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8/13/2019 PIEZOELECTRIC MATERIALS.pdf http://slidepdf.com/reader/full/piezoelectric-materialspdf 1/19 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
Transcript
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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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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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  T 

1

l;  D   Q

1 Dl

   RT 

1 R

1

l;  E   Q

1 E l

(2.29)

 D

1

cl RTR

1

1

e

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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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 

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 

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


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