A phenomenological cohesive model of ferroelectric fatigue
ABSTRACT We develop a phenomenological model of electromechanical ferroelectric fatigue based on a ferroelectric cohesive law that couples mechanical displacement and electricpotential discontinuity to mechanical tractions and surfacecharge density. The ferroelectric cohesive law exhibits a monotonic envelope and loadingunloading hysteresis. The model is applicable whenever the changes in properties leading to fatigue are localized in one or more planarlike regions, modelled by the cohesive surfaces. We validate the model against experimental data for a simple test configuration consisting of an infinite slab acted upon by an oscillatory voltage differential across the slab and otherwise stress free. The model captures salient features of the experimental record including: the existence of a threshold nominal field for the onset of fatigue; the dependence of the threshold on the appliedfield frequency; the dependence of fatigue life on the amplitude of the nominal field; and the dependence of the coercive field on the size of the component, or size effect. Our results, although not conclusive, indicate that planarlike regions affected by cycling may lead to the observed fatigue in tetragonal PZT. Peer Reviewed Postprint (author's final draft)
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ABSTRACT: This paper presents new finite elements with embedded strong discontinuities in the form of jumps in the mechanical displacement field as well as jumps in the electric potential for the modeling of failure in electromechanical coupled solids. Such materials which are commonly used in smart systems in the form of actuators or sensors are prone to defects in the form of cracks represented by such strong discontinuities. In this work a discrete account of the individual jumps is made with the possibility of their propagation through the individual finite elements without the need of remeshing or refinement strategies. Originated for purely mechanical based materials and commonly referred as the strong discontinuity approach an extension to account for the electromechanical coupling and the additional jumps in the electric potential is made in this work. The decomposition of that methodology into a global problem, representing here the electromechanical boundary value problem, and a local problem through which the strong discontinuities are introduced, serves convenient when developing the corresponding finite elements based on the requirement to avoid stress locking phenomena. This idea is extended in this work to develop new finite elements based on the incorporation of electrical separation modes directly into the discrete formulation, keeping its local and computational efficient property with regard to the storage and possible static condensation of the parameters used to describe the mechanical and electrical discontinuities. On a constitutive level, the mechanical traction separation law is extended to account for a softening type behavior in the relation between the normal component of the electric displacement field and the surface charge density. This is accomplished through the derivation of a localized electromechanical damage model based on the principle of maximum damage dissipation. The performance of the resulting new finite elements is illustrated with several numerical simulations including a compact tension test and a three point bending test of piezoelectric ceramics and compared with experimental results available in the literature.Computer Methods in Applied Mechanics and Engineering 01/2011; · 2.62 Impact Factor  SourceAvailable from: Xin Wang[show abstract] [hide abstract]
ABSTRACT: Synapses made by local interneurons dominate the intrinsic circuitry of the mammalian visual thalamus and influence all signals traveling from the eye to cortex. Here we draw on physiological and computational analyses of receptive fields in the cat's lateral geniculate nucleus to describe how inhibition helps to enhance selectivity for stimulus features in space and time and to improve the efficiency of the neural code. Further, we explore specialized synaptic attributes of relay cells and interneurons and discuss how these might be adapted to preserve the temporal precision of retinal spike trains and thereby maximize the rate of information transmitted downstream.Current Opinion in Neurobiology 07/2011; 21:726733. · 7.34 Impact Factor  SourceAvailable from: Jacob MuthuH NguyenVinh, I Bakar, MA Msekh, J H Song, J Muthu, G Zi, P Le, S Bordas, R Simpson, S Natarajan, T Lahmer, T Rabczuk[show abstract] [hide abstract]
ABSTRACT: We present an extended finite element formulation for dynamic fracture of piezoelectric materials. The method is developed in the context of linear elastic fracture mechanics. It is applied to mode I and mixed modefracture for quasisteady cracks. An implicit time integration scheme is exploited. The results are compared to results obtained with the boundary element method and show excellent agreement.Engineering Fracture Mechanics 01/2012; · 1.41 Impact Factor
Page 1
A phenomenological cohesive model of ferroelectric fatigue
I. Ariasb, S. Serebrinskya,*, M. Ortiza
aDivision of Engineering and Applied Science, Graduate Aeronautical Laboratories, California Institute of Technology,
1200 E. California Blvd., MS20545, Pasadena, CA 91125, USA
bLaboratori de Ca `lcul Nume `ric, Departament de Matema `tica Aplicada III, Universitat Polite `cnica de Catalunya, Jordi Girona 13, Barcelona E08034, Spain
Received 4 August 2005; received in revised form 5 October 2005; accepted 19 October 2005
Available online 5 January 2006
Abstract
We develop a phenomenological model of electromechanical ferroelectric fatigue based on a ferroelectric cohesive law that cou
ples mechanical displacement and electricpotential discontinuity to mechanical tractions and surfacecharge density. The ferroelectric
cohesive law exhibits a monotonic envelope and loading–unloading hysteresis. The model is applicable whenever the changes in prop
erties leading to fatigue are localized in one or more planarlike regions, modeled by the cohesive surfaces. We validate the model
against experimental data for a simple test configuration consisting of an infinite slab acted upon by an oscillatory voltage differential
across the slab and otherwise stress free. The model captures salient features of the experimental record including: the existence of a
threshold nominal field for the onset of fatigue; the dependence of the threshold on the appliedfield frequency; the dependence of
fatigue life on the amplitude of the nominal field; and the dependence of the coercive field on the size of the component, or size
effect. Our results, although not conclusive, indicate that planarlike regions affected by cycling may lead to the observed fatigue
in tetragonal PZT.
? 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Ferroelectricity; Fatigue; Fracture; Nucleation; Cohesive
1. Introduction
Ferroelectric materials are extensively used in a variety
of sensor and actuator applications, where the coupling
between mechanical and electrical fields is of primary
interest. They are also a promising set of materials for
improved dynamic as well as nonvolatile memory devices,
where only the electrical properties are directly exploited.
However, ferroelectrics are brittle, and their low fracture
toughness (in the order of 1 MPa m1/2) makes them sus
ceptible to cracking. In addition, ferroelectric materials
exhibit electrical fatigue (loss of switchable polarization)
under cyclic electrical loading and, due to the strong elec
tromechanical coupling, sometimes mechanical fatigue as
well. Conversely, the propagation of fatigue cracks hin
ders the performance of the devices and raises serious
reliability concerns.
Ferroelectric fatigue is caused by a combination of
electrical, mechanical and electrochemical processes,
each of which has been claimed to be responsible for
fatigue [1]. Several electrochemical mechanisms have
been posited as the likely cause of polarization fatigue
[2], but no general consensus appears to have emerged
as yet. Fatigue mechanisms variously include processes
of distributed damage over the bulk (see, e.g., Refs.
[3,4]) and processes of localized damage, including micro
cracks and ferroelectric–electrode interfaces (see, e.g.,
Refs. [5–7]). The proposed mechanisms include domain
wall pinning and inhibition of reversed domain nucle
ation [8,9,2]. The former mechanism is thought to play
a dominant role in the bulk, whereas the latter mecha
nism is thought to operate primarily at or near electrode
13596454/$30.00 ? 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
doi:10.1016/j.actamat.2005.10.035
*Corresponding author. Tel.: +1 626 395 3282; fax: +1 626 449 2677.
Email address: serebrin@caltech.edu (S. Serebrinsky).
www.actamatjournals.com
Acta Materialia 54 (2006) 975–984
Page 2
interfaces [10]. The relative roles of these and other
mechanisms may depend on the frequency of the applied
field [10]. At the atomic level, oxygen vacancies are likely
to promote fatigue, e.g., by migrating under the action of
the electric field to form extended defects that pin
domain walls [11–13]. Purely mechanical mechanisms,
such as microcracking, are also likely to play some role.
Thus, fatigued ceramic specimens often contain scattered
microcracks of size comparable to that of the grains
[14,15]. Profuse microcrack clouds have been observed
at the specimen edges and surrounding macroscopic
cracks [16]. Macroscopic crack patterns are present in
some fatigued ceramic specimens [17]. Severe cracking
was also observed in barium titanate single crystals sub
jected to cyclic bipolar electric load [18]. Modeling work
suggests that microcracking is indeed a cause of loss of
polarization [6].
Experiments on crack propagation have been reported
for samples loaded electrically, mechanically, or under
combined loading, cyclic or static [19–22]. It is not
uncommon for different experiments to lead to appar
ently contradictory conclusions [23], a testament to the
complexity of the phenomenon of ferroelectric fatigue.
On the other hand, a large body of experimental data
concerns smooth samples, or samples without an initial
precrack. In these experiments, the main property of
interest is the evolution of polarization, and often no
special attention is given to the mechanical integrity of
the sample. Epitaxially grown thin films constitute a
common configurationfor
samples [24].
Despite theserecent experimental
advances, the precise nature of the interactions between
fracture, deformation and defect structures underlying
ferroelectric fatigue is in need of further elucidation,
and a physicsbased multiscale model enabling the predic
tion of the fatigue life of ferroelectric devices is yet to
emerge. Therefore, there remains a need for phenomeno
logical and empirical models that can be experimentally
validated and used in engineering design. In this vein,
cohesive theories provide an effective means of modeling
fatiguecrack nucleation and growth for arbitrary crack
and specimen geometries and loading histories [25,26].
As noted by Nguyen et al. [25], the essential feature that
a cohesive law must possess in order to model fatigue
crack growth is loading–unloading hysteresis. By this sim
ple but essential device, cohesive models of mechanical
fatigue have been shown to account for deviations from
Paris’s law in metals such as are observed for short
cracks and overloads [25]; and to predict fatiguecrack
nucleation in smoothsurface metallic specimens [26].
The appealing feature of cohesive models of fatigue is
that a single mechanistic, albeit empirical, model applies
to nucleation and propagation, short and long cracks,
and arbitrary loading histories.
Conventional cohesive models of fracture seek to
describe the relation between cohesive tractions and open
tests involving smooth
andmodeling
ing displacements at the tip of a crack. Working by anal
ogy, Gao et al. [27–29] have proposed the use of
Dugdalelike models for the electric displacement in fer
roelectric materials. In this paper, we further extend the
concept of cohesive surface to ferroelectric materials by
understanding it to be a surface of both mechanical open
ing displacement and electricpotential discontinuity. Corre
spondingly, we extend the concept of cohesive law to
mean a general relation between the workconjugate
pairs: mechanical displacement and electricpotential dis
continuity; and mechanical tractions and surfacecharge
density. Following Nguyen et al. [25], in order to model
ferroelectric fatigue we endow the ferroelectric cohesive
law with loading–unloading hysteresis. In order to sim
plify modeling, following [30] we make the additional
assumption that the cohesive law derives from a potential
that depends on a scalar effective electromechanical
jump, which is a weighted norm of mechanical opening
displacements and electricpotential jump.
Using a simple Smith–Ferrante monotonic envelope
and an exponentialdecay law of loading–unloading hys
teresis, we show that the model is able to qualitatively
capture salient features of the experimental record includ
ing: the existence of a threshold nominal field for the
onset of fatigue; the dependence of the threshold on the
appliedfield frequency; the dependence of fatigue life
on the amplitude of the nominal field; and the depen
dence of the coercive field on the size of the component,
or size effect.
2. Electromechanical cohesive laws
The essential structure of cohesive laws in ferroelectrics
may be elucidated by recourse to a conventional Coleman–
Noll argument (cf., [30] for an application to mechanical
decohesion). In particular, the pairing between stresslike
and deformationlike variables is determined by the inter
nalpower identity or, equivalently, by the virtualwork
identity. For a dielectric solid, this identity is [31–33]
Z
¼
Xðr : d? ? D ? dEÞ dV
Z
where X is the spatial domain occupied by the solid and oX
is its boundary; r is the stress;
Xðb ? du ? qd/Þ dV þ
Z
oXðt ? du ? xfd/Þ dS;
ð1Þ
? ¼1
is the strain; D is the electric displacement;
2ðru þ ruTÞð2Þ
E ¼ ?r/
is the electric field; b is the body force; u is the displace
ment; q is the free charge density; / is the electric potential;
t = r Æ n is the surface traction; and xf= ?D Æ n is the free
surface charge per unit area. For a solid with a surface of
discontinuity C, or cohesive surface, we have
ð3Þ
976
I. Arias et al. / Acta Materialia 54 (2006) 975–984
Page 3
dWext¼
Z
Z
Z
þ r ? ðDd/Þ? dV þ
Xðb ? du ? qd/Þ dV þ
Z
Z
Z
oXðt ? du ? xfd/Þ dS
¼
Xðb ? du ? qd/Þ dV þ
oX½ðn ? rÞ ? du þ ðn ? DÞd/? dS
¼
Xðb ? du ? qd/Þ dV þ
X?½r ? ðr ? duÞ
Z
C½ðn ? rÞ ? sdut þ ðn ? DÞsd/t? dS
ð4Þ
and consequently
Z
þ ðn ? DÞds/t? dS.
For monotonic loading of the cohesive surface, these work
conjugacy relations naturally suggest a free energy per unit
surface, or cohesive potential, of the form U(sub,s/b,q)
such that
dWint¼
X?ðr : d? ? D ? dEÞ dV þ
Z
C½ðn ? rÞ ? dsut
ð5Þ
t ¼
oU
osut;
ð6aÞ
? xf¼
oU
os/t;
ð6bÞ
where q is some suitable set of internal variables. A possible
additional dependence of U on temperature is omitted for
notational convenience. Thus, U depends both on the dis
placement and electric potential jumps across C and acts
as a potential jointly for the mechanical tractions and the
surfacecharge density.
It bears emphasis that the ferroelectric cohesive law
((6a) and (6b)) allows for an arbitrary coupling of the
mechanical and electrical fields. It should also be care
fully noted that the ferroelectric cohesive law describes
the physics of mechanical or electrical decohesion and
does not presume a particular form of the constitutive
law governing the behavior in the bulk. In particular, it
is possible to apply the ferroelectric cohesive law ((6a)
and (6b)) in conjunction with Landau–Ginzburg–Devon
shire models of domain switching (cf., e.g., Ref. [34]).
The detailed boundary conditions on the crack faces are
thought to greatly affect the fracture behavior of electri
cally driven crack growth [35]. For instance, partial dis
charge or charge separation effects have been suggested
to play an important role in the vicinity of the crack
tip [36]. The ferroelectric cohesive law ((6a) and (6b)) pro
vides a useful framework for modeling those phenomena
as well. Moreover, the ferroelectric cohesive law encodes
the physics of decohesion, and thus can be tailored to
represent any of the localized mechanisms of ferroelectric
fatigue.
2.1. Ferroelectric fatigue cohesive law
As noted by Nguyen et al. [25], reversible cohesive laws
do not predict crack advance under cyclic loading and,
therefore, are insufficient for modeling fatigue. Instead,
for a cohesive law to predict fatigue it must be irreversible
and account for loading–unloading hysteresis. Loading–
unloading irreversibility may be built into a cohesive law
by means of the internal variable formalism (cf., e.g.,
Ref.[30]).Theloading–unloading
developed subsequently extends that of Nguyen et al. [25]
to ferroelectric fatigue.
The modeling process is greatly simplified by the
assumption that the cohesive potential U depends on the
displacement and electricpotential jumps only through
the effective electromechanical jump [37,30]
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
i.e., by assuming
hysteresismodel
d ¼
d2
nþ b2d2
sþ c2w2
q
;
ð7Þ
U ¼ Uðd;qÞ;
where we write
d ¼ sut;
dn¼ d ? n;
dn¼ dnn;
ds¼
ds¼ d ? dn¼ dss;
w ¼ s/t;
for the normal and tangential components of the opening
displacement and the electric potential jump, respectively.
The parameters b and c assign different relative weights
to normal and tangential opening displacements, thus dif
ferentiating between mode I and modes II and III of frac
ture; and to opening displacements and the electric
potential jump, thus differentiating between mechanical
and electrical fatigue.
An effective electromechanical flux may also be defined
as
r ¼oU
Using the chain rule, (6a) and (6b) evaluate to
s ¼r
ð8Þ
ð9aÞ
ð9bÞ
ð9cÞ
ð9dÞ
ð9eÞ
ð9fÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
d2? d2
n
q
;
od.
ð10Þ
d½ð1 ? b2Þdnn þ b2d? ¼r
dðdnn þ b2dssÞ ¼ snn þ sss;
ð11aÞ
ð11bÞ
? xf¼r
We also note that
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Next we further specialize the preceding framework along
the lines of the cohesive model of fatigue crack nucleation
and growth proposed by Serebrinsky and Ortiz [26] and by
Nguyen et al. [25]. The essential feature to include in the
model is loading–unloading hysteresis. The specific scheme
chosen to build hysteresis into the model is illustrated in
Fig. 1. Monotonic loading is characterized by a cohesive
dc2w.
r ¼
s2
nþ s2
s=b2þ x2
f=c2
q
.
ð12Þ
I. Arias et al. / Acta Materialia 54 (2006) 975–984
977
Page 4
law r(d), referred to as the monotonic envelope, which is
characterized by the critical electromechanical flux rc
and the critical electromechanical jump dc, and possibly
an initial threshold rth. In contrast, cyclic loading is hyster
etic and governed by the loading–unloading law:
(
_ r ¼
Ku_d
Kl_d
if_d < 0;
if_d > 0;
ð13Þ
where Kland Kuare the loading and unloading incremental
stiffnesses, respectively. Equivalently, we may write
_ r ¼ f1ð_d;Kl;KuÞ_d
with
ð14Þ
f1ð_d;Kl;KuÞ ¼ hð_dÞKlþ hð?_dÞKu;
where h is the Heaviside step function. For definiteness, we
assume unloading to take place towards the origin and Ku
to be constant during unloading, Fig. 1. The value of Ku
during an unloading event is determined by the values of
r and d at the unloading point. Upon reloading, we addi
tionally suppose that the reloading slope Kl= dr/dd de
creases with increasing electromechanical jump as a
result of interfacial degradation mechanisms occurring at
the microscale. During unloading, a partial recovery of
Klis also allowed for. For definiteness, we take
(
ð15Þ
_Kl¼
ðKl? KuÞ_d=da
?Kl_d=da
if_d < 0;
if_d > 0;
ð16Þ
where dais an intrinsic length of the material. Equivalently,
we may write
_Kl¼ f2ð_d;Kl;KuÞ_d=da
with
ð17Þ
f2ð_d;Kl;KuÞ ¼ ?hð_dÞKlþ hð?_dÞðKl? KuÞ.
Thus, q = (Ku,Kl) may be regarded as the internal variables
of the model. An appealing feature of the model is the
small number of parameters, namely: the constants defin
ing the monotonic envelope, such as tensile strength and
ð18Þ
toughness; the coupling constants b and c; and the intrinsic
length da.
The cyclic behavior predicted by the model just outlined
is shown schematically in Fig. 1 for cycling between con
stant maximum and minimum effective electromechanical
fluxes. Thus, for monotonic loading the cohesive surface
follows the monotonic envelope. Upon cyclic loading, the
cohesive stiffness of the surface degrades steadily and, after
a certain number of cycles, the (r(t),d(t)) curve meets the
monotonic envelope. We identify this event with the end
of the fatigue life of the material. Indeed, once the curve
(r(t),d(t)) meets the monotonic envelope, the material can
no longer sustain a loading cycle of the same amplitude
and, consequently, fails catastrophically under load control.
2.2. Bulk behavior
A closed set of governing equations may be obtained by
appending a suitable bulk energy density to the cohesive
model just described and considering the corresponding
gradient flow. Following Zhang and Bhattacharya [38],
we assume a bulk energy density of the form
W ðP;?Þ ¼ W1ðPÞ þ W2ð?Þ þ W3ðP;?Þ;
where the polarization energy density W1(P) is polynomial
of degree eight in the polarization P, with multiple wells
corresponding to the different variants of the material, and
ð19Þ
W2ð?Þ ¼1
W3ðP;?Þ ¼ ?1
are the strain and mixed energy densities, respectively. Fol
lowing Shu and Bhattacharya [34], we additionally assume
the total energy of the body to be of the form
Z
where W is the bulk energy density and ?0= 8.854 ·
10?12F/m is the permeability of vacuum and the energy
of domain walls is neglected for simplicity. A gradient flow
of this energy results in the timedependent Ginzburg–
Landau equations
2? : C : ?;
2? : ðP ? B ? PÞ;
ð20aÞ
ð20bÞ
E½P;u;/? ¼
X
W ðP;?Þ þ?0
2jr/j2
??
dX;
ð21Þ
l_P ¼ ?oW
r ? ðP þ ?0EÞ ¼ 0;
r ?oW
oPðP;?Þ þ E;
ð22Þ
ð23Þ
ð24Þ
o?ðP;?Þ ¼ 0;
where 1/l is the mobility. In these equations, we recognize
an equation of evolution for the polarization P, Gauss’s
law, and the equation of mechanical equilibrium, respec
tively. At equilibrium_P ¼ 0 and Eq. (22) reduces to
?oW
oPðP;?Þ þ E ¼ 0;
which is also obtained in the formal limit of l ! 0. Eqs.
(22)–(24), in conjunction with (3) and (2), define a closed
ð25Þ
cohesive
envelope
4
3
nucleation
×
+
Λ

Λ+
Λ = δ/δc
Σ = σ/σc
0123456
0
1
Σmin
Σmax
Fig. 1. Cyclic behavior predicted by the model and conventional
definition of fatigue initiation.
978
I. Arias et al. / Acta Materialia 54 (2006) 975–984
Page 5
initialboundaryvalue problem for the polarization P, the
electrostatic potential / and the mechanical displacement
u.
3. Experimental validation
In order to make contact with experiment and assess the
validity of the model, we consider a simple test configura
tion consisting of an infinite slab of thickness u acted upon
by an oscillatory voltage differential D/Tacross the slab
and otherwise stress free. The mid plane of the slab is a
weak interface governed by the cohesive model proposed
in Section 2. In order to simplify the analysis, we assume
all fields to be uniform outside the cohesive interface. This
assumption has the effect of reducing the initialboundary
value problem (22)–(24) to a simple set of ordinary differen
tial equations in time. Taking the cohesive plane to coincide
with the (x1,x2)coordinate plane and assuming uniaxial
strain conditions, ?11= ?22= ?12= ?13= ?23= 0, the vari
ous bulk energy densities reduce to
W1ðP3Þ ¼a1
W2ð?33Þ ¼c1
W3ðP3;?33Þ ¼ ?b1
2P2
2?2
3þa2
4P4
3þa3
6P6
3;
ð26aÞ
ð26bÞ
33;
2?33P2
3;
ð26cÞ
where a1, a2, a3, c1and b1are empirical constants. The
transverse stress follows from these expressions as:
r33¼ c1?33?b1
2P2
3.
ð27Þ
The stressfree condition r33= 0 then gives the transverse
strain as
?33¼b1
2c1P2
3.
ð28Þ
Using this identity, (22) reduces to
l_P3þ a1P3þ
a2?b2
1
2c1
??
P3
3þ a3P5
3¼ E3.
ð29Þ
From Eq. (23) and the boundary conditions, at the cohe
sive interface, we additionally have
? xf¼ P3þ ?0E3;
w ¼ E3u þ D/T.
ð30aÞ
ð30bÞ
Finally, the coupled electromechanical cohesive law spe
cializes to
_ r ¼ f1ð_d;Kl;r=dÞ_d;
_Kl¼ f2ð_d;Kl;r=dÞ_d;
c2P3u ? ?0D/T
ruc2? ?0d
ð31aÞ
ð31bÞ
??2
¼ 1;
ð31cÞ
where
d ¼ cjwj;
r ¼1
ð32aÞ
ð32bÞ
cjxj.
In calculations, we enforce the constraint (31c) in differen
tial form and solve the resulting system of ordinary differ
ential equations in time by means of a fourthorder
Runge–Kutta algorithm. In implementing this algorithm,
care must be exercised in order to resolve ambiguities in
the selection of branches and the handling of singularities
in the response functions as d tends to zero. In the sequel,
P = P3 and similarly for other vector quantities, and
? = ?33.
For sufficiently high applied field, the calculated response
exhibits characteristic hysteresis loops in a P–E plot, and
butterfly loops in a ?–E plot. These loops are shown in
Fig. 2 for several values of the peak electric field Epand
the appliedfield frequency f. Several aspects of the bulk
response are noteworthy. Thus, below a minimum value
of Epno switching occurs, and the sign of the polarization
E
a
b
P
0.040.0200.020.04
2
0
2
4
Stable equilibrium
Ep=0.04, f=2.15e4
Ep=0.02, f=2.15e4
Ep=0.01, f=2.15e4
Ep=0.004, f=2.15e4
Ep=0.04, f=1.08e4
Dimensionless variables
E
 0.04 0.0200.020.04
0
0.01
0.02
Stable equilibrium
Ep=0.04, f=2.15e4
Ep=0.02, f=2.15e4
Ep=0.01, f=2.15e4
Ep=0.004, f=2.15e4
Ep=0.04, f=1.08e4
Dimensionless variables
Fig. 2. Bulk response for different field amplitudes and frequencies. (a) P–
E hysteresis loops. (b) ?–E butterfly loops.
I. Arias et al. / Acta Materialia 54 (2006) 975–984
979
Page 6
remains constant in time. The ratedependency introduced
by the timedependent Ginzburg–Landau equation is evi
dent in Fig. 2. In the limit of l = 0 or, equivalently, of a
small frequency of the applied field, this rate dependency
is removed and an equilibrium loop is obtained. For non
zero l, the bulk response depends on the appliedfield
frequency.
The predictions of the model in the example just
described can now be compared with experimental data
by way of validation. In order to facilitate comparisons
for different materials, we introduce the following normal
ization constants for the variables: polarization, Pref; stress,
elastic constants, cref; permittivity, ?0ref¼ P2
field, Eref¼ Pref=?0ref; thickness, displacements, Lref= da;
potential, /ref= ErefLref; time tref¼ lP2
sity coefficients, bref¼ a1ref¼ cref=Pref, a2ref¼ a1ref=P2
a3ref¼ a1ref=P4
ness Kref= cref/Lref. Dimensionless variables are used in
the calculations, and the parameters and material constants
are summarized in Table 1. We have selected material con
stants for the bulk model readily available from the litera
ture [38], which correspond to BaTiO3. The initial
conditions are P30= 1, Kl0¼ 2311, and d0and r0are cho
sen to give a ratio Kl0=Ku0¼ 0:99.
Typical experimental data are given in the form
ref=cref; electric
ref=cref; energy den
ref,
ref; coefficient cref= Pref/cref; cohesive stiff
Px¼ PxðE?
where E?
which should be carefully differentiated from the peak ac
tual field Epin the material, N is the cycle number, and
Pxis a measure of the polarization state of the material,
typically the remanent polarization Prbut also sometimes
Psw, the switching polarization. From these data, a relation
E?
by solving the equation
p;N;uÞ;
p¼ ?D/p=u is the peak nominal electric field,
ð33Þ
fðN;uÞ, or the inverse relation NfðE?
p;uÞ, can be derived
PxðE?
PxðE?
where C is the loss of polarization due to fatigue. We shall
take these fatigue maps as the basis for the validation of the
model.
As expected, the predicted fatigue behavior depends crit
ically on whether switching occurs. In the presence of
switching, the interface degrades upon cycling and the peak
value of d increases monotonically with the number of
cycles, Fig. 3. The increase in d is initially slow and accel
erates markedly in the last stages of the fatigue life of the
material. The rate of damage accumulation increases with
the amplitude of the nominal field. Correspondingly, as
the interface degrades the switching capability and the
actuation strain are impaired.
p;N;uÞ
p;1;uÞ¼ 1 ? C;
ð34Þ
The calculated fatigue map, for small frequencies for
which ratedependency is negligible, is shown in Fig. 4(a).
It is immediately apparent from this map that for each
thickness u there is a threshold amplitude E?
fatigue does not occur. Specifically, fatigue occurs if the
nominal field E?
and fatigue does not occur otherwise. Thus, the fatigue
threshold E?
E?
fatigue threshold E?
Nfto shorten with increasing nominal field amplitude E?
This dependency is well approximated by the power law
Nf¼ AE?
with exponent ?n ? 2.8.
For bulklike samples and high frequencies, the fatigue
life has a steeper dependency on the applied field, as shown
in Fig. 5(a). It is readily verified that the transition from
nonswitching to switching bulk behavior, as the applied
field increases, appears smoother as the field frequency
increases, thus leading to the calculated fatigue behavior.
In addition, E?
Fig. 5(b) shows. It also indicates that the low frequency
limit is attained for the lowest frequency used.
Corresponding experimental data for smooth tetrago
nal PZT samples [39,1,40] are shown in Fig. 4(b) for pur
poses of comparison. Evidently, the predicted existence of
a fatigue threshold and the reduction in fatigue life with
nominal field amplitude are consistent with the data.
The nominal fatigue threshold E?
to be greater than or equal to the nominal coercive field
E?
Besides these experiments where no microcracking was
observed or analyzed, the relation between switching
and fatigue has also been established on indented and pre
cracked samples by the experiments of Zhu et al. [22].
Thus, switching appears as a necessary (though not suffi
f;thbelow which
pis sufficiently strong to cause switching,
f;thcoincides with the nominal coercive field
cvfor the virgin material. For nominal fields above the
f;th, the model predicts the fatigue life
p.
p
n;
ð35Þ
f;thcan be greatly in excess of E?
cv, as
f;this indeed observed
cv, as obtained from the respective hysteresis loops.
Table 1
Values of the model parameters
a1
a2
a3
b1
c1
?0
cl
?0.007
?0.009 0.02611.42821850.1318001
N
δp
10
2
10
3
10
4
10
5
10
6
10
7
10
8
10
9
10
10
0
1E06
2E06
Ep*=0.1, u=3.38e6
Ep*=0.01, u=3.38e6
Ep*=0.1, u=3.38e7
Dimensionless
variables
Fig. 3. Evolution with number of cycles of the peak effective opening
displacement dp.
980
I. Arias et al. / Acta Materialia 54 (2006) 975–984
Page 7
cient) condition for electrical fatigue, regardless of the
fatigue mechanism operating. It should be noted that
there are other methods, alternative to electric fatigue,
to suppress polarization (e.g., [41]). The experimental data
shown in Fig. 4(b) is also suggestive of powerlaw behav
ior (see Eq. (35)) with exponents in the range ?n ? 0.16–
0.24. The difference with the predicted value is due to the
choice bulk behavior, which has no dependence of the
actual coercive field on actual field amplitude for Ep> Ec,
and not to the cohesive model. The dependence of the
fatigue threshold on the appliedfield frequency, with high
frequencies delaying the onset of fatigue, was established
by Grossmann et al. [39].
It is worth mentioning that some alternatives were
developed to suppress polarization fatigue, including using
conducting oxide electrodes [42]. Such devices can be mod
eled by an interface with a large value of da, which would
lead to a minor fatigue effect. In the limit of da! 1, fati
gue would be completely absent.
In order to exhibit the size effect predicted by the model,
Fig. 6(a) plots specimen thickness against the minimum
value of E?
value of E?
this dependence may be derived by analyzing the equilib
rium case of_P ¼ 0 in Eq. (22). In this case, the governing
equations reduce to
?
P ¼ EðKuc2u ? ?0Þ þ Kuc2D/T.
For a virgin material, i.e., a material with high Ku, it is
readily shown that the nominal coercive field E?
for switching is of the form
pfor which switching occurs, and the maximum
pfor which switching does not occur. Insight into
E ¼ a1P þ
a2?b2
1
2c1
?
P3þ a3P5;
ð36aÞ
ð36bÞ
cvrequired
E?
cvðuÞ ¼ D/cðuÞ=u ? Eceqþ ðPceqþ ?0EceqÞ=ðKuc2uÞ.
where Eceq= 0.007129. This relation is plotted in Fig. 6(a)
along with the exact calculated values of E?
Eq. (36a). It is noteworthy that E?
cally with size u to a limiting value Eceqcharacteristic of
large components. Conversely, E?
small component thickness. This prediction of the model
is born out by the experimental data, Fig. 6, which is
ð37Þ
cvðuÞ from
cvdecreases monotoni
cvincreases as u?1for a
Ef,th*
f=2.15e4
u=3.4e6
f=1.08e3
f=4.30e3
f=2.15e2
N
Ef*
102
103
104
105
106
107
108
109
1010
102
101
Dimensionless
variables
No damage
(nonswitching
field)
Switching
(with degradation)
No switching (with
no degradation)
Ec,eq
frequency
Ef,th*
104
103
102
101
102
101
Dimensionless variables
a
b
Fig. 5. Effect of applied field frequency on fatigue properties. (a) Fatigue
life as a function of applied field frequency. (b) Effect of frequency on
nominal threshold field.
Ecv*
Ecv*≈Ef,th*
Ecv*
Ef,th*
No measured
loss
No measured
loss
Ef,th*
u=3.4e6
f=2.15e4
u=3.4e7
u=3.4e8
Dimensionless
variables
No damage
(nonswitching
field)
a
N
Ef* (kV/mm)
2
1010
3
104
105
106
107
108
109
1010
N
102
103
104
105
106
107
108
109
1010
101
100
Ef*
102
101
b
Fig. 4. Fatigue life as a function of nominal field and thickness. (a)
Calculated. In all cases, E?
loss = 10%. j: Grossmann et al. [39], .: Nuffer et al. [1], m: Mihara
et al. [40]. In all cases, E?
f;th¼ E?
cv. (b) Experimental (all in PZT),
f;thP E?
cv.
I. Arias et al. / Acta Materialia 54 (2006) 975–984
981
Page 8
consistent with the E?
the size effect above the nominal fatigue threshold E?
negligible, Fig. 4(a). This lack of sensitivity to the compo
nent size is also visible in the evolution of dp, Fig. 3. Thus,
the main effect of component size is on the ability of the
material to switch and, by extension, on the nominal fati
gue threshold E?
explanations for the size effect have been proposed, includ
ing depolarization fields [47,48], epitaxial stress effects
[49,45] and variants of a conductive layer next to the ferro
electric–electrode interface [50,51].
cv? u?1scaling behavior. By contrast,
f;this
f;th. It should be noted that alternative
4. Summary and concluding remarks
We have presented a model of electromechanical fer
roelectric fatigue based on the postulate of a ferroelectric
cohesive law that: couples mechanical displacement and
electricpotential discontinuity to mechanical tractions
and surfacecharge density; and exhibits a monotonic
envelope and loading–unloading hysteresis. In conjunc
tion with a constitutive model accounting for domain
switching, the electromechanical cohesive fatigue law is
able to induce ferroelectric fatigue by the following mech
anism: as degradation proceeds, the surface of electro
mechanical discontinuity absorbs an increasingly large
amount of the displacement and/or electric potential dif
ference, thereby unloading the bulk and hindering switch
ing. We identify the end of the fatigue life with the time
at which the material loses its ability to sustain loading/
applied field cycles of a certain constant amplitude. We
have compared selected predictions of the model with
experimental data for a simple test configuration consist
ing of an infinite slab acted upon by an oscillatory volt
age differential across the slab and otherwise stress free.
The model captures salient features of the experimental
record including: the existence of a threshold nominal
field for the onset of fatigue; the dependence of the
threshold on the appliedfield frequency; the dependence
of fatigue life on the amplitude of the nominal field;
and the dependence of the coercive field, and thus of
the fatigue threshold, on the size of the component, or
size effect. Our results seem to indicate that planarlike
regions affected by cycling may lead to the observed fati
gue in tetragonal PZT.
The ability of the model to predict the observed size
effect stems directly from the fact that cohesive laws intro
duce a characteristic or intrinsic material length scale.
Hence, in the present model the size effect is a material
property and a direct consequence of material behavior.
This is in contrast to other explanations of the size effect
found in the literature (e.g., [50,51,49]), where the length
scale is often structural.
In closing, we emphasize that the primary focus of this
study has been to investigate qualitative trends and no sys
tematic attempt has been made to optimize fit to the exper
imental data. It is conceivable that good quantitative
agreement with the data could be obtained by some exten
sions and careful calibration of the model, including full
finite element calculations; the use of more accurate bulk
constitutive relations; and a systematic identification of
aspects of the cohesive law such as the precise shape of
the monotonic envelope and of the loading–unloading law.
Acknowledgment
The financial support of the Army Research Office un
der MURI Grant No. DAAD190110517 is gratefully
acknowledged.
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