Since the phase transformations can be activated un-der very different conditions to obtain different effects, to have a picture of SMMs behavior, it is necessary to see how stress and t
Trang 2SHAPE MEMORY ALLOYS, TYPES AND FUNCTIONALITIES 961
important conclusion is that for most practical
constrain-ing conditions, only a small fraction of martensite is
ac-tually transforming during the constrained heating and
cooling Therefore, the observed hysteresis is much smaller
than the overall hysteresis, typically below 10 K for Ni–Ti,
below 5K for Ni–Ti–Cu, and below 2K for the R-phase
transformation (99,114)
As stated before, it is often assumed that stress changes
linearly as temperature changes Moreover, the stress rate
dσr/dT is often considered a material constant directly
de-rived from a Clausius–Clapeyron equation Such
descrip-tions should be considered very elementary simplificadescrip-tions
The results in Fig 5 show clearly that the stress rate
de-creases during constrained heating It has been also found
that the stress is affected by many other parameters,
in-cluding the thermomechanical history and the prestrain
(69,86)
Depending on the magnitude of the prestrain, either
a plastic upper limit or an elastic upper limit to σr
ex-ists At lower prestrains, the stress increases during
heat-ing until the reverse transformation is completed The
upper stress limit in this case is given by the strain
di-vided by Young’s modulus When the prestrain is
suffi-ciently high, the stress increases during heating until
plas-tic yield occurs at a temperature Md So, the upper stress
limit in this case is the plastic yield stressσy Evidently, in
cyclic actuation, the maximum temperature should be kept
below Md
In all cases discussed, a constraint prevents the SMA
el-ement from returning to the hot shape when heated Thus,
a more specific name would be “hot shape” recovery stress
It has been found in trained Cu-based SMA-elements that
stresses can also be generated when the TWME is
im-peded during cooling (69,83) Because the constraint in
this case prevents the sample from returning to the cold
shape when cooled, the generated stress was called “cold
shape” recovery stress to contrast with “hot shape”
recov-ery stress Practical applications have not been reported so
far
Quantitative comprehension of recovery stress
genera-tion presented in the literature is far below the
comprehen-sion of the other functional properties of shape-memory
al-loys Therefore, recovery stress generation was discussed
a bit more extensively than other functional properties
Considering the substantial research efforts in developing
hybrid composites that have embedded shape-memory
ele-ments, substantial progress in quantitatively
understand-ing recovery stress generation can be expected in the near
future
Work Output
One- and two-way memory effects can be used for free
re-covery applications in which the single function of the SMA
element is to cause motions without any biasing stress
Under constant strains, shape-memory elements can
generate substantial recovery stresses Between these two
extremes of free recovery and completely constrained
re-covery, shape-memory components can yield a wide
var-iety of combinations of strains and stresses As shown in
F T
Figure 6 The work output The sample is deformed at a
tem-perature at a below Mf(A →B), followed by unloading (B→C) and
loading again using a bias weight W (C→D) Shape recovery
oc-curs under an opposing force W during heating to a temperature
above Af (D →E) So work is done [from (69)].
Fig 6, a shape-memory element can be deformed by lowforce in the martensitic condition or during the forwardtransformation and can exert a substantially higher force
as it reverts to the hot shape when heated So, work up
to 5 J/g is done during heating This concept can be used
in thermal actuators in which the SMA element is vated by an increase in the environmental temperature,
acti-or in electrical actuatacti-ors in which the SMA element is ingeneral activated by direct Joule heating The work needed
to deform the SMA element is much lower than the workthat can be obtained during heating This has been the ba-sis of many prototypes of heat engines that convert heatinto useful work [see (117)]
SMA actuators offer distinct advantages comparedother types of actuators (118) The main advantage is that
by far SMA actuators offer the highest work and to-weight ratios of all available actuating technologies atlow levels of weight (119) These high-work and high-powerdensities enable a whole class of applications (e.g., in thefield of micro actuation) that are impossible to realize byusing other actuating technologies (120–123) SMA actua-tors can be reduced mostly to a single SMA element withoutauxiliary parts, resulting in simple compact and reliabledevices
power-Several important drawbacks that limit the use of SMAactuators to specific niches should also be considered Theconversion of heat into mechanical energy via SMA actua-tors was studied extensively 15 to 25 years ago Simplethermodynamic calculations showed that the maximumtheoretical efficiency of an SMA actuator is less than 10%(124) In practice, the conversion of heat into mechanicalwork is less efficient, and the result is that real efficiency iseven one order of magnitude smaller than the theoreticalvalue Another drawback is that the SMA actuator has to
be heated and cooled The low cooling rate, especially limitsthe use of SMA actuators to relatively low-frequency ap-plications It was discussed before that stresses in trainedCu-based SMA elements can also be generated when the
Trang 3TWME is impeded during cooling Similarly, it has been
shown that these trained SMA elements can do a small
amount of work during cooling (82,83)
High Damping Capacity
SMA elements have high damping capacity in the
austenitic and martensitic conditions Shape-memory
al-loys show strong amplitude-dependent internal friction in
the martensitic condition For impact loads, the specific
damping capacity can be as high as 90% Starting from
the austenitic condition, energy is dissipated during
su-perelastic cycling as a result of stress hysteresis between
superelastic loading and unloading, as explained before A
detailed analysis can be found in (125,126)
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SHAPE-MEMORY MATERIALS, MODELING
DAVIDEBERNARDINI Universit `a di Roma “La Sapienza”
Rome, Italy
THOMASJ PENCE Michigan State University East Lansing, MI
INTRODUCTION
An understanding of shape-memory behavior requiresknowledge of various physical processes that operate ondifferent length scales The crystallographic shifts thatare responsible for shape-memory behavior take place inunit cells of atomic dimension Here, however, these mi-croscopic length scales are not the focus, rather, this arti-cle considers the macroscopic scale modeling that allowsfor the engineering assessment of thermomechanical re-sponse (stress–strain–temperature) and energy balances(including damping) for devices such as connectors, actua-tors, vibration absorbers, and biomedical stents The macroscale is useful for primary design evaluation such as pre-dicting triggering forces and determining range of motion.Since the shape-memory material is typically incorporatedinto a larger engineered device or structure, there is also aneed for detailed computational simulation of the system
as a whole
BASIC MATERIAL BEHAVIOR AND MODELING ISSUES
The term shape-memory material (SMM) is meant to
en-compass a wide class of metallic alloys with the commonfeature that they exhibit, at the macroscopic scale, somepeculiar and useful functional properties such as pseu-doelasticity and shape memory Nickel-titanium (NiTi) isperhaps the best known and most widely used such ma-terial SMM functional properties derive from transforma-
tions between two different solid phases: austenite (A) and martensite (M) Aspects of the A↔ M phase transforma-
tion are essential for model development and tation In addition, certain intermediate phases may alsooccur, but these are neglected here because their effect onthe macroscopic response is small in comparison
implemen-The A↔ M transformation can be induced by a
vari-ety of energy inputs (mechanical, thermal, magnetic, trasonic, etc.), and it is influenced by grain boundaries,dislocations, inclusions, and other material defects Thisarticle considers the standard thermomechanical setting,
ul-namely the A↔ M transformation that is induced by
tem-perature T and stress σ In general, austenite is favored at
high temperatures and low stress, whereas martensite isfavored at low temperatures and high stress We will use a
boldface σ (and ε) to denote a general stress and strain
ten-sors with componentsσ i j(andε i j) Models involving only
a single stress componentσ will be developed more than
those involving the tensor σ.
Austenite is of higher crystallographic symmetry and socan transform into one or more martensite variants thatdiffer mainly by their orientation relation to the austenite
Trang 6SHAPE-MEMORY MATERIALS, MODELING 965
Figure 1 A phase diagram defines the zones of the (σ, T )-plane
where the various phase transformations can occur Each curve
represents the (σ, T )-points at which a transformation can either
be activated or else completed Hence there are two curves (start
and finish) for each transformation The figure shows a sketch of
a phase diagram that can arise when the SMM is modeled as a
mixture of austenite A and two martensite variants M+, M− Such
a diagram can be viewed as an unfolding of a conventional phase
diagram triple point so as to include the effects of phase mixing
and transformational hysteresis.
parent By contrast, all martensite variants tend to
trans-form into a single common austenite crystal structure
The transformation from the A structure to that of a
par-ticular M variant is characterized by a crystallographic
transformation strain Typically, an A material region
transforms into a martensitic microstructure with several
variants that combine in complicated twin arrangements
and plate morphologies These microstructures provide a
local transformation strain γ∗ This in turn gives a
macro-scopic transformation strain ε∗ at the engineering scale
This ε∗gives a potentially large strain in stress-induced
A→ M transformations It is negligible in cooling-induced
A→ M transformations because the resulting
microstruc-tures involve so-called self-accommodated martensite with
local strains γ∗that cancel each other
Since the phase transformations can be activated
un-der very different conditions to obtain different effects, to
have a picture of SMMs behavior, it is necessary to see
how stress and temperature differ with respect to A↔ M
transformation Figure 1 shows a stress and temperature
phase diagram of a single austenite phase A and two
fam-ilies of martensite variants M+ and M− The curves in
this diagram show the relation between stress and
tem-perature levels at which various phase transformations
begin and end This partitions the (σ, T )-plane into three
single phase regions, three double phase regions, and a
triple phase region
Purely Thermal Transformation
In the absence of stress, austenite is stable at high
tem-peratures, and martensite is stable at low temperatures
Stress-free cooling of austenite gives A → M conversion
beginning and concluding at temperatures M s and M f,
re-spectively (M f < M s) The resulting microstructure is an
unbiased martensite with a fine-scale arrangement of
vari-ant twins with opposing local transformation strainsγ∗
This produces a negligible engineering scale tion strainε∗= 0 Similarly, a temperature increase causes
transforma-M → A conversion, the start and finish temperatures
being A s and A f , with A f > A s The transformation
tem-peratures M f , M s , A s , and A fare the basic material eters for purely thermal transformations They are highlysensitive to the alloy concentrations and to the granularand defect structure as determined by heat treatment andcold work Once the material is ready for service, thesetemperatures are easily determined by various means in-cluding resistivity and calorimetry testing
param-Low-Temperature Martensite Reorientation
Austenite is not present at temperatures below M f.Nonzero stress at these low temperatures causes certainmartensite variants to be relatively more favored Thisfavoritism correlates with the value of the transforma-
tion work σ · γ∗= σ i j γ∗
i j In the important special case ofuniaxial tension/compression, the variants favored in ten-sion are those for whichγ∗projects onto the tensile axis as
a positive quantity It is convenient to group all of the
vari-ants favored in tension into an M+ variant family, and all variants favored in compression into an M− variant fam- ily Unbiassed martensite involves a mixture of both M+
and M− Sufficiently high tensile loading at temperatures
below M f causes movement of the internal boundariesseparating the martensite plates, which can be viewed
as a conversion from the M− family into the M+ family This M− → M+ transformation yields biased martensite,
namelyε∗= 0 Unloading does not cause the reverse
trans-formation (M+ → M−) so long as the load does not
be-come compressive Hence the transformation strain ε∗ islike a conventional plastic strain upon unloading (Fig 2)
The M− ↔ M+ transformation is referred to as
reorien-tation As a result, a plasticlike reorientation plateau is
observed on the isothermal stress–strain curve, with thetensile reorientation beginning and concluding at stresses
σ+
s andσ+
f (σ+
f > σ+
s > 0) These reorientation stresses are
relatively insensitive to temperature changes (there may
Low temperature reorientation
Figure 2 Sketch of the macroscopic effects of the various phase
transformations in the stress–strain–temperature (σ -ε-T) space
(left) Effect of the loading rate on the pseudoelastic behavior:
faster loads give rise to greater hardening and temperature
vari-ations (right, where t denotes time).
Trang 7decrease) Similar reorientation processes occur in
com-pressive loading due to the M+ → M− transformation,
with the start and finish of the reorientation plateau
Austenite can be present at temperatures above M f Such
austenite is converted to biased martensite upon
applica-tion of sufficiently high stress, either tensile (activating
A → M+) or compressive (activating A → M−).
Consider the tensile case A stress increase gives the
A→ M+ transformation, which again generates a plateau
on the stress–strain diagram This is referred to as a
pseu-doelastic plateau so as to distinguish it from the
reorien-tation plateau observed at the lower temperatures For
T > A f, the start and finish stresses for pseudoelasticity
are greater than the M− → M+ reorientation stresses
σ+
s andσ+
f They are also highly temperature sensitive, creasing with temperature at an approximately constant
in-rate However, if the temperature is close to Mf, then
little distinction can be made between pseudoelasticity and
reorientation because the loading plateau stresses match
the reorientation valuesσ+
s andσ+
f Unloading activates
the M+ → A transformation if T > As, resulting in an
un-loading plateau below the A→ M+ loading plateau The
unloading plateau rejoins the loading curve if M+ → A
goes to completion (T > A f), and so defines a hysteresis
loop (Fig 2) At temperatures A s < T < A f, the M+ → A
unloading conversion does not go to completion and the
un-loading plateau intersects the strain axis before reaching
the origin of the stress-strain diagram If T < A s, then the
M+ → A transformation is not even activated upon
unload-ing Thus, in all cases where T < A f, there is some residual
strain due to the presence of M+ martensite when
unload-ing is complete
Shape-Memory Effect
At all temperatures, where T < A f, after sufficiently high
load causing either A → M+ or M− → M+ transformation,
residual strain is present after unloading due to the
pres-ence of biased martensite Unlike conventional plastic flow
in metals (generated by dislocations) the SMM plasticlike
residual strain is recovered by heating above A f, because
this converts martensite to austenite Since this
austen-ite converts to unbiased martensausten-ite upon any later
stress-free cooling, the residual plastic strain does not return
(unless there is further loading/unloading) This
heat-ing/cooling elimination of an apparently “plastic” strain
due to previous loading/unloading is the shape-memory
effect.
While the preceding discussion has covered the basic
aspects of the material behavior that macroscopic models
should reproduce, SMM is often employed in situations
in-volving further effects that are important objectives for
useful modeling The most important of these are briefly
described next
Response to Complex Loading Paths
At constant temperature, loading reversals that interrupt
A → M and M → A before completion lead to internal
subloops within the major stress–strain hysteresis loopassociated with complete transformation Load paths in-
volving simulataneous change in T and σ generally
aug-ment or diminish transformation that would occur undereither T orσ alone This is critical for modeling the rate
effect that is described next
Rate Dependency due to Transformational Heating
If mechanical loads producing phase transformation arenot applied in a quasi-static way, then temperature varia-tion occurs in the sample and a rate-dependent response isobserved This is due to the exothermic and endothermic
nature of the A → M and M → A transformations,
re-spectively During A→ M the material self-heats and the
temperature rise works against the transformation
(con-versely during M→ A the sample self-cools) This might
involve a number of consequences: different onset stressesfor transformation plateaus, plateau steepening, and vari-ation in the shape of internal loops (Fig 2) The extent
of this effect is governed by the heat exchange with theenvironment: high rates of loading can cause significantdeparture from isothermal behavior High rates of loadingcan occur in both actuator and damping shape-memory de-vices
Tension/Compression Asymmetries
In uniaxial loading, significant differences in the stress–strain behavior have been observed between tension andcompression This is due to the different microstructures
that the formed In particular, the behavior of the M+
vari-ant family in tension is not the symmetric image of the
behavior of the M− variant family in compression Such
a phenomenon is modeled by a phase diagram that is symmetric with respects toσ
un-Three-Dimensional States
In the three-dimensional case involving tensor σ rather
than scalarσ, the experimental behavior is less well
un-derstood, and complex multivariant structures are to beexpected in most cases A key issue in three-dimensionalmodeling is the proper constitutive description of an appro-priate local transformation strain that transcends the lack
of information about the actual multivariant ture In view of the correlation of variant favoritism with
microstruc-the transformation work σ · γ∗= σ i j γ∗
i j, some sort of iality relations between stress and transformation strainare conjectured at modeling scales appropriate to a multi-variant microstructure These difficulties are compoundedunder nonproportional loading, since the transformationstrain then evolves as a consequence of both pseudoelastic
coax-A ↔ M processes and M ↔ M reorientation of existing
variants
STATE OF THE ART AND HISTORICAL DEVELOPMENTS
Modeling of the macroscopic behavior of SMM has beenthe subject of much activity since the beginning of the1980s, attracting the interest of engineers, applied mathe-maticians, and materials scientists This section surveysthe state of the art on the basis of the huge literature
Trang 8SHAPE-MEMORY MATERIALS, MODELING 967
available on the subject The survey is restricted to models
that directly relate to macroscopic modeling and does not
delve into the voluminous literature on metal physics and
purely microstructural development, even though much of
this literature provides enormous insight While an effort
has been made to be rather comprehensive, the survey is
still far from complete We have attempted to give
care-ful bibliographic references by selecting one
representa-tive paper for each approach Each of these models then
typically gives rise to refinements, generalizations,
verifi-cation studies, and implementation strategies For the sake
of conciseness, complete bibliographic references cannot be
given for all of these modeling extensions
The discussion of the previous section makes clear that
the behavior of SMM observable at the macroscopic scale
is the effect of several complex microstructural
pheno-mena This section is organized, as in the list below, with
respect to contributions that include an explicit model
for such microstructural phenomena and others that do
not Although certain models that will be discussed can
be viewed as spanning more than one such approach, the
classification given below aids in organizing the numerous
modeling approaches that have been proposed
Approaches modeling one or
more microstructural
phenomena
Lattice cell mechanics Interface nucleation and propagation
Approaches Modeling One or More
Microstructural Phenomena
Models included in this group are grounded in theories that
analyze the material at a scale in which the multiphase
na-ture of the material is rendered explicit and one or more
effects of phase transformations can be described by some
direct microstructural model Macroscopic behavior is then
recovered by some kind of averaging procedure In a
con-tinuum setting, this implies that each point belongs to one
phase and the first spatial derivatives of the displacement
and temperature fields can be discontinuous
Lattice Cell Mechanics In this approach, the
macro-scopic response of the material is determined by
study-ing the behavior of a collection of lattice cells that can
be in a particular phase or phase variant In response to
loads and temperature changes at the system boundary,
cell transitions between different phases can take place
Two approaches for the transition kinetics can be broadly
identified: statistical mechanics and strain energy
minimi-zation.
The statistical mechanics approach has roots in Muller
and Wilmanski (1) and has been further developed by
Achenbach (2) The cellular array is grouped as a stack of
layered aggregates of cells that can be found in one of the
Figure 3 The three-well energy function of the
Muller-Achenbach model without (left) or with (right) mechanical load.
Each minimum corresponds to a phase with different structure
(austenite A and two martensite variants M+, M− ) The ences among the ordinates of the various minima represent the en- ergy barriers that cells have to overcome to undergo phase trans- formations The right graph shows the effect of a mechanical load
differ-P that lowers the right minumum and causes the M+ phase to be the energetical favorite.
three phases: austenite A and two martensite variants
M+, M−; each characterized by a different cell length Cellsare in random thermal motion, and thermal fluctuationspermit them to transform from one phase to another Suchtransitions lead to variations in the stack compositionthat are monitored by the phase fractions ξA , ξ+, and ξ− A three-well potential energy φ whose minima are
each associated with one phase is the basic tive ingredient from which all material parameters arederived by statistical arguments (Fig 3) Macroscopicstrain and temperature are obtained respectively fromthe normalized length of the whole layer aggregate andfrom a measure of the thermal fluctuation The phasefraction evolution is governed by a system of ordinarydifferential equations expressing the transition ratebalance between layers on the basis of the probability ofovercoming the energy barriers that separate the minima
constitu-of φ This finally provides a complete model for uniaxial
stress pseudoelasticity and reorientation
A second approach for the description of the transitionsbetween lattice cells is based on strain energy minimiza-tion A model developed by Morris and his collaborators(3) involves a multidimensional lattice of cells with cor-responding multidimensional transformation strains foreach cell The total free energy is the sum of a temperature-dependent chemical free energy and a strain-dependentelastic strain energy The strain energy contribution ishighly dependent on cell location and choice of transforma-tion variant, due to the constraint of surrounding cells Thecomputation is based on isotropic elasticity, and the under-lying mathematical technique developed by Khachaturyanrequires equality of elastic constants in all phases For agiven change in temperature, the overall transformationprocess is simulated by a stepwise energy minimization
At each step, the particular cell that transforms is selected
as the one that most lowers the energy, and such mations continue so long as the overall energy is lowered.Complex microstructures and internal stress states occur.These complex states are the main focus of such modeling,
transfor-as opposed to providing a macroscopic model for overallstress–strain–temperature behavior
Trang 9Interface Nucleation and Propagation Large
deforma-tion theories of continuum mechanics have also been used
to treat martensitic phase transformation These
treat-ments stem from analysis by Ericksen of a uniaxial
con-tinuum with a nonconvex elastic strain energy density
(4) These energies give rise to nonmonotone stress–strain
curves Fundamental thermodynamic arguments harking
back to Gibbs and Maxwell indicate that strains on the
de-scending branches of these stress–strain curves are
unsta-ble in that they cannot be part of a deformation field that
minimizes energy Energy minimization naturally gives
rise to distinct material regions, each involving
continu-ous strain, that are separated from each other by
inter-faces across which the strain is discontinuous so as to
avoid unstable branches The connection to stress-induced
phase transformation follows by placing the distinct stable
branches of the stress–strain curve into correspondence
with distinct material phases Avoidance of the unstable
branches is then formally similar to spinodal
decomposi-tion One outgrowth of this work has focused on putting the
crystallographic theory of martensite on a rigorous
mathe-matical foundation so as to predict microstructure without
invoking the approximations inherent in the linear theory
of elasticity (5–7)
The fundamental nature of much of the work cited
im-mediately above renders it outside the scope of this
model-ing survey However, under suitable interpretation, certain
treatments of this type do provide a macroscopic model for
the thermomechanical behavior of shape-memory
materi-als In particular, SMM uniaxial response follows from a
thermoelastic free-energy density with either two or three
minima that each define a distinct phase Boundary value
problems give solutions in which the continuum is
subdi-vided via phase boundary interfaces into different phase
regions Phase transformation proceeds from the
nucle-ation of new interfaces or from the propagnucle-ation of the
ex-isting ones (8) The jump in the Gibbs free energy across
the interface follows from the Eshelby energy-momentum
tensor in the form of a generally nonzero driving traction
Phase boundary movement gives either energy dissipation
or energy accumulation as determined by the direction of
interface motion Standard boundary value problems do
not have a unique solution when phase boundaries are
present unless the constitutive theory is augmented with
both interface nucleation criteria and interface kinetic
mo-tion criteria These typically depend on the driving
trac-tion, and they can be formulated so that the overall load–
displacement–temperature relation reproduces
pseudoe-lastic and reorientation behavior Kinetic criteria can be
derived, for example, similarly to that of Achenbach and
Muller, so as to involve a probability of overcoming the
en-ergy barriers between phases on the basis of thermal
fluc-tuation Quasi-static and fully dynamic treatments follow
for isothermal, adiabatic, and heat conducting cases The
kinetic motion criteria can be extracted from more refined
theories in which the phase boundaries are regarded as
transition zones exhibiting additional physical effects (9)
Micromechanics In the models of this group each
mac-roscopic point is put in correspondence with a
represen-tative volume element (RVE) of a multiphase material in
P
RVE ΩMacroscopic scale
Microscopic scale
A
M
Figure 4 Macroscopic versus microscopic scale modeling Each
macroscopic point P corresponds to a microscopic region
the multiphase nature of the alloy (A, M) can be appreciated
ex-plicitly At the macroscopic scale the features of described by the internal variablesα.
which some regions are subjected to local transformationstrains due to the different crystal structure of the phases(Fig 4)
Under proper boundary conditions, a boundary valueproblem on the RVE that models the effects of the phasetransformations is obtained Local quantities are volumeaveraged over the RVE, giving macroscopic quantities thatgenerally retain a dependence on the microstructural fea-tures through some overall descriptorα (usually the phase
fractionsξ) The resulting equations for the macroscopic
behavior fit into the framework of internal variable els, as described later in this section Constitutive ingre-dients are a macroscopic free-energy function and a set
mod-of kinetic rate equations for the microstructural tors α The free energy provides, via partial differentia-
descrip-tion with respect to stress (or strain), equadescrip-tions for strain(or stress) while derivatives with respect to α give gen-
eralized forces that drive the phase transformations Thefree-energy functions are structured as the sum of an elas-tic strain energy and a chemical a free energy While thechemical contribution is specified mainly by standard ther-modynamic expressions, the elastic term varies consider-ably among such models as it follows from different mi-cromechanic assumptions on the accommodation processdue to phase interaction (10) Kinetic equations are de-rived from phase transformation criteria stating that thetransformations occur when the generalized driving forcesmeet experimentally determined threshold values.Patoor, Eberhardt, and Berveiller initiated such anapproach in 1987 by combining ideas from transforma-tion plasticity, continuum micromechanics, and crystallo-graphic theories of martensitic transformation (11) In itsrecent formulation, their model considers, at the single-crystal level, a linear elastic RVE consisting of an austen-ite matrix with inclusions of 24 martensite variants, eachexhibiting a local transformation strain computable fromthe lattice parameters Each variant is assumed to grow,mixed with austenite, in a well-defined cluster The inter-action energy describing the accommodation between pairs
of variants is then computed using the interfacial tor method of Hill The minimization of this interactionenergy determines the cluster orientation and the over-all free energy finally depends only on the variant frac-tions At the polycrystalline level, a second RVE consisting
opera-of nontextured assemblies opera-of spherical grains is ered and a self-consistent approach is used to derive the
Trang 10consid-SHAPE-MEMORY MATERIALS, MODELING 969
final macroscopic constitutive equations Large numerical
simulations involving the representation of each grain are
required Comparisons with experimental data are given,
mainly for uniaxial stress, and show good agreement even
in the prediction of tension–compression asymmetry
Re-cent developments also include the consideration of
non-isothermal behavior Promising applications of this model
have been proposed by Gall and Sehitoglu (1999) who used
experimentally determined grain orientation distribution
functions to simulate the effect of texture
In 1993, Sun and Hwang proposed to treat the
prob-lem focusing directly on the polycrystals with an RVE
con-sisting of grains that are each wholly in the austenite or
martensite phase The phase interaction energy is
com-puted using Mori-Tanaka theory by considering martensite
grains as randomly dispersed spherical inclusions within
the matrix of austenite grains While martensite is
con-sidered as a single phase without explicitly accounting
for the different variant orientations, neglect of the
mul-tivariant structure is overcome by proposing a direct
re-lation between the local transformation strain and the
average stress in the matrix so as to simulate the
bias-ing effect of stress in the variant selection process The
local transformation strain is therefore not strictly
crys-tallographic, and the resulting description is in terms of
an equivalent transformation strain Reorientation effects
are taken into account by introducing a second martensite
fraction Issues related to nonproportional loading are also
discussed (12)
Starting in the early 1990s, Levitas developed
mod-els from a somewhat different viewpoint (13) The SMM
is modeled as a dissipative material already at the
mi-croscale where, due to the phase transformations, relevant
field quantities vary between two values reflecting an
un-derlying two-phase model for the microstructure A
first-averaging procedure is performed over an internal time
scale representative of the transformation duration in
or-der to obtain an average dissipation rate and driving force
Different energetic transformation criteria are given: an
overall nucleation criterion results after integration over
the RVE while a criterion for interface propagation is given
after integration over the propagating interface An
ex-tremum principle with respect to the variation of the RVE
boundary conditions is invoked to determine the evolution
of the microstructural parameters
Goo and Lexcellent (14) proposed a model for single
crystals based on a free-energy function and a
dissipa-tion rate funcdissipa-tion The free-energy funcdissipa-tion is derived by
a self-consistent evaluation of internal stresses among the
phases The model allows for nonisothermal behavior,
re-orientation, and two-way shape-memory effect The
influ-ence of the interaction energy on the macroscopic modeling
is examined, and the comparisons with experimental data
under uniaxial stress show good agreement with the
mod-eling prediction
The analysis of Lu and Weng (15) treats each grain as
a mixture of austenite and a single martensite variant
whose local transformation strain is computed from lattice
parameters The particular variant is selected in analogy
with the Patel-Cohen criterion on maximum
transforma-tion work Polycrystals are then modeled by an assembly of
nontextured spherical grains, and a self-consistent method
is used to compute the macroscopic response As with thePatoor-Eberhardt-Berveiller model, this requires large nu-merical simulations involving the representation of eachgrain
Huang and Brinson (16) propose a different tural model at the single-crystal level An austenitematrix with martensite inclusions made of groups of self-accomodating variants, each exhibiting the local crystal-lographic transformation strain, is arranged in a wayreminiscent of the experimentally observed wedgelike mi-crostructure Free energy is then computed by assuming
microstruc-a rmicrostruc-andom distribution of such inclusions thmicrostruc-at microstruc-are tmicrostruc-aken to
be of spherical shape This idealization is shown to be ful in modeling thermally activated transformations andlow-temperature reorientation The model captures alsothe tension/compression asymmetry and the different re-sponse observed experimentally when the loading direc-tion varies with respect to crystal axes The model hasbeen extended to cover penny-shaped inclusions and poly-crystalline behavior by studying an assembly of nontex-tured spherical grains homogenized by a self-consistentmethod
use-Summarizing, micromechanic approaches incorporateseveral features into the modeling, including the effect
of a multiple-variant microstructure and the effect of itspolycrystalline texture This permits explanations for mostmacroscopically observed behaviors, though certain de-tailed issues remain under discussion Such issues includethe determination of the number of variants and the mod-eling of their arrangement (17), as well as the modeling ofnonproportional multi-axial loading; for recent experimen-tal studies, see (18,19)
Approaches Modeling Directly the Macroscopic Behavior
Direct modeling will be understood as including theorieswhere each point of the material, instead of being in anidentifiably distinct phase, is representative of a phasemixture whose microstructural features are described byone or more descriptive variables In a continuum setting,the associated strain and temperature gradient fields arecontinuous
Models without Internal Variables In such models the
material behavior is described by strain, stress, ature, and entropy without the introduction of quantitiesrepresenting the phase mixture Constitutive information
temper-is provided by a free-energy function whose partial tives provide constitutive equations for strain (or stress)and entropy
deriva-In 1980, Falk proposed a Landau-Devonshire type
of free-energy function based on the analogy betweenSMM uniaxial stress–strain curves and the electric field–magnetization curves of ferromagnetic materials (20).Nonmonotone stress–strain curves are obtained, and theunstable negative slope part is interpreted as the occur-rence of the phase transition The actual pattern followedduring transformation is assumed to proceed at constantstress The particular form of the Landau-Devonshire free-energy accounts for the temperature dependence of the
Trang 11ε
Figure 5 Nonmonotone stress–strain curves The negative slope
part is unstable and the dashed lines represent an assumed
trans-formation path.
isothermal stress–strain behavior Hysteresis arises as
consequence of the different stress levels of the extremal
points of the unstable region, as indicated in Fig 5
Under suitable interpretation, many aspects of this
model correspond to aspects of Ericksen’s treatment and
its subsequent extensions In particular, although the time
evolution of a phase transformation is not treated directly,
such information can be inferred by associating the extent
of the transformation with the strain distance on the
con-stant stress transitions
Hysteresis Models Hysteresis models seek to reproduce
experimentally observed curves that involve high
nonlin-earity and complex looping They have been widely used in
several fields, with that of magnetic materials being most
developed In this approach, constitutive equations are
pro-posed directly on the basis of their mathematical
proper-ties, often without explicit focus on their link with the
phys-ical phenomena of interest Reliability and the robustness
of the model are favorably matched to experiments, and the
resulting algorithm allows for the treatment of arbitrarily
complex driving input
Two main algorithm classes have received special
atten-tion in the context of SMM phase transformaatten-tion The first
one is based on tracking subdomain conversion/reversion
and lead to integral based algorithms The most common of
these is known as the Preisach algorithm and it has been
used to describe uniaxial isothermal pseudoelastic stress–
strain SMM response (21,22)
The second algorithm class involves differential
equa-tions with separate forms for driving input increase and
driving input decrease Differential equations of
Duhem-Madelung form have been used to model SMM phase
frac-tion evolufrac-tion during thermally induced transformafrac-tion
(23,24) This gives phase fraction subloops for temperature
histories obeying M f < T (t) < A f Under sustained
ther-mal cycling, these subloops collapse onto a final limiting
subloop, with the resulting shakedown behavior
register-ing the fadregister-ing influence of the initial phase-fraction state
By being formulated so as to link the internal variable of
phase fraction to the driving force variable of temperature,
such algorithms lend themselves to a wider internal
vari-able framework as is described next
Models with Internal Variables The key feature of this
approach is to introduce one or more internal variables
(order parameters) α describing the internal structure of
the material (see again Fig 4) A general thermodynamicaltreatment then proceeds by providing equations for theevolution of these internal variables (23,26) The first ap-plication of such an approach to SMM seems to be due
to Tanaka and Nagaki (27) where internal variables areemployed to describe the development of the underlying
phase mixture The internal variables α, along with a
set of mechanical and thermal control variables, then
de-fine a collection of state variables s Typical mechanical control variables are either strain ε or stress σ Typical
thermal control variables are either temperature T or
en-tropyη The internal variables α typically include one or more phase fractions ξ and/or macroscopic transformation
strains For example,
The temperature gradient∇T must also be incorporated
into s if heat conduction is considered The theory is
com-posed of the physical laws, the constitutive equations that characterize the features typical of each material, and ma- terial behavior requirements that ensure thermodynamical
process restrictions
Constitutive information is specified by two kinds of lations:
re-1 State equations for the entities that are conjugate
to the control variables These can be formulateddirectly or else obtained as partial derivatives of
a suitable free energy function after enforcing theClausius-Duhem inequality for every process The
Gibbs free-energy G is appropriate for Eq (1) and
constitu-heat flux q (usually the Fourier equation) is also
re-quired The relation between microscale phenomenaand the structure of the macroscale free-energy func-tions is discussed in (10)
2 A set of kinetic equations for the internal variables
α In view of phase transformation hysteresis, theseequations generally depend on the past history of thematerial Standard practice in most internal variablemodels is to specify this dependence through equa-tions relating the rates of the internal variables to thestate and its time derivatives The internal state thenfollows from the solution of differential equations intime:
F (s, ˙s) = 0 typically giving ˙α = f (s, ˙σ, ˙T ),
(3)and often linear in ˙σand ˙T as well The superposed
dot in Eq (3) denotes time differentiation
Trang 12SHAPE-MEMORY MATERIALS, MODELING 971
The full thermomechanical behavior of a system
involv-ing SMM is then described by a complete “initial value
problem”: given an initial state, an initial time, and an
arbitrary loading history, predict the state reached by the
system at subsequent times This initial value problem is
governed by the preceding constitutive equations together
with appropriate initial and boundary conditions and with
the physical laws of:
Energy conservation (first law of thermodynamics)
˙e = σ · ˙ε + ˙Q, (4)
Conservation of linear momentum (equation of motion)
Entropy balance and Clausius-Duhem inequality
(sec-ond law of thermodynamics)
= ˙η − Q˙
where e is the internal energy, ρ the mass density, ˙ Q the
rate of heat exchange with the environment (positive if
ab-sorbed by the SMM), u the displacement, b the body force
density, and the internal entropy density production rate.
The Gibbs free energy is G = e − Tη − σ · ε.
Although sometimes employing formalisms that are
quite different, several models fitting into this basic
frame-work have been proposed to describe SMM behavior
Irrespective of their derivation, they involve a
constitu-tive description prescribed via state equations and kinetic
equations Differences involve the choice and
interpreta-tion of the internal variables α and the form of the kinetic
equations The following survey proceeds in chronological
order, using a common notation that may depart from that
of the original work
One of the first explicit macroscopic models for SMM
has been given by Tanaka and his coworkers for
uniax-ial isothermal pseudoelasticity (28) This model for A↔ M
transformation employs one scalar internal variable, the
martensite fractionξ M, together with strain and
temper-ature as control variables so that α = {ξ M} and s=
{ε, T, ξ M} The constitutive equation for stress is given
asσ = L(ε − ε∗), where the overall transformation strain
is ε∗= γ∗ξ M Here L is the elastic modulus and γ∗ the
local transformation strain, both of which are regarded
as material parameters The kinetic equation for the
martensite fraction ξ M is derived from a dissipation
po-tential resulting in a form analogous to the exponential
Koistinen-Marburger relation used in metallurgy The
ki-netic equation is especially simple and so enables closed
form integration to give
β F = a M (M s − T) + b M σ,
β R = a A ( A s − T) + b A σ, where a M , b M , a A and b Aare material constants
In 1987 Fr´emond proposed a three-dimensional modelbased on the following state description:
s= {ε, T, ξ A , ξ+, ξ−, ∇ξ A , ∇ξ+, ∇ξ−},
where ξ A , ξ+, ξ− are the fractions of austenite and twomartensite variants and∇ denotes gradient (29) The freeenergy is the sum of the pure phase energies weighted bythe respective fractions plus a term aimed to enforce theconstraintξ A + ξ++ ξ .= 1 The evolution equations arederived from a dissipation potential The balance equa-tions governing the model are derived on the basis of theprinciple of virtual power taking into account explicitlythe contribution of the internal variables The model per-formances are discussed within simplified constitutive as-sumptions, and show the ability of the model to capture themain qualitative features of pseudoelasticy and low tem-perature reorientation
In 1988 Bondaryev and Wayman developed a surface plasticity theory for SMM to account for three-dimensional isothermal pseudoelasticity and reorienta-tion (30) This yields a three-dimensional framework thatwould allow for the generalization of many of the uniaxialstress models that are the major focus of this article The
multi-macroscopic transformation strain tensor ε∗is treated as
an internal variable on its own so that s= {σ, T, ε∗} Freeenergies are given for the austenite phase and for a gen-eral martensite phase in which the transformation strain is
of arbitrary orientation The free-energy difference amongthe phases initiates the transformation activity when re-sistive the thresholds are met This defines temperature-dependent threshold surfaces in stress space, which areanalogous to traditional yield surfaces These surfaces,
govern transformation strain accumulation (A→ M),
re-duction (M→ A), and reorientation (M → M) The change
in transformation strain proceeds according to a ity condition with respect to these threshold surfaces and
normal-so determines the orientation of a transformation strainincrement This gives:
A→ M: Transformation strain increment dε∗coaxialwith the stress deviator
M→ A: Transformation strain annihilated without
regard to stress orientation
M→ M: dε∗ oriented according to the difference tween the current stress deviator orientation and thecurrentε∗orientation
be-The magnitude of the transformation strain incrementfollows by analogy to traditional yield surface plasticity.Under hardening, continued plastic straining requires asustained increase in stress A consistency condition for re-maining on the evolving yield surface then determines themagnitude of the plastic strain increment In the absence
of hardening, the yield surface is fixed so that continuedplastic straining can be sustained under constant stress.The magnitude of the plastic strain increment is then de-termined from boundary conditions Bondaryev-Wayman
initially present A→ M and M → A thresholds that do not
harden (corresponding to M s = M ffor A→ M and A s = A f
Trang 13for M→ A) In contrast, the M → M reorientation
thresh-old is naturally dependent on the currentε∗and so exhibits
a changing form analogous to hardening At any instant
of time there may be (1) no transformation, (2) a single
transformation from among A→ M, M → A, M → M, or
(3) a multiple transformation consisting of M→ M in
con-cert with one of M→ A, A → M Case 3 requires the
de-termination of a transformation strain incremental
mul-tiplier for each simultaneous process Modifications are
then presented for A→ M and M → A threshold
harden-ing (M s > M f and A s < A f) The few model predictions that
are presented confirm the correct qualitative features of
this approach, but a detailed discussion of the full range of
model predictions is not given
Muller and his coworkers (31) proposed one of the first
models for stress–strain curve sublooping when phase
transformations do not go to completion The main
con-stitutive ingredient of the theory is an overall Helmholtz
free energy of the form
(ε A , ε M , T, ξ M)= (1 − ξ M)φ A(ε A , T ) + ξ M φ M(ε M , T )
M(1− ξ M),
whereφA, φM, εA, εM are the free energies and strains of
the pure phases andξ M is the martensite fraction
inter-nal variable The last term is a phase interaction energy,
with
ε = (1 − ξ M)ε A + ξ M ε M Values ofξ M , ε A, andε M are
deter-mined by minimizing under strain constraint As in the
models of Ericksen (4) and Falk (20), the resulting uniaxial
stress–strain response is nonmonotone with the negative
slope part being unstable Constant stress lines from the
maxima and minima define a stress–strain outer envelope
loop associated with complete transformations The
unsta-ble negative slope portion of the stress–strain response
then provides a triggering threshold for phase
transfor-mations within the interior of the stress-strain envelope
(Fig 6) The associated model for internal sublooping
deter-mines the evolution ofξ Mand so, roughly speaking, plays
a similar role to the kinetic evolution equations in other
internal variable models
Ortin and planes have developed a detailed
thermody-namic framework for SMM materials (32) that provides a
basis for thermomechanical modeling and the
experimen-tal determination of material characterization parameters
They develop a model uniaxial stress, describing the state
as s= {σ, T, ξ M} Energy balance during phase
transforma-tion generates a transformatransforma-tion kinetic in which the
free-energy differential during transformation is balanced by
the sum of an elastic energy storage rate differential and an
energy dissipation rate differential The evolution ofσ and
T then determines the evolution of ξ Monce the dissipation
rate is given a constitutive prescription Dissipation
func-tions can be constructed so as to ensure known qualitative
aspects of phase transformation hysteresis, including fine
sublooping features Experimental data fitting with
refer-ence to purely thermal transformation allows for explicit
functional forms Full stress–strain–temperature
depen-dence for uniaxial A↔ M transformation then follows.
In 1990, Liang and Rogers proposed a modification of
the Tanaka A↔ M transformation model so as to account
for the effect of phase fraction values at the beginning ofthe transformation in the event of an initial phase mixture(33) This allows for the treatment of internal subloops.They also replaced the exponential Koistinen-Marburgerkinetic equation with a trigonometric expression, giving
whereξ0is the value ofξ Mwhen the transformation process
is first activated andβ F , β Rare as given previously
In 1992, Raniecki, Tanaka, and Lexcellent proposed
a model for three-dimensional pseudoelasticity based on
the martensite fraction as internal variable so that s=
{ε, T, ξ M} (34) They proposed a Helmholtz free-energyfunction in the form
given as ε∗= γ∗ξ M The local transformation strain tensor
γ∗is assumed to be traceless and coaxial with the straindeviator In this context the phase equilibrium corresponds
to the vanishing of the quantity = ∂ /∂ξ M, which is
iden-tified as the driving force for A↔ M phase transformation.
The condition = 0 gives rise to a stress–strain curve with
a descending branch which, as in Muller’s model, triggersthe activation of certain transformations (Fig 6) Here,however, the transformation evolution is described by akinetic equation forξM A generalized expression for such
a kinetic equation is proposed and includes the exponentialform of Tanaka as a special case Subsequent development
of the model includes the incorporation of micromechanicalconsiderations into the derivation of the free energy and
the proposal of a modified relation between γ∗and the state
σ
ε
Figure 6 Modeling of subloops in a model of Muller (31) The
up-per and lower constant stress lines arise as in Fig 5 and bound an internal region in stress–strain space where subloops can occur.
If a transformation associated with these bounding lines is terrupted before completion due to load reversal, then the stress– strain path enters the internal region The reverse transformation
in-is only activated if the internal path encounters the negatively sloped line associated with unstable stress–strain response.
Trang 14SHAPE-MEMORY MATERIALS, MODELING 973
variables so as to capture an optimal variant arrangement
on the assumption that the actual value of γ∗minimizes
the free energy
In 1993, Brinson proposed a further extension of the
Tanaka-Liang-Rogers model in order to distinguish
be-tween stress-induced (oriented) martensite and thermally
induced (unbiased) martensite (35) Accordingly, ξ M=
ξ s + ξ Twhereξ sis the fraction of stress-induced martensite
resulting in a local transformation strainγ∗andξ Tis the
fraction of thermally induced self-accommodated
marten-site resulting in a zero transformation strain Hence α=
{ξ s , ξ T}, and the resulting model gives a true shape-memory
capability, formally absent in the earlier
Tanaka-Liang-Rogers model, in that transformation strain at zero stress
is annihilated on heating without reappearing on cooling
Kinetic equations for ξ s and ξ T then model both A↔ M
pseudoelasticity and low-temperature conversion of
ther-mally induced martensite to oriented martensite under
isothermal uniaxial stress Subsequent development
ex-plicitly correlates the kinetic equations to the (σ, T )-phase
diagram, with the resulting model permiting both
analyt-ical and numeranalyt-ical treatment of initial value problems for
uniaxial response (36)
In 1994, Ivshin and Pence proposed a model for uniaxial
pseudoelasticity based on s= {σ, T, ξ A }, where ξ A = 1 − ξ M
is the austenite phase fraction (24,37) State equations are
given for strain and entropy by assuming that the pure
phases are subject to a common overall stress Kinetic
equations are given in terms of the Duhem-Madelung
hys-teresis algorithm Both stress and temperature-induced
A↔ M transformation are treated in a unifying
thermody-namic framework by the identification of the proper driving
input via the Clausius-Clapeyron equation This allows
for a straightforward treatment of arbitrary
thermomecha-nical loading paths in (σ T )-space under uniaxial tension.
Load-cycling behavior and the resulting shakedown
re-sponse are then easily determined Unlike earlier
treat-ments, rate effects due to heating and cooling intrinsic
in the A↔ M transformation are systematically
investi-gated To treat uniaxial compression in the same
frame-work, Wu and Pence (38) decompose the martensite phase
into two martensite variant families M+ and M− each
characterized by its own transformation strain and,
respec-tively, favored under tension (M+) or compression (M−)
Accordingly, ξ M = ξ++ ξ−, leading to s= {σ, T, ξ+, ξ−} In
this setting, unbiased (thermally induced) martensite is
the specific mixture of the two variant families giving zero
overall transformation strain While maintaining all of the
features of the Ivshin-Pence formulation, the Wu-Pence
model provides a complete uniaxial description for SMM
not only at temperatures near A fbut also at temperatures
well below M f where reorientation applies
In Lubliner and Auricchio (39), a three-dimensional
model for isothermal pseudoelasticity is proposed on the
basis of s= {σ, T, ξ M , v}, where v defines the
orienta-tion of ε∗ via ε∗= γ∗v and γ∗ is a material parameter
Kinetic equations for v andξ Mare given by normality
con-ditions to proper loading functions specified for each type
of transformation A Drucker-Prager form is taken for v
so as to treat the dependence of phase transformation on
hydrostatic stress Pseudoelasticity and high-temperature
reorientation as induced by nonproportional loads are sidered In the uniaxial setting, the Brinson decomposition
con-ξ M = ξ s + ξ Tis introduced to account for low-temperaturereorientation The extension to finite kinematics is develo-ped Numerical implementation is given in the context offinite elements
Boyd and Lagoudas (40) present a three-dimensional
model for SMM behavior based on s= {σ, T, ξ M , ε∗} Afree energy with a structure similar to that of Muller isgeneralized so that the phase interaction term is speci-fied by a series expansion whose coefficients are left to ex-perimental identification The macroscopic transformationstrain rate is decomposed into the sum of a pseudoelastic
and a reorientation contribution, ˙ε∗= ˙ε pe + ˙ε re Similar toTanaka et al (28), dissipation potentials are used to de-rive kinetic equations for ˙ξ M , ˙ε re , while ˙ε pe is related to
the martensite fraction rate via ˙ε pe = Λ˙ξ M The
orienta-tion tensor Λ is assumed as coaxial with a modified stress deviator for A→ M transformation, whereas it is aligned with the ε∗ deviator for M→ A transformation Various
aspects of the model, such as sublooping, connection withmicromechanics, and numeral implementation, have beendeveloped in subsequent papers
Rajagopal and Srinivasa (41) apply the concept of rials with multiple natural configurations to uniaxial pseu-doelasticity with the martensite fractionξM acting as aninternal variable The Green-Naghdi approach to balanceand constitutive equations is used together with a principle
mate-of maximum dissipation This elegant treatment accountsfor nonisothermal behavior and finite deformations within
a rigorous and innovative framework of continuum momechanics
ther-Sittner, Stalmans, and Tokuda (18) have developed
a hysteretic model for uniaxial pseudoelasticity based
on a martensite fraction that is governed by an tion equation with a driving force that involves the con-cept of an effective equilibrium temperature This effec-tive equilibrium temperature, which generally differs fromthe actual material temperature, is formally dependent
evolu-on the martensite volume fractievolu-on Unlike other models,the macroscopic transformation strain is assumed in thefrom ε∗= χγ∗ξ M, where χ = χ(σ ) is a nonlinear stress-
dependent function The model is able to capture a widevariety of sublooping behavior including a notion of returnpoint memory
Within the setting of finite kinematics Govindjee andHall (42) present a model based on two martensite vari-ants M+ and M− A phase diagram approach is used,and the transformation kinetics are derived by argumentsreminiscent to those of Muller-Achenbach and Abeyaratne-Knowles The resulting model allows for both pseudoelas-ticity and reorientation Algorithmic issues specific to finiteelement implementation are carefully considered, and civilengineering scale applications are presented
A COMPREHENSIVE MODEL FOR UNIAXIAL STRESS
As seen from the previous section, SMM uniaxial behavior
is well understood This covers a wide range of applicationsincluding several kinds of actuators, vibration absorbers,
Trang 15and applications exploiting the material in the form of
wires Models available for this setting treat
reorienta-tion, shape memory, and pseudoelasticity under arbitrary
temperature-stress paths so as to reproduce
nonisother-mal behavior, transformational heating/cooling, and
inter-nal subloops Accordingly, this section summarizes a
prac-tical and complete one stress-component material model
It synthesizes aspects of the previous section’s discussion
with a focus on the complementary roles of state equations
and kinetic equations in generating a well-posed and
com-plete model The development is framed in the context of
tensile and compressive loading, although it also applies
to other choices of stress component, such as a
particu-lar shear stress component The equations that are
pre-sented in the final implementation follow those of Ivshin
and Pence (24,37) and Wu and Pence (38), although the
discussion is framed so as to permit alternative
implemen-tations
The SMM is treated as subject to mechanical loads
specified by histories of prescribed stressσ(t) or strain ε(t)
or a combination of both (e.g., representing bias springs in
actuators among elastic restraints) and thermal loads
spec-ified by histories of prescribed temperature or heat rate
Al-ternatively, under mechanical loads, the temperature can
be determined as a consequence of the heat exchange with
a known environmental temperature T E
The state in Eq (1) is given by s= {σ, T, α} with α =
+> 0, representing the maximum positive
macro-scopic transformation strain when an M+ microstructure
is maximally oriented with the tensile stress Similarly, the
M−family has an associated scalar transformation strain
State Equations for Strain and Entropy
The first group of constitutive equations are obtained from
a Gibbs free energy that is taken as
G(s) = ξ A g A(σ A , T) + ξ+g+(σ+, T) + ξ−g−(σ−, T), (8)
where g A , g+, g− andσ A , σ+, σ− are the free energies and
the stresses relative to the pure phase The phase stresses
depend on the microstructural phase arrangement This
arrangement is henceforth regarded as giving
In conditions different from Eq (9), an additional term
ex-pressing the interaction energy between the phases would
otherwise arise (10) In the present model, hysteresis
prop-erties that would be influenced by such an interaction
en-ergy are instead modeled with the aid of envelope functions
introduced below This permits easy specification of desired
thermal hysteresis properties
Standard forms for the pure phase free energies are
g A= −12
σ2
E+− γ∗ +σ − m+σ(T − T∗)+ C+
Here E A , E+, E− are the elastic moduli; m A , m+, m − are
the coefficients of thermal expansion, and C A , C+, C−
are the specific heats at constant stress of the ous phases The expressions (10) make use of a stress-temperature reference state (σ, T) = (O, T∗), which is re-garded as at the center of the multiphase region of Fig 1
vari-by taking T∗as the average of the four transformation peratures The constantsη A0, η+0, η−0 and g A0 , g+0, g−0arethe single-phase entropies and single-phase free energies
tem-at this reference sttem-ate Additional simplifictem-ation follows byassuming the following:
rA common specific heat C in martensite and austenite.
rA common reference state entropyη Moin all site variants
marten-rNegligible thermal expansion.
rNegligible slip plasticity.
None of these simplifications is essential, and the ciated generalizations are easily made According to thegeneral formulation, the macroscopic strain and entropyfollow from Eq (2) as
asso-ε = − ∂G ∂σ = Dσ + ε∗, η = − ∂G ∂T = CIn T
T∗ + η0, (11)where
the Reuss estimate of the effective compliance Introduce
η 0 = η A0 − η M0, whereuponη 0(ξ) = η M0 + ξ A η 0 Otherthan the baseline value ofη in Eq (11), the model depen-
dence onη M0andη A0is only via the basic material eterη 0 > 0 It is given by η 0 = H/T∗whereH is the
param-latent heat of the M→ A transformation as measured from
−, C These eleven parameters are
suffi-cient for modeling the basic SMM behavior with the
ex-ception of M− ↔ M+reorientation As will be explained inwhat follows, reorientation is modeled by including four
Trang 16SHAPE-MEMORY MATERIALS, MODELING 975
additional material parametersσ+
s , σ+
f , σ−
s , σ−
f All 15 terial parameters are taken to be positive by definition
ma-Phase Transformation Kinetics
The second group of constitutive equations give the phase
transformation kinetics The model derives such equations
from two constitutive ingredients that are, in principle,
both experimentally measurable:
rThe phase diagram that defines the loci of the points
in the stress-temperature plane in which the variousphase transformations can be activated (an example
of which is reported in Fig 1)
rThe envelope functions ζ M →A (T) and ζ A →M (T) that, as
explained below, determine the equations for the ternal variables thus describing how the phase trans-formations evolve once activated
in-Here, ζ A →M (T ) is the value of ξ A associated with σ = 0
and T decreasing from above A f to below M f Similarly,
ζ M →A (T ) is the value of ξ Aassociated withσ = 0 and T
in-creasing from below M f to above A f Each function has
a graph that monotonically increases from zero to one
as T increases over an appropriate interval: M f < T < M s
for ζ A →M (T ); A s < T < A f for ζ M →A (T ) In the absence of
detailed experimental data, these functions can be
proximated on their transition interval The simplest
ap-proximation is a linear function of T, but this gives a
slope change at the associated start and finish
tempera-tures Such slope changes give rise to sharp corners on the
model stress-strain curves at the beginning and end of the
transformation plateaus Smoother forms, which eliminate
such corners, involve hyperbolic or trigonometric functions
A simple and useful representation is the Liang-Rogers
For the sake of clarity of exposition, it is useful to discuss
the features of the phase transformation kinetics with
ref-erence first to purely thermal transformation and then to
combined stress and thermally induced transformation
Purely Thermal Transformation In this case the driving
input is the temperature history T(t) Times t of
temper-ature reversal are switching instants The martensite is
then unbiased in that it involves M+and M−in the ratio
−, λ−= γ+∗
γ∗ ++ γ∗
and ξ M = ξ++ ξ−= 1 − ξ A is the overall martensite tion The transformation kinetic reduces to the determina-tion ofξ A as T changes The graph of ξ A versus T will gener-
frac-ally involve complicated sublooping if there are numerous
switching instants obeying M f < T < A f that prevent the
M→ A and A → M transformations from going to
comple-tion However, no matter how complicated, this graph will
be contained between the two envelope functions so that
con-such as d ξ M /dT = H A →M(ξ A , T ) for A → M The
center-pieces of Eq (15) are the governing functions H M →A and
H A →M, which must be formulated so that the phase fractionobeys the envelope containment conditionζ M →A (T ) ≤ ξ A≤
ζ A →M (T ) Envelope coincidence must occur for
transforma-tions that begin from a pure M or a pure A state so long
as temperature reversal is avoided The following generalform for the governing functions ensures these properties:
flat-that enhanced experimental correlation is obtained for n near n = 3 Further refinements and modifications can be
invoked, and certain sublooping situations that are cult to describe with a D-M equation have been noted (44)
diffi-Some examples of (T , ξ A)-trajectories within the boundingenvelope functions are shown in Fig 7
The purely thermal process involves simultaneous
A→ M+and A→ M−transformation for temperature
de-crease, and simultaneous M+→ A and M−→ A
transfor-mation for temperature increase, always with ξ+= λ+ξ M
and ξ− = λ−ξM It follows thatξ M can be eliminated from
Eq (15) by rewriting them as
Trang 17Figure 7 Magnetic susceptibility versus T under purely
ther-mal loading for a NiTi thin film as determined by measurement
(above) Magnetic susceptibility correlates directly with austenite
phase fraction Both the outer hysteresis loops for cooling (upper
envelope) and heating (lower envelope) are shown The internal
curves leave the upper envelope if T is increased before A → M
transformation is complete This behavior is modeled (below) on
the basis of Eq (15), with a hyperbolic tangent envelope form and
H M →A governing function with n= 3.
These equations, in conjunction with Eq (7), determine
the evolution of the phase fraction variablesξ A , ξ+, ξ−for
purely thermal transformation Although unwieldy
com-pared to Eq (15), the alternative formulations (17) and
(18) recast the phase transformation kinetic in terms of
the internal variables α = {ξ A , ξ+, ξ−}
Pseudoelasticity While temperature is the driving input
in the purely thermal case, under simultaneous change in
temperature and stress, the functions
−12
1
1
E− − 1
E A
are the more generalized driving forces for the
respec-tive transformations M+↔ A, M−↔ M+, and M−↔ A.
Changes in (σ, T) that cause +A, −+, and−Ato increase
favor M+→ A, M−→ M+, and M−→ A, respectively.
Conversely, the decrease favors transformation in theopposite direction The associated transformation is acti-vated only if the current value of (σ, T) is also within the
corresponding transformation zone of the (σ, T)-phase
dia-gram (e.g., see Fig 1)
The M−↔ M+transformation zone does not extend into
T ≥ A f, in which case only transformations M+↔ A and
M−↔ A can occur Accordingly, for T ≥ A f attention is cused on+Aand−Awhich are renormalized as follows:
2 = (E A − E−)/(2E A E−ηo)≥ 0 Theinequalities follow from η o > 0, γ∗
+> 0, γ∗
−> 0, E A ≥ E+,
and E A ≥ E− Under this renormalizationβ+(0, T) = T For
σ = 0, the function β+(σ, T) behaves like a “stress-adjusted
temperature” that governs the M+↔ A transformation
for general changes in (σ, T) The function β−(σ, T)
plays a corresponding role with respect to M−↔ A
transformation
Extending the purely thermal algorithm with driving
input T(t) to the case where the driving input is given by
σ (t) in conjunction with T(t) ≥ A f amounts to replacing
T(t) in Eq (17) with β+(σ(t), T(t)) and to replacing T(t) in
Eq (18) withβ−(σ(t), T(t)) This gives
H A →M, H M →Aandζ A →M, ζ M →A, it follows that the A→ M+
and M+→ A transformations are activated and completed
for those values of (σ, T) such that β+= M s , β+= M f , β+=
A s , β+= A f, respectively Along with Eq (20), they definethe boundaries of the relevant zones of the (σ, T) phase
diagram A similar remark applies to the
transforma-tions A→ M−and M−→ A vis-`a-vis the function β−(σ, T).
The strain and entropy follow from Eqs (11) and (12)
The resulting model for T ≥ A faithfully predicts general
Trang 18SHAPE-MEMORY MATERIALS, MODELING 977
tension/compression asymmetry, pseudoelastic
transfor-mation, and pseudoelastic sublooping
Low-Temperature Reorientation If the temperature is
below A f, then two additional phenomena have to be taken
into account: The M−↔ M+reorientation transformation
and the possibility of multiple transformations An
exam-ple of the latter is provided in the purely thermal case
when A↔ M−and A↔ M+take place simultaneously The
model can be extended to cover such situations provided a
phase diagram is available that describes the activation
and the completion of all possible transformations
Equa-tions (21), (22) can then model the additional
transforma-tion possibilities provided that the functransforma-tionsβ+(σ, T) and
β−(σ, T) are modified from the specification (20) so as to
ac-count for the change in zonal geometry of the (σ, T )-phase
diagram when T < A f A modification that accomplishes
this purpose and so establishes such a phase diagram is
Contours of constantβ+and constantβ−give
continu-ous curves on the phase diagram, although each such curve
will have up to two sharp corners by virtue of the abrupt
formula changes in Eqs (23) and (24) Smooth contoursmore resembling those in Fig 1 can be obtained by a morecomplicated redefinition ofβ+, β−
Note that the phase diagram of Fig 1 is defined by sixcontinuous curves The present treatment provides simi-
lar such curves The four curves that continue into T < M f
as approximately constant stress curves parallel to the
T-axis are defined by β+= A s , β+= A f , β−= A s , and β−=
A f The zone A s < β+< A fbounded by two of these curves,
β+= A s andβ+= A f, is associated with transformations
that deplete M+ This depletion gives M+ → A for σ > 0.
For σ < 0, this depletion gives M+→ M− reorientation
provided that T is sufficiently low The low-temperature threshold for such pure reorientation is given by T=
M f − k−
1σ + k−
2σ2as specified in Eq (23) In particular, this
M+→ M− reorientation is activated in compression, ginning at σ = −σ−
2σ2 Here M+transforms into a mixture
of A and M− Similar comments hold with respect to the
zone A s < β−< A f provided that the roles of M+and M−
are interchanged Note that neither zone is specifically
as-sociated with depletion of A The zone of A depletion is
bounded by the remaining two curves on the phase
dia-gram These two curves pass through T = M s and T = M f
on the T-axis and are given by β+= M sand β+= M f on
σ > 0 and by β−= M sandβ−= M f onσ < 0 The
auste-nite depletion is via A→ M+ifσ is sufficiently tensile, and
is via A→ M−ifσ is sufficiently compressive For σ near
zero, the depletion of A is into a mixture of M+and M−, as
in the case of thermally induced martensite
In summary, Eqs (7), (21), and (22) with (23) and (24)
now describe general A↔ M+, A ↔ M−, M−↔ M+formation throughout the full extent of the (σ, T )-phase di-
trans-agram A more detailed discussion is given in Wu and Pence(38) for the special case of symmetric tension/compressionbehavior Equations (23), (24) also give the proper ten-sion/compression asymmetry if any of the equalitiesγ∗
dis-an unexpressed unstable austenite intermediary (45) gorithm (15) is recovered under purely thermal trans-
Al-formation At temperatures T < A f, isothermal tensionfollowed by compression generates open stress–straincurves that are sometimes referred to as ferroelasticresponse
Operative Equations for Various Driving Conditions
The set of equations required for the actual computation
of SMM response are summarized in the following Thisset may be different depending on the conditions in whichthe material is used and the way that the loading inputspecified
If T and σ in the material are prescribed, then the
de-termination ofξ A , ξ+, ξ−proceeds directly on the basis ofEqs (7), (21), and (22) In this case the strainε as provided
by Eq (11) decouples from the kinetics
Trang 19If T in the material is prescribed, but σ in the
mate-rial is not, then the first of Eqs (11) must be solved in
conjunction with Eqs (7), (21), (22) This includes cases
whereε is prescribed (various hard constraint situations)
or whenσ is related to ε by external constraint (e.g., a bias
spring)
If T in the material is not prescribed directly, then the
complete thermomechanical behavior requires
considera-tion of the material thermal balance and heat exchange
with the environment The heating rate ˙Q can be written
˙
Q= ˙Qrev+ ˙Qirrev (26)where, in view of Eq (6), the reversible contribution is
˙
The irreversible contribution ˙Qirrev = −T includes
dissi-pative effects that are intrinsic with the phase
transfor-mation, and by Eqs (2), (4), (8), (26), and (27), it can be
According to Ortin and Planes (46), ˙Qirrevcan be neglected
in a first approximation More generally,
Here ˙Q provides the heat exchange with the environment
and requires additional description to this effect
Impor-tant situations in which T is not prescribed include
adia-batic conditions and conditions of convective heat transfer
to an ambient temperature T E The adiabatic case is ˙Q= 0,
and involves no heat exchange with the environment It is
an appropriate model under sustained and rapid
mechani-cal loading The case of convective heat transfer may often
be described by
˙
whereκ > 0 is a known parameter Here κ → 0 gives the
adiabatic limit, whileκ → ∞ enforces T(t) = T E (t) If T is
in fact prescribed, then Eqs (29) and (30) determine
ei-ther the heat exchange ˙Q or the fluid/atmospheric media
temperature T Ethat is necessary to sustain the
prescrip-tion Table 1 summarizes the equations one uses for the
determination ofξ A , ξ+, ξ−
Table 1 Modeling Equation Summary
Prescribed Quantities Equations Stressσ Temperature T (7), (21), (22)
Strainε Temperature T (7), (11)1, (21), (22)
Stressσ Heat rate ˙Q (7), (21), (22), (29)
Strainε Heat rate ˙Q (7), (11)1, (21), (22), (29)
Stressσ Media temperature T E (7), (21), (22), (29), (30)
Strainε Media temperature T E (7), (11)1, (21), (22), (29), (30)
Table 2 Representative Values for Material Constants
erning functions (16), with n= 1 and tension/compressionsymmetry, which implies thatγ∗
circum-Subloops The subloop model provides internal stress–
strain subloops within the fuller stress–strain curve that
is associated with complete transformation Repeated cling between either fixed stresses or fixed strains causesthe subloops to converge onto a final limiting response Thisallows for the prediction of shakedown behavior associatedwith either repeated stress cycling or repeated strain cy-cling (Fig 8)
cy-Pseudoelasticity and Reorientation For T > A f, bined tension/compression loading gives transforma-
com-tion behavior that alternates: A→ M+→ A → M−→ A →
M+→ · · · As the temperature is lowered, the model gives
isothermal stress–strain behavior with plateau stressesthat decrease with temperature in the correct way Once
the temperature is lowered below A f, the isothermalstress–strain behavior under tension/compression begins
to involve A↔ M transformation in conjunction with
di-rect M−↔ M+reorientation The model Eqs (7), (21), (22)track this multiple transformation activity The model also
retrieves pure M−↔ M+reorientation with
temperature-independent plateau stresses when T < M f For T <
A f, isothermal tension/compression loading excursions
200 400
Strain
Figure 8 Stress cycling between 150 MPa and 400 MPa is
mod-eled at T = 315 K > A fusing the trigonometric envelope functions
(13) and D-M governing functions (16) with n= 1 Each new loop
is richer in M+and leaner in A than the previous loop After about
five transient loops, the response has converged to a repeating loop
that stabilizes the cycling between A and M+.
Trang 20SHAPE-MEMORY MATERIALS, MODELING 979
Figure 9 Stress cycling between−150 MPa and 150 MPa is
mod-eled at T = 235 K > M f using the same envelope and
govern-ing functions as in Fig 8 Hereσ+
s = −σ−
s = 120 MPa and σ+
f =
−σ−
f = 210 MPa Since 120 < 150 < 210 MPa, the cycling
gener-ates a sequence of transformations: M−→M+→M−→M+→ ,
all of which are incomplete The convergent stable loop is
symmet-ric, because the particular material parameter choice represents
a tension/compression symmetric material.
generate open stress–strain curves (ferroelasticity) As is
the case for high-temperature behavior, loop convergence
takes place under repeated cycling The convergence is
im-mediate after the first cycle if the stress or strain
magni-tude is sufficient to complete all of the transformations
If, however, the cycling magnitudes do not cause complete
transformation, then the curves again shake down to their
limiting response (Fig 9)
Nonisothermal Pseudoelasticity Rate dependency
fol-lows from this model under the common condition of
con-vective heat transfer as described by Eqs (29), (30) For
fixed κ, faster loading gives less phase transformation
because convection inefficiency gives heat retention that
works against the A↔ M transformation The adiabatic
limit is approached under very rapid loading The opposite
limit of isothermal transformation occurs under very slow
loading Figure 10 shows the connection among
isother-mal, convective, and adiabatic loading as predicted by this
modeling
Experimental Validation As an example of validation of
the model, we report a comparison of the model
predic-tions with experimental data Figure 11 shows the results
of mechanical loading tests with temperature
measure-ments performed on 1 mm near equiatomic commercial
grade NiTi wires (47) and the corresponding prediction of
the model after straightforward material parameter
iden-tification The comparison gives excellent agreement in
predicting sublooping behavior and temperature change
under the combined conditions of cyclic loading,
transfor-mational heating/cooling, and convective heat transfer
Further features of the model that also follow but are
not obvious from the previous figures include
temperature-dependent stress–strain curves due to the change in
pseu-doelastic yield stress with temperature, stress–free strain
upon unloading due to the presence of stress-induced
martensite, and the shape-memory effect
Figure 10 Three stress–strain simulations beginning at T=
323 K> A f (above) This simulation uses a hyperbolic tangent
envelope function and slightly changed material parameters
sug-gestive of a different NiTi material microstructure (e.g., A f =
315 K, E M= 20000 MPa) The extended stress–strain curve is the isothermal response The curve on the far left is the adiabatic res-
ponse, and the middle curve models heat transfer to T E= 323 K over 20 s loading and unloading intervals Nonisothermal loading
gives heating since A→ M+ is exothermic Nonisothermal
unload-ing gives coolunload-ing since M+→A is endothermic Under adiabatic
conditions, the cooling returns the material temperature back to
T= 323 K at the conclusion of unloading Under convective
con-ditions, there is temperature undershoot on unloading (below)
be-cause of net heat transfer to the generally cooler ambient.
SUMMARY AND CONCLUSIONS
The thermomechanical stress–strain–temperature ior of SMM can be modeled so as to predict shape memory,pseudoelasticity, and martensite reorientation In order toachieve broad engineering utility, it is also necessary topredict sublooping, shakedown, and nonisothermal behav-ior Current models can reliably capture these effects under
behav-350 700
294 303
Figure 11 Comparison of this modeling (red) to measured
res-ponse (black) in a NiTi wire at a loading rate for which convection
is important.
Trang 21simple states of stress such as uniaxial tension or simple
shear These settings cover a rather wide range of
appli-cations such as actuators, vibrations absorbers and other
devices Models for multi-axial loads are not yet as
thoro-ughly validated with experiment, especially with regard
to reorientation under nonproportional loading As was
indicated by the literature surveyed above, these models
differ principally in how refined a description of the
mi-crostructure is implicit in their framework Some models
explicitly treat interactions between microstructural
enti-ties, whereas others treat the effect of microstructural
evo-lution in a thermodynamic framework with consolidated
internal variables
A brief description of a generally useful internal
vari-able model for uniaxial stress and a variety of heating
ef-fects was also presented The internal variables describe
the microstructure in terms of local phase fractions for
austenite A and two variants of martensite M+ and M−
with transformation strains of opposite orientation The
internal variables evolve with temperature and stress
ac-cording to kinetic equations (7), (21), (22) Strain and
en-tropy follow from Eqs (11) in a way that is consistent with
their status as thermodynamic conjugates to stress and
temperature Equations (29), (30) treat the effect of
trans-formational heating/cooling and heat transfer, which
nat-urally provides a rate effect to the stress–strain behavior
The examples presented here have not addressed the effect
of inhomogeneous deformation or nonuniform temperature
within the SMM material These issues can be treated for
this and similar models with appropriate field equations,
such as Eqs (4) to (6), and Fourier’s law This approach in
turn allows for computational simulation of SMM
compo-nents within larger systems
BIBLIOGRAPHY
1 I Muller and K Wilmanski, Il Nuovo Cimento B57: 283–318
(1980).
2 M Achenbach, Int J Plast 5: 371–398 (1989).
3 P Xu and J.W Morris, Metall Trans A24: 1281–1294 (1993).
9 L Truskinovsky In P.M Duxbury and T.J Pence, eds.,
Dynamics of Crystal Surfaces and Interfaces, Plenum, New
York, 1997, pp 185–197.
10 D Bernardini, J Mech Phys Sol 49: 813–837 (2001).
11 E Patoor, A Eberhardt, and M Berveiller, Arch Mech 40:
755–794 (1988).
12 Q.P Sun and K.C Hwang (1994) In J Hutchinson and T.W.
Wu, eds., Advances in Applied Mechanics, Vol 31, Academic
Press, San Diego, CA, 1994, pp 249–298.
13 V.I Levitas, Int J Sol Struct 35: 889–940 (1998).
14 B.C Goo and C Lexcellent, Acta Mater 45: 727–737 (1997).
15 Z.K Lu and G.J Weng, Acta Mater 46: 5423–5433 (1998).
16 M Huang and L.C Brinson, J Mech Phys Sol 46: 1379–1409
20 F Falk, Acta Metall 28: 1773–1780 (1980).
21 Y Huo, Continuum Mech Thermodyn 1: 283–303 (1989).
22 J Ortin, J Appl Phys 71: 1454–1461 (1992).
23 A.A Likhacev and Y.N Koval Scripta Metall Mater 27: 223–
26 J.R Rice, J Mech Phys Sol 19: 433–455 (1971).
27 K Tanaka and S Nagaki, Ingenieur Archiv 51: 287–299
31 I Muller, Continuum Mech Thermodyn 1: 125–142 (1989).
32 J Ortin and A Planes, Acta Metall Mater 37: 1433–1441
35 L.C Brinson, J Intell Mater Syst Struct 4: 229–242 (1993).
36 A Bekker and L.C Brinson, J Mech Phys Sol 45: 949–988
45 R Wasilewski, Metall Trans 2: 2973–2981 (1971).
46 J Ortin and A Planes, J Physique IV, C4: C4–C13 (1991).
47 D Bernardini and F Brancaleoni, In Proc Manside Workshop,
January 28–29 1999, Rome, Italy, part II, pp 73–84.
48 K Gall and H Sehitoglu, Int J Plast 15: 69–92 (1999).
49 P Sittner, M Takakura, and M Tokuda, Mater Sci Eng A,
234–236:216–219 (1997).
Trang 22SHIP HEALTH MONITORING 981
SHIP HEALTH MONITORING
Although ships have been sailing for hundreds of years,
the field of ship health monitoring is relatively near Two
main forces have spurred the rise in the popularity and
importance of ship health monitoring systems First,
re-duced budgets and rere-duced manpower have resulted in
smaller crew sizes and less routine maintenance This
re-duced maintenance has brought into question the health
of many ships, their components, and the ability to detect
a problem quickly Clearly, a failure aboard any vessel will
bring a loss of revenue and productivity, if not worse The
second driver behind the growth of ship health
monitor-ing systems is the decrease in the expense of computers
and sensors coupled with an increase in their capabilities
and the associated data processing algorithms Until
re-cently, an automated health monitoring system was either
impossible to achieve or prohibitively expensive to install
and maintain Today, ship health monitoring systems are
becoming increasingly more common as their capabilities
are further demonstrated and understood by the maritime
community Nevertheless, many issues remain to be settled
as technology improves, resulting in continuously
expand-ing requrements
OVERVIEW OF SHIP HEALTH MONITORING
Modern ships, both commercial and military, are extremely
complicated machines, so that the aspects of the ship to
which a ship health monitoring system can be applied
are numerous To date, efforts have focused primarily on
the global hull response and diagnostic monitoring of the
propulsion system Although these two applications have
the greatest potential for financial and safety
improve-ments, other areas, such as local hull stresses and cargo
tanks, can benefit from monitoring (1) In military
appli-cations, many specialized monitoring applications can be
envisioned, such as weapon systems or ordinance
monitor-ing and battle damage estimates (2)
Potential Benefits
Ship health monitoring systems can be employed for many
reasons In almost all cases (except research vessels), the
primary reason is to reduce the overall cost of ship
op-erations In isolated cases, other reasons have dictated
the use and development of monitoring systems Another
readily apparent benefit of an installed system is failure
prevention Although this is rarely the primary factor for
ship monitoring, the safety benefit gained from avoiding a
catastrophic failure is a strong motivator for using a
sys-tem Finally, because of the advanced technology and
re-cent emergence of ship health monitoring systems, many
systems have been installed for research The research hasfocused on the capabilities, benefits, and logistics of long-term ship health monitoring system use
Financial Ultimately, for the field of ship health
mon-itoring to be viable, installed systems must decrease theoverall costs of operating a ship and/or provide a substan-tial performance benefit In both cases, a financial impetuswill exist for installating and using a ship monitoring sys-tem In general, cost savings will come from three sources:
a reduction in maintenance labor-hours, reduced ment costs, and increased readiness and uptime Unfortu-nately, of these three, only the cost savings from the reduc-tion in maintenance labor-hours can be easily calculated
equip-If a ship health monitoring system is operating properly,many equipment failures are likely to be detected earlierbefore they become disastrous Early detection will in turnhelp prevent one failure from affecting other componentsand lower the costs to fix a failure, resulting in a reduction
in equipment costs that is hard to quantify Furthermore,unscheduled maintenance will be reduced, resulting in anincrease in ship availability The increased availability willthen create higher revenues for the entire system
Safety An additional benefit of ship health monitoring is
the inherent safety provided by such a system If the cal health of critical areas is constantly monitored, catas-trophic failures are less likely to occur Most critical regions
physi-of the ship can be monitored for stress overloads Should
a stress overload occur, the monitoring system quickly forms the bridge and provides pertinent information to thecrew to lessen the amount of damage caused by the over-load If a failure has occurred, this information will also
in-be passed to the bridge In this case, although the systemwas unable to prevent a major failure, it will give the crewadditional time and information to deal with the problem.For military vessels, the safety of the crew can be protected
by the provision of real-time, accurate battle damage mates Again, this information will allow for a rapid dam-age assessment that could easily save lives onboard a crip-pled vessel
esti-Potential Dangers to Ships
Ocean and seagoing ships are subject to many tial hazards The following sections describe some of theoperational hazards in detail In general, these hazards arecaused by wind and waves, ice, and material (cargo) han-dling/storage In addition to the hazards faced by commer-cial vessels, military vessels face the obvious threats fromenemy actions However, this section discusses only haz-ards that are encountered during typical ship operations
poten-Wind and Waves Waves, both large and small, are
an ever-present hazard to shipping Waves stress a shipthrough several different phenomena that are adaptedfrom (3) and described here
Quasi-Static Hull Girder Stress (Global Stress). Hullgirder shears and moments are caused by the cyclic buoy-ancy of a wave that is superimposed on the ship’s geometry
in a quasi-static balance with the ship accelerations The
Trang 23moment values depend more on the projected wave length
superimposed on the hull (wave length/cosine of the
head-ing angle) than on the encounter frequency However, the
pitch and heave resonance (a function of the encounter
fre-quency versus the motion natural frefre-quency) can increase
the hull girder moments
Hull Girder Whipping (Global Stress) When a structure
is impacted, much of the impact energy is absorbed by the
structure as vibrational energy This vibration generally
forms as motion of the structure at its first natural
fre-quency When a ship is impacted, such as during a slam,
the ship hull vibrates in its fundamental bending modes
(vertical and lateral) This is termed hull girder whipping
Slams can occur on the bottom and on the flare of the ship’s
bow Bottom slams occur when the forefoot of the ship is
lifted clear of the sea by severe ship and wave motions
during rough seas A slam occurs as the ship reenters the
sea and the bow impacts the water Flare slamming may
occur as the result of relative motion between the vessel
and the sea, even without bow emergence, but can also
occur when there is little motion between the vessel and
the sea, if the wave is steep enough Bottom slams tend to
have shorter lengths and higher loading frequencies than
flare slams The dominant slam type depends on the ship
type High-speed containerships that have finer forward
lines and a flaring bow experience greater stresses from
a flare slam, but the opposite is true for full-form tankers
that have little flare Measurements on an aircraft carrier
have shown that the whipping moments are of the same
magnitude as the quasi-static moments during flare slam
(1) Although the whipping vibrations and energy
dissipa-tion mechanisms are not well understood for large complex
vessels, they are generally less severe in flexible (i.e., high
L/D ratio) ships (4) The whipping moment components are
usually small compared to the quasi-static moment, but the
whipping moments occur at higher frequencies Recent
in-vestigations suggest that whipping may increase fatigue
damage by 20% to 30% (5)
Springing (Global Stress) When a ship impacts waves
at a frequency that is at or close to the primary hull
res-onant frequency (the two-noded, vertical bending mode),
springing may occur Springing is steady-state resonance
of the ship at its natural frequency Although this is also a
low-frequency event, springing frequencies are of an order
of magnitude higher than quasi-static hull girder stresses
The resulting moment, especially when superimposed on
the quasi-static stresses, may be significant in long-term
fatigue damage Ships can experience springing in small
and moderate seas, as long as the encounter frequency
ap-proaches the ship’s natural frequency
Wave Refraction (Local Stress) Hull girder stresses are
generally caused by larger waves of the order of the ship’s
length But the lower stresses created by smaller waves
impacting on the sides of the ship can cause localized
long-term fatigue damage and may lead to cracks and crack
propagation This effect is intensified by wave reflection in
beam seas Localized fatigue cracking has been a problem
on some Trans Alaska Pipeline System (TAPS) trade
tankers where the local waves generally strike the
star-board side during the southern voyages and the port side
on the northerly routes
Slamming (Local Stress). The damage from forefootslamming has been mentioned previously as a cause ofwhipping In addition, the locally high stresses can causedamage to the bow structure Bottom slamming in shipsoften results in dishing of the bottom shell plate, and flareslamming results in dishing of the side shell and possiblythe loss of the flare strake
Ice In addition to the stresses caused by sea loading,
ice represents a significant danger to shipping The dangerfrom ice comes primarily from localized impact loading onthe hull However, global stresses can also become a factordue to hull girder whipping
Ice Transit (Local Stress) Because ship speed, heading,
and ice conditions can vary greatly, local ice loads on theship’s structure are complex The danger from ice dependsmore on local stresses than on the global hull forces Ship-board measurements have shown that typical hull girderstresses induced by ice transit are less than those induced
by opwater waves (3) The pressures and forces countered during ship–ice impacts are random and followlog-normal type probability distributions (6) Ice loads arenonuniform, and high loads are applied to small areas ofthe hull (e.g., 0.5 m2) In addition, these loads occur at manylocations along the hull, predominately on the bow The in-stantaneous area of the hull that is most highly loaded de-pends on the type of operation (ramming, turning, etc.) andthe geometry and strength of the hull structure Ice loadsare more difficult to measure than slamming loads because
en-of their high frequencies and random distributions ies indicate that strain rates for ice loading in the localstructure are similar to those for the global response andthat neither of these is significantly different from thoseexperienced from sea loading (3)
Stud-Hull Girder Whipping (Global Stress) Stud-Hull girder
whip-ping caused by slamming is described in the previoussection Because whipping is the vibration of the ship’sprimary bending modes due to impulsive loading, ice ram-ming can also induce hull girder whipping This effect isusually felt during initial ice impact and less during steadyice transit
Cargo As for ice, cargo and material handling can also
add additional stresses to a ship’s hull Cargo loads can becategorized into two forms
Static Hull Girder Stress (Global Stress)
Quasi-static hull girder stress was mentioned previously whencaused by long length waves superimposed on the geom-etry of the ship, causing high moment and stress val-ues For cargo, quasi-static girder stresses are caused
by differences in the loading distribution curve and theship’s buoyancy curve along the length of the ship.Care must be taken during cargo loading and unload-ing so that the maximum allowable stress values are notexceeded
Cargo Loads (Local Stress) Cargo loading anomalies can
often result in localized regions of high stress Two ples of loading anomalies are uneven loading in bulk ships(this, it has been hypothesized, is the cause for a number ofbulk ship failures) and unequal hydrostatic pressure heads
Trang 24exam-SHIP HEALTH MONITORING 983
across tank boundaries The American Bureau of Shipping
(ABS) SafeHull code specifically considers a checkerboard
loading pattern in cargo and ballast tanks as a worst case
scenario Hence, the loading sequence can result in
exces-sively high global and local stresses
Different Ship Types and Requirements
Ship Types Specific ship designs are susceptible to
var-ious hull responses Table 1, taken from (3), shows the
typical monitoring requirements for several common ship
types
Operating Environment Ships are designed for many
functions and operate in a wide variety of conditions The
size, type, and operating environment of a ship greatly
in-fluence the requirements placed on a health monitoring
system For example, because container ships that
oper-ate in calm, smooth woper-aters in the tropical zones are rarely
subjected to icy conditions or severe storms, whipping and
other wave/ice-induced fatigue stresses are not critical
fac-tors Therefore, a proper health monitoring system should
concentrate on cargo-induced loads and maximizing
oper-ational efficiency
On the other hand, TAPS trade tankers are constantly
subjected to severe storms, high waves, and very
direc-tional sea states (7) As cargo runs are made to the south,
waves primarily strike the starboard side of the ship On
the return trip to the north, waves strike the port side
pri-marily This pattern has resulted in localized fatigue
prob-lems and requires a high density of local stress sensors to
detect the onset of cracking In the North Sea, hull girder
bending, slamming, and green water are present because of
Table 1 Monitoring Requirements by Ship Type
Passenger ship Ship motion (roll)
Bow flare slam Tanker/products carrier Midship hull girder stress
Bow/amidships shell stiffeners Forefoot slam
Explosive environment Bulk ships Cargo loading hull girder stresses
Cargo hold frame stresses Stress concentrations at hatch corners Forefoot slam
Container ships Stress concentrations at hatch corners
Hull girder torsion Bow flare slam Green water Whipping/cargo accelerations LNG/internal tank Forefoot slam
Temperature/explosive environment Sloshing
Barges/platforms Towline/mooring tension
Motions and inertial forces Lateral motion
Naval combatant Bow flare slam
Fire control plane deflections
the very steep waves and require sensors similar to thosefor the TAPS trade tankers to detect any potential prob-lems Bulk ships operating on the Great Lakes are sus-ceptible to springing because of their high L/D ratios andrequire global hull bending sensors Ice breakers are sub-ject to high localized stresses, especially around the bow,but the global bending stresses are typically lower thanthose experienced by other oceangoing vessels
These basic generalizations apply only to small groups
of ships that continuously operate in a given environment.Many ships are not easy to classify; their health monitoringconsiderations must include all of the potential operatingconditions for a specific vessel Most military vessels areprime examples of ships that are commonly operated with
no fixed route and that experience many types of sea statesthroughout a voyage
Additional Capabilities
Industry has realized that ship operators expect a shiphealth monitoring system to do more than simply mea-sure the state of stress throughout the ship and monitormachinery health As mentioned previously, ship healthmonitoring systems must be financially attractive to becommercially acceptable It is clear that preventing the loss
of a ship or increasing the useful life of a ship by structuralmonitoring are financial benefits However, additional ben-efits can be gained by incorporating external (non-ship-based) sensor readings into a comprehensive monitoringsystem These external readings are currently composed oftwo primary types: weather avoidance/planning and routemonitoring
Weather Reports Current weather reports are available
to ships from a variety of environmental sensor platforms,including fixed land sensors, ocean buoys, other ships, air-craft, and satellites Collectively, these platforms can pro-vide the necessary information to a ship to help prepare
an optimal route to increase the efficiency of the route and
to lessen any damage that may be caused by ice or poorweather
Fixed Land Sensors Fixed land sensors are primarily
used to measure basic meteorological conditions such aswind speed and direction, temperature, and precipitation
Ocean Buoys The National Data Buoy Center (NDBC)
operates the Ocean Data Acquisition System (ODAS),which is a network of more than 60 buoys that are anchored
in deep ocean areas off of North America These buoys sendsatellite transmissions to the National Weather Service(NWS) that provide weather and oceanographic data fromtheir stations in the Atlantic, Pacific, Gulf of Mexico, andthe Great Lakes The wind speed and direction data, onceprocessed by NDBC, is reportedly accurate to within± 10◦and± 1 m/s
Ships NWS receives weather reports every 3 hours
from ships that participate in the U.S and World orological Organization (WMO) Voluntary Observing ShipProgram (VOS) These reports include basic meteorologi-cal conditions and best estimates of the current sea state,ice, and visibility These programs include 49 participatingcountries and approximately 7000 ships that provide about
Trang 25Mete-1000 reports a day The U.S program has existed as a
des-cendant of the U.S Coast Guard Ocean Weather Station
ships for several decades The data provided by the VOS is
commonly used for weather forecasting and is distributed
by the National Ocean Weather Service through the Global
Telecommunications System
Aircraft Aircraft are most commonly used to track
hur-ricanes in the Atlantic Ocean But they have also been
ex-perimentally used to track ice conditions in the polar
re-gions (8) and have been used to provide Synthetic Aperture
Radar (SAR) readings to estimate local sea states
Satellites In general, satellites provide a great deal of
information to weather forecasters Due to the advent of
modern satellite imaging technology, forecasters can
accu-rately predict the weather many days in advance In
ad-dition to the basic weather forecasting functions, several
satellite instruments have been used to aid in ship
naviga-tion
Advanced High-Resolution Radiation (AVHRR) sensors
are used to sense ocean temperatures and map sea
cur-rents Flown by NOAA since 1978, AVHRR sensors detect
infrared radiation to measure the sea surface temperature
This data is critical to helping oceanographers track ocean
currents Generally, two AVHRR satellites are in polar
or-bit on 24-hour cycles phased 12 hours apart to give both
day and night readings Although clouds interfere with
AVHRR readings, this interference has not significantly
affected their usefulness
Radar altimetry has been used to measure the distance
between a satellite and the ocean waves to estimate the sea
state Although the technology was first demonstrated
on-board the GEOS-3 in the 1970s, suitable accuracy was not
obtained until the recent launch of the TOPEX/Poseidon
in 1992 However, until additional satellites become
oper-ational, this technology will be used primarily for research
A final sensor technology is scatterometry
Scatterom-etry measures the scatter within a return pulse from a
radar altimeter to determine the roughness of the seas
Calm seas give a clear, concise radar reflection, whereas
rough seas tend to distort the return The sea state is then
related to the local wind speed through an empirical
cor-relation Again, scatterometry is a new technology that is
currently in the development stages and that will hopefully
be available to health monitoring systems in the future
Route Monitoring/Planning Adding the capability of
monitoring weather conditions to a ship health monitoring
system can improve the crew’s ability to plan an optimal
course through the weather Ideally, though, the
moni-tored weather conditions would be integrated with a route
planning system to provide an optimal route to the crew
automatically By integrating the local health monitoring
system with a real-time routing system, the ship’s
han-dling can also be adjusted to minimize danger to the ship
Using this type of system, it is possible to plan the best
route to reach a given destination while reducing time and
fuel consumption from unwanted ship responses Several
technologies needed for such a system are already in
ex-istence, including weather forecasting, the predicted ship
responses, and the local sensors to determine the actual
ship response However, an intelligent software product for
performing the optimization at real-time speeds has notbeen fully realized
The benefits of route planning were demonstrated in
1993 by ARCO Marine (9) Two sister TAPS trade tankerstraveled from San Francisco, California to Valdez, Alaskawith the same ballast condition One ship contained a voy-age planning system based on the predicted wave heightsand directions; the other did not The ships remainedwithin a narrow corridor and varied only the timing andspeed The ship that used route planning arrived approxi-mately 18 hours earlier than the sister ship, and the sistership suffered $400,000 in wave-induced damage
ENVIRONMENTAL ISSUES
For a ship health monitoring unit to be acceptable, it shouldnot adversely affect the operation of the ship or its crew.Furthermore, the system must be reliable and require verylow maintenance A primary driving benefit of a ship healthmonitoring system is a reduction in crew workload If thesystem is constantly in need of repairs, this benefit doesnot materialize Unfortunately, the maritime environment
is very harsh and unforgiving Every component of a healthmonitoring system must be considered for reliability andmaintenance The primary factor that reduces the life ofship components is the highly corrosive marine environ-ment Many other sensor location-specific factors may alsocontribute to component failures These include explosiveenvironments, inadvertent physical damage by the crew,and operational overloads
Corrosive Marine Environment
The marine environment is tremendously harsh and sive This environment quickly affects most exposed metalsand many other materials Therefore, a ship’s health mon-itoring system components must either be protected fromthis environment or constructed of materials that are notsubject to marine corrosion Although any material willeventually corrode, corrosion in the marine environment
corro-is more severe for several reasons First, most metals rode slowly at ambient temperatures and low humidity.Increasing the humidity provides water, which is neces-sary as an electrolyte for charge transfer In addition to theever-present water, the saline environment of ocean waterfurther increases corrosion by speeding up the localizedbreakdown of oxide films The chloride also increases theconductivity of seawater compared to freshwater, again in-creasing the corrosion of many metals A final contributor
cor-to corrosion is the low acidity of seawater The pH of water is usually 8.1 to 8.3
sea-Other
In addition to the highly corrosive marine environment,ship health monitoring sensors are placed in other types ofextreme environments Many modern ships are designedfor and dedicated to transporting large quantities of oil
or natural gas For these ships, it is desirable to sure the health of the fuel storage tanks This environ-ment places additional safety requirements on the sensor
Trang 26mea-SHIP HEALTH MONITORING 985
and any associated electronics due to the explosive nature
of the cargo A few sensors are intrinsically safe in an
explosive environment, but most sensors must be
encap-sulated or encased in an explosion-proof container, which
can substantially increase the cost and complexity of the
sensor
Another concern is to protect the health monitoring
sys-tem’s components from physical harm It is often necessary
to place sensors or wiring in locations that are susceptible
to physical damage Examples include deck-mounted
com-ponents that can be damaged by the crew’s activities or
forefoot-mounted pressure sensors that must be designed
to handle the high forces experienced during slams or
ice-breaking duties In some early systems, pressure sensors
failed most frequently of all equipment (3) Without the
proper protection, these sensors fail quickly
SENSOR TECHNOLOGY
The sensor network is crucial to providing real-time,
accu-rate information from the ship health monitoring unit to
the ship’s crew It is the distributed sensors that directly
measure the motions and health of the ship To provide a
detailed picture of the entire ship, these sensors must
com-prise many different types and must be placed throughout
the vessel Therefore, current and future health monitoring
systems will incorporate a wide variety of sensor types that
measure a diverse number of physical parameters Figure 1
illustrates a possible ship health monitoring sensor
ar-rangement In addition to choosing the physical
param-eters that must be measured, where they are to be
mea-sured, and the type of sensor to be used, one must make
decisions concerning the cost, reliability, and safety of the
specific sensor Current state-of-the art sensors can meet
these objectives, but novel sensor designs are continuously
being developed and must be considered as possible
im-provements over existing designs
Vertical acceleration
Long-base strain gauges
Data acquisition display and recorder
Trust power shaft speed Ship motions
roll and pitch vertical acceleration In-tank local
strain gauges
Measurands and Potential Sensors
As mentioned, a wide variety of parameters must be sured to obtain an adequate picture of a ship’s motionsand health The following sections describe many of thesemeasurands and the potential sensors that are currentlyavailable for the measurement
mea-Pressure Pressure gauges are most often used in a ship
health monitoring system to measure slamming pressures,forefoot emergence, and in-tank hydrostatic pressures.Multiple pressure gauges are often located longitudinallyalong the forefoot to detect the extent of emergence and
to determine the magnitude and extent of bottom impactpressures (10) To facilitate maintenance, it is important toensure that any sensors, especially forefoot pressure sen-sors, can be replaced or repaired without entering dry-dock.There are two primary types of pressure transducers:diaphragm and piezoelectric types Both types are com-monly available commercially at about the same cost, buteach has certain advantages and disadvantages
Diaphragm-Type Pressure Transducers There are two
types of diaphragm style pressure transducers: one has
a clamped circular plate, and the other employs a hollowcylinder However, the clamped plate design is more suitedfor the pressure ranges of ship health monitoring, and onlythey are discussed in this section The strain distribution
on a clamped circular plate of constant thickness has beensolved analytically and has been experimentally validated.Based on these results, a special purpose diaphragm straingauge has been designed to take advantage of this straindistribution Using this type of strain gauge arrangement,one finds that the pressure is proportional to the measuredstrain Typical strain gauge instrumentation can be used
to sample the data
Piezoelectric-Type Pressure Transducer. This type ofpressure transducer uses a piezoelectric crystal as both thediaphragm and sensor In general, the piezoelectric crystal
Trang 27Figure 2 Long-base strain
gauge.
(most commonly quartz) is placed inside a hollow
cylin-der Because of the piezoelectric effect, an applied
pres-sure generates an electrostatic charge that is proportional
to the pressure The piezoelectric crystal has a high output
impedance Therefore, a charge amplifier is commonly
em-ployed to convert the charge into an amplified voltage that
is read by a standard voltage recorder The low-frequency
response of the transducer depends on the time constant of
the amplifying circuit but can be designed to use
frequen-cies that are low enough for ship health monitoring The
primary disadvantage of this type of transducer is that the
charge amplifier electronics make the system less
intrinsi-cally safe than diaphragm-type pressure transducers
Global Strain Global strain is most commonly measured
using long-baseline strain gauges Long-baseline strain
gauges normally consist of a long rod (approximately 2
me-ters long) rigidly attached at one end to the hull The
sec-ond end is allowed to move freely through a set of guides
that ensure only axial movement The extent of movement
of the free end, as measured from a set “zero” strain
loca-tions, divided by the length of the rod gives the average
strain in the deck across a region of the hull that is the
length of the rod Figure 2 is an illustrative example of a
long-base strain gauge
Although the basic mechanism is simple and robust,
these devices are commonly placed on the deck of a ship
and must be appropriately protected from both physical
damage and the environment Three common techniques
are currently available for measuring the displacement of
the free end
Linear Potentiometer Linear potentiometers are
sim-ple resistors that have a varying resistance that is
pro-portional to the displacement The displacement is
mea-sured by sending a low voltage and current through the
po-tentiometer and measuring the resistance This method is
very inexpensive, and recent advances have made precision
linear potentiometers as accurate and repeatable as other
technologies One disadvantage is that the resistor’s life is
limited and it must be protected from the environment
Linear Variable Differential Transformers (LVDT) LVDTs
are the most popular variable-inductance sensor used for
displacement measurements In an LVDT, a magnetic core
moves through an insulated bobbin without physical tact Three symmetrically spaced coils are wound aroundthe bobbin The position of the magnetic core controls themutual inductance between the center primary coil andthe two outer secondary coils When a voltage is applied tothe primary coil, a voltage is set up in the two secondarycoils The secondary coils are wired 180◦out of phase witheach other Therefore, when the core is centered within thebobbin, the secondary voltages are of equal magnitude andcancel out However, a small movement of the core results
con-in a larger voltage con-in one of the secondary coils and hence
a sensor reading Because there is no contact between faces, LVDTs are free from friction and have very long lives.The response is also free from hysteresis The resolution of
sur-an LVDT is partially determined by the voltage recorder,which give LVDTs superb resolution and accuracy The twoprimary disadvantages of LVDTs are their higher cost com-pared to linear potentiometers and the relatively high volt-age (5 to 15 volts) that must be supplied to the primary coil
Linear Displacement Transducer A linear displacement
transducer is a magnetostrictive sensor that measures thetime between an interrogating magnetic pulse and a returnpulse that is generated by a magnet connected to the freeend of the rod As for an LVDT, this type of device has
no contacts and a long service life It is also intrinsicallysafe for hazardous environments but is much higher in costthan either of the other two options
Local Strain Local strain measurements are often used
in ship health monitoring systems They are applicable todetecting a wide variety of potential hazards They aremost commonly used along the bow, in cargo and ballasttanks, and along any other critical internal structures.Local strain gauges are also ideal for detecting crackingaround certain areas such as hatch corners and highlystressed welds In general, the number of local straingauges is limited by the cost of installation and data ac-quisition as opposed to the cost of the gauges or the desire
to measure more parameters Ideally, almost any failurecan be detected by a local strain gauge if it is installed inthe right location
By far the most common method of measuring localstrains is to use a resistance foil strain gauge Details of
Trang 28SHIP HEALTH MONITORING 987
the performance and use of resistance strain gauges can be
found in many texts The strain gauge operates by
detect-ing a change in resistance of metal wire as it is stretched
The change in resistance is caused by a decrease in the
con-ductive cross-sectional area due to Poisson’s ratio To
am-plify the change in resistance, one employs thin metal-foil
grids to maximize the amount of conductor within a given
region Because the change in resistance is very slight, a
Wheatstone bridge is usually used to convert the changing
resistance to a variable voltage Resistance strain gauges
can be purchased in a wide variety of sizes, sensitivities,
and geometries It is also possible to purchase strain gauge
rosettes that permit measuring all three surface stress
components simultaneously Because strain gauge
read-ings depend on measuring a changing resistance, they can
be made safe by using very low voltages
An important consideration is the temperature
sensi-tivity of strain gages As a material is heated or cooled, the
material expands or contracts, depending on the material’s
coefficient of thermal expansion If the strain gauge does
not similarly expand or contract, an apparent strain will be
seen by the strain gauge Two methods exist for
eliminat-ing this apparent strain The first method is ensureliminat-ing that
the coefficient of expansion is identical for both the
struc-ture and gauge Therefore, both expand at the same rate,
and no apparent strain is seen For this reason, commercial
vendors sell strain gauges made from a wide variety of
ma-terials The second approach is not as attractive for
practi-cal applications In this approach, a second gauge/material
combination is bonded together but is placed in a region
that is completely free of stress Through this technique,
the amount of apparent temperature-induced strain can
be calculated from the second gauge and can be subtracted
from the active gauge to provide the final mechanical
strain reading However, finding a suitable location for the
dummy gauge is not generally possible
The installation of strain gauges is also a
well-documented field that has a myriad of options The most
common techniques include adhesive bonding and welding,
but other methods are also available for packaged sensors
Generally, it is preferable to protect the strain gauge
af-ter installation Again, this is done by using a variety of
protective coatings and encapsulation techniques Many
strain gauges sold have the gage itself prepackaged into
an encapsulated enclosure for protection
Motion Ships’ motions are possible in all six degrees of
freedom The three translational degrees of freedom are
surge (longitudinal), sway (lateral), and heave (vertical)
Roll, pitch, and yaw are the three rotational degrees of
freedom Roll, pitch, and heave are generally considered
the three most important degrees of freedom for the
fol-lowing reasons:
Roll Roll generally affects the crew and cargo loads.
In addition to crew and passenger discomfort, rolling
cre-ates lateral cargo loads that must be resisted by horizontal
restraints In fluid tanks, rolling may increase the
hydro-static pressure head or induce sloshing A ship’s master
usually turns the ship into waves to reduce rolling, but
this increases hull girder stresses
Pitch Pitching has the same effect on fluid-filled cargo
tanks as rolling and results in an increased hydrostaticpressure head and sloshing In addition, the length of theship increases the distance between the ends of the shipsand the pitch axis, resulting in high pitch accelerationsthat may in turn result in slamming
Heave Heave is more pronounced than surge or sway
because of wave motions and the coupling with pitch tions Heave causes effects similar to roll and pitch in fluid-filled tanks In addition, the vertical accelerations requirevertical cargo restraints
mo-Ship motions are commonly measured by a variety ofsensor types The most common devices are accelerome-ters and gyros, although there are many variations of both.Table 2 lists the current state of the art in ship motionsensors (11)
Shaft Speed, Power, and Thrust The ship’s performance
parameters (shaft speed, power, and thrust) can providemeasurements of the propulsive efficiency relative to thecurrent environmental conditions These measurementscan also provide an indication of the health of the propul-sion equipment Most ships are currently configured withequipment to measure these parameters directly from thepropeller shaft In general, this information is alreadypassed into the engine control room and can be furtherrouted to the ship health monitoring system
Global Positioning Similar to the propulsion system
measurements, ships usually already have an installedGlobal Positioning Systems (GPS) As a rule, GPS unitsoutput a serial signal dictating the ship’s position (lati-tude and longitude), heading, and speed on a regular ba-sis Therefore, the ship health monitoring unit only needs
to read this information from the preexisting unit
Advanced Sensors Fiber-Optic Strain Gauges One promising new technol-
ogy is fiber optic strain gauges There are two primarytypes of fiber-optic strain gauges, Fabry–Perot and fiberBragg grating sensors (12) Both types offer a number ofadvantages over traditional resistance strain gauges Be-cause fiber-optic strain gauges use light as the sensing andtransmitting element, they are intrinsically safe and pose
no fire or explosive hazards Furthermore, fiber optics arevery resistant to corrosive elements because they have nometallic components and are covered by protective hermet-ically sealed coatings A final common advantage is thatfiber optics are immune to electromagnetic interference.Therefore, they can be placed in regions of high electric ormagnetic fields without any degradation of performance.Both types of fiber-optic strain gauges are also capable ofstrain resolutions equal to or greater than that of resis-tance strain gauges
Fabry–Perot Fiber-Optic Strain Gauges. Fabry–Perotstrain gauges are manufactured by placing a small air gap(or an internal mirror) within a fiber, followed by a reflec-tive surface, which can be either a micromirror or moreoptic fiber A broadband light wave is transmitted downthe length of the fiber At the first junction between the
Trang 29Table 2 Ship Motion Sensor Technology
Roll and pitch Vertical gyro Reliable, may be able to Drift, cost, power
use existing ship unit Magnetometer Moderate cost Calibration on steel ships Solid-state gyro Low cost and power, units Sensitive to external vibrations
packaged with integral rates and displacements
“Watson meter” Reliable, accurate for Moderate cost
pendulum-based design
Yaw (heading) Gyrocompass Current state of the art Expensive, frequently needs service
Solid-state Low-cost combination of rate Unproven, unknown life and reliability gyro (KVH) gyro and flux gate compass
Solid-state gyro New laser ring technology, Expensive, not yet commercialized (fiber optic) no moving parts for ship use
Flux gate Good for small ships once Difficult to use effectively unless compass compensated for, low cost able to swing the ship for compensation Magnetometer Moderate cost Calibration on steel ships
Piezoelectric Good for machinery Unsuitable for ship response accelerometer vibrational measurements frequencies
Surge Sway Heave Piezoresistive Low cost, good for short- Subject to temperature
accelerometer term ship motions cross-axis errors Servo-accelerometer Excellent stability, Expensive
accuracy, and reliability Capacitative Moderate cost, performance Cross-axis sensitivity higher accelerometer nearing that of a servo- than that of servoaccelerometers
accelerometer
fiber and the air gap (or mirror), some of the light is
re-flected back to the source, and some of the light is
trans-mitted into the gap At the mirror or second air/fiber
in-terface, light is again reflected and transmitted Now, two
separate light sources are reflecting light back along the
length of the fiber The length of the air gap dictates the
phase difference between the two waves Allowing this air
gap to expand or contract, based on the local strain, creates
a strain sensor Hence, a measurement of the phase offset
can be correlated to a strain measurement There are
ex-trinsic Fabry–Perot sensors, inex-trinsic Fabry–Perot sensors,
and in-line fiber etalon (ILFE) sensors All three are based
on the same principle, the differences lie in the choice of the
reflective medium Although intrinsic Fabry–Perot sensors
are sensitive to strain and temperature, extrinsic Fabry–
Perot sensors and ILFEs have very low thermal
sensitiv-ity One disadvantage of Fabry–Perot sensors, compared to
fiber Bragg grating sensors, is the difficulty in multiplexing
many sensors along a single fiber
Fiber Bragg Grating Strain Gauges Fiber Bragg gratings
are based on the photorefractive effect Bare fiber is
ex-posed to a hydrogen environment and then imprinted
us-ing an ultraviolet laser The imprintus-ing is done by one of
several methods, and it leaves a series of equally spaced
lines along a region of the fiber This series of lines is
called a Bragg grating; the lines are actually very small
regions that have a slightly different index of refraction
Bragg gratings can be fabricated through an
interferomet-ric (holographic) method or by using phase masks In a fiber
that has a Bragg grating, transmitted broadband light is
reflected back toward the source at a specific frequency thatcorresponds to the grating wavelength All other frequen-cies of light pass unaffected through the Bragg grating.Because the frequency of the reflected light is propor-tional to the spacing of the Bragg grating, a change in thespacing will result in a change in the reflected wavelength.Hence, a strain gauge can be made by bonding a Bragggrating of a specific wavelength to a structure As the struc-ture is strained, the Bragg grating will expand or contract,thereby changing the wavelength of the reflected light Bymeasuring the wavelength of the reflected light, one candeduce the strain at the location of the grating (13).One major advantage of using fiber Bragg gratings as lo-cal strain sensors is the capability of using wavelength and/
or time division multiplexing to place many Bragg gratings(strain sensors) along a single optical fiber (14) When abroadband light source is used with a Bragg grating, everywavelength, except the wavelength corresponding to thegrating, is transmitted through the grating Therefore, asecond Bragg grating, at a different wavelength, may beplaced further along the fiber This second Bragg gratingwill reflect a different wavelength back to the source Now,two separate strain readings can be taken by monitoringthe two reflected wavelengths This process can be repeatedmany times along the length of the fiber, which allows mak-ing many distributed local strain readings within a singlefiber-optic cable that also transmits all of the data back
to the control computer Similarly, time division ing can be achieved by monitoring the time of return ofthe Bragg grating wavelengths, enabling interrogation of
Trang 30multiplex-SHIP HEALTH MONITORING 989
multiple sensors along a fiber One disadvantage of Bragg
grating strain sensors is its strong thermal sensitivity
Nu-merous methods have been proposed to compensate for this
thermal sensitivity, but none have yet been commercially
successful
Other Fiber-Optic Gauges In addition to fiber-optic
strain gauges, Fabry–Perot and Bragg grating strain
sen-sors have been incorporated into other designs to enable
de-tecting pressure, temperature, or even chemical content In
general, these sensors have many of the same advantages
due to the nature of fiber optics as opposed to electrical
components But again, very few of these hybrid sensors
are currently available commercially These sensors will
probably become more available and less expensive as the
technology matures Using Bragg grating-type sensors, it
will also be possible to construct a series of varying sensors
that are connected to the same fiber-optic transmission
ca-ble For example, the fiber-optic cable from several forefoot
pressure sensors could be run up to the cargo tanks where
several Bragg grating strain sensors were located Each of
these sensors could be multiplexed together so that all of
them are interrogated by a single fiber-optic cable running
back to the control computer
Sensor Power
Another consideration for ship health monitoring sensor
systems is the source of the required power Most
com-mon sensors, including strain gages and accelerometers,
require a constant electrical input to operate This power
is usually provided by the control computer and is sent
through installed wiring to the individual sensors The
dis-advantage is that this approach often leads to additional
bulky cabling An alternative is to use the ship’s existing
power distribution network This approach is, however,
complicated because the existing power, especially near
the bow, is limited and of poor quality, and high voltage
spikes are common The control computer and critical
sen-sors must also be connected to an uninterruptable power
supply (UPS) to maintain operation in the event of a power
failure
DATA
The wealth of information obtained from the remote and
on-board sensors of a health monitoring system must be
transmitted and processed into a form that is both useful
and concise It is widely accepted that a useful health
mon-itoring system must have a bridge terminal to display all of
the pertinent information to the ship’s crew This first step
is to transmit the data to a central location, either on the
bridge or nearby After the data has been transmitted to
a central location, it is input into a computer system that
analyzes and formats it into easy-to-read displays In
ad-dition to displaying the real-time data to the crew during
the voyage, it is often desirable to store this data for future
analysis Each step in this process is an involved function,
and each is described in the following sections
fac-Hard Wiring fac-Hard wiring is the most common form of
data transmission in currently installed health monitoringsystems In ships that have protected longitudinal passage-ways, shielded and grounded cables offer the route thathas the lowest installation expense However, many ships,including tankers and product carriers, do not have thesepassageways and require more extensive cable routing andcost The use of armored cable is also recommended forany external cable routing to protect it from physical dam-age In explosive environments, it is extremely important
to ground all cables to reduce the risk of sparking; in factthis procedure is generally preferred in all applications toreduce noise
Radio Links Radio links between sensors and the
con-trol computer offer the advantage of simplicity of lation because no cables need to be run through the ship’shull Nevertheless radio links have increased costs because
instal-of the transmitter and antenna An additional advantagefor explosive environments is that radio links eliminatethe spark hazard found in hard wiring This type of signaltransmission becomes less attractive for large numbers oflocalized sensors When wiring, one can lay a multitude ofsensor cables at one time For radio transmission, multipletransmitters are required for additional sensors Althoughradio transmission does not degrade over the length of theship, signal interference is possible and can corrupt thedata with spurious signals
Fiber-Optic Network Fiber-optic networks represent a
good alternative to hard wiring and many advantages buthigher cost As proved by the telecommunications industry,fiber optics can easily transmit many signals hundreds offeet without any signal degradation Fiber optics are alsoinherently safe for explosive environments and do not suf-fer from electromagnetic interference For many of thesereasons, current naval vessels are being outfitted withwide area distributed fiber-optic networks (15) Similar tohard wiring, fiber-optic networks suffer the disadvantage
of needing a fiber-optic cable from the sensors to the trol computer In addition, most standard analog electricalsensors require expensive signal converters and decoders
con-to convert the data incon-to a corresponding light signal andback to electrical signals at the control computer However,for a ship that has a preexisting fiber optic network, thisform of transmission is extremely attractive and will beattractive for other systems as the cost of fiber-optic com-ponents continues to drop
Trang 31The analysis and display of the sensor data are the primary
objectives of the standard ship health monitoring unit It is
the responsibility of the control computer to perform data
acquisition from the individual sensors and to process this
data to determine whether the ship has been damaged or
faces any immediate danger The most common method
of analyzing the information is to monitor each sensor in
terms of the absolute magnitude of the sensor reading
Al-though the algorithm is relatively simple, the crew should
determine whether the ship has experienced a reading that
approaches or exceeds the maximum allowable level If an
overload has occurred, the crew must know immediately,
so that appropriate action may be taken This may dictate
a change in the ship’s heading, an adjusted cargo
load-ing pattern, or possibly a visual inspection of the sensor
location
More complicated analyses are also performed to
de-termine the overall fatigue experienced by the ship and
to locate any general trends in the data that might
indi-cate a potential failure These analyses include average
sensor levels, standard deviations, and peak values More
complicated signal processing techniques have been
devel-oped for machinery health monitoring, but such techniques
have not yet been used for ship health monitoring systems
Limited attempts have been made at this point to use ship
health monitoring information to predict the remaining
fa-tigue life of ship components
Display
A key component of the ship health monitoring system is
the bridge display The information displayed on at the
bridge is the primary interface between the monitoring
system and the ship’s crew To be easily used by the crew,
the information must meet many different requirements
Information The information given by the display is the
most important function of the entire system The crew
re-quires simple displays that can quickly inform them of any
potential dangers and the effect of various maneuvers on
the state of the ship Complicating the information
dis-play is the crew members’ needs for different types and
amounts of data The information that is required during
cargo loading is very different from the information needed
when traveling through rough seas Support personnel are
interested in different types of information as they
post-process the data To meet these many demands
simultane-ously, it is common practice for the bridge display to consist
of numerous (more than five) different screens
Experience has also shown which data formats and
types of information are the most helpful to bridge
person-nel For example, the ABS requires displaying hull girder
stresses over a relatively short period of time so that the
effect of speed or heading changes on the measured stress
levels can be evaluated Experience has also shown that
bridge crews want the stress information to be displayed as
a percentage of the maximum allowable stress as opposed
to actual stress or strain readings The actual values are
important and are generally saved for later use, but theimmediate needs do not require this information
Alarms Both audible and visual alarms are standard in
all ship health monitoring systems The alarms are needed
to inform the crew quickly of any potential dangers ever, it is very important to set the alarm sensitivity levelshigh enough so that the crew does not become frustrated
How-by the alarm system Excessively sensitive alarms have sulted in disconnection of the alarm system in previous ap-plications, especially because of ice-induced local stresses(3)
re-Color/Lighting The graphical display of the ship health
monitoring system must be capable of operating in twomodes During the daytime, it is desirable to have a brightscreen with obvious color clues to inform the crew quickly
of the ship’s status Standard danger colors such as redand yellow should be used to highlight high sensor levels
or overloads Similarly, cool colors should be used to cate that the ship is operating normally However, the dis-play must not interfere with the crew’s night vision duringevening hours Hence, the display must be able to switch to
indi-a second mode where lower intensity schemes cindi-an be used
to maintain night vision Intensity variations can then beused to signify danger as opposed to a color change
Storage
After the information has been displayed to the crew, thesensor information must be stored for later retrieval andanalysis Using modern storage media, it is not difficult
to store vast amounts of data in relatively little space.Nonetheless, a continuously operating ship health moni-toring system can generate huge amounts of data Severaloptions exist for storing data, including magnetic disks andtapes and optical disks The primary considerations for thestorage medium are the cost and capacity of the deviceversus the frequency with which the medium needs to bechanged by the ship’s crew during a voyage
COMMERCIAL SYSTEMS
To date, there are a few commercially available hensive ship health monitoring systems In addition, sev-eral manufacturers commercially produce Hull ResponseMonitoring Systems (HRMS) Although these systems arenot comprehensive health monitoring systems that en-compass the entire vessel, they provide detailed and ad-equate monitoring of the vessel’s hull structure and in-clude many other associated monitoring functions Thesesystems will surely form the basis for a comprehensive shiphealth monitoring system To regulate commercial HRMS,the ABS published, in 1995, classifications for Hull Con-dition Monitoring Systems Although these guidelines aregeneral, they do provide a minimum compliance level forall HRMS These requirements are listed in Table 3
Trang 32compre-SHIP HEALTH MONITORING 991
Table 3 ABS HRMS Requirements
Measurement Device Parameter Sensitivity
Accuracy ±0.01 g’s Frequency 3 × required response
Accuracy ± 5 µε
Frequency 5 Hz
Current Systems
Currently, there are approximately 10 commercial
manu-facturers of HRM systems Although many of the systems
focus on recording the same physical parameters, no two
systems are identical and almost every manufacturer will
specially design a system to the end user’s needs
Nonethe-less, it is helpful to give approximate capabilities of these
individual systems The values in Table 4 are given only as
a reference; individual or all values may be higher or lower
for any given manufacturer and system
Future Enhancements
The future of HRMS and comprehensive ship health
moni-toring systems is both exciting and dynamic As the cost
of new vessels continues to rise, there is an increasing
demand to maintain and extend the operational life of
new and existing vessels A ship health monitoring unit
is uniquely capable of extending the life of a vessel by
pro-viding the optimal course to the crew to avoid severe storms
and to limit the damage incurred by the ship The health
monitoring system may also allow aging ships to be kept in
service for longer periods of time by accurately identifying
any failures before they become catastrophic
Almost every aspect of future systems is likely to be
enhanced over the current state of the art within the next
decade These advancements include the following:
1 Improved Sensors Almost every type of sensor will
have more capabilities, reduced cost, and improvedsafety and reliability
2 Increased Sensor Density As the cost of individual
sensors and data acquisition hardware decreases, thenumber of sensors installed in a typical system willincrease This will bring a greater density of sensors
to a given region and will allow for more detailedmeasurements of local stresses
Table 4 Typical HRMS Features
Data storage capacity 1 GB
Average display length 5 min
No of display screens 5–10
3 Additional Monitoring Functions: In addition to
im-proving the ability to detect and locate any tial damage to the ship’s hull, advancements will al-low monitoring the health of additional regions of theship or ship components For example, researchers inrelated fields have demonstrated the capability of ac-curately detecting transmission faults before they be-come catastrophic Faults such as cracked or brokengear teeth, damaged bearing raceways, or misalignedshafts have all been successfully detected before fail-ure These systems generally use accelerometers tomeasure the vibrations close to the gears, bearings,and shafts in many transmission systems The mea-sured vibrational signals can be analyzed by a widevariety of methods of varying complexity and comput-ing power Although the global parameters of power,speed, and torque are measured in current systems,they can only inform the crew of a transmission faultafter it has occurred and has begun to affect the ship’soperation Therefore, this type of transmission healthmonitoring system is a logical addition to currentship health monitoring systems
poten-4 Increased Computing Power: The continued push for
faster computers will enable health monitoring tems to perform more detailed analysis in real time.Such advances will lead to more sophisticated andsensitive algorithms that can inform the crew of apotential problem before it occurs This is especiallyhelpful in machinery diagnostics
sys-5 Condition-Based Maintenance: Currently, ship
com-ponents are repaired or replaced based on one of twofactors Either the part is replaced on a time-dictatedschedule, to prevent the part statistically from everfailing, or it is repaired/replaced after failing Nei-ther of the current methods is optimal In the firstcase, healthy components are discarded only becausethey have been used for a set amount of time In thelatter case, the ship is potentially unavailable for ser-vice because of the required repairs An alternativeapproach, which may be provided by an advancedship health monitoring system, is condition-basedmaintenance (16–18) By monitoring the health of astructure, one can also monitor the remaining life
of that component For example, a crack may be tected, but the monitoring system may show that thepart has an additional 6 months of life before failure.This monitoring capability will ensure that the ship’savailability is not lost because of repairs or waitingfor a component to become available, and healthyparts will not be inadvertently wasted
de-6 Increased Data Storage: It is well known that the
cost of data storage is continuously dropping Thistrend will be very beneficial to ship health monitoringsystems as increased amounts of raw data will bestored for possible retrieval and postprocessing
7 Improved Weather Forecasting and Route Planning:
Although weather forecasting is not a direct part of aship health monitoring unit, the advances in weatherforecasting will give the system greater confidence in
Trang 33the predicted weather and will improve the ness of route planning activities This will be cou-pled to improved route planning algorithms that will
effective-be developed, as the relationships effective-between the ship’sresponse and weather conditions are more fully un-derstood
BIBLIOGRAPHY
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Architects, 1974.
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Symp III, Philadelphia, June 1999.
SMART PEROVSKITES
ZHONGL WANG
Georgia Institute of Technology
Atlanta, GA
Perovskite and perovskite-related structures are a class of
smart materials (1) Perovskite-structured materials have
important applications in ferroelectricity, piezoelectricity,
ferromagnetism, magnetoresistance, superconductivity,
ionic conductivity, and dielectricity Typical perovskite
materials of technological importance are piezoelectricPb(Zr,Ti)O3, electrostrictive Pb(Mg,Nb)O3, magnetoresis-tant (La,Ca)MnO3, and superconductive YBa2Cu3O7.Perovskite-related materials are versatile matrices forgenerating transition- and rare-earth metal oxides that ex-hibit a broad spectrum of properties and functions (2) thatare related to the following characteristics: (1) Nearly in-numerable combinations of metal cations can be accommo-dated within perovskite-related structural systems (2) Byreduction / reoxidation processes, nonstoichiometry (i.e.,controlled amounts of ordered oxygen vacancies) can beintroduced into the structure In turn, high oxygen ion mo-bility or modified electronic and magnetic features can beimplemented, and (3) the design of composite structuralsystems containing perovskite building units (perovskiteslabs of different thicknesses) allows fine-tuning electronicand magnetic properties
From the viewpoint of crystal structure, the ABO3typestructure, in which the cation A usually has valence 2+ andthe cation B has valence 4+, is the fundamental perovskite.The perovskite family is created by doping other types ofcations into the stoichiometry and /or introducing anion de-ficiency Understanding the structures and the relation-ships among the abundant structures in the perovskitesmay lead to some insights into the intrinsic connection be-tween structure and properties This article focuses on thestructure and structural evolution of perovskites and ex-plores the intrinsic linkages among the members of theperovskite family First, we introduce the “smart” proper-ties of perovskites Then, the intrinsic connection amongthe perovskites is explored Finally, the analysis of mixedvalences and oxygen deficiency is addressed
THE FAMILY OF PEROVSKITE–STRUCTURED MATERIALS Examples of Perovskite Structures
The most typical perovskite structure is BaTiO3(Fig 1a).The Ba atoms appear at the corners of the unit cell andoxygen atoms at the face centers Both the Ba and O make
up a face-centered lattice structure The octahedrally dinated titanium ion is located at the center of the unit cell.This structure can be generically written as ABO3, which
coor-is the fundamental structural configuration of perovskites.Materials that have perovskite-like structures are nu-merous The most typical are ceramic high-temperaturesuperconductors such as YBa2Cu3O7(Fig 1b) The unit cellcan be considered a stack of three perovskite units alongthe c-axis direction, where the cation lattice preserves that
of the perovskite, and oxygen vacancies are introduced Thedistortion in the oxygen lattice sites is due mainly to thevacancies The long periodicity of the c axis (e.g., super-structure) is the result of alternate distribution of the
Y and Ba cations and the ordered structure of oxygenvacancies
A comparison of BaTiO3 with YBa2Cu3O7 indicatesthe following In BaTiO3, the cations are screened by an-ions so that two cations are not directly face-to-face InYBa2Cu3O7, although the cation distribution is the same
as that in BaTiO , the Cu ions at the top layer of the
Trang 34SMART PEROVSKITES 993
(a)
Ba
TiO
Figure 1 Atomic structural models of (a) BaTiO3 and
(b) YBa2Cu3O7.
unit cell are face-to-face without the screening of anions,
whereas the Cu cations next to Y are well coordinated This
structural configuration is possible only if Cu ions have
different valence states at the two types of lattice sites
Therefore, perovskites have three major structural
char-acteristics: cation substitution, ordered oxygen vacancies,
and mixed valences of cations
Perovskite-Like Structures
In the ABO3 structure, the valences of the A
(12-coordi-nated) and B (6-coordi(12-coordi-nated) cations are usually 2+ and
4+, respectively The valence variation at the A cation
po-sition can cause distortion or displacement of the oxygen
anion array, possibly resulting in distortion in the
B-cation-centered octahedron The B cation must have the flexibility
to tolerate this effect, and the transition-metal elements
are candidates for filling the B-cation position because of
their multivalences and their special 3d and 4d electronic
configurations This is the reason that transition-metal
oxides have perovskite-type structures (3) Perovskite-like
structures can be sorted by the valence combination of the
A and B cations as follows (4):
1 A1 +B5 +O3 type, such as KNbO3, NaNbO3, LiNbO3
and KTaO3
2 A2 +B4 +O3type, in which the A2 +cations are
alkaline-earth ions such as cadmium or lead, and the B4 +ionscan be Ce, Fe, Pr, Pu, Sn, Th, Hf, Ti, Zr, Mo, and
U BaTiO3and PbTiO3are typical examples Thesetwo compounds are well known for their remarkableferroelectic properties (see later section)
3 A3 +B3 +O3 type, such as GdFeO3, YAlO3, PrVO3,
PrCrO3, NdGaO3and YScO3
4 A2 +(B3+0.67 B6+0.33)O3 type, such as Ba(Sc0.67 W0.33)O3
and Sr( Cr0.67Re0.33)O3
5 A2 +(B2+0.33B5+0.67)O3type, for example Ba(Sr0.33Ta0.67)
O , and Pb(Mg.33Nb .67)O
6 A2 +(B3+0.5 B5+0.5)O3, A2 +(B2+0.5 B6+0.5)O3, A2 +(B1+0.5 B7+0.5)
O3, and A3 +(B2+0.5B4+0.5)O3types, such as Ba(Sr0.5W0.5)
O3, Pb(Sc0.5Ta0.5)O3and Pb(Sc0.5Nb0.5)O3 The ounds, Pb(Mg0.33 Nb0.67)O3,Pb(Sc0.5 Ta0.5)O3 and Pb(Sc0.5Nb0.5)O3, are very important ferroelectric ma-terials, and they are usaully called “relaxors.”
comp-7 A2 +(B1+0.25 B5+0.75)O3type, such as Ba(Na0.25Ta0.75)O3and Sr(Na0.25Ta0.75)O3
8 A2 +(B2+0.5 B5+0.5)O2.75 and A2 +(B3+0.5 B4+0.5)O2.75 thatare anion deficient, such as Sr(Sr0.5 Ta0.5)O2.75 andBa(Fe0.5Mo0.5)O2.75
9 A2 +(B3+0.5B2+0.5)O2.25
It is apparent that the perovskite structures cover alarge group of materials Three questions are particularlyinteresting: What are the special properties of perovskites
as far as smart materials are concerned? What is the tionship between these structures, for example, the struc-tural evolution in perovskite, and what is the relationshipbetween the cation valence and its coordination? The fol-lowing analysis explores the answers to these questions
rela-STRUCTURES AND PROPERTIES Ferroelectricity
Ferroelectric materials are candidates for robust volatile memories (5) Figure 2 gives a high-resolutiontransmission electron microscopy (TEM) image of BaTiO3oriented along [100] (or [001]), where the cations are indark contrast and the contrast is directly related to the
non-Figure 2 High-resolution transmission electron microscopy
im-age of BaTiO3oriented along [100], showing the cation (in dark contrast) distribution in the crystal The atom types can be clearly identified At the top of the film, surface steps of one unit cell height are seen, and the termination layer is Ba–O.
Trang 35cc
atomic number The oxygen anions are not clearly resolved
in the image because of its weak scattering power At the
top of the film, the last ending layer is the Ba–O layer,
clearly indicating that the Ti atom strongly demands a
complete octahedral coordination even at the boundary of
the crystal The octahedral coordination of Ti is at the root
of ferroelectricity
The Ti ion is surrounded by six oxygen ions in an
octahe-dral configuration (Fig 1a) BaTiO3has a cubic structure
at T > 120◦C For 5 < T < 120◦C, it is tetragonal In the
low-temperature range of− 90 < T < 5◦C, it has an orthorhombic
structure, and for T <−90◦C, it is rhombohedral
There-fore, the structural transformation from centrosymmetric
to noncentrosymmetric occurs at 120◦C, and
ferroelectric-ity occurs at T < 120◦C Below the 120◦C transition
tem-perature, the oxygen and titanium ions are displaced to
new positions (Fig 3a,b), forming a tetragonal structure
where c/a = 1.01 (3) A unilateral displacement of the Ti4 +
ion against O2 − results in a dipole moment When all of
the dipoles of different domains point in the same
direc-tion, the material is ferroelectric If the dipoles have equal
strength but are aligned in an antiparallel configuration
so that they cancel each other and the material does not
exhibit a macroscopic dipole, it is antiferroelectric If these
dipoles cannot completely cancel each other, the residual
dipoles add up, forming a macroscopic dipole, which is
ferroelectricity.
The spontaneous alignment of dipoles that occurs at
the onset of ferroelectricity is often associated with a
crystallographic phase change from a centrosymmetric,nonpolar lattice to a noncentrosymmetric polar lattice If
an external electric field is applied to the crystal, the ulation of the domains whose polarizations are parallel tothe field increases, and those whose polarizations are an-tiparallel and not parallel to the field decrease If the ex-ternal electric field is removed, the domains cannot spon-taneously compensate for each other again, and a rema-
pop-nent polarization Prremains To remove the remanent larization, an oppositely oriented electric field whose field
po-strength is E c , called the coercive field, has to be applied to
the crystal The polarization hysteretic loop (Fig 3c) is thebasis of electric data storage using ferroelectric materials
An increasing number of materials have been foundthat demonstrate spontaneous polarization Lead titanate(PbTiO3), which has the same perovskite structure asBaTiO3, is ferroelectric Other examples includes Rochellesalt (potassium sodium tartrate tetrahydrate), KH2PO4,
KH2AsO4; perovskites NaCbO3, KCbO3, NaTaO3, andKTaO3; ilmenite structures, LiTaO3 and LiCbO3; andtungsten oxide, WO3
Domains and domain boundaries can be formed in
fer-roelectric materials The spontaneous polarization of the
Ti and oxygen ions creates an electrostatic polarization P
along the c axis This anisotropic structural configurationcan form 90 and 180◦domain boundaries defined with ref-
erence to the orientations of the c axes or the P vectors
that belong to the two crystal domains (Fig 3d,e) The 90◦domain boundary is just a (101) [or (011)] twin boundary of
... 28: 17 73? ?17 80 (19 80).21 Y Huo, Continuum Mech Thermodyn 1: 283–303 (19 89).
22 J Ortin, J Appl Phys 71: 14 54? ?14 61 (19 92). ... determination of the number of variants and the mod-eling of their arrangement (17 ), as well as the modeling ofnonproportional multi-axial loading; for recent experimen-tal studies, see (18 ,19 )
Approaches...
Perovskite-Like Structures
In the ABO3 structure, the valences of the A
(12 -coordi-nated) and B (6-coordi (12 -coordi-nated) cations are usually 2+ and
4+,