THERMOMECHANICAL BEHAVIOUR OF FUNCTIONALLY …...thermomechanical behaviour of functionally graded...
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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
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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
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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
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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.
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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:
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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].
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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: [email protected]
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: [email protected]
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: [email protected]
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.
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F.M.B. Fernandes, Thin Solid Films 519 (2010) 122-128.
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[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.
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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: [email protected]
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
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10.1016/j.jallcom.2011.12.151.
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[17] B.A.S. Shariat, M.R. Eslami, J. Thermal Stresses 28 (2005) 1183-1198.
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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.
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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: [email protected]
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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K.M.C. Cheung, P.K. Chu, Biomaterials, 32 (2011) 330-338.
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[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.
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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: [email protected]
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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[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-
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[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-
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Czech Republic, 2009.
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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.