress-induced martensitic transformaton can be adjusted by

varying the taper angle.

3. A mathematical model is established to describe the pseudoelastic behaviour of

tapered NiTi bars (wires). Closed-form solution for load-displacement (stress-strain)

relation of tapered pseudoelastic NiTi bars (wires) is derived. The mathematical

model agrees well with tensile testing of tapered pseudoelastic NiTi bar.

Acknowledgement

This work is partially supported by the Korea Research Foundation Global Network

Program Grant KRF-2008-220-D00061 and the French National Research Agency

Program N.2010 BLAN 90201.

References

[1] J. Van Humbeeck, Mater. Sci. Eng. A 273-275 (1999) 134-148.

111

[2] Y. Liu, Y. Liu, J.V. Humbeeck, Scripta Mater. 39 (1998) 1047-1055.

[3] G. Tan, Y. Liu, P. Sittner, M. Saunders, Scripta Mater. 50 (2004) 193-198.

[4] B. Samsam, Y. Liu, G. Rio, Mater. Sci. Forum 654-656 (2010) 2091-2094.

[5] A.S. Mahmud, Y. Liu, T.H. Nam, Phys. Scr. T129 (2007) 222-226.

[6] Y. Fu, H. Du, W. Huang, S. Zhang, M. Hu, Sensor. Actuat. A-Phys 112 (2004) 395-

408.

[7] R.M.S. Martins, N. Schell, H. Reuther, L. Pereira, K.K. Mahesh, R.J.C. Silva,

F.M.B. Fernandes, Thin Solid Films 519 (2010) 122-128.

[8] B.A.S. Shariat, M.R. Eslami, AIAA J. 44 (2006) 2929-2936.

[9] B.S. Shariat, M.R. Eslami, A. Bagri, Thermoelastic stability of imperfect

functionally graded plates based on the third order shear deformation theory, in: 8th

Biennial ASME Conference on Engineering Systems Design and Analysis, ASME,

Torino, Italy, 2006, pp. 313-319.

[10] A.J. Birnbaum, G. Satoh, Y.L. Yao, J. Appl. Phys. 106 (2009).

[11] A.S. Mahmud, Y. Liu, T.H. Nam, Smart Mater. Struct. 17 (2008) 1-5.

[12] S.-y. Yang, S.-w. Kang, Y.-M. Lim, Y.-j. Lee, J.-i. Kim, T.-h. Nam, J. Alloys

Compd. 490 (2010) L28-L32.

[13] M. Dolce, D. Cardone, Int. J. Mech. Sci. 43 (2001) 2657-2677.

[14] J. McCormick, J. Tyber, R. DesRoches, K. Gall, H.J. Maier, J. Eng. Mech. 133

(2007) 1019-1029.

[15] L. Bergmans, J. Van Cleynenbreugel, M. Beullens, M. Wevers, B. Van Meerbeek,

P. Lambrechts, International Endodontic Journal 35 (2002) 820-828.

[16] S.A. Alghamdi, Comput. Struct. 27 (1987) 565-569.

[17] X. Tang, Comput. Struct. 46 (1993) 931-941.

[18] K.K. Raju, G.V. Rao, Comput. Struct. 19 (1984) 617-620.

[19] T. Sakiyama, Comput. Struct. 23 (1986) 119-120.

[20] W.G. Smith, Comput. Struct. 28 (1988) 677-681.

[21] D. Das, P. Sahoo, K. Saha, Journal of Sound and Vibration 324 (2009) 74-90.

[22] J.M. Gere, B.J. Goodno, Mechanics of Materials, Cengage Learning, 2009, pp. 17-18.

112

113

Paper 4: Modelling and experimental investigation of geometrically

graded NiTi shape memory alloys

Bashir S. Shariata, Yinong Liua* and Gerard Riob

aLaboratory for Functional Materials, School of Mechanical and Chemical Engineering,

The University of Western Australia, Crawley, WA 6009, Australia

Tel: +61 8 64883132, Fax: +61 8 64881024, email: yinong.liu@uwa.edu.au

bLaboratoire d’Ingénierie des Matériaux de Bretagne, Université de Bretagne Sud,

Université Européenne de Bretagne, BP 92116, 56321 Lorient cedex, France

Smart Materials and Structures, In Press.

Abstract

To improve actuation controllability of a NiTi shape memory alloy component in

application, it is desirable to create a wide stress window for the stress-induced

martensitic transformation in the alloy. One approach is to create functionally graded

NiTi with a geometric gradient in the actuation direction. This geometric gradient leads

to transformation load and displacement gradients in the structure. This paper reports a

study of the pseudoelastic behaviour of geometrically graded NiTi by means of

mechanical model analysis and experimentation using three types of sample geometry.

Closed-form solutions are obtained for nominal stress-strain variation of such

components under cyclic tensile loading and the predictions are validated with

experimental data. The geometrically graded NiTi samples exhibit distinctive positive

stress gradient for the stress-induced martensitic transformation and the slope of the

stress gradient can be adjusted by sample geometry design.

Keywords: Shape memory alloy; NiTi; Martensitic transformation; Pseudoelasticity;

Functionally graded material

1. Introduction

NiTi shape memory alloys (SMAs) show a variety of remarkable engineering

properties, most notably the shape memory effect and pseudoelasticity [1, 2]. They are

used in a wide range of applications including actuation mechanisms [3, 4]. The

actuating force in these mechanisms is the result of shape change along the direction of

114

force due to solid state transformation induced by either stress or temperature. Thus, this

force can be used to perform work. To have an optimum control of displacement over

controlling parameters, i.e. temperature for thermally induced actuation or stress for

stress-induced actuation, the shape memory component is required to act in a wide

range of these parameters. It is known that tensile deformation behaviour of this alloy is

characterised by nucleation and propagation of localised transformation bands [5-8].

However, in the case of stress-induced transformation of a typical NiTi, the large

transformation strain occurs over a constant value of stress due to Lüders-type

deformation under tensile loading, creating a situation of mechanical instability [9-11].

This sudden deformation weakens the controllability of the actuating component over

the stress plateau with respect to the controlling parameter. Therefore, it is necessary to

widen the controlling interval of the NiTi element to enhance its controllability. One

way to achieve this is to create transformation stress gradient along the actuation

direction. This may be realised by creating functional gradient in either the

microstructure [12, 13] or geometry of the SMA component in that direction [14].

Mechanics of functionally graded materials have been extensively studied by

researchers in the past two decades [15, 16]. In typical functionally graded plates and

shells, the material composition is graded through the thickness of the structure,

providing the desired gradient of material properties [17-19]. In SMAs, functionally

grading has been mostly focused on NiTi-based thin films [20]. The common type is to

gradually change the NiTi composition in the thickness direction [21]. This composition

gradient through the film thickness can be effectively achieved by controlling the Ni/Ti

sputtering ratio [22]. Another example is to add a layer of TiN to improve tribological

characteristics and load bearing capacity of the structure without sacrificing

pseudoelasticity and/or shape memory effect of the NiTi film [21]. In addition to NiTi

thin films, a few studies have been reported on microstructurally grading NiTi wires by

a designed annealing temperature gradient [23, 24]. As the transformation properties are

highly sensitive to heat treatment conditions [25-27], the anneal temperature profile

imposes graded transformation properties along wire length or plate thickness.

Another way to impose the transformation stress gradient is to structurally or

geometrically grade a NiTi component. By changing the cross-sectional area

corresponding to the loading direction, different cross sections of the component

experience varied values of stress, causing transformation stress gradient along the

115

loading direction. For the case of NiTi strips (and plates), this can be achieved by

creating variation in strip width as the thickness remains constant. In NiTi bars (and

wires), transformation stress gradient can be generated by uniformly tapering the

components. This paper introduces an idea of geometrically grading NiTi structures to

obtain a stable response from NiTi which is originally unstable during forward and

reverse martensitic transformations. Unique closed-form solutions are presented for

stress-strain relations of three types of 2D geometrically graded NiTi structures and the

results are validated by tensile testing.

2. Definition of transformation parameters of pseudoelastic NiTi SMAs

For the analysis, we define the transformation parameters based on an idealised

pseudoelastic behaviour of a typical NiTi element undergoing the stress-induced

martensitic transformation as shown in Fig. 1. Six distinctive stages of deformation are

in the pseudoelastic stress–strain hysteresis loop. The forward and reverse

transformation stresses are denoted tσ and tσ ′ , respectively. The forward and reverse

transformation strains are denoted tε and tε ′ , respectively. EA and EM are the apparent

elastic moduli of the austenite and martensite phases, respectively [28, 29]. Of the six

parameters defined, five are independent. The reverse transformation strain tε ′ can be

expressed in terms of the other parameters from the geometrical relation defined in Fig.

1, as:

( )1 1t t t t

M AE Eε ε σ σ

′ ′= − − −

(1)

Fig. 1. Definition of transformation parameters and deformation stages of a typical

pseudoelastic NiTi alloy

tε

tσ ′

tσAE ME

tε ′

σ

ε

(1)

(2)

(6)

(4)

(3)

(5)

116

3. Analytical modelling

In this section, we analyse NiTi strips with constant thickness and varying width,

creating 2D geometrically graded SMA structures.

3.1. Pseudoelastic NiTi strip with linearly varying width

A uniformly tapered NiTi strip of length L and thickness h is considered as shown in

Fig. 2. The width ( )w x increases linearly from the minimum value a at the top end to

the maximum value b at the bottom end of the sample. The length variable x is

measured from point O where the sides of the strip meet. The strip is assumed to be

under axial tensile load F. We define the nominal stress σ as the axial force divided by

the maximum initial cross-sectional area at the bottom end:

F

bhσ = (2)

Fig. 2. Linearly tapered NiTi strip under tensile loading; (a): during forward

transformation, (b): during reverse transformation

We define the nominal strain of the NiTi strip under tensile loading as the total

elongation TotL∆ divided by the initial length L [30]:

TotL

Lε ∆= (3)

The width ratio α is defined as:

a

b

L

L1

L2

O

x

dx

w(x)

A

M

xA-M

A-MBoundary

F

Ө

(a)

a

b

L

L1

L2

O

x

dx

w(x)

A

M

xA-M

A-MBoundary

F

Ө

(b)

117

a

bα = (4)

The taper angle Ө is assumed to be small (i.e., L is large relative to a and b). Because of

the martensitic transformation involved, we need to consider separate stages of

deformation to obtain the load-displacement relation. The load-displacement relation of

each stage is derived according to the procedure developed in Appendix A. Then, the

nominal stress-strain relations corresponding to different stages of the loading cycle are

established using Eqs. (2), (3) and (4) as following:

Stage (1): 0 tσ ασ≤ ≤ and Stage (6): 0 tσ ασ ′≤ <

ln( )

(1 )AE

αε σα

−=−

(5)

Stage (2): t tασ σ σ< ≤

1 2 3lnc c cε σ σ σ= + + (6)

( )1 2

1

1 ln

d

d c c

σε σ

=+ +

(7)

Stage (3): tσ σ> and Stage (4): tσ σ ′≥

ln( ) 1 1

(1 )t

tM A M tE E E

εαε σ σα σ

−= + − + − (8)

Stage (5): t tασ σ σ′ ′≤ <

1 2 3lnc c cε σ σ σ′= + + (9)

( )1 2

1

1 ln

d

d c c

σε σ

=′ + +

(10)

where

1

2 3

1

ln 1 ln( ) 11

1

1 1 1 1 1,

1 1

1 1

ln ln( )1

1

t t t

A M t

t t

M A M A t

t tM At t

A M t

cE E

c cE E E E

E Ec

E E

σ ασ εα σ

ασ εα α σ

σ εσ ασ

α σ

+ += − + −

= − = − − − −

− − ′ ′ ′ = − −

′−

(11)

118

Eqs. (5), (6), (8), (9) and (11) form a set of equations which describes the nominal

stress-strain variation of a pseudoelastic NiTi strip with linearly varying width at

different stages of the loading cycle. It is understood that the stress-strain relations

depend on the width ratio α as the only geometrical parameter.

In above analysis, it is assumed that the A→M transformation has been completed

throughout the entire structure during Stage (2), thus the deformations at Stages (3) and

(4) are in fully martensite phase. For the case of partial loading cycle, where the sample

is stretched up to the loading level below tσ prior to unloading, the structure is partially

transformed to martensite and the apparent modulus of elasticity upon unloading is a

composite modulus of the austenite and the martensite, in series connection. In this

case, the deformation of the sample at Stage (2) is described by Eq. (6), ending at the

partial loading level p tσ σ< . Upon unloading, the sample is elastically deformed in

combined austenite and martensite phases over Stage (4p). The load-displacement

relation of this stage is derived in Appendix A. Using this relation and Eqs. (2) and (3),

the nominal stress-strain relation of this stage is presented as:

Stage (4p): t pp

t

σ σσ σ

σ′

≤ <

1 1 1 1 1ln ln

1 1p p tt t

M t A p A M tE E E E

σ σ ασσ εε σα ασ σ α σ −

= + + − + − − (12)

At t pt

t

σ σασ σ

σ′

′ ≤ < of Stage (5), the martensite part of the sample undergoes M→A

transformation and the overall deformation follows Eq. (9).

To illustrate the analytical solution, we assume a pseudoelastic NiTi strip with linearly

varying width and following specifications:

2400 , 200 , 0.06, 90 , 30 ,

3t t t A MMPa MPa E GPa E GPaσ σ ε α′= = = = = = (13)

Using the set of descriptive equations, the nominal stress-strain behaviour of a linearly

graded NiTi strip with the above parameters can be computed, as shown in Fig. (3). The

numbers on the solid curve correspond to the stages defined in the aforementioned

sections. Stage (4p) is related to the partial loading cycle according to Eq. (12) with

119

335p MPaσ = . The dash-line curve is the stress-strain behaviour of the original strip

with uniform width. Positive stress gradients are evident in Stages (2) and (5), as

described by nonlinear Eqs. (6) and (9), respectively. These gradients improve

controllability of the SMA component over forward and reverse transformation stages.

Despite the presence of nonlinear terms in Eqs. (6) and (9), the result is close to a linear

evolution of stress over strain during the stress-induced martensitic transformation,

providing easy control of stress-strain variation during transformation. It is observed

that the transformation strain in the tapered NiTi strip is slightly larger than that of the

uniform strip.

Fig. 3. Nominal stress-strain diagram of a pseudoelastic NiTi strip with linearly varying

width

Fig. 4(a) shows the effect of the width ratio α on the nominal stress-strain behaviour

while other parameters are kept constant as defined in Eqs. (13). As observed, by

decreasing α , both the transformation strain and the stress slope increase. Fig. 4(b)

demonstrates the effect of EM variation on the deformation behaviour of a tapered NiTi

strip. As EM decreases, the curve slope in Stage (2) progressively decreases, which

results in the increase of the overall plateau strain. It is noticed that the stress-strain

curves corresponding to all values of EM pass through the intersection point I at the

following stress and strain values:

(1 ),

ln( )t t

tAE

α σ σσ ε εα

−= = +−

(14)

0 0.02 0.04 0.06 0.080

100

200

300

400

500

Nominal Strain

No

min

al S

tres

s (M

Pa)

(1)

(2)

(6)

(4)

(3)

(5)

F

F

2

3α =

(4p)

120

Fig. 4. The effects of parameters variations on deformation behaviour of a pseudoelastic

NiTi strip with linearly varying width; (a): variation of the width ratio, (b): variation of

the martensite modulus of elasticity

3.2. Pseudoelastic NiTi strip with parabolic sides

In Sec. 3.1, we analysed a uniformly tapered NiTi strip in which the cross-sectional area

varies linearly along the length. In this section, we consider NiTi strips with convex or

concave parabolic sides that provide quadratic variation of cross-sectional area with

respect to the axis passing through their mid length.

3.2.1. Concave strip

A NiTi strip of length L and thickness h with concave parabolic sides is subjected to the

tensile load F as shown in Fig. 5(a). The structure is symmetric about x and y axes. The

width ( )w x varies quadratically from the minimum value a at the middle to the

maximum value b at the top and bottom ends of the sample as:

0 0.02 0.04 0.06 0.080

100

200

300

400

500

Nominal Strain

No

min

al S

tres

s (M

Pa)

F

F

0.9α =

0.7α =

0.5α = 0.3α =

(a)

0 0.02 0.04 0.06 0.080

100

200

300

400

500

Nominal Strain

No

min

al S

tres

s (M

Pa)

10ME GPa=

30ME GPa=

60ME GPa=

F

F

2

3α =

I

(b)

121

22

4( )( )

b aw x a x

L

−= + (15)

Similar to the case of the linearly tapered NiTi strip studied in Sec. 3.1, we need to

consider distinct stages of deformation to establish the analytical stress-strain relations.

The load-displacement relations of these stages can be derived using the procedure

described in Appendix B. Using these relations and considering Eqs. (2), (3) and (4), the

following stress-strain relations are obtained:

Stage (1): 0 tσ ασ≤ ≤ and Stage (6): 0 tσ ασ ′≤ <

1cε σ= (16)

Stage (2): t tασ σ σ< ≤

11 2 3tan 1

(1 )t

t t

c c cσ ασσε σ σ

ασ α σ− −= + − +

− (17)

Stage (3): tσ σ> and Stage (4): tσ σ ′≥

1 3c cε σ′= + (18)

Stage (5): t tασ σ σ′ ′≤ <

11 2 3tan 1

(1 )t

t t

c c cσ ασσε σ σ

ασ α σ− ′−= + − +

′ ′− (19)

where

1

1

2

3

1

1

1tan

(1 )

1 1

(1 )

1 1

1tan

(1 )

A

M A

tt

A M t

M

cE

E Ec

cE E

cE

αα

α α

α α

εσσ

αα

α α

−

−

−

=−

−=

−

= − +

−

′ =−

(20)

Eqs. (16), (17), (18), (19) and (20) describe the nominal stress-strain behaviour of a

pseudoelastic NiTi strip with concave parabolic sides at different stages of deformation.

122

Fig. 5. Pseudoelastic NiTi strips with parabolic sides during forward transformation; (a):

concave strip, (b): convex strip

3.2.2. Convex strip

A NiTi strip of length L and thickness h with symmetric convex parabolic sides relative

to x and y axes is placed under a tensile load F as shown in Fig. 5(b). The width ( )w x

changes quadratically from the minimum value a at both ends to the maximum value b

at the mid length giving the following relation:

22

4( )( )

b aw x b x

L

−= − (21)

As the loading level increases, the A→M transformation initiates at both ends and

propagates toward the middle of the strip (see Fig. 5(b)). The deformation stages can be

analysed using the same approach of Sec. 3.2.1 for concave strip resulting in the

following nominal stress-strain relations which describe the deformation behaviour of

the whole cycle:

Stage (1): 0 tσ ασ≤ ≤ and Stage (6): 0 tσ ασ ′≤ <

1cε σ= (22)

Stage (2): t tασ σ σ< ≤

xA-M

2

L

x ydx

a

b

b

M

A

A

F

F

A-MBoundaries

22

2( )

2

a b ay x

L

−= +

(a)

xA-M

2

L x

y

b

a

a

M

A

F

F

A-MBoundaries

M

A

22

2( )

2

b b ay x

L

−= −

(b)

123

1 2 3

1 1 / 1 /ln 1

11 1 /t t

t

c c cσ σ σ σε σ σ

ασ σ

+ − −′= + + − −− − (23)

Stage (3): tσ σ> and Stage (4): tσ σ ′≥

1 3c cε σ′= + (24)

Stage (5): t tασ σ σ′ ′≤ <

1 2 3

1 1 / 1 /ln 1

11 1 /t t

t

c c cσ σ σ σε σ σ

ασ σ

′+ − ′−′= + + − −′− − (25)

where

1

1

2

3

1 1 1ln

2 1 1 1

1 1 1ln

2 1 1 1

1 1

2 1

1 1

A

M

A M

tt

A M t

cE

cE

E Ec

cE E

αα α

αα α

αεσσ

+ −= − − −

+ −′ = − − −

−=

−

= − +

(26)

3.2.3. Analytical illustration

Using materials parameters defined in Eqs. (13) and the descriptive equations given in

Sections 3.2.1 and 3.2.2, the nominal stress-strain curves of pseudoelastic NiTi strips

with concave and convex sides can be obtained. The results are plotted as solid curves

in Fig. 6(a) for a concave strip and Fig. 6(b) for a convex strip. The numbers on the

curves correspond to the relevant stages of deformation. The dash-line curve is the

stress-strain behaviour of the uniform strip with constant width. Nonlinear stress

gradients are observed in Stages (2) and (5) of each sample, as expressed by Eqs. (17)

and (19) for the concave strip and Eqs. (23) and (25) for the convex strip.

124

Fig. 6. Nominal stress-strain diagram of a pseudoelastic NiTi strip with parabolic sides;

(a): concave strip, (b): convex strip

Fig. 7 illustrates the effect of α variation from 0.3 to 0.9 on the stress-strain behaviour

of the parabolic NiTi strips. The material properties are defined above in Eqs. (13). By

decrease of the a/b width ratio, the average slopes of the stress-strain curve of the stress-

induced martensitic transformation increase in both cases. Also, the overall

transformation strains increase, more considerably in the case of concave strip.

0 0.02 0.04 0.06 0.080

100

200

300

400

500

Nominal Strain

Nom

inal

Str

ess

(MP

a)

(1)

(2)

(6)

(4)

(3)

(5)

F

F

2

3α =

(a)

0 0.02 0.04 0.06 0.080

100

200

300

400

500

Nominal Strain

Nom

inal

Str

ess

(MP

a)

(1)

(2)

(6)

(4)

(3)

(5)

F

F

2

3α =

(b)

125

Fig. 7. The effect of width ratio on deformation behaviour of pseudoelastic NiTi strip

with parabolic sides; (a): concave strip, (b): convex strip

Fig. 8 shows the effect of EM variation on deformation behaviour of NiTi strips with

parabolic sides. It is seen that decrease of EM (relative to EA) leads to the increase of the

forward transformation strain and the decrease of the reverse transformation strain. The

stress-strain behaviour of the concave strip is more sensitive to EM variation than the

convex strip, especially during the forward transformation.

0 0.02 0.04 0.06 0.080

100

200

300

400

500

Nominal Strain

No

min

al S

tres

s (M

Pa) 0.9α =

0.7α =

0.5α =

0.3α =

(a)

F

F

0 0.02 0.04 0.06 0.080

100

200

300

400

500

Nominal Strain

No

min

al S

tres

s (M

Pa)

0.9α =

0.7α =

0.5α =0.3α =

(b)

F

F

126

Fig. 8. The effect of EM variation on deformation behaviour of pseudoelastic NiTi strip

with parabolic sides; (a): concave strip, (b): convex strip

4. Experiments

In this section, tensile testing results are presented for 2D geometrically graded

structures. Ti-50.8at%Ni sheets of 0.1 mm thickness were used for fabricating the

geometrically structured samples. The transformation behaviour of the alloy was

measured by differential scanning calorimetry shown in Fig. 9. The transformation

temperatures are also shown in this figure. The samples were prepared by electric

discharge machining. The gauge length of each sample is 30 mm. The tensile tests were

carried out at the strain rate of 2.8µ10-4/sec using an Instron machine implementing a

displacement-controlled procedure. Fanned air cooling was used to maintain the

isothermal condition.

0 0.02 0.04 0.06 0.080

100

200

300

400

500

Nominal Strain

No

min

al S

tres

s (M

Pa)

10ME GPa=

30ME GPa=

60ME GPa=

2

3α =

I

(a)

F

F

1

(1 ),

1tan

ttt

A

IE

σ α ασ εα

α−

− = + −

0 0.02 0.04 0.06 0.080

100

200

300

400

500

Nominal Strain

Nom

inal

Str

ess

(MP

a)

10ME GPa=

30ME GPa=

60ME GPa=2

3α =

I

(b)

F

F

2 1,

1 1ln

1 1

t tt

A

IE

σ σ αεαα

− = + + − − −

127

Fig. 9. Thermal transformation behaviour of the Ti-50.8at%Ni alloy

4.1. Tensile tests on NiTi strips with constant width

Fig. 10 shows the tensile stress-strain behaviour of straight strip samples at 290 K

(Figure (a)) and 300 K (Figure (b)). The material exhibits good pseudoelastic behaviour,

with a full deformation recovery of 8%, which is ~2% beyond the end of the stress-

induced transformation. The transformation strain (tε ) is ~5.8% for the forward A-M

transformation. The stress for the forward transformation is ~280 MPa at 290 K and

~320 MPa at 300 K.

175 200 225 250 275 300 325 350 375H

eat

Flo

w

Temperature (K)

278M AT K→ =

276A RT K→ =217R MT K→ =

0

200

400

600

800

1000

0 0.02 0.04 0.06 0.08 0.1 0.12

Str

ess

(MP

a)

Strain

F

F

(a)

T = 290 K

128

Fig. 10. Tensile stress-strain diagrams of Ti-50.8at%Ni alloy; (a): at 290 K, (b): at 300K

4.2. Tensile tests on NiTi strips with linearly varying width

Strip samples with tapered gauged section similar to that shown in Fig. 2 were used for

tensile testing. A schematic of the sample geometry is shown in Fig. 11(a). The shaded

ends represent the sections fixed within the clamps of tensile testing. The dimensions

are b = 6 mm and a = 2, 3 and 4 mm 1 1 2

( , and )3 2 3

α = for different samples. Fig.

11(a) shows the stress-strain behaviour of the alloy with 2

3α = at 300 K. Positive stress

gradients can be observed over forward and reverse processes of the stress-induced

martensitic transformation. The maximum strain recovered upon unloading is 8.7%

after deforming to 9.5%. Figs. 11(b) and (c) illustrate tensile test results at 290 K for

1

2α = and

1

3α = , respectively. The average slopes of stress-strain curves at the

forward and reverse transformation stages are measured for the three samples shown in

Fig. 11. The results are illustrated in Fig. 12. Also shown in Fig. 12 are the variations of

d

d

σε

with respect to the width ratio α calculated from Eqs. (7) and (10) for the mean

value of nominal stress during forward and the reverse transformations. The material

properties required for analytical calculation are adopted from Fig. 10(a) as:

290 , 60 , 0.058, 45 , 25t t t A MMPa MPa E GPa E GPaσ σ ε′= = = = = (27)

0

200

400

600

800

1000

0 0.02 0.04 0.06 0.08 0.1 0.12

Str

ess

(MP

a)

Strain

F

F

(b)

T = 300 K

129

The right vertical axis in Fig. 12 shows the variation of stress window, i.e., the stress

difference of Stage (2) defined as d

d

σσ εε

∆ = ∆ with 0.06ε∆ = . It is evident that by

reduction of α , the slopes over the forward and reverse transformations and the stress

windows increase. It is also evident that the analytical solutions agree well with the

experimental measurements.

Fig. 11. Deformation behaviour of NiTi strips with linearly varying width under cyclic

loading; (a): 2

3α = tested at 300 K, (b):

1

2α = tested at 290 K, (c):

1

3α = tested at 290K

0

100

200

300

400

500

600

700

0 0.02 0.04 0.06 0.08 0.1

No

min

al S

tres

s (M

Pa)

Nominal Strain

T = 300 K2

3α =

F

F

(a)ab

0

50

100

150

200

250

300

350

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

No

min

al S

tre

ss (

MP

a)

Nominal Strain

T = 290 K1

2α =

F

F

(b)

0

50

100

150

200

250

300

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

No

min

al S

tres

s (M

Pa)

Nominal Strain

T = 290 K1

3α =

F

F

(c)

130

Fig. 12. Variation of stress-strain slope over forward and reverse transformation stages

based on experiments and analytical computations

4.3. Tensile tests on NiTi strips with parabolic sides

Concave and convex NiTi strips as those shown in Fig. 5 were tested under tensile

loading. The end sections for gripping are rectangular extensions of the same width as

the ends of the gauge section (as for the tapered samples shown in Fig. 11). The

dimensions of the gauge section are b = 6 mm and a = 2, 3 and 4 mm. Fig. 13(a) depicts

the nominal stress-strain behaviour of a concave strip of 2

3α = . Fig. 13(b) shows that

result for a convex strip of the same width ratio. It is observed that the parabolic

samples show curved stress gradient for the stress-induced martensitic transformation,

as predicted by the analytical model shown in Fig. 6. It is also evident that the concavity

and convexity of the strip affect the curvature direction of stress gradient for the stress-

induced martensitic transformation. In comparison, the concave strip shows a definitive

onset for the stress-induced transformation with a clear upper-lower yielding whereas

the convex strip exhibits a clear end of the forward transformation. It is clearly related

to the differences in the location of the initiation and termination of the stress-induced

martensitic transformation and the direction of propagation of the transformation front

between the two samples. For the concave sample, the stress-induced transformation

initiates from the centre and propagates towards the ends, whereas for the convex

sample the transformation starts at the ends and propagates towards the centre.

0

30

60

90

120

150

180

0

0.5

1

1.5

2

2.5

3

0.20 0.30 0.40 0.50 0.60 0.70 0.80

Str

ess

Win

dow

(M

Pa)

Ave

rag

e S

tres

s-S

trai

n S

lop

e (G

Pa)

Width Ratio (α)

Forward Transformation

Reverse Transformation

131

Fig. 13. Deformation behaviour of NiTi strips with parabolic sides (2

3α = ) under cyclic

loading at 300 K; (a): concave strip, (b): convex strip

Fig. 14 demonstrates the deformation behaviour of concave and convex strips of 1

2α =

and 1

3α = tested at 290 K. As the width ratio changes from

1

2 to

1

3, the average stress-

strain slopes over the forward and reverse transformations increase for both the concave

and convex strips. As observed in Figs. 14(a) and (b), the concave strips of 1

2α = and

1

3α = can fully recover 6% and 5% of nominal strain respectively upon unloading.

Increasing the nominal strain to 6.5% leads to plastic deformation in the narrow areas at

the middle of strips, causing residual strain upon unloading. The amount of residual

strains after the last cycle are 0.7% and 1.2% for 1

2α = and

1

3α = , respectively. We

can observe similar phenomena for the case of convex strips (Figs. 14(c) and (d)).

0

100

200

300

400

500

600

0 0.02 0.04 0.06 0.08 0.1N

omin

al S

tres

s (M

Pa)

Nominal Strain

T = 300 K

2

3α =

(a)

F

F

0

100

200

300

400

500

600

0 0.02 0.04 0.06 0.08 0.1

No

min

al S

tres

s (M

Pa)

Nominal Strain

T = 300 K

2

3α =

(b)

F

F

132

Fig. 14. Deformation behaviour of NiTi strips with parabolic sides under cyclic loading

at 290 K; (a): concave strip (1

2α = ), (b): concave strip (

1

3α = ), (c): convex strip (

1

2α = ), (d): convex strip (

1

3α = )

0

50

100

150

200

250

300

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

Nom

inal

Str

ess

(MP

a)

Nominal Strain

T = 290 K1

2α =

(a)

F

F

0

50

100

150

200

250

300

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

No

min

al S

tres

s (M

Pa

)

Nominal Strain

T = 290 K

(b)

F

F

1

3α =

0

50

100

150

200

250

300

350

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

No

min

al S

tres

s (M

Pa)

Nominal Strain

T = 290 K1

2α =

(c)

F

F

0

50

100

150

200

250

300

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

No

min

al S

tre

ss (

MP

a)

Nominal Strain

T = 290 K1

3α =

(d)

F

F

133

4.4. Comparison of the analytical model with experiments

To plot the corresponding stress-strain curves based on the analytical solutions given in

Sec. 3, we need to define the material and transformation parameters from the original

deformation behaviour of NiTi alloy presented in Fig. 10(b) at 300 K, as given below:

2320 , 80 , 0.058, 45 , 25 ,

3t t t A MMPa MPa E GPa E GPaσ σ ε α′= = = = = = (28)

Fig. 15(a) shows the comparison of the actual stress-strain curves of linearly tapered,

concave and convex strips extracted from Figs. 11(a), 13(a) and 13(b). Fig. 15(b) shows

the corresponding calculated curves based on the analytical solutions given in Sec. 3

and the parameters defined in Eqs. (28). It is apparent the analytical solutions match

well with the experimental observations.

Fig. 15. Comparison of deformation behaviour of NiTi strips with linear and parabolic

variation of width (2

3α = ) tested at 300 K; (a): experiments, (b): analytical modelling

0

100

200

300

400

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

No

min

al S

tres

s (M

Pa)

Nominal Strain

T = 300 K2

3α =

(a)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

100

200

300

400

Nominal Strain

Nom

inal

Str

ess

(MP

a)

T = 300 K2

3α =

(b)

134

5. Conclusions

1. This study presents an analytical model to describe the deformation behaviour of

geometrically graded pseudoelastic NiTi shape memory alloys. Closed-form

solutions for nominal stress-strain relations of geometrically structured NiTi strips

with linear and parabolic variations of width under tensile loading are derived.

2. Geometrically graded 2D NiTi samples were fabricated with linear and parabolic

variations of width along the axial direction. The samples were tested under partial

loading cycles. The experimental results can suitably validate the analytical model.

3. The linear and quadratic variations of NiTi strip width create stress gradients along

the length of the sample during tensile loading. Under such conditions, samples

exhibit positive stress slopes and widened stress windows over the range of the

stress-induced martensitic transformation. The slope and widened stress window

improve the controllability of SMA component for stress-induced actuation in

applications.

4. The average slope and the stress window for stress-induced martensitic

transformation can be adjusted by changing the width ratio. For strips of width ratios

with large deviation from unity, care needs to be taken to limit total deformation to

avoid plastic deformation in the smallest area of the strip in order to maintain full

pseudoelasticity.

Appendix A

In this section, the derivations of load-displacement relations of pseudoelastic NiTi strip

with linearly varying width are presented.

A.1. Stage (1): 0 tσ ασ≤ ≤

At this stage, the entire structure is in austenite phase since the applied load is below the

required value to induce martensitic transformation at the top end of the strip where the

width is the smallest. The total elongation of the strip is determined by integration of

elastic deformation along the length of the sample:

2

1( )

L

TotAL

FdxL

E w x h∆ = ∫ (A.1)

135

where L1 and L2 are distances from the origin point to the top and the bottom ends,

respectively, as shown in Fig. 2, and are described in terms of the geometrical

dimensions of the strip as:

1

2

aLL

b abL

Lb a

=−

=−

(A.2)

The width ( )w x is written as:

1 2

( )ax bx

w xL L

= = (A.3)

Using Eqs. (A.1), (A.2) and (A.3), the load-displacement relation of this stage is found

as:

ln( )

(1 )TotA

LL F

E bh

αα

−∆ =−

(A.4)

A.2. Stage (2): t tασ σ σ< ≤

As the cross-sectional area varies in the length direction, the strip undergoes varying

stresses along its length at a constant load. This stage starts with the initiation of

austenite to martensite transformation at the top end of the strip with the highest stress

level. The transformation progressively propagates downwards as the loading level

increases. The structure consists of both austenite and martensite regions, marked by A

and M, respectively in Fig. 2(a). The displacement of the moving A-M boundary

relative to the origin O is defined by variable -A Mx as shown in Fig. 2(a). This stage ends

when the A-M boundary reaches to the bottom end of the strip ( - 2A Mx L= ). At an

instance when the A-M boundary is at -A Mx , the following equation can be written:

- -2

( ) tt A M A M

bhF w x h x

L

σσ= = (A.5)

The displacement of the loading point related to the Stage (2) can be described as:

A ML L L∆ = ∆ + ∆ (A.6)

where AL∆ and ML∆ are the elongations of the austenite and martensite regions related to

the current stage. AL∆ is written as:

136

( ) ( )2

-

1 2

-

ln( )

A M

Lt t

AA A A Mx

F ah dx F ah L LL

E w x h E ah x

σ σ− − ∆ = =

∫ (A.7)

ML∆ includes the elastic elongation of the martensite region during this stage and the

displacement due to martensitic transformation. We consider a differential element dx

located at x in the martensite area M as shown in Fig. 2(a). The elastic elongation of this

element can be separated in two parts. One part is related to the austenite period of the

element from the start of this stage to the instant when the loading level reaches to the

critical transformation load of the element ( )tw x hσ . The other part is related to the

remaining period of this stage where the element is in martensite phase.

( ) ( ) ( )

( )

- -

1 1

- 1

-1 - 1

1

( ) ( )

( ) ( )

1 1ln

A M A Mx xt t t

M A M tA ML L

t tA Mt A M

M A A M t

w x h ah dx F w x h dxL x L

E w x h E w x h

xFL x L

E ah E L E E

σ σ σε

σ εσσ

− −∆ = + + −

= − + − + −

∫ ∫ (A.8)

To obtain the total elongation of the strip TotL∆ , we add the total displacement at the end

of stage (1) ( tF ahσ= ), determined using Eq. (A.4), to L∆ related to the current stage

using Eqs. (A.6), (A.7) and (A.8). The variable A Mx − can be eliminated using Eq. (A.5).

Applying Eqs. (A.2), the load-displacement relation of Stage (2) is then established in

the following form:

ln( ) 1 ln( ) 1

( )

1 1 1 1ln

( )

t t tTot

A M t

t t

M A M A t

bh ahLL F

b a h E E

aLLF F

b a h E E b a E E

σ σ εσ

σ εσ

+ +∆ = − + −

+ − + − − − −

(A.9)

A.3. Stage (3): tσ σ>

At this stage, the structure has entirely transformed to the product phase, martensite, and

is assumed to undergo an elastic deformation with martensite modulus of elasticity. L∆

of the strip related to this stage is found by the following relation:

( ) ( )2

1

ln( )

( ) (1 )

Lt

tM ML

F bh dx LL F bh

E w x h E bh

σ α σα

− −∆ = = −−∫ (A.10)

137

The total elongation with respect to the initial (unloaded) condition is obtained by

adding the total displacement at the end of Stage (2), given by Eq. (A.9) for tF bhσ= ,

to L∆ expressed by Eq. (A.10):

ln( ) 1 1

(1 )t

Tot tM A M t

LL F L

E bh E E

εα σα σ

−∆ = + − + − (A.11)

A.4. Stage (4): tσ σ ′≥

This stage is related to the elastic unloading period where the structure is in fully

martensite phase. The structure deforms according to Eq. (A.11) till the stress at the

bottom end of the NiTi strip reaches the reverse transformation stress and the next stage

starts.

A.5. Stage (5): t tασ σ σ′ ′≤ <

At this stage, the reverse M→A transformation begins at the wider end of the strip

where the stress level is the lowest. As the loading level decreases, the A-M boundary

continuously moves upward as seen in Fig. 2(b). This stage ends when the A-M

boundary reaches to the top end of the strip (- 1A Mx L= ). At an instance where the A-M

boundary is at -A Mx , the following relation can be written:

- -2

( ) tt A M A M

bhF w x h x

L

σσ′′= = (A.12)

The displacement of the loading point during this stage can be expressed by Eq. (A.6)

where ML∆ is written as:

( ) ( )-

1

1 -

1

ln( )

A Mxt t A M

MM ML

F bh dx F bh L xL

E w x h E ah L

σ σ′ ′− − ∆ = =

∫ (A.13)

and AL∆ is the displacement of the reversely transformed area A, as marked in Fig.

2(b), and takes into account the elastic deformation of each differential element at

martensite and austenite periods and the overall reverse transformation strain:

( ) ( ) ( )

( )

2 2

- -

2 A-M

21 2 A-M

-

( ) ( )

( ) ( )

1 1ln

A M A M

L Lt t t

A tM Ax x

t tt

A M A M A M t

w x h bh dx F w x h dxL L x

E w x h E w x h

LFL L x

E ah E x E E

σ σ σε

σ εσα σ

′ ′ ′− −′∆ = + − −

′ ′ ′= − − − + − ′

∫ ∫ (A.14)

138

The load-displacement relation with respect to the start of loading is derived using Eqs.

(1), (A.6), (A.13) and (A.14) and considering the total displacement at the end of Stage

(4) (at tσ σ ′= ):

1 1

ln( ) ln( )

( )

1 1 1 1ln

( )

t tM At t

TotA M t

t t

M A M A t

E Ebh ahLL F

b a h E E

aLLF F

b a h E E b a E E

σ εσ σ

σ

σ εσ

− − ′ ′ ∆ = − −

′−

+ − + − − − −

(A.15)

A.6. Stage (6): 0 tσ ασ ′≤ <

All structure has returned to the parent phase, austenite, and recovers elastically to the

original shape according to Eq. (A.4).

A.7. Stage (4p): t pp

t

σ σσ σ

σ′

≤ <

This stage describes the elastic unloading in combined austenite and martensite phases

of a partial loading cycle prior to entering Stage (5). The structure has been loaded at

the nominal stress of p tσ σ< . The elongation of the sample at this point of Stage (2) is

obtained using Eq. A.9 with pF bhσ= . The total displacement of the sample during

Stage (4p) is obtained by summation of the elastic deformation of martensite and

austenite regions during this stage and the net elongation of the entire structure at the

end of partial loading at Stage (2):

( ) ( )2

2

21

(at the end of partial loading)( ) ( )

1 1 1 1= ln ln

(1 ) 1

p

t

p

t

L

Lp p

TotLM AL

p p tt t

M t A p A M t

F bh dx F bh dxL L

E w x h E w x h

LF L

bh E E E E

σσ

σσ

σ σ

σ σ ασσ εα ασ σ α σ

− −∆ = + + ∆

− + + − + − −

∫ ∫

(A.16)

Appendix B

In this section, the derivation method of load-displacement relations of a concave

pseudoelastic NiTi strip is described.

139

B.1. Stage (1): 0 tσ ασ≤ ≤ and Stage (6): 0 tσ ασ ′≤ <

In these two stages the structure is entirely in austenite. The total elongation is

determined to be:

2

0

2( )

L

TotA

FdxL

E w x h∆ = ∫ (B.1)

B.2. Stage (2): t tασ σ σ< ≤

At the beginning of this stage, the martensitic transformation initiates at the middle of

the strip which has the minimum cross-sectional area. As the load increases, the A-M

boundaries symmetrically move toward the ends of the strip as shown in Fig. 5(a). At an

instance when the upper A-M boundary is at xA-M, we can write:

2- 2

4( )( )t A M t A M

b aF w x h h a x

Lσ σ −

− = = +

(B.2)

The elongations of the austenite and martensite regions related to this stage is written

as:

( ) ( )

( ) ( )-

- -

21 1

-

-

0 0

1-

22 tan tan

( ) ( )

( ) ( )2 2 2

( ) ( )

2tan

2( )

A M

A M A M

L

t tA A M

Ax A

x xt t t

M A M tA M

A M

tt

M A

F ah dx F ah L b a b aL x

E w x h a L aE a b a h

w x h ah dx F w x h dxL x

E w x h E w x h

b aL x

L aaF

E h E a b a

σ σ

σ σ σε

σ σ

− −

−

− − − −∆ = = − −

− −∆ = + +

−

= − + −

∫

∫ ∫

-

1 1 tA M

A M t

xE E

εσ

− +

(B.3)

Using Eqs. (A.6), (B.2) and (B.3) and considering the total displacement of the strip at

the end of the previous stage obtained from Eq. (B.1), the load-displacement relation is

derived.

B.3. Stage (3): tσ σ> and Stage (4): tσ σ ′≥

At these stages, the structure is entirely in martensite. The elastic elongation of this

stage is written as:

140

2

0

( )2

( )

L

t

M

F bh dxL

E w x h

σ−∆ = ∫ (B.4)

The total elongation relative to the start of loading is found by adding the total

displacement at the end of Stage (2) to the above equation.

B.4. Stage (5): t tασ σ σ′ ′≤ <

At this stage, the reverse transformation starts at both ends of the structure. Austenite

and martensite phases coexist and the A-M boundaries move toward the mid length of

the strip. By defining the relevant AL∆ and ML∆ and following the derivation process of

Sec. B.2 with the consideration of the reverse transformation parameters, the load-

displacement relation is obtained.

References

[1] Y. Liu, S.P. Galvin, Acta Mater. 45 (1997) 4431-4439.

[2] J.V. Humbeeck, Adv. Eng. Mater. 3 (2001) 837-849.

[3] J. Van Humbeeck, Mater. Sci. Eng. A 273-275 (1999) 134-148.

[4] Y. Bellouard, Mater. Sci. Eng. A 481-482 (2008) 582-589.

[5] J.A. Shaw, S. Kyriakides, Acta Mater. 45 (1997) 683-700.

[6] Q.-P. Sun, Z.-Q. Li, Int. J. Solids Struct. 39 (2002) 3797-3809.

[7] P. Sittner, Y. Liu, V. Novak, J Mech. Phys. Solids 53 (2005) 1719-1746.

[8] G. Murasawa, S. Yoneyama, T. Sakuma, Smart Mater. Struct. 16 (2007) 160.

[9] J.A. Shaw, S. Kyriakides, J Mech. Phys. Solids 43 (1995) 1243-1281.

[10] G. Tan, Y. Liu, P. Sittner, M. Saunders, Scripta Mater. 50 (2004) 193-198.

[11] B. Samsam, Y. Liu, G. Rio, Mater. Sci. Forum 654-656 (2010) 2091-2094.

[12] Q. Meng, Y. Liu, H. Yang, B.S. Shariat, T.-h. Nam, Acta Mater. 60 (2012) 1658–

1668.

[13] B.S. Shariat, Y. Liu, G. Rio, J. Alloys Compd. 541 (2012) 407-414.

[14] B.S. Shariat, Y. Liu, G. Rio, J. Alloys Compd., In Press, doi:

10.1016/j.jallcom.2011.12.151.

[15] B.A.S. Shariat, R. Javaheri, M.R. Eslami, Thin Wall. Struct. 43 (2005) 1020-1036.

[16] B.A.S. Shariat, M.R. Eslami, AIAA J. 44 (2006) 2929-2936.

[17] B.A.S. Shariat, M.R. Eslami, J. Thermal Stresses 28 (2005) 1183-1198.

[18] B.A.S. Shariat, M.R. Eslami, Int. J. Solids Struct. 43 (2006) 4082-4096.

[19] B.A.S. Shariat, M.R. Eslami, Compos. Struct. 78 (2007) 433-439.

141

[20] Y. Fu, H. Du, W. Huang, S. Zhang, M. Hu, Sensors and Actuators A: Physical 112

(2004) 395-408.

[21] S. Miyazaki, Y.Q. Fu, W.M. Huang, in: S. Miyazaki, Y.Q. Fu, W.M. Huang (Eds.)

Thin Film Shape Memory Alloys : Fundamentals and Device Applications, Cambridge

University Press, Cambridge, 2009, pp. 48-49.

[22] R.M.S. Martins, N. Schell, H. Reuther, L. Pereira, K.K. Mahesh, R.J.C. Silva,

F.M.B. Fernandes, Thin Solid Films 519 (2010) 122-128.

[23] A.S. Mahmud, Y. Liu, T.H. Nam, Phys. Scr. T129 (2007) 222-226.

[24] A.S. Mahmud, Y. Liu, T.H. Nam, Smart Mater. Struct. 17 (2008) 1-5.

[25] D.A. Miller, D.C. Lagoudas, Mater. Sci. Eng. A 308 (2001) 161-175.

[26] D. Favier, Y. Liu, L. Orgéas, A. Sandel, L. Debove, P. Comte-Gaz, Mater. Sci.

Eng. A 429 (2006) 130-136.

[27] Y. Zheng, F. Jiang, L. Li, H. Yang, Y. Liu, Acta Mater. 56 (2008) 736-745.

[28] Y. Liu, H. Xiang, J. Alloys Compd. 270 (1998) 154-159.

[29] Y. Liu, H. Yang, Mater. Sci. Eng. A 260 (1999) 240-245.

[30] J.M. Gere, B.J. Goodno, Mechanics of Materials, Cengage Learning, 2009, pp. 17-

18.

142

143

Supplement 1 to paper 4: Additional experimental results for geometrically graded

NiTi strips with a wide range of width ratio

In addition to tensile experimentation presented in Paper 4, complementary tests have

been conducted for linearly and parobolicly tapered NiTi samples with

1 1 2 5, , ,

3 2 3 6α α α α= = = = ( 6b = and 2,3,4,5a = ) at the same temperature (303 K)

to compare the deformation behaviour of samples over a wider range of α . The

thickness is 0.1 mm. The gage length is 30 mm unless otherwise stated. Also, several

experiments have been carried out at higher temperature levels to see the effect of

testing temperature on deformation behaviour.

1. Linearly tapered strip

0

100

200

300

400

0 0.02 0.04 0.06 0.08 0.1

No

min

al S

tres

s (M

Pa

)

Nominal Strain

F

F

303

1

3

T K

α

=

=

0

100

200

300

400

500

0 0.02 0.04 0.06 0.08 0.1

No

min

al S

tres

s (M

Pa)

Nominal Strain

F

F

303

1

2

T K

α

=

=

0

100

200

300

400

500

0 0.02 0.04 0.06 0.08

No

min

al S

tres

s (M

Pa)

Nominal Strain

F

F

303

2

3

T K

α

=

=

0

100

200

300

400

500

0 0.02 0.04 0.06 0.08

Nom

inal

Str

ess

(MP

a)

Nominal Strain

F

F

303

5

6

T K

α

=

=

144

0

100

200

300

400

500

0 0.02 0.04 0.06 0.08 0.1

No

min

al S

tres

s (M

Pa)

Nominal Strain

303

1

2

T K

α

=

=

F

F0

100

200

300

400

500

0 0.02 0.04 0.06 0.08 0.1

No

min

al S

tres

s (M

Pa)

Nominal Strain

303

1

2

T K

α

=

=

0

100

200

300

400

0 0.02 0.04 0.06 0.08

No

min

al S

tre

ss (

MP

a)

Nominal Strain

F

F303T K=

1α =

5 6α =

2 3α =1 2α =

1 3α =

0

40

80

120

160

200

240

0

0.5

1

1.5

2

2.5

3

3.5

4

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Str

ess

Win

dow

(M

Pa)

Str

ess-

Str

ain

Slo

pe (

GP

a)

Width Ratio (α)

Forward Transformation

Reverse Transformation

303T K=

0

100

200

300

400

500

0 0.02 0.04 0.06 0.08

Nom

inal

Str

ess

(MP

a)

Nominal Strain

F

F

2

3α =303T K=

311T K=318T K=

145

2. Concave strip

0

100

200

300

400

0 0.02 0.04 0.06 0.08 0.1

No

min

al S

tres

s (M

Pa)

Nominal Strain

303

1

3

T K

α

=

=

F

F

0

100

200

300

400

500

0 0.02 0.04 0.06 0.08 0.1

Nom

inal

Str

ess

(MP

a)

Nominal Strain

F

F

303

1

2GL 30

T K

mm

α

=

=

=

0

100

200

300

400

500

0 0.02 0.04 0.06 0.08

Nom

ina

l Str

ess

(MP

a)

Nominal Strain

303

2

3

T K

α

=

=

F

F

0

100

200

300

400

500

0 0.02 0.04 0.06 0.08

Nom

ina

l Str

ess

(MP

a)

Nominal Strain

303

5

6

T K

α

=

=

F

F

0

100

200

300

400

500

0 0.02 0.04 0.06 0.08 0.1

No

min

al S

tres

s (M

Pa

)

Nominal Strain

303

1

2GL 20

T K

mm

α

=

=

=

F

F0

100

200

300

400

500

0 0.02 0.04 0.06 0.08 0.1

No

min

al S

tres

s (M

Pa

)

Nominal Strain

GL = 20mm

GL = 30mm

F

F

303

1

2

T K

α

=

=

0

100

200

300

400

0 0.02 0.04 0.06 0.08

No

min

al S

tres

s (M

Pa)

Nominal Strain

303T K=

1α =

5 6α =

2 3α =

1 2α =

1 3α =F

F0

100

200

300

400

500

0 0.02 0.04 0.06 0.08

No

min

al S

tres

s (M

Pa)

Nominal Strain

2

3α =303T K=

311T K=318T K=

F

F

146

3. Convex strip

4. Comparison of the three geometry types for 1

2α =

0

100

200

300

400

0 0.02 0.04 0.06 0.08 0.1

No

min

al S

tres

s (M

Pa)

Nominal Strain

F

F

303

1

3

T K

α

=

=

0

100

200

300

400

500

0 0.02 0.04 0.06 0.08 0.1

Nom

ina

l Str

ess

(MP

a)

Nominal Strain

F

F

303

1

2

T K

α

=

=

0

100

200

300

400

500

0 0.02 0.04 0.06 0.08

No

min

al S

tres

s (M

Pa)

Nominal Strain

F

F

303

2

3

T K

α

=

=

0

100

200

300

400

500

0 0.02 0.04 0.06 0.08

No

min

al S

tres

s (M

Pa)

Nominal Strain

F

F

303

5

6

T K

α

=

=

0

100

200

300

400

0 0.02 0.04 0.06 0.08

No

min

al S

tres

s (M

Pa)

Nominal Strain

303T K=

1α =5 6α =

2 3α =

1 2α =

1 3α =F

F

0

100

200

300

400

500

0 0.02 0.04 0.06 0.08

Nom

inal

Str

ess

(MP

a)

Nominal Strain

303T K=

318T K= 311T K=

F

F

2

3α =

0

100

200

300

400

0 0.02 0.04 0.06 0.08

Nom

ina

l Str

ess

(MP

a)

Nominal Strain

303

1

2

T K

α

=

=

147

5. Comparison of the experimental results (2

3α = ) with the analytical model

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

100

200

300

400

Nominal Strain

Nom

inal

Str

ess

(MP

a)

2

3α =

(a)

F

F

T = 303 K

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

100

200

300

400

Nominal Strain

Nom

inal

Str

ess

(MP

a)

2

3α =

(b)

F

F

T = 303 K

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

100

200

300

400

Nominal Strain

Nom

inal

Str

ess

(MP

a)

2

3α =

(c)

F

F

T = 303 K

148

149

Supplement 2 to paper 4: Finite element simulation of geometrically graded NiTi

strips with experimental validation

Here, we use finite element method and elastohysteresis model with the procedure

described in Sec. 4.3 of Chapter 1 to numerically model the global and local

deformation behaviour of geometrically graded NiTi strips with linear and parabolic

sides. The simulation results are compared with experiments. The 11 hyperelastic and

hysteresis parameters of the model, defined in Figs. 13 and 14 of Chapter 1, are

determined from a tensile test on a uniform NiTi sample as following:

1 2 3 1 2

0

270000, 247, 0.065, 12000, 200, 8500, =0.001, =0.005

10500, 100, 2s eK Q Q

Q np

µ µ µ α αµ

= = = = = == = =

0

100

200

300

400

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

No

min

al S

tres

s (M

Pa)

Nominal Strain

F

F

303T K=

Simulation

Experiment

0

100

200

300

400

500

0 0.02 0.04 0.06 0.08

No

min

al S

tre

ss (

MP

a)

Nominal Strain

F

F

303T K= Simulation

0

100

200

300

400

0 0.02 0.04 0.06 0.08

No

min

al S

tre

ss (

MP

a)

Nominal Strain

303

1

2

T K

α

=

=

Experiment

SimulationF

F0

100

200

300

400

0 0.02 0.04 0.06 0.08

No

min

al S

tres

s (M

Pa)

Nominal Strain

303

1

2

T K

α

=

=

Experiment

Simulation

F

F

0

100

200

300

400

0 0.02 0.04 0.06 0.08

No

min

al S

tres

s (M

Pa

)

Nominal Strain

303

1

2

T K

α

=

=

Simulation

Experiment

F

F

0

100

200

300

400

0 0.02 0.04 0.06 0.08

No

min

al S

tres

s (M

Pa)

Nominal Strain

303

1

2

T K

α

=

=

Simulation

150

0

100

200

300

400

0 0.02 0.04 0.06 0.08

No

min

al S

tres

s (M

Pa)

Nominal Strain

303

1

2

T K

α

=

=

F

F

Simulation

0

100

200

300

400

0 0.02 0.04 0.06 0.08

Nom

inal

Str

ess

(MP

a)

Nominal Strain

303

1

2

T K

α

=

=

F

FSimulation

0

100

200

300

400

0 0.02 0.04 0.06 0.08

No

min

al S

tres

s (M

Pa)

Nominal Strain

303

2

3

T K

α

=

=

Experiment

SimulationF

F

0

100

200

300

400

0 0.02 0.04 0.06 0.08

No

min

al S

tre

ss (

MP

a)

Nominal Strain

303

2

3

T K

α

=

=

Experiment

Simulation

F

F

0

100

200

300

400

0 0.02 0.04 0.06 0.08

No

min

al S

tres

s (M

Pa

)

Nominal Strain

303

2

3

T K

α

=

=

Simulation

Experiment

F

F

0

100

200

300

400

0 0.02 0.04 0.06 0.08

No

min

al S

tres

s (M

Pa)

Nominal Strain

303

2

3

T K

α

=

=

Simulation

0

100

200

300

400

0 0.02 0.04 0.06 0.08

Nom

inal

Str

ess

(MP

a)

Nominal Strain

303T K=

1α =

5 6α =

2 3α =

1 2α =

1 3α =F

FSimulation

151

Chapter 4

Perforated NiTi plates under tensile loading

Paper 5: Pseudoelastic behaviour of perforated NiTi shape memory plates under tension Paper 6: Numerical modelling of pseudoelastic behaviour of NiTi porous plates

152

153

Paper 5: Pseudoelastic behaviour of perforated NiTi shape memory

plates under tension

Bashir S. Shariata, Yinong Liua* and Gerard Riob

aLaboratory for Functional Materials, School of Mechanical and Chemical Engineering,

The University of Western Australia, Crawley, WA 6009, Australia

Tel: +61 8 64883132, Fax: +61 8 64881024, email: yinong.liu@uwa.edu.au

bLaboratoire d’Ingénierie des Matériaux de Bretagne, Université de Bretagne Sud,

Université Européenne de Bretagne, BP 92116, 56321 Lorient cedex, France

Smart Materials and Structures, Under Review.

Abstract

This paper reports the tensile deformation behaviour of near-equiatomic NiTi plates

with circular and noncircular holes. The investigation is done both experimentally and

by mathematical modelling. It is found that the nominal stress-strain curve of such

structures deviates from typical stress-strain variation of NiTi with flat stress plateaus

over the forward and reverse transformations. Such mechanical behaviour is

advantageous for better controllability for shape memory actuation and sensing. This

deviation is explained by means of mathematical expressions. The effects of hole size

and numbers along the loading direction on pseudoelastic behaviour are discussed.

Keywords: Shape memory alloy (SMA); NiTi; Martensitic phase transformation;

Pseudoelasticity

1. Introduction

Near-equiatomic NiTi alloy deforms via stress-induced martensitic transformation,

which exhibits hystoelastic mechanical behaviour with large recoverable nonlinear

deformation, known as the pseudoelasticity [1-3]. The pseudoelasticity of NiTi often

manifests in a Lüders-type manner over a stress plateau. This unique property has

facilitated many engineering applications. In most applications, NiTi is used in thin wire

forms under tensile loading [4, 5], to benefit from the high force output in tension and

the rapid actuation due to fast thermal conduction. Other common forms include helical

springs and thin walled tubes. However, some applications require the use of more

154

complex shapes and under more complex loading conditions, such as perforated plates

[6] or holed structures [7], NiTi thin films [8, 9], functionally graded NiTi structures

[10, 11], cellular structures [12] or surgical stents [13], woven NiTi layers embedded in

composite structures [14], and porous NiTi [15]. The complex geometry alters the

mechanics conditions, thus mechanical behaviour of the components. For such

conditions, standard uniaxial tensile testing of straight forms (e.g., wires and strips) is

inadequate to describe or predict the mechanical behaviour of the component. This

study deals with the pseudoelastic behaviour of perforated NiTi plate under tension.

Perforated plates are generally used as heat exchangers [16], sound absorbers [17],

screens and filters [18]. The perforated or porous SMAs, in particular, can be used as

orthopaedic devices, such as artificial bone implants, spinal vertebrae spacers [19] and

skull plates (fixation plates for craniotomy operation) [13]. The existence of hole(s) in a

SMA plate can be considered a geometrical defect to provoke transformation

localisation [20]. Because of variations in geometry, different locations in a holey plate

experience variations in stress state. This inhomogeneous stress field induces local

martensitic transformation in the structure at different loading levels, complicating the

global deformation behaviour of the plate [21].

Perforated plates also provide a study case as a 2D model for porous structures. The

deformation behaviour of porous SMA structures has been mainly studied under

compression [22-27]. The effect of porosity on global stress-strain variation has been

discussed. The numerical modellings of such structures mostly implement

micromechanical averaging techniques or periodic patterns of pores [28]. In this article,

we explore the effect of holes on deformation behaviour of perforated NiTi plates

(strips) under tension. Mathematical expressions are derived to describe the nominal

stress-strain variation of such components.

2. Experimental investigation

Ti-50.8at%Ni sheets of 0.1 mm thickness were used for fabricating NiTi samples with

holes. The samples were prepared by electric discharge machining. The tensile

experiments were carried out at the strain rate of 2.4µ10-4/s and 303 K with fanned air

cooling to maintain the isothermal condition. Fig. 1 presents the stress-strain diagram of

a uniform strip (4 mm µ 30 mm) of such material under tension. Partial deformation

cycles were carried out. The material exhibited good pseudoelastic behaviour, with a

155

full deformation recovery of up to 8%, which is ~2% beyond the end of the stress

plateau. The critical stress for the forward A→M transformation is ~380 MPa and that

for the reverse M→A transformation is ~140 MPa. Here, A and M refer to the austenite

and martensite phases, respectively. Also shown in this figure is the thermal

transformation behaviour of the alloy as measured by differential scanning calorimetry.

Fig.1. Thermomechanical properties of the Ti-50.8at%Ni alloy

Fig. 2 shows the deformation behaviour of NiTi plates of 9 mm µ 35 mm (gauge

section) with circular holes. The holes were created symmetrically along the loading

axis of the samples. Figs. 2(a), (b) and (c) show the nominal stress-strain variations of

the samples with one, two and three holes of 3 mm in diameter, respectively. The

nominal stress is defined as the axial load F divided by the initial full cross-sectional

area of the plate (at where without holes), and the nominal strain is calculated as the

total elongation of the plate TotL∆ divided by the initial length L [29]:

, TotLF

bh Lσ ε ∆= = (1)

where b and h are the width and the thickness of the plate, respectively. All samples

exhibit good pseudoelastic behaviour. However, the nominal stress-strain curves over

A↔M transformations deviated from the flat stress plateaus observed in Fig. 1. Due to

geometrical heterogeneity caused by the holes, the samples experienced normal stress

variation across the loading direction in the course of tensile loading. The A→M

transformation initiated in the regions at the sides of the holes, where the cross-sectional

area is the smallest and the real stress is the highest, and gradually propagated into

regions of higher cross sections as the load increases. As the transformation fronts

0

100

200

300

400

500

600

0 0.02 0.04 0.06 0.08

Str

ess

(MP

a)

Strain

F

F303T K=

175 225 275 325 375

He

at F

low

Temperature (K)

156

reached to the solid part of the plate (with cross-sectional area bh), the nominal stress

remained relatively constant until the entire structure was transformed to martensite.

The stress variation during stress-induced martensitic transformation may be divided

into two regimes, as indicated in Fig. 2(a). The first regime is obviously related to the

section of the strip sample affected by the hole(s) and the second regime is related to the

solid section(s) of the strip sample. Taking sample (a) for example, the length ratio

between the hole section (the diameter of the hole in the length direction) and the total

gauge length is 0.09hη = , whereas the ratio of the transformation strain of regime I ( Iε

) to the total transformation strain (tε ) is 0.27sη = . It is obvious that h sη η< . This

implies that the effect of the hole expanded beyond the diameter of the hole, as

schematically indicated by the shaded section in sample (a) in Fig. 2(a). With the same

argument, the hole-affected lengths are also marked in samples (b) and (c). It is seen

that the hole affected length is approximately triple the diameter of the hole and that

when hole diameter covers 33% of the gauge length, the entire sample behaves like a

functionally graded material. This information is useful for material design for

functionally graded NiTi components.

Fig. 2(d) replots the full deformation cycles of the three perforated NiTi plate samples

together in addition to that of a solid plate sample. It is seen that the forward

transformation started at practically the same value of nominal stress (~250 MPa). This

loading level provided mean local stress of ~380 MPa at the minimum cross-sectional

areas of the plates to induce A→M martensitic transformation, which is practically

identical to the critical stress for inducing the martensitic transformation in the solid

plate (Fig. 1). By increasing the number of holes from 0 (solid plate) to 3, the flat stress

plateaus over A↔M transformations are progressively substituted by stress gradients

over stress-induced martensitic transformations. It can be inferred that the perforated

plates have better controllability of load-displacement over portions of A↔M

transformations with stress gradients. As observed, the overall transformation strain

increases by increase of number of holes.

157

Fig. 2. Deformation behaviour of perforated NiTi plates under tension; (a): one hole,

(b): two holes, (c): three holes, (d): comparison of the three samples with solid plate

0

100

200

300

400

0 0.02 0.04 0.06 0.08

No

min

al S

tre

ss (

MP

a)Nominal Strain

(a)

303

3

T K

d mm

==

I

IIIε

tε

F

F

0

100

200

300

400

0 0.02 0.04 0.06 0.08

No

min

al S

tres

s (M

Pa)

Nominal Strain

(b)

303

3

T K

d mm

==

F

F

0

100

200

300

400

0 0.02 0.04 0.06 0.08

Nom

inal

Str

ess

(MP

a)

Nominal Strain

303

3

T K

d mm

==

(c)

F

F

0

100

200

300

400

500

0 0.02 0.04 0.06 0.08

Nom

inal

Str

ess

(MP

a)

Nominal Strain

303

3

T K

d mm

==

Solid

1 hole 2 holes

3 holes

(d)

158

Fig. 3(a) illustrates the effect of hole size on stress-strain variation of the NiTi plate

with one circular hole under uniaxial loading. In comparison with the solid plate, the

forward transformation started at a lower loading level (as also seen in Fig. 2). This

critical stress for initiating the stress-induced martensitic transformation decreased with

increase of hole diameter. Fig. 3(b) shows dependence of the critical stress on hole

diameter. The horizontal axis is presented as the ratio of the hole diameter and the plate

width. The diagram shows a rather linear decrease of the nominal critical stress with

respect to the increase of d/b ratio. Fig. 3(c) depicts the variation of affected length to

hole diameter ratio against d/b ratio. It is understood that the ratio of affected length to

hole diameter, which is practically s hη η , remains at a nearly constant value of 3 as d/b

ratio varies over studied range.

0

100

200

300

400

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

No

min

al S

tres

s (M

Pa)

d/b

(b)

159

Fig. 3. The effect of hole size on deformation behaviour of perforated NiTi plates under

tension; (a): full deformation cycle, (b): critical nominal stress for A→M

transformation, (c): the ratio of affected length to hole diameter (the samples with

square and elliptical holes are also shown)

Fig. 4 shows the deformation behaviour of NiTi plates (9 mm µ 35 mm) with non-

circular holes, with sample (a) having an elliptical hole and sample (b) having a square

hole. The elliptical hole in sample (a) had a minor axis of 3 mm along the width

direction and a major axis of 5 mm along the length direction of the strip. The square

hole in (b) had a side length of 3 mm. It is seen that in both cases the A→M

transformation started at the nominal stress of ~250 MPa, as the minimum cross-

sectional area is equal for two samples. For the plate with the elliptical hole, the

transformation developed over a continuous stress gradient, as expected of the hole

geometry. For the plate with the square hole, the transformation firstly evolved over a

partial stress plateau relative to the holed region with a constant cross-sectional area.

Then, the loading level increased progressively to that of the second stress plateau

where transformation is mostly completed in the solid part. This deformation behaviour

is divided into three regions, as indicated in the figure. Based on the transformation

strain fractions of the three regions, the corresponding affected lengths within the gauge

length are marked in the sample shown in Fig. 4(b). It is interesting to note that the

deformation behaviour exhibited a gradient stress in region II despite that the sample

cross-section areas remains unchanged. In Fig. 3(c), the s hη η ratios of these samples

are demonstrated with square and triangular markers, respectively for the plate with a

square hole and the one with an elliptical hole. It is observed that this ratio is higher for

the sample with a square hole and lower for that with an elliptical hole comparing with

a sample with a circular hole.

0

1

2

3

4

5

0.2 0.3 0.4 0.5 0.6 0.7d/b

(c)

s hη η

Square hole

Circular hole

Elliptical hole

160

Fig. 4. Deformation behaviour of NiTi plates with a noncircular hole under tension; (a):

elliptical hole, (b): square hole

3. Analytical modelling

To analytically describe the deformation behaviour of a holed NiTi plate, we use a

mathematical approach to obtain the nominal stress-strain curve of such a structure at

different stages of loading cycle. Fig. 5(a) shows a NiTi plate of length L, width b and

thickness h with a circular hole of diameter d, positioned symmetrically with respect to

x and y axes. We define α as the hole diameter divided by the plate width, i.e.

/d bα = . The plate is under tensile load F along x axis, uniformly applied over its top

and bottoms edges. The forward and reverse transformation stresses and the forward

transformation strain (plateau length) of the plate material are denoted as tσ , tσ ′ and tε

, respectively. The elastic moduli of austenite and martensite phases are AE and ME ,

respectively. To find an analytical solution to the problem, we assume that the stress is

uniform within each cross section of the plate along the loading direction. While

0 (1 )tσ σ α≤ < − , the structure is fully austenite and TotL∆ is found as:

/2

0

( )2

( )

d

TotA A

Fdx F L dL

E w x E bh

−∆ = +∫ (2)

0

100

200

300

400

500

0 0.02 0.04 0.06 0.08

No

min

al S

tres

s (M

Pa)

Nominal Strain

303T K=

(a)

F

F

0

100

200

300

400

0 0.02 0.04 0.06 0.08

Nom

inal

Str

ess

(MP

a)

Nominal Strain

303T K=

(b)

II

III

I

F

F

161

where ( )w x is the width of plate at x in section II marked in Fig. 5(a), and is written as:

22( ) 2

4

dw x b x= − − (3)

Using Eqs. (1), (2) and (3), the nominal stress-strain relation for this stage is found:

12 1tan

1 2(1 )(1 )A

b L

E L b

σ α πε ααα α

− += − + − −+ − (4)

During (1 )t tσ α σ σ− ≤ < , section II is transformed to martensite with A→M

transformation initiating at the middle and propagating toward the top and bottom ends

of this section. Variable A Mx − expresses the displacement of the A-M boundaries within

the plate. It corresponds to the angle variable θ ranging from 0 to / 2π as A Mx − varies

from 0 to / 2d . L∆ of this stage includes the elongation of austenite and martensite

regions as marked in Fig. 5(a) during this stage. Using similar approach reported for

geometrically graded NiTi structures [30, 31], L∆ during this period can be written as:

0 0

/2

( ( )) ( ( ) ( ) )2 2 2

( ) ( )

( ( ) ) ( ( ) )( )2

( )

A M A M

A M

x x

t t tA M t

M A

dt t

A Ax

F w x dx w x b d h dxL x

E w x E w x

F b d h dx F b d h L d

E w x E bh

σ σ σ ε

σ σ

− −

−

−− − −∆ = + +

− − − − −+ +

∫ ∫

∫ (5)

Using Eqs. (1), (3) and (5) and considering the strain produced at the end of previous

stage (using Eq. (4)), the nominal stress-strain relation of this stage is obtained as a set

of two equations:

( )

1

1 1

(1 )2 1 1tan tan

1 2 2(1 )(1 )

2 (1 )sin( ) 1 1

1 1 1 1tan tan tan

1 1 2 2 2(1 )(1 )

t

M A

tt t

A M A

b

L E E

bb

L E E E L

L

b

σ ασ α θ θεαα α

σ σ αα θ σ ε

α α θ πα θα αα α

−

− −

− += − − −+ −

− − + − + + ×

+ + − + − + − − −+ −

( )

1(1 ) 2 1tan

1 2(1 )(1 )

1 cos

t

A

t

b L

E L b

σ α α π ααα α

σ σ α θ

−

− ++ − + − −+ −

= −

(6)

162

In the above equations, σ and ε are expressed in terms of the common variable θ

changing from 0 to / 2π .

At tσ σ= , Sections I and III of the plate, denoted in Fig. 5(a), are transformed to

martensite, adding ( ) tL d ε− to the ε at the end of transformation of section II. When

tσ σ> , the plate is elastically deforming in fully martensite phase. The strain produced

during this stage can be obtained from Eq. (4) by substituting AE by ME . The

unloading stress-strain relations can be established by substituting tσ by tσ ′ and tε by

(1/ 1/ )( )t M A t tE Eε σ σ ′− − − in the above equations and following the procedure of the

loading period.

Fig. 5(b) shows the deformation behaviour of the plate based on the developed

analytical solution for plates with one hole of different diameters. The material

parameters used in the model are obtained from the actual experimental results shown in

Fig. 1, as:

380 , 140 , 0.057, 45 , 25t t t A MMPa MPa E GPa E GPaσ σ ε′= = = = = (7)

It is seen that the analytical solution shows of the stress-strain variations exhibit

qualitative agreement with the experimental observations shown in Fig. 3(a). It is

obvious that this model is a simple 1D model which does not take into account the

stress concentration near the hole. In this regard, the qualitative agreement is

satisfactory as a guide for predicting the deformation behaviour of perforated NiTi

plates in uniaxial tension.

(a) b

L/2I

II

III

d

Ө

x

xA-My

A

M

A

163

Fig. 5. NiTi plate with a circular hole during martensitic transformation; (a): schematic

design, (b): analytical stress-strain diagram

4. Conclusions

The deformation behaviour of perforated pseudoelastic NiTi plates is investigated

through experimental and analytical approaches. The main conclusions of the study may

be summarised as following.

1. The nominal stress-strain curves of perforated pseudoelastic NiTi plates for stress-

induced martensitic transformation exhibit a distinctive region with a stress gradient

associated with the deformation within the hole-affected section. The gradient stress

for the transformation renders the material better controllability for stress-initiated

actuation.

2. The expanse of the affected deformation region with stress gradient is dependent on

the geometry and number of holes, and the expanse of affected region is larger than

the dimension of the holes in the direction of loading. For the sample geometry

tested, 33% of total hole length for uniformly distributed circular holes is enough to

result in totally gradient deformation for stress-induced martensitic transformation.

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[3] G. Tan, Y. Liu, P. Sittner, M. Saunders, Scripta Mater., 50 (2004) 193-198.

[4] J. Van Humbeeck, Mater. Sci. Eng. A, 273-275 (1999) 134-148.

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[6] I.Y. Khmelevskaya, E.P. Ryklina, S.D. Prokoshkin, G.A. Markossian, E.P. Tarutta,

E.N. Iomdina, Mater. Sci. Eng. A, 481–482 (2008) 651-653.

[7] M.H. Elahinia, M. Hashemi, M. Tabesh, S.B. Bhaduri, Prog. Mater Sci., 57 (2012)

911-946.

164

[8] X. Wang, Y. Bellouard, J.J. Vlassak, Acta Mater., 53 (2005) 4955-4961.

[9] K.P. Mohanchandra, G.P. Carman, in: S. Miyazaki, Y.Q. Fu, W.M. Huang (Eds.)

Thin film shape memory alloys: fundamentals and device applications, Cambridge

University Press, 2009, pp. 321-345.

[10] Q. Meng, Y. Liu, H. Yang, B.S. Shariat, T.-h. Nam, Acta Mater., 60 (2012) 1658–

1668.

[11] B.S. Shariat, Y. Liu, G. Rio, J. Alloys Compd., 541 (2012) 407-414.

[12] D.S. Grummon, J.A. Shaw, J. Foltz, Mater. Sci. Eng. A, 438–440 (2006) 1113-

1118.

[13] T. Duerig, A. Pelton, D. Stöckel, Mater. Sci. Eng. A, 273–275 (1999) 149-160.

[14] R.-x. Zhang, Q.-Q. Ni, A. Masuda, T. Yamamura, M. Iwamoto, Compos. Struct.,

74 (2006) 389-398.

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K.M.C. Cheung, P.K. Chu, Biomaterials, 32 (2011) 330-338.

[16] K. Krishnakumar, G. Venkatarathnam, Cryogenics, 43 (2003) 101-109.

[17] Y.Y. Lee, E.W.M. Lee, C.F. Ng, J. Sound Vib., 287 (2005) 227-243.

[18] G. Gan, S.B. Riffat, Therm Fluid Sci., 14 (1997) 160-165.

[19] P.K. Kumar, D.C. Lagoudas, in: D.C. Lagoudas (Ed.) Shape memory alloys:

modeling and engineering applications, Springer, 2008.

[20] A. Duval, M. Haboussi, T. Ben Zineb, Int. J. Solids Struct., 48 (2011) 1879-1893.

[21] B. Samsam, Y. Liu, G. Rio, Mater. Sci. Forum, 654-656 (2010) 2091-2094.

[22] B.-Y. Li, L.-J. Rong, Y.-Y. Li, Intermetallics, 8 (2000) 643-646.

[23] C. Greiner, S.M. Oppenheimer, D.C. Dunand, Acta Biomater., 1 (2005) 705-716.

[24] Y. Zhao, M. Taya, Y. Kang, A. Kawasaki, Acta Mater., 53 (2005) 337-343.

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[26] M. Barrabés, P. Sevilla, J.A. Planell, F.J. Gil, Mater. Sci. Eng. C, 28 (2008) 23-27.

[27] Z. Guo, H. Xie, F. Dai, H. Qiang, L. Rong, P. Chen, F. Huang, Mater. Sci. Eng. A,

515 (2009) 117-130.

[28] M. Panico, L.C. Brinson, Int. J. Solids Struct., 45 (2008) 5613-5626.

[29] J.M. Gere, B.J. Goodno, Mechanics of Materials, Cengage Learning, 2009.

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http://dx.doi.org/10.1016/j.jallcom.2011.12.151.

[31] B.S. Shariat, Y. Liu, G. Rio, Modelling and experimental investigation of

geometrically graded NiTi shape memory alloys, Smart Mater. Struct., In Press.

165

Supplement to Paper 5: Numerical modelling of perforated NiTi plates based on

elastohysteresis model and finite element method

Fig. 1. Comparison of the numerical and experimental results for deformation behaviour

of perforated NiTi plates under tension; (a): circular hole, (b): square hole

0

100

200

300

400

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

Nom

inal

Str

ess

(MP

a)

Nominal Strain

F

F303T K=

Simulation

Experiment

Ti – 50.8at.%Ni(a)

0

100

200

300

400

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

No

min

al S

tres

s (M

Pa

)

Nominal Strain

303T K=

F

F

Simulation

Experiment

Ld

b

35

9

3

L mm

b mm

d mm

===

(b)

0

100

200

300

400

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

No

min

al S

tre

ss (

MP

a)

Nominal Strain

303T K=

F

F

(c)

Simulation

Experiment

166

Fig. 2. Numerical modelling of deformation behaviour of perforated NiTi plates under

tension; (a): effect of hole size, (b): effect of number of holes along loading direction

The finite element numerical results of Fig. 2 are in consistent with the actual

experimental results presented in paper 5 (see Figs. 2(d) and 3(a) of Paper 5).

The fact that we see some “residual strain” after unloading in numerical results

regardless of experiments is due to the definition of elastohysteresis model. In this

model, the total deformation is the superposition of hyperelastic and hysteresis

contributions. The first one can be completely reversible, but the second one is “always”

irreversible, giving the total deformation an amount of irreversible behaviour (i.e. non-

zero residual strain). This residual strain can be minimised to some extent, but cannot be

eliminated.

0

100

200

300

400

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

No

min

al S

tres

s (M

Pa

)

Nominal Strain

F

F

Solid

3d mm=

4d mm=

2d mm=

Simulation

(a)

0

100

200

300

400

500

0 0.02 0.04 0.06 0.08

Nom

inal

Str

ess

(MP

a)

Nominal Strain

3d mm=

Solid

1 hole 2 holes

3 holesF

FSimulation

(b)

167

Paper 6: Numerical modelling of pseudoelastic behaviour of NiTi

porous plates

Bashir S. Shariata, Yinong Liua and Gerard Riob,*

aLaboratory for Functional Materials, School of Mechanical and Chemical Engineering, The

University of Western Australia, Crawley, WA 6009, Australia

bLaboratoire d’Ingénierie des Matériaux de Bretagne, Université de Bretagne Sud, Université

Européenne de Bretagne, BP 92116, 56321 Lorient cedex, France

Tel: +33 2 97 87 45 73, Fax: +33 2 97 87 45 72, Email: gerard.rio@univ-ubs.fr

Journal of Intelligent Material Systems and Structures, Under Review.

Abstract

This study presents a computational model for deformation behaviour of near-

equiatomic NiTi holey plates using finite element method. Near-equiatomic NiTi alloy

deforms via stress-induced A↔M martensitic transformation, which exhibits a typical

hystoelastic mechanical behaviour over a stress plateau, known as the pseudoelasticity.

In this model, the transformation stress is decomposed into two components: the

hyperelastic stress, which describes the main reversible aspect of the deformation

process, and the hysteretic stress, which describes the irreversible aspect of the process.

It is found that with increasing the level of porosity (area fraction of holes), the apparent

elastic modulus before and after the stress plateau decrease, the nominal stresses for the

A↔M transformation decrease and the strain increases, and the pseudoelastic stress

hysteresis decreases. In particular, the transformation strain increases by about 25% by

introducing 32% porosity. Upon loading, the strain in a holey plate made of NiTi is

more uniformly distributed than in a steel plate of the same geometry. The majority of

steel plate remains in a low strain range, with a small portion highly strained. In NiTi, a

large volume fraction of the plate undergoes moderate strains.

Keywords: Shape Memory Alloy, NiTi, Stress-induced martensitic transformation,

Pseudoelasticity

168

1. Introduction

Owing to the participation of the martensitic transformation, deformation behaviour of

shape memory alloys (SMAs) is very different from that of conventional metallic

materials. The common metal exhibits typically a smooth stress-strain curve involving

elastic and plastic deformations, with certain level of strain hardening during plastic

deformation. In contrast, the SMA, e.g., NiTi, exhibits a large stress plateau prior to

proceeding to the more conventional elastic and plastic deformations similar to those of

the common metal [1, 2]. The stress plateau is due to either stress-induced martensitic

transformation or martensite reorientation, and exhibits no strain hardening. This unique

behaviour of NiTi has attracted much attention in the past for researchers to understand

the mechanisms and to characterise the mechanical responses of this material. One

aspect that is relatively less understood is the mechanical behaviour of SMAs in

complex loading.

Complex loading condition can be found in many engineering applications of NiTi

alloys. This study investigates the deformation behaviour of NiTi plates with holes

under uniaxial tension. Due to variations in geometry, different locations in the holey

plate experience variations in both the level and state of stress. This inhomogeneous

stress field induces local martensitic transformation in the structure at different loading

levels, complicating the global deformation behaviour of the plate [3, 4].

Holey plates provide a study case as a 2D model for porous structures. Several

researchers have attempted to model the mechanical behaviour of porous SMAs [5-9].

In general, these models aim to predict the global response of a porous SMA under

compressive loading as a function of pore volume fraction. They are mainly based on

either using micromechanical averaging techniques [5-8] or assuming a periodic

distribution of pores [5, 9]. In micromechanical averaging method, the porous SMA is

considered as a composite medium consisting of parent phase as the matrix and

randomly distributed pores as the inclusions. In this method, the local variation of stress

and strain cannot be investigated, although the irregular distribution of pores is a near-

to-fact assumption. In contrast, the second method assumes a regular and periodic

distribution of pores through the porous structure. This arrangement deviates

considerably from real porous SMAs, but allows numerical analysis to be reduced to

that of a unit cell with appropriate boundary conditions. The unit cell approach provides

the possibility of obtaining the approximate values of local quantities in periodic unit

169

cells as presented by Qidwai et al. [5]. Recently, Olsen and Zhang [10] have studied the

influence of micro-voids on the fracture behaviour of shape memory alloys with low

void volume fractions (<10%). They concluded that introducing micro-voids lowers the

average stress levels for both phase transformation and plastic yielding and also lowers

the stress hysteresis of the pseudoelasticity.

Because of the approximation involved in the two aforementioned methods, the exact

deformation of holey plates with desired geometry cannot be predicted by general

porous structure modelling. Also, the very few previous studies on the mechanical

analysis of perforated plates made of SMAs are limited to those with one hole. Birman

[11] reported closed-form and exact solutions for the stresses in an infinite SMA plate

with a circular hole subjected to bi-axial tensile loading. He subdivided the plate into

three regions of pure austenite, mixed austenite and martensite, and pure martensite, and

obtained the stresses for each region with some assumptions. Recently, Vieille et al.

[12] have validated the numerical model of SMAs pseudoelasticity through tensile tests

on CuAlBe perforated strips. The in-plane strain fields of the CuAlBe sample with a

circular hole under tensile loading are obtained by image correlation technique and

computational simulation. They concluded that the in-plane strain components other

than the one along the loading direction show negligible heterogeneity in their

distribution. This conclusion seems to be true for a sample with one hole, but as the

number of holes increases the in-plane shear component can present a considerable

inhomogeneous distribution due to occurrence of transformation in shear direction in

some areas.

In this paper, the mechanical behaviour of pseudoelastic NiTi holey plates is simulated

using an in-house finite element software in which the elastohysteresis model is

implemented. The unique feature of this model is to decompose the mechanical

response of NiTi into hysteretic and hyperelastic constituents that account for the

irreversible and reversible parts of NiTi deformation, respectively. Plates with different

numbers and sizes of holes are analysed using this numerical code. The deformation

behaviour of NiTi plates is compared with that of steel plates.

2. The Elastohysteresis Model

The thermomechanical behaviour of SMAs shows simultaneous reversible and

irreversible phenomena. These are expressed in the elastohysteresis model [13] by

decomposing the total stress σ in two parts as:

170

hr σσσ += (1)

where rσ is the hyperelastic stress component and hσ is the hysteretic stress

component.

The hyperelastic stress is calculated from the hyperelastic potential proposed by Orgeas

et al. [14], as expressed in the following general form [15]:

εεαεαασ ⋅++= 210gr (2)

where g represents the metric tensor and iα are functions of the three invariants of the

Almansi strain tensor ε that are the relative variation of volume, the norm of the

deviatoric part of deformation and the direction angle of deformation tensor in the

deviatoric plane with respect to the principal directions.

The hysteresis behaviour is related to the deviatoric part of the stress tensor and is

described by the following relation:

[ ] htrh

tr D

tσβµσ Φ∆+=∆

∂∂

2 (3)

where D and Φ denote the deviatoric strain rate tensor and the intrinsic dissipation

rate, respectively. Parameter µ corresponds to the Lame’s coefficient and β is a

function of the Masing parameter [14]. The operator tr∆ denotes the variation of the

quantity at time t with respect to the reference or initial state.

The elastohysteresis model, which is briefly described by Eqs. (1)-(3), depends on 11

independent parameters related to hyperelastic and hysteresis responses that can be

determined by simple tension and shear tests on a 1D NiTi structure [16].

3. Numerical Modelling

The above constitutive equations have been implemented in a finite element code,

named Herezh++ [17]. Using this computational code, simulations of mechanical

behaviour of several pseudoelastic NiTi holey plates have been carried out. Also, the

deformation of a mild steel plate of identical geometry is simulated for comparison.

Circular holes of different diameters are introduced in the plates. Linear pentahedron

elements are used to generate 3D finite element meshes with finer elements around the

holes. The top and bottom edges of the plates are assumed to be clamped, while other

edges are free. Tensile loading and unloading are applied along the length of each

171

sample by controlling the displacement of the upper edge of the plate, while the lower

edge is fixed. The nominal stress-strain (S-S) curves of the pseudoelastic NiTi and the

mild steel plates used in the simulations are illustrated in Fig. 1. The experimental

curves in this Figure are obtained from actual tensile tests on a Ti-50.8 at.%Ni strip and

a mild steel sample [16, 18]. The essential parameters to define the material behaviour

of NiTi and steel plates are adopted from these original curves and utilised in the

numerical code.

Fig. 1. Stress-strain curves of NiTi and steel solid plates under uniaxial tension based on

experiment and simulation

3.1. Global Behaviour

3.1.1. Effect of porosity

A series of plates (60µ25µ0.11 mm) with different numbers of holes of 5 mm in

diameter are designed. These plates provide a range of porosities starting from 0 (solid

plate) to 32%. The deformation of each sample is simulated while undergoing

longitudinal loading up to 8% overall elongation and then unloading to zero load. Fig.

2(a) shows the design of two samples with 16% porosity. Plate (A) contains 12 holes in

regular distribution and plate (B) contains the same number of holes but in uniform

random distribution. Fig. 2(b) shows the nominal stress-strain (S-S) curves of the two

plates by numerical modelling. It is seen that the global behaviour of plate (B) deviates

only slightly from that of plate (A). To see the effect of porosity on the global behaviour

of the designed plates, for simplicity, we neglect the effect of irregularity of holes on

global behaviour for the rest of discussion in global behaviour.

172

Fig. 2. Deformation behaviour of two NiTi holey plates with 16% porosity and regular

and random hole arrangements. (a) schematic of the design of the samples. (b) nominal

stress-strain curves.

Fig. 3(a) shows comparison of numerical results for the nominal S-S relations of the

plates of different porosities with regular distribution of holes. As observed, the nominal

S-S relation of a solid NiTi plate changes by introducing porosity to the structure. In

order to evaluate the changes in the global behaviour, several parameters are determined

from these curves, as indicated in Fig. 3(b). In this figure, AE1 and AE2 are the apparent

moduli of elasticity determined at 0~0.5% and 0.5~1% of the global strain, respectively.

ME1 and ME2 are determined at 6~7%. 1S and 2S are S-S slopes over the stress plateau,

defined at 3% of global strain, during forward austenite to martensite (A→M) and

reverse M→A transformations, respectively. The transformation strain over the stress

plateau is defined bytε . 1σ and 2σ are the critical stresses for A→M and M→A

0

100

200

300

400

500

600

700

800

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Nom

ina

l Stre

ss (M

Pa

)

Nominal Strain

Regular Distribution of Holes (A)

Random Distribution of Holes (B)

(b)

173

transformation, respectively. Also shown in the figure is the stress hysteresis (H ) of the

pseudoelastic loop, which is the difference between 1σ and 2σ .

Fig. 3. Nominal stress-strain curves of NiTi holy plates of different porosities (a) and

determination of characteristic parameters from these curves (b)

Fig. 4 shows the effect of porosity on the transformation stresses and strain. As seen in

Fig. 4(a), both 1σ and 2σ decrease continuously with increasing porosity, so does the

stress hysteresis. Fig. 4(b) shows the effect of porosity on the transformation strain (tε ).

It is seen that the transformation strain increases progressively with increasing porosity,

by 25% with 32% porosity compared to a solid NiTi plate.

0

200

400

600

800

1000

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

No

min

al S

tress

(MP

a)

Nominal Strain

Solid plate

4% porosity

16% porosity

25% porosity

32% porosity

(a)

(b)

2S

1S

1ME

2ME

1AE

2AE

H

tε

1σ

2σ

174

Fig. 4. Variation of transformation properties versus porosity: (a) forward and reverse

transformation stresses and hysteresis; (b) transformation strain

Fig. 5 shows the effect of porosity on the characteristic moduli AE1 , AE2 , ME1 and ME2

and the slopes of the stress plateaus 1S and 2S . It is seen in Fig. 5(a) that all the elastic

moduli decrease with increasing porosity. Fig. 5(b) shows the effect of porosity on the

S-S slopes over stress plateau during the forward and the reverse transformations. It is

seen that 1S increases rapidly with increasing porosity whereas 2S remains unchanged.

Fig. 5. Variation of slopes in global stress-strain relation: (a) apparent elastic modulus;

(b) the slope over stress plateau

3.1.2. Effect of hole size

In this section, we compare the deformation behaviour of holey plates with the same

porosity (area fraction of holes) but different number of holes, or hole size. Fig. 6(a)

shows the design of six NiTi plates of 60µ25µ0.11 mm in dimension. The number of

holes along the loading direction varies from 1 to 6, all giving 16% of porosity. Fig.

6(b) illustrates the numerical results for nominal stress-strain variations of those

samples. As the number of holes decreases, i.e., the hole size increases, the minimum

0

50

100

150

200

250

300

350

0 5 10 15 20 25 30 35

Nom

inal

Str

ess

(MP

a)

Porosity (%)

1σ

2σ

H

(a)

0

1

2

3

4

5

6

0 5 10 15 20 25 30 35

Tran

sfor

mat

ion

Str

ain

(%)

Porosity (%)

(b)

tε

0

10

20

30

40

50

0 5 10 15 20 25 30 35

App

aren

t Ela

stic

Mod

ulus

(Gpa

)

Porosity (%)

1AE

1ME

2ME

2AE

(a)

0

1

2

3

4

5

0 5 10 15 20 25 30 35

S-S

Slo

pe O

ver

Pla

teau

(G

Pa)

Porosity (%)

(b)

1S

2S

175

cross-sectional area(s) of the plate reduces. Therefore, the A→M transformation starts

at lower loading levels, and the stress-strain slopes over the forward and reverse

transformations increase.

Fig. 6. NiTi holey plates with same porosity and different number of holes along the

loading direction (a) and their nominal stress-strain curves (b)

Fig. 7(a) shows four sample designs of square plates of 50µ50 mm in dimension with

similar porosity of 16% and different number of holes. The holes are uniformly

distributed along the length and width directions. Fig. 7(b) illustrates the nominal stress-

strain curves of those samples under uniaxial tensile loading. It is observed that by

increase of the number of holes, the stress-strain slopes over forward and reverse

transformations remain relatively constant, but the plateau level progressively increases.

0

100

200

300

400

500

600

700

800

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Nom

inal

Str

ess

(MP

a)

Nominal Strain

1 hole2 holes3 holes4 holes5 holes6 holes

(b)

176

Fig. 7. NiTi square plates with same porosity, uniform distribution and different number

of holes (a) and their nominal stress-strain curves (b)

3.2. Local Behaviour

A NiTi and mild steel holey plate with the same geometry are simulated for comparison

of local deformation behaviours. Circular holes of 8 and 10 mm in diameter are

introduced in each plate. Fig. 8 shows the distribution of the normal strain component

along the loading direction (yyε ) in NiTi and steel plates at 6% of global strain. The

strain range in each plate is divided into eight equal strain windows. As seen in this

figure, the maximum local strain found in steel holey plate (18.1%) is significantly

higher than that found in NiTi holey plate (9.5%). Furthermore, the strain is more

evenly distributed in NiTi than in steel. In steel, a large volume fraction remains in low

strains, and a very small fraction experiences high strains. In NiTi, on the other hand, a

larger volume fraction of the plate undergoes moderate strains as compared to the steel

plate.

0

100

200

300

400

500

600

700

800

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Nom

inal

Str

ess

(MP

a)

Nominal Strain

1 hole

9 holes

36 holes

81 holes

(b)

177

Fig. 8. Distribution of normal strain component along the loading direction in NiTi plate

(a) and steel plate (b) at 6% of global strain

The major difference in strain distribution of NiTi and steel illustrated in Fig. 8 is due to

the fact that the stress-strain behaviour of NiTi is very different from that of steel as

shown in Fig. 1. The narrow areas in NiTi reach the transformation stress before other

areas due to stress concentrations and transform to martensite with the increase of

loading level. These transformed areas undergo a more elastic behaviour with rather

steep stress-strain slope after transformation. This practically prevents further

deformation of the transformed areas and forces the transformation into other

untransformed areas with increasing loading level. Thus, a larger area fraction of the

plate shares the global strain. In contrast, the mild steel has a very low strain hardening

coefficient, making the deformation very easy to propagate. Thus, the deformation

behaviour is much more dominated by the geometry of the sample, i.e. stress

concentration. The few small areas with highest stress concentrations will deform first

and then continue to deform plastically to high strain levels with very limited spreading

into adjacent areas, providing the most of the global strain.

178

Fig. 9 shows comparisons of volume fraction (VF) distribution of certain mean strain

levels for the NiTi and steel plates. For both the steel and NiTi samples, a global strain

of 6% was applied. In Fig. 9(a), the full strain range of each plate is divided by equal

1% strain windows, each of which is represented by the average strain of the window on

the x-axis. The y-axis shows the volume fraction of each strain window. It is seen that

the steel sample showed a large population of areas with a narrow strain range of

<1.5%, but a long tail of low volume fraction extends to very high strain. In

comparison, the NiTi plate showed a much more uniform distribution with a wider

strain range between 0.5 and 6.5% and a much shorter tail after that. Obviously, the

wider distribution of volume fraction in the intermediate strain range is due to the nil

strain hardening coefficient over the stress plateau associated with the stress-induced

transformation and the very short tail to higher strain levels is due to the high strain

hardening after the stress plateau.

Fig. 9. Volume fraction-strain variations of NiTi and steel plates at 6% of global strain

Fig. 9(b) shows the accumulated volume fraction of different strain levels. In this plot,

the entire strain range of each sample is divided into eight equal strain windows and

each window is expressed on the x-axis as its mean strain to the mean strain of the last

window with the highest strain level. The volume fractions of the strain windows are

0

10

20

30

40

-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

Vol

ume

Fra

ctio

n (%

)

Average Strain

Equal window size comparison

Steel at 6% of global strain

NiTi at 6% of global strain

(a)

0

10

20

30

40

50

60

70

80

90

100

-0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Cum

ulat

ive

Vol

ume

Fra

ctio

n (%

)

Relative Average Strain

Equal window number comparison

NiTi at 6% of global strain

Steel at 6% of global strain

(b)

179

then accumulated from low to high mean strain levels. It is seen that 41% of the steel

plate is in the lowest 1/8 of the entire strain range whereas only 0.6% of the NiTi plate

is in this range. In addition, 72% of the steel plate is in the lowest 1/4 of the whole strain

range while this value for the NiTi plate is only 16%. These observations are different

expressions of the same effects explained above.

Fig. 10 shows the VF-strain curves of the samples deformed to different global strain

levels. The strain window size is 1%. Fig. 10(a) shows the results for the steel plate. It is

seen that VF distribution profile remained the same, with the peak position being at

0.5% of local strain for all the three loading levels. As the loading level is increased, the

height of VF-strain curve decreases. However, the largest volume fraction of steel

remains in 0~1% strain window as the narrow areas continue to carry the most of global

displacement. At 10% of global strain, these areas are locally strained up to 24%.

Fig. 10. Volume fraction-strain variations of NiTi and steel plates at different loading

levels of global strain

Fig. 10(b) shows the results for the NiTi plate. It is seen that at the low global strain of

3%, the VF distribution is similar to that of steel, with ~54% in volume deforming to

0

10

20

30

40

50

60

-0.04 0 0.04 0.08 0.12 0.16 0.2 0.24

Vol

ume

Fra

ctio

n (%

)

Average Strain

Equal window size comparison

Steel at 3% of global strain

Steel at 6% of global strain

Steel at 10% of global strain

(a)

0

10

20

30

40

50

60

-0.03 -0.01 0.01 0.03 0.05 0.07 0.09 0.11 0.13

Vol

ume

Fra

ctio

n (%

)

Average Strain

Equal window size comparison

NiTi at 3% of global strain

NiTi at 6% of global strain

NiTi at 10% of global strain

(b)

180

<1%. Increasing the global strain level to 6% caused averaging of local strains to a

wider range corresponding to the stress plateau. Further increasing the global strain to

above the limit for the stress plateau, the distribution curve is pushed to higher strain

levels above 5.5%. This implies that more areas of the NiTi plate have been transformed

to the martensite.

It should be noted that the local strains are sensitive to mesh density in finite element

calculations, in particular, where there is localisation phenomenon. In this study, the

mesh density was limited by computational time.

Acknowledgement

This work is partially supported by the French National Research Agency Program

N.2010 BLAN 90201.

4. Conclusions

The global and local deformations of pseudoelastic NiTi holey plates are simulated

using elastohysteresis concept and finite element method and compared with those of

steel holey plates. The study considers real structural geometries and boundary

conditions and not classical infinite periodic materials. The deformation behaviour is

found to be dependent on porosity as well as hole size. To study the effect of each factor

on global behaviour, the other factor is kept constant. In this context, the study presents

the following conclusions which are linked to the typical behaviour of NiTi:

1. The apparent elastic modulus of austenite and martensite during loading and

unloading decrease by increasing porosity level, with that before the forward

transformation tends to reduce more rapidly.

2. The stress-strain slope over stress plateau during forward transformation

continuously increases with the increase of porosity whereas the one during reverse

transformation remains constant.

3. The strain is more uniformly distributed in NiTi than steel. In steel, the major volume

fraction remains in a very low strain range while a small portion of the plate is highly

strained. In NiTi, a large volume fraction of the plate undergoes moderate strains.

4. The peak of VF-strain curve for NiTi moves from low to high strains by increase of

loading level, while that for steel remains at a constant strain range.

181

References

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[2] G. Tan, Y. Liu, P. Sittner, M. Saunders, Scripta Mater. 50 (2004) 193-198.

[3] B.S. Shariat, Y. Liu, G. Rio, J. Alloys Compd. (2012).

[4] B.S. Shariat, Y. Liu, G. Rio, J. Alloys Compd. 541 (2012) 407-414.

[5] M.A. Qidwai, P.B. Entchev, D.C. Lagoudas, V.G. DeGiorgi, Int. J. Solids Struct. 38

(2001) 8653-8671.

[6] P.B. Entchev, D.C. Lagoudas, Mech. Mater. 34 (2002) 1-24.

[7] P.B. Entchev, D.C. Lagoudas, Mech. Mater. 36 (2004) 893-913.

[8] S. Nemat-Nasser, Y. Su, W.-G. Guo, J. Isaacs, J Mech. Phys. Solids 53 (2005) 2320-

2346.

[9] M. Panico, L.C. Brinson, Int. J. Solids Struct. 45 (2008) 5613-5626.

[10] J.S. Olsen, Z.L. Zhang, Int. J. Solids Struct. 49 (2012) 1947-1960.

[11] V. Birman, Int. J. Solids Struct. 36 (1999) 167-178.

[12] B. Vieille, J.F. Michel, M.L. Boubakar, C. Lexcellent, Int. J. Mech. Sci. 49 (2007)

280-297.

[13] D. Favier, P. Guelin, P. Pegon, Mater. Sci. Forum 56-58 (1990) 559-564.

[14] L. Orgéas, D. Favier, G. Rio, Revue Européenne des éléments finis 7 (1998) 111-

136.

[15] G. Rio, P.Y. Manach, D. Favier, Arch. Mech. 47 (1995) 537-556.

[16] V. Grolleau, H. Louche, A. Penin, P. Chauvelon, G. Rio, Y. Liu, D. Favier, Bulge

tests on ferroelastic and superelastic NiTi sheets with full field thermal and 3D-

kinematical measurements: experiments and modelling, in: ESOMAT 2009, Prague,

Czech Republic, 2009.

[17] G. Rio, Herezh++: FEM software for large deformations in solids, in, Laboratoire

d’ingénierie des matériaux de Bretagne (UEB-UBS), dépôt APP (Agence pour la

Protection des Programmes) - Certification

IDDN.FR.010.0106078.000.R.P.2006.035.20600, 2011.

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13 (2008) 139-148.

182

183

Chapter 5

Closing Remarks

The specific technical conclusions of each chapter are presented in the corresponding

papers. Here, I present the summary of the main research findings as closing remarks:

(1) Geometrical or microstructural grading of a shape memory alloy component

imposes gradient stress plateaus over forward and reverse stress-induced

transformations. This increases the stress windows over martensitic

transformations, providing improved controllability of the SMA element. The

typical magnitude of stress windows achievable are 80 ~ 300 MPa.

(2) This thesis provides closed-form solutions for load-displacement relations of 1D

and 2D SMA structures that are geometrically or microstructurally graded. The

analytical solutions provide engineering tools for designing such functionally

graded components and prediction of stress-strain behaviour.

(3) In geometrically graded NiTi, the stress-strain slope and the stress window size

over A↔M transformations can be controlled by geometrical design. As the sample

geometrical deviation from axial uniform shape increases, the stress-strain slope

over transformation increases, but the plastic deformation of narrow areas can

hinder the alloy to exhibit an increased stress window with full pseudoelastic

behaviour. For Ti-50.8at%Ni strips tested at 303 K, the maximum achieved stress-

strain slopes with maintaining full pseudoelasticity are ~3.7 GPa and ~2 GPa for

forward and reverse transformations, respectively. The corresponding stress

windows are ~220 MPa and ~120 MPa, respectively. In this case, the nominal

transformation strain is increased by ~2% for the linearly tapered strip and ~3% for

the concave strip, compared with the uniform sample.

184

(4) In microstructurally graded NiTi, the property gradient can be obtained by

application of gradient anneal over the length of the NiTi wire. Increase of the

temperature range increases the slopes of the stress plateaus associated with the

stress-induced martensitic transformations. But, there exists an effective annealing

temperature range depending on the testing temperature and the yield strength of

the material. For Ti-50.5at%Ni tested at 313 K, this range was found to be 630-810

K. In this case, partial pseudoelastic behaviour is observed with the stress window

of ~300 MPa over A→M transformation. To achieve full pseudoelasticity at this

testing temperature, the higher annealing temperature limit has to reduce to ~720 K.

The stress-strain slope over reverse transformation is higher than that of forward

transformation unlike the behaviour of geometrically graded NiTi samples.

(5) For 2D NiTi structures, the microstructural gradient can be achieved in the

thickness direction by laser scan anneal, which imposes gradient annealing

temperature across the thickness. In this case, the stress-strain curve over stress-

induced martensitic transformations is partially deviated from the flat stress plateau

configuration. After tensile loading, the sample retains residual curvature. It

demonstrates complex shape change upon recovery of transformation deformation

during heating.

(6) A perforated NiTi plate, which provides a study case of 2D porous structure, can be

considered as a geometrically graded SMA structure, which creates gradient stress

over stress-induced martensitic transformation. The area fraction of holes (porosity)

considerably affects the global stress-strain behaviour. By increase of porosity, the

stress-strain slope over forward transformation continuously increases. The

apparent elastic moduli of austenite and martensite during loading and unloading

decrease by increase of porosity, with that before the forward transformation tends

to reduce more rapidly. During tensile loading, the strain is more uniformly

distributed in a perforated NiTi plate in comparison with a steel plate of the same

geometry. In NiTi, a large volume fraction of the plate undergoes moderate strains.

In steel, the major volume fraction remains in a very low strain range, while a small

portion of the plate is highly strained.

185

Suggestion for future work

The concept of functionally grading SMA structures can be addressed to a wide range of

design and architecture of shape memory alloys. In this thesis, I have explored specific

types of functionally graded SMAs which provide desired functionalities. The analytical

and numerical approaches used to model the deformation behaviour of studied

structures under complex transformation field can be applied to other examples of SMA

structures with the complexity of phase transformation. Also, more research is required

for fabrication of functionally graded SMAs and experimentation of their

transformation and mechanical behaviours. Some suggested topics for future research

are:

(1) Bending of SMA beams and tubes

Due to bending moment, normal stress gradient is created across the thickness direction

from maximum tension to maximum compression, which provokes progressive

transformation initiation as the moment increases. The magnitude of the stress in beam

layers is limited by the level of the stress plateau. As the forward transformation is

initiated in each layer, the local stress stops increasing, while the strain continues to

grow with the increase of bending moment. It is known that the transformation stress of

NiTi is higher in magnitude in compression compared to tension. This asymmetric

tension and compression feature of NiTi causes the neutral axis to move from the

central axis toward the compression side. This issue needs to be included in the

analysis. The moment-curvature relations can be established for different stages of the

loading cycle.

(2) Torsion of SMA bars

In the course of pure torsion, the shear stress is varying proportionally with the distance

from the centre of the bar. The variation of shear stress within the bar creates complex

transformation field throughout the structure. The transformation initiates at the outer

surface of the bar and progressively propagates toward the centre as the torque level

increases. Several stages of deformation can be defined during torsion cycle. With

similar analytical approach introduced in this thesis, the variation of angle of twist

versus torque can be established over transformation stages.

A subtopic in this area is a SMA bar under nonuniform torsion. Nonuniform torsion can

be generated by either application of constant torque to a bar with continuously varying

186

cross sections or applying varying torque to a uniform bar. In both cases, the

transformation field is more complicated than uniform torsion, as the transformation

initiation gradients exist in both radial and axial directions.

(3) Bulging test of SMA plates

An effective method to study the complex deformation behaviour of SMAs is bulging

test. In this test, the SMA sample is subjected to hydrostatic pressure and experiences a

wide spectrum of biaxial stress field, which provides unique complex transformation

field. The challenge is to find an effective model to correlate the global deformation

behaviour, i.e. the lateral displacement, to the applied pressure. Also, local strain and

stress fields are required to be obtained through modelling and experimental

investigations.

(4) 3D porous SMA structures

The numerical method developed based on elastohysteresis model can be extended to

analyse the deformation behaviour of 3D porous structures. Periodic distribution of

pores and homogenisation technics are required to be applied in the calculus. The

challenge is to find the proper size of the representative volume element. Also, a

realistic material behaviour should be implemented in the developed code, which takes

in to account the deformation behaviour of SMA under tension, compression and shear

and plastic deformation.