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2-1
2. PIEZOELECTRIC MATERIALS
Among the actions of different kinds into which electricity has conventionally been subdivided, there is, I think,
none of which excels, or even equals in importance that called Induction - Michael Faraday.
2.1 Introduction
mong the many types of smart or adaptive materials that are available to be
integrated into smart structures, this thesis will only use piezoelectric materials for the reasons
mentioned in Chap. 1 and primarily because theoretical analysis of this material has been so well
developed. This chapter provides information about the nature of the piezoelectric effect to the
extent of which it is sufficient for the understanding and appreciation of its applications in shape
control of smart structures in this thesis. The linear piezoelectric constitutive equations that will
be used in later chapters will also be developed here. For more extensive information, the
interested reader should consult literature dedicated to the field of piezoelectric such as in Cady
(1964), Tiersten(1969) and the IEEE Standard on Piezoelectricity (1988).
The direct piezoelectric effect was discovered when electric charges were created by
mechanical stress on the surface of tourmaline crystals. This discovery was not by chance; rather
such an effect was anticipated by the Curie brothers from consideration of crystal structure and
the pyroelectric phenomena (thermo-electric coupling effect) (Cady, 1964). However, it was
through thermodynamic reasoning which led Lippman to predict the converse piezoelectric effect
that prompted the Curie brothers to discover it shortly after. The first generation of piezoelectric
materials were crystals such as tourmaline, Rochelle salt (tartaric acid) and quartz. The
macroscopic/phenomenological theory of piezoelectricity, based on thermodynamic principles,
can be traced back to Lord Kelvin. However, it was Voigt who made significant contribution to
the theory as we know it today. The first major practical application of piezoelectric materials
came in the Great War where it was used as resonators for ultrasound sources in sonar devices.
Since then development of the materials has led to new and better types of piezoelectric materials
such as Barium Titanate and recently in the field of smart structures, piezoceramics and
piezopolymers. Even more recently, breakthrough in single crystal growth technique has enabled
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2. Piezoelectric Materials
2-2
ij c
ijkl
kl i, j,k ,l1,2,3 (2.1)
1
xx,
2
yy,
3
zz
4
yz,
5
zx,
6
xy
(2.2)
ij
1
2
[u
j
xi
ui
x j
] (2.3)
the development of high strain and high electric breakdown piezoceramics.
In §2.2, the individual mechanical, electrical and piezoelectric effects are described
mathematically, and then combined together from thermodynamic principles to obtain a coupled
set of constitutive equations. This thermodynamic approach reveals the reversibility and the
equivalence of the piezoelectric constants of the direct and converse piezoelectric effects. In
addition, it clarifies why certain formulations are preferred over others. The crystal structure is
briefly mentioned and a more general constitutive equation that includes rotation is presented.
In §2.3, issues concerning non-linear effects are discussed while §2.4 surveys some piezoelectric
materials specifically for smart structure applications.
2.2 Linear Constitutive Equations
2.2.1 Mechanical Elasticity - Notation
The general theory of linear elasticity is assumed to be well-known by the reader. This
section focuses on describing the notation that will be used throughout this thesis rather than
developing any mechanical relationships. Tensor notation will be used whenever convenient
while matrices are used when explicit description is required.
The mechanical stress (σij) and mechanical strain(εij) are second rank tensors and are
related to the stiffness tensor (cijkl) by Hooke’s Law Eq. (2.1).
The symmetry of the stress tensor enables the nine stress components to be reduced to six
independent stress components. This also enables the tensor notation to be contracted into a
pseudo tensor form as:
tensor index: 11 22 33 23,32 13,31 12,21
contracted index: 1 2 3 4 5 6where the contracted stress is related to the common engineering stress notation as in Eq. (2.2).
The strain of Eq. (2.1) is defined as Cauchy’s infinitesimal strain tensor in Eq. (2.3).
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2. Piezoelectric Materials
2-3
1
xx,
2
yy,
3
zz
42
yz
yz,
52
zx
zx,
62
xy
xy
(2.4)
i c
ij j
i, j1,....,6 (2.5)
However the contracted notation for the shear strain involves a factor of 2 in order to correspond
to the common engineering strain notation as in Eq. (2.4).
With the contracted stress defined as in Eq. (2.2) and the contracted strain defined as in Eq. (2.4),
Hooke’s Law is rewritten as in Eq. (2.5) where the 4th-rank stiffness tensor (cijkl) is reduced to a
two dimensional matrix(cij). It is the coefficients of this stiffness matrix which are commonly
available as manufacturer’s data or tabulated in literature.
To maintain consistency with the contracted notation, the stress and strain will be called the
stress vector and the strain vector respectively while the stiffness will be called the stiffness
matrix, though they are in fact higher rank tensors. In this contracted notation, the indices 1,2,3
refer to the normal and the indices 4,5,6 refer to the shear quantities. The stiffness matrix is
symmetrical and can have up to a maximum of 21 independent coefficients, in which case the
material is called anisotropic.
2.2.2 Dielectric Materials and Polarization
When an electric field is applied on a particular material, there are three possible
responses which are determined by their electrical properties. Firstly, electric current may flow
freely in the material due to the presence of free charged particles in the material (free electrons
in the case of metals) which could move easily under the influence of the electric field. Such
materials, which include most metals, are called conductors. Secondly, current may only flow
at certain conditions when some electrons within the material receive enough energy to overcome
the local binding energy. These materials are known as semiconductors. Thirdly, no current flows
within the material because there are no free charged particles within the material to conduct the
current. These materials are dielectrics and are commonly known as insulators (Cheng, 1989).
Although dielectrics do not conduct current, they may be polarized under the influence
of an externally applied electric field. Within dielectrics, there are bound charges arising from
the simple fact that all matter consist subatomic particles which are charged (i.e. protons and
electrons). Some dielectrics are made of polar molecules in which, due to the geometry, one end
of the molecule has a slightly more positive charge whereas the other end has a slightly more
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2. Piezoelectric Materials
2-4
P Lim
v 0
N
k 1 p
k
v
(2.6)
p P
i,i (2.7)
E i,i 1
o
( p) i1,2,3 (2.8)
Di,i D
i
o E
iP
i (2.9)
negative charge even tough the whole molecule is neutral. Such molecules possess a dipole
moment and are also known as electric dipoles. However, since the molecules are randomly
oriented in the material, macroscopically the material is also neutral. But when an external
electric field is applied, all the polar molecules will align themselves in the direction of the field
and the material is said to be polarized . For dielectrics that are composed of non-polar molecules,
there are no intrinsic dipole moments. But the application of an external electric field modifies
the distribution of charges in each molecule such that dipole moments are induced. In other
words, whether the molecules are polar or non-polar, the presence of an external field will
polarize the material by aligning the electric dipoles.
The polarization vector P, is defined as the volume density of the electric dipole moment
p and is defined in Eq. (2.6).
where N is the number of electric dipoles and v is the volume.
When the dielectric material is polarized, the aligned electric dipoles produces an
equivalent volume charge density, ρ p which affects the electric field. The volume charge density
can be shown to be related to the polarization as in Eq. (2.7). This leads to a modification of
Gauss’ Divergence Theorem to incorporate ρ p as shown in Eq. (2.8) and hence to a definition of
a new quantity called the electric displacement, D in Eq. (2.9).
where the Einstein convention of summing over indices is used and the comma representing
differentiation with respect to the index that follows.
where χo = electric permittivity in vacuo = 8.854x10-12 F/m
ρ = free charge volume density.
For dielectric materials in general, except for a type of materials known as Electrets,
polarization only exist in the presence of an external electric field. The relationship between
electric field E and the polarization P is taken to be linear (Eq. (2.10)) and is related by a constant
tensor κ , known as the electric susceptibility (with sincere apologies to my fellow physicists for
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2. Piezoelectric Materials
2-5
Pi
oij E
j (2.10)
Di
o(1
ij) E
j
ij E
j i, j1,2,3 (2.11)
Pi d
ijk
jk i, j,k 1,2,3 (2.12)
Di d
ijk
jk i, j,k 1,2,3
Di d
il
l i1,2,3 l1,2,...6
(2.13a)
(2.13b)
not using the conventional symbols for susceptibility and permittivity to avoid a clash with other
symbols such as mechanical strain).
Using the polarization of Eq. (2.10), the electric displacement can be expressed solely as
a function of the electric field as in Eq. (2.11) and is related by the material property known as
absolute electric permittivity, χ. In the following work concerning dielectrics such as
piezoelectric materials, there would no longer be any need to use the polarization vector in
analysis since it has been absorbed into the electric displacement vector. The χ tensor is
symmetric and thus for anisotropic material it will have at most 6 independent entries.
The subscript indices 1,2,3 correspond to the Cartesian directions x,y,z respectively.
2.2.3 Piezoelectricity and the Third Rank Tensor
By definition, the direct piezoelectric effect is the creation of polarization caused by
mechanical stress on the dielectric material. Dielectric material with this capability is known as
piezoelectric material. The direct effect may be formulated as a linear relation where each of the
three components of polarization(Pi) is a linear combination of all 9 components of the stress(σ jk )
tensor (Nye, 1985). This is related via the material property known as the piezoelectric strain
constant (d ijk ) which is a third rank tensor as shown in Eq. (2.12).
When the polarization is due solely to the mechanical stress, then the electricdisplacement of Eq.(2.9) can be re-written for the direct piezoelectric effect, as Eq. (2.13a).
In general, the piezoelectric strain tensor has 27 components. But due to the symmetry of the
stress tensor which allow for the contraction of the stress, the d ijk tensor can also be contracted
correspondingly. Thus the jk indices of d ijk is contracted exactly as the indices of the stress tensor.The contracted direct piezoelectric equation is shown in Eq. (2.13b) where the contracted
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2. Piezoelectric Materials
2-6
jk
d ijk
E i
i, j,k 1,2,3
l d
il E
i i1,2,3 l1,2,...6
(2.14a)
(2.14b)
piezoelectric strain constant is a 3x6 matrix.
The converse piezoelectric effect can be defined as the strain on the material caused by
the application of an external electric field. This effect can be formulated as another linear
relationship using the same piezoelectric strain constant as shown in Eq. (2.14). The tensor form
of Eq. (2.14a) has been contracted to Eq. (2.14b) by contracting the strain tensor and the
piezoelectric strain tensor as described before. Note that the order of the indices in Eq. (2.14b)
indicate that when written in matrix form the piezoelectric strain matrix is the transpose of its
counterpart in Eq. (2.13b). The d 31 constant for example, represents the electric displacement in
the 3 or z direction created by a stress applied in the 1 or x direction. Conversely it is the strain
generated in the x direction due to an electric field applied in the z direction.
From the equations above, it is clear that the piezoelectric effect has a directional nature
and will manifest only if the structure is deformed or an electric field is applied for the direct and
converse effects respectively. By reversing the direction of the electric field, the deformation will
also be reversed and vice versa. This feature provides useful directional authority in its
application to smart structures compared with other materials. So far, the mechanical, electrical
and the piezoelectric constitutive equations have been described separately. In the next section,
the full electro-mechanically coupled constitutive equations will be developed from
thermodynamic formulations. This will reveal the equivalence of the piezoelectric strain constant
between the direct and the converse piezoelectric equations and the fact that the piezoelectric
effect is reversible.
2.2.4 Electro-mechanical Constitutive Equations via Thermodynamic - Energy
Considerations
For materials that are non piezoelectric, the mechanical behavior and the electrical
behavior are independent from each other. As for piezoelectric materials, their electrical and
mechanical behaviors are said to be coupled where the mechanical variables of stress, σ and
strain, ε are related to each other as well as to the electrical variables of electric field, E and
electric displacement, D. The coupled constitutive equations can be taken empirically as the
linear combination of the pure mechanical or pure electrical effect with the piezoelectric effect.However, it is the intention of this section to provide a more rigorous development from energy
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2. Piezoelectric Materials
2-7
dU EdD d (2.15)
H U EDF U
G U ED(2.16)
dH d DdE H H (, E )dF EdD d F F (, D)dG d DdE G G(, E )
(2.17)
considerations, such that the constitutive equations will emerge naturally from the derivations.
In addition to revealing the intrinsic nature of the electro-mechanical coupling, some results of
practical importance will also emerge. It should be noted that the full thermodynamic derivation
should link mechanical, electrical and thermal effects, where the thermo-electric coupling give
rise only to the pyroelectric effect. However, since this thesis will not focus on pyroelectricity
and that all the coupling effects are assumed to be linear, the thermal influence can be safely
neglected.
For a general piezoelectric material, the total internal energy density is the sum of the
energy due to the mechanical and electrical work done (neglecting thermal effects), shown in Eq.
(2.15) in differential form, (IEEE, 1988). Note that the equations in this section contain vectors
and tensors but their indices will not be shown to prevent overcrowding of symbols.
The internal energy U(ε, D) of Eq. (2.15) is a thermodynamic potential and is a function
of the natural variables of strain and electric displacement. In different applications, other sets
of variables might be more convenient for analysis. In a process called Legendre transformation
(Callen, 1960), three other thermodynamic potentials (Eq. (2.16)) can be defined in terms of the
internal energy in order to transform other variables into natural variables.
The thermodynamic potentials of Eq. (2.16) are analogous to H - enthalpy, F - Helmholtz
free energy and G - Gibbs free energy in the notation of gaseous thermodynamics theory.
Expressing H, F and G in differential form and using Eq. (2.15), the differential forms of the
three thermodynamic potentials are shown in Eq. (2.17).
The G potential, having σ and E as its natural variables will lead into the strain
formulation of the piezoelectric constitutive equation. This is the preferred formulation in
analysis seeking to find exact solutions (Chan & Hagood, 1994; Crawley & deLuis, 1987;Meressi & Paden, 1993; Thompson & Loughlan, 1995). For finite element work, the preferred
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2. Piezoelectric Materials
2-8
(, E ) ( H
) E ; D(, E ) (
H
E )
; (2.18)
d (, E ) (
) E d (
E )
dE ; dD(, E ) ( D
) E d (
D
E )
dE ; (2.19)
(
i
j
) E c
E
ij ; ( D
i
E j
)
ij (2.20)
(i
E j
) ( H 2
E j
i
) ; ( Di
j
) ( H 2
j E
i
) (2.21)
natural variables are strain and electric field which is provided by the H potential (Allik &
Hughes, 1970; Tzou & Tseng, 1988, 1990; Bent & Hagood, 1995). Hence the four
thermodynamic potentials will facilitate four different sets of piezoelectric constitutive
formulations.
The H potential will lead into the piezoelectric stress formulation of the constitutive
equations that will be used in later chapters. Note that from Eq. (2.17), the stress and electric
displacement can be defined as derivatives of H as in Eq. (2.18).
where the superscript variables denote that those variables are kept constant during the variation
of other variables.
Since the stress and the electric displacements are functions of the natural variables -
strain and electric field - their total derivatives can be expressed as Eq. (2.19). This is in fact the
differential form of the electro-mechanically coupled piezoelectric constitutive equations.
The material properties of mechanical stiffness and electric permittivity can be readily
identified from Eq. (2.19) using Eq. (2.5) & (2.11) as shown in Eq. (2.20).
The two remaining partial derivatives of Eq. (2.19) can be taken as the derivatives of Eq.
(2.18) thus leading to the mixed second derivatives of H , as shown in Eq. (2.21). However, these
two terms are numerically identical because of the equivalence of the mixed second partial
derivatives of the H potential. In fact, these two terms can be used to define the piezoelectric
stress constant eij (contracted from a third rank 3x3x3 tensor to a 3x6 matrix), as shown in Eq.
(2.22). Note that the piezoelectric strain constant defined in §2.2.3 is obtained from a different
formulation, as will be explained later. Thus thermodynamic reasoning has led to the reversibility
of the direct and converse piezoelectric effect and the equivalence in magnitude of the associated
piezoelectric constant.
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2. Piezoelectric Materials
2-9
(
j
E i
) ( D
i
j
) eij (2.22)
i(, E ) c
E
ij j eki E
k ; D
i(, E ) e
ij j
ik E k
(2.23)
U (, D): i(, D) c
D
ij j hki D
k ; E
i(, D) h
ij j
ik Dk
F (
, D
): i
(
, D
)
s
D
ij j
gki Dk ;
E i(
, D
)
gij j
ik Dk
G(, E ): i(, E ) s
E
ij j d ki E
k ; D
i(, E ) d
ij j
ik E k
(2.25)
(
j
Di
) ( E
i
j
) hij
; (
j
Di
) ( E
i
j
) gij
; (
j
E i
) ( D
i
j
) d ij (2.26)
1
2
3
4
5
6
D1
D2
D3
c E
11 c
E
12 c
E
13 c
E
14 c
E
15 c
E
16
e11
e21
e31
c E
21 c
E
22 c
E
23 c
E
24 c
E
25 c
E
26
e12
e22
e32
c E
31 c
E
32 c
E
33 c
E
34 c
E
35 c
E
36
e13
e23
e33
c E
41 c
E
42 c
E
43 c
E
44 c
E
45 c
E
46
e14
e24
e34
c E
51 c
E
52 c
E
53 c
E
54 c
E
55 c
E
56
e15
e25
e35
c E
61 c
E
62 c
E
63 c
E
64 c
E
65 c
E
66
e16
e26
e36
e11
e12
e13
e14
e15
e16
11
12
13
e21
e22
e23
e24
e25
e26
21
22
23
e31 e32 e33 e34 e35 e36
31
32
33
1
2
3
4
5
6
E 1
E 2
E 3
(2.24)
Upon linearising Eq. (2.19) and substituting the material properties in Eq. (2.20) & (2.22),
the electro-mechanically coupled constitutive equation of the piezoelectric stress formulation in
contracted notation is given in Eq. (2.23) and its matrix form in Eq. (2.24)
The three other thermodynamic potential give rise to three formulations as shown in Eq.
(2.25), where sij is the elastic compliance and βij is equivalent to the inverse of the electrical
permittivity. Their corresponding piezoelectric constants are defined in Eq. (2.26).
Of the four different constitutive formulations, only two of them are commonly used in
the field of smart structures. As described before, FE work usually adopts the formulation based
on the H(ε , E) potential while the analytical work of finding exact solutions uses the G(σ , E)
potential formulation. The practical reason is that both of these potentials have natural variables
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2. Piezoelectric Materials
2-10
eijs
E
jk d ik
; d ijc
E
jk eik
; (c E
ij ) 1
s E
ij ;
ij
ij eik
d jk
(2.27)
of σ, ε and E which are directly measurable whereas the other two potentials U and F contain D
as a natural variable which is not experimentally measurable. The material constants found in all
four formulations can be related algebraically by manipulating the four constitutive equations.
In practice, the two most common formulations mentioned above uses the piezoelectric stress
constant eij and the piezoelectric strain constant d ij and their relation are shown in Eq. (2.27)
among other material properties.
2.2.5 Internal Structure of Piezoelectric Materials
This section intends to provide a brief overview of the influence of the crystal structure
on the piezoelectric nature of materials, it is by no means a detailed study on the crystallography
of piezoelectric materials.
In dielectric materials, the regular repetitive arrangement of atom, ions or molecules in
a lattice is called the crystal structure. The presence of the piezoelectric phenomenon in materials
depends on the internal crystal structure of the dielectric; unlike electrostriction which is
independent of the internal structure and thus can occur in liquids and gases (Bottcher, 1952).
In particular, the piezoelectric effect can only occur in crystals which do not possess a center of
symmetry. From the direct piezoelectric effect point of view, when the material is elastically
deformed, the center of gravity of the positive and negative charges are displaced and the lack
of symmetry prevents a net electrical cancellation. The summation of these electric dipoles lead
to a macroscopic non-zero polarization field. From a mathematical point of view, when the
crystal possess a center of symmetry then the piezoelectric coefficients which are third rank
tensors, must cease to exist (Nye, 1985). Therefore a material with a center of symmetry cannot
exhibit piezoelectric effects.
In the study of crystallography, there exist a total 32 different crystal classes out of which
20 possess the piezoelectric capability. The 32 classes is divided into seven groups which are
triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal and cubic. These groups are
also associated with the elastic nature of the material where triclinic represents anisotropic
material, orthorhombic represents orthotropic material and cubic are usually isotropic materials.
The choice of crystal structure of the piezoelectric to be adopted in this work were based on the
following factors: the generality of the piezoelectric material to accommodate a greater variety
of directional actuation, avoiding over-generalization of having more independent material
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2. Piezoelectric Materials
2-11
c E
11 c E
12 c E
13 0 0 0 0 0 e31
c E
12 c E
22 c E
23 0 0 0 0 0 e32
c E 13 c E
23 c E 33 0 0 0 0 0 e33
0 0 0 c E
44 0 0 0e
240
0 0 0 0 c E
55 0 e15
0 0
0 0 0 0 0 c E
66 0 0 0
0 0 0 0 e15
0 11 0 0
0 0 0 e24
0 0 0 22 0
e31
e32
e33
0 0 0 0 0 33
(2.28)
coefficients than is necessary, the availability of numerical data from manufacturer or literature.
Having considered this, the chosen crystal structural model for this research work is that of the
"orthorhombic - class mm2".
The piezoelectric material and the non-piezoelectric material (substrate) considered in
this work will be, at most, orthotropic. This means that it can be anything up to orthotropic;
including isotropic and transversely isotropic. In particular, two types of piezoelectric materials
widely used in smart structure applications are piezo-ceramics which is usually poly-crystalline
materials such as (Pb(Zr,Ti)O3) and piezo-polymers such as polyvinyl fluoride (PVF). Due to
polarization methods (to initiate the piezoelectric properties), the piezoceramics often adopt the
mm6 crystal structure whereas the piezopolymers are of the mm2 crystal structure (Zelenka,
1986). The former can be considered as a degenerate subset of the latter crystal structure. Thus
the choice of mm2 structure for the present work is general enough to cover the piezoelectric
materials often associated with smart structures. The elasto-piezo-dielectric material matrix that
correspond to the mm2 crystal structure is shown in Eq. (2.28).
2.2.6 Effect on Rotation of Matrices
In the most general case, this work will be able to model laminated composite structures
where each layer can be rotated about the transverse (z) axis. Hence any part of the material
which constitute the structure will have the option of being rotated. Although the material is
taken to be orthorhombic as described in §2.2.5, rotation of the material will change the material
matrix (Eq. 2.28) and the coefficients which are zero in the local coordinate system {c16 , c26 , c36 ,
c45 , e14 , e36 , 12
} will now be non-zero. From a practical point of view, these extra material
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2. Piezoelectric Materials
2-12
T
1
l; D Q
1 Dl
RT
1 R
1
l; E Q
1 E l
(2.29)
D
T
1
cl RTR
1
T
1
e
T
l Q
Q
1el RTR
1 Q
1
lQ
E
cg
e2g
e1g
g
E (2.30)
properties will give the smart structure additional actuation authority.
The stress and strain, being 2nd rank tensors are rotated using a different transformation
tensor than that used by vector quantities such as the electric displacement and the electric field.
The relation of these quantities between the local coordinate system and the global coordinate
system using the transformation tensors (see Appendix A) is given in Eq. (2.29).
where T = 2nd rank tensor transformation tensor (contracted notation)
Q = 1st rank tensor (vector) transformation tensor
R = engineering strain transformation matrix
By transforming the local quantities of the constitutive equation, the general piezoelectric
stress formulation of the coupled constitutive equation taking account of rotation is given in Eq.
(2.30). The global material properties are now functions of the rotation angle about the lateral
x-y plane. It is this equation that will be used in later work. Note that the magnitudes of the global
piezoelectric stress e1g and e2g are identical.
where l = local coordinate system, g = global coordinate system
The transformation matrices found in Appendix A, are used to derive the equivalent
material matrices with respect to the global structural coordinates. The local and global material
property matrices are found in Appendix B.
The transformation technique can also be applied to the special case when the
piezoelectric material is flipped with respect to the polarization direction. It can be shown the
stiffness and the electric permittivity matrices are unchanged while each of the piezoelectric
stress constants, e, changes sign.
2.3 Non-Linear Constitutive Models
Most models of piezoelectric material used as sensors and actuators in intelligent
structures consider only the linear piezoelectric constitutive equations (LPCE). The LPCE is
partly developed from the electromagnetic theory which states: Di =χij E j. However, this assumes
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2. Piezoelectric Materials
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P j a jk E k
b jkl E k E l .............................. (2.31)
G 1
2s
E
ij i
j d
mi
i E
m
1
2
mn E m E
n (2.32)
1
3s
E
ijk i
j
k d
mij
i
j E
m r
mni
i E
m E
n
1
3
mnp E m E
n E
p (2.33)
that the polarization (P) of the dielectric material is directly, linearly proportional to the electric
field (E) (Cheng, 1989). But for more complex material, this linear relationship might not hold,
instead Jackson (1962) suggested the form in Eq. (2.31)
Non-linear effects become significant in applications involving high electric fields, and
cyclic fields results in hysterisis (Ehlers & Weisshaar, 1990). It was briefly noted that in §2.2.4
that in reality there is interaction between the mechanical, electrical and thermal effects on
materials. In high temperature conditions or cryogenic conditions that might be typical of deep
space environment, the material properties no doubt will be different to that of room temperature
conditions. In general, the coupling between the mechanical and electrical effects give rise to
Non-linear Piezoelectric Constitutive Equations (NPCE). There has been some research into non-
linear piezoelectric constitutive models but it is only until very recently that the applications of
non-linear models in piezoelectric intelligent structures were investigated (Chan & Hagood,
1994; Gaudenzi & Bathe, 1995; Ghandi & Hagood, 1996). Such methods are often based on
adding higher order terms in the derivation to obtain a non-linear piezoelectric constitutive
model. For example, the strain formulation would be derived from the G potential / free energy
of Eq. (2.32):
By adding the higher order terms as in Eq. (2.33) to Eq. (2.32), it is then possible to derive a set
of non-linear constitutive equations with stress and electric field as the natural variables, (Beige
& Schmidt, 1985). Depending on the physical model, it is then possible to neglect some of the
terms.
Polarization reversal brings out the hysterisis and non-linear behavior of piezoceramics
(Chan & Hagood, 1994) while other causes of non-linear behavior include large induced stresses
due to large electric field or complex geometries needed for high actuation strain and anisotropic
strain. In the constitutive equations, the non-linearity manifests as extra terms due to Taylor’s
expansion. Non uniform electric fields arising from the geometry and positioning of the actuators
leads to non-linear relations between the strain and electric field. A FE approach was adopted by
Ghandi & Hagood. (1996) to model such cases and solutions was obtained via iteration. The
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iterations minimized the difference between the internal loads and charges with their external
counterparts in order to find the equilibrium values of displacements and charges. The dielectric
hysterisis curve ( D vs. E ) and the strain-field curve (butterfly loop) which is characteristic of non-
linear effects could also be produced numerically. However, these models were limited by
various assumptions. For example, assumptions in the Chan & Hagood (1994) model included
each crystalline being single-domain and it did not address the situation in which the
piezoelectric material is restraint by or embedded in the substrate, which is usually an anisotropic
medium.
Another way of looking at the non-linearity in intelligent structures is that the non-
linearity occurs in the piezoelectric coupling constants d ij or eij because these values could be
affected by the induced strain. In practical applications where the strains are a source for control,
the term "Actuation Strain" refers to the strain other than that caused by stresses, such as thermal,
magnetic or piezoelectric effects. Thus the actuation strain is the term Λi = d ji E j. For an
unconstrained, unstressed piezoelectric material, the induced strain would be the same as
actuation strain. However, if the piezoelectric material is constrained, being embedded or bonded
with another material, then the actuation strain is clearly different to the induced strain. Since the
induced strain is influenced by the actuation strain and the actuation strain is determined by the
coupling term which depends on the induced strain, an accurate calculation of the actuation strain
will involve an iteration process after the inclusion of non-linear terms in the coupling coefficient
d as performed by Crawley & Lazarus (1991). Koconis, et. al. (1994a) also followed a similar
approach by expanding the coefficient d as a function of applied voltage and total in-plane
strains.
Although several approaches exist, up until now there has not been a standard method for
working with these non-linearities. However, the linear formulation has been accepted by many
as being adequate, compared to the extra complexity that will arise when using non-linear
formulations. It was also pointed out by Jackson(1962) that the linear approximation of the
polarization is quite adequate for fields and temperatures in laboratory conditions and so too with
the other quantities. Lee & Moon (1989) reported experimental results in which the applied
voltage on a piezoelectric bimorph reached 600V yet without any hysterisis effects.
Acknowledging the complexities of non-linear effects and its importance in certain conditions,
the current work will however proceed with using the linear constitutive equations in the
following structural models. But note that the constitutive equations developed until Eq. (2.19)
has not made any linear assumptions and thus can be adopted for non-linear analysis.
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2.4 Usage of Piezoelectric Materials in Smart Structures
2.4.1 Piezoelectric Poling
To maximize the use of the piezoelectric effect, a high value of the piezoelectric constant
is desirable. Piezoelectric materials are often "poled" along one direction, conventionally taken
to be along the x3 (or z) axis. The poled direction is determined during the poling process,
following fabrication, when the piezoelectric material is subjected to a high electric field in the
chosen direction, under high temperature conditions, to create the permanent piezoelectric
property. Poling is analogous to the magnetization of a permanent magnet, in this case the
piezoelectric crystal structure will become slightly distorted and the dipoles aligned. However,
applying a high electric field opposite to the poling direction may cause the material to be de-
poled or become accidentally poled in the opposite direction. At even higher voltages, electric
breakdown occurs and the material will lose all its piezoelectric properties. Or if the operating
temperature is above a certain temperature called the Curie temperature then the piezoelectric
properties will also be destroyed (Chaudhry & Rogers, 1995).
2.4.2 Attributes of Piezoelectric Materials
The advantages of piezoelectric materials being used as sensors and actuators, include
ease of integration into existing structures, easily controlled by voltage, low weight, low power
requirements, low-field linearity and high bandwidth (allowing large range of applications). In
general, piezoelectric materials can be broadly classified into two groups: piezoceramics and
piezopolymers. The most common piezoceramic is Lead Zirconate Titanate (PZT) (chemical
notation: Pb(Zr, Ti)O3 ) and its piezopolymer counterpart is Polyvinylidene Fluoride (PVDF).
PZT ceramic, like any other ceramic has high stiffness while the PVDF polymer is more flexible,
has low stiffness and high damping. The high stiffness of the PZT makes it a suitable actuator
because of its high actuation authority and fast actuation response. In contrast, the flexibility and
low stiffness of PVDF makes it a better sensor. From these two types of piezoelectric material,
there exist a variety of configurations in which they can be manufactured to be used as sensors
and actuators. In terms of handling and practicality, the brittleness of piezoceramics places a
restriction on its minimum thickness. In addition, the attachment of piezoelectric materials to
passive structures are non-trivial, with the need to address issues such as electrical insulation and
the attachment of wires to the electrodes on the surface. However, these are not seriously
detracting characteristics as there are already various methods and solutions documented
throughout the literature (e.g. Rogers & Hagood, 1995; Safari et.al. 1996)
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Different grades of these two materials have been used by various researchers in this field
as sensors and actuators. The material data such as the piezoelectric coefficients can be found in
texts such as the Landolt-Bornstein Numerical Data (1979). The interested reader is also directed
to texts such as Nye (1985) and Jaffe et. al. (1971) for a more detailed discussion on material
structure and their properties. Manufacturing techniques have a significant effect on the quality
of piezoelectric material and composites. Some of the difficulties in manufacturing may have
several solutions while others are still unresolved. However it is not the intention of the present
work to discuss manufacturing processes but the interested reader is referred to Bent et.al. (1995),
Hagood & Bent (1993), Rodgers & Hagood (1995) and Safari, et. al. (1996) for the
manufacturing of piezoelectric fibers or ceramics in composites.
Finally it should be noted that the properties of manufactured piezoelectric materials
discussed above will no doubt be improved upon in the future, as technology advances, thereby
addressing current issues of concern regarding the use pf piezoelectric materials. As an example,
technological breakthrough in crystal growth methods has led to the development of single
crystal piezoelectrics which are now commercially available (e.g. TRS Ceramics). This new
material boasts an increase in field induced strain by an order of magnitude due to increased d 33
(> 2000 pC/N ) and electric breakdown (>150 kV/cm).
2.4.3 Different Forms of Piezoelectric Materials
Depending on the specific application, different physical forms of piezoelectric materials
have advantages over the others. In particular, some have even been designed to enhance certain
overall piezoelectric properties.
2.4.3.1 Distributed PVDF Layers and Monolithic Piezoelectrics
The simplest form of piezoelectric material that can be used as sensors and actuators is
a layer of material such as monolithic piezoceramic-PZT or PVDF films with uniform properties;
e.g. Ha et. al.(1992) used 0.13mm thick piezoceramics while Tzou & Tseng (1990) used 40µm
thick PVDF layers. The material is usually poled in the normal (x3) direction and assumed to be
transversely isotropic, resulting in a simple constitutive model. Discrete monolithic pieces of
piezoelectric material can also be bonded to the top and bottom of the substrate (Crawley &
Lazarus, 1991 and Thompson & Loughlan, 1995) or it can be embedded within the substrate
(Crawley & deLuis, 1987). Two usual assumptions are i) unless an electric field is applied, the
presence of the piezoelectric material on or in the substrate does not alter the overall structural
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properties significantly, ii) the bonding adhesives cause negligible property changes.
In structural control investigations, for instance, distributed layers of PVDF were bonded
and covered the entire top and bottom of the substrate (non-piezoelectric part of structure) as in
the works of Batra & Liang, 1996; Ray et. al., 1993; Ray et. al., 1993b; Tzou & Tseng, 1988,
1990; Tzou et. al., 1990; Tzou & Ye, 1996; Hwang & Park, 1993 and Hwang et. al., 1993). The
extreme form of distributed piezoelectric films is in the form of a piezoelectric paint (Egusa &
Iwasawa, 1994). This have the added advantages that it can be easily applied (painted) to curved
surfaces and does not require adhesives.
2.4.3.2 Piezoelectric Rod 1-3 Composites
The main material parameters of a piezoelectric composite are significantly superior than
a single phase monolithic material, (Newnham et. al., 1980).The "1-3 rod composites" with PZT
ceramic fibers embedded within an epoxy resin matrix in 1-3 connectivity, combine the
properties of high stiffness and flexibility. In the 1-3 connectivity, parallel fibers are embedded
in a matrix in the longitudinal ( x3) direction. A more complicated configuration such as 3-3
connectivity, involve interlocking of two phases in three dimensional networks and usually
provide greater strength and flexibility. A composite with PZT volume fraction of 40% can have
a value of d 33 almost the same as the PZT ceramic itself. The fabrication processes also include
poling of the PZT fibers/rods; the composite can be poled at the very last stage or the fibers can
be prepoled before embedding in the matrix. Some to the methods are casting the polymer around
the aligned PZT rods, the "lost wax" method, the "dice-and-fill" technique and a lamination
process (Smith, 1989), while PZT fibers may be produced by sol-gel processing, the relic process
and the Viscous Suspension Spinning Process (Safari et. al., 1996).
The material parameters such as compliance, stiffness, permittivity and piezoelectric
constants for composites are obviously dependent on the arrangement of the matrix and
fibers/rods in the composite. The reason for using composites is to enable several important
parameters to be optimized together whereas the use of single phase material might require a
trade-off between optimizing some parameters (e.g. stiffness against brittleness). Calculation of
effective composite parameters as a function of volume fraction is based on the Rules of Mixture
(Chan & Unsworth, 1989 and Smith et. al., 1985). Parameters such as permittivity, piezoelectric
constants and stiffness vary linearly with volume fraction for low to medium volume fractions.
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2.4.3.3 Piezoelectric Fiber Composites and Inter-Digitated Electrode
Piezoelectric Fiber Composite (PFC) were designed to enhance orthotropic and
anisotropic actuation capability thus allowing more independent and direct control of twist, bend
and extension by using layers of PFCs in laminated composites. The basic configuration of PFCs
in layer form contain piezoelectric fibers all aligned in the x1 direction. The PFC layer is along
the x1-x2 plane and is relatively thin; and hence can be considered as planar structures. The
effective properties of the basic PFC with unidirectional fibers can be modeled using the Rules
of Mixture with the basic assumptions such as equal fiber and matrix strain when the loading is
in the fiber direction and uniform mechanical and electrical fields (Hagood & Bent, 1993). A
typical PFC setup is having the active layer or the "electroceramic fibre composite" layer
sandwiched between two "porous interlaminar electrode" layers which is then sandwiched
between two "host composite material plies" (Hagood & Bent, 1993).
Conventionally, monolithic piezoceramics were actuated in the poling/normal
direction( x3), hence the piezoelectric effect in the transverse direction( x1) is less because |d 33| >
|d 31|. Hagood et. al. (1993) developed something similar to a circuit layer with electrodes, called
Inter-Digitated Electrodes (IDE), that will be placed at the top and bottom of the piezoelectric
layer. This allows the electric field to be applied in the transverse( x1) direction of a piezoceramic,
thus maximizing the transverse actuation. The two technologies of IDE and PFC have been
combined, (Bent & Hagood, 1995), where a PFC layer is placed between the IDE. This further
improves structural actuation since the IDE will produce electric fields in the direction ( x1) of
the piezoelectric fibers in the PFC.
2.5 Notation
The notation for material properties and physical variables associated with the
piezoelectric constitutive equations, along with their SI units, are listed in Table 2.1 below. Note
that some quantities are denoted with contracted notation as explained in the preceding sections
of this chapter. The indices of the symbols reveal the rank and size of the matrices or vector
quantities. (Definition of the symbols from FE formulation and shape control algorithms are
found in the "Symbols" section).
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Table 2.1 The main quantities of the electro-mechanically coupled piezoelectric effect and their units.
Symbol Quantity SI Unit IndicescE
ij 1 elastic stiffness N/m2 i,j=1...6
sEij
1 elastic compliance m2 /N i,j=1...6
σimechanical stress N/m2 i=1...6
εimechanical strain 1 i=1...6
χijε 2 absolute permittivity at constant strain C2 /(Nm2) or
F/m
i,j=1...3
χijσ 3 absolute permittivity at constant stress C2 /(Nm2) or
F/m
i,j=1...3
Di electric displacement or electric flux density C/m2 i=1...3
Ei electric field strength N/C or V/m i=1...3
Pi electric polarization C / m2 i=1..3
U,H,G,F thermodynamic energy potentials J / m3 ------
dij piezoelectric field-strain constant m/V or C/N i=1...3, j=1..6
eij piezoelectric field-stress constant C/m2 or
N/(Vm)
i=1...3, j=1..6
ui displacement m i=1...3
1. superscript E denotes constant electric field condition
2. superscript ε denotes constant strain condition
3. superscript σ denotes constant stress condition