a brief review of shape memory alloys and introduction to...

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.

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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.

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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

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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.

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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

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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

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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

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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

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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

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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

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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.

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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

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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

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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.

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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

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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.

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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

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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

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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].

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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).

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