INSTITUTE OF PHYSICS PUBLISHING
SMART MATERIALS AND STRUCTURES
Smart Mater. Struct. 13 (2004) 552–561
PII: S0964-1726(04)76267-9
The work production of shape memory
alloy
Yong Liu
School of Mechanical and Production Engineering, Nanyang Technological University,
639798, Singapore
E-mail: mliuy@ntu.edu.sg
Received 7 November 2003, in final form 2 February 2004
Published 26 April 2004
Online at stacks.iop.org/SMS/13/552
DOI: 10.1088/0964-1726/13/3/014
Abstract
This paper is to evaluate the work production and thermal efficiency of
shape memory alloy (SMA). NiTi wire under pure tension mode is chosen
for the work production and several bias components are evaluated including
a linear elastic component, a non-linear superelastic SMA wire and an SMA
with well developed two-way memory effect (TWME). The calculated
thermal efficiency shows good agreement with the reported experimental
data. The estimation of the thermal efficiency is further discussed with
respect to the latest results on SMAs, especially the transformation enthalpy
change. Ways of increasing both the work production and the thermal
efficiency of SMAs are further suggested.
1. Introduction
The discovery of shape memory alloy (SMA) has inspired
high enthusiasm in developing solid engines to convert low
grade thermal energy to mechanical energy and further to
electrical energy. As a result, various types of solid engine
have been constructed or proposed, for example, by Banks [1],
Smith [2], Ginell et al [3], Golestaneh [4], Wayman [5],
Wang [6, 7], Johnson [8], Goldstein [9], Otsuka [10] etc. The
underlying mechanism is a solid-state phase transformation
that converts heat into motion through self-rearrangement of
atoms in a piece of metal, known as shape memory effect.
Heat supply is required to trigger the rearrangement of atoms
(phase transformation) and to result in dimensional change
of the metal. Since thermal energy is one of the least
mechanically recoverable forms of energy, the advantage of
such a mechanism of energy conversion is that the low grade
thermal energy can be utilized conveniently and cleanly. Such
thermal energy is abundant in nature, including especially
solar thermal energy, geothermal energy and industrial exhaust
warm water. However, as soon as people realized that the
use of SMAs for energy conversion is limited by several
intrinsic and extrinsic factors, the enthusiasm was cooled
down. Among them, the energy conversion efficiency and the
fatigue property are among the most important factors, that
were also found to be interrelated [11]. Although there are
still some trials today [12, 13], the number of publications
in this field has reduced significantly in recent years. The
0964-1726/04/030552+10$30.00 © 2004 IOP Publishing Ltd
most important theoretical works in this field were published
more than a decade ago by, for example, Ahlers [14], Tong
and Wayman [15], Golestaneh [16], Wollants et al [17–19],
Mukherjee [20], Salzbrenner [21], McNichols and Cory [22],
Jardine [23] and Duerig et al [11]. However, the published
theoretical thermal efficiencies were found to vary from 2.8%
to 29%, which are unacceptably widely scattered values for
engineering practice. Some of the estimated values were later
found to be either under-estimated or over-estimated.
Re-examination of the previous works shows that the
difference in the theoretical results is due largely to the different
research approaches adopted and due partly to the selection
of the input experimental data. Some of the previous works
suggested that the thermal efficiency is intrinsically determined
by the transformation enthalpy change, the equilibrium
temperature between martensite and austenite phases and
the specific heat of the material, while the change in shape
recovery properties was not reflected in the calculation. This
clearly contradicts the experimental observations. Another
controversial parameter in estimating the thermal efficiency
was the transformation enthalpy change. Some researchers
have used the experimental values determined from fully
annealed specimens, while in some other cases the effect of
stress on the transformation enthalpy change was proposed by
assuming that the entropy change during phase transformation
is unaffected by the application of the stress. Meanwhile,
confusion between chemical transformation enthalpy change
Printed in the UK
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Work production of shape memory alloy
Figure 1. A schematic diagram illustrates the set-up for energy
conversion using SMAs. The SMA wire under tension mode is used
for converting thermal energy to mechanical work. The biasing
component in chamber I can be either a simple weight, an elastic
component, a superelastic SMA wire or an SMA wire identical to
that in chamber II. Initial conditions: martensitic SMA II
pre-strained to ε is balanced by a bias component at a stress level σ .
and measured transformation latent heat has been found in the
calculation, thus leading to unrealistic estimated results.
Since our understanding of the SMAs has been improved
tremendously during the last 10 years, it is time for us to
re-evaluate the design of the solid engine and the thermal
efficiency of SMAs by taking into account the latest progress
in this field. Theoretical results will likely provide guidelines
for the engineering practices towards more effective use of the
materials. It is hoped that the present paper will re-evaluate
the major factors affecting both the work production and the
thermal efficiency of the solid engine and to provide guidelines
for effective use of SMAs. Since NiTi SMAs have the best
shape recovery property and much better fatigue resistance
than other SMAs discovered so far, and since they have been
used in most engine designs, the analysis will be performed
based on the data of NiTi SMAs.
The present work will begin by evaluating the work output
of a proposed design of solid engine by using different biasing
components, namely (1) a bias component of linear elasticity,
(2) a superelastic SMA wire and (3) a SMA wire identical to
that for work production. An SMA with two-way memory
effect (TWME) will be further evaluated. The paper then
proceeds to evaluate the thermal efficiency of the SMA and
its influencing factors. Finally, ways of increasing both the
work output and the thermal efficiency are suggested.
2. Design evaluations
A simple design which uses the SMA wires under a tension
mode is schematically illustrated in figure 1. This simple
design is likely the best design in terms of fully exploiting
the capacity of SMA wires. Polycrystalline SMA wire
under tension mode should provide the best shape recovery
characteristic, thus the highest kinetic output per unit volume
if comparing with other loading modes, namely, compression,
torsion and bending. One reason is that the martensite
deformation and shape recovery is found to be asymmetric
between tension and compression (see for example, [24–
27]) due to different deformation mechanisms. For textured
polycrystalline SMAs, tension gives rise to a stress plateau
during martensite reorientation and a higher recovery strain
upon reverse transformation, while compression leads to a
lower recovery strain [28, 29]. Another consideration is that
Figure 2. Stress–strain curves of superelastic and martensitic
SMAs. For a system incorporating an SMA and an elastic bias
component, the stress in the elastic component at temperatures
between As and Af can be described by line AM. The slope of the
line is related to the modulus of the elastic component. The stress
value at point M is the martensite stress in the initial condition and
the stress value at point A is the austenite stress after heating to Af .
The austenite stress varies depending on the modulus of the elastic
component. Both stresses are functions of temperature.
the material under compression load requires a larger diameter
in order to avoid buckling. This will result in slow response
due to low cooling rate, thus a low output power. In the case
of shear deformation, the distribution of the shear strain across
the diameter of the SMA wires or bars will significantly reduce
the work output per unit volume of SMAs. As in this case, the
strain in the central axis is zero, while that on the outer surface
is limited by the maximum shape recovery strain achievable
by different types of SMA. Similarly, in bending mode, one
side of the SMA is under tension while the other side is under
compression. In the central line, the strain is zero. Thus, both
the deformation and shape recovery are unevenly distributed
through the thickness of the materials, resulting in ineffective
use of the materials.
In figure 1, the temperature chamber is separated by a
piston movable horizontally in one dimension. This onedimensional motion can be conveniently translated into a
rotary motion through a proper design, which is, however,
beyond the scope of the present paper. For convenience in
description, we label the left-hand chamber chamber I and the
right-hand chamber chamber II. The initial condition is that
the SMA wire in chamber II is in martensitic state and is under
tension to a strain slightly over the stress-plateau region in
the martensite stress–strain curve as shown in figure 2. In
chamber I, the resetting system could be a simple weight, a
linear elastic component, a superelastic SMA wire of smaller
diameter as compared to that of SMA II or an SMA wire of
the same type as that in chamber II. In the first three cases, the
temperature of chamber I is kept constant during the operation.
Thus, the stress in the bias component is either a constant
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Y Liu
Figure 3. Schematic diagram showing the estimation of the work
output of the SMA during one thermal cycle. Line AM is due to the
constraint from an elastic bias component. The energy conversion
device employs the SMA under tension mode as shown in figure 1.
value or a function of strain. For the third design, the cross
sectional area of the superelastic SMA in chamber I is chosen
in such a way that its bias force will lead to the yield of the
martensitic SMA II, and is yielded when SMA II is heated
to the austenitic state. In the fourth design, the temperature
in chamber I changes during operation, identical to that in
chamber II. The stress–strain relations of SMA wire in both
martensitic and austenitic states are schematically illustrated
in figure 2 together with that of the elastic bias component. For
convenience of discussion, we name the device incorporating
two identical SMAs in both chambers the coupled SMA system.
The unit work output that an SMA is able to produce is
approximately shown by a square area between the stress–
strain curves of martensite and austenite (figure 3) and can be
estimated by
wsma =
ε1M
ε2A
σH dε −
ε1M
ε2A
σL dε ≈
(σr − σ1M )(ε1M
− ε2A )
(1)
where σH and σL are stresses at high (> Af ) and low (<Mf )
temperatures, respectively. σ1M is the martensite detwinning
stress and σr is the recovery stress when heated.
When a bias elastic component is added, part of the work
will be used to overcome the resistance due to the bias force.
According to equation (1), in order to maximize the work
output of SMA, one needs to maximize the combination of
recovery stress and strain. Meanwhile, in order to maximize
the work output of the solid engine, one needs to minimize the
work done against the bias component. In the following, the
work output of the device will be evaluated by using different
bias components.
2.1. Elastic bias component for resetting
A schematic diagram of a complete thermal cycle of the energy
conversion device is illustrated in figure 4. In the initial
condition at T1 (<Mf ), the SMA II is in the martensitic state and
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Figure 4. A schematic diagram illustrates the set-up for energy
conversion using the SMA under tension. From condition 1 to 3, a
complete thermal cycle is performed. The SMA wire in chamber II
was initially in martensitic state (initial condition) and, when heated
to above Af , transformed to austenitic state (condition 2).
Meanwhile, the piston is moved to the right-hand side due to shape
recovery of SMA II. Work is done by SMA II against force F0 and
the resistance of bias component. Upon subsequent cooling to
below Mf (condition 3), the system resets to its initial state.
is deformed under tension to a strain ε1M by the bias component
and is balanced at a stress σ1M . In condition 2, the SMA II is
heated to a temperature T2 (>Af ) and is recovered to a strain
ε2A under the constraint of the bias component and is balanced at
a stress σ2A . ε2A can also be considered to be the residual strain
under a constraint condition. Thus, the recovery strain is given
by εr = ε1M −ε2A . See also figure 2 for the stress–strain relation.
In condition 3, the SMA II is again cooled to T1 and the bias
component deforms it to ε1M and is balanced at a stress σ1M .
F0 is the external force to be overcome in order to produce
work output. This external force is different from design to
design. In order to maximize the work production, one needs to
maximize this force through proper design within the allowable
Work production of shape memory alloy
range of recovery stress and strain. In the present work, we
discuss the simplest case where F0 is assumed to be a constant.
If, however, the force is a function of displacement, one needs
to substitute the function into the deduced formula. In fact,
for any distribution function of the force, the work output can
be calculated by using an average force value multiplying the
displacement. Further, the nature of this force is so designed
that it exists only when there is a movement, i.e., F0 exists
only in condition 2 in which the shape recovery of SMA
II takes place but is negligibly small during resetting of the
system (conditions 1 and 3). In order to generate motion, the
maximum recovery force achievable by SMA II must be higher
than the sum of F0 and bias force. After passing through a
shape recovery process by overcoming the force F0 , the forces
experienced by SMA II at T1 and T2 are F1 and F2 , respectively.
The corresponding stresses experienced by SMA II in the
balanced condition are σ1M and σ2A (σ2A > σ1M ). Under the
application of F0 , the recovery stress needed to overcome the
resistance is much higher than that in a balanced condition, i.e.,
σr > σ2A > σ1M . The stresses experienced by the elastic bias
component are respectively σ1 and σ2 , and the corresponding
strains are ε1 and ε2 .
Based on figure 3, for a device using a linear elastic bias
component, we have the estimated limit of the unit work output:
wout ≈ (σr − σ1M )(ε1M − ε2A ) − 12 (σ2A − σ1M )(ε1M − ε2A ) (2)
or
wout ≈ [σr − 12 (σ1M + σ2A )](ε1M − ε2A ).
(3)
From the above equation, it is clear that, in order to increase
the work output, σ2A should be reduced. Its lowest limit is σ1M
and, in practice, it should always be σ2A > σ1M . However, if
SMA II is trained to possess a stable TWME, F1 and σ1M can
then be reduced to negligibly small within the TWME strain.
In this case,
wout ≈ [σr − 12 σ2A ](ε1M − ε2A ).
(4)
Thus, in order to increase the work output, one can reduce σ2A
as much as possible.
Based on figure 4, it can be envisaged that the actual work
output of the device (Wout ) during one thermal cycle can be
calculated by
(5)
Wout = F0 L 2 .
Assuming F0 is negligibly small during resetting, the work
production is zero during resetting. Thus, the total work output
is
(6)
Wout = F0 L 2 = F0 (ε1M − ε2A )L 0 .
Combining equations (3) and (6) results in
Wout = F0∗ (ε1M − ε2A )L 0 ≈ [σr − 21 (σ1M + σ2A )](ε1M − ε2A )A0 L 0 .
(7)
Thus,
(8)
F0∗ ≈ [σr − 12 (σ1M + σ2A )]A0
where A0 is the initial cross sectional area of SMA wire and
F0∗ is the average force.
The amount of unit work production that an SMA can
deliver is limited by its achievable combination of recovery
stress and strain. Clearly, in order to increase the work output
of the device, one needs to reduce the work compensated by
deforming the bias component. One way is to reduce the force
applied by the bias component as much as possible. Another
way is to train the SMA to possess TWME. Using SMA having
TWME, equation (8) becomes
F0∗ ≈ [σr − 12 σ2A ]A0 .
(9)
2.2. Superelastic SMA for resetting
To further reduce the amount of work due to deforming the
bias component, a superelastic SMA wire can be used as the
bias component. Such superelastic wire should possess a low
hysteresis and a flat stress plateau. Thus, the area below the
AM line of figure 3 can be significantly reduced. The selection
of the superelastic wire should consider at least the following
two points.
(1) The length of stress plateau associated with stress-induced
martensitic transformation should be the same as (or
longer than) that of the TWME strain (or the plateau strain
associated with martensite detwinning if the SMA has only
one-way memory effect). Satisfaction of this condition
results in a constant bias force throughout the complete
shape recovery process.
(2) The diameter of the superelastic wire (or the cross section
area) should be carefully selected so that the bias force is
just enough to bring back the system to its original position
in condition 3, and does not exert a significantbias force
in condition 2.
In such case, equations (8) and (9) still hold for the one-way
memory effect and TWME, respectively. The only difference
now is that the constraint stress in figure 2 is a constant, and
it can be set to as close to σ1M as possible for the one-way
memory effect and as low as possible for TWME.
2.3. Identical SMA for resetting
This design is to use identical SMA wires for both actuation and
resetting as illustrated in figure 5. The advantage of this design
is that the work done against bias component is minimized.
Meanwhile, the work output of the device is doubled. In
figure 5, chamber I and chamber II are identical. During the
operation, two chambers are to have different temperatures,
i.e., when the temperature in chamber I is above Af , the
temperature in chamber II should be below Mf , and vice versa.
In the initial condition, the martensitic SMA I is deformed to
a strain slightly over the stress-plateau and a residual strain ε0M
exists. In chamber II the same SMA wire without pre-strain
is connected. Temperature in both chambers is below Mf at
the starting point. In this case, no stress exists, assuming also
the same nature of the force F0 . When the SMA I is heated
to above Af as illustrated in condition 2, its shape recovery
will lead to the straining of SMA II to ε1M which is slightly
less than ε0M due to deformation of austenitic wire I (ε0A ). In
condition 3, the situation is reversed. SMA I is cooled to
below Mf and SMA II is heated to above Af . In this case,
shape recovery of SMA II leads to the straining of SMA I to
ε1M and is balanced by a stress σ1M . A small residual strain (ε0A )
exists in SMA II due to austenite deformation. By reversing
the temperatures in both chambers (condition 4), the situation
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In the coupled SMA system shown in figure 5, the actual
work produced by a single SMA wire is estimated by
ε2M
Wsma = Wout +
σ1M A0 L 0 dε.
(10)
0
The work output Wout is given by equation (5). Equation (8)
now has the form
F0∗ ≈ [σr − 21 (σ1M + σ1M )]A0 = (σr − σ1M )A0 .
(11)
We can design the coupled SMA system in such a way
that the recovery strain is within the range of the plateau
strain. Based on various experimental observations, we can
reasonably assume that the stress plateau is flat for the selected
SMA wire in martensitic state, thus, the balance stress σ1M is
constant throughout the recovery process, neglecting the minor
complication at the beginning of the deformation. Thus, we
further have the maximum work output of SMA wire in each
chamber
max
≈ (σr − σ1M )ε2M A0 L 0 .
(12)
Wsma
From this simple equation, it is clear that in order to maximize
the work output of SMA, a material approach is to increase
the difference between the recovery stress and the martensite
yield stress as well as to increase the shape recovery strain.
The maximum work output of the coupled system is doubled
max
Wout
≈ 2(σr − σ1M )ε2M A0 L 0 .
(13)
By further using SMA wires with well developed TWME, the
bias stress can be reduced to zero. In this case, the maximum
work output of the coupled SMA system is equal to that of the
maximum work output achievable by SMA wire.
max
max
≈ 2Wsma
= 2σr A0 εr L 0
Wout
Figure 5. A schematic diagram illustrates the coupled SMA system
for work production by using two identical SMA wires under
tension mode. In the initial condition, both SMA wires are in
martensitic state, and one of them was pre-strained. Upon heating,
the pre-strained wire I recovers and leads to the deformation of the
SMA wire in chamber II (condition 2). By reversing the
temperatures in the two chambers, SMA II recovers and deforms
SMA I to the same strain magnitude (condition 3). Further reversing
the temperatures (condition 4), the situation in condition 2 is
repeated. During one complete thermal cycle, SMA wires in both
chambers produce work against the external force F0 .
is reversed and is identical to that of condition 2. After one
complete thermal cycle, SMA wires in both chambers have
produced work against the external force F0 .
556
(14)
where σr is the recovery stress, A0 is the initial cross sectional
area of the SMA wire, L 0 is the initial length and εr is the
recovery strain that should have the same magnitude of ε0M .
Thus, in order to increase the work output of the coupled
SMA device, one needs to maximize the recovery stress and
recoverable strain within the allowable range, and to increase
both the cross-sectional area and the initial length of the
SMA wires. For NiTi SMA, the recovery stress is up to
600 MPa and the achievable TWME strain is up to 5–6%.
Due to the relatively small strain amplitude, we neglect the
small difference between the true strain and engineering strain
and use only the engineering strain in the following analysis.
Increasing the cross sectional area does not necessarily imply
increasing the diameter of the SMA wire. One can simply
increase the number of wires in each chamber so that the total
area of cross sections is increased. By doing so, the operational
speed will not be significantly affected. The maximum F0
achievable is the recovery stress times the total area of the
cross sections of all the SMA wires in one chamber. The work
output per unit volume of SMA is
wsma =
σr A0 εr L 0
(J cm−3 ).
V0
(15)
The work output per unit mass of SMA material is
wsma =
σr A0 εr L 0
σr εr
=
(J g−1 ).
V0 ρ
ρ
(16)
Work production of shape memory alloy
If reasonably taking 500 MPa for recovery stress and 5% for
TWME strain, and the density of NiTi ρ = 6.45 g cm−3 , the
unit work output of NiTi is estimated to be 3.88 J g−1 . In the
ideal case, if we use 1 kg NiTi wire, the total work production
of SMA is
Wsma ≈ 3.88 J g−1 × 1000 g kg−1 = 3.88 kJ cycle−1 kg−1 .
(17)
If the frequency of thermal cycling f = 0.5 Hz, the work
production for 24 h would be
Wsma ≈ wsma × f × t
= 3.88 J g−1 × 0.5 s−1 × 86 400 s day−1
= 167 616 kJ kg−1 day−1 .
(18)
The generated power is
Psma =
Wsma
= 1.94 kW kg−1 .
t
(19)
In order to increase the power, more SMA wires are needed
to build a rather large device. Increasing the frequency by
reducing the wire diameter and enhancing the heat convection
is another option. Doubling the frequency will double the
work output and power. Utilizing solar thermal energy in
tropical areas in combination with low temperature of the
seawater would be an ideal case. Geothermal energy in
combination with cold air in frigid regions would be another
ideal condition. Estimation based on a combination of
recovery stress of 500 MPa and a recovery strain of 5% is
reasonable. These values are already widely reported and
can be achieved conveniently. Progress in the research of
SMAs is likely to achieve a better combination of recovery
stress and strain through further understanding the controlling
factors and through developing new SMAs. A combination
of 600 MPa recovery stress and 6% recovery strain will result
in an estimated unit work output of 5.58 J g−1 . This ‘slight’
improvement in shape recovery characteristic will not only
increase the output power of the SMA, but also increase
the thermal efficiency that will be discussed in the following
sections.
3. Thermal efficiency
In order to use the SMA more energy efficiently, we would
like to heat the material to a temperature just high enough
to generate enough force for work production and to cool
it to a temperature just low enough to be re-deformed
to the required strain amplitude. Obviously, these two
critical temperatures are Af and Mf and, according to the
Clausius–Clapeyron equation, both are functions of applied
stress. One needs to pay attention to the accuracy of these
temperatures determined by using different techniques under
stress-free conditions. The most used technique is DSC
in which the transformation temperatures are estimated by
using a slope-line-extension method based on the experimental
curves. Thus, the transformation temperatures determined by
using DSC do not represent the actual starting and finishing
temperatures. For the energy production, we like to have 100%
martensite/austenite phase at target temperatures. This will
increase the work output in terms of increasing displacement
and force. Fully transformed SMA during operation may also
help to increase the fatigue resistance since the martensite–
austenite interface is likely the weakest place of the materials.
Thus, it is recommended to include a few more degrees to
the experimentally determined transformation temperatures in
both operation and calculation. During heating part of the input
thermal energy is converted into endothermic transformation
latent heat denoted as HM→A , while during cooling part of
the driving force is converted to the exothermic transformation
latent heat, H A→M .
The SMA only produces work through reverse transformation from martensite to austenite. It does not produce work
upon cooling. Thus, input thermal energy (Hin ) is needed only
to heat the SMA to above Af . This thermal energy transforms
into work output (wout ), the work done against bias component (wbias ) and the wasted heat that does not produce any
work (Hw ).
(20)
Hin = wout + wbias + Hw .
In the case of the two-way memory effect, wbias = 0. The
input thermal energy includes only the heat (HMf →Af ) needed
to increase the temperature of the SMA from Mf to Af , and
the transformation enthalpy change of the reverse endothermic
σ
.
transformation HM→A
σ
Hin = HMf →Af + HM→A
.
(21)
For simplicity, we only consider the heat needed by the
SMA wire and do not take into account the heat required to
heat up the chamber. The latter is an engineering problem
and varies from design to design and, in practice, it should
σ
can be
be minimized. For an adiabatic process, HM
f →Af
estimated from the specific heat of the SMA. The reported
values of specific heat of NiTi vary significantly, being
0.46 J g−1 ◦ C−1 [11], 0.2 cal g−1 ◦ C−1 (0.8368 J g−1 ◦ C−1 ,
Shape Memory Applications), 0.49 J g−1 ◦ C−1 (AMT,
Belgium) etc. Since 0.460 J g−1 ◦ C−1 is close to the mostly
reported range from independent sources, it will be used in
the present calculations. The value selected significantly
influences the calculated results. During heating from Mf to
Af , the total heat required is
σ
Hin = c p (Af − Mf ) + HM→A
(J g−1 ).
(22)
According to the Clausius–Clapeyron equation, the stress and
the transformation temperatures are interrelated:
dσ
H
= −ρ
dT
εT0
(23)
where σ is the stress imposed on the transforming constituent
and ε, ρ and H are the amount of strain along the applied
stress direction, the density of the alloy and the transformation
enthalpy change, respectively. T0 (σ ) is the equilibrium
temperature between two phases under stress and can be
approximated by
Af (σ ) + Ms (σ )
.
2
Equation (23) can also be written as
σ − σ0
dσ
H
=
= −ρ
Ms (σ ) − Ms (σ0 )
εT0
dT Ms
T0 (σ ) =
(24)
(25)
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transformation, the latent heat is required from the energy
source. For NiTi, transformation latent heat is also a function
of annealing temperature after cold work. Our research shows
that it reaches the maximum value when annealed at about
σ =0
600 ◦ C, where HM→A
≈ 25 J g−1 . It is about 20 J g−1
when annealed at 300 or 1000 ◦ C after cold work. Combining
equations (16), (28) and (30), we have the expression of thermal
efficiency
−1 σr εr
dσ
wsma
c p Af (0) − Mf (0) +
=
σ
η=
Hin
ρ
dT Af
ε A (0) + M (0) + dσ −1 σ −1
f
s
dσ
dT Af
. (31)
+
dT Af
2ρ
Figure 6. Schematic diagram showing the thermal cycle of one
SMA component in the coupled SMA system shown in figure 5.
Reverse transformation is under constrained stress while the forward
transformation is under a stress-free condition due to the two-way
memory effect.
dσ
where the constant ( dT
) Ms is the slope of the stress–
temperature line.
Its reported values vary from 4 to
20 MPa ◦ C−1 [11]. Rearranging equation (25), we have
−1
dσ
Ms (σ ) − Ms (0)
=
σ
dT Ms
−1
Mf (σ ) − Mf (0)
dσ
=
σ
dT Mf
(26)
−1
As (σ ) − As (0)
dσ
=
σ
dT As
−1
Af (σ ) − Af (0)
dσ
.
=
σ
dT Af
According to figures 5 and 6, the SMA is under stress F0
during reverse transformation, while it is under a stressfree condition during forward transformation. Thus, the
temperature difference between fully martensitic (Mf (0)) and
fully austenitic (Af (σ )) states is expressed as
−1
dσ
Af (σ ) − Mf (0) = Af (0) − Mf (0) +
σ.
(27)
dT Af
Thus,
Hin = c p
Af (0) − Mf (0) +
dσ
dT
−1 σ
Af (0) + Ms (0) +
dσ −1
Af (σ ) + Ms (0)
dT Af
=
2
2
dσ εT0
σ
=−
HM→A
dT ρ
 dσ −1 
(0)
+
M
(0)
+
σ
ε
A
f
s
dT Af
dσ
.

=−
dT Af
2ρ
T0 (σ ) =
(28)
σ
(29)
(30)
The minus sign in the above equation only tells us whether
the latent heat is released or absorbed. In the case of reverse
558
Ms (0) = 40 ◦ C (313 K)
Mf (0) = 30 ◦ C (303 K)
As (0) = 60 ◦ C (333 K)
Af (0) = 70 ◦ C (343 K)
dσ
= 10 MPa ◦ C−1
dT
ρ = 6.45 g cm−3
σr = 500 MPa
εr = 5%
c p = 0.460 J g−1 ◦ C−1 .
The estimated value of the thermal efficiency of NiTi wire
under tension mode is
η=
Af
σ
+ HM→A
(J g−1 )
It needs to be noted that Af (σ ) − Mf (0) represents the
operation temperature range of the solid engine. If the
actual operation temperature range is wider than the range of
Af (σ ) − Mf (0), one should use the actual temperature range
to estimate the thermal efficiency. Both the transformation
hysteresis and the transformation intervals are affected by
various factors including especially alloying content and
internal stress associated with processing routes. We take the
following input values from the published data for an ideal
case.
Af (0) − Mf (0) = 40 ◦ C
wsma
3.88
=
≈ 5.64%.
Hin (J g−1 )
68.76
(32)
This value is much lower than that estimated by Tong and
Wayman [15], Mukherjee [20] and Jardine [23]. The estimated
value is, however, much higher than that the experimental
result of Salzbrenner [21]. The very low efficiency (1.1–1.4%)
reported by Salzbrenner is likely due to the wide operation
temperature range between the low (25 ◦ C) and the high
(190 ◦ C) setting temperatures. According to the condition
given in Salzbrenner’s report (∼685 MPa recovery stress and
∼1.5% recovery strain), the thermal efficiency estimated by
using equation (31) is 1.36%. If taking a value of 25 J g−1 for
latent heat, the estimated thermal efficiency would be 1.2%.
Work production of shape memory alloy
Both values are within the range of the measured value (1.1–
1.4%) reported by Salzbrenner [21]. This shows that, to
some extent, equation (31) gives a reasonable estimation of
the thermal efficiency. Nevertheless, factors that affect the
estimation of thermal efficiency will be further discussed in
the next section. Taking a recovery stress of 600 MPa and
a recovery strain of 6%, the thermal efficiency estimated by
using equation (31) would be 7%.
4. Discussion of the estimated thermal efficiency
In the estimation of thermal efficiency in both the present work
and the works previously reported, one uncertainty is the effect
of stress on the transformation enthalpy change. Wollants et al
[17–19] suggested that the transformation enthalpy change is a
function of stress and it increases with increasing stress. Such
a suggestion was based on the assumption that the entropy
change during phase transformation is independent of the
applied stress. This assumption, however, needs to be further
evaluated. Our recent understanding shows that the martensite
variant formed under bias stress is aligned to the stress
direction in a more orderly way than that formed under stressfree conditions. Thus, the entropy of such ‘stress-assisted’
martensite could be smaller than that of self-accommodated
martensite. This may further result in the decrease of the
entropy difference (S) between martensite and austenite.
When the gradient of the stress–temperature line is fixed (being
typically 7–10 MPa ◦ C−1 ), whether H increases or decreases
under stress will depend on the value of S, that needs to be
carefully examined.
On the other hand, Salzbrenner and Cohen [30] have
examined the enthalpy change of thermoelastic martensitic
transformation by using CuAlNi single crystals having either
a single transformation interface or multiple transformation
interfaces. The results show that the observed latent heat
of samples having a single transformation interface is higher
than those having multiple interfaces. They further proposed
that part of the chemical driving force was stored in the
multi-interfaced samples in the form of elastic strain energy
due to microstructural constraint. Thus, the experimentally
determined transformation enthalpy change decreases under
constraint. The stored elastic energy will likely promote the
reverse transformation, thus, less chemical driving force is
needed. In this context, to calculate the thermal efficiency, the
stored elastic energy should be excluded from the chemical
transformation enthalpy change. Experiments of Salzbrenner
and Cohen [30] have provided a useful indication of the nature
of the observed latent heat, while further research is needed to
accurately determine the relation between the observed latent
heat and the applied stress. During such research, one needs
to pay special attention to whether the SMA is under stresscontrolled or strain-controlled mode, since this will affect the
microstructural mechanism which in turn alters the origin of
the observed latent heat change.
The decrease in latent heat under constraint was also
observed recently [31] by using pre-strained SMA wire
embedded in epoxy. This is a partially strain-controlled mode
and the authors attributed the decrease in latent heat to the
incomplete phase transformation. As the sample has also had
a shape change upon reverse transformation, the mechanism
proposed by Salzbrenner and Cohen [30] may have been
partially operative. Nevertheless, the effect of constrained
stress on the transformation latent heat is not concretely
concluded. Further investigation from both theoretical and
experimental approaches is needed.
Recent results [32, 33] have also shown that, after TWME
training, the observed latent heat decreases. This result might
be explained by the mechanism proposed by Salzbrenner and
Cohen [30], i.e., part of the chemical transformation enthalpy
was transformed into elastic energy as a result of training.
Such elastic strain energy results in the preferential growth
of martensite variants responsible for the observed TWME.
The stored elastic energy will be released during reverse
transformation, thus less chemical driving force is needed.
This is responsible for the decrease of the observed latent heat.
The decrease in the latent heat after TWME training was also
reported independently by another group [34]. In some cases,
the decrease was very significant; the latent heat peak nearly
disappeared after TWME training. Taking these observations
into account, for SMA possessing well developed TWME,
equation (30) is no longer valid; the actual thermal efficiency
can be much higher than that estimated by using equation (31).
In this case, equation (31) can be further written into a more
general form
σr εr
wsma
ρ
=
dσ −1 σ
Hin
c p Af (0) − Mf (0) + dT
σ + HTWME
Af
(33)
σ
is the transformation latent heat of SMA
where HTWME
possessing well developed TWME. According to previous
σ
is influenced by both internally
discussion, HTWME
developed stress and externally applied stress. Taking the
value of 17.86 J g−1 reported by Da Silva [33] for NiTi with
TWME, the estimated efficiency would be increased from
5.64% (equation (31)) to 6.55% (equation (33)). Considering
the result of Miller and Lagoudas [34] and taking an estimated
value of 10 J g−1 for the latent heat after TWME training,
the resulted thermal efficiency will be further increased to
σ
7.55%. Clearly, selection of the value of HTWME
significantly
affects the estimated thermal efficiency. By further reducing
the transformation temperature range (Af (0) − Mf (0)) to
20 ◦ C while other conditions remain unchanged, the thermal
efficiency could be increased to as high as 9.2%. If further
increasing the recovery stress to 600 MPa and recovery strain
to 6%, the estimated thermal efficiency would be 10% for a
transformation range of 40 ◦ C and 11.93% for a transformation
range of 20 ◦ C.
Reducing transformation hysteresis by adding Cu is often
accompanied by the decrease in recovery stress. Edwards
and Perkins [35] have reported that the recovery stress (σr )
is closely related to the yield stress of austenite (the stress for
inducing martensite, σ A→M ).
η=
σr |ε,T ≈ 0.9σ A→M |ε,T .
(34)
According to this formula, one can increase the recovery stress
by reducing the grain size, forming precipitates, adding solid
solution etc. However, the effect on the shape recovery strain
should be investigated as well.
Table 1 lists the calculated thermal efficiencies of NiTi
under several selected conditions including the effect of shape
559
Y Liu
Table 1. Calculated thermal efficiencies of NiTi with TWME for
several assumed conditions.
Recovery
stress
(MPa)
Recovery
strain
(%)
Af − Mf
(◦ C)
H M→ A
(J g−1 )
Thermal
efficiency
(%)
300
300
300
300
500
500
500
500
600
600
600
600
685
685
5
5
5
5
5
5
5
5
6
6
6
6
1.5
1.5
40
40
40
20
40
40
40
20
40
40
40
20
165
165
Equation (30)
17.86
10
10
Equation (30)
17.86
10
10
Equation (30)
17.86
10
10
Equation (30)
25
3.96
4.65
5.51
7.05
5.64
6.55
7.55
9.2
7.04
8.78
9.96
11.93
1.36
1.2
recovery property, latent heat and transformation temperatures.
In the optimal condition, the thermal efficiency is higher than
10%. Certainly, reaching such a high value in engineering
practice is technically challenging. NiTi-based SMAs are
excellent materials but still have their limits. Developing
new SMAs with low specific heat and better shape recovery
property is very meaningful.
We need to bear in mind that, during the calculation, the
thermal energy needed to heat the chamber was not considered.
Thus, the actual energy conversion efficiency of the device is
lower than the estimated value of thermal efficiency. One needs
to pay great attention to the design of the device for higher
energy conversion rate. Nevertheless, since the solar thermal
energy is so abundant in nature, it is likely a feasible way to
convert solar heat to mechanical or even electrical energy by
using SMAs. Collection and concentration of the solar energy
is needed in order to heat the SMA more effectively. The
most attractive aspects of such devices are their cleanliness,
simplicity in design and low production and maintenance cost.
For a solid engine utilizing the temperature difference between
that below the sea surface and that above the seawater, once
it begins to operate, in principle, it may be like a ‘perpetual
machine’ as long as there is a sun in the sky and as long as there
is water in the sea. Another important aspect for successful
application of a solid engine is the stability of SMAs especially
the stability of the shape recovery stress and strain, which is
under extensive investigation by various groups.
A further interesting consideration is the utilization
of the latent heat released during the forward martensitic
transformation provided that such latent heat is a significant
amount. Since the work production practice is a continuous
operation involving a large number of thermal cycles rather
than a single thermal cycle, if we are able to utilize the
exothermic latent heat through proper design, we can further
increase the thermal efficiency. This can be understood from
the formula below.
σ
σ
Hin∗ = c p (Af − Mf ) + HM→A
− H A→M
(J g−1 ).
(1) Use SMA in tension mode. This will fully utilize the
capacity of SMA through a uniform distribution of the
deformation and recovery strain. For especially textured
SMAs (wires, plates, sheets, rods, thin foils), tension
deformation leads to higher shape memory strain than that
of compression and torsion. In addition, it also has lower
martensite detwinning stress, thus less energy dissipation
upon martensite deformation.
(2) Use textured SMAs rather than non-textured since textured
SMAs possess an optimized shape recovery strain along
specific directions. If a textured SMA sheet or thin
foil is used, tension along the rolling direction provides
much better performance than tension along the transverse
direction. This is due to the deformation and shape
recovery anisotropy due to texture distribution, see for
example [36–38].
(3) Train the SMA to have TWME so that the work done on
deforming the bias component is minimized. A coupled
SMA system provides an optimal work output and thermal
efficiency.
(4) Reduce the endothermic transformation latent heat
through TWME training and utilize the exothermic latent
heat if it is significant.
(5) Carefully select the operation temperature range. Upon
cooling the temperature should be just low enough to
form 100% martensite, and it is just high enough to form
100% austenite upon heating (with slight offset). This
will increase the work output and thermal efficiency and
likely increase the service life of SMAs as well.
(35)
Thus, for continuous thermal cycling, the thermal efficiency
can be written as
wsma
(36)
η=
σ
σ
c p (Af (σ ) − Mf (σ )) + HM→A
− µH A→M
560
where µ is a coefficient representing the utilization rate of the
exothermic latent heat. It is related to design. When µ =
σ
0, H A→M
is not utilized, and equation (36) becomes (31)
σ
is fully
(for design shown in figure 5). When µ = 1, H A→M
utilized. Utilization of latent heat is technically challenging.
It is considerable only when the amount of latent heat is
significant. If one is able to reduce the amount of latent heat
through various means, e.g. TWME training, utilization of the
exothermic latent heat is less important.
As long as high thermal efficiency is required, designs
using SMA helical springs as actuation media should be
avoided. The reason is that, in the helical spring, the SMA wire
is under mainly shear deformation. Both recovery stress and
recovery strain are not evenly distributed across the diameter of
the wire and their average values are rather small as compared
to the same SMA under pure tension. Thus, the unit work
output of SMA is low and much more material is needed to
achieve the same amount of work production as that of SMA
under tension. Although an SMA helical spring may have
longer operation life due to less deformation amplitude, it is
not practical for real work productions due to the very low
thermal efficiency.
In summary, in order to use SMA effectively, the following
considerations are recommended.
5. Conclusions
The present research evaluates the work production of solid
engine and the thermal efficiency of shape memory alloy.
SMA under tension mode is recommended and evaluated.
Work production of shape memory alloy
Reducing the energy dissipation due to deformation of the bias
component through proper design can effectively increase the
work output of the solid engine. A coupled SMA system using
identical SMA wires having TWME for both actuation and
resetting is recommended for achieving higher work output
and thermal efficiency.
Analysis shows that the thermal efficiency is strongly
related to the shape recovery property of SMA; it increases with
increasing shape recovery stress and strain. The calculated
thermal efficiency is in good agreement with the reported
experimental data. When estimating the thermal efficiency,
attention should be paid to the difference between chemical
transformation enthalpy change and the measured latent heat
which is lowered after TWME training.
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