Chapter
1 A Brief Review of Shape Memory Alloys and Introduction to the Theory of Elasticity
1.1 Introduction
Smart materials are systems that integrate the functions of sensing,
actuation, logic and control to respond adaptively to changes in their conditions or
the environment to which they are exposed, in a useful and repetitive manner. The
concept of smart materials is most general when sensors and actuators are included
with external macro scale control and logic circuits. Embedding such systems in
structures such as a composite aircraft wing, bridge, dam, building or computer
disk drives results in a smart structure.
Smart systems are the results of a design philosophy that emphasizes
predictive, adaptive and repetitive system responses. Systems entering the market
today are competitive because of evolutionary improvements in materials and
processes. Recent availability of cost-effective digital signal processors and
microcontroller chips has been a major accelerating influence. Research
breakthroughs are still required to achieve competitive prices for most smart
systems. Smart materials also include shape memory alloys, optical fibres and
conducting polymers.
2
Considerable effort is being made to develop smart materials and
structures. The technological benefits of such systems have begun to be identified
and, demonstrators are under construction for a wide range of applications from
space technology to civil engineering and domestic products. In many of these
applications, the cost benefit analyses of such systems have yet to be fully
demonstrated.
The concept of engineering materials and structures which respond to their
environment, including their human owners, is an important concept. It is therefore
not only important that the technological and financial implications of these
materials and structures are addressed, but also issues associated with public
understanding and acceptance.
1.1.1 Shape Memory Effect and Pseudo-elasticity
Shape memory effect (SME) is discovered in an Au-Cd alloy in 1951 by
Otsuka and Wayman [1]. Research became more active after the effect is discovered in
Ti-Ni alloy in 1963 by Delaey [2]. According to Planes and Manosa [3] the term
‘shape memory alloy’ (SMA) is applied to that group of materials which have the
peculiar property of being able to recover from large deformations when subjected
to the appropriate thermo-mechanical procedure. The physical mechanism behind
this effect is a diffusionless, first-order structural transition, usually referred to as
the martensitic transition. Recently, Lin et.al. and Zhang et.al. [4,5] have reported
that shape memory alloys have been attracting keen attention as smart materials,
since they can function as sensors and actuators simultaneously because of their
unique properties pseudo-elasticity (PS) and SME.
3
SMA exhibit on cooling, or under pressure, a first-order diffusionless
structural phase transition known as martensitic transformation (MT). Generally
speaking, during the MT the alloy suffers a transition from an open structure
towards a close-packed structure. In Cu based alloys, the MT takes place between
the high-temperature high symmetric phase known as austenite and a low-
temperature low symmetric structure known as martensite as shown in Fig. 1.1. A
suitable way for describing the lattice distortion associated with the transformation
is in terms of a combination of two homogeneous shears. The technological
interest underlying the MT referred to above has encouraged scientists to obtain a
better comprehension of all the features of the transition, including precursor
effects such as anomalies in the elastic constants and in the phonon dispersion
curves. Thus research on SMA is becoming a target of physicists as well as
material scientists and engineers, and martensitic transformations are being studied
by various approaches, both experimental and theoretical. For these reasons,
Ostuka and Kakeshita [6] mentioned that rapid progress has been made recently in
both fundamental studies and its applications. We believe that the side-by-side
development of basic sciences and its applications are crucial for the future success
of materials science and engineering, and the science and technology of SMA are
good prototypes of this kind of development.
The two unique properties described above are made possible through a
solid state phase change, that is a molecular rearrangement, which occurs in the
shape memory alloy. A solid state phase change is similar to that of a molecular
rearrangement, but the molecules remain closely packed. In most shape memory
4
alloys, a temperature change of about 10°C is necessary to initiate this phase
change. The two phases, which occur in shape memory alloys, are Martensite, and
Austenite.
Martensite is the relatively soft and easily deformed phase of shape
memory alloys, which exists at lower temperatures. The molecular structure in this
phase is twinned. Upon deformation this phase takes on the second form.
Austenite, the stronger phase of shape memory alloys, occurs at higher
temperatures. The shape of the austenite structure is generally cubic-based. The un
deformed martensite phase is of the same size and shape as the cubic austenite
phase on a macroscopic scale, so that no change in size or shape is visible in shape
memory alloys until the martensite is deformed.
Temperatures at which each of the martensite and austenite phases begin
and finish forming are represented by Ms, Mf, As and Af respectively. The amount
of loading placed on a piece of shape memory alloy increases the values of these
variables. The initial values of these variables are also affected by the chemical
composition of the wire. SME is observed when the temperature of a piece of
shape memory alloy is cooled below the temperature Mf. At this stage the alloy is
completely composed of martensite which can be easily deformed. After distorting
the SMA the original shape can be recovered simply by heating the wire above the
temperature Af. The heat transferred to the wire is the power driving the molecular
rearrangement of the alloy, similar to heat melting ice into water, but the alloy
remains solid. The deformed martensite is now transformed to the cubic austenite
phase, which is configured in the original shape of the wire.
5
Pseudo-elasticity occurs in shape memory alloys when the alloy is completely
composed of austenite. Unlike the shape memory effect, pseudo-elasticity occurs
without a change in temperature. The load on the shape memory alloy is increased
until the austenite becomes transformed into martensite simply due to the loading. The
loading is absorbed by the softer martensite, but as soon as the loading is decreased the
martensite begins to transform back to austenite since the temperature of the wire is
above Af, and the wire springs back to its original shape.
Figure 1. Schematic illustration of the mechanism of the shape memory effect and super-elasticity
Super-elasticity path
Shape memory path
Austenite
6
There are still some difficulties with SMA that must be overcome before
they can live up to their full potential. These alloys are still relatively expensive to
manufacture and machine compared to other materials such as steel and aluminum.
The phenomenon of the SME is clearly shown in the photographs in figure 1d for a
Ti-Ni wire, which is a typical and practical SMA. The wire in the martensitic state
(1), whose shape is the same as in the parent phase, is deformed at ambient
temperature (2). However, it will revert to its original shape by means of the
reverse transformation (3)–(5) if it is heated to a temperature above Af. The
mechanism of this phenomenon is explained in figures 1a–1c in a simplified
manner. When the parent phase in figure 1a is cooled below Mf, martensite
variants are formed side by side, as shown in figure 1b, as a result of self
accommodation. If a stress is applied, deformation proceeds by twin boundary
movement from figures 1b to 1c. If, however, the sample is heated to a temperature
above Af, the martensite variants rearranged under stress revert to their original
orientation in the parent phase (if the transformation is thermoelastic, since the
thermoelastic martensitic transformation is crystallographically reversible). When
the sample is stressed at a temperature above Af , we will get a result similar to that
shown in the graph (inset) in figure 1 for a Cu-Al-Ni single crystal, which shows a
recoverable strain exceeding 10%. This is super-elasticity, whose mechanism can
be explained using figures 1a and 1c. Since the martensitic transformation occurs
by a shear-like mechanism, it is possible to induce it even above Ms if we apply
stress. This is a stress-induced martensitic transformation. It is possible to induce a
transformation above Af if slip does not occur under the applied stress. However,
the martensite is completely unstable at a temperature above Af in the absence of
7
stress. The reverse transformation should occur during unloading, and the strain
completely recovers in the thermo-elastic transformation, if slip is not involved in
the process. This indicates that a high critical stress for slip is important for the
realization of super-elasticity; it is in fact possible to increase the critical stress for
slip by thermo-mechanical treatments. It is clear that both the shape memory effect
and super-elasticity occur in a SMA, and which phenomenon occurs depends upon
its temperature.
Since the discovery of SME in Au-Cd alloy in 1951, it has provoked much
interest with its peculiar characteristics and its relationship to martensitic
transformation. According to Otsuka and Wayman [1] elaborate studies on the
theory and application of the SME have begun when it is observed in the
equiatomic percent of Ti-Ni alloy. In addition to these alloys, it has been known
that non-ferrous alloys such as Ag-Cd, Cu-Zn, Cu-Zn-X (X=Al, Si, Sn, Ga),
Cu-Al-Ni, Ni-Al and ferrous alloys such as Fe-Pt, Fe-Ni-Co-Ti, Fe-Ni-C, Fe-Mn-
Si also show SME. But only Ti-Ni alloys and a few Cu based alloys like Cu-Al-Ni
and Cu-Al-Zn are currently used and Fe-based alloys have been recently studied
for application although many alloys have the SME. Due to their outstanding SME
property, Ti-Ni alloys make it possible for the SMA to be studied for practical
applications. Until the beginning of the 1980’s, the funental understanding of the
SME has not been available because of the problems in making Ti-Ni single
crystals. In the 1980’s, however, the progress in manufacturing processes like the
vacuum melting technique made basic studies on the deformation mechanism,
SME mechanism, crystallography and phase diagrams possible. It is reported that
8
Cu-Al-Ni also show the SME, and widespread studies on Cu based shape memory
alloys focusing on Cu-Al-Ni and Cu-Al-Zn alloys are carried out to replace Ti-Ni
alloys. As a result it has been possible to understand the SME mechanism of
Cu based alloys and the crystallography of stress-induced martensitic transformation.
But it is difficult to use Cu based alloys in the polycrystalline state. Following the
discovery of the SME in Fe-Mn alloys, the SME also found in many other ferrous
alloys such as Fe-Mn-Si and Fe-Co-Ti. Fe-based shape memory alloys have better
machinability and lower manufacturing cost than non-ferrous shape memory alloys.
Kim [7] reports that further studies on the improvement of the SME and corrosion
resistance of Fe-based alloys are necessary. Although a relatively wide variety of
alloys and compounds are known to exhibit the shape memory effect, only those that
can recover substantial amounts of strain are of commercial interest.
1.1.2 Cu-Al-Ni
Being one of the most studied alloy in the group of Cu based SMA,
Cu-Al-Ni is special for its possible use at temperatures near 473 K as given by
Landaz’abal et al. [8]. Highest SMEs are observed in these alloys with Al content
close to 14 mass% and with a varying Ni content. However, since the alloys become
brittle with increasing Ni content, the optimum compositions lie around Cu-14~14.5
Al-3~4.5 Ni. Structural studies on Cu-Al-Ni single crystal have been conducted by
Nakata et al. [9] and Landazabal et al. [8] using neutron powder diffraction which
showed L21 ordering with Fm3m symmetry. Comas et al. [10] have measured the
changes in the ultrasonic sound velocity induced by the application of uniaxial stress
9
and showed that the application of a stress in the [001] direction, reduces the
material resistance to a (110)[1ī0] shear and thus favours the martensitic transition.
Acoustic properties of single crystalline Cu-Al-Ni are investigated by
Landa et al. [11] at room temperature in the austenite phase and the complete set of
second- and third-order elastic constants are determined. The second-order elastic
constants of bcc-based austenite and 2H orthorhombic martensite of Cu-Al-Ni are
determined by Sedlak et al. [12] using ultrasonic pulse echo technique. The elastic
constants and the Debye temperature of Cu-Al-Ni have been experimentally
determined by Recarte et al. [13].
The effect of homogenization heat treatment and hot rolling, on the
transformation temperatures of these alloys is investigated by Veloso et al. [14] using
differential scanning calorimetry and observed that the transformation temperatures
increase with long homogenization times, and also by hot rolling, and this
displacement is smaller for alloys with 4.0% of Nickel. Based on high-sensitivity
adiabatic calorimetry, the distribution density of the elastic energy states in the
martensitic phase is directly derived from the specific heat data of Rodriguez et al.
[15]. Nikolaev et al. [16] have studied the generation and relaxation of reactive
stresses in Cu-Al-Ni shape memory alloy single crystals during a single cycle of
temperature variation in the range 293-800 K. Pseudo-elastic, isothermal
mechanical cycling of Cu-Al-Ni shape memory alloy single crystals has been
carried out by Kannarpady et al. [17]. Stress-free transformation temperatures at
the end of 1000 stress cycles do not change, whereas the transformation stresses
10
decreases by 10%. The mechanical behaviour and fracture characteristics resulting
from thermal cycling treatments under different applied loads are investigated in a
monocrystal alloy by Matlakhova et al. [18] and found that the thermal cycling
treatments promote significant changes in the structure due to a reversible
martensitic transformation. The temperature memory effect exhibited by Cu-Al-Ni
shape memory alloy has been studied by Rodriguez et al. [19] using adiabatic
calorimetry and microscopic observations and the specific heat of the specimen
measured from 140 to 350 K throughout the phase transition region.
Seiner et al. [20] observed the growth mechanisms (i.e. mechanisms of
nucleation and growth of the twinned structures) in the alloy and analyzed them
using optical methods. In Cu-Al-Ni shape memory alloy, the influence of
deformation and thermal treatments on the microstructure and mechanical
properties under the compression test have been studied by Scanning Electron
Microscopy (SEM) and Differential Scanning Calorimetry (DSC). The mechanical
properties of the alloy can be enhanced by heat treatment. Sari and Kirindi [21]
stated that the alloy exhibits good mechanical properties with high ultimate
compression strength and ductility after annealing at high temperature. The
interaction between dislocations generated during cycling and the twinned
structure of the martensite is studied by Gastien et al. [22] on the basis of stacking
faults. A neutron single crystal diffraction method for inspecting the quality of
martensite single crystals is introduced by Molnar et al. [23] to detect and
distinguish the presence of individual lattice correspondence variants of the 2H
orthorhombic martensite phase in Cu-Al-Ni as well as to follow the activity of
11
twinning processes during the deformation test on the martensite variant single
crystals. Structural observations by Tunneling Electron Microscopy (TEM) are
performed by Zarubova et al. [24] and have analyzed in-situ strained foils of a Cu-
Al-Ni shape memory alloy at room temperature.
Thin films of Cu-Al-Ni are grown by Lovey et al. [25] using d.c. magnetron
sputtering from the alloy target previously melted in an induction furnace and
microstructures of the as grown films have been analyzed by TEM. Electrical
resistance and elastic modulus measurements are quantified by Kamal et al. [26],
which allow the crystal’s sensitivity to the thermal treatments, that change the relative
stability of both the parent phases existing prior to the transformation and the
martensite. Creuziger et al. [27] investigated the possibilities of applying a focused ion
beam for the preparation and characterization of Cu-Al-Ni melt-spun ribbons. The
experiment is made to study the influence of Platinum deposition on the quality of
3D-cross sections. The DSC thermograms of the aged samples show increase in
transformation temperatures as well as transformation hysteresis with aging as given
by Suresh and Ramamurthy [28]. Landa et al. [29] have studied the temperature
dependence of the elastic constants of the cubic austenite and orthorhombic 2H
martensite phases of the Cu-Al-Ni alloy in the transformation temperature range
detected by resonance ultrasonic spectroscopy. The elastic anisotropy of both
phases significantly increases when approaching transformation temperatures. The
anisotropy factor of martensite increases about ten times more strongly than in the
case of austenite.
12
The decomposition of early phases and its influence on the MT has been
analysed in Cu-Al-Ni by Recarte et al. [30]. Suresh and Ramamurthy [31] have
conducted dynamic mechanical analysis of β-phase aged single crystal Cu-Al-Ni
alloy to investigate the changes in the internal friction through the thermo-elastic
MT temperature regime. Studies of grain refinement and thermo-mechanical
processing by Sampath [32] on Cu-Al-Ni showed that they could improve the
shape memory characteristics and ductility of these alloys. Ding et al. [33] find, in
a Cu-Al-Ni single crystal, softening of a particular elastic mode with increasing
applied stress along the [001] direction, until a stress-induced martensitic
transformation occurs at the critical stress. This gives direct evidence for the
existence of lattice softening prior to a stress-induced martensitic transformation.
Their results lead to the conclusion that the softening of elastic constant C’ is a
common feature prior to both temperature- and stress-induced martensitic
transformations. Sade et al. [34] have studied the effects of repeated pseudo-elastic
fatigue in Cu-Al-Ni. We study on the single crystal specimen of Cu-14.3wt% Al-
4.1wt% Ni of which the reported results by Landa et al. [29] are available.
1.1.3 Cu-Al-Zn
Cu-Al-Zn is a promising candidate because of its efficient shape memory
and super-elastic properties which explore its application both in the field of
sensors and actuators. They have good ductility and other mechanical properties.
Their operating temperature is 373 K by which it can replace Ni-Ti alloy in some
selected applications as stated by Landaz’abal [24]. Lanzini et al. [35] have
presented the order-disorder temperatures for bcc Cu-Al-Zn shape memory alloy
13
using calorimetric and resistometric techniques. Sade et al. [34] have studied the
effects of repeated pseudo-elastic fatigue in Cu-Al-Zn. Ciatto et al. [36] and
Asanovic et al. [37] have investigated the structure of the austenite phase in Cu-Al-
Zn shape memory alloys by a combined X-ray absorption and diffraction
analysis. Evidence is presented for the existence of an ordered B2-like structure
different from the L21 one recently proposed by neutron diffraction. However,
some partial L21 ordering is present at room temperature. The stress-strain
hysteresis loops during the martensitic transition of a Cu-Al-Zn single crystal have
been studied by Bonnot et al. [38]. The transition is either stress-driven or strain-
driven. The comparison between the two mechanisms shows significant
differences in the transformation path. The behaviour of helical springs made from
lamellar specimens of Cu-Al-Zn shape memory alloy, which are martensitic at
room temperature, is analyzed by Dia et al. [39] and the tensile behaviour of the
springs on loading between the austenite-start and the austenite-finish temperatures
shows a stiffening tendency with increasing both elongation and temperature. By
means of DSC, x-ray diffraction and optical and electron microscopy, Bujoreanu
et al. [40] have shown that the effects of the stress-state induced in austenite
persists even after martensitic transformation and tempering at 773 K.
The effect of thermal cycling in the martensitic phase transitions of Cu-Al-
Zn single crystal shape memory alloy has been studied by monitoring the acoustic
activity generated during the transition. Results indicate that for the whole
transition there is a learning process in which the system seeks an optimal path of
its internal variables that connect the parent and the martensitic phases. According
14
to Bonnot et. al. [41] the learning process in the central part of the transition
dominates the global behaviour and is different from the behavior in the early- and
late-stages of the transition. Planes et al. [42] have reported specific heat
measurements from 2 to 300 K in two Cu-Al-Zn SMA. One remains in the parent
phase, while the other is in the martensitic phase over the full measurement range.
Data confirms the existence of a boson peak in the parent L21-phase of the studied
material. The mechanical properties and shape memory capacity of thin sheets of
Cu-Al-Zn SMA are studied by Asanovic et al. [43] and in quenched specimens,
martensitic structure as well as small quantity of DO3 phase are observed. As
modification of Cu-Al-Zn alloys, the characteristics of alloys containing different
amounts of rare earth Gd are investigated by Xu et al. [44] using optical microscopy,
SEM and x-ray diffraction, and found that there is no effect of Gd addition on the
martensitic transformation type and martensitic transformation temperatures of
Cu-Al-Zn alloys. Experiments aimed at comparing the hysteresis response of a
Cu-Al-Zn alloy single crystal undergoing a martensitic transition under strain-driven
and stress-driven conditions are reported by Bonnot et al. [45] and significant
differences in the hysteresis loops are observed. According to Cai et al. [46] ultra-
violet photoelectron spectroscopy shows a reversible change in the apparent work
function during transformation, presumably due to the contrasting surface electronic
structures of the martensitic and austenitic phases. Specific-heat measurements on
isoelectronic Cu-Al-Zn SMA in the parent cubic phase and in the close-packed
martensitic phase are compared by Lashley et al. [47]. Measurements are made by
thermal-relaxation calorimetry over the temperature range 1.9-300 K.
15
Kustov et al. [48] have reported analysis of pinning and reordering
processes on a microscopic scale, using experimental data on non-linear
anelasticity. An extensive investigation on the elastic and anelastic properties of a
variety of binary and ternary Cu based alloys forming faulted martensites shows
the existence of a strong anelastic relaxation over a wide temperature range as
given by Kustov et al. [49]. Photoelectron emission microscopy observations of
the thermal martensitic transformation in a Cu-Al-Zn shape memory alloy are
reported by Xiong et al. [50] is marked by a sharp change in photoelectron
intensity as well as a significant displacement and reorientation of surface features.
The surface of single crystal austenite Cu-Al-Zn is irradiated by 170 keV Cu ions
at room temperature and their morphology, structure, and composition are
analyzed by Zelaya et al. [51]. The effect of dislocations, produced by plastic
deformation of 18R Cu-Al-Zn single crystals, on the thermally induced martensitic
transformation has been analyzed by calorimetric measurements. The changes of
the transformation temperatures, the heat of transformation, and the two-way shape
memory effect are also discussed by Cunibertiin [52]. The shape memory effects
are studied by Hopulele [53] using dilatometric analysis and the effect of small
variations in the Zn and Al compositions on the critical points is revealed.
Kim et al. [54] have examined the two-way shape memory effect in Cu-Al-
Zn alloys with thermo-mechanical cycling and observed the new martensite phases
in addition to the existing martensite in Cu-Al-Zn alloys. The pseudo-elastic
cycling of Cu-Al-Zn single crystals leads to surface and bulk defects with
consequences on the mechanical behavior and fracture properties of these alloys as
16
given by Damiani et al. [55]. Indarto et al. [56] and Wang et al. [57] have reported
a series of methanol synthesis catalyst containing Cu-Al-Zn which are prepared
using co-precipitation method and applied for partial oxidation of methane into
methanol using dielectric barrier discharge.
Nagasawa et al. [58] have conducted ultrasonic pulse echo overlapping
measurements and calculated the complete set of second- and third-order elastic
constants. The temperature dependence of elastic constants is studied by Verlinden
et al. [59]. The influence of a shear strain on single crystal Cu-Al-Zn is also studied
by Virlinden and Delaey [60]. The effects of grain size and thermal stability
induced by cold rolling are investigated using dilatometry, optical microscopy,
differential scanning calorimetry and electrical conductivity measurements by
Wang et al. [61]. Structural analysis by Planes et al. [3,62] and by neutron
diffraction experiments and Pujari et al. [63] by positron annihilation technique
showed various cubic orders with Fm3m symmetry. Neutron scattering studies on
the effects of uniaxial stresses and aging on the bcc-based Cu-Al-Zn are conducted
by Nagasawa et al. [64]. The temperature dependence of sound velocities for
various modes has been measured by Comas et al. [65, 66] in a single crystal of
Cu-Al-Zn monoclinic 18R martensite and calculated the second-order elastic
constants. Stress-induced martensitic transformations proceeding by the formation
of internally faulted martensite plates are studied by Stupkiewicz [67] in the
martensitic phase of Cu-Al-Zn. Present study is made on the single crystalline
specimen of Cu-66.5at% Al-12.7at% Zn of which the reported results by Virlinden
and Delaey [60] are available.
17
1.1.4 Cu-Al-Be
Among Cu based shape memory alloys, Cu-Al-Be is the one that has been
developed more recently and is unique because of its adaptability for high as well
as low temperature actuator applications as stated by Belkahla and Guenin [68].
Somoza et al. [69] have suggested that this potentiality is the result of the drastic
effect of a small addition of Be to the Cu-Al system, which strongly reduces the
martensitic transition temperature and leaves practically unaltered low temperature
limit of stability of the bcc-based phase, as well as the order-disorder transition
line. Zhang et al. [5] have examined the suitability of super-elastic Cu-Al-Be alloy
wires for the seismic protection in cold regions motivated from the recent use of
SMA for bridge restrainers subject to harsh winter conditions. Humbeeck et al.
[70] designed damping elements utilizing pseudo-elastic hysteresis, transient
damping effects and damping capacity of the martensitic phase of Cu-Al-Be alloy.
Cyclic loading and unloading tensile tests are carried out by Lu et al. [71] and
Sulpice et al. [72] on the super-elastic behavior of Cu-Al-Be alloy wires and showed
that both modes of loading and unloading improve the pseudo-elasticity. A
polycrystalline Cu-Al-Be shape memory alloy has been studied by Mussot et al. [73]
to detect the influence of the process temperature on the microstructure evolution
and on the mechanical properties of the wire. Sapozhnikov et al. [74] have studied
the diffusion mobility of point-like defects in the martensitic phase of a Cu-Al-Be
using acoustic technique operating at a frequency of about 100 kHz. Xie et al. [75]
performed bending fatigue test and tensile test to study the super-elastic property
of Cu-Al-Be alloy wires and showed that the alloy has strong recoverable strain
18
capacity, which is not lower than 20%, and the maximum strain is up to 40%. Song
et al. [76] have designed a single crystal directional furnace to prepare single
crystal Cu-Al-Be shape memory alloy to be used for industrial application.
Perez et al. [77] have conducted studies of stress-induced martensitic
transformation of polycrystalline Cu-Al-Be alloys. The effect of grain size on the
pseudo-elastic behaviour of the alloy is studied by Montecinos et al. [78] and show
that the stress-induced martensite is effectively stabilized. The Cu-Al-Be super-
elastic wires are produced by heated mould continuous casting process and tested
by Huang et al. [79] using cyclic bending, tensile, and rolling experiments to study
their fatigue property, cold working properties and microstructure after cold
working. The efficiency of shape memory alloy as damper and/or standard actuator
is truly enhanced when the material is cycled without any relevant accumulation of
the permanent deformation as stated by Sepulveda et al. [80]. Studies on quenched
martensitic Cu-Al-Be alloys by Dunne et al. [81] showed that the initial quenching
conditions exert an ongoing influence on the characteristics of the normalized
reversible martensitic transformation, probably through the presence of residual
quenching stresses. The results of anodic polarization test done by Kuo et al. [82]
show that the anodic dissolution rates of alloys decreased slightly with increasing
the concentrations of aluminium or beryllium.
Polycrystalline Cu-Al-Be shape memory alloy specimens have been subjected
to both X-ray diffraction stress analysis and optical microscopy during stress-induced
martensitic transformation at room temperature by Kaouache et al. [83] and have
observed that the austenitic stress state in the grains depends mainly on martensitic
19
volume fraction, number and arrangement of variants. Thermal aging treatments in
a polycrystalline Cu-Al-Be shape memory alloy performed by Lara et al. [84, 85]
and the formation of various phases are studied with optical microscopy, TEM, and
X-ray diffraction techniques.
Comas et al. [86] have measured the higher-order elastic constants of the Cu-
Al-Be alloy as a function of temperature in order to quantify the anharmonicity of
this alloy on approaching their MT. Manosa et al. [87] used neutron scattering
technique to study the premartensitic state of a family of Cu-Al-Be alloys which
transform from bcc phase to 18R martensitic structure. Austenite phase has Fm3m
symmetry as stated by Planes and Manosa [3]. Planes et al. [88] have used the
experimental data from ultrasonic and inelastic neutron scattering measurements to
analyze the effect of composition on shape memory characteristics. Jurado et al.
[89] have quantified the vibrational anharmonicity using high pressure ultrasonic
study and calculated the complete set of second- and third-order elastic constants
and mode Gruneisen parameters of Cu-Al-Be. Planes et al. [90] also have studied
the martensitic transformation of a family of Cu-Al-Be alloys by measuring the
entropy change of the transition using a high sensitivity calorimeter and the elastic
anisotropy of the high temperature phase as a function of temperature by ultrasonic
method. At present, we report the work on the single crystalline specimen of Cu-
74.1at% Al-23.1at % Be along with the results by Jurado et al. [89].
20
1.1.5 Cu-Al-Pd
Materials like Ni-Ti, Cu-Al-Ni and Cu-Al-Zn lack the high transformation
temperatures and long term thermal stability desired in many commercial
applications. Cu-Al-Pd alloys are found to have an austenite transformation range
of 388 K- 643 K depending on their composition, heat treatment and working
process. They have excellent workability and exhibit fatigue properties comparable
with Ni-Ti alloy. Lin et al. [91] report an optimal shape memory property for a
composition of Cu-13.1wt% Al-2.4wt% Pd which has a transformation
temperature of 453 K and a recoverable strain of 4.8%. In addition this alloy is
expected to have higher grain boundary adhesion compared to other members in
the Cu based SMA, thus improving the mechanical properties. The elastic
properties, phase compositions, microstructure and specific volume changes in the
Cu-Al-Pd alloy mixed with Zr and Ni have been studied by Luzgin et al. [92] using
ultrasonic technique, x-ray diffraction, TEM and Archimedean methods and have
showed that the structural changes are accompanied by decrease in specific
volume, bulk modulus, Lame parameter and Poisson's ratio. Nagasawa et al. [93]
used ultrasonic pulse echo method and X-ray and neutron diffraction methods to
investigate the lattice instability of premartensitic phase of Cu-Al-Pd alloy under
uniaxial pressure. Higher-order elastic constants and the mode Gruneisen
parameters show that the austenite phase is strongly anharmonic. Present study is
made on the single crystalline specimen of Cu-67.5at% Al-27at% Pd. This alloy
has DO3 type structure with Fm3m symmetry and the lattice constant a = 0.53 nm
at room temperature.
21
1.2 Applications
As actuators these materials can be used for active deployment of a host of
devices including antennae and solar panels. As a super-elastic flexure or constant
force spring, it can be used for passive movement of instrument cover doors and
hinges. Ray chem. Corp. has succeeded in making fasteners and tube couplings. In
the application of SMA as a thermal actuator a temperature sensitive SMA spring
is used. One of the draw backs in this case is their slow response which is restricted
by heat conduction. But according to Uchino [94], the ferromagnetic SMA can be
effectively used here which is driven by magnetic fields. Search on micro actuators
leads to the fabrication of a robot hand, similar in size to a human hand, with
thirteen degrees of freedom and also the active endoscope with multiple degrees of
freedom. Smart Composites are also developed which include carbon fibre
reinforced plastics and SMA wires as given by Rogers [95]. Vibration control
using a smart composite is another important target, since the elastic constants can
be varied continuously through the transformation range. The SMA may also be
manufactured with a very low transition temperature opening an entirely new
application area in cryogenics. Cryogenic valves have applications in missiles and
satellites that carry sophisticated instruments such as sensors and cameras that need
to be cryogenically cooled. The super-elastic properties may be used in catheter
guide wires and laparoscopic instruments for medical applications. Much smaller
electrical connectors, switches, and circuit breakers used in test equipments by the
computer industry can be manufactured. And generally, it can replace materials in
many existing commercial applications such as eye glass frames, cellular phone
antennae, and thermostatic valves. The SME is currently used in space shuttle,
22
thermostats, vascular stents and hydraulic fittings. SMA is important as high
damping materials, since twin boundary movements in martensites contribute
greatly to internal friction as stated by Humbeeck and Kustov [70].
Though Ni-Ti SMA is used extensively in a variety of engineering and
medical applications because of their attractive shape memory characteristics, they
still suffer from certain draw backs such as low transformation temperatures,
difficulty in production and processing and high cost of raw materials. Copper
based alloys have, therefore, come as an alternative to Ni-Ti alloys. They are
comparatively easier to produce and process and are also less expensive. In the
present work we study the elastic and thermal behaviour of selected Cu based
SMA such as Cu-Al-Ni, Cu-Al-Zn, Cu-Al-Be and Cu-Al-Pd. These alloys have
received, in recent years, much attention concerning their development and
commercial exploitation. In the present context these alloys are especially
appealing because the electronic contribution to their entropy is negligibly small.
Cu based alloys are Hume-Rothery alloys as given by Planes and Manosa [3] for
which the phase stability is mainly controlled by the average number of valence
electrons per atom. The point of view adopted here for the study of this class of
solids connects the MT with the loss of stability of the high temperature bcc phase.
In general there is more vibrational entropy in the bcc phase than in the close-
packed structure. Manosa et al. [96] have investigated the different contributions
to the entropy change at the transition of various families of Cu- based shape
memory alloys and they found that the stability of the bcc phase is mostly due to
the harmonic vibrations of the lattice. Stipcich et al. [97] show that the entropy
23
difference originates in the low-energy TA2 vibrational modes. Influence of the
external applied stress on the elastic shear constants in the various {110} planes
are studied by Verlinden et al. [98]. The relation between the elastic constant C′
and the atomic order state in binary alloys is also analysed. Cu based shape
memory films are also investigated by Pletea et al. [99]. In addition a number of
experimental investigations including ultrasonic and neutron scattering
measurements are also done to analyze the elastic properties of Cu based SMA by
different authors [10-12].
1.3 Mechanism and Precursor effects of Martensitic Transformation
It is acknowledged that vibrational anharmonicity plays an essential role in
the mechanisms, leading to the MT of shape memory materials. Generally
precursor or pretransition effects are phenomena announcing that a system is
preparing for a phase transition before it actually occurs as stated by Planes and
Manosa [3]. They are related to possible anticipatory visits of the system into the
approaching phase and are characteristic of systems that undergo second-order
transitions. However, among the wide class of materials undergoing MT, precursor
phenomena are observed in several cases. Thus understanding precursor effects can
throw more light on the real mechanism of MT leading to shape memory and
super-elastic effects.
The problem can be better understood when it is noticed that all systems
showing premonitory phenomena have a common characteristic: restoring forces in
specific lattice directions are weak. Therefore the vibrational properties of these
systems are highly anisotropic. They show that these systems are incipiently
24
unstable with respect to specific long- and short-wavelength acoustic modes.
Interestingly, the lattice distortion resulting from the MT is precisely related to
such anomalous acoustic modes. Moreover, there is experimental evidence that in
Cu based alloys the transition occurs at a fixed value of the elastic anisotropy. It
has been shown that for these alloys, anisotropy factor (A) is essentially
determined by the ratio of a combination of second-order elastic constants
(SOECs) associated with a shear mechanism. This coupling between different
vibrational modes and other well known effects such as temperature or stress
dependence of the elastic constants, are due to the anharmonicity of the interatomic
potential. In Cu based alloys, there is no particular phonon on the TA2 branch with
a peculiar behaviour; instead it is the whole branch which softens with
temperature. Hence, a good approach to the vibrational anharmonicity of these
alloys is gained by the investigation of the vibrational anharmonicity of long
wavelength acoustic modes, quantified by higher-order elastic constants. We have
obtained the complete set of SOECs and third-order elastic constants (TOECs) of
Cu-Al-Ni, Cu-Al-Zn, Cu-Al-Be and Cu-Al-Pd single crystals at room temperature.
In order to assess quantitatively the anharmonic effects it is quite useful to compute
the acoustic mode Gruneisen parameters, which give the strain dependence of the
long wavelength acoustic modes.
1.4 Finite Strain Theory of Elasticity
Many of the physical properties of crystals involve the frequencies of the
normal modes of vibration of the structure. There are number of methods which
provide information about modes having a particular wave number or covering a
25
limited range of wave numbers. The velocity of ultrasonic waves through the
crystal can be measured which give the frequencies of long wave length acoustic
modes. Measurements of infrared and Raman line spectra give the frequencies of
optic-modes for which also the wavelength is long compared with the unit cell
dimensions. The relation between the intensity of thermal diffuse scattering of X-
rays and the frequencies of lattice vibrations can be used to obtain the frequencies
of normal modes. The most powerful method is the one of slow neutron
spectroscopy. When a neutron beam interacts with a crystal in a one phonon
process, the change in the energy and momentum can be used to get the frequency
of any normal mode.
In the present work we follow the method of homogeneous deformation
given by Born and Huang [100]. Consider an elastic medium where the co-
ordinates of any point can be denoted as (a1, a2, a3). Choose a set of orthonormal
vectors e1, e2, e3 as the basis vectors for the co-ordinate system and denote the kth
component of the stress acting on the plane ei = 0 by σik where i and k are the
component indices. Consider the equilibrium of a small element centered at the
point ai and bounded by the plane ai + ½ dai. Let ui denote the elastic displacement
of the point ai and ρ the density at this point. The equation of state of this volume
element can be derived by considering the total force acting on the volume
element. If we ignore the body forces, the equations of motion for an elastic solid
can be written as,
ρüi = ∂σik/∂ak (1.1)
26
where the stress tensor σik is given by
σik = ∂φ/∂εik (1.2)
where φ is the crystal potential and εik are the components of the strain tensor given by
εik = ½∂∂
∂∂
ua
ua
k
i
i
k
+
(1.3)
σik and εik are symmetric tensors of second rank. According to Hooke’s law
σik = Ciklm εlm (1.4)
The constants Ciklm form a fourth rank tensor with 81 components.
From equations (1.2) and (1.4), we have
Ciklm = ∂σ∂ε
ik
lm
= ∂ φ∂ε ∂ε
2
lm ik
= ∂ φ∂ε ∂ε
2
ik lm
= Clm ik (1.5)
Hence the elastic constants Ciklm are multiple strain derivatives of the stress
functions and since the strains εlm are symmetric, the elastic constants possess
complete Voigt’s symmetry. Thus,
Ciklm = Ckilm = Cikml = Clmik (1.6)
These quantities are symmetric with respect to interchange of the
subscripts. It will be convenient to abbreviate the double subscript notation to the
single subscript Voigt notation running from 1 to 6, according to the following
scheme:
11→1; 22→2; 33→3; 23→4; 13→5 and 12→6.
27
Hence the matrix of elastic constants Ciklm would contain a 6 × 6 array of
36 independent quantities in the most general case. This number is, however,
reduced to 21 by the requirements that the matrices be symmetric on interchange of
double indices. The number of independent elastic constants will be further
reduced by the symmetry operations of the respective crystal classes. The cubic
compounds have three independent elastic constants [101]. The elastic constant
matrix for this class of compounds is given by
=
44
44
44
111212
121112
121211
000000000000000000000000
CC
CCCCCCCCCC
Cij (1.7)
In the equation of motion for an elastic medium, the forces on an element of
volume are given by the divergence of the stress field.
Using equations (1.3) and (1.4), the equation (1.1) can be written as
ρüi = ∂∂
∂∂
∂∂a
Cua
uaj
ijklk
l
l
k
12
+
(1.8)
For an elastic plane wave, we have
uk = Ak exp i(ωt – k.a) (1.9)
28
where Ak are the components of the amplitude of vibration, ω is the angular
frequency and k is the wave vector corresponding to the wavelength λ = 2π/k. The
resulting equations of motion from equation (1.8) are
(ρω2δim – Cijlm kj kl) um = 0 (1.10)
Substituting k = k $n , where $n is the unit vector, we get
(Tijlm njnl – v2δim) um = 0 (1.11)
where Tijlm = Cijlm/ρ are the reduced elastic constants and v is the phase velocity
given by v = ω/k. The components of second rank tensor Λ are given by
Λil = Tijkl njnl (1.12)
Hence equation (1.11) can be written as
(Λ – v2)u = 0 (1.13)
This shows that u is the eigen vector of tensor Λ where eigen value is v2. Hence v2
is the root of the equation
Λ – v2 = 0 (1.14)
The theory of elastic waves generally reduces to find u and v for all plane
waves propagating in an arbitrary direction for crystals possessing different
symmetries. In this situation, all terms in equation (1.11) that involves
differentiation with respect to co-ordinates other than that along the propagation
direction drop out.
29
A more fundamental significance to the elastic constants is implied by their
appearance as the second derivatives of elastic energy with respect to strains. It
should be noted here that the stored elastic energy is only a part of the complete
thermodynamic potential of the crystal, since it depends on many other variables.
Also, one can introduce elastic constants as a constitutive, local relation between
stress and strain for materials in which long-range atomic forces are unimportant.
Let the position co-ordinates of a material particle in the unstrained state be
ai (i = 1, 2, 3). Let the co-ordinates of the material particle in the strained state be
xi. Consider two material particles located at ai and ai + dai. Let their co-ordinates
in the deformed state be xi and xi + dxi. The elements dxi are related to dai by the
equation.
dxi = ∂∂xa
dai
ii
= ( )∑ +=j
ij ij ida1
3δ ε (1.15)
The convention that repeated indices indicate summation over the indices
will be followed here. δij is the Kronecker delta and εij are the deformation
parameters. The Jacobian of the transformation
J = Det∂∂xa
i
i
(1.16)
is taken to be positive for all real transformations. If dVa is a volume element in
the natural state and dVx its volume after deformation
30
dVdV
x
a
= ρρ
0 = J (1.17)
where ρ0 and ρ are the densities in the natural and strained states respectively. Let
the square of the length of arc from ai to ai + dai be 20dl in the unstrained state and
2dl in the strained state. Then
2dl – 20dl = dxidxi – daidai
= dxda
dxda
i
j
i
kik−
δ dajdak
= 2ηjk dajdak (1.18)
where ηjk are the Lagrangian strain components which are symmetric with respect
to the interchange of the indices j and k. In terms of εik,
ηjk = ½ ε ε ε εjk kji
ij ik+ + ∑
(1.19)
The internal energy function U (S,ηjk) for the material is a function of the
entropy S and Lagrangian strain components [104]. U can be expanded in powers
of the strain parameters about the unstrained state as
U = U0 + 12!
∂∂η ∂η
η η2
0
U
ij kl S
ij kl
,
+ 13!
∂∂η ∂η ∂η
η η η3
0
U
ij kl mn S
ij kl mn
,
+ . (1.20)
The linear term in strain is absent because the unstrained state is one where
U is minimum. We shall define the elastic constants of different orders referred to
the unstrained state as
31
Cij kls
, = ∂∂η ∂η
2
0
U
ij kl S
,
(1.21)
and
smnklijC ,, = ∂
∂η ∂η ∂η
3
0
U
ij kl mn S
,
(1.22)
Here the derivatives are to be evaluated at equilibrium configuration and constant
entropy. Cij klS
, and Cij kl mnS
, , are the adiabatic elastic constants of second- and
third-orders respectively. They are tensors of fourth and sixth ranks. The number
of independent second-and third-order elastic constants for different crystal classes
have been tabulated by Bhagavantam (101).
References
1. K. Otsuka, C. M. Wayman: Shape Memory Materials (Cambridge University
Press) (1998).
2. L. Delaey: Phase Transformations in Materials, edited by Haasen P, VCH,
Weinheim, (1991).
3. A. Planes and L. Manosa: Solid State Physics 55 (2001) 159.
4. Z. C. Lin, W. Yu, R. H. Zee and B. A. Zin: Intermetallics 8 (2000) 605.
5. Y. Zhang, J. A. Camilleri and S. Zhu: Smart Mater. Struct. 17 (2008) 025008.
6. K. Ostuka and T. Kakeshita: MRS Bulletin, Feb. (2002) 91.
32
7. S. J. Kim: The Minerals, Metals & Materials Society (1992) Edited by
Changxu Shi, Hengde Li and Alexander Scott. p59
8. J. I. P´erez-Landaz´abal, V. Recarte, R. B. P´erez-S´aez, M. L. N´o, J.
Campo and J. S. Juan: Appl. Phys. Lett. 81 (2002) 1794.
9. Y. Nakata, T. Tadaki and K. Shimizu: Materials Trans. JIM 31 (1990) 652.
10. A. Gonzalez-Comas and L. Manosa: Phys. Rev. B 54 (1996) 6007.
11. M. Landa, V. Novac, P. Sedlak and P. Sittner: Ultrasonics 42 (2004) 519.
12. P. Sedlak, H. Seiner, M. Landa, V. Novak, P. Sittner and L. Manosa: Acta
Materialia 53 (2005) 3643.
13. V. Recarte, J. I. P´erez-Landaz´abal, and V. S. Alarcos: Appl. Phys. Lett.
81 (2002) 1794.
14. A. C. R. Veloso, R. M. Gomes and D. B. D. Santos, I. C. E. G. Lima,S. J.
G. Lima and T. A. A. Melo: State-of-the-Art Research and Application of
SMA Technologies (2008) 59 92.
15. J. A. Rodriguez, I. R. Larrea, M. L. No and A. L. Echarri and J. San Juan:
Acta Materialia 56 (2008) 6283.
16. V. I. Nikolaev, S. A. Pul'nev, G. A. Malygin, V. V. Shpeizman and S. P
Nikanorov, Physics of the Solid State 50 (2008) 2170.
33
17. G. K. Kannarpady, A. Bhattacharyya, M. Wolverton, D. W. Brown, S. C.
Vogel and S. Pulnev: Acta Materialia 56 (2008) 4724.
18. L. A. Matlakhova, E. C. Pereira, A. N. Matlakhov, S. N. Monteiro and R.
Toledo: Materials Characterization 59 (2008) 1630.
19. J. A. Rodriguez, I. R. Larrea, M. L. No, A. Lopez-Echarri, San Juan and J.
Maria: Acta Materialia 56 (2008) 3711.
20. H. Seiner, P. Sedlak and M. Landa: Phase Transitions 81 (2008) 537.
21. U. Sari, T. Kirindi: Materials Characterization 59 (2008) 920.
22. R. Gastien, M. Sade and F. C. Lovey: Materials Science & Engineering, A:
Structural Materials: Properties, Microstructure and Processing A481
(2008) 518.
23. P. Molnar, P. Sittner, V. Novak and P. Lukas: Materials Science &
Engineering, A: Structural Materials: Properties, Microstructure and
Processing A481 (2008) 513.
24. N. Zarubova, J. Gemperlova, V. Gaertnerova and A. Gemperle: Materials
Science & Engineering, A: Structural Materials: Properties, Microstructure
and Processing A481 (2008) 457.
25. F. C. Lovey, A. M. Condo, J. Guimpel and M. J. Yacaman: Materials
Science & Engineering, A: Structural Materials: Properties,
Microstructure and Processing A481 (2008) 426.
34
26. M. Kamal and S. Gouda: Radiation Effects and Defects in Solids 163 (2008) 237.
27. A. Creuziger, A.; Crone and W. C. Acta Materialia 56 (2007) 518.
28. N. Suresh and U. Ramamurty: Journal of Alloys and Compounds 449 (2008) 113.
29. M. Landa, P. Sedlak, P. Sittner, H. Seiner and V. Novak: Materials Science
& Engineering, A: Structural Materials: Properties, Microstructure and
Processing A462 (2007) 320.
30. V. Recarte, J. I. P´erez-Landaz´abal, and V. S. Alarcos: J. Phys. Condens.
Matter 17 (2005) 4223.
31. N. Suresh and U. Ramamurty: Smart Mater. Struct. 14 (2005) N47.
32. V. Sampath: Smart Mater. Struct. 14 (2005) S253.
33. X. Ding, J. Zhang, Y. Wang, Y. Zhou, T. Suzuki, J. Sun, K. Otsuka and X.
Ren: Phys. Rev. B 77 (2008) 174103.
34. M. Sade, C. Damiani, R. Gastien, F. C. Lovey, J. Malarrıa and A. Yawny:
Smart Mater. Struct. 16 (2007) S126.
35. F. Lanzini, R. Romero, M. Stipcich, and M. L. Castro: Phys. Rev. B 77
(2008) 134207.
36. G. Ciatto, P. L. Solari, S. De Panfilis, A. L. Fiorini, S. Amadori, L. Pasquini
and E. Bonetti: Applied Physics Letters 92 (2008) 241903.
37. V. Asanovic, K. Delijic and N. Jaukovic: Scripta Materialia 58 (2008) 599.
35
38. E. Bonnot, R. Romero, M. Morin, E. Vives, L. Manosa and A. Planes:
Journal of Materials Science 43 (2008) 3832.
39. V. Dia, L .G. Bujoreanu, S. Stanciu and C. Munteanu: Materials Science &
Engineering, A: Structural Materials: Properties, Microstructure and
Processing A481 (2008) 697.
40. L. Bujoreanu: Materials Science & Engineering, A: Structural Materials:
Properties, Microstructure and Processing A481 (2008) 395.
41. E. Bonnot, E. Vives, L. Manosa and A. Planes: Materials Science &
Engineering, A: Structural Materials: Properties, Microstructure and
Processing A481 (2008) 223.
42. A. Planes, L. Manosa, R. Romero, M. Stipcich, J. C. Lashley and J. L.
Smith: Materials Science & Engineering, A: Structural Materials:
Properties, Microstructure and Processing A481 (2008) 194.
43. V. Asanovic and K. Delijic: Metalurgija (Belgrade, Serbia) 13 (2007) 59.
44. J. Xu : Journal of Alloys and Compounds 448 (2008) 331.
45. E. Bonnot, R. Romero, I.X. Ricardo, L. Manosa, A. Planes, Antoni and E.
Eduard: Physical Review B: 76 (2007) 064105.
46. M. Cai, S.C. Langford, J.T. Dickinson, G. Xiong, T. C. Droubay, A. G. Joly,
K.M. Beck and W. P. Hess: Journal of Nuclear Materials 361 (2007) 306.
36
47. J. C. Lashley, F. R. Drymiotis, D. J. Safarik, J. L. Smith, R. Romero, R. A.
Fisher, A. Planes and L. Manosa: Physical Review B: 75 (2007) 064304.
48. S. Kustov, M. Corro, J. Pons, E. Cesari and J. Van Humbeeck: Materials
Science & Engineering, A: Structural Materials: Properties, Microstructure
and Processing A438 (2006) 768.
49. S. Kustov, E. Cesari, S. Golyandin, K. Sapozhnikov, J. Van Humbeeck, G.
Gremaud and R. De Batist: Materials Science & Engineering, A: Structural
Materials: Properties, Microstructure and Processing A432 (2006) 34.
50. G. Xiong, A.G. Joly, K.M. Beck, W.P. Hess, M. Cai, S.C. Langford and
J.T. Dickinson: Applied Physics Letters 88 (2006) 091910.
51. E. Zelaya, A. Tolley, A.M. Condo, F.C. Lovey, P.F.P. Fichtner, P.B.
Bozzano : Scripta Materialia 53 (2005) 109.
52. A. Cuniberti, R. Romero : Scripta Materialia 51 (2004) 315.
53. I. Hopulele, S. Istrate, S. Stanciu and G. Calugaru: Journal of Optoelectronics
and Advanced Materials 6 (2004) 277.
54. H.Kim: Journal of Materials Processing Technology 146 (2004) 326.
55. C.Damiani, M. Sade and F.C. Lovey: Journal de Physique IV:
Proceedings 112 (2003) 623.
56. A. Indarto, D. Yang, J. Palgunadi, J. Choi, H. Lee and H. Song: Chemical
Engineering and Processing 47 (2008) 780.
37
57. T. Wang, J. Chang, Y. Fu, Q. Zhang and Y. Li: Korean Journal of
Chemical Engineering 24 (2007) 181.
58. A. Nagasawa, T. Makita and Y. Takagi: J. Phys. Soc. Jpn. 51 (1982) 3876.
59. A. Verlinden, T. Suzuki, L. Delaey and G. Guenin: Scripta Met. 18 (1984) 975.
60. B. Virlinden and L. Delaey: J. Phys. F: Met. Phys 16 (1986) 1391.
61. J. J. Wang, T. Omori, Y. Sutou, R. Kainuma and K. Ishida: Journal of
Electronic Materials, 33 (2004) 1098.
62. A. Planes, L. Manosa, E. Vives, J.R. Carvajal, M. Morin, G. Guenin, and J.
L. Macqueron: J. Phys.: Condens. Matter 4 (1992) 553.
63. P. K. Pujari, T. Datta, K. Madangopal and J. Singh: Phys. Rev. B 47 (1993) 11677.
64. A. Nagasawa, M. Yamada and Y. Morii: J. Phys. Soc. Jpn. 65 (1996) 778.
65. A. G. Comas, L. Manosa, A. Planes, F. C. Lovey, J. L. Pelegrina and G.
Guenin: Phys. Rev. B 56 (1997) 5200.
66. A. G. Comas, L. Manosa and M. Cankurtaran: J. Phys. Condens. Matter 10
(1998) 9737.
67. S. Stupkiewicz: European J. Mech. A/ Solids 23 (2004) 107.
68. S. Belkahla and G. Guenin: J. Phys. IV 1 (1991) C4.
69. A. Somoza, R. Romero, L. Manosa and A. Planes: J. Appl. Phys. 85 (1999) 130.
38
70. J. V. Humbeeck and S. Kustov: Smart Mater. Struct. 14 (2005) S171.
71. S. Lu, J. Chen and L. Chen: Advanced Materials Research 47 (2008)886.
72. L. Saint-Sulpice, S. Chirani and S. Calloch: Materials Science & Engineering,
A: A481 (2008) 174.
73. G. Mussot-Hoinard, E. Patoor and A. Eberhardt: Materials Science &
Engineering, A: A481 (2008) 538.
74. K. Sapozhnikov, S. Golyandin, S. Kustov and E. Cesari: Materials Science
& Engineering, A: A481 (2008) 532.
75. L. Xie and Y. Yu: Tezhong Zhuzao Ji Youse Hejin 27 (2007) 407.
76. G. Song, X. Mao, H. Xu, C. Li, M. Hong and X. Lu: Tezhong Zhuzao Ji
Youse Hejin 27 (2007) 161.
77. J. Cortes-Perez, A. Souza Jimenez, G. A. Lara Rodriguez, L. A. Ferrer and
H. Flores Zuniga: Materials Science Forum 561 (2007) 1485.
78. S. Montecinos, A. Cuniberti and A. Sepulveda: Materials Characterization 59
(2008) 117.
79. W. Huang, W. Li and Y. Yu: Cailiao Yanjiu Yu Yingyong 1 (2007) 118.
80. A. Sepulveda, R. Munoz, F. C. Lovey, C. Auguet, A. Isalgue and V.
Torra: Journal of Thermal Analysis and Calorimetry 89(2007) 101.
39
81. D. Dunne, N. Stanford, M. Morin, C. Gonzalez and G. Guenin: Philosophical
Magazine Letters 87 (2007) 483.
82. H. H. Kuo, W.H. Wang, Y.F. Hsu and C.A. Huang: Corrosion Science 48
(2006) 4352.
83. B. Kaouache, K. Inal, S. Berveiller, A. Eberhardt and E. Patoor: Materials
Science & Engineering, A: A438 (2006) 773.
84. M.T. Ochoa-Lara, H. Flores-Zuniga and D. Rios-Jara: Journal of Materials
Science 41 (2006) 5455.
85. M. T. Ochoa-Lara, H. Flores-Zuniga, D. Rios-Jara and G. Lara-Rodriguez:
Journal of Materials Science 41 (2006) 4755.
86. A. G. Comas, L. Manosa, A. Planes and M. Morin: Phys. Rev. B 59 (1999) 246.
87. L. Manosa, J. Zarestkey, T. Lograsso, D. W. Delaney and C. Stasis: Phys.
Rev. B 48 (1993) 15708.
88. A. Planes, L. Manosa and E. Vives: Phys. Rev. B 53 (1996) 3039.
89. M.A. Jurado, M. Cankurtaran, L. Manosa and G.A. Saunders: Phys. Rev. B
46 (1992) 14174.
90. A. Planes, L. Manosa, D. R. Jara and J. Ortin: Phys. Rev. B 45 (1992) 7633.
91. Z. C. Lin, W. Yu, R. H. Zee and B. A. Chin: Intermetallics 8 (2000) 605.
40
92. L. Luzgin, V. Dmitri, M. Fukuhara and A. Inoue: Acta Materialia 55
(2007)1009.
93. A. Nagasawa, Y. Kuwabara, K. Morri, Fuchizaki and S. Funahashi: Mater.
Trans. JIM 33 (1992) 203.
94. K. Uchino: Shape Memory Materials, edited by K. Otsuka and C. M.
Wayman (Cambridge University Press, Cambridge, 1998) p. 184.
95. C. A. Rogers, Smart Materials, Structures, and Mathematical Issues
(Technomic, Lancaster, PA, 1989).
96. L. Manosa, A. Planes and J. Ortin: Phys. Rev. B 48 (1993) 3611.
97. M. Stipcich, J. Marcos, L. Manosa and A. Planes: Phys. Rev. B 68 (2003)
214302.
98. B. Verlinden and L. Delaey: Metallurgical Trans. A 19A (1998) 207.
99. M. Pletea, H. Wendrock, R. Kaltofen, O. G. Schmidt and R. Koch: J.
Phys.: Condens. Matter 20 (2008) 255215.
100. M. Born and K. Huang: Dynamical Theory of Crystal Lattice, Oxford
University Press, N.Y. 1962.
101. S. Bhagavantam: Crystal symmetry and physical properties, Academic
Press Inc. London 1966.
102. F. D. Murnaghan: Finite Deformation of an elastic solid, Wiley, New York 1951.
Top Related