Advances in Mechanics and Mathematics 9
Jiashi Yang
An Introduction
to the Theory of
Piezoelectricity
Second Edition
Advances in Mechanics and Mathematics
Volume 9
Series Editors
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Enrique Zuazua, Universidad Autónoma de Madrid and DeustoTech
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More information about this series at http://www.springer.com/series/5613
Jiashi Yang
An Introduction to the Theory
of Piezoelectricity
Second Edition
Jiashi Yang
Department of Mechanical and Materials Engineering
University of Nebraska-Lincoln
Lincoln, NE, USA
ISSN 1571-8689
ISSN 1876-9896 (electronic)
Advances in Mechanics and Mathematics
ISBN 978-3-030-03136-7
ISBN 978-3-030-03137-4 (eBook)
https://doi.org/10.1007/978-3-030-03137-4
Library of Congress Control Number: 2018960763
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Preface
The previous edition of this book, published in 2005, was based on the lecture notes
for a graduate course offered at the former Department of Engineering Mechanics of
the University of Nebraska–Lincoln since 1998. The course was intended to provide
graduate students performing research in electromechanical materials and devices
with the basic theoretical aspects of the continuum modeling of electroelastic
interactions in solids. As such, the scope of the book was, and still is, somewhat
limited. However, this second edition includes multiple additions and removals of
sections and has gone through a reorganization of chapters and sections. In particular, the derivation of the electric body force, couple, and power, which is one of the
fundamental building blocks for the construction of the nonlinear theory of
electroelasticity, is presented using both integral and differential approaches based
on Tiersten’s two-continuum model. They are also derived using the polarized
particle model and the effective polarization charge model in Appendices 1 and
2, respectively.
Although most of the book is devoted to the linear theory of piezoelectricity, the
book begins with a chapter on the nonlinear theory of electroelasticity. It is hoped that
this will be helpful for a deeper understanding of the theory of piezoelectricity, which
is the linearization of the nonlinear theory about a natural state with zero fields. The
presentation of the linear theory of piezoelectricity begins with Chap. 2, and readers
who are not interested in nonlinear electroelasticity can begin with Chap. 2 directly.
Chapters 3, 4, and 5 are all based on the linear theory of piezoelectricity, which
assumes infinitesimal deviations from an ideal reference state in which there are no
pre-existing mechanical and/or electric fields (initial or biasing fields). The presence of
biasing fields makes a material behave like a different one and renders the linear theory
of piezoelectricity invalid. The behavior of electroelastic bodies under biasing fields
can be described by the linear theory for infinitesimal incremental fields superposed on
finite biasing fields in Chap. 6, which is the linearization of the nonlinear theory of
electroelasticity about a finite bias. Chapter 7 discusses a few effects that are external
to the general theoretical framework developed in Chap. 1. Chapter 8 presents a few
examples of the application of the theory of piezoelectricity.
v
vi
Preface
For the linear theory of piezoelectricity, the notation of IEEE Standard on
Piezoelectricity is followed; however, notation varies among researchers regarding
the nonlinear theory of electroelasticity. In this book, the notation of H. F. Tiersten is
followed. The literature on the theory of piezoelectricity is abundant, so only those
that are directly used in the book are listed as references.
This book mainly covers the exact three-dimensional theory of piezoelectricity.
For approximate one- and two-dimensional theories of thin piezoelectric structures
of beams, plates, and shells as well their applications in piezoelectric devices, please
see the following books by this author published by World Scientific: The Mechanics of Piezoelectric Structures, 2006; Analysis of Piezoelectric Devices, 2006;
Antiplane Motions of Piezoceramics and Acoustic Wave Devices, 2010; and Vibration of Piezoelectric Crystal Plates, 2013.
Lincoln, NE, USA
September, 2018
Jiashi Yang
Contents
1
Nonlinear Theory of Electroelasticity . . . . . . . . . . . . . . . . . . . . . . .
1.1
Continuum Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2
Polarization and Piezoelectric Effects . . . . . . . . . . . . . . . . . . . .
1.3
Conservation of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4
Two-Continuum Model for Polarization . . . . . . . . . . . . . . . . . .
1.5
Conservation of Linear Momentum . . . . . . . . . . . . . . . . . . . . .
1.6
Conservation of Angular Momentum . . . . . . . . . . . . . . . . . . . .
1.7
Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.8
Balance Laws of Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . .
1.9
Summary of Balance Laws . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.10 Maxwell Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.11 Material Form of Balance Laws . . . . . . . . . . . . . . . . . . . . . . . .
1.12 Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.13 Jump Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.14 Displacement-Potential Formulation . . . . . . . . . . . . . . . . . . . . .
1.15 Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.16 Third-Order Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1
1
11
12
14
16
19
24
27
29
30
32
35
38
44
45
47
51
2
Linear Theory of Piezoelectricity . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1
Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2
Displacement-Potential Formulation . . . . . . . . . . . . . . . . . . . . . .
2.3
Principle of Superposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4
Poynting’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5
Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6
Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7
Four-Vector Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8
Matrix Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.9
Polarized Ceramics and Crystals of Class 6mm . . . . . . . . . . . . . .
2.10 Quartz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
53
58
59
60
61
62
65
66
68
72
vii
viii
Contents
2.11 Lithium Niobate and Lithium Tantalate . . . . . . . . . . . . . . . . . . .
2.12 Curvilinear Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
77
79
3
Static Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
Extension of a Ceramic Rod . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2
Thickness Stretch of a Ceramic Plate . . . . . . . . . . . . . . . . . . . .
3.3
Thickness Shear of a Ceramic Plate . . . . . . . . . . . . . . . . . . . . .
3.4
Capacitance of a Ceramic Plate . . . . . . . . . . . . . . . . . . . . . . . .
3.5
Capacitance of a Quartz Plate . . . . . . . . . . . . . . . . . . . . . . . . .
3.6
Torsion of a Ceramic Cylinder . . . . . . . . . . . . . . . . . . . . . . . . .
3.7
Azimuthal Shear of a Ceramic Cylinder . . . . . . . . . . . . . . . . . .
3.8
Capacitance of a Ceramic Cylinder . . . . . . . . . . . . . . . . . . . . .
3.9
Circular Hole Under Axisymmetric Loads . . . . . . . . . . . . . . . .
3.10 Circular Hole Under Shear . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.11 Circular Cylinder in an Electric Field . . . . . . . . . . . . . . . . . . . .
3.12 Surface Distribution of Electric Potential . . . . . . . . . . . . . . . . .
3.13 Screw Dislocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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81
81
84
87
89
91
95
97
98
101
103
105
108
109
111
4
Waves in Unbounded Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
Plane Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
Reflection and Refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3
Surface Waves on a Ceramic Half Space . . . . . . . . . . . . . . . . .
4.4
Ceramic Half Space with a Mass Layer . . . . . . . . . . . . . . . . . .
4.5
Ceramic Half Space with a Fluid . . . . . . . . . . . . . . . . . . . . . . .
4.6
Interface Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7
Waves in a Ceramic Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8
Waves Through the Joint of Two Plates . . . . . . . . . . . . . . . . . .
4.9
Trapped Waves in an Inhomogeneous Plate . . . . . . . . . . . . . . .
4.10 Waves in a Plate on a Half Space . . . . . . . . . . . . . . . . . . . . . . .
4.11 Gap Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.12 Waves on a Circular Cylindrical Surface . . . . . . . . . . . . . . . . .
4.13 Scattering by a Circular Cylinder . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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113
113
117
122
125
127
130
132
135
139
143
146
150
152
155
5
Vibrations of Finite Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1
Thickness Stretch of a Ceramic Plate Through e33 . . . . . . . . . . .
5.2
Thickness Shear of a Ceramic Plate Through e15 . . . . . . . . . . . .
5.3
Thickness Shear of a Quartz Plate Through e26 . . . . . . . . . . . . .
5.4
Thickness Shear of a Quartz Plate Through e36 . . . . . . . . . . . . .
5.5
Mass Sensitivity of Thickness-Shear Frequency . . . . . . . . . . . .
5.6
Azimuthal Shear of a Ceramic Cylinder . . . . . . . . . . . . . . . . . .
5.7
Axial Shear of a Ceramic Cylinder . . . . . . . . . . . . . . . . . . . . . .
5.8
Extension of a Ceramic Rod Through e33 . . . . . . . . . . . . . . . . .
5.9
Extension of a Ceramic Rod Through e31 . . . . . . . . . . . . . . . . .
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157
157
161
165
170
172
174
177
181
183
Contents
ix
5.10 Radial Vibration of a Circular Ceramic Plate . . . . . . . . . . . . . . . 186
5.11 Orthogonality of Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
6
Linear Theory for Small Fields on a Finite Bias . . . . . . . . . . . . . . . .
6.1
Nonlinear Spring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2
Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3
Linearization About a Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4
Boundary-Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5
Effects of a Small Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6
Frequency Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.7
Linearization Using Total Stress . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
193
193
195
197
200
201
203
204
205
7
Other Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1
Nonlinear Thermal and Dissipative Effects . . . . . . . . . . . . . . . . .
7.2
Linear Thermal and Dissipative Effects . . . . . . . . . . . . . . . . . . .
7.3
Semiconduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4
Extension of a Semiconducting Fiber . . . . . . . . . . . . . . . . . . . . .
7.5
Nonlocal Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6
Gradient Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.7
Electromagnetic Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.8
Quasistatic Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
207
207
210
212
214
216
223
228
233
234
8
Piezoelectric Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1
Generator or Energy Harvester . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2
Transducer for Acoustic Wave Generation . . . . . . . . . . . . . . . . .
8.3
Power Delivery Through a Wall . . . . . . . . . . . . . . . . . . . . . . . . .
8.4
Transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.5
Gyroscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.6
Elastic Rod with Piezoelectric Actuator/Sensor . . . . . . . . . . . . . .
8.7
Elastic Beam with Piezoelectric Actuator/Sensor . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
237
237
242
244
248
256
260
263
267
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
Chapter 1
Nonlinear Theory of Electroelasticity
In this chapter we develop the nonlinear theory of electroelasticity for large
deformations and strong electric fields. Readers who are only interested in the
linear theory of piezoelectricity may skip this chapter and begin with Chap. 2, Sect.
2.2. This chapter uses the two-point Cartesian tensor notation, the summation
convention for repeated tensor indices, and the convention that a comma followed
by an index denotes partial differentiation with respect to the coordinate associated
with the index.
1.1
Continuum Kinematics
This section is on the kinematics of a deformable continuum. The section is not
meant to be a complete treatment of the subject. Only results needed for the rest of
the book are presented.
Consider a deformable continuum which, in the reference configuration at time t0,
occupies a region V with a boundary surface S (see Fig. 1.1). N is the unit exterior
normal of S. In this state the body is free from deformation and fields. The position of
a material point in this state may be denoted by a position vector X¼XKIK in a
rectangular coordinate system XK. XK denotes the reference or material coordinates
of the material point. They are a continuous labeling of material particles so that they
are identifiable. At time t, the body occupies a region v with a boundary surface s and
an exterior normal n. The current position of the material point associated with X is
given by y¼ykik which denotes the present or spatial coordinates of the material
point.
Since the coordinate systems are othogonal,
ik il ¼ δkl ,
IK IL ¼ δKL ,
ð1:1Þ
where δkl and δKL are the Kronecker delta. In matrix notation,
© Springer Nature Switzerland AG 2018
J. Yang, An Introduction to the Theory of Piezoelectricity, Advances in Mechanics
and Mathematics 9, https://doi.org/10.1007/978-3-030-03137-4_1
1
2
1 Nonlinear Theory of Electroelasticity
S
N
Reference
state
s
V
u
v
Present
state
X
X3
y
y1
I3
I1
n
O
I2
i1
y3
o
i3
i2
X2
X1
y2
Fig. 1.1 Motion of a continuum and coordinate systems
2
1
½δkl ¼ ½δKL ¼ 4 0
0
0
1
0
3
0
0 5:
1
ð1:2Þ
The transformation coefficients between the two coordinate systems are denoted by
ik IL ¼ δkL ¼ δLk :
ð1:3Þ
In the rest of this book the two coordinate systems are chosen to be coincident, that is,
o ¼ O,
i 1 ¼ I1 ,
i2 ¼ I2 ,
i3 ¼ I3 :
ð1:4Þ
Then δkL becomes the Kronecker delta. A vector can be resolved into rectangular
components in different coordinate systems. For example, we can also write
y ¼ yK IK ,
ð1:5Þ
yM ¼ δMi yi :
ð1:6Þ
with
The motion of the body is described by
yi ¼ yi ðX; t Þ:
The displacement vector u of a material point is defined by
ð1:7Þ
1.1 Continuum Kinematics
3
y ¼ X þ u,
ð1:8Þ
yi ¼ δiM ðX M þ uM Þ:
ð1:9Þ
or
A material line element dX at t0 deforms into the following line element at t:
dyi jt
fixed
¼ yi, K dX K ,
ð1:10Þ
where the deformation gradient
yk, K ¼ δkK þ uk, K
ð1:11Þ
is a two-point tensor. The following determinant is called the Jacobian of the
deformation:
J ¼ det yk, K ¼ εijk yi, 1 y j, 2 yk, 3
1
¼ εKLM y1, K y2, L y3, M ¼ εklm εKLM yk, K yl, L ym, M ,
6
ð1:12Þ
where εklm and εKLM are the permutation tensor, and
εijk ¼ ii i j ik
8
< 1
¼ 1
:
0
i, j, k ¼ 1, 2, 3;
i, j, k ¼ 3, 2, 1;
otherwise:
2, 3, 1;
2, 1, 3;
3, 1, 2,
1, 3, 2,
ð1:13Þ
The following relationship exists (ε-δ identity) [1]:
εijk εpqr
δip
¼ δ jp
δkp
δiq
δ jq
δkq
δir δ jr :
δkr ð1:14Þ
As a special case, when i ¼ p, then
εijk εiqr ¼ δ jq δkr δ jr δkq :
ð1:15Þ
With Eq. (1.14), it can be shown from Eq. (1.12) that [2]
"
3 #
1 ∂yK ∂yL ∂yM
∂yK ∂yL ∂yM
∂yM
2
J¼
3
þ
:
6 ∂X L ∂X M ∂X K
∂X K ∂X M ∂X L
∂X M
It can be verified that for all L, M, and N the following is true:
ð1:16Þ
4
1 Nonlinear Theory of Electroelasticity
εijk yi, L y j, M yk, N ¼ JεLMN :
ð1:17Þ
From Eq. (1.17) the following can be shown:
εijk y j, M yk, N ¼ JεLMN X L, i ,
εijk yk, N ¼ JεLMN X L, i X M , j :
ð1:18Þ
Proof: Multiplying both sides of Eq. (1.17) by XL,r, we have
εijk yi, L X L, r y j, M yk, N ¼ JεLMN X L, r :
ð1:19Þ
εijk δir y j, M yk, N ¼ JεLMN X L, r ,
ð1:20Þ
εrjk y j, M yk, N ¼ JεLMN X L, r :
ð1:21Þ
Then
Replacing the index r by i gives Eq. (1.18)1. Similarly, Eq. (1.18)2 can be proved by
multiplying both sides of Eq. (1.18)1 by XM,p.
The following relationship can then be derived:
εijk εLMN y j, M yk, N ¼ 2JX L, i :
ð1:22Þ
Proof: Multiply both sides of Eq. (1.18)1 by εPMN
εijk εPMN y j, M yk, N ¼ JεLMN εPMN X L, i ¼ JεMNL εMNP X L, i
¼ J ðδNN δLP δNP δLN ÞX L, i ¼ J ð3δLP δLP ÞX L, i
¼ 2JX P, i ,
ð1:23Þ
where Eq. (1.15) has been used. Replacing the index P by L gives Eq. (1.22).
The derivative of the Jacobian with respect to one of its elements is
∂J
¼ JX L, i :
∂yi, L
ð1:24Þ
Proof: From Eq. (1.12)
6
∂J
¼ εijk εLMN δip δLQ y j, M yk, N
∂yp, Q
þ εijk εLMN yi, L δ jp δMQ yk, N þ εijk εLMN yi, L y j, M δkp δNQ
¼ εpjk εQMN y j, M yk, N þ εipk εLQN yi, L yk, N þ εijp εLMQ yi, L y j, M
¼ 3εpjk εQMN y j, M yk, N ¼ 3 2JX Q, p ,
ð1:25Þ
1.1 Continuum Kinematics
5
where Eq. (1.22) has been used.
With Eq. (1.22), we can also show that
ðJX K , k Þ, K ¼ 0:
ð1:26Þ
Proof: The differentiation of both sides of Eq. (1.22) with respect to XL gives
2ðJX L, i Þ, L ¼ εijk εLMN y j, ML yk, N þ y j, M yk, NL ¼ 0,
ð1:27Þ
because
εLMN y j, ML ¼ 0,
εLMN yk, NL ¼ 0:
ð1:28Þ
Similarly, the following is true:
J 1 yk, K
,k
¼ 0:
ð1:29Þ
The lengths of a material line element before and after deformation are given by
ðdLÞ2 ¼ dX K dX K ¼ δKL dX K dX L ,
ð1:30Þ
ðdlÞ2 ¼ dyi dyi ¼ yi, K dX K yi, L dX L ¼ C KL dX K dX L ,
ð1:31Þ
and
where CKL is the deformation tensor
CKL ¼ yk, K yk, L ¼ C LK :
ð1:32Þ
1
C 1
KL ¼ X K , k X L, k ¼ C LK ,
1
C KL CLM ¼ δKM :
ð1:33Þ
detðC KL Þ ¼ J 2 ,
ð1:34Þ
Its inverse is
From Eq. (1.32),
which defines J as a function of C.
It can then be shown that
6
1 Nonlinear Theory of Electroelasticity
∂J
1
¼ JC 1 :
∂CKL 2 LK
ð1:35Þ
∂J
∂J ∂C KM
∂J
∂ ¼
¼
y j, K y j, M
∂yi, L ∂C KM ∂yi, L
∂CKM ∂yi, L
∂J δ ji δKL y j, M þ y j, K δ ji δML
¼
∂C KM
∂J δKL yi, M þ yi, K δML
¼
∂C KM
∂J
∂J
yi, M þ
yi , K
¼
∂C
∂C
LM
KL
∂J
∂J
¼
þ
y ¼ JX L, i ,
∂CLK ∂C KL i, K
ð1:36Þ
Proof:
where Eq. (1.24) has been used, and the components of C are treated as if they were
independent in the partial differentiation. Equation (1.36) implies that
∂J
∂J
þ
¼ JX L, i X K , i ¼ JC1
LK :
∂C LK ∂C KL
ð1:37Þ
If J is written as a symmetric function of C in the sense that
∂J
∂J
¼
,
∂CLK ∂CKL
ð1:38Þ
then Eq. (1.35) is true.
The derivative of C-1 with respect to C is given by
∂C 1
1
1
1 1
KN
¼ C 1
KL C MN þ C KM C LN :
2
∂C LM
ð1:39Þ
Proof: From Eq. (1.33)2, for a small variation of C,
∂C1
KL
δ C 1
δC PQ þ C1
KL C LM ¼ C LM
KL δC LM ¼ 0,
∂C PQ
ð1:40Þ
where the components of C are treated as if they were independent in the partial
differentiation. Multiply Eq. (1.40) by C 1
MN :
1.1 Continuum Kinematics
7
∂C 1
1
KN
δCPQ þ C 1
KL C MN δC LM ¼ 0,
∂C PQ
ð1:41Þ
or
1
∂C1
∂C KN
1 1
1 1
KN
þ CKL CMN δCLM ¼
þ C K1 C 1N δC 11
∂CLM
∂C11
1
1
∂C KN
∂C KN
1 1
1 1
þ
þ C K1 C 2N þ
þ C K2 C 1N δ C12 þ ¼ 0:
∂C12
∂C 21
ð1:42Þ
Hence
∂C 1
∂C1
1
1 1
KN
KN
þ
¼ C 1
KL C MN C KM C LN :
∂C LM ∂C ML
ð1:43Þ
Equation (1.39) follows when C-1 is written as a symmetric function of C similar to
(1.38).
From Eqs. (1.30) and (1.31):
ðdlÞ2 ðdLÞ2 ¼ ðC KL δKL ÞdX K dX L ¼ 2E KL dX K dX L ,
ð1:44Þ
where the finite strain tensor EKL is defined by
E KL ¼ ðC KL δKL Þ=2 ¼ yi, K yi, L δKL =2
¼ ðuK , L þ uL, K þ uM , K uM , L Þ=2 ¼ E LK :
ð1:45Þ
The unabbreviated form of Eq. (1.45) is given below:
E11
E22
E33
E23
E31
E12
¼ u1, 1 þ ðu1, 1 u1, 1 þ u2, 1 u2, 1 þ u3, 1 u3, 1 Þ=2,
¼ u2, 2 þ ðu1, 2 u1, 2 þ u2, 2 u2, 2 þ u3, 2 u3, 2 Þ=2,
¼ u3, 3 þ ðu1, 3 u1, 3 þ u2, 3 u2, 3 þ u3, 3 u3, 3 Þ=2,
¼ ðu2, 3 þ u3, 2 þ u1, 2 u1, 3 þ u2, 2 u2, 3 þ u3, 2 u3, 3 Þ=2,
¼ ðu3, 1 þ u1, 3 þ u1, 3 u1, 1 þ u2, 3 u2, 1 þ u3, 3 u3, 1 Þ=2,
¼ ðu1, 2 þ u2, 1 þ u1, 1 u1, 2 þ u2, 1 u2, 2 þ u3, 1 u3, 2 Þ=2:
ð1:46Þ
ð1Þ
At the same material point consider two non-collinear material line elements d X
ð2Þ
ð1Þ
ð2Þ
and d X which deform into d y and d y . The area of the parallelogram spanned by
ð 1Þ
ð2Þ
ð1Þ
ð2Þ
d X and d X , and that by d y and d y , can be represented by the following vectors,
respectively:
8
1 Nonlinear Theory of Electroelasticity
ð1Þ
ð2Þ
N L dS ¼ dSL ¼ εLMN d X M d X N ,
ð1Þ
ð1:47Þ
ð2Þ
ni ds ¼ dsi ¼ εijk d y j d yk :
ð1:48Þ
dsi ¼ JX L, i dSL :
ð1:49Þ
They are related by
Proof:
ð1Þ
ð1Þ
ð2Þ
ð2Þ
dsi ¼ εijk d y j d yk ¼ εijk y j, M d X M yk, N d X N
ð1Þ
ð2Þ
ð1Þ
ð 2Þ
¼ εijk y j, M yk, N d X M d X N ¼ JX L, i εLMN d X M d X N
¼ JX L, i dSL ,
ð1:50Þ
where Eq. (1.18) has been used.
At the same material point consider three non-coplanar material line elements
ð 1Þ
ð2Þ
ð3Þ
ð1Þ
ð2Þ
ð3Þ
d X , d X , and d X which deform into d y , d y , and d y . The volume of the
ð1Þ
ð2Þ
ð3Þ
ð1Þ
ð2Þ
ð3Þ
parallelepiped spanned by d X , d X , and d X , and that by d y , d y , and d y , are
related by
dv ¼ JdV:
ð1:51Þ
Proof:
ð1Þ
ð2Þ
ð3Þ
ð1Þ ð2Þ ð3Þ
dv ¼ d y d y d y ¼ εijk d yi d y j d yk
ð1Þ
ð2Þ
ð3Þ
¼ εijk yi, L d X L y j, M d X M yk, N d X N
ð1Þ
ð2Þ
ð3Þ
¼ εijk yi, L y j, M yk, N d X L d X M d X N
ð1Þ
ð2Þ
ð1:52Þ
ð3Þ
¼ JεLMN dX L d X M d X
N
ð1Þ
ð2Þ
ð 3Þ
¼ J d X d X d X
¼ J dV,
where Eq. (1.17) has been used.
The velocity and acceleration of a material point are given by the following
material time derivatives:
1.1 Continuum Kinematics
9
dyi
∂yi ðX; t Þ
¼ y_ i ¼
vi ¼
,
∂t X fixed
dt
2
dvi ∂ yi ðX; t Þ
¼
v_ i ¼
:
∂t 2 dt
ð1:53Þ
X fixed
The deformation rate tensor dij and the spin tensor ωij are introduced by
decomposing the velocity gradient into symmetric and anti-symmetric parts, that is,
∂i v j ¼ v j, i ¼ dij þ ωij ,
1
d ij ¼ v j, i þ vi, j ,
2
1
ωij ¼ v j, i vi, j :
2
ð1:54Þ
d
d ∂yi
∂ dyi
ðdy Þ ¼
dX K ¼
dX K
dt i
dt ∂X K
∂X K dt
∂
¼
ðvi ÞdX K ¼ vi, K dX K ¼ vi, j y j, K dX K :
∂X K
ð1:55Þ
We also have
The strain rate and the deformation rate are related by
E_ KL ¼ dij yi, K y j, L :
ð1:56Þ
1
1
E_ KL ¼ y_ i, K yi, L þ yi, K y_ i, L ¼ vi, K yi, L þ yi, K vi, L
2
2
1
¼ vi, j y j, K yi, L þ yi, K vi, j y j, L
2
1
¼ v j, i yi, K y j, L þ yi, K vi, j y j, L
2
1
¼ v j, i þ vi, j yi, K y j, L ¼ dij yi, K y j, L :
2
ð1:57Þ
Proof:
The material derivative of the Jacobian is given by
J_ ¼ Jvk, k :
Proof: From Eq. (1.12),
ð1:58Þ
10
1 Nonlinear Theory of Electroelasticity
1
J_ ¼ εklm εKLM vk, K yl, L ym, M þ yk, K vl, L ym, M þ yk, K yl, L vm, M
6
1
1
¼ εklm εKLM vk, K yl, L ym, M ¼ vk, K εklm εKLM yl, L ym, M
2
2
1
¼ vk, K 2JX K , k ¼ Jvk, k ,
2
ð1:59Þ
where Eq. (1.22) has been used.
The following expression will be useful later in the book:
d
ðdvÞ ¼ vk, k dv:
dt
ð1:60Þ
d
d
dJ
ðdvÞ ¼ ðJdV Þ ¼ dV ¼ Jvk, k dV ¼ vk, k dv,
dt
dt
dt
ð1:61Þ
Proof:
where Eq. (1.58) has been used. In addition,
d
X L, j ¼ vi, K X K , j X L, i :
dt
ð1:62Þ
yi, K X K , j ¼ δij ,
ð1:63Þ
Proof: Since
we have, upon differentiating both sides,
y_ i, K X K , j þ yi, K
d
X K , j ¼ 0:
dt
ð1:64Þ
Then
yi, K
d
X K , j ¼ vi, K X K , j :
dt
ð1:65Þ
The multiplication of both sides of Eq. (1.65) by XL,i gives
d
X L, j ¼ vi, K X K , j X L, i :
dt
ð1:66Þ
1.2 Polarization and Piezoelectric Effects
1.2
11
Polarization and Piezoelectric Effects
When a dielectric is placed in an electric field, the electric charges in its molecules
redistribute themselves microscopically, resulting in a macroscopically polarized
state. The microscopic charge redistribution occurs in different ways (see Fig. 1.2).
At the macroscopic level the distinctions among different polarization mechanisms do not matter. A macroscopic polarization vector per unit present volume,
P
P ¼ lim
Δv
Δv!0
PMicro
Δv
ð1:67Þ
,
is introduced which describes the macroscopic polarizing state of the material. For a
finite dielectric body, the effect of the polarization vector P can be represented by the
following effective volume polarization charge density ρ p and surface polarization
charge density σ p [3]:
ρp ¼ Pi, i ,
σ p ¼ ni P i ,
ð1:68Þ
where n is the unit outward normal of the boundary surface of the body. We have
R
R
dvR þ s σ p ds
R
¼ Rv ðPi, i Þdv þ s ni Pi ds
¼ v ½ðPi, i Þ þ Pi, i dv ¼ 0,
vρ
p
ð1:69Þ
which shows that the total polarization charge is zero, and
R
R
þ s x j σ p ds R
¼ v xh j ðPi, i Þdv þ s x j ni Pii ds
R
¼ v x j ðPi, i Þ þ x j Pi , i dv
R
¼ Rv x j ðPi, i Þ þ δij Pi þ x j Pi, i dv
¼ v P j dv,
p
v x j ρ Rdv
ð1:70Þ
E
E
E
PMicro
PMicro
PMicro
(a)
(b)
(c)
Fig. 1.2 Microscopic polarizations: (a) electronic, (b) ionic, and (c) orientational
12
1 Nonlinear Theory of Electroelasticity
Fig. 1.3 Macroscopic
piezoelectric effects. (a)
Direct effect. (b) Converse
effect
F
+
P
-
F
(a) Direct effect.
(b) Converse effect.
P
(a) Undeformed.
(b) Deformed.
Fig. 1.4 Origin of the direct piezoelectric effect. (a) Undeformed. (b) Deformed
which shows that the first-order electric moment of the total polarization charge is
the total polarization vector. These are as expected.
Experiments show that in certain materials polarization can also be induced by
mechanical loads. Figure 1.3a shows such a phenomenon called the direct piezoelectric effect. The induced polarization can be at an angle, for example, perpendicular to the applied load, depending on the anisotropy of the material. When the load
is reversed, so is the induced polarization. When a voltage is applied to a material
possessing the direct piezoelectric effect, the material deforms. This is called the
converse piezoelectric effect (see Fig. 1.3b).
Whether a material is piezoelectric depends on its microscopic charge distribution. For example, the charge distribution in Fig. 1.4a, when deformed into Fig. 1.4b,
results in a polarization.
1.3
Conservation of Mass
The total mass of the material body is a constant, which is the mass in the reference
state, that is
1.3 Conservation of Mass
13
Z
Z
ρdv ¼
v
ρ0 dV,
ð1:71Þ
V
where ρ0 is the mass density in the reference state. For the left-hand side of
Eq. (1.71), we apply the usual change of integration variables to the reference state
using Eq. (1.51). Then the conservation of mass takes the following form:
Z
ρJ ρ0 dV ¼ 0:
ð1:72Þ
V
Equation (1.72) holds for any v. Assuming a continuous integrand, we conclude
from Eq. (1.72) that
ρ0 ¼ ρJ:
ð1:73Þ
Alternatively, since the total mass is conserved, we have
d
dt
Z
Z
Z
d
d
ðρJ ÞdV
ρdv ¼
ρJdV ¼
v R dt V RV dt
¼ RV ρ_ J þ ρJ_ dV ¼ V ρ_ J þ ρJvi, i dV
¼ v ρ_ þ ρvi, i dv ¼ 0,
ð1:74Þ
where Eq. (1.58) has been used. Then
ρ_ þ ρvi, i ¼ 0:
ð1:75Þ
From Eqs. (1.60) and (1.75), we can also obtain
d
d
ðρdvÞ ¼ ρ_ dv þ ρ ðdvÞ
dt
dt
¼ ρvi, i dv þ ρvi, i dv ¼ 0:
ð1:76Þ
In fact, for a differential material element of the body, the conservation of mass can
be written as
ρdv ¼ ρ0 dV:
ð1:77Þ
With Eqs. (1.58) and (1.75) it can be shown that
d
dt
Z
Z
ρ½ dv ¼
v
ρ
v
d½ dv,
dt
ð1:78Þ
where the square brackets represent a tensor field. Proof: With the change of
integration variables to the reference state, we have
14
1 Nonlinear Theory of Electroelasticity
d
dt
1.4
Z
Z
Z
d
d
ðρ½ J ÞdV
ρ½ dv ¼
ρ½ JdV ¼
dt
dt
v
V
V
R
d½ _
¼ V ρ_ ½ J þ ρ
J þ ρ½ J dV
dt
R
d½ J þ ρ½ Jvi, i dV
¼ V ρvi, i ½ J þ ρ
Z dt
R d½ d½ JdV ¼ ρ
dv:
¼ Vρ
dt
dt
v
ð1:79Þ
Two-Continuum Model for Polarization
This section gives a presentation of the two-continuum model for polarization in
[4]. Consider a deformable and polarizable body consisting of a lattice continuum
and an electronic charge continuum (see Fig. 1.5). In the reference state at time t0, the
two continua coincide with each other and occupy a spatial region V. The position of
a typical material point of the lattice continuum is denoted by X or XK. In the
reference state, the lattice comtinuum has a mass density ρ0(X) and a positive charge
density 0μl(X). The electronic continuum is massless and has a negative charge
density 0μe(X). Initially the body is assumed to be electrically neutral with
0μ
l
ðXÞ þ 0 μe ðXÞ ¼ 0:
ð1:80Þ
At time t, the lattice continuum occupies a spatial region v. The current position of
the material point of the lattice continuum associated with X is given by y ¼ y(X, t)
Lattice
continuum
v, r
V,r0
3
h
X
v´= v
y
y+h
2
1
Fig. 1.5 Motion and polarization of an electroelastic body
Electronic charge
continuum
1.4 Two-Continuum Model for Polarization
15
or yk ¼ yk(X, t). In this state the mass density of the lattice continuum is ρ and its
charge density is μl. The charge density of the electronic continuum is μe. The
electronic continuum is permitted to displace with respect to the lattice continuum
by an infinitesimal displacement field η(y, t) which accounts for the polarization. It is
assumed that η(y, t) preserves the volume of the electronic continuum, that is,
η k , k ð yÞ ¼
∂ηk ðyÞ
¼ 0:
∂yk
ð1:81Þ
An η(y, t) satisfying Eq. (1.81) is sufficient for describing polarization and Eq. (1.81)
is needed to obtain the proper electric charge equation [5], which will be shown later
in this book. As a consequence of Eqs. (1.80) and (1.81), at time t we have the
following charge neutrality condition
μl ðyÞ þ μe ðy þ ηÞ ¼ 0:
ð1:82Þ
With η(y, t), the dipole density per unit volume or the polarization vector can be
written as
P ¼ μl ðyÞðηÞ ¼ μe ðy þ ηÞη ffi μe ðyÞη:
ð1:83Þ
The conservation of mass is given by Eq. (1.75):
dρðyÞ
þ ρðyÞvk, k ¼ 0:
dt
ð1:84Þ
We also have the following equations from conservation of charge for the lattice and
electronic continua, respectively,
dμl ðyÞ
þ μl ðyÞvk, k ¼ 0,
dt
ð1:85Þ
dμe ðy þ ηÞ
þ μe ðy þ ηÞðvk þ ηk Þ, k ¼ 0:
dt
ð1:86Þ
Since ηk,k ¼ 0, from Eqs. (1.84), (1.85), and (1.86) we obtain the following
relationship which will be useful later:
μ_ l ðyÞ μ_ e ðy þ ηÞ ρ_ ðyÞ
¼ vk, k :
¼
¼
μl ðyÞ μe ðy þ ηÞ ρðyÞ
ð1:87Þ
The two-continuum model for polarization in [4] is global or integral, for the
whole body. Its local or differential version was introduced in [6]. For the lattice
continuum occupying v at time t, consider a differential element centered at y in the
16
1 Nonlinear Theory of Electroelasticity
Fig. 1.6 Current state of the
differential elements of the
lattice and electronic0
continua (dv and dv ) as a
composite particle
dy2
dv
dy3
dy1
3
dv'
y
y+h
1
2
form of a rectangular cuboid (see Fig. 1.6). Its edges dy1, dy2, and dy3 are along the
coordinate axes. Its volume is dv¼dy1dy2dy3. The corresponding differential
0
element of the electronic continuum dv is centered at y + η. It may become a
0
parallelepiped while preserving the same volume of dv ¼ dv. The two differential
elements can be viewed together as a composite element or particle which will be
useful later.
1.5
Conservation of Linear Momentum
When a deformable and polarizable material is subjected to an electric field, a
differential element of the material carries a small polarization vector and thus
experiences a body force and a body couple under the electric field. When such a
material deforms and polarizes, the electric field also does work to the material.
Fundamental to the development of the equations of electroelasticity is the derivation of the electric body force, couple, and power due to the electric field. This can be
done by averaging fields associated with charged and interacting particles or particles with internal degrees of freedom [7, 8]. It can also be done using the
two-continuum model [4] for polarization in the previous section. In [4], the basic
laws of physics were applied to each continuum and the resulting equations were
combined for the two continua together as a composite body to obtain the expressions for the electric body force fE, couple cE, and power wE as well as the equations
for nonlinear electroelasticity. Below we follow the approach in [9] to apply the
relevant physical laws directly to the combined continuum.
For the lattice continuum occupying v at time t and the associated electronic
continuum together (see Fig. 1.5), the integral form of the balance of linear momentum can be written as
1.5 Conservation of Linear Momentum
d
dt
17
Z
Z
ρvdv ¼ t ds
v R s
þ v ρ f þ μl ðyÞEðyÞ þ μe ðy þ ηÞEðy þ ηÞ dv,
ð1:88Þ
where f is the mechanical body force per unit mass, t is the surface traction per unit
area, and E is the Maxwellian electric field. In [4] the balance of linear momentum
was written separately for the lattice and electronic continua using a local electric
field for the interaction between the two continua in addition to the Maxwellian
electric field and then combined together. Different from [4], in Eq. (1.88) we
directly applied the balance of linear momentum to the combined continuum.
Since the electronic continuum is massless, the Maxwellian electric field and the
local electric field in [4] acting on the electronic continuum balance each other. In
other words, the local electric field and the Maxwellian electric field acting on the
electronic continuum are equal in magnitude and opposite in direction, one being the
negative vector of the other. In Eq. (1.88), the electric body forces due to the
Maxwellian electric field are acting on both the lattice continuum and the electronic
continuum through their charges at their corresponding locations, but the interaction
between the lattice and electronic continua is an internal interaction for the composite continuum and therefore does not appear in Eq. (1.88). Since η(y, t) is assumed to
be infinitesimal, we write the electric field E at y + η through Taylor’s expansion as
E j ðy þ ηÞ ffi E j ðyÞ þ ηi E j, i ðyÞ:
ð1:89Þ
With the use of Eqs. (1.82), (1.83), and (1.89), those terms in Eq. (1.88) that are
related to the electric body force can be further written in component form as
μl ðyÞE j ðyÞ þ μe ðy þ ηÞE j ðy þ
ηÞ
ffi μl ðyÞE j ðyÞ þ μe ðy þ
η
Þ
E j ðyÞ þ ηi E j, i ðyÞ
¼ μl ðyÞ þ μe ðy þ ηÞ E j ðyÞ þ μe ðy þ ηÞηi E j, i ðyÞ
¼ 0 þ Pi E j, i ¼ f Ej ,
ð1:90Þ
where we have introduced the body foce vector as
f kE ¼ Pi E k, i ,
f E ¼ P ∇E:
ð1:91Þ
We note that fE is nonlinear in the unknown electric and polarization fields and thus
disappears in the linear theory of piezoelectricity. The Cauchy stress tensor τji can be
introduced by [4]
t i ¼ n j τ ji ,
ð1:92Þ
18
1 Nonlinear Theory of Electroelasticity
through the usual tetrahedron argument [10]. Then, from Eq. (1.88), with the use of
Eqs. (1.79), (1.90), (1.92) and the divergence theorem, the balance of linear
momentum becomes
Z
Z
R dvi
E
dv
¼
ρ
ρf
þ
f
t i ds
dv
þ
i
i
v dt
v
s
R
R
¼ v ρf i þ f iE dv þ s τ ji n j ds
R
R
¼ v ρf i þ f iE dv þ v τ ji, j dv:
ð1:93Þ
τ ji, j þ ρf i þ f iE ¼ ρv_ i :
ð1:94Þ
Hence
The above is the integral derivation of the linear momentum equation and the
expression of the electric body force. Alternatively, these can be obtained using a
differential approach [6] by applying the conservation of linear momentum to the
differential elements in Fig. 1.6, which is presented in the following. The charges of
the differential elements of the lattice and the electronic continua, μl(y)dv and μe
0
(y + η)dv , are under the action of the Maxwellian electric field E. Their electrical
interaction with each other is an internal force for the composite particle. Therefore,
in the following equation of the balance of linear momentum of the composite
particle, we only need to consider contributions from the usual mechanical body
force per unit mass f which acts on dv, mechanical tractions on the six faces of the
rectangular cuboid of the lattice continuum in Fig. 1.6 represented by the Cauchy
0
stress tensor τkl, and the Maxwellian electric fields on dv and dv ¼ dv:
d
ðvk ρdvÞ ¼ ρf k dv
dt
1
1
þ τ1k y þ dy1 i1 dy2 dy3 τ1k y dy1 i1 dy2 dy3
2
2
1
1
þ τ2k y þ dy2 i2 dy3 dy1 τ2k y dy2 i2 dy3 dy1
2
2
1
1
þ τ3k y þ dy3 i3 dy1 dy2 τ3k y dy3 i3 dy1 dy2
2
2
þ μl ðyÞE k ðyÞdv þ μe ðy þ ηÞEk ðy þ ηÞdv:
ð1:95Þ
The conservation of mass implies that d(ρdv)/dt ¼ 0 (see Eq. (1.76)). Hence
d
dvk
ρdv:
ðvk ρdvÞ ¼
dt
dt
ð1:96Þ
For an infinitesimal dy1, with Taylor’s expansions at y, we have, approximately,
1.6 Conservation of Angular Momentum
19
1
1
τ1k y þ dy1 i1 dy2 dy3 τ1k y dy1 i1 dy2 dy3
2
2
1
ffi τ1k ðyÞ þ τ1k, 1 ðyÞ dy1 dy2 dy3
2
1
τ1k ðyÞ τ1k, 1 ðyÞ dy1 dy2 dy3
2
¼ τ1k, 1 ðyÞdv,
ð1:97Þ
and, similarly,
1
1
τ2k y þ dy2 i2 dy3 dy1 τ2k y dy2 i2 dy3 dy1 ffi τ2k, 2 ðyÞdv,
2
2
1
1
τ3k y þ dy3 i3 dy1 dy2 τ3k y dy3 i3 dy1 dy2 ffi τ3k, 3 ðyÞdv:
2
2
ð1:98Þ
ð1:99Þ
For small η, with the use of Eqs. (1.90) and (1.96), (1.97), (1.98) and (1.99),
Eq. (1.95) becomes
ρ
dvk
¼ τik, i þ ρf k þ f kE ,
dt
ð1:100Þ
which is the same as Eq. (1.94).
It will be shown in Appendix 2 that the electric body force fE can also be
calculated from the effective volume and surface polarization charges in Eq. (1.68)
through:
Z
Z
ρ E j dv þ
σ p E j ds:
p
v
1.6
ð1:101Þ
s
Conservation of Angular Momentum
The integral form of the balance of angular momentum for the combined lattice–
electronic continuum in Fig. 1.5 can be directly written as [9]
d
dt
Z
Z
y ρvdv ¼ y tds
v R s
þ v y ρf þ y μl ðyÞEðyÞ þ ðy þ ηÞ μe ðy þ ηÞEðy þ ηÞ dv:
ð1:102Þ
20
1 Nonlinear Theory of Electroelasticity
For small η, those terms in Eq. (1.102) that are related to the electric body couple can
be written as
y μl ðyÞEðyÞ þ ðy þ ηÞ μe ðy þ ηÞEðy þ ηÞ
ffi y μl ðyÞEðyÞ
þ y μe ðy þ ηÞ½EðyÞ þ η ∇EðyÞ þ η μe ðy þ ηÞEðy þ ηÞ
¼ y μl ðyÞEðyÞ þ y μe ðy þ ηÞEðyÞ
þ y μe ðy þ ηÞ½η ∇EðyÞ þ η μe ðy þ ηÞEðy þ ηÞ
ffi y μl ðyÞ þ μe ðy þ ηÞ EðyÞ þ y ½P ∇EðyÞ þ P EðyÞ
¼ 0 þ y f E þ cE ,
ð1:103Þ
where Eqs. (1.82), (1.83), (1.89) and the definition of fE in Eq. (1.91) have been
used, and we have introduced the electric body couple cE by
cE ¼ P E,
ciE ¼ εijk P j Ek :
ð1:104Þ
Obviously, cE is unknown and nonlinear, and therefore is irrelevant in the linear
theory of piezoelectricity. With Eq. (1.103), the balance of angular momentum in
Eq. (1.102) takes the following form:
d
dt
Z
εijk y j ρvk dv
R
R
¼ v εijk y j ρf k þ f kE þ ciE dv þ s εijk y j t k ds:
v
ð1:105Þ
With Eq. (1.79), the term on the left-hand side of Eq. (1.105) can be written as
d
dt
Z
Z
d
εijk y j ρvk dv ¼ ρεijk y j vk dv
v R
v
dt
¼ v ρεijk y_ j vk þ y j v_ k dv
R
R
¼ v ρεijk v j vk þ y j v_ k dv ¼ v ρεijk y j v_ k dv:
ð1:106Þ
The last term on the right-hand side of Eq. (1.105) can be written as
R
R
¼ s εijk y j τlk nl ds
R
¼ v εijk y j τlk , l dv ¼ v εijk δ jl τlk þ y j τlk, l dv
R
¼ v εijk τ jk þ y j τlk, l dv:
s εijk yRj t kds
ð1:107Þ
Substituting Eqs. (1.106) and (1.107) back into Eq. (1.105), we obtain
R
R
¼ v εijk y j ρf k þ f kE þ ciE dv
τ jk þ y j τlk, l dv,
dv
v ρεijkRy j v_ k
þ
or
v εijk
ð1:108Þ
1.6 Conservation of Angular Momentum
R
ρv_ ρf k f kE τlk, l dv
k
¼ v εijk τ jk þ ciE dv:
v εijk yRj
21
ð1:109Þ
The left-hand side of Eq. (1.109) vanishes because of the linear momentum equation
in Eq. (1.100). Hence,
Z
v
εijk τ jk þ ciE dv ¼ 0,
ð1:110Þ
εijk τ jk þ ciE ¼ 0:
ð1:111Þ
which implies that
An important implication of Eq. (1.111) is that the Cauchy stress tensor τkl is
asymmetric.
The angular momentum equation In Eq. (1.111) can also be obtained from a
differential approach [6]. For the composite particle in Fig. 1.6, the angular momentum equation about the origin of the reference frame is
d
εijk y j vk ρdv ¼ εijk y j ρf k dv
dt
1
1
þ εijk y j þ dy1 δ1 j τ1k y þ dy1 i1 dy2 dy3
2
2
1
1
εijk y j dy1 δ1 j τ1k y dy1 i1 dy2 dy3
2
2
1
1
þ εijk y j þ dy2 δ2 j τ2k y þ dy2 i2 dy3 dy1
2
2
1
1
εijk y j dy2 δ2 j τ2k y dy2 i2 dy3 dy1
2
2
1
1
þ εijk y j þ dy3 δ3 j τ3k y þ dy3 i3 dy1 dy2
2
2
1
1
εijk y j dy3 δ3 j τ3k y dy3 i3 dy1 dy2
2
2 þ εijk y j μl ðyÞEk ðyÞdv þ εijk y j þ η j μe ðy þ ηÞEk ðy þ ηÞdv,
ð1:112Þ
where the moment of inertia of ρdv about its own mass center has been neglected as a
higher-order infinitesimal. For the terms in Eq. (1.112), from the left-hand side to the
right-hand side, we have
d
d
εijk y j vk ρdv ¼ εijk y j vk ρdv
dt
dt
dvk
dvk
¼ εijk v j vk þ y j
ρdv ¼ εijk y j ρdv,
dt
dt
ð1:113Þ
22
1 Nonlinear Theory of Electroelasticity
1
1
εijk y j þ dy1 δ1 j τ1k y þ dy1 i1 dy2 dy3
2
2
1
1
εijk y j dy1 δ1 j τ1k y dy1 i1 dy2 dy3
2
2
1
1
ffi εijk y j þ dy1 δ1 j τ1k ðyÞ þ τ1k, 1 ðyÞ dy1 dy2 dy3
2
2
1
1
εijk y j dy1 δ1 j τ1k ðyÞ τ1k, 1 ðyÞ dy1 dy2 dy3
2
2
ffi εijk y j τ1k, 1 ðyÞdv þ εijk δ1 j τ1k ðyÞdv,
1
1
εijk y j þ dy2 δ2 j τ2k y þ dy2 i2 dy3 dy1
2
2
1
1
εijk y j dy2 δ2 j τ2k y dy2 i2 dy3 dy1
2
2
ffi εijk y j τ2k, 2 ðyÞdv þ εijk δ2 j τ2k ðyÞdv,
1
1
εijk y j þ dy3 δ3 j τ3k y þ dy3 i3 dy1 dy2
2
2
1
1
εijk y j dy3 δ3 j τ3k y dy3 i3 dy1 dy2
2
2
ffi εijk y j τ3k, 3 ðyÞdv þ εijk δ3 j τ3k ðyÞdv,
εijk y j μl ðyÞEk ðyÞdv þ εijk y j þ η j μe ðy þ ηÞE k ðy þ ηÞdv
ffi εijk y j f kE dv þ ciE dv,
ð1:114Þ
ð1:115Þ
ð1:116Þ
ð1:117Þ
where Eqs. (1.76), (1.103), and Taylor’s expansions have been used. The substitution of Eqs. (1.113), (1.114), (1.115), (1.116) and (1.117) into Eq. (1.112) yields
dvk
ρ
dt
¼ εijk y j ρf k þ εijk y j τmk, m ðyÞ þ εijk δmj τmk ðyÞ þ εijk y j f kE þ ciE ,
εijk y j
ð1:118Þ
or
dvk
τmk, m ρf k f kE ¼ εijk τ jk ðyÞ þ ciE ,
εijk y j ρ
dt
ð1:119Þ
which reduces to
εijk τ jk þ ciE ¼ 0:
Equation (1.120) is the same as Eq. (1.111).
ð1:120Þ
1.6 Conservation of Angular Momentum
23
The angular momentum equation can also be written with respect to the center
of mass of the composite particle in Fig. 1.6 which is the same as the center of mass
of the differential element of the lattice continuum ρdv because the electronic
continuum is massless. In this case the moment of inertia of ρdv about its own mass
center is a higher-order infinitesimal and can be neglected. In addition, for a
differential element, the mechanical body force on the differential element of the
lattice continuum essentially go through the mass center of ρdv and does not
contribute to the moment about its mass center. Hence the angular momentum
equation is simply
1
1
εijk dy1 δ1 j τ1k y þ dy1 i1 dy2 dy3
2
2 1
1
εijk dy1 δ1 j τ1k y dy1 i1 dy2 dy3
2
2
1
1
þ εijk dy2 δ2 j τ2k y þ dy2 i2 dy3 dy1
2
2
1
1
εijk dy2 δ2 j τ2k y dy2 i2 dy3 dy1
2
2
1
1
þ εijk dy3 δ3 j τ3k y þ dy3 i3 dy1 dy2
2
2
1
1
εijk dy3 δ3 j τ3k y dy3 i3 dy1 dy2
2
2
þ εijk η j μe ðy þ ηÞE k ðy þ ηÞdv ¼ 0:
ð1:121Þ
With the use of the Taylor’s expansions of the stress components and Eq. (1.83), we
can write Eq. (1.121) as
εijk δ1 j τ1k dv þ εijk δ2 j τ2k dv þ εijk δ3 j τ3k dv þ εijk P j E k dv
¼ εijk δmj τmk dv þ
εijk P j Ek dv
¼ εijk τ jk þ ciE dv ¼ 0,
ð1:122Þ
which gives Eq. (1.120). It is worth mentioning that in the derivation of the angular
momentum equation about the center of mass of ρdv in Eq. (1.122), the linear
momentum equation in Eq. (1.100) was not used. This is different from the derivation of Eq. (1.120).
It will be shown in Appendix 2 that the electric body couple cE can also be
calculated from the effective volume and surface polarization charges in Eq. (1.68)
through:
Z
Z
εijk x j ρ E k dv þ
εijk x j σ p Ek ds:
p
v
s
ð1:123Þ
24
1.7
1 Nonlinear Theory of Electroelasticity
Conservation of Energy
Let the internal energy of the body per unit mass be ε. For the conservation of energy
of the combined lattice–electronic continuum in Fig. 1.5, we can directly write the
integral form of the conservation of energy as [9]
d
dt
Z
1
ρ v v þ ε dv ¼ t vds
v R 2
s
þ v ρf v þ μl ðyÞEðyÞ vðyÞ
þ μe ðy þ ηÞEðy þ ηÞ ½vðyÞ þ η_ dv:
Z
ð1:124Þ
For small η, those terms in Eq. (1.124) that are related to the electric body power
become
μ1 ðyÞE k ðyÞvk ðyÞ þ μe ðy þ ηÞEk ðy þ ηÞ½vk ðyÞ þ η_ k ffi μ1 ðyÞE k ðyÞvk ðyÞ
þ μe ðy þ ηÞ½E k ðyÞ þ E k, i ðyÞηi ½vk ðyÞ þ η_ k ffi μ1 ðyÞE k ðyÞvk ðyÞ
þ μe ðy þ ηÞ½E k ðyÞvk ðyÞ þ E k, i ðyÞηi vk ðyÞ þ E k ðyÞη_ k ð1:125Þ
¼ ½μ ðyÞ þ μ ðy þ ηÞE k ðyÞvk ðyÞ
1
e
þ μe ðy þ ηÞEk, i ðyÞηi vk ðyÞ þ μe ðy þ ηÞEk ðyÞη_ k
¼ 0 þ Pi Ek, i ðyÞvk ðyÞ þ μe ðy þ ηÞE k ðyÞη_ k
¼ f kE vk ðyÞ þ wE ,
where we have used Eqs. (1.182), (1.183), (1.189), (1.191) and introduced the
electric body power wE by
wE ¼ μe ðy þ ηÞE k ðyÞη_ k
d e
dμe ðy þ ηÞ
ηk
¼ E k ð yÞ
½μ ðy þ ηÞηk dt
dt
¼ Ek ðyÞ P_ k μ_ e ðy þ ηÞηk
μ_ e ðy þ ηÞ
¼ Ek P_ k μe ðy þ ηÞ e
η E k ð yÞ
μ ðy þ η Þ k
e
μ_ ðy þ ηÞ
ρ_
Pk Ek ¼ E k P_ k Pk E k
¼ Ek P_ k e
μ ðy þ ηÞ
ρ
¼ ρ_ π k þ ρπ_ k E k ρ_ π k E k ¼ ρE k π_ k :
ð1:126Þ
In obtaining Eq. (1.126), Eqs. (1.183) and (1.87) have been used and we have
introduced the polarization per unit mass as
1.7 Conservation of Energy
25
π i ¼ Pi =ρ:
ð1:127Þ
With Eq. (1.125), the conservation of energy in Eq. (1.124) becomes
d
dt
1
ρ vi vi þ ε dv
v R 2
R
¼ v ρf k þ f kE vk þ wE dv þ s t k vk ds:
Z
ð1:128Þ
With Eq. (1.79), the left-hand side of Eq. (1.128) can be written as
d
dt
Z
1
d 1
vi vi þ ε dv
ρ vi vi þ ε dv ¼ ρ
v R 2
v dt 2
¼ v ρ vi v_ i þ ε_ dv:
Z
ð1:129Þ
Using Eq. (1.92) and the divergence theorem, the last term on the right-hand side of
Eq. (1.128) can be written as
R
R
¼ s τlk nl vk dsR
¼ v ðτlk vk Þ, l dv ¼ v ðτlk, l vk þ τlk vk, l Þdv:
s t k vk da
R
ð1:130Þ
The substitution of Eqs. (1.129) and (1.130) back into Eq. (1.128) gives
R
R E
E
v ρ vkRv_ k þ ε_ dv ¼ v ρf k þ f k vk þ w dv
þ v ðτlk, l vk þ τlk vk, l Þdv,
ð1:131Þ
or
Z
vk ρv_ k ρf k v
f kE
Z
τlk, l dv ¼
τlk vk, l þ wE ρε_ dv:
ð1:132Þ
v
The left-hand side of Eq. (1.132) vanishes because of the linear momentum equation
in Eq. (1.100). Hence
Z
τlk vk, l þ wE ρε_ dv ¼ 0,
ð1:133Þ
v
which implies that
ρε_ ¼ τij v j, i þ wE :
ð1:134Þ
Equation (1.134) can also be derived using a differential approach [6]. For the
composite particle in Fig. 1.6, the energy equation can be written as
26
1 Nonlinear Theory of Electroelasticity
d 1
vk vk ρdv þ ερdv ¼ ρf k vk dv
dt 2 1
1
þ τ1k y þ dy1 i1 vk y þ dy1 i1 dy2 dy3
2
2
1
1
τ1k y dy1 i1 vk y dy1 i1 dy2 dy3
2
2
1
1
þ τ2k y þ dy2 i2 vk y þ dy2 i2 dy3 dy1
2
2
1
1
τ2k y dy2 i2 vk y dy2 i2 dy3 dy1
2
2
1
1
þ τ3k y þ dy3 i3 vk y þ dy3 i3 dy1 dy2
2
2
1
1
τ3k y dy3 i3 vk y dy3 i3 dy1 dy2
2
2
þ μl ðyÞEk ðyÞvk ðyÞdv þ μe ðy þ ηÞE k ðy þ ηÞ½vk ðyÞ þ η_ k dv:
ð1:135Þ
For the terms in Eq. (1.135), from the left-hand side to the right-hand side, we have
d 1
d 1
dvk
dε
vk vk ρdv þ ερdv ¼ ρdv
vk vk þ ε ¼ vk
ρdv þ ρdv,
dt 2
dt 2
dt
dt
1
1
τ1k y þ dy1 i1 vk y þ dy1 i1 dy2 dy3
2
2
1
1
τ1k y dy1 i1 vk y dy1 i1 dy2 dy3
2
2
dy1
dy
ffi τ1k ðyÞ þ τ1k, 1 ðyÞ
vk ðyÞ þ vk, 1 ðyÞ 1 dy2 dy3
2
2
dy1
dy1
τ1k ðyÞ τ1k, 1 ðyÞ
vk ðyÞ vk, 1 ðyÞ
dy2 dy3
2
2
ffi τ1k ðyÞvk, 1 ðyÞdv þ τ1k, 1 ðyÞvk ðyÞdv,
1
1
τ2k y þ dy2 i2 vk y þ dy2 i2 dy3 dy1
2
2
1
1
τ2k y dy2 i2 vk y dy2 i2 dy3 dy1
2
2
ffi τ2k ðyÞvk, 2 ðyÞdv þ τ2k, 2 ðyÞvk ðyÞdv,
1
1
τ3k y þ dy3 i3 vk y þ dy3 i3 dy1 dy2
2
2
1
1
τ3k y dy3 i3 vk y dy3 i3 dy1 dy2
2
2
ffi τ3k ðyÞvk, 3 ðyÞdv þ τ3k, 3 ðyÞvk ðyÞdv,
ð1:136Þ
ð1:137Þ
ð1:138Þ
ð1:139Þ
1.8 Balance Laws of Electrostatics
27
μl ðyÞEk ðyÞvk ðyÞdv þ μe ðy þ ηÞE k ðy þ ηÞ½vk ðyÞ þ η_ k dv
ffi f kE vk dv þ wE dv,
ð1:140Þ
where Eqs. (1.76), (1.91), and (1.125) have been used. The substitution of
Eqs. (1.136),(1.137), (1.138), (1.139) and (1.140) into Eq. (1.135) gives
dvk
dε
E
ρ
τmk, m ρf k f k vk dv þ ρ dv
dt
dt
¼ τmk vk, m dv þ wE dv:
ð1:141Þ
The first term on the left-hand side of Eq. (1.141) vanishes because of the linear
momentum equation in Eq. (1.100). Then Eq. (1.141) reduces to
ρ
dε
¼ τmk vk, m þ wE ,
dt
ð1:142Þ
which is the same as Eq. (1.134).
It will be shown in Appendix 2 that the electric body power wE can also be
calculated from the effective volume and surface polarization charges in Eq. (1.68).
1.8
Balance Laws of Electrostatics
Let l be a closed curve. For electrostatics, the integral form of Faraday’s law or more
generally the Maxwell–Faraday equation is
Z
E dl ¼ 0:
ð1:143Þ
l
With Stoke’s theorem, the line integral along l can be converted to a surface integral
over an area s whose boundary is l:
Z
Z
E dl ¼
l
ð∇ EÞ ds ¼ 0,
ð1:144Þ
s
which implies that
∇ E ¼ εijk Ek, j ii ¼ 0:
Let s be a closed surface. The integral form of Gauss’s law states that
ð1:145Þ
28
1 Nonlinear Theory of Electroelasticity
Z
Z
ε0 E ds ¼
s
ðρp þ ρe Þdv,
ð1:146Þ
v
where ε0 is the permittivity of free space and ρe is the free charge density per unit
volume. For dielectrics ρe ¼ 0. Since free charges will be relevant later in this book
when metal electrodes are involved, ρe is kept here. From Eq. (1.146), using the
divergence theorem, we can write
Z
Z
ni ε0 Ei ds ¼
Z
ðρp þ ρe Þdv,
ð1:147Þ
½ε0 Ei, i ðρp þ ρe Þdv ¼ 0,
ð1:148Þ
s
v
v
which implies that
ε0 E i, i ¼ ρp þ ρe :
ð1:149Þ
From Eq. (1.68), ρ p ¼ Pi,i. Therefore, Eq. (1.149) can be further written as
ðε0 Ei þ Pi Þ, i ¼ Di, i ¼ ρe ,
ð1:150Þ
where we have introduced the electric displacement vector D through
D i ¼ ε0 E i þ P i :
ð1:151Þ
Alternatively, from the two-continuum model of polarization, Gauss’s law takes
the following form [5]
Z
Z
l
μ ðyÞ þ μe ðyÞ dv,
ε0 E ds ¼
s
ð1:152Þ
v
where the free charge density is assumed to be zero. The localization of Eq. (1.152)
gives
ε0 Ei, i ¼ μl ðyÞ þ μe ðyÞ ¼ μe ðy þ ηÞ þ μe ðyÞ
¼ μe ðyÞ μ,ei ðyÞηi þ μe ðyÞ
¼ ½μe ðyÞηi , i þ μe ðyÞηi, i
ffi Pi, i ,
ð1:153Þ
where Eqs. (1.81), (1.82), and (1.83) have been used. Eq. (1.153) can be written as
Di, i ¼ 0,
which is the same as Eq. (1.150).
ð1:154Þ
1.9 Summary of Balance Laws
1.9
29
Summary of Balance Laws
For convenience, we gather from Eqs. (1.74), (1.143), (1.146), (1.88) or (1.93),
(1.102) or (1.105), and (1.124) or (1.128) the balance laws from various origins in
integral forms below.
d
dt
Z
Z
ρdv ¼ 0,
ð1:155Þ
E dl ¼ 0,
ð1:156Þ
v
l
Z
Z
D ds ¼
d
dt
s
Z
Z
ρvdv ¼
v
ρe dv,
v
ρ f þ f E dv þ
v
ð1:157Þ
Z
tds,
ð1:158Þ
s
Z
Z
Z
d
y ρvdv ¼
y ρ f þ f E þ cE dv þ y t ds,
dt v
v
s
Z Z
Z
d
1
ρ v v þ ε dv ¼
ρ f þ f E v þ wE dv þ t vds,
dt v 2
v
s
ð1:159Þ
ð1:160Þ
where, from Eqs. (1.91), (1.104), and (1.126),
f kE ¼ Pi E k, i , f E ¼ P ∇E,
ciE ¼ εijk P j E k , cE ¼ P E,
wE ¼ ρE k π_ k ,
ð1:161Þ
which are nonlinear and unknown. In addition, from Eqs. (1.151), (1.127), and (1.92),
Di ¼ ε0 E i þ Pi ,
π i ¼ Pi =ρ,
t i ¼ n j τ ji :
ð1:162Þ
The corresponding differential forms of the balance laws are taken from Eqs. (1.75),
(1.145), (1.150), (1.94), (1.111), and (1.134):
ρ_ þ ρvi, i ¼ 0,
ð1:163Þ
εijk Ek, j ¼ 0,
ð1:164Þ
Di, i ¼ ρe ,
ð1:165Þ
30
1 Nonlinear Theory of Electroelasticity
ρv_ k ¼ τik, i þ ρf k þ f kE ,
ð1:166Þ
εijk τ jk þ ciE ¼ 0,
ð1:167Þ
ρε_ ¼ τmk vk, m þ wE :
ð1:168Þ
Since the electric field is curl free, an electrostatic potential can be introduced such
that
E i ¼ φ, i :
ð1:169Þ
Then Eq. (1.164) is automatically satisfied. A free energy χ can be introduced
through the following Legendre transform from the internal energy [4]:
χ ¼ ε Ei π i ,
ð1:170Þ
χ_ ¼ ε_ E_ i π i E i π_ i :
ð1:171Þ
Then the energy equation in Eq. (1.168) becomes
ρχ_ ¼ τij v j, i Pi E_ i :
1.10
ð1:172Þ
Maxwell Stress
It will be proven convenient to introduce the electrostatic Maxwell stress tensor T ijE
[11] whose divergence yields the electric body force through
T ijE, i ¼ f Ej :
ð1:173Þ
For the existence of such a T ijE consider
1
T ijE ¼ Di E j ε0 Ek Ek δij
2
1
¼ Pi E j þ ε0 E i E j E k E k δij :
2
ð1:174Þ
We have, when ρe ¼ 0,
T ijE, i ¼ Di, i E j þ ε0 E i E j, i E i, j þ Pi E j, i ,
¼ ρe E j þ 0 þ Pi E j, i ¼ f Ej ,
ð1:175Þ
1.10
Maxwell Stress
31
where Eqs. (1.164), (1.165), and (1.91) have been used. We also have
εijk T Ejk ¼ εijk P j Ek ¼ ciE :
ð1:176Þ
It can be easily verified that
Z
v
Z
f iE dv ¼
s
T Eji n j ds:
ð1:177Þ
We note that T ijE as defined by Eq. (1.173) is not unique in the sense that there are
other tensors that also satisfy Eq. (1.173). For example, adding a second rank tensor
with a zero divergence to the T ijE in Eq. (1.174) does not affect its satisfaction of
Eq. (1.173).
With T ijE , the balance of linear momentum, Eq. (1.166), can be written as
τij þ T ijE þ ρf j ¼ ρv_ j :
,i
ð1:178Þ
With Eq. (1.104), the balance of angular momentum, Eq. (1.167), can be written as
εijk τ jk þ P j E k ¼ 0,
ð1:179Þ
which implies that τjk + PjEk is a symmetric tensor and that εijk τ jk þ T Ejk ¼ 0, or
the sum of the Cauchy stress tensor τij and T ijE is also symmetric. We introduce the
following total stress tensor and denote it by tij [12]:
t ij ¼ τij þ T ijE
1
¼ τij þ Pi E j þ ε0 E i E j Ek E k δij ¼ t ji :
2
ð1:180Þ
tij is symmetric and can be decomposed into the sum of a symmetric tensor τijS and the
symmetric Maxwell stress tensor T ijES as follows [12]:
t ij ¼ τijS þ T ijES ,
τijS ¼ τij þ Pi E j ¼ τ Sji ,
1
T ijES ¼ ε0 E i E j E k E k δij ¼ T ES
ji :
2
ð1:181Þ
In terms of the total stress tensor tij, the linear and angular momentum equations in
Eqs. (1.166) and (1.167) become
32
1 Nonlinear Theory of Electroelasticity
t ij, i þ ρf j ¼ ρv_ j ,
εijk t jk ¼ 0:
1.11
ð1:182Þ
Material Form of Balance Laws
Up to this point, the field equations in Eqs. (1.163), (1.164), (1.165), (1.166), (1.167)
and (1.168) have been written in terms of the present coordinates yi in the sense that
all spatial derivatives are taken with respect to y. Since the reference coordinates of
material points are known while the present coordinates are not, it is essential to have
the equations written in terms of the reference coordinates XK. For this purpose we
introduce the material potential gradient WK, the material electric field E L , the
material polarization P L , the material electric displacement D L , and the free charge
density per unit reference volume ρE as:
W K ¼ φ, K ¼ φ, i yi, K ¼ E i yi, K ,
E L ¼ JX L, i E i ¼ JX L, i X K , i W K ¼ JC1
LK W K ,
P L ¼ JX L, i Pi ,
D L ¼ JX L, i Di ¼ ε0 E L þ P L ,
ρE ¼ ρe J,
ð1:183Þ
where J ¼ det (yk,L). Some of their inversions are
E i ¼ J 1 yi, L E L ,
Pi ¼ J 1 yi, L P L ,
Di ¼ J 1 yi, L D L :
ð1:184Þ
Similarly, for the mechanical fields, we introduce the two-point forms of the
symmetric stress tensor τijS , the symmetric Maxwell stress T ijES , and the total stress
tij through
F Lj ¼ JX L, i τijS ,
M Lj ¼ JX L, i T ijES ,
K Lj ¼ JX L, i t ij ¼ F Lj þ M Lj :
ð1:185Þ
The inverse expressions of Eq. (1.185) are
τijS ¼ J 1 yi, L F Lj ,
T ijES ¼ J 1 yi, L M Lj ,
t ij ¼ J 1 yi, L K Lj :
ð1:186Þ
1.11
Material Form of Balance Laws
33
FLj, MLj, and KLj are two-point tensors corresponding to the first Piola-Kirchhoff
S
stress in nonlinear elasticity. For τijS , we also introduce its material form T KL
through
S
¼ JX K , i X L, j τijS ,
T KL
S
S
τij ¼ J 1 yi, K y j, L T KL
:
ð1:187Þ
S
corresponds to the second Piola–Kirchhoff stress in nonlinear elasticity.
T KL
The material forms of the balance laws can be obtained by modifying
Eqs. (1.163), (1.164), (1.165), (1.166), (1.167) and (1.168) properly. They can
also be obtained from the integral forms of the balance laws in Eqs. (1.155),
(1.156), (1.157), (1.158), (1.159) and (1.160). The material form of the conservation
of mass is simply (see Eq. (1.73))
ρ0 ¼ ρJ:
ð1:188Þ
From Eq. (1.156), by change of variables to the reference configuration, we have
R
R
R
dl
R ¼ lE i dli ¼ L Ei yi, KRdLK
¼ S εIJK Ei yi, K , J dSI ¼ S εIJK W K , J dSI ¼ 0,
lE
ð1:189Þ
which implies that
εIJK W K , J ¼ 0:
ð1:190Þ
Equation (1.190) is automatically satisfied by WK ¼ φ,K.
From Eq. (1.157),
Z
Z
Di dsi ¼
s
ρe dv:
ð1:191Þ
v
Changing the integration back to the reference configuration, using Eqs. (1.49) and
(1.51), we obtain
Z
Z
Di JX L, i dSL ¼
S
ρe JdV,
ð1:192Þ
V
or
Z
Z
Di JX L, i N L dS ¼
S
With the divergence theorem,
ρe JdV:
V
ð1:193Þ
34
1 Nonlinear Theory of Electroelasticity
R
R
ðD JX Þ dV ¼ ρe JdV,
RV i L, i , LR E V
V D L, L dV ¼ V ρ dV,
ð1:194Þ
where Eq. (1.83)4,5 have been used. Equation (1.194) implies that
D L, L ¼ ρ E :
ð1:195Þ
In terms of the total stress tij in Eq. (1.181)1, the integral form of the balance of
linear momentum in Eq. (1.158) takes the following form:
R
R
R
E
_
ρ
v
dv
¼
ρf
þ
f
dv þ s t j ds
j
j
j
v
v
R
R
¼ v ρf j þ T ijE, i dv þ s t j ds
R
R
¼ v ρf j dv þ s τij ni þ T ijE ni ds
R
R
¼ Rv ρf j dv þ Rs t ij ni ds
¼ v ρf j dv þ s t ij dsi :
ð1:196Þ
Using Eq. (1.49) and the divergence theorem, the last term on the right-hand side of
Eq. (1.196) can be written as
R
R
R
i ¼
s t ij dsR
S t ij JXL, i dSL ¼ R S t ij JX L, i N L dS
¼ V t ij JX L, i , L dV ¼ V K Lj, L dV:
ð1:197Þ
Substituting Eq. (1.197) into Eq. (1.196), using Eq. (1.51), we obtain
Z
Z
ρv_ j JdV ¼
V
Z
ρf j JdV þ
V
K Lj, L dV,
ð1:198Þ
V
which implies that
K Lj, L þ ρ0 f j ¼ ρ0 v_ j :
ð1:199Þ
The conservation of angular momentum in Eq. (1.182)2 does not have a spatial
derivative and remains the same:
εijk t jk ¼ 0:
ð1:200Þ
Equation (1.200) will be automatically satisfied later when tjk is a symmetric tensor
given by proper constitutive relations.
For the conservation of energy, from Eq. (1.172),
1.12
Constitutive Relations
35
d
ρχ_ ¼ τij v j, i Pi E_ i ¼ τijS v j, i Pi E j v j, i Pi ðX L, i W L Þ
dt
d
¼ τijS dij þ ωij Pi E j v j, i Pi ðX L, i ÞW L þ X L, i W_ L
dt
d
S
¼ τij dij Pi E j v j, i Pi ðX L, i ÞW L Pi X L, i W_ L
dt
S
¼ J 1 yi, K y j, L T KL
dij Pi E j v j, i
Pi v j, M X M , i X L, j W L J 1 yi, K P K X L, i W_ L
S _
E KL Pi E j v j, i þ Pi v j, i E j J 1 P K W_ K
¼ J 1 T KL
1 S _
¼ J T KL E KL J 1 P K W_ K ,
ð1:201Þ
where Eqs. (1.181)2, (1.183)1, (1.54)1, (1.187)2, (1.62), (1.184)2, and (1.56) have
been used. Hence,
S _
ρ0 χ_ ¼ T KL
E KL P K W_ K :
ð1:202Þ
In summary, from Eqs. (1.188), (1.190), (1.195), (1.199), (1.200), and (1.202),
the balance laws in material form are
ρ0 ¼ ρJ,
ð1:203Þ
εIJK W K , J ¼ 0,
ð1:204Þ
D K , K ¼ ρE ,
ð1:205Þ
K Lk, L þ ρ0 f k ¼ ρ0 v_ k ,
ð1:206Þ
εkij t ij ¼ 0,
ð1:207Þ
S _
E KL P K W_ K ,
ρ0 χ_ ¼ T KL
ð1:208Þ
where the spatial derivatives are taken with respect to X.
1.12
Constitutive Relations
The field equations in Eq. (1.203), (1.204), (1.205), (1.206), (1.207) and (1.208)
represent basic laws of physics valid for electroelastic materials in general. The
specific electromechanical behavior of a particular material is specified by additional
equations called constitutive relations. For constitutive relations of electroelastic
materials we begin with the following forms as suggested by Eq. (1.208):
36
1 Nonlinear Theory of Electroelasticity
χ ¼ χ ðE KL ; W K Þ,
S
S
T KL
¼ T KL
ðE KL ; W K Þ,
P K ¼ P K ðEKL ; W K Þ:
ð1:209Þ
The substitution of Eq. (1.209) into Eq. (1.208) gives
T
S
KL
∂χ
0 ∂χ
_
E KL P K þ ρ
W_ K ¼ 0:
ρ
∂E KL
∂W K
0
ð1:210Þ
Since Eq. (1.210) is linear in E_ KL and W_ K , for the equation to hold for any E_ KL
¼ E_ LK and W_ K , we must have
1 ∂χ
∂χ
∂χ
þ
,
¼ ρ0
2 ∂EKL ∂ELK
∂E KL
∂χ
P K ¼ ρ0
,
∂W K
S
T KL
¼ ρ0
ð1:211Þ
where the partial derivatives are taken as if the strain components were independent,
and the free energy is written as a symmetric function of the strain tensor similar to
Eq. (1.38). Then, from Eq. (1.185)3,
K Lj ¼ JX L, i t ij ¼ JX L, i τijS þ T ijES
S
þ JX L, i T ijES
¼ JX L, i J 1 yi, K y j, M T KM
S
ES
¼ y j, M T ML þ JX L, l T lk δkj
S
¼ y j, M T ML
þ JX L, l T lkES y j, M X M , k
S
ES
¼ y j, M T ML þ y j, M T ML
ES
0 ∂χ
¼ y j, K ρ
þ T KL
∂ E KL
¼ F Lj þ M Lj ,
ð1:212Þ
where Eqs. (1.181)1, (1.187)2, and (1.211)1 have been used. We have also introduced
ES
ES
¼ JX M , k X L, l T klES ¼ T LM
:
T ML
ð1:213Þ
From the derivation of Eq. (1.212), it can be seen that
S
F Lj ¼ y j, K T KL
¼ y j, K ρ0
ES
M Lj ¼ y j, K T KL
:
∂χ
,
∂ E KL
From Eq. (1.211)2, the constitutive relation for the electric displacement is
ð1:214Þ
1.12
Constitutive Relations
37
D K ¼ ε 0 E K þ P K ¼ ε 0 E K ρ0
∂χ
:
∂ WK
ð1:215Þ
From Eq. (1.211), using Eqs. (1.187)1 and (1.183)3, we can also write
∂χ
,
∂E KL
∂χ
JX K , i Pi ¼ ρ0
:
∂W K
JX K , i X L, j τijS ¼ ρ0
ð1:216Þ
Then, with Eq. (1.181)2, we have
τijS ¼ τij þ Pi E j ¼ J 1 yi, K y j, L ρ0
Pi ¼ J 1 yi, K ρ0
∂χ
,
∂W K
∂χ
¼ τ Sji ,
∂EKL
ð1:217Þ
where the symmetry of τijS is guaranteed by the constitutive relations. Then, with
Eqs. (1.73) and (1.127),
∂χ
∂χ
y þ ρyi, K
E j,
∂E KL j, L
∂W K
∂χ
π i ¼ yi, K
:
∂W K
τij ¼ ρyi, K
ð1:218Þ
In applications, the displacement gradient uj,K is often used instead of the
deformation gradient yj,K. For example, from Eqs. (1.214)1 and (1.11)
S
S
¼ δ jK þ u j, K T KL
,
F Lj ¼ y j, K T KL
ð1:219Þ
which, in the unabbreviated notation, takes the following form:
F 11
F 12
F 13
F 21
F 22
F 23
F 31
F 32
F 33
S
S
S
¼ ð1 þ u1, 1 ÞT 11
þ u1, 2 T 21
þ u1, 3 T 31
,
S
S
S
¼ u2, 1 T 11 þ ð1 þ u2, 2 ÞT 21 þ u2, 3 T 31
,
S
S
S
¼ u3, 1 T 11
þ u3, 2 T 21
þ ð1 þ u3, 3 ÞT 31
,
S
S
S
¼ ð1 þ u1, 1 ÞT 12
þ u1, 2 T 22
þ u1, 3 T 32
,
S
S
S
¼ u2, 1 T 12
þ ð1 þ u2, 2 ÞT 22
þ u2, 3 T 32
,
S
S
S
¼ u3, 1 T 12
þ u3, 2 T 22
þ ð1 þ u3, 3 ÞT 32
,
S
S
S
¼ ð1 þ u1, 1 ÞT 13
þ u1, 2 T 23
þ u1, 3 T 33
,
S
S
S
¼ u2, 1 T 13
þ ð1 þ u2, 2 ÞT 23
þ u2, 3 T 33
,
S
S
S
¼ u3, 1 T 13
þ u3, 2 T 23
þ ð1 þ u3, 3 ÞT 33
:
ð1:220Þ
38
1.13
1 Nonlinear Theory of Electroelasticity
Jump Conditions
The field equations in Eqs. (1.163), (1.164), (1.165), (1.166), (1.167) and (1.168)
were obtained by applying the integral forms of the balance laws in Eqs. (1.155),
(1.156), (1.157), (1.158), (1.159) and (1.160) to a material body in which the fields
are continuous. When there are discontinuities of the fields at for example an
interface between two materials, the implications of the balances laws need to be
re-examined. Consider the current state of an interface between medium 1 and
medium 2 (see Fig. 1.7). The unit normal n of the interface points from 1 to 2.
A mechanical traction load t j acts on the interface. The free charge density per unit
area on the interface is σ e . σ e is zero at an interface between two dielectrics. It is
nonzero when there is an interface electrode.
In this book, at an interface the displacement vector u and the electric potential φ
are assumed to be continuous. Below we examine the implications of the balance
laws at the interface. We construct a pillbox on the interface as shown in Fig. 1.7.
The application of the integral form of the balance of linear momentum in
Eq. (1.182)1 to the pillbox yields:
ð2Þ
ð1Þ
t ij ni t ij ni þ t j ¼ 0:
ð1:221Þ
Similarly, applying Gauss’s law of electrostatics in Eq. (1.157) to the pillbox results in
ð2Þ
ð1Þ
D i ni D i ni ¼ σ e :
ð1:222Þ
From Eqs. (1.49), (1.212), and (1.183)4, we have
t ij ni ds ¼ t ij dsi ¼ t ij JX L, i dSL ¼ K Lj dSL ¼ K Lj N L dS,
ð1:223Þ
Di ni ds ¼ Di dsi ¼ Di JX L, i dSL ¼ D L dSL ¼ D L N L dS:
ð1:224Þ
and
Then Eqs. (1.221) and (1.222) can be written in the material form as
Fig. 1.7 A material
interface
n
1
2
1.13
Jump Conditions
39
ð2Þ
ð1Þ
N L K Lj N L K Lj þ Tj ¼ 0,
ð2Þ
ð1Þ
ð1:225Þ
N LD L N LD L ¼ σE ,
ð1:226Þ
ds
Tj ¼ t j ,
dS
ð1:227Þ
where
σE ¼ σe
ds
:
dS
In the case when the interface is between a dielectric body and free space and the free
space electric field is considered, because of the different mappings from the present
state to the reference state in the material and free space, some special treatment is
needed for the jump conditions [2].
For some details of the electromechanical jump conditions, we consider a few
examples below. As a preparation, consider two charged planes at x3 ¼ h in a
vacuum as shown Fig. 1.8. We apply Coulomb’s law,
F¼
q1 q2
,
4πε0 r 2
ð1:228Þ
to differential areas of the two charged planes. Through integration in cylindrical
coordinates, it can be found that the two charged planes interact with each other with
the following force along x3 per unite area:
1 ð1Þ ð2Þ
σ σ :
2ε0
ð1:229Þ
The electrostatic force per unit area in Eq. (1.229) may be repelling or attractive
depending on the whether the charges on the two planes are of the same or
opposite signs. This force can also be obtained by considering a differential area
of one charged plane in the electric field of the other charged plane obtained by
Gauss’s law.
As an example of electromechanical jump conditions, consider the isotropic and
hence nonpiezoelectric dielectric plate in Fig. 1.9. It carries free charges σ e on its
two surfaces at x3 ¼ h. There are no fields in the free space above and below the
plate. The electric, polarization, and electric displacement fields in the plate are along
the x3 axis. From electrostatics the electric displacement in the plate is
D3 ¼ σ e :
Hence the electric field and the polarization field in the plate are
ð1:230Þ
40
1 Nonlinear Theory of Electroelasticity
Fig. 1.8 Two charged
planes in a vacuum
x3
s
(2)
x1
2h
s (1)
Fig. 1.9 An isotropic
dielectric plate
x3
n
-s + s = -s = s
e
p
(2)
x1
2h
s e - s p = s = s (1)
1
σe
E 3 ¼ D3 ¼ ,
ε
ε
σe ε0 e
P3 ¼ D3 ε0 E 3 ¼ σ e ε0 ¼ 1 σ:
ε
ε
ð1:231Þ
The surface polarization charge density at x3¼h is given by
ε0 e
σ p ¼ P3 ¼ 1 σ:
ε
ð1:232Þ
The total surface charge density at x3¼h is denoted by
σ ¼ σ e þ σ p ¼ ε0 e
σ:
ε
ð1:233Þ
Correspondingly, the surface polarization charge density and the total charge density
at x3 ¼ h are σ p and σ, respectively. From Eq. (1.174), the nonzero components
of the Maxwell stress tensor in the plate are
E
E
T 11
¼ T 22
¼
ε0 2
E ,
2 3
E
T 33
¼ P3 E 3 þ
ε0 2
E :
2 3
ð1:234Þ
Consider the upper surface of the plate. Since the electric field in the upper free space
is zero, the Maxwell stress tensor and the total stress tensor there are also zero. There
is no mechanical load at the plate surface. In this case, along x3, the jump condition in
Eq. (1.221) leads to
ε0
E
þ 0 ¼ τ33 P3 E3 E 23 ¼ 0:
0 τ33 þ T 33
2
ð1:235Þ
1.13
Jump Conditions
41
Then the 33 component of the Cauchy stress tensor in the plate is given by
τ33
ε0 2
ε0 σ e 2
¼ P3 E 3 E 3 ¼ P3 E 3 2
2 ε
1 ε0 σ e 2
¼ P3 E 3 2ε0 ε
σ2
1 ð1Þ ð2Þ
¼ ðσ p ÞE3 ¼ ðσ p ÞE3 þ
σ σ :
2ε0
2ε0
ð1:236Þ
Physically, Eq. (1.236) represents the balance of linear momentum of the pillbox at
the plate upper surface in Fig. 1.9. The bottom of the pillbox has a surface polarization charge density σ p which experiences a force of (σ p)E3 per unit area. The
last term in Eq. (1.236) is due to the interaction of the two charged surfaces with total
charge densities of σ at x3 ¼ h. The last term in Eq. (1.236) shows a
compressive stress because of the opposite signs of the charges on the plate surfaces.
For another example of the electromechanical jump conditions, consider the
two-layer plate of different isotropic dielectric materials as shown in Fig. 1.10. It
carries free charges σ e on its two surfaces at x3 ¼ h. There are no fields in the
free space above and below the two-layer plate. The fields in the lower layer are
ð1Þ
D3 ¼ σ e ,
1 ð1Þ
σe
ð1Þ
E 3 ¼ ð1Þ D3 ¼ ð1Þ ,
ε
ε
ð1Þ
P3
¼
ð1Þ
D3
ð1Þ
ε0 E 3
σe
¼ σ ε0 ð1Þ ¼
ε
e
ε0
1 ð1Þ σ e :
ε
ð1:237Þ
The surface polarization charge density and total charge density at x3 ¼ h are
given by
ε0
ð1Þ
σ pð1Þ ¼ P3 ¼ 1 ð1Þ σ e ,
ε
pð1Þ ε0 e
ð1Þ
e
¼ ð1Þ σ :
σ ¼ σ þ σ
ε
ð1:238Þ
From Eq. (1.174), the nonzero components of the Maxwell stress tensor in the lower
layer are
ε0 ð1Þ 2
E ð1Þ
E ð1Þ
E3
T 11 ¼ T 22 ¼ ,
2
E ð1Þ
ð1Þ ð1Þ
T 33 ¼ P3 E 3 þ
Similarly, the fields in the upper layer are
ε0 ð1Þ 2
E3
:
2
ð1:239Þ
42
1 Nonlinear Theory of Electroelasticity
Fig. 1.10 A two-layer plate
of isotropic dielectrics
x3
-s e + s p (2) = s (2)
(2)
e ,h
x1
s p = s p (1) - s p (2)
e (1), h
s e - s p (1) = s (1)
ð2Þ
D3 ¼ σ e ,
1 ð2Þ
σe
ð2Þ
E 3 ¼ ð2Þ D3 ¼ ð2Þ ,
ε
ε
ð2Þ
P3
¼
ð2Þ
D3
ð2Þ
ε0 E 3
σe
¼ σ e ε0 ð2Þ ¼
ε
ε0
1 ð2Þ σ e :
ε
ð1:240Þ
The surface polarization charge density and the total charge density at x3¼h are
given by
ð2Þ
σ pð2Þ ¼ P3 ¼
ε0
σe,
εð 2 Þ
ε0
¼ ð 2Þ σ e :
ε
1
σ ð2Þ ¼ σ e þ σ pð2Þ
ð1:241Þ
The nonzero components of the Maxwell stress tensor in the upper layer are
ε0 ð2Þ 2
E ð2Þ
E ð2Þ
E3
T 11 ¼ T 22 ¼ ,
2
E ð2Þ
ð2Þ ð2Þ
T 33 ¼ P3 E 3 þ
ε0 ð2Þ 2
E3
:
2
ð1:242Þ
The total surface polarization charge density at the interface between the two layers
where x3¼0 is
ð2Þ
σ p ¼ σpð1Þ σ p
ε0
ε0
ε0
ε0
¼ 1 ð1Þ σ e 1 ð2Þ σ e ¼ ð2Þ ð1Þ σ e :
ε
ε
ε
ε
ð1:243Þ
At the lower surface of the two-layer plate, along x3, the jump condition in
Eq. (1.221) leads to
ε0 ð1Þ 2
ð 1Þ
E ð1Þ
ð1Þ
ð1Þ ð1Þ
τ33 þ T 33
E3
¼ 0:
0 þ 0 ¼ τ33 þ P3 E 3 þ
2
ð1:244Þ
Then the 33 component of the Cauchy stress tensor in the lower layer is given by
1.13
Jump Conditions
43
ð1Þ
ε0 ð1Þ 2
E
2 3
1 ð1Þ p
1 ð1Þ ð2Þ
ð1Þ
¼ σ pð1Þ E 3 þ
σ σ þ
σ σ :
2ε0
2ε0
ð1Þ ð1Þ
τ33 ¼ P3 E 3 ð1:245Þ
Physically, Eq. (1.245) represents the balance of linear momentum for the pillbox at
the lower surface of the plate as shown in Fig. 1.10. The top of the pillbox has a
ð1Þ
surface polarization charge density σ p(1) which experiences a force of σ pð1Þ E3 per
unit area. The last two terms in Eq. (1.245) are due to the interactions of the three
charged surfaces. Similarly, for the upper surface of the two-layer plate, along x3, the
jump condition in Eq. (1.221) leads to
ε0 ð2Þ 2
ð 2Þ
E ð2Þ
ð2Þ
ð2Þ ð2Þ
0 τ33 þ T 33
E3
¼ 0:
þ 0 ¼ τ33 P3 E 3 2
ð1:246Þ
Then the 33 component of the Cauchy stress tensor in the upper layer is given by
ε0 ð2Þ 2
ð 2Þ
ð2Þ ð2Þ
τ33 ¼ P3 E 3 E
2 3
pð2Þ ð2Þ
1 ð2Þ p
1 ð2Þ ð1Þ
E3 þ
¼ σ
σ σ þ
σ σ :
2ε0
2ε0
ð1:247Þ
Equation (1.247) represents the balance of linear momentum for the pillbox at the
upper surface of the plate as shown in Fig. 1.10. The bottom of the pillbox has a
ð2Þ
surface polarization charge density σ p(2) which experiences a force of σ pð2Þ E 3
per unit area. The last two terms in Eq. (1.247) are due to the interactions of the three
charged surfaces. Finally, for the interface between the two layers, along x3, the jump
condition in Eq. (1.221) leads to
ð1Þ
E ð1Þ
τ33 þ T 33
þ0
2
ε
ε0 ð1Þ 2
0
ð2Þ
ð2Þ ð2Þ
ð2Þ
ð1Þ
ð1Þ ð1Þ
E3
E
¼ τ33 þ P3 E3 þ
τ33 P3 E 3 ¼ 0:
2
2 3
ð2Þ
E ð2Þ
τ33 þ T 33
ð1:248Þ
Then the jump of the 33 component of the Cauchy stress tensor across the interface is
given by
ε0 ð1Þ 2
ε0 ð2Þ 2
ð2Þ ð2Þ
E3
E
P3 E 3 2
2 3
1 p ð2Þ
1 p ð 1Þ
ð1Þ
ð2Þ
¼ σ pð1Þ E 3 σ pð2Þ E 3 þ
σ σ σ σ :
2ε0
2ε0
ð2Þ
ð1Þ
ð1Þ ð1Þ
τ33 τ33 ¼ P3 E3 þ
ð1:249Þ
Equation (1.249) represents the balance of linear momentum for the pillbox at the
interface of the plate as shown in Fig. 1.10. The top and bottom of the pillbox have
surface polarization charge densities σ p(2) and σ p(1) which experience forces of
44
1 Nonlinear Theory of Electroelasticity
ð2Þ
ð1Þ
σ pð2Þ E 3 and σ pð1Þ E3 per unit area, respectively. The last two terms in Eq. (1.249)
are due to the interactions of the three charged surfaces. Equation (1.249) can also be
obtained from the difference of Eqs. (1.247) and (1.245).
1.14
Displacement-Potential Formulation
Since WK ¼ φ,K, the curl-free condition of the electric field in Eq. (1.204) is
automatically satisfied. The angular momentum equation in Eq. (1.207) is also
automatically satisfied because tij is symmetric according to the constitutive relation.
The energy equation in Eq. (1.208) is also satisfied by the constitutive relations.
From Eqs. (1.205) and (1.206) we have
K Lj, L þ ρ0 f j ¼ ρ0 v_ j ,
D K , K ¼ 0,
ð1:250Þ
where, from Eqs. (1.212) and (1.215),
∂χ
1
þ JX L, i ε0 Ei E j E k E k δij ,
∂ EKL
2
∂χ
D K ¼ ε 0 E K ρ0
,
∂ WK
K Lj ¼ y j, K ρ0
ð1:251Þ
and, from Eqs. (1.45) and (1.183)1,
E KL ¼ yi, K yi, L δKL =2,
W K ¼ φ, K :
ð1:252Þ
With successive substitutions from Eqs. (1.251) and (1.252), we can write
Eq. (1.250) as four equations for the four unknowns yi(X, t) and φ(X, t). Once
yi(X, t) and φ(X, t) are known, the current mass density ρ can be obtained from the
conservation of mass in Eq. (1.203). Thus all of Eqs. (1.203), (1.204), (1.205),
(1.206), (1.207), and (1.208) are satisfied. Other fields can also be calculated from
yi(X, t) and φ(X, t).
For mechanical boundary conditions, the boundary surface S is partitioned into Sy
and ST on which motion (or displacement) and traction are prescribed, respectively.
Electrically, S is partitioned into Sφ and SD. Sφ is electroded, with a prescribed
electric potential. SD is unelectroded and surface free charge is assumed to be zero.
We also assume that
Sy [ ST ¼ Sφ [ SD ¼ S,
Sy \ ST ¼ Sφ \ SD ¼ 0:
ð1:253Þ
1.15
Variational Formulation
45
Usual boundary value problems for an electroelastic body consist of Eqs. (1.250),
(1.251), (1.252) and the following boundary conditions:
yi ¼ yi on Sy ,
φ ¼ φ on Sφ ,
N L K Lk ¼ Tk on ST ,
N K D K ¼ 0 on SD ,
ð1:254Þ
where yi and φ are the prescribed boundary motion and potential, and Ti is the
surface traction per unit undeformed area. The electric field in the surrounding free
space is neglected. For dynamic problems, initial conditions need to be added.
1.15
Variational Formulation
A variational formulation in terms of the internal energy density ε for an elastic
dielectric in spatial form can be found in [13]. To be consistent with the material
formulation in this book, we are interested in a variational formulation in material
form in terms of the free energy χ. From the two continuum model in Sect. 1.4, we
construct the following spatial Lagrangian density L and material Lagrangian density
L:
1
1
L ¼ ρy_ i y_ i ρε þ ε0 E i Ei
2
2
μl ðyÞφðyÞ μe ðy þ ηÞφðy þ ηÞ þ ρf i yi ρe φ
1
1
ffi ρy_ i y_ i ρε þ ε0 Ei E i
2
2
μl ðyÞφðyÞ μe ðy þ ηÞ φðyÞ þ φ, i ηi þ ρf i yi ρe φ
1
1
¼ ρy_ i y_ i ρε þ ε0 Ei E i þ Pi E i þ ρf i yi ρe φ
2
2
1
1
¼ ρy_ i y_ i ρχ þ ε0 E i E i þ ρf i yi ρe φ ¼ J 1 L,
2
2
ð1:255Þ
where Eqs. (1.82), (1.83), and (1.170) have been used. Then consider the following
variational functional [14–16]:
R t1
R
1 0
ρ y_ i y_ i ρ0 χ ðEKL ; W K Þ
2
þπRðEKL ;RW K Þ þ ρ0 f i yRi ρRE φdV
t
t
þ t01 dt ST Ti yi dS t01 dt SD σ E φdS,
Πðy; φÞ ¼
t0
dt
V
where the electric field energy density π is given by
ð1:256Þ
46
1 Nonlinear Theory of Electroelasticity
1
1
π ðEKL ; W K Þ ¼ ε0 JEk Ek ¼ ε0 Jφk φ, k
2
2
1
1
¼ ε0 JX L, k X M , k φ, L φ, M ¼ ε0 JC 1
MN W M W N ,
2
2
ð1:257Þ
and, from Eqs. (1.45) and (1.183)1,
C KL ¼ δKL þ 2EKL , E KL ¼ yi, K yi, L δKL =2,
W L ¼ φ, L :
ð1:258Þ
In Eq. (1.256), ρE and σ E are in fact zero for dielectrics and are only kept formally.
The admissible yi and φ for Π satisfy the following conditions at t0 and t1 as well as
essential boundary conditions on Sy and Sφ:
δyi jt¼t0 ¼ 0; δyi jt¼t1 ¼ 0 in V,
yi ¼ yi on Sy , t 0 < t < t 1 ,
φ ¼ φ on Sφ , t 0 < t < t 1 :
ð1:259Þ
By straightforward differentiation, it can be verified that
∂π
¼ ε0 JC 1
KM W M ¼ JX K , k ε0 E k ¼ ε0 E K :
∂W K
ð1:260Þ
From Eqs. (1.35) and (1.39), the other partial derivative of π is equal to
∂π
∂π
1
∂J 1
∂C 1
MN
¼2
¼2
C MN þ J
ε0
WMWN
∂ E KL
∂CKL
2
∂C KL
∂CKL
1
1 1 1
1
1 1
¼ ε0 JC 1
WMWN
KL C MN J C MK C LN þ C ML C KN
2
2
1 1
1 1
1 1
¼ ε0 J C 1
KL C MN C MK C LN C ML C KN W M W N
2
1 1
1 1
¼ ε0 J C 1
KL C MN 2C MK C LN W M W N
2
ES
¼ JX K , k X L, l ε0 ðE k El E m E m δkl =2Þ ¼ T KL
:
ð1:261Þ
Then the first variation of Π can be found as
R t R δΠ ¼ t01 dt V K Li, L þ ρ0 f i ρ0€yi δyi þ ðD L, L ρE Þδφ dV
Rt R t01 dt ST N L K Li Ti δyi dS
R t1 R t0 dt SD N L D L þ σ E δφdS:
ð1:262Þ
Therefore, the stationary condition of Π implies the following field equations and
natural boundary conditions:
1.16
Third-Order Theory
47
K Lk, L þ ρ0 f k ¼ ρ0€yk in V,
D K , K ¼ ρE in V,
N L K Lk ¼ Tk on ST ,
N K D K ¼ σ E on SD :
ð1:263Þ
A more compact variational formulation results if we introduce the following
total free energy density per unit mass:
ρ0 ψ ðE KL ; W K Þ ¼ ρ0 χ ðEKL ; W K Þ π ðE KL ; W K Þ:
ð1:264Þ
Then, from Eqs. (1.211), (1.260), and (1.261), the constitutive relations take the
following form:
ρ0
ρ0
∂ψ
∂χ
∂π
S
ES
¼ ρ0
¼ T KL
þ T KL
¼ T KL ,
∂ E KL
∂ E KL ∂ E KL
∂ψ
∂χ
∂π
¼ ρ0
¼ P K ε0 E K ¼ D K ,
∂ WK
∂ WK ∂ WK
ð1:265Þ
ð1:266Þ
where we have introduced a total stress tensor T KL in material form. In terms of the
two-point total stress tensor KLj, the constitutive relations become
K Lj ¼ y j, K T KL ¼ y j, K ρ0
∂ψ
,
∂ EKL
ð1:267Þ
∂ψ
:
∂W K
ð1:268Þ
D K ¼ ε0 E K þ P K ¼ ρ0
The variational functional in Eq. (1.256) then takes the following form:
Rt
R
1 0
ρ y_ i y_ i ρ0 ψ ðEKL ; W K Þ
2
þρR0 f i yi R ρE φdV R
R
t
t
þ t0 dt ST Ti yi dS t0 dt SD σ E φdS:
Πðy; φÞ ¼
t0
dt
V
ð1:269Þ
The stationary condition of the functional in Eq. (1.269) is still given by Eq. (1.263).
1.16
Third-Order Theory
The free energy χ that determines the constitutive relations of nonlinear
electroelastic materials may be written as [2]
48
1 Nonlinear Theory of Electroelasticity
ρ0 χ ðE KL ; W K Þ
1
1
W AW B
E AB E CD eABC W A E BC χ
¼ c
2 2 ABCD
2 2 AB
1
1
þ c
E AB E CD E EF þ d
W A E BC EDE
6 3 ABCDEF
2 1 ABCDE
1
1
bABCD W A W B E CD χ W A W B W C
2
6 3 ABC
1
1
c
E AB ECD EEF E GH þ d
W A EBC E DE E FG
24 4 ABCDEFGH
6 2 ABCDEFG
1
1
þ a
W A W B ECD E EF þ d
W A W B W C EDE
4 1 ABCDEF
6 3 ABCDE
1
χ
W A W B W C W D þ ,
24 4 ABCD
þ
ð1:270Þ
where the linear terms describing initial stain and initial electric field are dropped.
The material constants in Eq. (1.270),
c
2 ABCD
c
,
3 ABCDEF
c
eABC ,
,
4 ABCDEFGH
χ ,
2 AB
d
1 ABCDE
,
d
,
bABCD ,
2 ABCDEFG
,
a
χ
,
3 ABC
1 ABCDEF
,
ð1:271Þ
d
3 ABCDE
,
χ
,
4 ABCD
are called the second-order elastic, piezoelectric, second-order electric permeability,
third-order elastic, first odd electroelastic, electrostrictive, third-order electric permeability, fourth-order elastic, second odd electroelastic, first even electroelastic,
third odd electroelastic, and fourth-order electric permeability, respectively. These
material constants are called the fundamental material constants. The second-order
constants are mainly responsible for linear material behaviors. The third- and higherorder material constants are mainly related to nonlinear behaviors of materials. The
structure of χ(EKL, WK) depends on material symmetry. Integrity bases for
constructing a scalar function of a symmetric tensor and a vector for all crystal
classes are known [17].
For weak nonlinearity, the higher-order terms in Eq. (1.270) can be neglected. By
a third-order theory we mean that effects of all terms up to the third powers of the
displacement and potential gradients or their products are included [2]. To obtain the
third-order theory by expansions and truncations from the nonlinear theory, we
proceed as follows [2]. From
X M ¼ δMj y j uM ,
ð1:272Þ
by repeated use of the chain rule of differentiation, we obtain, to the second order in
products of the derivative of uM,
1.16
Third-Order Theory
X M , i ¼ δMi uM , L X L, i ¼ δMi uM , L ðδLi uL, K X K , i Þ
¼ δMi uM , L δLi þ uM , L uL, K X K , i
ffi δMi uM , L δLi þ uM , L uL, K δKi :
49
ð1:273Þ
From Eq. (1.16), retaining terms up to the second order in the derivative of uM, we
find
1
1
J ffi 1 þ uK , K þ ð u K , K Þ 2 u K , L u L, K :
2
2
ð1:274Þ
From Eqs. (1.274) and (1.273),
JX L, i ffi δLi uL, R δRi þ δLi uR, R þ uL, K uK , R δRi
1
1
uR, R uL, K δKi þ δLi uK , K uR, R δLi uK , R uR, K :
2
2
ð1:275Þ
Then, from the free energy in Eq. (1.270), the constitutive relations in Eqs. (1.214)
and (1.215), and Eq. (1.275), retaining terms up to the third-order of the small
displacement and potential gradients, we obtain [2]
1
uA, B þ eALM φ, A þ c
uK , A uK , B
2 2 LMAB
1
þc
uM , K uA, B þ c
uA, B uC, D þ eALK uM , K φ, A
2 LKAB
2 3 LMABCD
1
1
d
uB, C φ, A bABLM φ, A φ, B þ c
u M , R uK , A uK , B
1 ABCLM
2
2 2 LRAB
1
1
ð1:276Þ
þ c
uM , K uA, B uC, D þ c
uA , B uK , C uK , D
2 3 LKABCD
2 3 LMABCD
1
þ c
uA , B uC , D uE , F d
uB , C uM , K φ , A
1 ABCLK
6 4 LMABCDEF
1
1
d
u K , B u K , C φ, A d
uB, C uD, E φ, A
2 1 ABCLM
2 2 ABCDELM
1
1
1
bABLK uM , K φ, A φ, B þ a
u C , D φ , A φ, B þ d
φ φ φ ,
2
2 1 ABCDLM
6 3 ABCLM , A , B , C
F Lj ffi δ jM c
2 LMAB
1
P L ffi eLBC uB, C χ φ, A þ eLBC uK , B uK , C
2
2 AL
1
1
d
uB, C uD, E bALCD uC, D φ, A þ χ φ, A φ, B
2 1 LBCDE
2 3 ABL
1
1
d
uB, C uK , D uK , E d
uB, C uD, E uF, G
2 1 LBCDE
6 2 LBCDEFG
1
1
bALCD uK , C uK , D φ, A þ a
u C , D u E , F φ, A
2
2 1 ALCDEF
1
1
þ d
u D , E φ, A φ, B χ
φ φ φ ,
2 3 ABLDE
6 4 ABCL , A , B , C
ð1:277Þ
50
1 Nonlinear Theory of Electroelasticity
1
M Lj ffi ε0 δ jM φ, L φ, M φ, K φ, K δLM φ, K φ, M uK , L
2
φ, K φ, M u L, K þ φ, L φ, M u K , K φ , L φ, K u K , M
1
1
þφ, K φ, R uR, K δLM þ φ, K φ, K uL, M φ, R φ, R uK , K δLM ,
2
2
ð1:278Þ
ε0 E L ffi ε0 φ, L þ φ, K uL, K φ, L uK , K þ φ, K uK , L
1
φ, M u L, K u K , M þ φ, K u M , M u L, K φ , L u K , K u M , M
2
1
þ φ, L uK , M uM , K φ, M uL, K uM , K þ φ, M uM , L uK , K φ, M uM , K uK , L :
2
ð1:279Þ
and
Notice that the fourth-order material constants are needed for a complete description
of the third-order effects.
As a special case of the third-order theory, if we keep up to the second-order terms
of the small gradients only, we obtain the second-order theory below as
1
uA, B þ eALM φ, A þ c
u K , A uK , B
2 2 LMAB
1
þc
uM , K uA , B þ c
uA , B uC , D
2 LKAB
2 3 LMABCD
1
þeALK uM , K φ, A d
uB, C φ, A bABLM φ, A φ, B ,
1 ABCLM
2
F Lj ffi δ jM c
2 LMAB
1
P L ffi eLBC uB, C χ φ, A þ eLBC uK , B uK , C
2
2 AL
1
1
d
uB, C uD, E bALCD uC, D φ, A þ χ φ, A φ, B ,
2 1 LBCDE
2 3 ABL
1
M Lj ffi ε0 δ jM φ, L φ, M φ, K φ, K δLM ,
2
ε0 E L ffi ε0 φ, L þ φ, K uL, K φ, L uK , K þ φ, K uK , L :
ð1:280Þ
ð1:281Þ
ð1:282Þ
ð1:283Þ
The first-order or the linear theory is given by
F Lj ffi δ jM c
2 LMAB
uA, B þ eALM φ, A ,
ð1:284Þ
P L ffi eLBC uB, C χ φ, A ,
ð1:285Þ
M Lj ffi 0,
ð1:286Þ
ε0 E L ffi ε0 φ, L :
ð1:287Þ
2 AL
References
51
References
1. I.S. Sokolnikoff, Tensor Analysis, 2nd edn. (Wiley, New York, 1964)
2. H.F. Tiersten, Nonlinear electroelastic equations cubic in the small field variables. J. Acoust.
Soc. Am. 57, 660–666 (1975)
3. L.D. Landau, E.M. Lifshitz, Electrodynamics of Continuous Media, 2nd edn. (ButterworthHeinemann, Linacre House, Jordan Hill, Oxford, 1984)
4. H.F. Tiersten, On the nonlinear equations of thermoelectroelasticity. Int. J. Eng. Sci. 9, 587–604
(1971)
5. H.G. de Lorenzi, H.F. Tiersten, On the interaction of the electromagnetic field with heat
conducting deformable semiconductors. J. Math. Phys. 16, 938–957 (1975)
6. J.S. Yang, Differential derivation of the momentum and energy equations in electroelasticity.
Acta Mechanica Solida Sinica 30, 21–26 (2017)
7. A.C. Eringen, G.A. Maugin, Electrodynamics of Continua, vol I (Springer, New York, 1990)
8. D.F. Nelson, Electric, Optic, and Acoustic Interactions in Dielectrics (Wiley, New York, 1979)
9. J.S. Yang, On the derivation of electric body force, couple and power in an electroelastic body.
Acta Mechanica Solida Sinica 28, 613–617 (2015)
10. A.C. Eringen, Mechanics of Continua (Robert E. Krieger, Huntington, 1980)
11. H.F. Tiersten, A Development of the Equations of Electromagnetism in Material Continua
(Springer, New York, 1990)
12. J.C. Baumhauer, H.F. Tiersten, Nonlinear electroelastic equations for small fields superposed
on a bias. J. Acoust. Soc. Am. 54, 1017–1034 (1973)
13. R.A. Toupin, The elastic dielectric. J. Rational Mech. Anal. 5, 849–915 (1956)
14. F. Bampi, A. Morro, A Lagrangian density for the dynamics of elastic dielectrics. Int. J.
Non-Linear Mech. 18, 441–447 (1983)
15. G.A. Maugin, M. Epstein, The electroelastic energy-momentum tensor. Proc. R. Soc. Lond. A
433, 299–312 (1991)
16. J.S. Yang, R.C. Batra, Mixed variational principles in non-linear electroelasticity. Int. J.
Non-Linear Mech. 30, 719–725 (1995)
17. G.F. Smith, M.M. Smith, R.S. Rivlin, Integrity bases for a symmetric tensor and a vector-the
crystal classes. Arch. Rat. Mech. Anal. 12, 93–133 (1963)
Chapter 2
Linear Theory of Piezoelectricity
In this chapter we reduce the nonlinear equations in Chap. 1 for large deformations
and strong fields to the case of infinitesimal deformations and weak fields, which
results in the linear theory of piezoelectricity. A few theoretical aspects of the linear
theory are also discussed.
2.1
Linearization
To reduce the nonlinear electroelastic equations in the previous chapter to the linear
theory of piezoelectricity for infinitesimal deformation and weak fields, consider
small amplitude motions of an electroelastic body around its reference state due to
small mechanical and electrical loads. It is assumed that under a certain norm, for
example, kui,Kk ¼ max j ui,Kj, the displacement gradient is infinitesimal in the
following sense:
kui, K k << 1:
ð2:1Þ
It is also assumed that the electric potential gradient φ,K is infinitesimal. We want to
obtain equations linear in ui,M and φ,K, and neglect their products and higher powers.
The linear terms themselves are also dropped in comparison with any finite quantity
such the Kronecker delta or 1. For example, under Eq. (2.1), we have, using
Eq. (1.11),
∂ui
∂ui
∂ui
∂ui
¼
y ¼
ðδkK þ uk, K Þ ffi
δkK ,
∂X K ∂yk k, K ∂yk
∂yk
φ, K ¼ φ, i yi, K ffi φ, i δiK ,
© Springer Nature Switzerland AG 2018
J. Yang, An Introduction to the Theory of Piezoelectricity, Advances in Mechanics
and Mathematics 9, https://doi.org/10.1007/978-3-030-03137-4_2
ð2:2Þ
53
54
2 Linear Theory of Piezoelectricity
which implies that, to the first order of approximation, the displacement and potential gradients calculated from the material and spatial coordinates are approximately
equal. The material time derivative of an infinitesimal field f (y,t)¼f[y(X, t), t] is
approximately equal to:
df ∂ f ¼ dt ∂t X
¼ ∂∂tf ¼ ∂∂tf fixed
y
∂f
þ ∂y
i
y fixed
fixed
t fixed
∂f
þ ∂y
vi ffi
i
∂yi t fixed ∂t X fixed
∂f
:
∂t y fixed
ð2:3Þ
For the finite strain tensor, from Eq. (1.45), we have, approximately,
1
1
E KL ¼ ðuL, K þ uK , L þ uM , K uM , L Þ ffi ðuL, K þ uK , L Þ:
2
2
ð2:4Þ
In the notation of the linear theory of piezoelectricity [1, 2], the capital X in Chap. 1
is replaced by the lowercase x, and only lowercase indices are used. Therefore,
following the notation in [1, 2] for the linear theory of piezoelectricity, the infinitesimal strain tensor is denoted by
1
Skl ¼ ðul, k þ uk, l Þ ¼ Slk :
2
ð2:5Þ
The material electric potential gradient in Eq. (1.183)1 becomes
W K ¼ E i yi, K ffi Ei δiK ! Ek ¼ φ, k :
ð2:6Þ
Similarly, from Eqs. (1.174), (1.181), (1.185), (1.187), and (1.183), we have
T ijE ffi 0, T ijES ffi 0, τij ffi τijS ffi τ ji ,
S
M Lj ffi 0, K Lj ffi F Lj ffi δLi τij , T KL
ffi δKi δLj τij ,
P K ! Pk , D K ! Dk :
ð2:7Þ
Since the stress tensors are either approximately zero (quadratic in the infinitesimal
potential gradient) or about the same in the linear theory, we use Tij [1, 2] to denote
the infinitesimal Cauchy stress tensor, that is,
τij ffi τijS ! T ij ,
K Lj ffi F Lj ffi δLi τij ! T lj ,
S
T KL
ffi δKi δLj τij ! T kl :
ð2:8Þ
For small fields the total free energy given by Eqs. (1.264) and (1.257) is
approximated by
2.1 Linearization
55
1
ρ0 ψ ðEKL ; W K Þ ¼ ρ0 χ ðE KL ; W K Þ ε0 JEk Ek
2
1
1
1
ffi c
E AB ECD eABC W A EBC χ W A W B ε0 JE k E k
2 2 ABCD
2 2 AB
2
1 E
1
! cijkl
Sij Skl eijk E i S jk εijS Ei E j ¼ H ðSkl ; E k Þ,
2
2
ð2:9Þ
where J ffi 1 has been used.
εijS ¼ χ þ ε0 δij
ð2:10Þ
2 ij
is the dielectric tensor. H is called the electric enthalpy in the theory of linear
E
piezoelectricity. The superscript E in the elastic constants cijkl
indicates that the
independent electric constitutive variable is the electric field E. The superscript S in
the dielectric constants εijS indicates that the mechanical constitutive variable is the
strain tensor S. The linear constitutive relations generated by H are
∂H
E
¼ cijkl
Skl ekij E k ,
∂Sij
∂H
Di ¼ ¼ eikl Skl þ εikS E k :
∂Ei
T ij ¼
ð2:11Þ
Hence T, D, and P are linear in the infinitesimal displacement and potential
gradients. The material constants in Eq. (2.11) have the following symmetries:
E
E
E
cijkl
¼ cjikl
¼ cklij
,
ekij ¼ ekji ,
εijS ¼ ε Sji :
ð2:12Þ
We also assume that the elastic and dielectric material tensors are positive-definite in
the following sense:
E
cijkl
Sij Skl 0 for any Sij ¼ S ji ,
E
Sij Skl ¼ 0 ) Sij ¼ 0,
and cijkl
S
εij Ei E j 0 for any E i ,
and εijS E i E j ¼ 0 ) E i ¼ 0:
ð2:13Þ
The total internal energy density U per unit volume in the linear theory can be
obtained from H by a Legendre transform as follows:
U ðS; DÞ ¼ H ðS; EðS; DÞÞ þ EðS; DÞ D:
ð2:14Þ
56
2 Linear Theory of Piezoelectricity
Constitutive relations in the following form then follow:
∂U
,
∂D
ð2:15Þ
D
T ij ¼ cijkl
Skl hkij Dk ,
E i ¼ hikl Skl þ βikS Dk :
ð2:16Þ
T¼
∂U
,
∂S
E¼
or
It can be shown that U is positive-definite:
U ¼ H þ E i Di
1 E
1
¼ cijkl
Sij Skl eijk E i S jk εijS Ei E j þ E i eikl Skl þ εikS E k
2
2
1 E
1 S
¼ cijkl Sij Skl þ εij E i E j 0:
2
2
ð2:17Þ
For small fields, the total internal energy density U per unit volume and the internal
energy density ε per unit mass in Chap. 1 are related by
U ¼ H þ E i D i ¼ ρ0 ψ þ E i D i
1
¼ ρ0 χ π þ E i D i ¼ ρ0 χ ε 0 E i E i þ E i D i
2
1
¼ ρ0 ε E i P i ε 0 E i E i þ E i ð ε 0 E i þ P i Þ
2
1
¼ ρ0 ε þ ε0 E i E i ¼ ρ0 ε þ π,
2
ð2:18Þ
where Eqs. (2.14), (2.9), (1.264), (1.257), and (1.170) have been used.
Similar to Eqs. (2.11) and (2.16), linear constitutive relations can also be written
as [2]
E
Sij ¼ sijkl
T kl þ d kij E k ,
Di ¼ d ikl T kl þ εikT E k ,
ð2:19Þ
D
Sij ¼ sijkl
T kl þ gkij Dk ,
Ei ¼ gikl T kl þ βikT Dk :
ð2:20Þ
and
In the notation of the linear theory, the equations of motion and the charge
equation of electrostatics in Eq. (1.263)1,2 are written as
2.1 Linearization
57
T ji, j þ ρf i ¼ ρ€ui ,
Di, i ¼ ρe :
ð2:21Þ
In Eq. (2.21), the body force f and the body free charge density ρe are both
infinitesimal. ρ is understood to be the known reference mass density ρ0 in
Chap. 1. In Eq. (2.21)2, D is infinitesimal. The difference between the reference
and present free charge densities is neglected, that is, ρe ffi ρE and usually they are
both zero. The surface loads are also infinitesimal. We have
Tj ffi t j ,
σE ffi σe ,
ð2:22Þ
and
N L K Lj ffi ni T ij ,
N L D L ffi ni D i :
ð2:23Þ
We note that the six strain components in Eq. (2.5) are derived from three
displacement components. Therefore it is natural to expect that some relationships
among the strain components exist whether they are linear or nonlinear. It can be
directly verified that the following equations can be satisfied by the linear strain
components:
S11, 23 ¼ ðS31, 2 þ S12, 3 S23, 1 Þ, 1 ,
S22, 31 ¼ ðS12, 3 þ S23, 1 S31, 2 Þ, 2 ,
S33, 12 ¼ ðS23, 1 þ S31, 2 S12, 3 Þ, 3 ,
2S23, 23 ¼ S22, 33 þ S33, 22 ,
2S31, 31 ¼ S33, 11 þ S11, 33 ,
2S12, 12 ¼ S11, 22 þ S22, 11 :
ð2:24Þ
Equation (2.24) is called the compatibility condition of strains. The compatibility
condition is a necessary condition for the six strain components derived from three
displacement components. They are also sufficient in the sense that for six strain
components satisfying Eq. (2.24), there exist three displacement components from
which the six strain components are derivable according to Eq. (2.5). The sufficiency
of Eq. (2.24) is true over a simply connected domain only. For a multiply connected
domain, some additional conditions are needed. The compatibility condition is
useful when solving a problem using the stress components as the primary
unknowns. In more compact form, Eq. (2.24) can be written as [3]
Sij, kl þ Skl, ij Sik, jl S jl, ik ¼ 0,
ð2:25Þ
εijk εlmn Sil, jm ¼ 0:
ð2:26Þ
or [3]
58
2.2
2 Linear Theory of Piezoelectricity
Displacement-Potential Formulation
In summary, the linear theory of piezoelectricity consists of the equations of motion
and charge from Eq. (2.21):
T ji, j þ ρf i ¼ ρ€ui ,
Di, i ¼ ρe ,
ð2:27Þ
the constitutive relations from Eq. (2.11):
T ij ¼ cijkl Skl ekij Ek ,
Di ¼ eijk S jk þ εij E j ,
ð2:28Þ
the strain–displacement relation and electric field–potential relation from Eqs. (2.5)
and (2.6):
Sij ¼ ui, j þ u j, i =2,
E i ¼ φ, i ,
ð2:29Þ
where u is the mechanical displacement vector, T is the stress tensor, S is the strain
tensor, E is the electric field, D is the electric displacement, φ is the electric potential,
ρ is the known reference mass density, ρe is the body free charge density which is
usually zero, and f is the body force per unit mass. The coefficients cijkl, ekij and εij
are the elastic, piezoelectric and dielectric constants. We have neglected the superscripts in the material constants in Eq. (2.11). With successive substitutions from
Eqs. (2.29) and (2.28), Eq. (2.27) can be written as four equations for u and φ
cijkl uk, lj þ ekij φ, kj þ ρf i ¼ ρ€ui ,
eikl uk, li εij φ, ij ¼ ρe :
ð2:30Þ
Let the region occupied by the piezoelectric body be V and its boundary surface
be S as shown in Fig. 2.1. Let the unit outward normal of S be n. For boundary
conditions we consider the following partitions of S:
Su [ ST ¼ Sφ [ SD ¼ S,
Su \ ST ¼ Sφ \ SD ¼ 0,
ð2:31Þ
where Su is the part of S on which the mechanical displacement is prescribed, and ST
is the part of S where the traction vector is prescribed. Sφ represents the part of
S which is electroded where the electric potential is no more than a function of time,
and SD is the unelectroded part. For mechanical boundary conditions we have
ui ¼ ui on Su ,
and
ð2:32Þ
2.3 Principle of Superposition
59
x2
Fig. 2.1 A piezoelectric
body and partitions of its
surface
SD
Sj
ST
i2
x3
i1
x1
i3
n
ni T ij ¼ t j on ST :
Su
ð2:33Þ
Electrically, on the electroded part of S,
φ ¼ φ on Sφ ,
ð2:34Þ
where φ does not vary spatially. On the unelectroded part of S, the charge
condition can be written as
n j D j ¼ σ e on SD ,
ð2:35Þ
where σ e is the free charge density per unit surface area which is usually zero. In
the above formulation, we assumed a very thin electrode and the mechanical effects
of the electrode were neglected. We also neglected the electric field in the surrounding free space.
On an electrode Sφ, the total free charge Qe can be calculated from
Z
Q ¼
ni Di dS:
e
ð2:36Þ
Sφ
Then the electric current flowing out of the electrode is given by
I ¼ Q_ e :
ð2:37Þ
Sometimes there are two (or more) electrodes on a piezoelectric body which are
connected to an electric circuit. Then circuit equation(s) usually need to be
considered.
2.3
Principle of Superposition
The linearity of Eq. (2.30) allows for the superposition of solutions. Suppose the
solutions under two different sets of loads of {f(1), ρe(1)} and {f(2), ρe(2)} are {u(1), φ(1)}
and {u(2), φ(2)}, respectively. Then, under the combined load of {f(1) + f(2), ρe(1) + ρe(2)},
60
2 Linear Theory of Piezoelectricity
the solution to Eq. (2.30) is {u(1) + u(2), φ(1) + φ(2)}. This is called the principle of
superposition and can be shown though
ð1Þ
ð2Þ
cijkl uk þ uk
þ ekij φð1Þ þ φð2Þ , kj
, lj
2
∂ ð 1Þ
ð1Þ
ð2Þ
ð2Þ
þ ρ fi þ fi
ρ 2 ui þ ui
∂t
ð1Þ
ð2Þ
ð1Þ
ð2Þ
¼ cijkl uk, lj þ cijkl uk, lj þ ekij φ, kj þ φ, kj
ð1Þ
ð2Þ
ð1Þ
ð2Þ
þ
ρf i þ ρf i ρ€ui ρ€ui
ð1Þ
ð1Þ
ð1Þ
ð1Þ
¼ cijkl uk, lj þ ekij φ, kj þ ρf i ρ€ui
ð2Þ
ð2Þ
ð2Þ
ð2Þ
þ cijkl uk, lj þ φ, kj þ ρf i ρ€ui
¼ 0 þ 0 ¼ 0,
ð2:38Þ
and
ð1Þ
ð2Þ
eikl uk þ uk
εij φð1Þ þ φð2Þ , ij ρeð1Þ þ ρeð2Þ
ð1Þ
, li
ð2Þ
ð1Þ
ð2Þ
¼ eikl uk, li þ eikl uk, li εij φ, ij εij φ, ij ρeð1Þ ρeð2Þ
ð1Þ
ð1Þ
ð2Þ
ð2Þ
¼ eikl uk, li εij φ, ij ρeð1Þ þ eikl uk, li εij φ, ij ρeð2Þ
¼ 0 þ 0 ¼ 0:
ð2:39Þ
The above principle of superposition can be generalized to include boundary loads.
2.4
Poynting’s Theorem
We begin with the rate of change of the total internal energy density in Eq. (2.14):
∂U _
∂U _
S ij þ
U_ ¼
Di
∂Sij
∂Di
¼ T ij S_ ij þ E i D_ i ¼ T ij u_ i, j φ, i D_ i
¼ T ij u_ i , j T ij, j u_ i φD_ i , i þ φD_ i, i
¼ T ij u_ i , j ρ€ui ρf i u_ i φD_ i , i þ φρ_ e
∂ 1
ρu_ i u_ i þ ρf i u_ i φD_ i , i þ φρ_ e ,
¼ T ji u_ j , i ∂t 2
ð2:40Þ
where Eqs. (2.15), (2.29), and (2.27) have been used. Hence,
∂
ðT þ U Þ ¼ ρf i u_ i þ φρ_ e φD_ i T ji u_ j , i ,
∂t
where
ð2:41Þ
2.5 Uniqueness
61
1
T ¼ ρu_ i u_ i
2
ð2:42Þ
is the kinetic energy density, and φ D_ i T ji u_ j is the Poynting vector describing
energy flux in linear piezoelectricity. Equation (2.41) may be considered as
Poynting’s theorem in linear piezoelectricity.
The integration of Eq. (2.41) over V gives
∂
∂t
Z
ðT þ U ÞdV
R R ¼ RV ρf i u_ i þ φρ_ e dV RS ni φD_ i T ji uR_ j dS
¼ V ρ f i u_ i þ φρ_ e dV þ Su ni T ji u_ j dS þ ST t j u_ j dS
R
R
Sφ ni D_ i φdS þ SD σ_ e φdS,
V
ð2:43Þ
where Eqs. (2.32) through (2.35) have been used. Integrating Eq. (2.43) further from
t0 to t, we obtain
R
R
þ U Þjt dV ¼ V ðT þ U Þjt0 dV
Rt R þ t0 dt V ρ f i u_ i þ φρ_ e dV
Rt R
Rt R
þ t0 dt Su ni T ji u_ j dS þ t0 dt ST t j u_ j dS
Rt R
Rt R
dt ni D_ i φdS þ dt σ_ e φdS:
V ðT
t0
Sφ
t0
ð2:44Þ
SD
Equation (2.44) is called the energy integral which states that the energy at time t is
equal to the energy at time t0 plus the work done by the body and boundary sources
from t0 to t.
2.5
Uniqueness
Consider two solutions to the following initial boundary value problem of linear
piezoelectricity with governing equations
T ji, j þ ρf i ¼ ρ€ui in V, t > t 0 ,
Di, i ¼ ρe in V, t > t 0 ,
T ij ¼ cijkl Skl ekij E k in V, t > t 0 ,
Di ¼ eijk S jk þ εijE j in V, t > t 0 ,
Sij ¼ ui, j þ u j, i =2 in V, t > t 0 ,
Ei ¼ φ, i in V, t > t 0 ,
boundary conditions
ð2:45Þ
62
2 Linear Theory of Piezoelectricity
ui ¼ ui on Su , t > t 0 ,
n j T ji ¼ t i on ST , t > t 0 ,
φ ¼ φ on Sφ , t > t 0 ,
ni Di ¼ σ e on SD , t > t 0 ,
ð2:46Þ
and initial conditions
ui ¼ u0i in V,
u_ i ¼ v0i in V,
φ ¼ φ0 in V,
t ¼ t0 ,
t ¼ t0 ,
t ¼ t0 :
ð2:47Þ
From the principle of superposition, the difference of the two solutions satisfies the
homogeneous version of Eqs. (2.45),(2.46) and (2.47). Let u∗, φ∗, S∗, T∗, E∗, and
D∗ denote the differences of the corresponding fields and apply the energy integral
in Eq. (2.44) to them. The initial energy and the external work for the difference
fields are zero. Then Eq. (2.44) implies that, for the difference fields, at any t > t0,
Z
V
ðT ∗ þ U ∗ Þjt dV ¼ 0,
t > t0 :
ð2:48Þ
Since both T and U are nonnegative,
U ∗ ¼ 0,
T ∗ ¼ 0 in
V,
t > t0 :
ð2:49Þ
From the positive-definiteness of T and U,
S∗ ¼ 0,
D∗ ¼ 0,
u_ ∗ ¼ 0 in
V,
t > t0 :
ð2:50Þ
Hence the strain, electric displacement, stress and electric field of the two solutions
are identical. The displacement fields of the two solutions may still differ by a rigid
body displacement.
2.6
Variational Formulation
The governing equations, some or all of the boundary conditions of linear piezoelectricity in Eqs. (2.30), (2.32), (2.34) and (2.35) can be derived from various
variational principles. Consider Hamilton’s variational principle below [1]:
Rt R 1
Πðu; φÞ ¼ t01 dt V ρu_ i u_ i H ðS; EÞ þ ρf i ui ρe φ dV
2 R
R
Rt R
t
þ t01 dt ST t i ui dS t01 dt SD σ e φdS,
ð2:51Þ
2.6 Variational Formulation
63
where
Sij ¼ ui, j þ u j, i =2,
E i ¼ φ, i :
ð2:52Þ
u and φ are variationally admissible if they are smooth enough and satisfy
δui jt0 ¼ δui jt1 ¼ 0 in V,
ui ¼ ui on Su , t 0 < t < t 1 ,
φ ¼ φ on Sφ , t 0 < t < t 1 :
ð2:53Þ
The first variation of Π is
δΠ ¼
R t1 R ρ€ui δui þ ðDi, i ρe Þδφ dV
t 0Rdt V R Tji, j þ ρf i Rt R t
t01 dt ST n j T ji t i δui dS t01 dt SD ni Di þ σ e δφdS,
ð2:54Þ
where we have denoted
T¼
∂H
,
∂S
D¼
∂H
:
∂E
ð2:55Þ
Therefore, the stationary condition of Π is
T ji, j þ ρf i ¼ ρ€ui in V, t 0 < t < t 1 ,
Di, i ¼ ρe in V, t 0 < t < t 1 ,
ð2:56Þ
n j T ji ¼ t i on ST , t 0 < t < t 1 ,
ni Di ¼ σ e on SD , t 0 < t < t 1 ,
ð2:57Þ
in the body and
on the boundary. Hamilton’s principle can be stated as: Among all the admissible
{u,φ} satisfying Eq. (2.53), the one that also satisfies Eqs. (2.56) and (2.57) makes
the Π in Eq. (2.51) stationary. The Π in Eq. (2.51) is in terms of the electric enthalpy
H(S, E). Variational principles based on the internal energy U(S, D) or other Legendre transforms of H (counterparts of the complementary energy density in elasticity)
can also be established [4].
If the variational functional in Eq. (2.51) is viewed to be dependent on u, φ, S, and
E, then Eq. (2.52) should be considered as constraints among the independent
variables. These constraints, along with the boundary data in Eq. (2.53)2,3, can be
removed by the method of Lagrange multipliers. Then the following variational
functional results [5]:
64
2 Linear Theory of Piezoelectricity
Π∗ ðu; φ; S; T; E; DÞ
Rt R 1
1
¼ t01 dt V ρu_ i u_ i H ðS; EÞ þ T ij Sij ui, j þ u j, i
2 2
Di Ei þ φ, i þ ρf i ui ρe φ dV
Rt R
Rt R
þ t01 dt Su n j T ji ui ui dS þ t01 dt Sφ ni Di φ φ dS
R t1 R
R t1 R e
þ t dt S t i ui dS t dt S σ φdS:
0
T
0
ð2:58Þ
D
u, φ, S, T, E, and D are admissible if they are smooth enough and satisfy
δui jt0 ¼ δui jt1 ¼ 0
in V:
ð2:59Þ
The first variation of Π is
δΠ∗ ¼
T ji, j þ ρf i ρ€ui δui þ ðDi, i ρe Þδφ
1
þ Sij ui, j þ u j, i δT ij Ei þ φ, i δDi
2
∂H
∂H
þ T ij δSij Di þ
δE i gdV
∂ERi
ij
R t1 R ∂S
R t1
þ t0 dt Su ui ui δT ji n j dS þ t0 dt Sφ φ φ δDi ni dS
Rt R Rt R t 1 dt S n j T ji t i δui dS t 1 dt S ni Di þ σ e δφdS:
R t1
t0
dt
0
R V
T
0
ð2:60Þ
D
Therefore the stationary condition of Π∗ is
T ji, j þ ρf i ¼ ρ€ui, Di, i ¼ ρe in V, t 0 < t < t 1 ,
Sij ¼ ui, j þ u j, i =2, Ei ¼ φ, i in V, t 0 < t < t 1 ,
∂H
∂H
T ij ¼
, Di ¼ in V, t 0 < t < t 1 ,
∂Sij
∂E i
ð2:61Þ
in the body and
ui ¼ ui on Su , t 0 < t < t 1 ,
n j T ji ¼ t i on ST , t 0 < t < t 1 ,
φ ¼ φ on Sφ , t 0 < t < t 1 :
ni Di ¼ σ e on SD , t 0 < t < t 1 ,
ð2:62Þ
on the boundary. Hence, among all the admissible {u, φ, S, T, E, D} satisfying
Eq. (2.59), the one that also satisfies Eqs. (2.61) and (2.62) makes Π∗(u, φ, S, T, E,
D) stationary. Π∗ has all of the u, φ, S, T, E, and D fields as independent fields and
corresponds to the generalized or mixed variational principles in elasticity.
2.7 Four-Vector Formulation
2.7
65
Four-Vector Formulation
Let us define the four-dimensional coordinate system [6]
xp ¼ fxi ; t g,
ð2:63Þ
U p ¼ fui ; φ g,
ð2:64Þ
and the four-dimensional vector
where subscripts p, q, r, s are assumed to run from 1 to 4. Also define the secondrank four-dimensional tensor
2
ρpq ¼
ρδpq ,
0,
ρ
60
p, q ¼ 1, 2, 3,
¼6
40
p, q ¼ 4,
0
0
ρ
0
0
0
0
ρ
0
3
0
07
7,
05
0
ð2:65Þ
and the fourth-rank four-dimensional tensor Mpqrs, where
M ijkl ¼ cijkl , M 4 jkl ¼ e jkl , M ijk4 ¼ ekij ,
M 4 jk4 ¼ ε jk , M p44s ¼ ρps ,
ð2:66Þ
and all other components of Mpqrs ¼ 0. Then
U p, q M pqrl , r ¼ U i, j M ijrl þ U 4, j M 4 jrl þ U i, 4 M i4rl þ U 4, 4 M 44rl , r
¼ U i, j M ijkl þ U 4, j M 4 jkl þ U i, 4 M i4kl þ U 4, 4 M 44kl , k
þ U i, j M ij4l þ U 4, j M 4 j4l þ U i, 4 M i44l þ U 4, 4 M 444l , 4
¼ ui, j cijkl þ φ, j e jkl , k þ u_ i ρil , 4
¼ cijkl ui, jk þ e jkl φ, jk ρ€ul ,
ð2:67Þ
and
U p, q M pqr4 , r
¼ U i, j M ijr4 þ U 4, j M 4 jr4 þ U i, 4 M i4r4 þ U 4, 4 M 44r4 , r
¼ U i, j M ijk4 þ U 4, j M 4 jk4 þ U i, 4 M i4k4 þ U 4, 4 M 44k4 , k
þ U i, j M ij44 þ U 4, j M 4 j44 þ U i, 4 M i444 þ U 4, 4 M 4444 , 4
¼ ui, j ekij φ, j ε jk , k
¼ ui, jk ekij φ, jk ε jk :
ð2:68Þ
66
2 Linear Theory of Piezoelectricity
Therefore,
U p, q M pqrs
,r
¼0
ð2:69Þ
yields the homogeneous equation of motion and the charge equation of electrostatics
in Eq. (2.30).
2.8
Matrix Notation
We now introduce a compact matrix notation [1, 2]. This notation consists of
replacing pairs of tensor indices ij or kl by single matrix indices p or q, where i, j,
k, and l take the values of 1, 2, and 3, and p and q take the values of 1, 2, 3, 4, 5, and
6 according to
ij or kl :
p or q :
11
1
22 33
2 3
23 or 32
4
31 or 13
5
12 or 21
:
6
ð2:70Þ
Thus
cijkl ! cpq ,
eikl ! eip ,
T ij ! T p ,
ð2:71Þ
according to Eq. (2.70). For the strain tensor, we introduce Sp such that
S1 ¼ S11 , S2 ¼ S22 , S3 ¼ S33 ,
S4 ¼ 2S23 , S5 ¼ 2S31 , S6 ¼ 2S12 :
ð2:72Þ
Then the constitutive relations in Eq. (2.11) can then be written as
E
T p ¼ cpq
Sq ekp E k ,
Di ¼ eiq Sq þ εikS E k :
In matrix form, Eq. (2.73) becomes
ð2:73Þ
2.8 Matrix Notation
67
8 9 0 E
18 9
E
E
E
E
E
T1 >
c11 c12
c13
c14
c15
c16
>
>
>
>
>
> S1 >
>
> B c E c E c E c E c E c E C>
>
> T2 >
> S2 >
>
>
>
21
22
23
24
25
26 C>
>
>
>
>
B
< = B E
E
E
E
E
E C< S =
T3
c
c
c
c
c
c
3
31
32
33
34
35
36
C
B
¼B E
E
E
E
E
E C
T4 >
>
>
>
>
> S4 >
B c41 c42 c43 c44 c45 c46 C>
>
>
>
E
E
E
E
E
E A>
>
>
>
@
T
S5 >
>
>
>
>
c
c
c
c
c
c
5
51
52
53
54
55
56
>
>
>
;
: ;
: >
E
E
E
E
E
E
T6
S6
c61 c62 c63 c64 c65 c66
1
0
e11 e21 e31
B e12 e22 e32 C8 9
C
B
B e13 e23 e33 C< E 1 =
C E2 ,
B
B
C
B e14 e24 e34 C: E 3 ;
@ e15 e25 e35 A
e16 e26 e36
8 9
S1 >
>
>
>
> >
8 9 2
3>
S2 >
>
>
e11 e12 e13 e14 e15 e16 >
=
< >
< D1 =
S
3
D2 ¼ 4 e21 e22 e23 e24 e25 e26 5
> S4 >
: ;
>
D3
e31 e32 e33 e34 e35 e36 >
>
>
>
S >
>
>
>
;
: 5>
S6
0 S
18 9
S
S
ε11 ε12
ε13
< E1 =
S
S
S A
þ @ ε21
E :
ε22
ε23
: 2;
S
S
S
E3
ε31
ε32
ε33
ð2:74Þ
ð2:75Þ
Similarly, Eqs. (2.16), (2.19), and (2.20) can also be written in matrix forms. The
matrices of the material constants in various expressions are related by [1, 2]
E E
D D
cpr
sqr ¼ δp , cpr
sqr ¼ δp ,
S S
T T
βik ε jk ¼ δijq , βik ε jk ¼ δijq ,
D
E
D
E
¼ cpq
þ ekp hkq , spq
¼ spq
dkp gkq ,
cpq
T
S
T
S
εij ¼ εij þ diq e jq , βij ¼ βij giq h jq ,
E
, d ip ¼ εikT gkp ,
eip ¼ diq cpq
T
D
gip ¼ βik dkp , hip ¼ giq cqp
:
ð2:76Þ
As an example, some of the relations in Eq. (2.76) are shown below. In matrix-vector
notation, Eq. (2.73) can be written as
fT g ¼ ½cE fSg ½eT fE g,
fDg ¼ ½efSg þ ½εS fEg:
From Eq. (2.77)1,
ð2:77Þ
68
2 Linear Theory of Piezoelectricity
cE fSg ¼ fT g þ ½eT fE g:
ð2:78Þ
The multiplication of both sides of Eq. (2.78) by the inverse of [cE] yields
fSg ¼ c E
1
fT g þ c E
1
½eT fE g:
ð2:79Þ
The substituting of Eq. (2.79) into Eq. (2.77)2 gives
1
1
fDg ¼ ½e ½cE fT g þ ½cE ½eT fE g þ ½εS fE g
1
1
¼ ½e½cE fT g þ ½e½cE ½eT þ ½εS fE g:
ð2:80Þ
Comparing Eqs. (2.79) and (2.80) with Eq. (2.19) which is rewritten in matrix form
below:
fSg ¼ ½sE fT g þ ½dT fE g,
fDg ¼ ½dfT g þ ½εT fEg,
ð2:81Þ
we identify
1
1
½sE ¼ ½cE , ½d ¼ ½e½cE ,
1
½εT ¼ ½εS þ ½e½cE ½eT :
2.9
ð2:82Þ
Polarized Ceramics and Crystals of Class 6mm
Polarized ceramics are transversely isotropic. Let a, a constant unit vector, represent
the direction of the axis of rotational symmetry or the poling direction of the
ceramics. For linear constitutive relations we need a quadratic electric enthalpy
function H. For transversely isotropic materials, a quadratic H is a function of the
following invariants of degrees one and two [7, 8] (higher degree invariants are not
included):
I 1 ¼ a S a, I 2 ¼ trS, I 3 ¼ a E,
II 1 ¼ a S2 a, II 2 ¼ trS2 ,
II 3 ¼ E E, II 4 ¼ a S E þ E S a,
ð2:83Þ
where tr is for the trace of a second-rank tensor. A complete quadratic function of the
above seven invariants can be written as [7]
2.9 Polarized Ceramics and Crystals of Class 6mm
69
H ¼ c1 I 21 þ c2 I 22 þ c3 I 1 I 2 þ c4 II 1 þ c5 II 2
þε1 I 23 þ ε2 II 3
þe1 I 1 I 3 þ e2 I 2 I 3 þ e3 II 4 ,
ð2:84Þ
where c1, c2, c3, c4, and c5 are elastic constants. ε1 and ε2 are dielectric constants. e1,
e2, and e3 are piezoelectric constants. Differentiating Eq. (2.84), according to
Eq. (2.11), we obtain
∂H ∂H
∂H
∂H
¼
aaþ
1þ
ða S a þ a S aÞ
∂S ∂I 1
∂I 2
∂II 1
∂H
∂H
þ2
Sþ
ð a E þ E aÞ
∂II 2
∂II 4
¼ ð2c1 I 1 þ c3 I 2 þ e1 I 3 Þa a þ ð2c2 I 2 þ c3 I 1 þ e2 I 3 Þ1
þ c4 ða S a þ a S aÞ þ 2c5 S þ e3 ða E þ E aÞ,
T¼
ð2:85Þ
and
∂H
∂H
∂H
∂H
¼
a2
E2
Sa
∂E
∂I 3
∂II 3
∂II 4
¼ ð2ε1 I 3 þ e1 I 1 þ e2 I 2 Þa 2ε2 E 2e3 S a,
D ¼
ð2:86Þ
where is for the tensor product between two vectors or a vector and a second-rank
tensor. In a Cartesian coordinate system, let a ¼ i3 and rearrange Eqs. (2.85) and
(2.86) in the form of Eqs. (2.74) and (2.75), we arrive at the following matrices:
0
c11
B c21
B
B c31
B
B 0
B
@ 0
0 0
0
@ 0
e31
c12
c11
c31
0
0
0
0
0
e31
c13
c13
c33
0
0
0
0
0
e33
0
0
0
c44
0
0
0
e15
0
0
0
0
0
c44
0
e15
0
0
1
0
0 C
C
0 C
C,
0 C
C
0 A
c661 0
0
ε11
0 A, @ 0
0
0
ð2:87Þ
0
ε11
0
1
0
0 A,
ε33
where c66 ¼ (c11–c12)/2. The elements of the above matrices are related to the
material constants in Eq. (2.84) linearly. The matrices in Eq. (2.87) have the same
structures as those of hexagonal crystals in class 6mm or C6v [9]. Therefore, within
the theory of linear piezoelectricity, crystals in class 6mm behave the same as
polarized ceramics. With Eq. (2.87), the constitutive relations of ceramics poled in
the x3 direction take the following form:
70
2 Linear Theory of Piezoelectricity
¼ c11 u1, 1 þ c12 u2, 2 þ c13 u3, 3 þ e31 φ, 3 ,
¼ c12 u1, 1 þ c11 u2, 2 þ c13 u3, 3 þ e31 φ, 3 ,
¼ c13 u1, 1 þ c13 u2, 2 þ c33 u3, 3 þ e33 φ, 3 ,
¼ c44 ðu2, 3 þ u3, 2 Þ þ e15 φ, 2 ,
¼ c44 ðu3, 1 þ u1, 3 Þ þ e15 φ, 1 ,
¼ c66 ðu1, 2 þ u2, 1 Þ,
ð2:88Þ
D1 ¼ e15 ðu3, 1 þ u1, 3 Þ ε11 φ, 1 ,
D2 ¼ e15 ðu2, 3 þ u3, 2 Þ ε11 φ, 2 ,
D3 ¼ e31 ðu1, 1 þ u2, 2 Þ þ e33 u3, 3 ε33 φ, 3 :
ð2:89Þ
T 11
T 22
T 33
T 23
T 31
T 12
and
For source-free problems with ρe¼0 and f ¼ 0, the equations of motion and charge
are
c11 u1, 11 þ ðc12 þ c66 Þu2, 12 þ ðc13 þ c44 Þu3, 13 þ c66 u1, 22
þ c44 u1, 33 þ ðe31 þ e15 Þφ, 13 ¼ ρ€
u1 ,
c66 u2, 11 þ ðc12 þ c66 Þu1, 12 þ c11 u2, 22 þ ðc13 þ c44 Þu3, 23
þ c44 u2, 33 þ ðe31 þ e15 Þφ, 23 ¼ ρ€
u2 ,
c44 u3, 11 þ ðc44 þ c13
Þu
þ
c
u
þ ðc13 þ c44 Þu2, 23
1
,
31
44
3
,
22
þ c33 u3, 33 þ e15 φ, 11 þ φ, 22 þ e33 φ, 33 ¼ ρ€u3 ,
e15 u3, 11 þ ðe15 þ e31
Þu1, 13 þ e15u3, 22 þ ðe15 þ e31 Þu2, 32
þ e31 u3, 33 ε11 φ, 11 þ φ, 22 ε33 φ, 33 ¼ 0:
ð2:90Þ
For the special case of motions independent of x3, Eq. (2.90) reduces to:
c11 u1, 11 þ ðc12 þ c66 Þu2, 12 þ c66 u1, 22 ¼ ρ€u1 ,
c66 u2, 11 þ ðc12 þ c66 Þu1, 12 þ c11 u2, 22 ¼ ρ€u2 ,
c44 ðu3, 11 þ u3, 22 Þ þ e15 φ, 11 þ φ, 22 ¼ ρ€u3 ,
e15 ðu3, 11 þ u3, 22 Þ ε11 φ, 11 þ φ, 22 ¼ 0:
ð2:91Þ
Equations (2.91)1,2 show that u1 and u2 are coupled but they do not interact with the
electric field. They form the usual plane-strain problem of linear elasticity. The
motion described by u3 is the so-called antiplane or shear-horizontal (SH) motion in
elasticity and it is coupled to φ.
For antiplane problems of polarized ceramics, let
u1 ¼ u2 ¼ 0,
u3 ¼ uðx1 ; x2 ; t Þ,
φ ¼ φðx1 ; x2 ; t Þ:
ð2:92Þ
Corresponding to Eq. (2.92), from Eq. (2.29), the nonzero components of the strain
tensor Sij and electric field Ei are
2.9 Polarized Ceramics and Crystals of Class 6mm
71
S5
2S31
¼
¼ — u,
S4 2S23
E1
¼ — φ,
E2
ð2:93Þ
where — ¼ i1 ∂1 þ i2 ∂2 is the two-dimensional gradient operator. Then, from
Eqs. (2.88) and (2.89), the nontrivial components of the stress tensor Tij and the
electric displacement vector Di are
T5
T 31
¼
¼ c— u þ e— φ,
T4 T 23
D1
¼ e— u ε— φ,
D2
ð2:94Þ
where we have denoted the relevant elastic, piezoelectric, and dielectric constants by
c ¼ c44 ,
e ¼ e15 ,
ε ¼ ε11 :
ð2:95Þ
The relevant equation of motion and the charge equation in Eqs. (2.91)3,4 can be
written as
c— 2 u þ e— 2 φ ¼ ρ€u,
e— 2 u ε— 2 φ ¼ 0,
2
ð2:96Þ
2
where — 2 ¼ ∂1 þ ∂2 is the two-dimensional Laplacian.
Equation (2.96) can be decoupled [10]. We introduce
e
ψ ¼ φ u:
ε
ð2:97Þ
Then, in terms of u and ψ, from Eq. (2.94),
T 31 ¼ cu, 1 þ eψ , 1 ,
T 23 ¼ cu, 2 þ eψ , 2 ,
D1 ¼ εψ , 1 ,
D2 ¼ εψ , 2 :
ð2:98Þ
v2T — 2 u ¼ u€ ,
— 2 ψ ¼ 0,
ð2:99Þ
Equation (2.96) becomes
where
72
2 Linear Theory of Piezoelectricity
c
v2T ¼ ,
ρ
c ¼ c þ
e2
¼ c 1 þ k2 ,
ε
k2 ¼
e2
:
εc
ð2:100Þ
For later use we also introduce
k2 ¼
k2
,
1 þ k2
ε ¼ ε 1 þ k2 :
ð2:101Þ
Sometimes a piezoelectric device is heterogeneous, with ceramics poled in
different directions in different parts. In this case it is impossible to always orient
the x3 axis along different poling directions unless a few local coordinate systems are
introduced. Therefore, material matrices of ceramics poled along other axes are
useful. They can be obtained from the matrices in Eq. (2.87) by tensor transformations. For ceramics poled in the x1 direction, we have
0
c33
B c13
B
B c13
B
B 0
B
@ 0
0 0
e33
@ 0
0
c13
c11
c12
0
0
0
e31
0
0
c13
c12
c11
0
0
0
e31
0
0
1
0
0
0
0
0
0 C
C
0
0
0 C
C,
c66 0
0 C
C
0 c44 0 A
0
0 c441 0
0 0
0
ε33
0 0 e15 A, @ 0
0
0 e15 0
ð2:102Þ
0
ε11
0
1
0
0 A:
ε11
For ceramics poled in the x2 direction,
0
c11
B c13
B
B c12
B
B 0
B
@ 0
0 0
0
@ e31
0
2.10
c13
c33
c13
0
0
0
0
e33
0
c12
c13
c11
0
0
0
0
e31
0
0
0
0
c44
0
0
0
0
e15
1
0
0
0
0 C
C
0
0 C
C,
0
0 C
C
c66 0 A
0 c441 0
0 e15
ε11
0 0 A, @ 0
0
0 0
ð2:103Þ
0
ε33
0
1
0
0 A:
ε11
Quartz
Quartz belongs to the trigonal crystal class of 32 or D3. It has one axis of trigonal
symmetry and, in the plane at right angles, three digonal axes 120 apart. With
respect to the crystallographic axes (X,Y,Z ) corresponding to (1,2,3) where Z is
2.10
Quartz
73
the trigonal or c axis and X a digonal axis, the material matrices have the
following structures [1]:
0
c11
B c21
B
B c13
B
B c14
B
@ 0
0 0
e11
@ 0
0
c12
c11
c13
c14
0
0
e11
0
0
1
c13 c14
0
0
c13 c14 0
0 C
C
c33
0
0
0 C
C,
0
c44
0
0 C
C
0
0
c44 c14 A
0
0
c14 c66 1 0
0 e14
0
0
ε11
0 0 e14 e11 A, @ 0
0
0 0
0
0
ð2:104Þ
0
ε11
0
1
0
0 A,
ε33
where c66 ¼ (c11 c12)/2. The independent material constants are 6 + 2 + 2 ¼ 10.
Quartz plates taken out of a bulk crystal at different orientations are referred to as
plates of different cuts. A particular cut is specified by two angles, ϕ and θ, with
respect to (X,Y,Z ) (see Fig. 2.2) and is called a doubly rotated cut. For example, the
stress-compensated SC-cut has (ϕ, θ)¼(22.5 ,34.3 ) [11]. Plates of different cuts
have different material matrices with respect to the coordinate system (x1,x2,x3) in
and normal to the plane of the plates. One class of cuts of quartz plates, called rotated
Y-cuts, is particularly useful in device applications. Rotated Y-cut quartz plates have
ϕ ¼ 0 (see Fig. 2.3) and they are called singly rotated cuts. Rotated Y-cut quartz
plates exhibit monoclinic symmetry of class 2 or C2 in (x1,x2,x3), with x1 the digonal
axis. Rotated Y-cuts include a few common cuts, for example, AT-cut with
(ϕ, θ) ¼ (0 ,35.25 ) [11] and BT-cut with (ϕ, θ) ¼ (0 ,–49 ) [11] as special cases.
Therefore we list the equations for monoclinic crystals of class 2 below which are
useful for studying rotated Y-cut quartz plates. For these crystal plates, with the
digonal axis along the x1 axis, the material matrices are
0
c11
B c21
B
B c31
B
B c41
B
@ 0
0 0
e11
@ 0
0
c12
c22
c32
c42
0
0
e12
0
0
c13
c23
c33
c43
0
0
e13
0
0
c14
c24
c34
c44
0
0
e14
0
0
The constitutive relations are
0
0
0
0
c55
c65
0
e25
e35
1
0
0 C
C
0 C
C,
0 C
C
c56 A
c66 1 0
0
ε11
e26 A, @ 0
e36
0
ð2:105Þ
0
ε22
ε32
1
0
ε23 A:
ε33
74
2 Linear Theory of Piezoelectricity
Z
x3
Fig. 2.2 A quartz plate of a
doubly rotated cut
q
x2
Y
X
́
x1
Z
x3
Fig. 2.3 A rotated Y-cut
quartz plate (singly-rotated
cut)
q
x2
Y
X
x1
¼ c11 u1, 1 þ c12 u2, 2 þ c13 u3, 3 þ c14 ðu2, 3 þ u3, 2 Þ þ e11 φ, 1 ,
¼ c12 u1, 1 þ c22 u2, 2 þ c23 u3, 3 þ c24 ðu2, 3 þ u3, 2 Þ þ e12 φ, 1 ,
¼ c13 u1, 1 þ c23 u2, 2 þ c33 u3, 3 þ c34 ðu2, 3 þ u3, 2 Þ þ e13 φ, 1 ,
¼ c14 u1, 1 þ c24 u2, 2 þ c34 u3, 3 þ c44 ðu2, 3 þ u3, 2 Þ þ e14 φ, 1 ,
¼ c55 ðu3, 1 þ u1, 3 Þ þ c56 ðu1, 2 þ u2, 1 Þ þ e25 φ, 2 þ e35 φ, 3 ,
¼ c56 ðu3, 1 þ u1, 3 Þ þ c66 ðu1, 2 þ u2, 1 Þ þ e26 φ, 2 þ e36 φ, 3 ,
ð2:106Þ
D1 ¼ e11 u1, 1 þ e12 u2, 2 þ e13 u3, 3 þ e14 ðu2, 3 þ u3, 2 Þ ε11 φ, 1 ,
D2 ¼ e25 ðu3, 1 þ u1, 3 Þ þ e26 ðu1, 2 þ u2, 1 Þ ε22 φ, 2 ε23 φ, 3 ,
D3 ¼ e35 ðu3, 1 þ u1, 3 Þ þ e36 ðu1, 2 þ u2, 1 Þ ε23 φ, 2 ε33 φ, 3 :
ð2:107Þ
T 11
T 22
T 33
T 23
T 31
T 12
and
The equations of motion and charge are
2.10
Quartz
75
c11 u1, 11 þ ðc12 þ c66 Þu2, 12 þ ðc13 þ c55 Þu3, 13 þ ðc14 þ c56 Þu2, 13
þ ðc14 þ c56 Þu3, 12 þ 2c56 u1, 23 þ c66 u1, 22 þ c55 u1, 33
þ e11 φ, 11 þ e26 φ, 22 þ ðe36 þ e25 Þφ, 23 þ e35 φ, 33 ¼ ρ€u1 ,
c56 u3, 11 þ ðc56 þ c14 Þu1, 13 þ ðc66 þ c12 Þu1, 12 þ c66 u2, 11
þ c22 u2, 22 þ ðc23 þ c44 Þu3, 23 þ 2c24 u2, 23 þ c24 u3, 22
þ c34 u3, 33 þ c44 u2, 33 þ ðe26 þ e12 Þφ, 12 þ ðe36 þ e14 Þφ, 13 ¼ ρ€u2 ,
c55 u3, 11 þ ðc55 þ c13 Þu1, 13 þ ðc56 þ c14 Þu1, 12 þ c56 u2, 11
þ c24 u2, 22 þ 2c34 u3, 23 þ ðc44 þ c23 Þu2, 23 þ c44 u3, 22
þ c33 u3, 33 þ c34 u2, 33 þ ðe25 þ e14 Þφ, 12 þ ðe35 þ e13 Þφ, 13 ¼ ρ€u3 ,
ð2:108Þ
e11 u1, 11 þ ðe12 þ e26 Þu2, 12 þ ðe13 þ e35 Þu3, 13 þ ðe14 þ e36 Þu2, 13
þ ðe14 þ e25 Þu3, 12 þ ðe25 þ e36 Þu1, 23 þ e26 u1, 22
þ e35 u1, 33 ε11 φ, 11 ε22 φ, 22 2ε23 φ, 23 ε33 φ, 33 ¼ 0:
ð2:109Þ
In rotated Y-cut quartz, the following shear-horizontal (SH) or antiplane motions
with only one displacement component, u1, are allowed and are useful in device
applications. Consider
u1 ¼ u1 ðx2 ; x3 ; t Þ,
φ ¼ φðx2 ; x3 ; t Þ:
u2 ¼ u3 ¼ 0,
ð2:110Þ
Corresponding to Eq. (2.110), from Eq. (2.29), the nonzero components of the strain
tensor and the electric field are
S5 ¼ u1, 3 , S6 ¼ u1, 2 ,
E2 ¼ φ, 2 , E3 ¼ φ, 3 :
ð2:111Þ
From Eqs. (2.106) and (2.107),
T 31 ¼ c55 u1, 3 þ c56 u1, 2 þ e25 φ, 2 þ e35 φ, 3 ,
T 21 ¼ c56 u1, 3 þ c66 u1, 2 þ e26 φ, 2 þ e36 φ, 3 ,
D2 ¼ e25 u1, 3 þ e26 u1, 2 ε22 φ, 2 ε23 φ, 3 ,
D3 ¼ e35 u1, 3 þ e36 u1, 2 ε23 φ, 2 ε33 φ, 3 :
ð2:112Þ
The equations left to be satisfied by u1 and φ are Eqs. (2.108)1 and (2.109):
c66 u1, 22 þ c55 u1, 33 þ 2c56 u1, 23
þ e26 φ, 22 þ e35 φ, 33 þ ðe25 þ e36 Þφ, 23 ¼ ρ€u1 ,
e26 u1, 22 þ e35 u1, 33 þ ðe25 þ e36 Þu1, 23
ε22 φ, 22 ε33 φ, 33 2ε23 φ, 23 ¼ 0:
ð2:113Þ
76
2.11
2 Linear Theory of Piezoelectricity
Lithium Niobate and Lithium Tantalate
Consider lithium niobate and lithium tantalate which have stronger piezoelectric
couplings than quartz. For these two crystals the crystal class is C3v ¼ 3m, a different
type of trigonal crystal than quartz. When x3 is the trigonal axis like in quartz and x1 is
normal to a mirror plane in these crystals, the material matrices are [1].
0
c11 c12
B c21 c11
B
B c13 c13
B
B c14 c14
B
@ 0
0
0
0
0
0
0
@ e22 e22
e31 e31
c13
c13
c33
0
0
0
0
0
e33
1
c14
0
0
c14 0
0 C
C
0
0
0 C
C,
c44
0
0 C
C
0
c44 c14 A
0
c14 c66 1 0
0 e15 e22
ε11
e15 0
0 A, @ 0
0
0
0
0
ð2:114Þ
0
ε11
0
1
0
0 A:
ε33
When a rotated Y-cut is formed by rotating about x1, the material apparently has
m-monoclinic symmetry with the following matrices [1]:
0
c11
B c21
B
B c31
B
B c41
B
@ 0
0 0
0
@ e21
e31
c12
c22
c32
c42
0
0
0
e22
e32
c13
c23
c33
c43
0
0
0
e23
e33
c14
c24
c34
c44
0
0
0
e24
e34
0
0
0
0
c55
c65
e15
0
0
1
0
0 C
C
0 C
C,
0 C
C
c56 A
c66 1 0
e16
ε11
0 A, @ 0
0
0
ð2:115Þ
0
ε22
ε23
1
0
ε23 A:
ε33
The corresponding constitutive relations are
T 11
T 22
T 33
T 23
T 31
T 12
and
¼ c11 u1, 1 þ c12 u2, 2 þ c13 u3, 3 þ c14 ðu2, 3 þ u3, 2 Þ þ e21 φ, 2 þ e31 φ, 3 ,
¼ c12 u1, 1 þ c22 u2, 2 þ c23 u3, 3 þ c24 ðu2, 3 þ u3, 2 Þ þ e22 φ, 2 þ e32 φ, 3 ,
¼ c13 u1, 1 þ c23 u2, 2 þ c33 u3, 3 þ c34 ðu2, 3 þ u3, 2 Þ þ e23 φ, 2 þ e33 φ, 3 ,
¼ c14 u1, 1 þ c24 u2, 2 þ c34 u3, 3 þ c44 ðu2, 3 þ u3, 2 Þ þ e24 φ, 2 þ e34 φ, 3 ,
¼ c55 ðu3, 1 þ u1, 3 Þ þ c56 ðu1, 2 þ u2, 1 Þ þ e15 φ, 1 ,
¼ c56 ðu3, 1 þ u1, 3 Þ þ c66 ðu1, 2 þ u2, 1 Þ þ e16 φ, 1 ,
ð2:116Þ
2.12
Curvilinear Coordinates
77
D1 ¼ e15 ðu1, 3 þ u3, 1 Þ þ e16 ðu1, 2 þ u2, 1 Þ ε11 φ, 1 ,
D2 ¼ e21 u1, 1 þ e22 u2, 2 þ e23 u3, 3 þ e24 ðu2, 3 þ u3, 2 Þ ε22 φ, 2 ε23 φ, 3 ,
D3 ¼ e31 u1, 1 þ e32 u2, 2 þ e33 u3, 3 þ e34 ðu2, 3 þ u3, 2 Þ ε23 φ, 2 ε33 φ, 3 :
ð2:117Þ
The equations of motion and charge are
c11 u1, 11 þ ðc12 þ c66 Þu2, 12 þ ðc13 þ c55 Þu3, 13 þ ðc14 þ c56 Þu2, 13
þ ðc14 þ c56 Þu3, 12 þ 2c56 u1, 23 þ c66 u1, 22 þ c55 u1, 33
þ ðe21 þ e16 Þφ, 12 þ ðe31 þ e15 Þφ, 13 ¼ ρ€u1 ,
c56 u3, 11 þ ðc56 þ c14 Þu1, 13 þ ðc66 þ c12 Þu1, 12 þ c66 u2, 11 þ c22 u2, 22
þ ðc23 þ c44 Þu3, 23 þ 2c24 u2, 23 þ c24 u3, 22 þ c34 u3, 33 þ c44 u2, 33
þ e16 φ, 11 þ e22 φ, 22 þ ðe32 þ e24 Þφ, 23 þ e34 φ, 33 ¼ ρ€u2 ,
c55 u3, 11 þ ðc55 þ c13 Þu1, 13 þ ðc56 þ c14 Þu1, 12 þ c56 u2, 11 þ c24 u2, 22
þ 2c34 u3, 23 þ ðc44 þ c23 Þu2, 23 þ c44 u3, 22 þ c33 u3, 33 þ c34 u2, 33
þ e15 φ, 11 þ e24 φ, 22 þ ðe34 þ e23 Þφ, 23 þ e33 φ, 33 ¼ ρ€u3 ,
ð2:118Þ
ðe15 þ e31 Þu1, 31 þ e15 u3, 11 þ ðe16 þ e21 Þu1, 12 þ e16 u2, 11 þ e22 u2, 22
þ ðe23 þ e34 Þu3, 23 þ ðe24 þ e32 Þu2, 23 þ e24 u3, 22 þ e33 u3, 33 þ e34 u2, 33 ð2:119Þ
ε11 φ, 11 ε22 φ, 22 2ε23 φ, 23 ε33 φ, 33 ¼ 0:
2.12
Curvilinear Coordinates
Cylindrical and spherical shapes are often used in piezoelectric devices. To analyze
these devices, it is usually convenient to use cylindrical or spherical coordinates.
The cylindrical coordinates (r,θ, z) are defined by
x1 ¼ r cos θ,
x2 ¼ r sin θ,
x3 ¼ z:
ð2:120Þ
In cylindrical coordinates we have the following strain–displacement relation:
1
ur
Sθθ ¼ uθ, θ þ , Szz ¼ uz, z ,
r
r
1
uθ
1
2Srθ ¼ uθ, r þ ur, θ , 2Sθz ¼ uz, θ þ uθ, z ,
r
r
r
2Szr ¼ ur, z þ uz, r :
Srr ¼ ur, r ,
ð2:121Þ
The electric field-potential relation is given by
E r ¼ φ, r ,
1
E θ ¼ φ, θ ,
r
Ez ¼ φ, z :
ð2:122Þ
78
2 Linear Theory of Piezoelectricity
The equations of motion are
∂T rr 1 ∂T θr ∂T zr T rr T θθ
þ
þ
þ
þ ρf r ¼ ρ€ur ,
r ∂θ
∂r
∂z
r
∂T rθ 1 ∂T θθ ∂T zθ 2
þ
þ
þ T rθ þ ρf θ ¼ ρ€uθ ,
r ∂θ
r
∂r
∂z
∂T rz 1 ∂T θz ∂T zz 1
þ
þ
þ T rz þ ρf z ¼ ρ€uz :
r ∂θ
r
∂r
∂z
ð2:123Þ
The electrostatic charge equation is
1
1
ðrDr Þ, r þ Dθ, θ þ Dz, z ¼ ρe :
r
r
ð2:124Þ
The spherical coordinates (r,θ,ϕ) are defined by
x1 ¼ r sin θ cos ϕ,
x2 ¼ r sin θ sin ϕ,
x3 ¼ r cos θ:
ð2:125Þ
In spherical coordinates we have the following strain–displacement relation:
∂ur
1 ∂uθ ur
, Sθθ ¼
þ ,
r ∂θ
∂r
r
1 ∂uϕ ur uθ
Sϕϕ ¼
þ þ cot θ,
r sin θ ∂ϕ
r
r
∂uθ 1 ∂ur uθ
2Srθ ¼
þ
,
r ∂θ
∂r
r
1 ∂uϕ
1 ∂uθ uϕ
þ
cot θ,
2Sθϕ ¼
r ∂θ r sin θ ∂ϕ
r
1 ∂ur ∂uϕ uϕ
2Sϕr ¼
þ
:
r sin θ ∂ϕ
∂r
r
Srr ¼
ð2:126Þ
The electric field-potential relation is
Er ¼ ∂φ
,
∂r
The equations of motion are
Eθ ¼ 1 ∂φ
,
r ∂θ
Eϕ ¼ 1 ∂φ
:
r sin θ ∂ϕ
ð2:127Þ
References
79
∂T rr 1 ∂T θr
1 ∂T ϕr
þ
þ
r ∂θ
r sin θ ∂ϕ
∂r
1
þ 2T rr T θθ T ϕϕ þ T θr cot θ þ ρf r ¼ ρ€ur ,
r
∂T rθ 1 ∂T θθ
1 ∂T ϕθ
þ
þ
r ∂θ
r sin θ ∂ϕ
∂r
1
þ 3T rθ þ T θθ T ϕϕ cot θ þ ρf θ ¼ ρ€uθ ,
r
∂T rϕ 1 ∂T θϕ
1 ∂T ϕϕ
þ
þ
r ∂θ
r sin θ ∂ϕ
∂r
1
þ 3T rϕ þ 2T θϕ cot θ þ ρf z ¼ ρ€uϕ :
r
ð2:128Þ
The electrostatic charge equation is
r2
∂ 2 1 ∂
1 ∂
r Dr þ
Dϕ ¼ ρe :
ðDθ sin ϕÞ þ
∂r
r sin θ ∂θ
r sin θ ∂ϕ
ð2:129Þ
References
1. H.F. Tiersten, Linear Piezoelectric Plate Vibrations (Plenum, New York, 1969)
2. A.H. Meitzler, H.F. Tiersten, A.W. Warner, D. Berlincourt, G.A. Couqin, F.S. Welsh III, IEEE
Standard on Piezoelectricity (IEEE, New York, 1988)
3. Q.H. Du, S.W. Yu, Z.H. Yao, Theory of Elasticity (Science Press, Beijing, 1986)
4. H.-L. Zhang, On variational principles of a piezoelectric body. Acta Acustica 10, 223–230
(1985)
5. J. S. Yang, Mixed variational principles for piezoelectric elasticity, in: Developments in
Theoretical and Applied Mechanics (Proc. of the 16th Southeastern Conference on Theoretical
and Applied Mechanics), B. Antar, R. Engels, A. A. Prinaris and T. H. Moulden, ed., vol. XVI,
pp. II.1.31–38, The University of Tennessee Space Institute, 1992
6. R. Holland, E.P. EerNisse, Design of Resonant Piezoelectric Devices (MIT Press, Cambridge,
MA, 1969)
7. V.V. Varadan, J.-H. Jeng, V.K. Varadan, Form invariant constitutive relations for transversely
isotropic piezoelectric materials. J. Acoust. Soc. Am. 82, 337–341 (1987)
8. G.F. Smith, M.M. Smith, R.S. Rivlin, Integrity bases for a symmetric tensor and a vector-the
crystal classes. Arch. Rat. Mech. Anal. 12, 93–133 (1963)
9. B.A. Auld, Acoustic Fields and Waves in Solids, vol 1 (Wiley, New York, 1973)
10. J.L. Bleustein, A new surface wave in piezoelectric materials. Appl. Phys. Lett. 13, 412–413
(1968)
11. V.E. Bottom, Introduction to Quartz Crystal Unit Design (Van Nostrand Reinhold, New York,
1982)
Chapter 3
Static Problems
In this chapter, some static solutions to the equations of linear piezoelectricity are
presented. A few simple deformations useful in piezoelectric devices are discussed.
The concept of electromechanical coupling factor is introduced. In Particular, Sects.
3.3, 3.4, 3.8, 3.9, 3.10, 3.11, 3.12, and 3.13 are on antiplane deformations of
polarized ceramics [1].
3.1
Extension of a Ceramic Rod
Consider a cylindrical rod of length L made from polarized ceramics with axial
poling (see Fig. 3.1). The cross section of the rod can be arbitrary. The lateral surface
of the rod is traction free and is unelectroded. The electric field in the surrounding
free space is neglected. The two end faces are under a uniform normal traction p, but
there is no tangential traction. Electrically, the two end faces are electroded with a
circuit between the electrodes, which can be switched on or off. Two cases of open
and shorted electrodes will be considered.
From Eqs. (2.27), (2.19), (2.26), and (1.145), the boundary-value problem
consists of:
T ji, j ¼ 0, Di, i ¼ 0 in V,
E
Sij ¼ sijkl
T kl þ Dkij Ek , Di ¼ dikl T kl þ εikT E k
εijk εlmn Sil, jm ¼ 0, εijk Ek, j ¼ 0 in V,
in
V,
ð3:1Þ
and
© Springer Nature Switzerland AG 2018
J. Yang, An Introduction to the Theory of Piezoelectricity, Advances in Mechanics
and Mathematics 9, https://doi.org/10.1007/978-3-030-03137-4_3
81
82
3 Static Problems
x1
Electrode
Electrode
p
p
P
x3
x2
Traction-free, unelectroded
Fig. 3.1 An axially poled ceramic rod
n j T ji ¼ 0, ni Di ¼ 0 on the lateral surface,
T 31 ¼ 0, T 32 ¼ 0, T 33 ¼ p, E 1 ¼ E2 ¼ 0, x3 ¼ 0, L,
φðx3 ¼ 0Þ ¼ φðx3 ¼ LÞ, if the end electrodes are shorted,
or D3 ¼ 0, x3 ¼ 0, L, if the end electrodes are open:
ð3:2Þ
For uniaxial extension, it can be expected that many stress components vanish.
Therefore it is convenient to consider the stress tensor directly and use constitutive
relations with T as the independent mechanical variable. In this formulation the
compatibility condition on strains has to be satisfied. Consider the following T and
D fields:
T 33 ¼ p, all other T ij ¼ 0,
D3 ¼ constant, D1 ¼ D2 ¼ 0,
ð3:3Þ
which satisfy the equation of motion and the charge equation of electrostatics. Since
the T and D fields are constants, the constitutive relations imply that the S and
E fields are also constants. Therefore the compatibility conditions on S and the curlfree condition on E are satisfied. Equation (3.3) also satisfies the boundary conditions on the lateral surface and the mechanical boundary conditions on the end faces.
From the constitutive relations, we have
S23 ¼ S31 ¼ S12 ¼ 0,
E
E
S33 ¼ s33
p þ d33 E 3 , S11 ¼ S22 ¼ s13
p þ d31 E 3 ,
T
E1 ¼ E 2 ¼ 0, D3 ¼ d33 p þ ε33
E3 :
ð3:4Þ
Hence the electrical boundary conditions of E1 ¼ E2 ¼ 0 (constant electric potential
on an electrode) on the end electrodes are also satisfied. We consider two cases of
shorted and open electrodes separately below.
When the electrodes are shorted, there is no potential difference between the end
electrodes. Since E3 is constant along the rod, we must have
3.1 Extension of a Ceramic Rod
83
E3 ¼ 0,
ð3:5Þ
which implies that
D3 ¼ d 33 p,
E
S33 ¼ s33
p:
ð3:6Þ
The mechanical work done to the rod per unit volume during the static extensional
process is
1
1 E 2
W 1 ¼ T 33 S33 ¼ s33
p :
2
2
ð3:7Þ
When the electrodes are open, there is no net charge on the end electrodes. Since
D3 is constant over a cross section, we must have
D3 ¼ 0,
ð3:8Þ
which implies that
E3 ¼ S33 ¼
d33
T p,
ε33
E
s33
p
d 33
d233
E
d33 T p ¼ s33 1 T E p:
ε33
ε33 s33
ð3:9Þ
In this case the mechanical work done to the rod per unit volume is
1
1 E
d 233
W 2 ¼ T 33 S33 ¼ s33 1 T E p2 :
2
2
ε33 s33
ð3:10Þ
d233
T s E > 0,
ε33
33
ð3:11Þ
Since
the extensional strain S33 given by Eq. (3.6) is larger than that given by Eq. (3.9),
and, as a consequence,
W 1 > W 2:
ð3:12Þ
Therefore the rod appears to be stiffer when the electrodes are open and an axial
electric field is produced. This is called the piezoelectric stiffening effect. Graphically W1, W2, and their difference are represented by the areas in Fig. 3.2.
84
3 Static Problems
T33
p
W1
S33
psE33
W2
Fig. 3.2 Work done to the ceramic rod per unit volume along different paths
The following ratio is called the longitudinal electromechanical coupling factor
for the extension of a ceramic rod with axial poling, and is denoted by [2]
l
k33
2
¼
W1 W2
d2
¼ T 33E :
W1
ε33 s33
ð3:13Þ
As a numerical example, for PZT-5H, a common ceramic, from the material
constants in Appendix 4,
2
593 1012
¼ 0:56,
¼
3400 8:85 1012 20:7 1012
¼ 0:75,
l 2
k33
l
k33
ð3:14Þ
which is typical for polarized ceramics.
3.2
Thickness Stretch of a Ceramic Plate
Consider an unbounded ceramic plate poled in the thickness direction as shown in
Fig. 3.3. The top and bottom surfaces of the plate are under a normal traction p and
are electroded. Two cases of shorted and open electrodes will be considered.
From Eqs. (2.27), (2.28), and (2.29), the boundary-value problem consists of:
T ji, j ¼ 0, Di, i ¼ 0 in V,
T ij ¼ cijkl Skl ekij Ek , Di ¼ eikl Skl þ εik E k
Sij ¼ ui, j þ u j, i =2, E i ¼ φ, i in V,
and
in V,
ð3:15Þ
3.2 Thickness Stretch of a Ceramic Plate
Fig. 3.3 An electroded
ceramic plate under normal
stress
85
T33 = p, T31 = T32 = 0
x3
2h
P
x1
T33 = p, T31 = T32 = 0
n j T ji ¼ pδ3i , x3 ¼ h,
φðx3 ¼ hÞ ¼ φðx3 ¼ hÞ, if the electrodes are shorted,
or D3 ðx3 ¼ hÞ ¼ 0, if the electrodes are open:
ð3:16Þ
Because of the thickness poling of the ceramic plate, we consider the possibility of
the following displacement and potential fields:
u3 ¼ u3 ðx3 Þ,
u1 ¼ u2 ¼ 0,
φ ¼ φðx3 Þ,
ð3:17Þ
which is called the thickness stretch or thickness extension [3] of the plate. The
nontrivial components of strain, electric field, stress, and electric displacement are
S33 ¼ u3, 3 ,
E3 ¼ φ, 3 ,
ð3:18Þ
and
T 11 ¼ T 22 ¼ c13 u3, 3 þ e31 φ, 3 ,
T 33 ¼ c33 u3, 3 þ e33 φ, 3 ,
D3 ¼ e33 u3, 3 ε33 φ, 3 :
ð3:19Þ
The equation of motion and the charge equation require that
T 33, 3 ¼ c33 u3, 33 þ e33 φ, 33 ¼ 0,
D3, 3 ¼ e33 u3, 33 ε33 φ, 33 ¼ 0:
ð3:20Þ
Hence
u3, 33 ¼ 0,
φ, 33 ¼ 0,
ð3:21Þ
unless
k 233 ¼
e233
¼1
ε33 c33
ð3:22Þ
86
3 Static Problems
which we do not consider because usually k 233 < 1. Equation (3.21) implies that the
strain, stress, electric field, and electric displacement components are constants.
When the electrodes are shorted, since the potential at the two electrodes are equal
and E3 is a constant, we must have
E 3 ¼ 0:
ð3:23Þ
The mechanical boundary conditions require that T33 ¼ p. Then
S3 ¼ u3 , 3 ¼
p
,
c33
T 11 ¼ T 22 ¼
c13
p,
c33
D3 ¼
e33
p:
c33
ð3:24Þ
The work done to the plate per unit volume is
1
p2
:
W 1 ¼ T 33 S33 ¼
2
2c33
ð3:25Þ
On the other hand, if the electrodes are open, the boundary conditions require that
T 33 ¼ c33 u3, 3 þ e33 φ, 3 ¼ p,
D3 ¼ e33 u3, 3 ε33 φ, 3 ¼ 0:
ð3:26Þ
Equation (3.26) implies that
p
,
c33 1 þ k233
e
33
p:
E3 ¼ φ, 3 ¼ ε33 c33 1 þ k233
S3 ¼ u3 , 3 ¼
ð3:27Þ
In this case the work done to the plate per unit volume is given by
1
p2
:
W 2 ¼ T 33 S33 ¼
2
2c33 1 þ k 233
ð3:28Þ
The same as the case in the previous section, the open-circuit strain is smaller than
the short-circuit strain and, correspondingly,
W 2 < W 1:
ð3:29Þ
The electromechanical coupling factor for the thickness stretch of a ceramic plate
poled in the thickness direction is denoted by
3.3 Thickness Shear of a Ceramic Plate
87
t 2 W 1 W 2
1
k 233
k 33 ¼
¼1
¼
:
W1
1 þ k233 1 þ k233
ð3:30Þ
For PZT-7A, from the material constants in Appendix 4,
ð9:50Þ2
¼ 0:33,
235 8:85 1012 13:1 1010
¼ 0:58:
ðk33 Þ2 ¼ k33
3.3
ð3:31Þ
Thickness Shear of a Ceramic Plate
Consider an unbounded ceramic plate with in-plane poling along the x3 axis which is
determined from the x1 and x2 axes by the right-hand rule (see Fig. 3.4). The surfaces
of the plate are under a shear stress τ and are electroded. Two cases of shorted and
open electrodes will be considered.
This is an antiplane problem. We follow the notation in Sect. 2.9. For static
problems Eq. (2.96) reduces to
c∇2 u þ e∇2 φ ¼ 0,
e∇2 u ε∇2 φ ¼ 0:
ð3:32Þ
When cε + e2 6¼ 0 which we always assume, Eq. (3.32) is equivalent to
∇2 u ¼ 0,
∇2 φ ¼ 0:
ð3:33Þ
The boundary-value problem is:
∇2 u ¼ 0, ∇2 φ ¼ 0, j x2 j< h,
T 23 ¼ τ, x2 ¼ h,
φðx2 ¼ hÞ ¼ φðx2 ¼ hÞ, if the electrodes are shorted,
or D2 ðx2 ¼ hÞ ¼ 0, if the electrodes are open:
Fig. 3.4 An electroded
ceramic plate under shear
stress
x2
T23 = t
2h
T23 = t
ð3:34Þ
x1
88
3 Static Problems
For a ceramic plate with in-plane poling, we consider the possibility of the following
displacement and potential fields:
u ¼ uðx2 Þ,
φ ¼ φðx2 Þ,
ð3:35Þ
which is called the thickness shear of the plate [3]. The nontrivial components of
strain, electric field, stress, and electric displacement are
S4 ¼ 2S23 ¼ u, 2 ,
E2 ¼ φ, 2 ,
ð3:36Þ
and
T 4 ¼ T 23 ¼ cu, 2 þ eφ, 2 ,
D2 ¼ eu, 2 εφ, 2 :
ð3:37Þ
The equation of motion and the charge equation reduce to
u, 22 ¼ 0,
φ, 22 ¼ 0:
ð3:38Þ
Equation (3.38) implies that the strain, stress, electric field, and electric displacement
components are constants.
When the electrodes are shorted, since the potential at the two electrodes are equal
and E2 is a constant, we must have
E 2 ¼ 0:
ð3:39Þ
The mechanical boundary conditions require that T23 ¼ τ. Then
τ
S4 ¼ u , 2 ¼ ,
c
e
D2 ¼ τ:
c
ð3:40Þ
The mechanical work done by the surface shear stress to the plate per unit volume is
1
τ2
W 1 ¼ T 4 S4 ¼ :
2
2c
ð3:41Þ
In the case when the electrodes are open, the boundary conditions require that
T 4 ¼ cu, 2 þ eφ, 2 ¼ τ,
D2 ¼ eu, 2 εφ, 2 ¼ 0,
which implies that
ð3:42Þ
3.4 Capacitance of a Ceramic Plate
89
τ
,
S4 ¼ u, 2 ¼ c 1 þ k2
e
τ,
E 2 ¼ φ, 2 ¼ εc 1 þ k2
ð3:43Þ
where k is given by Eq. (2.100). The work done to the plate per unit volume is
1
τ2
:
W 2 ¼ T 4 S4 ¼ 2
2c 1 þ k2
ð3:44Þ
From Eqs. (3.41) and (3.44), clearly,
W 2 < W 1:
ð3:45Þ
Therefore, the electromechanical coupling factor for the thickness-shear deformation
of a ceramic plate with in-plane poling is
W1 W2
1
k2
¼1
¼
¼ k2 ,
W1
1 þ k2 1 þ k2
ð3:46Þ
where k is given by Eq. (2.101).
3.4
Capacitance of a Ceramic Plate
Consider the ceramic plate with in-plane poling along the x3 axis which is determined from the x1 and x2 axes by the right-hand rule in Fig. 3.5. The surfaces of the
plate are electroded. A voltage V is applied across the plate thickness. Two cases of
mechanical boundary conditions will be considered.
This is also an antiplane problem and the notation follows that in Sect. 2.9. The
boundary-value problem is:
∇2 u ¼ 0, ∇2 φ ¼ 0, j x2 j< h,
φ ¼ V=2, x2 ¼ h,
T 23 ¼ 0, x2 ¼ h, if the surfaces are traction-free,
or u ¼ 0, x2 ¼ h, if the surfaces are clamped ðfixedÞ:
Fig. 3.5 A ceramic plate
under a voltage
ð3:47Þ
x2
j =V / 2
2h
j = −V / 2
x1
90
3 Static Problems
Equation (3.38) still holds and the strain, stress, electric field, and electric displacement components are constants. In particular,
E 2 ¼ φ, 2 ¼ V
:
2h
ð3:48Þ
In the case when the plate surfaces are free, we have
T 4 ¼ cu, 2 þ eφ, 2 ¼ 0:
ð3:49Þ
eV
S4 ¼ u, 2 ¼ ,
c2h
V
eV
V
ε ¼ 1 þ k2 ε :
D2 ¼ e
c2h
2h
2h
ð3:50Þ
Then
The free charge per unit area on the electrode at x2 ¼ h is
V
σ e ¼ D2 ¼ 1 þ k 2 ε :
2h
ð3:51Þ
Hence the capacitance of the plate per unit area is
ε
σe :
¼ 1 þ k2
2h
V
ð3:52Þ
Equation (3.52) shows that the effect of piezoelectric coupling described by k2
enhances the capacitance. The electrical energy stored in the plate capacitor per
unit area is
V2 1 V2
1
1
U 1 ¼ σ e V ¼ 1 þ k2 ε ¼ ε ,
2
2
2h 2 2h
ð3:53Þ
where ε is given by Eq. (2.101).
In the case when the place surfaces are clamped, the strain S4 is still a constant and
u must be a linear function of x2. According to the displacement boundary conditions, this linear function must vanish at the plate surfaces. Hence
u ¼ 0,
which implies that
ð3:54Þ
3.5 Capacitance of a Quartz Plate
91
S4 ¼ u, 2 ¼ 0,
T4 ¼ e
V
,
2h
D2 ¼ ε
V
:
2h
ð3:55Þ
The free charge per unit area on the electrode at x2 ¼ h is
σ e ¼ D2 ¼ ε
V
:
2h
ð3:56Þ
Hence the capacitance of the plate per unit area is
σe
ε
¼ :
2h
V
ð3:57Þ
The electric energy stored in the capacitor per unit area is
1
1 V2
U2 ¼ σe V ¼ ε :
2
2 2h
ð3:58Þ
The electromechanical coupling factor for a ceramic plate with in-plane poling in
thickness shear can be calculated from
U1 U2
k2
¼
¼ k2 ,
U1
1 þ k2
ð3:59Þ
which is the same as Eq. (3.46).
3.5
Capacitance of a Quartz Plate
Consider a plate of rotated Y-cut quartz. The surfaces of the plate are electroded and
traction free. A voltage V is applied across the plate thickness (see Fig. 3.6). Two
cases of mechanical boundary conditions will be considered.
The boundary-value problem consists of:
T ji, j ¼ 0, Di, i ¼ 0 in V,
T ij ¼ cijkl Skl ekij Ek , Di ¼ eikl Skl þ εik E k
Sij ¼ ui, j þ u j, i =2, E i ¼ φ, i in V,
in V,
ð3:60Þ
and
φ ¼ V=2, x2 ¼ h,
n j T ji ¼ 0, x2 ¼ h, if the surfaces are traction-free,
or u j ¼ 0, x2 ¼ h, if the surfaces are clamped ðfixedÞ:
ð3:61Þ
92
3 Static Problems
x2
Fig. 3.6 A rotated Y-cut
quartz plate under a voltage
j =V / 2
2h
j
x1
= −V / 2
Because of the specific piezoelectric couplings in rotated Y-cut quartz plates in
Eq. (2.105), we consider the possibility of the following displacement and
potential fields:
u1 ¼ u1 ðx2 Þ,
u2 ¼ u3 ¼ 0,
φ ¼ φðx2 Þ,
ð3:62Þ
which is called the thickness shear of the plate [3]. The nontrivial components of the
strain and the electric field are
2S12 ¼ u1, 2 ,
E2 ¼ φ, 2 :
ð3:63Þ
The nontrivial components of stress and electric displacement are
T 31 ¼ c56 u1, 2 þ e25 φ, 2 , T 12 ¼ c66 u1, 2 þ e26 φ, 2 ,
D2 ¼ e26 u1, 2 ε22 φ, 2 , D3 ¼ e36 u1, 2 ε23 φ, 2 :
ð3:64Þ
The equation of motion and the charge equation require that
T 21, 2 ¼ c66 u1, 22 þ e26 φ, 22 ¼ 0,
D2, 2 ¼ e26 u1, 22 ε22 φ, 22 ¼ 0:
ð3:65Þ
Hence
u1, 22 ¼ 0,
φ, 22 ¼ 0,
ð3:66Þ
unless
k226 ¼
e226
¼ 1,
ε22 c66
ð3:67Þ
which we do not consider because usually k 226 < 1. Equation (3.66) implies that the
strain, stress, electric field, and electric displacement components are constants. In
particular,
3.5 Capacitance of a Quartz Plate
93
E2 ¼ V
:
2h
ð3:68Þ
When the plate surfaces are traction free, we have, at x2 ¼ h,
T 12 ¼ c66 u1, 2 þ e26 φ, 2 ¼ 0:
ð3:69Þ
Then
e26 V
,
c66 2h
e26 V
V
c56 V
þ e25 ¼ e25 e26
,
¼ c56
2h
c66 2h
c66 2h
S12 ¼ u1, 2 ¼ T 31
V
e26 V
V
ε22 ¼ 1 þ k226 ε22 ,
2h
2h
c66 2h
e26 V
V
e36 e26 V
D3 ¼ e36
ε23 ¼ ε23 þ
:
2h
c66 2h
c66 2h
D2 ¼ e26
ð3:70Þ
ð3:71Þ
ð3:72Þ
The free charge per unit area on the electrode at x2 ¼ h is
V
σ e ¼ D2 ¼ 1 þ k 226 ε22 :
2h
ð3:73Þ
Hence the capacitance of the plate per unit area is
ε22
σe ¼ 1 þ k 226
:
V
2h
ð3:74Þ
Equation (3.74) shows that the piezoelectric coupling enhances the capacitance by a
portion of k226 . The electrical energy stored in the plate capacitor per unit area is
V2
1
1
U 1 ¼ σ e V ¼ 1 þ k226 ε22 :
2
2
2h
ð3:75Þ
In the case when plate surfaces are clamped, the strain S12 is still a constant and u1
must be a linear function of x2. According to the displacement boundary conditions,
this linear function must vanish at the plate surfaces. Hence
u1 ¼ 0,
which implies that
ð3:76Þ
94
3 Static Problems
S12 ¼ u1, 2 ¼ 0,
V
,
2h
V
D2 ¼ ε22 ,
2h
T 31 ¼ e25
V
,
2h
V
D3 ¼ ε23 :
2h
T 12 ¼ e26
ð3:77Þ
ð3:78Þ
ð3:79Þ
In this case the free charge per unit area on the electrode at x2 ¼ h is
σ e ¼ D2 ¼ ε22
V
:
2h
ð3:80Þ
Hence the static capacitance of the plate per unit area is
σ e ε22
¼
:
V
2h
ð3:81Þ
The electric energy stored in the capacitor per unit area is
1
1 V2
U 2 ¼ σ e V ¼ ε22 :
2
2 2h
ð3:82Þ
The electromechanical coupling factor for a rotated Y-cut quartz plate in thickness shear can be calculated from
U1 U2
k226
k226 ¼
¼
:
U1
1 þ k226
ð3:83Þ
A rotated Y-cut of θ ¼ 35.25 is called an AT-cut [4] and is widely used in devices.
For AT-cut quartz plates, from the material constants in Appendix 4,
0:0952
¼ 0:0078,
39:8 1012 29:0 109
¼ 0:088,
k226 ¼ k26
ð3:84Þ
which is much smaller than that of polarized ceramics. Quartz is often used for
signal generation or processing in telecommunication or sensing rather than for
power handling. Therefore a small electromechanical coupling coefficient is
usually sufficient.
3.6 Torsion of a Ceramic Cylinder
3.6
95
Torsion of a Ceramic Cylinder
Consider a hollow circular cylinder of length L, inner radius a, and outer radius b as
shown in Fig. 3.7. The cylinder is made of ceramics with circumferential poling. We
choose (r,θ,z) to correspond to (2,3,1) so that the poling direction corresponds to
3. The cylinder is with 0 < z < L. The lateral cylindrical surfaces are traction free and
are unelectroded. The free space electric field is neglected as usual. The end faces are
electroded. The end electrodes can be either open or shorted. A torque M is applied.
Over any cross section, the stress distribution produces the same torque.
The boundary-value problem consists of:
T ji, j ¼ 0, Di, i ¼ 0 in V,
T ij ¼ cijkl Skl ekij Ek , Di ¼ eikl Skl þ εik E k
Sij ¼ ui, j þ u j, i =2, E i ¼ φ, i in V,
ð3:85Þ
in V,
and
n j T ji ¼ 0, ni Di ¼ 0, r ¼ a, b,
the stress distribution is statically equivalent to M, z ¼ 0, L,
φ ¼ constant, z ¼ 0, L,
φ
R ðz ¼ 0Þ ¼ φðz ¼ LÞ, if the electrodes are shorted,
z ¼ 0, L, if the electrodes are open:
a<r<b Dz dA ¼ 0,
ð3:86Þ
Consider the possibility of the following displacement and potential fields:
uθ ¼ Arz,
ur ¼ uz ¼ 0,
φ ¼ Bz,
ð3:87Þ
where A and B are undetermined constants. The nontrivial components of strain,
electric field, stress, and electric displacement are
S5 ¼ 2Sθz ¼ Ar,
E1 ¼ E z ¼ B,
ð3:88Þ
T 5 ¼ T θz ¼ c44 Ar e15 B,
D1 ¼ Dz ¼ e15 Ar þ ε11 B,
Fig. 3.7 A ceramic circular
cylinder in torsion
ð3:89Þ
M
1
P
2
3
b
M
a
96
3 Static Problems
thus the boundary conditions on the lateral surfaces are satisfied. The equation of
motion and the charge equation are trivially satisfied. At the end faces the φ given by
Eq. (3.87) is a constant. It is easy to verify that the stress distribution in Eq. (3.89)1
does not produce any net extensional or shear force and bending moment over a
cross section. The torque at a cross section is given by
Rb
a
Rb
T zθ ð2πrdr Þr ¼ a ðc44 Ar e15 BÞ2πr 2 dr
Rb
¼ a ðc44 A2πr 3 e15 B2πr 2 Þdr
2π 3
b a3 ¼ M,
¼ c44 AI p e15 B
3
ð3:90Þ
where
Ip ¼
π 4
b a4
2
ð3:91Þ
is the polar moment of inertia of the cross section about its center. The total charge
on the electrode at z ¼ 0 is given by
Rb
Rb
Qe ¼ a Dz 2πrdr ¼ a ðe15 Ar þ ε11 BÞ2πrdr
Rb
¼ a ðe15 A2πr 2 þ ε11 B2πr Þdr
2π 3
b a3 þ ε11 Bπ b2 a2 :
¼ e15 A
3
ð3:92Þ
When the two end electrodes are shorted, we have B ¼ 0 and
A¼
M
,
c44 I p
ð3:93Þ
which is the same as the solution from the theory of elasticity. There is no electric
field in the cylinder. However, Dz does exist, so the solution is not purely elastic.
If the end electrodes are open, we have Qe ¼ 0. From Eqs. (3.90) and (3.92) we
obtain
A¼
c44
M
,
2 ðb3 a3 Þ2
I p þ k215 2π
3 π ðb2 a2 Þ
ð3:94Þ
where
k215 ¼
e215
:
ε11 c15
ð3:95Þ
3.7 Azimuthal Shear of a Ceramic Cylinder
97
The terms in the square brackets of the denominator of the right-hand side of
Eq. (3.94) represents a piezoelectrically stiffened torsional rigidity. The open circuit
torsional rigidity in Eq. (3.94) is larger than the short-circuit torsional rigidity of
c44Ip in Eq. (3.93) without piezoelectric stiffening.
3.7
Azimuthal Shear of a Ceramic Cylinder
Consider an infinite circular cylinder of inner radius a and outer radius b (see
Fig. 3.8). The cylinder is made of ceramics with circumferential poling. We choose
(r,θ,z) to correspond to (2,3,1) so that the poling direction corresponds to 3. The
lateral cylindrical surfaces are unelectroded. The free space electric field is
neglected. r ¼ a is fixed. r ¼ b is under a shear stress Trθ ¼ τ.
The boundary-value problem consists of:
T ji, j ¼ 0, Di, i ¼ 0 in V,
T ij ¼ cijkl Skl ekij Ek , Di ¼ eikl Skl þ εik E k
Sij ¼ ui, j þ u j, i =2, E i ¼ φ, i in V,
in V,
ð3:96Þ
and
ui ¼ 0, r ¼ a,
T rr ¼ 0, T rθ ¼ τ, T rz ¼ 0,
Dr ¼ 0, r ¼ a, b:
r ¼ b,
ð3:97Þ
Consider the possibility of the following displacement and potential fields:
uθ ¼ uθ ðr Þ,
ur ¼ uz ¼ 0,
φ ¼ φðr Þ:
ð3:98Þ
The nontrivial components of strain, electric field, stress, and electric displacement are
S4 ¼ 2Srθ ¼
Fig. 3.8 A ceramic circular
cylinder under a shear stress
duθ uθ
,
dr
r
E2 ¼ Er ¼ 2
dφ
,
dr
ð3:99Þ
b
Fixed
3
a
P
Under a
shear
stress τ
98
3 Static Problems
duθ uθ
dφ
T 4 ¼ T rθ ¼ c44
þ e15 ,
dr
dr
r
duθ uθ
dφ
D2 ¼ Dr ¼ e15
ε11 :
dr
dr
r
ð3:100Þ
The stress components Trr and Trz vanish everywhere and on the lateral surfaces in
particular. The equation of motion and the charge equation to be satisfied are
dT rθ 2
þ T rθ ¼ 0,
r
dr
1
ðrDr Þ, r ¼ 0,
r
ð3:101Þ
C2
,
r
ð3:102Þ
which can be integrated to give
T rθ ¼
C1
,
r2
Dr ¼
where C1 and C2 are integration constants. Hence
duθ uθ
dφ C 1
þ e15 ¼ 2 ,
dr
dr
r
r
duθ uθ
dφ C 2
e15
ε11 ¼ :
dr
dr
r
r
c44
ð3:103Þ
For Dr to vanish at r ¼ a and/or b, we must have C2 ¼ 0, and hence Dr ¼ 0
everywhere. With C2 ¼ 0, we can eliminate φ from Eq. (3.103) and obtain a simple
equation for uθ. We are mainly interested in the stress distribution. For the shear
stress in Eq. (3.102) to satisfy the traction boundary condition at r ¼ b, we have
C 1 ¼ τ b2 :
ð3:104Þ
Therefore the only nonzero stress component is
T rθ ¼ τ
3.8
b2
:
r2
ð3:105Þ
Capacitance of a Ceramic Cylinder
Consider a hollow circular ceramic cylinder poled in the x3 direction with an inner
radius R1 and an outer radius R2 (see Fig. 3.9). The inner and outer surfaces are
electroded, with the electrodes shown by the thick lines in the Fig. A voltage V is
applied across the thickness of the cylinder. Mechanically the boundary surfaces are
3.8 Capacitance of a Ceramic Cylinder
99
x2
Fig. 3.9 A ceramic circular
cylinder under a voltage V
R2
R1
x1
j =0
j =V
Ceramic
either traction free or fixed. The problem is axisymmetric. It belongs to the antiplane
problems of Sect. 2.9. Therefore the notation for this section follows that in Sect. 2.9.
The boundary-value problem is:
∇2 u ¼ 0, ∇2 φ ¼ 0, R1 < r < R2 ,
φ ¼ 0, V, r ¼ R1 , R2 ,
T rz ¼ 0, r ¼ R1 , R2 , if the surfaces are traction-free,
or u ¼ 0, r ¼ R1 , R2 , if the surfaces are fixed:
ð3:106Þ
In polar coordinates, we have
2
2
∂ u 1 ∂u 1 ∂ u
þ
þ
¼ 0,
∂r 2 r ∂r r 2 ∂θ2
2
2
∂ φ 1 ∂φ 1 ∂ φ
þ
þ
¼ 0:
∂r 2 r ∂r r 2 ∂θ2
ð3:107Þ
For axisymmetric problems, Eq. (3.107) reduces to
d2 u 1du
¼ 0,
þ
dr 2 r dr
2
d φ 1dφ
¼ 0:
þ
dr 2 r dr
ð3:108Þ
The general solution is
u ¼ A1 ln r þ A2 ,
φ ¼ B1 ln r þ B2 ,
ð3:109Þ
where A1, A2, B1, and B2 are undetermined constants. The stress and electric
displacements are
100
3 Static Problems
1
T rz ¼ ðcA1 þ eB1 Þ ,
r
1
Dr ¼ ðeA1 ε B1 Þ :
r
ð3:110Þ
Consider traction-free surfaces below with the following boundary conditions:
1
1
¼ 0, ðcA1 þ eB1 Þ ¼ 0,
R1
R2
B1 ln R1 þ B2 ¼ 0, B1 ln R2 þ B2 ¼ V,
ð3:111Þ
e
V
,
cln ðR2 =R1 Þ
V
V
, B2 ¼ ln R1 :
B1 ¼
ln ðR2 =R1 Þ
ln ðR2 =R1 Þ
ð3:112Þ
ðcA1 þ eB1 Þ
which determine
A1 ¼ Hence
V
r
ln
φ¼
,
ln ðR2 =R1 Þ R1 e V
r
ln
þ C,
u¼
cln R2 =R1
R1
V
1
,
T rz ¼ 0, Dr ¼ ε
ln ðR2 =R1 Þ r
ð3:113Þ
where C is an arbitrary constant representing a rigid-body displacement
(corresponding to the A2 in Eq. (3.109)). The surface charge density on the electrode
at r ¼ R2 is given by
σ e ¼ Dr ðr ¼ R2 Þ ¼ ε
V
1
:
ln ðR2 =R1 Þ R2
ð3:114Þ
The capacitance per unit length of the cylinder is
2π R2 σ e
2πε
¼
:
ln ðR2 =R1 Þ
V
ð3:115Þ
Like in Sects. 3.4 and 3.5, Eq. (3.115) shows that the effect
ofpiezoelectric coupling
on the capacitance is on the order of k2 through ε ¼ ε 1 þ k 2 when the surfaces are
traction free. Let R2 ¼ R1 + 2 h. In the limit when R1 ! 1 and 2h is fixed, the
capacitance in Eq. (3.115) (when divided by 2πR2) approaches ε=ð2hÞ, the capacitance per unit area of a free plate given by Eq. (3.52).
3.9 Circular Hole Under Axisymmetric Loads
3.9
101
Circular Hole Under Axisymmetric Loads
Consider a circular hole of radius R in an unbounded domain of ceramics poled
along x3 (see Fig. 3.10). The hole surface at r ¼ R is electroded, with the electrode
shown by the thick line in the figure. On the hole surface we apply a shear stress
Trz ¼ τ. We consider the case that the electrode is not connected to other objects.
The surface free charge density on the electrode is fixed and is given to be σ e. The
problem is axisymmetric. It is an antiplane problem and the notation follows that
in Sect. 2.9.
The boundary-value problem consists of:
∇2 u ¼ 0,
∇2 φ ¼ 0,
r > R,
ð3:116Þ
r ¼ R,
r ! 1:
ð3:117Þ
and
T rz ¼ τ, Dr ¼ σ e ,
T rz ! 0, Dr ! 0,
Since the problem is axisymmetric, Eqs. (3.108) and (3.109) also apply here. The
general solution is
u ¼ A1 ln r þ A2 ,
φ ¼ B1 ln r þ B2 ,
ð3:118Þ
where A1, A2, B1, and B2 are undetermined constants. A2 and B2 represent a rigidbody displacement and a constant in the electric potential and are immaterial to the
problem we are considering. The stress and electric displacement are
1
T rz ¼ ðcA1 þ eB1 Þ ,
r
1
Dr ¼ ðeA1 εB1 Þ ,
r
ð3:119Þ
which satisfy the boundary conditions at infinity. On the hole surface,
Fig. 3.10 A circular hole in
polarized ceramics under
axisymmetric loads
x2
x
x
R
τ
x
x
x
x
x
Ceramic
x1
102
3 Static Problems
1
ðcA1 þ eB1 Þ ¼ τ,
R
1
ðeA1 εB1 Þ ¼ σ e ,
R
ð3:120Þ
1e
τ σ e R:
ε c
ð3:121Þ
which determines
A1 ¼
1
e
τ þ σ e R,
c
ε
B1 ¼
Hence
1
e
r
τ þ σ e R ln þ C1 ,
c ε
R
1
e
r
φ ¼ σ e þ τ R ln þ C 2 ,
c
R
ε
R
R
T rz ¼ τ , Dr ¼ σ e ,
r
r
u¼
ð3:122Þ
where C1 and C2 are arbitrary constants.
Now consider the limit of R ! 0. At the same time we let τ ! 1 and σ e ! 1 such
that
τ2πR ! F,
σ e 2πR ! Qe :
ð3:123Þ
Then
1 e
F þ Qe ln r þ C 01 ,
2π c
ε
1 e
F Qe ln r þ C 02 ,
φ¼
2πε c
F
Qe
, Dr ¼
:
T rz ¼
2π r
2π r
u¼
ð3:124Þ
Equation (3.124) describes the fields of a line force and a line charge at the origin.
Mathematically they represent the fundamental solution to the following problem:
e
c∇2 u þ F Qe δðxÞ ¼ 0,
ε
e
ε∇2 φ ¼ Qe F δðxÞ,
c
ð3:125Þ
where δ is the Dirac delta function. The stress and electric displacement in
Eq. (3.124) are unbounded when r ! 0. This is a typical failure of continuum
mechanics in problems of bodies under concentrated loads with a zero characteristic
dimension. Continuum mechanics is valid only when the characteristic dimension in
a problem is much larger than the microstructural characteristic length of matter.
Equation (3.124) is valid sufficiently far away from the origin. At a point very close
3.10
Circular Hole Under Shear
103
to the origin, the loads or sources can no longer be treated as line sources and their
size or distribution has to be considered.
3.10
Circular Hole Under Shear
Consider a circular cylindrical hole of radius R in an unbounded ceramic domain
poled along x3. The hole surface is electroded and the electrode is grounded. The
hole is under a uniform shear stress τ at x2 ¼ 1 (see Fig. 3.11). Electrically
x2 ¼ 1 may be shorted or open and we will consider the latter in the rest of this
section. This is an antiplane problem and our notation follows that in Sect. 2.9.
The boundary-value problem is:
∇2 u ¼ 0, ∇2 φ ¼ 0, r > R,
T rz ¼ 0, φ ¼ 0,
r ¼ R,
u, φ ! given far fields, r ! 1:
ð3:126Þ
First we need to determine the far fields in polar coordinates when r is very large. For
mechanical fields we have
T 23 ¼ cu, 2 þ eφ, 2 ¼ τ:
ð3:127Þ
When x2 ¼ 1 are electrically open, for large |x2|,
T23 = τ
Fig. 3.11 A circular hole in
polarized ceramics under
shear
x2
R
x1
j
=0
Ceramic
x
x
T23 = τ
x
x
104
3 Static Problems
D2 ¼ eu, 2 εφ, 2 ¼ 0:
ð3:128Þ
Based on Eqs. (3.127) and (3.128), we consider the case when the far fields are
given to be
τ
τ
u ¼ x2 þ C 1 ¼ r sin θ,
c
c
φ¼
eτ
eτ
x2 þ C2 ¼ r sin θ,
εc
εc
ð3:129Þ
where C1 and C2 are arbitrary constants and have been set to zero.
For problems periodic in θ, by separation of variables, the general solution for
u and φ satisfying the Laplace equations in Eq. (3.126) is [1]
u ¼ A0 ðP0 þ Q0 ln r Þ
1
X
þ ðAn cos nθ þ Bn sin nθÞðPn r n þ Qn r n Þ,
n¼1
φ ¼ C 0 ðR0 þ S0 ln r Þ
1
X
þ ðC n cos nθ þ Dn sin nθÞðRn r n þ Sn r n Þ,
ð3:130Þ
n¼1
where An, Bn, Cn, Dn, Pn Qn, Rn and Sn are undetermined constants.
In view of the above far-field solution and the general solution in Eqs. (3.129) and
(3.130), we look for a solution to the present problem governed by Eq. (3.126) in the
following form:
u¼
A2
A1 r þ
sin θ,
r
φ¼
B2
B1 r þ
sin θ,
r
ð3:131Þ
where A1, A2, B1, and B2 are undetermined constants. For Eq. (3.131) to match the
applied fields at infinity, we must have
τ
A1 ¼ ,
c
B1 ¼
eτ
:
ε c
ð3:132Þ
The stress and electric displacement components corresponding to Eq. (3.131) are
1
T rz ¼ ðcA1 þ eB1 Þ ðcA2 þ eB2 Þ 2 sin θ,
r 1
T θz ¼ ðcA1 þ eB1 Þ þ ðcA2 þ eB2 Þ 2 cos θ,
r
ð3:133Þ
3.11
Circular Cylinder in an Electric Field
105
1
Dr ¼ ðeA1 εB1 Þ ðeA2 ε B2 Þ 2 sin θ,
r 1
Dθ ¼ ðeA1 εB1 Þ þ ðeA2 ε B2 Þ 2 cos θ:
r
ð3:134Þ
At r ¼ R the boundary conditions require that
1
T rz ðRÞ ¼ ðcA1 þ eB1 Þ ðcA2 þ eB2 Þ 2 sin θ ¼ 0,
R
B2
φðRÞ ¼ B1 R þ
sin θ ¼ 0,
R
ð3:135Þ
which implies that
A2 ¼
1
1 þ 2k2 R2 τ,
c
B2 ¼ e 2
R τ:
cε
ð3:136Þ
Hence the displacement and potential fields are
eτ
R2
r
φ¼
sin θ,
cε
r
2
τ
2 R
u ¼ r þ 1 þ 2k
sin θ:
c
r
ð3:137Þ
Then the other fields can be calculated. In particular, the stress field is given by
R2 1
T rz ¼ τ 1 2 2 sin θ,
r r
R2
T θz ¼ τ 1 þ 2 cos θ:
r
ð3:138Þ
It can be seen that, on the hole surface where r ¼ R, Trz vanishes as dictated by the
boundary condition in Eq. (3.126) but Tθz ¼ 2τ cos θ. Obviously |Tθz| assumes its
maxima 2τ when θ ¼ 0 and θ ¼ π. This is the co-called stress concentration in the
theory of elasticity.
3.11
Circular Cylinder in an Electric Field
Consider an infinite ceramic circular cylinder of radius R in an electric field uniform
at infinity where E0 ¼ E0i1 (see Fig. 3.12). This is an antiplane problem and the
notation follows that in Sect. 2.9. The problem is symmetric about x2 ¼ 0 and is
antisymmetric about x1 ¼ 0.
106
3 Static Problems
x2
Fig. 3.12 A ceramic
circular cylinder in an
electric field
E0
R
x1
Ceramic
Free space
The boundary-value problem consists of
∇2 u ¼ 0,
∇2 φ ¼ 0,
∇2 φ ¼ 0,
r > R,
r < R,
ð3:139Þ
and
u and φ are bounded, r ¼ 0,
T rz ðr ¼ RÞ ¼ 0,
φðr ¼ R Þ ¼ φðr ¼ Rþ Þ,
Dr ðr ¼ R Þ ¼ Dr ðr ¼ Rþ Þ,
0
E ! E , r ! 1:
ð3:140Þ
For large r, the far fields are given to be
φ ¼ E 0 x1 ¼ E0 r cos θ,
E r ¼ E 0 cos θ, E θ ¼ E 0 sin θ:
ð3:141Þ
In view of the far-field solution and the general solution in Eqs. (3.141) and (3.130),
we look for solutions in the following form for the fields in the free space:
1
φ ¼ C 1 r þ C 2 cos θ,
r 1
E r ¼ C 1 C 2 2 cos θ,
r
1
E θ ¼ C1 þ C2 2 sin θ,
r
ð3:142Þ
where C1 and C2 are undetermined constants. For the electric field in Eq. (3.142) to
be equal to the applied field for large r, we must have
C 1 ¼ E 0 :
ð3:143Þ
3.11
Circular Cylinder in an Electric Field
107
Inside the cylinder we look for solutions in the following form according to
Eq. (3.130):
u¼
A2
A1 r þ
cos θ,
r
φ¼
B1 r þ
B2
cos θ,
r
ð3:144Þ
where A1, A2, B1, and B2 are undetermined constants. For the boundedness of u and φ
at the origin, we must have
A2 ¼ 0,
B2 ¼ 0:
ð3:145Þ
φ ¼ B1 r cos θ ¼ B1 x1 :
ð3:146Þ
Hence
u ¼ A1 r cos θ ¼ A1 x1 ,
Then the stress and electric displacement fields in the cylinder are
T rz ¼ ðcA1 þ eB1 Þ cos θ,
T θz ¼ ðcA1 þ eB1 Þ sin θ,
ð3:147Þ
Dr ¼ ðeA1 ε B1 Þ cos θ,
Dθ ¼ ðeA1 ε B1 Þ sin θ:
ð3:148Þ
At r ¼ R, the traction-free condition and the continuity of φ and Dr require that
T rz ðr ¼ RÞ ¼ ðcA1 þ eB1 Þ cos θ ¼ 0,
φðr ¼R Þ ¼ RB1 cos
θ
C2
¼ E 0 R þ
cos θ ¼ φðr ¼ Rþ Þ,
R
D r ðr ¼ R Þ ¼ ðeA1 ε B1 Þ cos θ
C2
¼ ε0 E 0 þ 2 cos θ ¼ Dr ðr ¼ Rþ Þ,
R
ð3:149Þ
which determines
C2 ¼
ε ε0 0 2
E R ,
ε þ ε0
B1 ¼ 2ε0 0
E ,
ε þ ε0
A1 ¼
e 2ε0 0
E :
c ε þ ε0
ð3:150Þ
Then the electric potential and field in the free space are given by
φ¼
ε ε 0 R2 0
r þ
E cos θ,
ε þ ε0 r
ð3:151Þ
108
3 Static Problems
ε ε0 R2 0
Er ¼ 1 þ
E cos θ,
ε þ ε0 r 2 ε ε0 R2
E θ ¼ 1 þ
E 0 sin θ:
ε þ ε0 r 2
ð3:152Þ
The fields inside the cylinder are
2ε0 0
E r cos θ,
εþε
e 2ε0 0
u¼
E r cos θ,
c ε þ ε0
φ¼
T θz ¼ 0,
2ε0 0
E cos θ,
Dr ¼ ε
ε þ ε0
2ε0 0
Dθ ¼ ε
E sin θ:
ε þ ε0
ð3:153Þ
T rz ¼ 0,
ð3:154Þ
We note that the strain, stress, electric field, and electric displacement inside the
cylinder are uniform. This may be unexpected but in fact uniform fields appear in
general in the so-called inclusion problems of the theory of elasticity when an
ellipsoid is imbedded in an unbounded domain of another material under uniform
far fields.
3.12
Surface Distribution of Electric Potential
Consider a ceramic half-space poled along the x3 direction as shown in Fig. 3.13.
The surface at x2 ¼ 0 is traction free and a periodic potential is applied. x2 ¼ + 1 is
mechanically fixed and electrically grounded. This is an antiplane problem and the
notation follows that in Sect. 2.9.
The boundary-value problem is:
∇2 u ¼ 0, ∇2 φ ¼ 0, x2 > 0,
T 23 ¼ 0, φ ¼ V cos ξ x1 , x2 ¼ 0,
u, φ ! 0, x2 ! þ1,
ð3:155Þ
where ξ is considered given and is assumed to be positive. Consider the possibility of
the following fields:
u ¼ Aexpðξx2 Þ cos ξ x1 ,
φ ¼ Bexpðξ x2 Þ cos ξ x1 ,
ð3:156Þ
3.13
Screw Dislocation
109
Fig. 3.13 A ceramic half
space under a surface
potential
Free space
Traction free, j =Vcosξ x1
x1
Ceramic half space
x2
which already satisfy the Laplace equations and the boundary conditions at infinity
in Eq. (3.155). For the boundary conditions at x2 ¼ 0 we need the following
expression:
T 23 ¼ cu, 2 þ eφ, 2 ¼ ðcAξ eBξÞexpðξx2 Þ cos ξx1 :
ð3:157Þ
The boundary conditions at x2 ¼ 0 require that
cðξÞA cos ξx1 þ eðξÞB cos ξx1 ¼ 0,
B cos ξx1 ¼ V cos ξx1 :
ð3:158Þ
Equation (3.158) determines
e
A ¼ V,
c
B ¼ V:
ð3:159Þ
Hence
e
u ¼ Vexpðξx2 Þ cos ξx1 ,
c
φ ¼ Vexpðξx2 Þ cos ξx1 :
ð3:160Þ
The displacement and potential fields in Eq. (3.160) decay exponentially from the
surface of the half space while varying trigonometrically along the surface.
3.13
Screw Dislocation
Consider a screw dislocation at θ ¼ π in a polar coordinate system in an otherwise
unbounded domain of ceramics poled along x3 (see Fig. 3.14). This is an antiplane
problem and the notation follows that in Sect. 2.9.
The boundary-value problem is:
110
3 Static Problems
x2
Fig. 3.14 A screw
dislocation in polarized
ceramics
r
θ
∇2 u ¼ 0, r > 0, π < θ < π,
∇2 φ ¼ 0, r > 0, π < θ < π,
uðr; π Þ uðr; π Þ ¼ δ,
φðr; π Þ φðr; π Þ ¼ V:
x1
ð3:161Þ
We look for a solution in the following form:
uðr; θÞ ¼ uðθÞ,
φðr; θÞ ¼ φðθÞ:
ð3:162Þ
The substitution of Eq. (3.162) into the Laplace equations in Eq. (3.161) gives:
2
2
2
∂ u 1 ∂u 1 ∂ u 1 ∂ u
þ
þ
¼
¼ 0,
∂r 2 r ∂r r 2 ∂θ2 r 2 ∂θ2
2
1∂ φ
∇2 φ ¼ 2 2 ¼ 0:
r ∂θ
∇2 u ¼
ð3:163Þ
The general solution is given by
u ¼ A1 θ þ A2 ,
φ ¼ B1 θ þ B2 ,
ð3:164Þ
where A1, A2, B1, and B2 are undetermined constants. From the boundary conditions
in Eq. (3.161), we have
A1 ¼
δ
,
2π
B1 ¼
V
:
2π
ð3:165Þ
Hence
u¼
δ
θ þ A2 ,
2π
φ¼
V
θ þ B2 ,
2π
ð3:166Þ
References
111
2Srz ¼ 0,
E r ¼ 0,
1δ
,
2Sθz ¼
r 2π
1V
,
Eθ ¼ r 2π
ð3:167Þ
T rz ¼ cu, r þ eφ, r ¼ 0,
1
1
δ
V
þe
,
T θz ¼ c u, θ þ e φ, θ ¼ c
r
r
2π r
2π r
ð3:168Þ
Dr ¼ eu, r εφ, r ¼ 0,
1
1
δ
V
ε
:
D θ ¼ e u, θ ε φ, θ ¼ e
r
r
2π r
2π r
ð3:169Þ
and
The singularity of the fields at the origin is an indication of the failure of continuum
mechanics in problems with a zero characteristic length. The solution is valid
sufficiently far away from the origin.
References
1. J.S. Yang, Antiplane Motions of Piezoceramics and Acoustic Wave Devices (World Scientific,
Singapore, 2010)
2. A.H. Meitzler, H.F. Tiersten, A.W. Warner, D. Berlincourt, G.A. Couqin, F.S. Welsh III, IEEE
Standard on Piezoelectricity (IEEE, New York, 1988)
3. R. D. Mindlin, An Introduction to the Mathematical Theory of Vibrations of Elastic Plates, ed. by
J. S. Yang (World Scientific, Singapore, 2006).
4. V.E. Bottom, Introduction to Quartz Crystal Unit Design (Van Nostrand Reinhold, New York,
1982)
Chapter 4
Waves in Unbounded Regions
This chapter is on waves in regions unbounded in at least one direction. These waves
can be propagating or stationary waves. They are nontrivial solutions of homogeneous differential equations and boundary conditions. Sections 4.2, 4.3, 4.4, 4.5, 4.6,
4.7, 4.8, 4.9, 4.10, 4.11, 4.12, and 4.13 are on antiplane problems of polarized
ceramics for which the notation in Sect. 2.9 is followed.
4.1
Plane Waves
First consider waves in an infinite region without a boundary. The waves are
governed by the following homogeneous equations only without boundary
conditions:
cijkl uk, lj þ ekij φ, kj ¼ ρ€ui ,
eikl uk, li εij φ, ij ¼ 0:
ð4:1Þ
In this section we are interested in the following plane wave:
uk ¼ Ak f ðn x vt Þ,
φ ¼ Bf ðn x vt Þ,
ð4:2Þ
where A, B, n, and v are constants, and f is an arbitrary function. n x vt is the
phase of the wave. v is the phase velocity. n x vt ¼ constant determines a wave
front which is a plane with a normal n. Differentiating Eq. (4.2), we obtain
uk, l ¼ Ak f 0 nl , uk, li ¼ Ak f 00 nl ni ,
φ, k ¼ Bf 0 nk , φ, ki ¼ Bf 00 nk ni :
€uk ¼ Ak f 00 v2 ,
© Springer Nature Switzerland AG 2018
J. Yang, An Introduction to the Theory of Piezoelectricity, Advances in Mechanics
and Mathematics 9, https://doi.org/10.1007/978-3-030-03137-4_4
ð4:3Þ
113
114
4 Waves in Unbounded Regions
The substitution of Eq. (4.3) into Eq. (4.2) yields the following linear homogeneous
equations for A and B:
cijkl Ak nl ni þ ekij Bnk ni ¼ ρv2 A j ,
eikl Ak nl ni εik Bnk ni ¼ 0:
ð4:4Þ
For nontrivial solutions of A and/or B, the determinant of the coefficient matrix of
Eq. (4.4) has to vanish. Equivalently, we proceed as follows. From Eq. (4.4)2:
B¼
eikl nl ni
Ak :
εpq np nq
ð4:5Þ
Substituting Eq. (4.5) into Eq. (4.4)1, we have
cijkl Ak nl ni þ ersj nr ns
eikl Ak nl ni
¼ ρv2 A j ,
εpq np nq
ð4:6Þ
which can be written as
Γ jk ρv2 δ jk Ak ¼ 0,
ð4:7Þ
where
Γ jk ¼ cijkl nl ni þ
ersj nr ns eilk nl ni
¼ Γkj
εpq np nq
ð4:8Þ
is the (piezoelectrically stiffened) acoustic tensor or Christoffel tensor. Equation
(4.7) is an eigenvalue problem of the acoustic tensor. For nontrivial solutions of A,
we must have
det Γ jk ρv2 δ jk ¼ 0,
ð4:9Þ
which is a polynomial equation of degree three for v2. Since the acoustic tensor is
real and symmetric, there exist three real eigenvalues:
vð1Þ , vð2Þ , vð3Þ ,
ð4:10Þ
and three corresponding eigenvectors that are orthogonal:
ð1Þ
ð2Þ
ð3Þ
Aj , Aj , Aj :
ð4:11Þ
Graphically, given a propagation direction n, there exist three plane waves with
velocities v(1), v(2), and v(3). Their displacement vectors are perpendicular to each
4.1 Plane Waves
115
Fig. 4.1 Plane waves
propagating in the direction
of n
u(1)
n
u(3)
u(2)
other as shown in Fig. 4.1. In anisotropic materials, usually one of the waves, e.g.,
the one with u(1), is roughly aligned with n. This is called the quasi-longitudinal
wave. The other two waves with u(2) and u(3) have their displacement vectors
roughly perpendicular to n and are called quasi-transverse waves. In materials
with high symmetries, for example isotropic materials, one wave is exactly longitudinal and the other two are exactly transverse.
As an example, consider plane waves propagating in the x3 direction of polarized
ceramics. Equation (2.90) directly reduces to
c44 u1, 33 ¼ ρ€u1 ,
c44 u2, 33 ¼ ρ€u2 ,
c33 u3, 33 þ e33 φ, 33 ¼ ρ€u3 ,
e31 u3, 33 ε33 φ, 33 ¼ 0:
ð4:12Þ
Equations (4.12)1,2 show two shear or transverse waves that are not coupled to the
electric field. We are interested in the extensional or longitudinal wave in Eq. (4.12)3
which is electrically coupled. Eliminating the electric potential from Eq. (4.12)3,4,
we obtain
c33 u3, 33 ¼ ρ€u3 ,
ð4:13Þ
e2
c33 ¼ c33 1 þ 33
¼ c33 1 þ k 233 ,
ε33 c33
ð4:14Þ
where
and the definition of k233 in Eq. (3.22) has been used. c33 is a piezoelectrically
stiffened elastic constant. Equation (4.13) is the standard wave equation treated in
a typical textbook
pffiffiffiffiffiffiffiffiffiffiffiof graduate mathematics. It describes extensional waves with a
velocity of c33 =ρ.
For another example, consider plane waves propagating in the x2 direction of
rotated Y-cut quartz. Equations (2.108) and (2.109) directly reduce to
116
4 Waves in Unbounded Regions
c66 u1, 22 þ e26 φ, 22 ¼ ρ€u1 ,
c22 u2, 22 þ c24 u3, 22 ¼ ρ€u2 ,
c24 u2, 22 þ c44 u3, 22 ¼ ρ€u3 ,
e26 u1, 22 ε22 φ, 22 ¼ 0:
ð4:15Þ
Equations (4.15)2,3 show a shear wave and an extensional wave coupled together but
are not coupled to the electric field. We are interested in the shear wave in Eq. (4.15)1
which is electrically coupled. Eliminating the electric potential from Eq. (4.15)1,4,
we obtain
c66 u1, 22 ¼ ρ€u1 ,
ð4:16Þ
e226
c66 ¼ c66 1 þ
¼ c66 1 þ k 226 ,
ε22 c66
ð4:17Þ
where
and the definition of k226 in Eq. (3.67) has been used. c26 is a piezoelectrically stiffened
shear
elastic constant. Equation (4.16) describes shear waves with a velocity of
p
ffiffiffiffiffiffiffiffiffiffiffi
c66 =ρ.
For the antiplane problem of polarized ceramics in Sect. 2.9, wave motions are
governed by Eq. (2.99)
v2T ∇2 u ¼ €u,
∇2 ψ ¼ 0:
ð4:18Þ
u ¼ Af ðn x vt Þ,
ψ ¼ Bf ðn x vt Þ:
ð4:19Þ
For plane waves we look for
Differentiating Eq. (4.19), we obtain
u, l ¼ Af 0 nl , u, ll ¼ Af 00 nl nl ¼ Af 00 , €u ¼ Af 00 v2 ,
ψ , k ¼ Bf 0 nk , ψ , kk ¼ Bf 00 nk nk ¼ Bf 00 :
ð4:20Þ
The substitution of Eq. (4.20) into Eq. (4.18) yields the following linear equations
for A and B:
v2T Af 00 ¼ v2 Af 00 ,
Bf 00 ¼ 0:
ð4:21Þ
4.2 Reflection and Refraction
117
For nontrivial solutions of A and/or B, the determinant of the coefficient matrix of
Eq. (4.21) has to vanish:
v2 f 00 v2 f 00
T
0
0 ¼ 0:
f 00 ð4:22Þ
00
We are interested in the case when f 6¼ 0. Equation (4.22) implies that B¼0 and
v ¼ vT :
ð4:23Þ
u ¼ Af ðn x vT t Þ ¼ Af ðn1 x1 þ n2 x2 vT t Þ,
ψ ¼ 0:
ð4:24Þ
Then the plane waves take the form
The corresponding electric potential and electric displacement are
e
φ ¼ u,
ε
D1 ¼ D2 ¼ 0:
ð4:25Þ
For harmonic waves, we use the usual complex notation with the real parts
representing the physical fields of interest and write
u ¼ Aexp½iξðn1 x1 þ n2 x2 vT t Þ,
ψ ¼ 0,
e
φ ¼ u:
ε
ð4:26Þ
Ceramics poled along x3 are isotropic in the x1-x2 plane. Waves propagating in all
directions have the same behavior. For waves propagating along x1, Eq. (4.18)1
reduces to
v2T u, 11 ¼ €u,
ð4:27Þ
pffiffiffiffiffiffiffi
which describes shear waves with a velocity vT ¼ c=ρ where c is a piezoelectrically stiffened shear elastic constant (see Eq. (2.100)).
4.2
Reflection and Refraction
Consider the reflection of plane waves at the boundary of a semi-infinite domain of
polarized ceramics first (see Fig. 4.2). The boundary is traction free and is
unelectroded. The electric field in the free space is neglected. For a semi-infinite
118
4 Waves in Unbounded Regions
Fig. 4.2 Incident and
reflected waves at a plane
boundary
x2
Reflected
wave
Incident
wave
b
a
x1
Ceramic
Free space
T2 j = 0, D2 = 0
medium, a wave solution needs to satisfy boundary conditions in addition to the
differential equations in Eq. (4.18).
The incident and reflected waves together must satisfy the following equations
and boundary conditions:
v2T ∇2 u ¼ u€ , ∇2 ψ ¼ 0, x2 > 0,
T 2 j ¼ 0, D2 ¼ 0, x2 ¼ 0:
ð4:28Þ
From Eq. (4.26) the incident wave can be written as
u ¼ Aexp½iξðx1 sin α x2 cos α vT t Þ,
ψ ¼ 0,
ð4:29Þ
where, comparing Eq. (4.29) with Eq. (4.26), we identify
n1 ¼ sin α,
n2 ¼ cos α:
ð4:30Þ
The incident wave represented by Eq. (4.29) is considered known. It already satisfies
the differential equations in Eq. (4.28). The electric displacement and the stress
component needed for the boundary conditions are
D2 ¼ 0,
T 23 ¼ cu, 2 þ eψ , 2
¼ cðξ cos αÞAexp½iξðx1 sin α x2 cos α vT t Þ:
ð4:31Þ
Similarly, we write the reflected wave as
u ¼ Bexp½iξðx1 sin β þ x2 cos β vT t Þ,
ψ ¼ 0,
ð4:32Þ
which satisfies the differential equations in Eq. (4.28) and is considered unknown.
For boundary conditions, we need
4.2 Reflection and Refraction
119
D2 ¼ 0,
T 23 ¼ cðiξ cos βÞBexp½iξðx1 sin β þ x2 cos β vT t Þ:
ð4:33Þ
The incident and reflected waves already satisfy the governing equations individually and so does their sum. At the boundary, the sum of the incident and reflected
waves together has to satisfy the boundary conditions in Eq. (4.28). D2 ¼ 0 is
trivially satisfied. We are left with
cðiξ cos αÞAexp½iξðx1 sin α vT t Þ
þ cðiξ cos βÞBexp½iξðx1 sin β vT t Þ ¼ 0,
ð4:34Þ
for any x1 and any t. This implies that
β ¼ α,
B ¼ A,
ð4:35Þ
and thus determines the reflected wave. At the boundary surface, the total traction is
zero. The total displacement is
u ¼ Aexp½iξðx1 sin α vT t Þ þ Bexp½iξðx1 sin β vT t Þ
¼ 2Aexp½iξðx1 sin α vT t Þ:
ð4:36Þ
Thus at a traction-free boundary the total displacement is twice that of the
incident wave.
Next consider the reflection and refraction at the interface between two semiinfinite media (see Fig. 4.3).
The incident and reflected waves together must satisfy the following equations:
cA ∇2 u ¼ ρA €u,
∇2 ψ ¼ 0,
x2 > 0:
ð4:37Þ
∇2 ψ ¼ 0,
x2 < 0:
ð4:38Þ
The refracted wave must satisfy
cB ∇2 u ¼ ρB €u,
The interface continuity conditions are
uðx2 ¼ 0þ Þ ¼ uðx2 ¼ 0 Þ,
T 23 ðx2 ¼ 0þ Þ ¼ T 23 ðx2 ¼ 0 Þ,
φðx2 ¼ 0þ Þ ¼ φðx2 ¼ 0 Þ,
D2 ðx2 ¼ 0þ Þ ¼ D2 ðx2 ¼ 0 Þ:
The incident wave can be written as
ð4:39Þ
120
4 Waves in Unbounded Regions
Fig. 4.3 Incident, reflected,
and refracted waves at an
interface
x2
Reflected
wave
Incident
wave
a
b
Ceramic A
x1
Ceramic B
g
u ¼ Aexp½iξA ðx1 sin α x2 cos α vA t Þ,
eA
cA
ψ ¼ 0, φ ¼ u, v2A ¼ , D2 ¼ 0,
εA
ρA
T 23 ¼ cA ðiξA cos αÞAexp½iξA ðx1 sin α x2 cos α vA t Þ,
Refracted
wave
ð4:40Þ
which is considered known. The reflected wave can be written as
u ¼ Bexp½iξA ðx1 sin β þ x2 cos β vA t Þ,
eA
ψ ¼ 0, φ ¼ u, D2 ¼ 0,
εA
T 23 ¼ cA ðiξA cos βÞBexp½iξA ðx1 sin β þ x2 cos β vA t Þ,
ð4:41Þ
which is considered unknown. We write the refracted wave as
u ¼ Cexp½iξB ðx1 sin γ x2 cos γ vB t Þ,
eB
cB
ψ ¼ 0, φ ¼ u, v2B ¼ , D2 ¼ 0,
εB
ρB
T 23 ¼ cB ðiξB cos γ ÞCexp½iξB ðx1 sin γ x2 cos γ vB t Þ,
ð4:42Þ
which is also unknown. For simplicity consider a special case when
eA eB
ffi :
εA εB
ð4:43Þ
An exact treatment of an interface without using the approximation in Eq. (4.43)
will be given later in Sect. 4.8. When Eq. (4.43) is approximately true, Eq. (4.39)3
becomes the same as Eq. (4.39)1. Equations (4.37), (4.38), and (4.39)4 are already
satisfied. Substituting Eqs. (4.40), (4.41), and (4.42) into Eq. (4.39)1,2, we have
4.2 Reflection and Refraction
121
Aexp½iξA ðx1 sin α vA t Þ þ Bexp½iξA ðx1 sin β vA t Þ
¼ Cexp½iξB ðx1 sin γ vB t Þ,
cA ðiξA cos αÞAexp½iξA ðx1 sin α vA t Þ
þ cA ðiξA cos βÞBexp½iξA ðx1 sin β vA t Þ
¼ cB ðiξB cos γ ÞCexp½iξB ðx1 sin γ vB t Þ:
ð4:44Þ
Equation (4.44) can be satisfied if the following two sets of equations are satisfied:
exp½iξA ðx1 sin α vA t Þ
¼ exp½iξA ðx1 sin β vA t Þ
¼ exp½iξB ðx1 sin γ vB t Þ,
ð4:45Þ
A þ B ¼ C,
cA ξA A cos α cA ξA B cos β ¼ cB ξB C cos γ:
ð4:46Þ
and
Equations (4.45) and (4.46) are analyzed separately below. For Eq. (4.45) to be true
for all t and all x1, we have
ξ A vA ¼ ξ B v B ,
ξA sin α ¼ ξA sin β ¼ ξB sin γ:
ð4:47Þ
Hence ξB is determined from
rffiffiffiffiffiffiffiffiffiffi
ρB cA
,
ρA cB
ð4:48Þ
sin γ ξA vB
¼ ¼ ,
sin α ξB vA
ð4:49Þ
ξ B vA
¼ ¼
ξ A vB
and β and γ are given by
β ¼ α,
which is Snell’s law. Then from Eq. (4.46) we solve for B and C:
B ρA sin 2α ρB sin 2γ
¼
,
A ρA sin 2α þ ρB sin 2γ
C
2ρA sin 2α
¼
:
A ρA sin 2α þ ρB sin 2γ
ð4:50Þ
Thus the reflected and refracted waves are fully determined. As a special case,
consider normal incidence with α¼0. From Eq. (4.49) β ¼ γ¼0. Equation (4.50)
implies that
122
4 Waves in Unbounded Regions
B ρA v A ρB v B
¼
,
A ρA v A þ ρB v B
C
2ρA vA
¼
,
A ρ A v A þ ρB v B
ð4:51Þ
where ρAvA or ρBvB is called the acoustic impedance. When ρAvA ¼ ρBvB, the incident
wave does not feel the interface and is not reflected, and total transmission occurs.
4.3
Surface Waves on a Ceramic Half Space
This section is on waves propagating near the surface of a piezoelectric half space.
These waves have been used extensively to make surface acoustic wave (SAW)
devices. As an introductory book, we discuss the well-known and relatively simple
Bleustein–Gulyaev wave only [1, 2]. Some Japanese researchers also contributed to
the discovery of the wave (see the review in [3]). Consider a ceramic half space as
shown in Fig. 4.4. The surface at x2 ¼ 0 is traction free and may be electroded or
unelectroded.
The governing equations for the ceramic half space are
v2T ∇2 u ¼ €u, x2 > 0,
∇2 ψ ¼ 0, x2 > 0,
e
φ ¼ ψ þ u, x2 > 0:
ε
ð4:52Þ
Consider the possibility of the following solution:
u ¼ Aexpðξ2 x2 Þ cos ðξ1 x1 ωt Þ,
ψ ¼ Bexpðξ1 x2 Þ cos ðξ1 x1 ωt Þ,
ð4:53Þ
T 23 ¼h ½cAξ2 expðξ2 x2 Þ þ eBξ1 expiðξ1 x2 Þ cos ðξ1 x1 ωt Þ,
e
φ ¼ Bexpðξ1 x2 Þ þ Aexpðξ2 x2 Þ cos ðξ1 x1 ωt Þ,
ε
D2 ¼ εξ1 Bexpðξ1 x2 Þ cos ðξ1 x1 ωt Þ,
ð4:54Þ
where A and B are undetermined constants, and ξ2 should be positive for decaying
behavior away from the surface. Equation (4.53)2 already satisfies Eq. (4.52)2. For
Eq. (4.53)1 to satisfy Eq. (4.52)1 we must have
Fig. 4.4 A ceramic half
space
Free space
x1
Ceramic
x2
Propagation
direction
4.3 Surface Waves on a Ceramic Half Space
123
c ξ21 ξ22 ¼ ρω2 ,
ð4:55Þ
which determines
ξ22 ¼ ξ21 ρω2
v2
¼ ξ21 1 2 > 0,
c
vT
v2 ¼
ω2
:
ξ21
ð4:56Þ
First we consider the case when the surface of the half space is electroded and the
electrode is grounded. The corresponding boundary conditions are
T 23 ¼ 0, x2 ¼ 0,
φ ¼ 0, x2 ¼ 0,
u, φ ! 0, x2 ! þ1,
ð4:57Þ
T 23 ¼ cu, 2 þ eψ , 2 ¼ 0, x2 ¼ 0,
e
ψ þ u ¼ 0, x2 ¼ 0,
ε
u, ψ ! 0, x2 ! þ1:
ð4:58Þ
or, in terms of u and ψ,
Substituting the relevant expressions in Eqs. (4.53) and (4.54) into Eq. (4.58)1,2:
cAξ2 þ eBξ1 ¼ 0,
e
A þ B ¼ 0:
ε
ð4:59Þ
e2
eξ1 ¼ cξ2 ξ1 ¼ 0,
1
ε
ð4:60Þ
For nontrivial solutions,
cξ2
e=ε
or
ξ2 ¼ k2 ξ1 ,
ð4:61Þ
e2
k2 ¼ :
ε c
ð4:62Þ
where
The substitution of Eq. (4.55) into Eq. (4.61) yields
124
4 Waves in Unbounded Regions
c ξ21 k4 ξ21 ¼ ρω2 ,
ð4:63Þ
from which the surface wave velocity can be determined as
v2 ¼
ω2 c ¼ 1 k4 ¼ v2T 1 k4 < v2T :
2
ξ1 ρ
ð4:64Þ
When k ¼ 0, we have ξ2 ¼ 0 and the wave is no longer a surface wave. Hence the
wave is truly piezoelectric in the sense that the existence of the wave relies on
piezoelectric coupling. It does not have an elastic counterpart.
If the surface of the half space is unelectroded, electric fields can also exist in the
b , we have
free space of x2 < 0. Denoting the electric potential in the free space by φ
b ¼ 0,
∇2 φ
x2 < 0:
ð4:65Þ
The boundary and continuity conditions are
b 2,
b , D2 ¼ D
T 23 ¼ 0, φ ¼ φ
u3 , φ ! 0, x2 ! þ1,
b ! 0, x2 ! 1,
φ
x2 ¼ 0,
ð4:66Þ
b,
or, in terms of u, ψ, and φ
T 23 ¼ cu3, 2 þ eψ , 2 ¼ 0, x2 ¼ 0,
e
b , x2 ¼ 0,
ψþ u¼φ
ε
b , 2 , x2 ¼ 0,
εψ , 2 ¼ ε0 φ
u, ψ ! 0, x2 ! þ1,
b ! 0, x2 ! 1:
φ
ð4:67Þ
From Eq. (4.65), in the free space,
b ¼ Cexpðξ1 x2 Þ cos ðξ1 x1 ωt Þ,
φ
b , 2 ¼ ε0 ξ1 Cexpðξ1 x2 Þ cos ðξ1 x1 ωt Þ:
D2 ¼ ε0 φ
ð4:68Þ
In Eq. (4.68), C is an undetermined constant. Substituting the relevant expressions in
Eqs. (4.53), (4.54), and (4.68) into Eq. (4.67)1,2,3:
cðAξ2 Þ þ eðBξ1 Þ ¼ 0,
εðξ1 BÞ ¼ ε0 Cξ1 ,
e
A þ B ¼ C:
ε
ð4:69Þ
4.4 Ceramic Half Space with a Mass Layer
125
For nontrivial solutions,
cξ2
0
e=ε
eξ1
ε
1
0 e2
cξ2 ε þ ε0 ξ1 cξ2 ε0 ¼ 0,
ε0 ¼ ε
1 ð4:70Þ
or
ξ2 ¼ k2 ξ1
1
:
1 þ ε=ε0
ð4:71Þ
The substitution of Eq. (4.55) into Eq. (4.71) yields the surface wave velocity as
"
#
"
#
2
4
4
ω
k
k
c
1
v2 ¼ 2 ¼
¼ v2T 1 < v2T :
ξ1 ρ
ð1 þ ε=ε0 Þ2
ð1 þ ε=ε0 Þ2
ð4:72Þ
We point out that if the electric field in the free space is neglected and D2¼0 is
prescribed as a boundary condition on the surface of the half space, no surface wave
solution can be obtained.
4.4
Ceramic Half Space with a Mass Layer
In this section we study the effect of a very thin layer of another material on the
propagation of the Bleustein–Gulyaev surface wave in the previous section (see
Fig. 4.5). This problem is the theoretical foundation for mass sensors based on
surface waves. The layer is assumed to be very thin compared to the wavelength we
are considering. Only the inertial effect of the layer is considered. Its stiffness is
neglected. We consider the case when the ceramic half space is with a grounded
electrode only. The case of an unelectroded half space can be found in [4].
The governing equations for the fields in the ceramic half space are
v2T ∇2 u ¼ ρ€u, x2 > 0,
∇2 ψ ¼ 0, x2 > 0,
e
φ ¼ ψ þ u, x2 > 0:
ε
ð4:73Þ
The fields in the ceramic half space are taken from Eqs. (4.53) and (4.54):
u ¼ Aexpðξ2 x2 Þ cos ðξ1 x1 ωt Þ,
ψ ¼ Bexpðξ1 x2 Þ cos ðξ1 x1 ωt Þ,
ð4:74Þ
126
4 Waves in Unbounded Regions
Fig. 4.5 A ceramic half
space with a thin mass layer
Free space
Mass layer
x1
Ceramic
Propagation
direction
x2
T 23 ¼h ½cAξ2 expðξ2 x2 Þ þ eBξ1 expiðξ1 x2 Þ cos ðξ1 x1 ωt Þ,
e
φ ¼ Bexpðξ1 x2 Þ þ Aexpðξ2 x2 Þ cos ðξ1 x1 ωt Þ,
ε
D2 ¼ εξ1 Bexpðξ1 x2 Þ cos ðξ1 x1 ωt Þ,
ð4:75Þ
where A and B are undetermined constants, and
c ξ21 ξ22 ¼ ρω2 ,
ð4:76Þ
which determines
ξ22
¼
ξ21
ρω2
v2
2
¼ ξ1 1 2 > 0,
c
vT
v2 ¼
ω2
:
ξ21
ð4:77Þ
ξ2 should be positive for decaying behavior away from the surface. Equation (4.74)
satisfies Eq. (4.73)1,2.
In the case when the surface of the half space is electroded and the electrode is
grounded, the boundary conditions are
T 23 ¼ ρ0 h0 €u, x2 ¼ 0,
φ ¼ 0, x2 ¼ 0,
0
ð4:78Þ
0
where ρ and h are the density and thickness of the layer, respectively. In terms of
u and ψ, Eq. (4.78) becomes
T 23 ¼ cu, 2 þ eψ , 2 ¼ ρ0 h0 €u,
e
φ ¼ ψ þ u ¼ 0, x2 ¼ 0:
ε
x2 ¼ 0,
ð4:79Þ
Substituting Eqs. (4.74) and (4.75) into Eq. (4.79), we obtain
cAξ2 þ eBξ1 ¼ ρ0 h0 ω2 A,
e
A þ B ¼ 0:
ε
ð4:80Þ
4.5 Ceramic Half Space with a Fluid
127
For nontrivial solutions,
cξ ρ0 h0 ω2
2
e=ε
e2
eξ1 ¼ cξ2 ρ0 h0 ω2 ξ1 ¼ 0,
1
ε
ð4:81Þ
or
ξ2 ρ0 h0 ω2 2
¼ k ξ1 ,
c
e2
k2 ¼ :
ε c
ð4:82Þ
The substitution of Eq. (4.77) into Eq. (4.82) yields
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ρω2 ρ0 h0 ω2 2
ξ21 ¼ k ξ1 ,
c
c
ð4:83Þ
or, in terms of the wave velocity v ¼ ω/ξ1,
sffiffiffiffiffiffiffiffiffiffiffiffiffi
v2
ρ0 v 2
1 2 ¼ k2 þ ξ1 h0 2 ,
ρ vT
vT
ð4:84Þ
which determines the surface velocity v. In the special case when the mass layer is
0 0
not present, that is, ρ h ¼ 0, Eq. (4.84) reduces to
sffiffiffiffiffiffiffiffiffiffiffiffiffi
v2
1 2 ¼ k2 ,
vT
ð4:85Þ
which is the same as Eq. (4.64) as expected. Equation (4.84) shows that the mass
layer inertia lowers the wave velocity. The difference between the wave velocities in
0 0
Eqs. (4.84) and (4.85) can be used to make mass sensors for measuring ρ h . Notice
that Eq. (4.84) determines a dispersive wave while Eq. (4.85) determines a
nondispersive wave.
4.5
Ceramic Half Space with a Fluid
In this section we study the effect of a linear viscous fluid on the propagation of the
Bleustein–Gulyaev surface wave in Sect. 4.3 (see Fig. 4.6). This problem is the
theoretical foundation for fluid sensors based on surface waves. We consider the case
when the ceramic half space is with a grounded electrode only. The case of an
unelectroded half space can be found in [4].
128
4 Waves in Unbounded Regions
Fig. 4.6 A ceramic half
space in contact with a fluid
Fluid
x1
Ceramic
Propagation
direction
x2
For the ceramic half space the governing equations are
v2T ∇2 u ¼ €u, x2 > 0,
∇2 ψ ¼ 0, x2 > 0,
e
φ ¼ ψ þ u, x2 > 0:
ε
ð4:86Þ
u ¼ Aexpðξ2 x2 Þexp½iðξ1 x1 ωt Þ,
ψ ¼ Bexpðξ1 x2 Þexp½iðξ1 x1 ωt Þ,
ð4:87Þ
The relevant fields are
T 23 ¼h ½cAξ2 expðξ2 x2 Þ þ eBξ1 expiðξ1 x2 Þexp½iðξ1 x1 ωt Þ,
e
φ ¼ Bexpðξ1 x2 Þ þ Aexpðξ2 x2 Þ exp½iðξ1 x1 ωt Þ,
ε
D2 ¼ εξ1 Bexpðξ1 x2 Þexp½iðξ1 x1 ωt Þ,
ð4:88Þ
where the complex notation has been used. A and B are undetermined constants. ξ2
should have a positive real part for decaying behavior away from the surface.
Equation (4.87)2 already satisfies Eq. (4.86)2. For Eq. (4.87)1 to satisfy Eq. (4.86)1
we must have
c ξ21 ξ22 ¼ ρω2 ,
ð4:89Þ
which determines
ξ22
¼
ξ21
ρω2
v2
2
¼ ξ1 1 2 ,
c
vT
v2 ¼
ω2
:
ξ21
ð4:90Þ
For the fluid, let v3(x1, x2, t) be the velocity field. The governing equation for v3
and the relevant stress component for the interface continuity condition are
∂v3
,
μðv3, 11 þ v3, 22 Þ ¼ ρ0
∂t
T 23 ¼ μv3, 2 ,
ð4:91Þ
4.5 Ceramic Half Space with a Fluid
129
0
where μ is the viscosity of the fluid and ρ is the fluid density. The following fields
are allowed by Eq. (4.91):
v3 ¼ Cexpðλx2 Þexp½iðξ1 x1 ωt Þ,
T 23 ¼ μλCexpðλx2 Þexp½iðξ1 x1 ωt Þ,
ð4:92Þ
where C is an arbitrary constant, λ has a positive real part and is determined by
μ λ2 ξ21 ¼ ρ0 iω:
ð4:93Þ
When the surface of the half space is electroded and the electrode is grounded, the
boundary and continuity conditions at the interface are
u_ ðx2 ¼ 0þ Þ ¼ v3 ðx2 ¼ 0 Þ,
T 23 ðx2 ¼ 0þ Þ ¼ T 23 ðx2 ¼ 0 Þ,
φ ¼ 0, x2 ¼ 0:
ð4:94Þ
Substituting Eqs. (4.87), (4.88), and (4.92) into Eq. (4.94), we have
iωA
¼ C,
cAξ2 þ eBξ1 ¼ μλC,
e
A þ B ¼ 0:
ε
ð4:95Þ
For nontrivial solutions,
iω
cξ2
e=ε
0
eξ1
1
1 cξ
μλ ¼ 2
e=ε
0 iω
eξ1 μλ
e=ε
1
0 ¼ 0,
1
ð4:96Þ
or
iωμλ
,
ξ2 ¼ k2 ξ1 þ
c
ð4:97Þ
which determines the surface wave frequency or velocity. In terms of the wave
velocity v ¼ ω/ξ1, Eq. (4.97) becomes
sffiffiffiffiffiffiffiffiffiffiffiffiffi
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
v2 2 iμξ1 v
ρ0 v
1 2 ¼k þ
1i
:
c
μξ1
vT
When μ ¼ 0, Eq. (4.98) reduces to
ð4:98Þ
130
4 Waves in Unbounded Regions
sffiffiffiffiffiffiffiffiffiffiffiffiffi
v2
1 2 ¼ k2 ,
vT
ð4:99Þ
which is the same as Eq. (4.64) as expected. Equation (4.98) can be used to design fluid
0
sensors for measuring μ or ρ through the difference between Eqs. (4.98) and (4.99).
4.6
Interface Waves
Consider two ceramic half spaces as shown in Fig. 4.7. The ceramics are both poled
along the x3 directions. We are interested in the possibility of a wave traveling near
the interface between the two half spaces [5].
The governing equations are
cA ∇2 uA ¼ ρA €uA , x2 > 0,
∇2 ψ A ¼ 0, x2 > 0,
eA
φA ¼ ψ A þ uA , x2 > 0,
εA
ð4:100Þ
cB ∇2 uB ¼ ρB u€B , x2 < 0,
∇2 ψ B ¼ 0, x2 < 0,
eB
φB ¼ ψ B þ uB , x2 < 0,
εB
ð4:101Þ
where the subscripts are not tensor indices and they indicate the fields in ceramic
A and ceramic B, respectively. For an interface wave, we require that
uA , ψ A ! 0,
uB , ψ B ! 0,
x2 ! þ1,
x2 ! 1:
ð4:102Þ
For x2 > 0,, the solution to Eq. (4.100) that satisfies Eq. (4.102)1 can be written as
uA ¼ U A expðηA x2 Þ cos ðξx1 ωt Þ,
ψ A ¼ ΨA expðξx2 Þ cos ðξx1 ωt Þ,
ð4:103Þ
where
Fig. 4.7 Two ceramic half
spaces
Ceramic B
x1
Ceramic A
Propagation
direction
x2
4.6 Interface Waves
131
η2A
ρ A ω2
v2
2
¼ξ ¼ ξ 1 2 > 0,
cA
vA
2
ð4:104Þ
and
v¼
ω
,
ξ
v2A ¼
cA
:
ρA
ð4:105Þ
Similarly, for x2 < 0, the solution to Eq. (4.101) that satisfies Eq. (4.102)2 is given by
uB ¼ U B expðηB x2 Þ cos ðξx1 ωt Þ,
ψ B ¼ ΨB expðξx2 Þ cos ðξx1 ωt Þ,
ð4:106Þ
where
η2B ¼ ξ2 ρ B ω2
v2
¼ ξ2 1 2 > 0,
cB
vB
v2B ¼
cB
:
ρB
ð4:107Þ
ð4:108Þ
For interface continuity conditions, we consider two situations separately.
When the interface is a grounded electrode we have, at x2¼0,
uA ¼ uB , T A ¼ T B ,
eA
eB
ψ A þ uA ¼ 0, ψ B þ uB ¼ 0,
εA
εB
ð4:109Þ
where T ¼ T23. The substitution of Eqs. (4.103) and (4.106) into Eq. (4.109) results
in a system of linear homogenous equations for UA, ΨA, UB, and ΨB. For nontrivial
solutions, the determinant of the coefficient matrix of the equations has to vanish,
which yields
cA
v2
1 2
vA
1=2
1=2
v2
e2 e2
þ cB 1 2
¼ Aþ B:
εA εB
vB
ð4:110Þ
Equation (4.110) determines the velocity of the interface wave. As a special case,
when medium B is free space with
cB ¼ 0,
Eq. (4.110) reduces to
eB ¼ 0,
εB ¼ ε0 ,
ð4:111Þ
132
4 Waves in Unbounded Regions
"
v ¼
2
v2A
e2A
1
εA cA
2 #
ð4:112Þ
,
which is the velocity of the corresponding Bleustein–Gulyaev wave given by
Eq. (4.64).
When the interface is unelectroded, the continuity conditions at x2¼0 are
uA ¼ uB ,
T A ¼ T B,
eA
eB
ψ A þ uA ¼ ψ B þ uB ,
εA
εB
DA ¼ DB ,
ð4:113Þ
where D ¼ D2. Substituting Eqs. (4.103) and (4.106) into Eq. (4.113) and setting the
determinant of the coefficient matrix of the equations to zero, we obtain
1=2
1=2
v2
v2
ðeA =εA eB =εB Þ2
cA 1 2
þ cB 1 2
¼
:
1=εA þ 1=εB
vA
vB
ð4:114Þ
As a special case, when medium B is the free space described Eq. (4.111),
Eq. (4.114) reduces to
"
v ¼
2
v2A
e2A
1
εA cA ð1 þ ε0 =εA Þ
2 #
ð4:115Þ
,
which is the velocity of the corresponding Bleustein–Gulyaev wave given in
Eq. (4.72).
4.7
Waves in a Ceramic Plate
In this section we analyze propagating waves in a ceramic plate. Plate waves are
widely used to make bulk acoustic wave (BAW) devices. Consider an electroded
plate with shorted and grounded electrodes (see Fig. 4.8) [6]. We assume very thin
Fig. 4.8 A ceramic plate
with shorted electrodes
T23 = 0, φ=0
Ceramic
2h
T23 = 0, φ=0
x2
x1
4.7 Waves in a Ceramic Plate
133
electrodes. Their mechanical effects are negligible. Therefore the surfaces of the
plate are treated as traction free.
The governing equations are
c∇2 u ¼ ρ€u, h < x2 < h,
∇2 ψ ¼ 0, h < x2 < h,
e
φ ¼ ψ þ u, h < x2 < h:
ε
ð4:116Þ
The boundary conditions are
T 23 ¼ 0, x2 ¼ h,
φ ¼ 0, x2 ¼ h,
ð4:117Þ
cu, 2 þ eψ , 2 ¼ 0, x2 ¼ h,
e
ψ þ u ¼ 0, x2 ¼ h:
ε
ð4:118Þ
or, in terms of u and ψ,
The waves we are considering can be classified as symmetric and antisymmetric
ones. We discuss them separately below.
For antisymmetric waves we consider the possibility of
u ¼ A sin ξ2 x2 cos ðξ1 x1 ωt Þ,
ψ ¼ Bsinhξ1 x2 cos ðξ1 x1 ωt Þ,
ð4:119Þ
where A and B are constants. For Eq. (4.119) to satisfy Eq. (4.116), we must have
2
ρ ω2
v
ξ21 ¼ ξ21 2 1 ,
c
vT
2
ω
c
v2 ¼ 2 , v2T ¼ :
ρ
ξ1
ξ22 ¼
ð4:120Þ
The substitution of Eq. (4.119) into Eq. (4.118) leads to
cAξ2 cos ξ2 h þ eBξ1 coshξ1 h ¼ 0,
e
A sin ξ2 h þ Bsinhξ1 h ¼ 0,
ε
ð4:121Þ
which is a system of linear homogeneous equations for A and B. For nontrivial
solutions the determinant of the coefficient matrix must vanish. This yields
134
4 Waves in Unbounded Regions
tan ξ2 h ξ2 2
¼ k :
tanhξ1 h ξ1
ð4:122Þ
Equation (4.122) can be written as
1=2 2
1=2
tan π2 Ω2 Z 2
Ω Z2
¼
,
tanhπ2Z
k2 Z
ð4:123Þ
where the dimensionless frequency and the dimensionless wave number in the x1
direction are defined by
π 2 c
Ω ¼ω =
,
4ρh2
2
2
Z ¼ ξ1 =
π
2h
:
ð4:124Þ
In the limit when Z ! 0, Eq. (4.124) reduces to
π
πΩ
tan Ω ¼ 2 :
2
2k
ð4:125Þ
The frequencies determined by Eq. (4.125) are called the cutoff frequencies of the
waves.
For symmetric waves we consider
u ¼ A cos ξ2 x2 cos ðξ1 x1 ωt Þ,
ψ ¼ Bcoshξ1 x2 cos ðξ1 x1 ωt Þ:
ð4:126Þ
For Eq. (4.126) to satisfy Eq. (4.116), we still have Eq. (4.120). The substitution of
Eq. (4.126) into Eq. (4.118) leads to
cAξ2 sin ξ2 h þ eBξ1 sinhξ1 h ¼ 0,
e
A cos ξ2 h þ Bcoshξ1 h ¼ 0:
ε
ð4:127Þ
For nontrivial solutions of A and/or B, we must have
tan ξ2 h
ξ
¼ k2 1 ,
tanhξ1 h
ξ2
ð4:128Þ
1=2
tan π2 Ω2 Z 2
k2 Z
¼ 1=2 :
π
tanh2Z
Ω2 Z 2
ð4:129Þ
or
4.8 Waves Through the Joint of Two Plates
135
Fig. 4.9 Dispersion curves
of SH waves in a ceramic
plate. Solid lines: symmetric
waves. Dash lines:
antisymmetric waves
The first few branches of the dispersion curves determined from Eqs. (4.123) and
(4.129) are plotted in Fig. 4.9. Waves in plates are dispersive in general. The lowest
branch represents a wave whose displacement has no nodal points (zeros) along the
plate thickness and is called the face-shear wave (FS). Higher-order waves have
nodal points along the plate thickness and are called the thickness-twist waves (TT).
The intersections of the dispersion curves with the vertical frequency axis are the
cutoff frequencies bellow which the corresponding waves cannot propagate.
The case of an unelectroded plate was also treated in [6] with consideration of the
free-space electric field. The dispersion relations of antiplane waves are determined
by the following equations for antisymmetric and symmetric waves, respectively:
1=2 2
1=2
tan π2 Ω2 Z 2
Ω Z2
¼
,
tanhπ2Z þ εε0
k2 Z
1=2
1 þ εε0 tanhπ2Z tan π2 Ω2 Z 2
tanhπ2Z
4.8
¼ k2 Z
Ω2 Z 2
ð4:130Þ
1=2 :
ð4:131Þ
Waves Through the Joint of Two Plates
In this section we analyze the propagation of thickness-twist waves through the
joint between two semi-infinite piezoelectric plates of different ceramics (see
Fig. 4.10) [7]. The plate is unelectroded. The electric field in the free space is
136
4 Waves in Unbounded Regions
x2
Fig. 4.10 Two semiinfinite plates and their joint
x1
2h
PZT-5
PZT-6B
neglected. We need to analyze the two halves separately and then apply interface
continuity conditions.
The equations for u, ψ, and φ are:
c∇2 u ¼ ρ€u,
∇2 ψ ¼ 0,
e
φ ¼ ψ þ u:
ε
ð4:132Þ
At the plate surfaces we consider traction-free boundary conditions with
T 23 ¼ 0,
D2 ¼ 0,
x2 ¼ h,
x2 ¼ h,
ð4:133Þ
or, equivalently, in terms of u and ψ,
u, 2 ¼ 0, ψ , 2 ¼ 0, x2 ¼ h:
ð4:134Þ
The above equations are valid for both parts of the plate when the corresponding
material constants are used. At the joint, the continuity of u, T13, φ, and D1 need to be
imposed.
Consider the left half first with incident waves coming from x1 ¼ 1. The waves
may propagate through the joint and may also get reflected there. Expressions of
these waves can be obtained by separation of variables. It can be verified by direct
substitution that the solutions to Eqs. (4.132) and (4.134) can be classified into
waves symmetric or antisymmetric in x2, and they are given by:
u ¼ cos ξ2 x2 ½A1 expiðξ1 x1 Þ þ A2 expiðξ1 x1 Þexpðiωt Þ,
ψ ¼ cos ξ2 x2 Bexpðξ2 x1 Þexpðiωt Þ,
mπ
, m ¼ 0, 2, 4, . . . ,
ξ2 ¼
2h
and
ð4:135Þ
4.8 Waves Through the Joint of Two Plates
137
u ¼ sin ξ2 x2 ½A1 expiðξ1 x1 Þ þ A2 expiðξ1 x1 Þexpðiωt Þ,
ψ ¼ sin ξ2 x2 Bexpðξ2 x1 Þexpðiωt Þ,
mπ
, m ¼ 1, 3, 5, . . . ,
ξ2 ¼
2h
ð4:136Þ
respectively, where
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffirffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 c 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ρ ω2
ρ
ξ1 ¼
ξ22 ¼
ω2 ω2m ,
ω2 mπ
¼
2h
ρ vT
c
rcffiffiffi
:
mπ 2 c
c
2
vT ¼
, ωm ¼ 2h
ρ
ρ
ð4:137Þ
A1 is for the incident wave (right-traveling) and is considered known. A2 and B are
undetermined constants. A2 is for the reflected wave (left-traveling). B represents a
nonpropagating field localized near the joint. The localized behavior of B or ψ is
dictated by Eq. (4.132)2. ωm is the cutoff frequency of thickness-twist waves.
For the right half of the plate with x1 > 0, we use a prime to indicate its material
constants. Similar to Eqs. (4.135), (4.136), and (4.137), we have the following
symmetric and antisymmetric fields and waves (right-traveling) satisfying the
governing equations and the boundary conditions at x2 ¼ h:
u ¼ A0 cos ξ2 x2 exp i ξ01 x1 ωt ,
ψ ¼ B0 cos ξ2 x2 expðξ2 x1 Þexpðiωt Þ,
mπ
, m ¼ 0, 2, 4, . . . ,
ξ2 ¼
2h
ð4:138Þ
u ¼ A0 sin ξ2 x2 exp i ξ01 x1 ωt ,
ψ ¼ B0 sin ξ2 x2 expðξ2 x1 Þexpðiωt Þ,
mπ
, m ¼ 1, 3, 5, . . . ,
ξ2 ¼
2h
ð4:139Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ffi
ρ 0 ω2
1
2
2
¼
ξ2 ¼ 0 ω ω0m ,
0
vT
sffiffiffiffic
0
0 mπ 2 c0
c
,0
ωm ¼ 2h 0 :
v0T ¼
ρ
ρ
ð4:140Þ
and
where
ξ01
138
4 Waves in Unbounded Regions
At the joint, for symmetric waves, the continuity of u, T13, φ and D1 requires that
A1 þ A2 ¼ A0 ,
cðA1 iξ1 A2 iξ1 Þ þ eBξ2 ¼ c0 A0 iξ01 þ e0 B0 ðξ2 Þ,
e
e0
B þ ðA1 þ A2 Þ ¼ B0 þ 0 A0 ,
ε
ε
εBξ2 ¼ ε0 B0 ðξ2 Þ:
ð4:141Þ
Equation (4.141) can be solved symbolically, which yields:
A2
A1
A0
A1
B
A1
B0
A1
h
i
2
¼ c0 ε0 εξ01 ðε0 þ εÞ þ cε0 εξ1 ðε0 þ εÞ þ iξ2 ðeε0 e0 εÞ Δ1 ,
¼ 2
cε0 εξ1 ðε0 þ εÞΔ1 ,
ð4:142Þ
¼ 2
cε0 ξ1 ðeε0 e0 εÞΔ1 ,
¼ 2
cεξ1 ðeε0 þ e0 εÞΔ1 ,
where
Δ ¼ c0 ε0 εξ01 ðε0 þ εÞ þ cε0 εξ1 ðε0 þ εÞ iξ2 ðeε0 e0 εÞ :
2
ð4:143Þ
Given the incident wave A1, Eq. (4.142) determines the reflected wave A2, the
0
0
transmitted wave A , and the local ψ fields B and B at both sides of the joint.
For numerical results consider the case when the two semi-infinite plates are
PZT-5 and PZT-6B, respectively. For this particular choice of materials, ω0m > ωm .
The face-shear wave with m ¼ 0 does not have a cutoff frequency. For thicknesstwist waves with m 1, we choose a specific case of m ¼ 2 to show the basic
behavior of these waves.
When ω > ω0m > ωm , the wave numbers ξ1 and ξ01 in the two halves of the plate
are both real. The u field is shown in Fig. 4.11. We have propagating waves on both
sides of the joint.
When ω2 < ω < ω02 , ξ1 is real, but ξ01 is imaginary. The u field for this case is
shown in Fig. 4.12. We have propagating waves in the left half of the plate. In the
right half u is exponentially decaying from the joint (cutoff).
For antisymmetric waves the results are similar except that m assumes odd
numbers.
4.9 Trapped Waves in an Inhomogeneous Plate
139
Fig. 4.11 u field near the
joint when ω > ω02 > ω2 ,
m¼2
Fig. 4.12 u field near the
joint when ω2 < ω < ω02 ,
m¼2
4.9
Trapped Waves in an Inhomogeneous Plate
In this section we study thickness-twist waves in an unbounded, inhomogeneous
ceramic plate in which the material of the central part is different from that of the rest
of the plate (see Fig. 4.13) [8]. The plate is unelectroded and the free-space electric
field is neglected. The surfaces of the plate are traction free. It will be shown that
trapped waves with their motion confined in the central part of the plate may exist
when the material properties of the plate satisfy certain conditions.
Due to the material inhomogeneity, we need to obtain solutions in different
regions and apply interface conditions. First consider the central part with |x1| < a.
The governing equations for u, ψ and φ are:
140
4 Waves in Unbounded Regions
x2
Fig. 4.13 An
inhomogeneous ceramic
plate
x1
2h
2a
c∇2 u ¼ ρ€u,
∇2 ψ ¼ 0,
e
φ ¼ ψ þ u:
ε
ð4:144Þ
For the unelectroded and traction-free surfaces at the top and bottom, we have
T 23 ¼ 0,
D2 ¼ 0,
x2 ¼ h,
ð4:145Þ
ψ , 2 ¼ 0,
x2 ¼ h:
ð4:146Þ
or, equivalently, in terms of u and ψ,
u, 2 ¼ 0,
The stationary wave solutions to Eqs. (4.144) and (4.146) can be classified into
waves symmetric or antisymmetric in x2, and they are given by
u ¼ ðA1 cos ξ1 x1 þ A2 sin ξ1 x1 Þ cos ξ2 x2 expðiωt Þ,
ψ ¼ ðB1 coshξ2 x1 þ B2 sinhξ2 x1 Þ cos ξ2 x2 expðiωt Þ,
mπ
, m ¼ 0, 2, 4, . . . ,
ξ2 ¼
2h
ð4:147Þ
u ¼ ðA1 cos ξ1 x1 þ A2 sin ξ1 x1 Þ sin ξ2 x2 expðiωt Þ,
ψ ¼ ðB1 coshξ2 x1 þ B2 sinhξ2 x1 Þ sin ξ2 x2 expðiωt Þ,
mπ
, m ¼ 1, 3, 5, . . . ,
ξ2 ¼
2h
ð4:148Þ
and
respectively, where
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffirffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
mπ 2 c
1
ρ ω2
ρ
2
2
¼
ω2 ω2m ,
ξ1 ¼
ξ2 ¼
ω 2h
c
ρ vT
c
rffiffiffi
mπ 2 c
c
vT ¼
, ω2m ¼
:
2h ρ
ρ
ð4:149Þ
A1, A2, B1, and B2 are undetermined constants. ωm is the cutoff frequency of the
thickness-twist waves.
For the outer regions of the plate where |x1| > a, we use a prime to indicate the
relevant material constants. Similar to Eqs. (4.147), (4.148), and (4.149), we have the
following fields satisfying the governing equations and the boundary conditions at
x2 ¼ h:
4.9 Trapped Waves in an Inhomogeneous Plate
141
A0 exp ξ01 ðx1 aÞ cos ξ2 x2 expðiωt Þ, x1 > a,
A00 exp ξ01 ðx1 þ aÞ cos ξ2 x2 expðiωt Þ, x1 < a,
B0 exp½ξ2 ðx1 aÞ cos ξ2 x2 expðiωt Þ, x1 > a,
ψ¼
B00 exp½ξ2 ðx1 þ aÞ cos ξ2 x2 expðiωt Þ, x1 < a,
mπ
, m ¼ 0, 2, 4, . . . ,
ξ2 ¼
2h
ð4:150Þ
A0 exp ξ01 ðx1 aÞ sin ξ2 x2 expðiωt Þ, x1 > a,
A00 exp ξ01 ðx1 þ aÞ sin ξ2 x2 expðiωt Þ, x1 < a,
B0 exp½ξ2 ðx1 aÞ sin ξ2 x2 expðiωt Þ, x1 > a,
ψ¼
B00 exp½ξ2 ðx1 þ aÞ sin ξ2 x2 expðiωt Þ, x1 < a,
mπ
, m ¼ 1, 3, 5, . . . ,
ξ2 ¼
2h
ð4:151Þ
u¼
u¼
where
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ρ0 ω 2
1 0 2
2
ωm ω 2 ,
¼ ξ2 0 ¼ 0
vT
c
sffiffiffiffi
0 2 mπ 2 c0
c0
,0
ωm ¼ 2h 0 :
v0T ¼
ρ
ρ
ξ01
ð4:152Þ
In Eqs. (4.150) and (4.151), the fields in the outer regions are exponentially decaying
from the interfaces at x1 ¼ a. Waves having this behavior are called trapped
waves and are particularly useful for acoustic wave device applications in which
device mounting can be designed in the outer regions where there is little motion.
The solutions in the central and outer regions have to satisfy the continuity
conditions of u, T13, φ, and D1 at the interfaces where x1 ¼ a. We have
A1 cos ξ1 a þ A2 sin ξ1 a ¼ A0
cðA1 ξ1 sin ξ1 a þ A2 ξ1 cos ξ1 aÞ þ eðB1 ξ2 sinhξ2 a þ B2 ξ2 coshξ2 aÞ
¼ c0 A0 ξ01 þ e0 ðB0 ξ2 Þ,
e
e0
B1 coshξ2 a þ B2 sinhξ2 a þ ðA1 cos ξ1 a þ A2 sin ξ1 aÞ ¼ B0 þ 0 A0 ,
ε
ε
εðB1 ξ2 sinhξ2 a þ B2 ξ2 coshξ2 aÞ ¼ ε0 ðB0 ξ2 Þ,
A1 cos ξ1 a A2 sin ξ1 a ¼ A00
cðA1 ξ1 sin
ξ1 aþ A2 ξ1 cos ξ1 aÞ þ eðB1 ξ2 sinhξ2 a þ B2 ξ2 coshξ2 aÞ
¼ c0 A00 ξ01 þ e0 ðB00 ξ2 Þ,
e
e0
B1 coshξ2 a B2 sinhξ2 a þ ðA1 cos ξ1 a A2 sin ξ1 aÞ ¼ B00 þ 0 A00 ,
ε
ε
εðB1 ξ2 sinhξ2 a þ B2 ξ2 coshξ2 aÞ ¼ ε0 ðB00 ξ2 Þ,
ð4:153Þ
which is true for waves symmetric or antisymmetric about x2. Equation (4.153) are
0
0
00
00
eight linear homogeneous equations for A1, A2, B1, B2, A , B , A , and B . For
nontrivial solutions, the determinant of the coefficient matrix has to vanish. This
142
4 Waves in Unbounded Regions
yields the frequency equation for the waves. These waves can be further separated
into waves symmetric and antisymmetric in x1. We discuss them separately below.
For modes that are symmetric in x1, we have
A2 ¼ 0,
B2 ¼ 0,
A00 ¼ A0 ,
B00 ¼ B0 :
ð4:154Þ
Then the last four equations in Eq. (4.153) become the same as the first four for A1,
0
0
B1, A , and B . We have
2
cos ξ1 a
6 c
ξ1 sin ξ1 a
6 1
4 eε cos ξ1 a
0
0
eξ2 sinhξ2 a
coshξ2 a
εξ2 sinhξ2 a
1
c0 ξ01
e0 =ε0
0
38 9
A1 >
0 >
>
=
< >
e0 ξ 2 7
7 B10 ¼ 0:
5
1 >
A >
>
;
: 0>
ε0 ξ 2
B
ð4:155Þ
Setting the determinant of the coefficient matrix of Eq. (4.155) to zero, we obtain
0 0
c ξ1 cξ1 tan ξ1 a ðε0 þ εtanhξ2 aÞ
0 2
¼ ξ2 εε0 eε eε0 tanhξ2 a:
ð4:156Þ
With the expressions for ξ1 and ξ01 in Eqs. (4.149) and (4.152), we can write
Eq. (4.156) as an equation for ω. For trapped waves to exist, we must have
mπ
2h
2
2 mπ
c
¼ ω2m < ω2 < ω0m ¼
2h
ρ
2c
0
ρ0
,
ð4:157Þ
or c=ρ < c0 =ρ0 which is equivalent to vT < v0T , that is, the shear wave velocity in the
central region must be smaller than that in the outer regions. The face-shear wave
corresponding to m ¼ 0 is not a trapped wave. As a numerical example consider the
case when m ¼ 2. We choose PZT-5 for the central region and PZT-6B for the outer
regions, respectively, so that Eq. (4.157) is satisfied. The plate thickness is
h ¼ 1 mm. a/h ¼ 10. Equation (4.156) is solved numerically. Figure 4.14 shows
the u field for the lowest two frequencies from Eq. (4.156). Clearly, these waves are
trapped waves decaying rapidly away from the interfaces.
Similarly, for trapped waves that are antisymmetric in x1, we have
A1 ¼ 0,
B1 ¼ 0,
A00 ¼ A0 ,
B00 ¼ B0 :
ð4:158Þ
Again the last four equations in Eq. (4.153) become the same as the first four for A2,
0
0
B2, A , and B :
4.10
Waves in a Plate on a Half Space
143
Fig. 4.14 Displacement field of the first two trapped waves
2
sin ξ1 a
6 cξ1 cos ξ1 a
6 1
4 eε sin ξ1 a
0
0
1
eξ2 coshξ2 a
c0 ξ01
sinhξ2 a
e0 =ε0
εξ2 coshξ2 a
0
38 9
A2 >
0 >
>
=
< >
e0 ξ 2 7
7 B20 ¼ 0:
1 5>
A >
>
;
: 0>
ε0 ξ 2
B
ð4:159Þ
In this case the frequency equation is given by
0 0
c ξ1 þ cξ1 cot ξ1 a ðε0 þ εcothξ2 aÞ
0 2
¼ ξ2 εε0 eε eε0 cothξ2 a:
4.10
ð4:160Þ
Waves in a Plate on a Half Space
This section is on propagating waves in an elastic plate on a piezoelectric half space
[9]. In the theory of elasticity, the corresponding waves are called Love waves.
Specifically, consider the metal plate and the ceramic half space in Fig. 4.15.
The governing equations are
c∇2 u ¼ ρ€u, x2 > 0,
∇2 ψ ¼ 0, x2 > 0,
e
φ ¼ ψ þ u, x2 > 0,
ε
ð4:161Þ
144
4 Waves in Unbounded Regions
Fig. 4.15 A metal plate on
a ceramic half space
Traction free
h
Metal
x1
Ceramic
half space
x2
Propagation
direction
and
b
ρ €u,
c ∇2 u ¼ b
h < x2 < 0,
ð4:162Þ
where b
ρ and b
c are the mass density and shear elastic constant of the metal plate. We
look for solutions satisfying
u,
ψ ! 0,
x2 ! þ1:
ð4:163Þ
For x2 > 0, the solutions to Eq. (4.161) satisfying Eq. (4.163) can be written as
u ¼ Aexpðξ2 x2 Þ cos ðξx1 ωt Þ,
ψ ¼ Bexpðξ1 x2 Þ cos ðξx1 ωt Þ,
ð4:164Þ
where A and B are constants. For Eq. (4.164)1 to satisfy Eq. (4.161)1, the following
must be true:
ξ22 ¼ ξ21 ρω2
v2
¼ ξ21 1 2 > 0,
c
vT
ð4:165Þ
where
v2 ¼
ω2
,
ξ21
c
v2T ¼ :
ρ
ð4:166Þ
The electric potential and the stress component needed for the boundary and
continuity conditions are
T 23 ¼ cu, 2 þ eψ , 2
¼ ½cAξ2 expðξ2 x2 Þ þ eBξ1 expðξ1 x2 Þ cos ðξx1 ωt Þ,
e
φ ¼ψ þ u
he ε
i
¼ Aexpðξ2 x2 Þ þ Bexpðξ1 x2 Þ cos ðξx1 ωt Þ:
ε
For h <x2 < 0, we write
ð4:167Þ
4.10
Waves in a Plate on a Half Space
145
b cos b
b sin b
u¼ A
ξ 2 x2 þ B
ξ 2 x2 cos ðξ1 x1 ωt Þ,
ð4:168Þ
2
b
ρ ω2
2
2
2 v
b
ξ1 ¼ ξ1 2 1 ,
ξ2 ¼
b
c
b
vT
ð4:169Þ
b
c
b
v 2T ¼ :
b
ρ
ð4:170Þ
where
and
For boundary and continuity conditions, we need
c u, 2
T 23 ¼ b
bb
bb
¼b
c A
ξ 2 sin b
ξ 2 x2 þ B
ξ 2 cos b
ξ 2 x2 cos ðξ1 x1 ωt Þ:
ð4:171Þ
The continuity and boundary conditions require that
e
φð0þ Þ ¼ A þ B ¼ 0,
ε
b ¼ uð0 Þ,
uð 0þ Þ ¼ A ¼ A
bb
cAξ eBξ1 ¼ b
cB
ξ 2 ¼ T 23 ð0 Þ,
T 23 ð0þ Þ ¼ 2
b
b
b
b ξ 2 sin ξ 2 h þ B
b ξ 2 cos b
c A
ξ 2 h ¼ 0:
T 23 ðhÞ ¼ b
ð4:172Þ
Using Eq. (4.172)1,2 to eliminate A and B, we obtain
b b
b 2 þ eeAξ
bb
cAξ
cB
ξ 2 ¼ 0,
ε 1
bb
bb
A
ξ 2 sin b
ξ2h þ B
ξ 2 cos b
ξ 2 h ¼ 0:
ð4:173Þ
¼ 0,
b
b
ξ 2 cos ξ 2 h ð4:174Þ
For nontrivial solutions,
e
cξ 2 þ e ξ 1
ε
b
ξ2h
ξ 2 sin b
b
cb
ξ2
or
ξ2 b
ξ
cb
2 tan b
ξ 2 h ¼ k2 :
ξ1 c ξ1
Substituting from Eqs. (4.165) and (4.169), we obtain
ð4:175Þ
146
4 Waves in Unbounded Regions
sffiffiffiffiffiffiffiffiffiffiffiffiffi
sffiffiffiffiffiffiffiffiffiffiffiffiffi
" sffiffiffiffiffiffiffiffiffiffiffiffiffi#
v2 b
v2
c v2
1 2 1
tan
ξ
h
1 ¼ k2 ,
1
b
vT c b
v 2T
v 2T
ð4:176Þ
which determines the surface wave velocity v as a function of the wave number ξ1.
Clearly, the waves determined by Eq. (4.176) are dispersive.
When the electromechanical coupling factor k ¼ 0, Eq. (4.176) reduces to the
equation that determines the dispersion relation for the well-known Love wave in an
elastic layer on an elastic half space. The dispersion relations for Love waves are real
and multi-valued when b
v 2T < v2 < v2T , for which the shear wave velocity of the layer
has to be smaller than that of the half space. In other words, the half space is more
rigid than the layer.
When b
c ¼ 0 or h¼0, Eq. (4.176) reduces to Eq. (4.64) for the Bleustein–Gulyaev
wave in a ceramic half space with an electroded surface.
4.11
Gap Waves
Consider two ceramic half spaces with a gap between them (see Fig. 4.16). The
surfaces of the half spaces at x2 ¼ h are traction free and unelectroded. Acoustic
waves in the half spaces can be coupled by the electric field in the gap. We consider
waves propagating near the surfaces of the half spaces at x2 ¼ h [10].
The governing equations are
cA ∇2 uA ¼ ρA €uA , x2 > h,
∇2 ψ A ¼ 0, x2 > h,
eA
φA ¼ ψ A þ uA , x2 > h,
εA
ð4:177Þ
∇2 φ ¼ 0,
ð4:178Þ
h < x2 < h,
and
Fig. 4.16 Two ceramic half
spaces with a gap
Propagation
direction
Ceramic B
Gap
Ceramic A
x1
2h
x2
4.11
Gap Waves
147
cB ∇2 uB ¼ ρB €uB , x2 < h,
∇2 ψ B ¼ 0, x2 < h,
eB
φB ¼ ψ B þ uB , x2 < h,
εB
ð4:179Þ
where subscripts A and B are not tensor indices and they indicate the fields in ceramic
A and ceramic B. For waves near the gap, we require that
uA , ψ A ! 0,
x2 ! þ1,
ð4:180Þ
uB , ψ B ! 0,
x2 ! 1:
ð4:181Þ
For x2 > h, the solution to Eq. (4.177) that satisfies Eq. (4.180) can be written as
uA ¼ U A expðηA x2 Þ cos ðξx1 ω t Þ,
ψ A ¼ ΨA expðξx2 Þ cos ðξx1 ωt Þ,
ð4:182Þ
where UA and ΨA are undetermined constants,
η2A
ρ A ω2
v2
2
¼ξ ¼ ξ 1 2 > 0,
cA
vA
2
ð4:183Þ
and
v¼
ω
,
ξ
v2A ¼
cA
:
ρA
ð4:184Þ
For continuity conditions, we need T23 and D2 in ceramic A, denoted by TA and DA:
T A ¼ cA uA, 2 þ eA ψ A, 2
¼ ½cA ηA U A expðηA x2 Þ þ eA ξΨA expðξx2 Þ cos ðξx1 ωt Þ,
DA ¼ εA ψ A, 2
¼ εA ξ ΨA expðξx2 Þ cos ðξx1 ωt Þ:
ð4:185Þ
Similarly, for x2 < h, the solution to Eq. (4.179) that satisfies Eq. (4.181) is
uB ¼ U B expðηB x2 Þ cos ðξx1 ωt Þ,
ψ B ¼ ΨB expðξx2 Þ cos ðξx1 ωt Þ,
ð4:186Þ
where UB and ΨB are undetermined constants,
η2B ¼ ξ2 ρ B ω2
v2
¼ ξ2 1 2 > 0,
cB
vB
ð4:187Þ
148
4 Waves in Unbounded Regions
and
v2B ¼
cB
:
ρB
ð4:188Þ
For continuity conditions, we need T23 and D2 in ceramic B, denoted by TB and DB:
T B ¼ cB uB, 2 þ eB ψ B, 2
¼ ½cB ηB U B expðηB x2 Þ þ eB ξΨB expðξx2 Þ cos ðξx1 ωt Þ,
DB ¼ εB ψ B, 2
¼ εB ξΨB expðξx2 Þ cos ðξx1 ωt Þ:
ð4:189Þ
The fields in the gap can be written as
φ ¼ ðΦ1 coshξx2 þ Φ2 sinhξx2 Þ cos ðξx1 ωt Þ,
ð4:190Þ
where Φ1 and Φ2 are undetermined constants and Eq. (4.178) is already satisfied. For
continuity conditions, we need D2 in the gap:
D2 ¼ ε0 φ, 2
¼ ε0 ðξΦ1 sinhξx2 þ ξΦ2 coshξx2 Þ cos ðξx1 ωt Þ:
ð4:191Þ
For interface continuity conditions, we impose
T A ¼ 0,
T B ¼ 0,
eA
uA ¼ φ,
εA
eB
ψ B þ uB ¼ φ,
εB
ψA þ
DA ¼ D2 ,
x2 ¼ h,
DB ¼ D2 ,
x2 ¼ h,
ð4:192Þ
which implies that
cA ηA U A expðηA hÞ þ eA ξΨA expðξhÞ ¼ 0,
eA
ΨA expðξhÞ þ U A expðηA hÞ ¼ Φ1 coshξh þ Φ2 sinhξh,
εA
εA ξΨA expðξhÞ ¼ ε0 ðξΦ1 sinhξh þ ξΦ2 coshξhÞ,
cB ηB U B expðηB hÞ þ eB ξΨB expðξhÞ ¼ 0,
eB
ΨB expðξhÞ þ U B expðηB hÞ ¼ Φ1 coshξh Φ2 sinhξh,
εB
εB ξΨB expðξhÞ ¼ ε0 ðξΦ1 sinhξh þ ξΦ2 coshξhÞ:
ð4:193Þ
For nontrivial solutions the determinant of the coefficient matrix of Eq. (4.193) has
to vanish, which yields the frequency equation that determines the dispersion
relations of the waves. We examine the special case when the two ceramic half
spaces are of the same material below, that is,
4.11
Gap Waves
149
cA ¼ cB ¼ c, eA ¼ eB ¼ e,
εA ¼ εB ¼ ε, ρA ¼ ρB ¼ ρ,
vA ¼ vB ¼ vT :
ð4:194Þ
Then the waves can be separated into symmetric and antisymmetric ones.
For symmetric waves we consider
UA ¼ UB,
ΨA ¼ ΨB ,
Φ2 ¼ 0,
ð4:195Þ
for which the last three of Eq. (4.193) become identical to the first three, and the
frequency equation assumes the following simple form:
cη
e
ξ
ε
0
eξ
1
ε
0
coshξh ¼ 0,
ε0 sinhξh
ð4:196Þ
or
cη eξ eξ e
þ ε0 sinhξh
1 ε
ε
e2
¼ ε
cηcoshξh þ ε0 cη ξ
sinhξh ¼ 0:
ε
cη
coshξh
0
ð4:197Þ
Equation (4.197) can be further written as
tanhξh ¼
ε
cη
n2 η
¼ 2
,
ξk η
cη
2
ε0 ξeε
ð4:198Þ
where we have denoted
n2 ¼ ε=ε0 :
ð4:199Þ
With Eq. (4.183), we can write Eq. (4.198) as:
qffiffiffiffiffiffiffiffiffiffiffi2ffi
1 vv2
T
ffi:
qffiffiffiffiffiffiffiffiffiffiffi
tanhξh ¼
2
2
k 1 vv2
n2
T
Equation (4.200) shows that the wave is dispersive.
ð4:200Þ
150
4 Waves in Unbounded Regions
For antisymmetric waves we consider
U A ¼ U B ,
ΨA ¼ ΨB ,
Φ1 ¼ 0:
ð4:201Þ
Then the last three of Eq. (4.193) become identical to the first three, and the
frequency equation assumes the following simple form:
tanhξh ¼
2
ε0 ξeε cη
ε
cη
¼
ξ k2 η
,
n2 η
ð4:202Þ
or
0
qffiffiffiffiffiffiffiffiffiffiffiffi2 11
1 vv2
B
T C
ffiA :
qffiffiffiffiffiffiffiffiffiffiffi
tanhξh ¼ @
2
2
k 1 vv2
n2
ð4:203Þ
T
The right-hand side of Eq. (4.203) is the inverse of that of Eq. (4.200). Therefore, for
v < vT, one of the right-hand sides of Eq. (4.200) and Eq. (4.203) is within (1,1).
Since the range of a hyperbolic tangent function is between 1 and 1, a root for ξ can
always be found from either Eq. (4.200) or Eq. (4.203).
4.12
Waves on a Circular Cylindrical Surface
Consider waves propagating on the surface of a ceramic circular cylinder [11] as
shown in Fig. 4.17. We choose (r,θ,z) to correspond to (1,2,3) so that the poling
direction corresponds to 3.
The governing equations are
x2
Fig. 4.17 A ceramic
circular cylinder
b
x1
a
Ceramic
4.12
Waves on a Circular Cylindrical Surface
151
c∇2 u ¼ ρ€u, a < r < b,
∇2 ψ ¼ 0, a < r < b,
e
φ ¼ ψ þ u, a < r < b:
ε
ð4:204Þ
The stress and electric displacement components relevant to boundary conditions are
T rz ¼ cu, r þ eψ , r ,
Dr ¼ εψ , r :
ð4:205Þ
From Eqs. (4.204)1,2, in polar coordinates, we have
2
v2T
2
∂ u 1 ∂u 1 ∂ u
þ
þ
∂r 2 r ∂r r 2 ∂θ2
2
!
¼ €u,
2
∂ ψ 1 ∂ψ 1 ∂ ψ
þ
þ
¼ 0,
∂r 2 r ∂r r 2 ∂θ2
ð4:206Þ
where v2T ¼ c=ρ. For waves propagating in the θ direction we consider
uz ðr; θ; t Þ ¼ uðr Þ cos ðνθ ωt Þ,
ψ ðr; θ; t Þ ¼ ψ ðr Þ cos ðνθ ωt Þ,
ð4:207Þ
where ν is allowed to assume any real value. The substitution of Eq. (4.207) into
Eq. (4.206) results in
2
∂ u 1 ∂u
ν2
2
þ ξ 2 u ¼ 0,
þ
∂r 2 r ∂r
r
2
2
∂ ψ 1 ∂ψ ν
ψ ¼ 0,
þ
∂r 2 r ∂r r 2
ð4:208Þ
where we have denoted
ξ¼
ω
:
vT
ð4:209Þ
Equation (4.208)1 can be written as Bessel’s equation of order ν. Then the general
solution can be written as
uz ¼ ½C 1 J ν ðξ r Þ þ C 2 Y ν ðξ r Þ cos ðνθ ωt Þ,
ψ ¼ ½C3 r ν þ C 4 r ν cos ðνθ ωt Þ,
ð4:210Þ
152
4 Waves in Unbounded Regions
where Jν and Yν are the ν-th order Bessel functions of the first kind and second kind,
and C1–C4 are undetermined constants. The following expressions are needed for
boundary conditions:
n
o
e
φ ¼ C3 r ν þ C 4 r ν þ ½C1 J ν ðξ r Þ þ C2 Y ν ðξ r Þ cos ðνθ ωt Þ,
ε
0
T rz ¼ c C1 ξJ ν ðξ r Þ þ C2 ξY 0ν ðξ r Þ
þe½C3 vr v1 C 4 νr ν1 cos ðνθ ωt Þ,
Dr ¼ εðC 3 vr v1 C4 νr ν1 Þ cos ðνθ ωt Þ,
ð4:211Þ
where a superimposed prime indicates a differentiation with respect to the whole
argument of a function. Consider a solid cylinder (a ¼ 0 in Fig. 4.17). The surface at
r ¼ b is traction free and is electroded. We restrict ourselves to the case when ν is
greater than or equal to 1. In this case, since Yν and rν1 are singular at the origin, the
terms associated with C2 and C4 are dropped. Trz and φ should both vanish at r ¼ b.
This leads to the following two linear homogeneous equations for C1 and C3:
cC 1 ξJ 0ν ðξ bÞ þ eC3 vbv1 ¼ 0,
e
C 3 bν þ C 1 J ν ðξ bÞ ¼ 0:
ε
ð4:212Þ
For nontrivial solutions the determinant of the coefficient matrix of the two linear
equations in Eq. (4.212) has to vanish, which results in the following equation for ξ:
cξJ 0 ðξ bÞ evbv1 e ν
bν J ν ð ξ bÞ
ε
e2
¼ cξJ 0ν ðξ bÞbν J ν ðξ bÞνbv1 ¼ 0,
ε
ð4:213Þ
ξbJ 0ν ðξ bÞ 2
¼k :
νJ ν ðξ bÞ
ð4:214Þ
or,
4.13
Scattering by a Circular Cylinder
In this section we examine the scattering of plane waves by a circular cylinder in an
unbounded ceramic body poled along x3 (see Fig. 4.18). To be simple let the cylinder
be a rigid conductor which is grounded.
The incident wave is known and is given by
4.13
Scattering by a Circular Cylinder
153
Fig. 4.18 Plane waves
incident upon a circular
cylinder
x2
R
Incident wave
x1
Rigid
conductor
Ceramic
uI ¼ Aexp½iξðx1 vT t Þ,
ψ I ¼ 0,
e
φI ¼ u I ,
ε
ð4:215Þ
which satisfies
v2T ∇2 uI ¼ €u,
∇2 ψ I ¼ 0,
r > R:
ð4:216Þ
The incident and scattered waves together must satisfy the following equations and
boundary conditions:
v2T ∇2 u ¼ €u, ∇2 ψ ¼ 0,
u ¼ 0, φ ¼ 0, r ¼ R:
r > R,
ð4:217Þ
Let
u ¼ uI þ uS ,
ψ ¼ ψ I þ ψ S,
φ ¼ φI þ φS :
ð4:218Þ
From Eqs. (4.216) and (4.217) we obtain the following equations and boundary
conditions for the scattered uS and ψ S:
v2T ∇2 uS ¼ u€S , r > R,
∇2 ψ S ¼ 0, r > R,
uI þ uS ¼ Aexp½iξðx1 vT t Þ þ uS ¼ 0, r ¼ R,
e
e
φI þ φS ¼ uI þ ψ S þ uS ¼ ψ S ¼ 0, r ¼ R:
ε
ε
ð4:219Þ
In addition, the scattered wave must be outgoing at infinity (radiation condition).
From Eq. (4.219)2,4, ψ S 0. uS has a time dependence of exp[iξ(vTt)] which is
dropped below for convenience. Then, from Eq. (4.219)1,3,
154
4 Waves in Unbounded Regions
∇2 uS þ ξ2 uS ¼ 0, r > R,
uS þ Aexpðiξ x1 Þ ¼ 0, r ¼ R:
ð4:220Þ
In polar coordinates Eq. (4.220)1 takes the following form:
2
2
∂ uS 1 ∂uS 1 ∂ uS
þ
þ
þ ξ2 uS ¼ 0:
r ∂r r 2 ∂θ2
∂r 2
ð4:221Þ
Let
uS ðr; θÞ ¼ uS ðr ÞexpðinθÞ,
n ¼ 0, 1, 2, :
ð4:222Þ
Then Eq. (4.221) becomes
2
∂ uS 1 ∂uS
n2 S
2
þ ξ 2 u ¼ 0,
þ
r ∂r
∂r 2
r
ð4:223Þ
2
∂ uS
1 ∂uS
n2
þ 1 2 2 uS ¼ 0:
þ
ξ2 ∂r 2 ξ2 r ∂r
ξ r
ð4:224Þ
or
Equation (4.224) is Bessel’s equation of order n. For scattered waves diverging
toward r ¼ 1, we have
uS ¼
1
X
C n H ðn1Þ ðξr ÞexpðinθÞ,
ð4:225Þ
n¼1
where Cn are undetermined constants. H ðn1Þ is the Hankel function of the first kind of
order n. Substituting Eq. (4.225) into the boundary condition in Eq. (4.220)2, we
have
1
X
Cn H ðn1Þ ðξRÞexpðinθÞ þ AexpðiξR cos θÞ ¼ 0:
ð4:226Þ
n¼1
Equation (4.226) is a system of linear algebraic equations for Cn. Cn can be obtained
from Eq. (4.226) using the orthogonality of exp(inθ).
References
155
References
1. J.L. Bleustein, A new surface wave in piezoelectric materials. Appl. Phys. Lett. 13, 412–413
(1968)
2. Y.V. Gulyaev, Electroacoustic surface waves in solids. JETP Lett. 9, 37–38 (1969)
3. F.S. Hickernell, Shear horizontal BG surface acoustic waves on piezoelectrics: a historical note.
IEEE Trans. Ultrason. Ferroelectr. Freq. Contr. 52, 809–811 (2005)
4. J.S. Yang, Antiplane Motions of Piezoceramics and Acoustic Wave Devices (World Scientific,
Singapore, 2010)
5. C. Maerfeld, P. Tournois, Pure shear elastic surface wave guided by the interface of two semiinfinite media. Appl. Phys. Lett. 19, 117–118 (1971)
6. J.L. Bleustein, Some simple modes of wave propagation in an infinite piezoelectric plate.
J. Acoust. Soc. Am. 45, 614–620 (1969)
7. J.S. Yang, Z.G. Chen, Y.T. Hu, Propagation of thickness-twist waves through a joint between
two semi-infinite piezoelectric plates. IEEE Trans. Ultrason. Ferroelectr. Freq. Contr. 54,
888–891 (2007)
8. J.S. Yang, Z.G. Chen, Y.T. Hu, Trapped thickness-twist modes in an inhomogeneous piezoelectric plate. Philos. Mag. Lett. 86, 699–705 (2006)
9. R.G. Curtis, M. Redwood, Transverse surface waves on a piezoelectric material carrying a metal
layer of finite thickness. J. Appl. Phys. 44, 2002–2007 (1973)
10. Y.V. Gulyaev, V.P. Plesskii, Acoustic gap waves in piezoelectric materials. Sov. Phys. Acoust.
23, 410–413 (1977)
11. C.L. Chen, On the electroacoustic waves guided by a cylindrical piezoelectric surface. J. Appl.
Phys. 44, 3841–3847 (1973)
Chapter 5
Vibrations of Finite Bodies
In this chapter we study vibrations of finite piezoelectric bodies. In some cases, for
example, thickness vibrations of unbounded plates, although the in-plane
dimensions of the plates are infinite, what matters is the finite plate thickness.
Sects. 5.2 and 5.7 are on antiplane problems of polarized ceramics for which the
notation in Sect. 2.9 is followed. The solutions in Sects. 5.1, 5.2, 5.3, 5.4, 5.5, 5.6,
and 5.7 are exact. Those in Sects. 5.8, 5.9, and 5.10 are approximate.
5.1
Thickness Stretch of a Ceramic Plate Through e33
Thickness vibrations plates are motions spatially varying along the plate thickness
only. Solutions to thickness vibrations of piezoelectric plates can be obtained in a
general manner [1]. In this section we discuss a special case, that is, the thicknessstretch or thickness-extensional vibration of a ceramic plate. Consider the ceramic
plate poled along the x3 axis in Fig. 5.1. The plate is bounded by two planes at
x3 ¼ h which are traction free and electroded. A time-harmonic voltage is applied
across the plate thickness.
For polarized ceramics the governing equations are those in Sect. 2.9. The
boundary conditions are:
T 3 j ¼ 0, x3 ¼ h,
φðx3 ¼ hÞ φðx3 ¼ hÞ ¼ Vexpðiω t Þ:
ð5:1Þ
Consider thickness-stretch motions described by
u3 ¼ u3 ðx3 Þexpðiω t Þ,
φ ¼ φðx3 Þexpðiω t Þ:
u1 ¼ u2 ¼ 0,
© Springer Nature Switzerland AG 2018
J. Yang, An Introduction to the Theory of Piezoelectricity, Advances in Mechanics
and Mathematics 9, https://doi.org/10.1007/978-3-030-03137-4_5
ð5:2Þ
157
158
5 Vibrations of Finite Bodies
x3
Fig. 5.1 An electroded
ceramic plate with thickness
poling
2h
P
x1
The nontrivial components of strain and electric field are
S33 ¼ u3, 3 ,
E3 ¼ φ, 3 ,
ð5:3Þ
where the time-harmonic factor has been dropped. The nontrivial stress and electric
displacement components are
T 11 ¼ T 22 ¼ c13 u3, 3 þ e31 φ, 3
T 33 ¼ c33 u3, 3 þ e33 φ, 3 ,
D3 ¼ e33 u3, 3 ε33 φ, 3 :
ð5:4Þ
In terms of the displacement and potential, the equations of motion and charge are
c33 u3, 33 þ e33 φ, 33 ¼ ρω2 u3 ,
e33 u3, 33 ε33 φ, 33 ¼ 0:
ð5:5Þ
Equation (5.5)2 can be integrated to yield
φ¼
e33
u3 þ B1 x3 þ B2 ,
ε33
ð5:6Þ
where B1 and B2 are integration constants, and B2 is immaterial. Substituting
Eq. (5.6) into the expressions for T33, D3, and Eq. (5.5)1, we obtain
T 33 ¼ c33 u3, 3 þ e33 B1 ,
D3 ¼ ε33 B1 ,
ð5:7Þ
c33 u3, 33 ¼ ρ ω2 u3 ,
ð5:8Þ
where
c33 ¼ c33 1 þ k233 ,
k233 ¼
e233
:
ε33 c33
ð5:9Þ
The general solution to Eq. (5.8) and the corresponding expression for the electric
potential are
5.1 Thickness Stretch of a Ceramic Plate Through e33
u3 ¼ A1 sin ξx3 þ A2 cos ξx3 ,
e33
φ ¼ ðA1 sin ξx3 þ A2 cos ξx3 Þ þ B1 x3 þ B2 ,
ε33
159
ð5:10Þ
where A1 and A2 are integration constants, and
ξ2 ¼
ρ 2
ω:
c33
ð5:11Þ
The expression for stress is then
T 33 ¼ c33 ðA1 ξ cos ξx3 A2 ξ sin ξx3 Þ þ e33 B1 :
ð5:12Þ
The boundary conditions in Eq. (5.1) require that
c33 A1 ξ cos ξh c33 A2 ξ sin ξh þ e33 B1 ¼ 0,
c33 A1 ξ cos ξh þ c33 A2 ξ sin ξh þ e33 B1 ¼ 0,
e33
2 A1 sin ξh þ 2B1 h ¼ V:
ε33
ð5:13Þ
Adding Eq. (5.13)1,2 and subtracting them from each other, we can rewrite Eq. (5.13)
as
c33 A1 ξ cos ξh þ e33 B1 ¼ 0,
c33 A2 ξ sin ξh ¼ 0,
e33
2 A1 sin ξh þ 2B1 h ¼ V:
ε33
ð5:14Þ
Equation (5.14) decouples into two sets of equations. One is Eq. (5.14)2 alone. The
other consists of Eq. (5.14)1,3.
Consider free vibrations with V ¼ 0 first. Equation (5.14)2 determines what may
be called antisymmetric modes for which
c33 A2 ξ sin ξh ¼ 0:
ð5:15Þ
sin ξh ¼ 0,
ð5:16Þ
Nontrivial solutions may exist if
or
ξðnÞ h ¼
nπ
,
2
n ¼ 0, 2, 4, 6, . . . ,
ð5:17Þ
160
5 Vibrations of Finite Bodies
which determines the resonance frequencies as
ω
ðnÞ
nπ
¼
2h
rffiffiffiffiffiffi
c33
,
ρ
n ¼ 0, 2, 4, 6, . . . :
ð5:18Þ
Equation (5.16) implies that B1 ¼ 0 and A1 ¼ 0. The corresponding modes are
ðnÞ
u3 ¼ cos ξðnÞ x3 ,
φðnÞ ¼
e33
cos ξðnÞ x3 ,
ε33
ð5:19Þ
where n ¼ 0 is a rigid-body mode. Equation (5.14)1,3 determines symmetric modes
for which
c33 A1 ξ cos ξh þ e33 B1 ¼ 0,
e33
2 A1 sin ξh þ 2B1 h ¼ 0:
ε33
ð5:20Þ
The resonance frequencies are determined by
c33 ξ cos ξh
e33
ε sin ξh
33
e33 e233
¼
c
ξh
cos
ξh
sin ξh ¼ 0,
33
h ε33
ð5:21Þ
or
ξh
tan ξh ¼ 2 ,
k 33
ð5:22Þ
t 2
e2
e2
k 233
33 2 ¼
¼ k33
:
k233 ¼ 33 ¼
2
ε33 c33 ε33 c33 1 þ k33
1 þ k 33
ð5:23Þ
where
Equations (5.22) and (5.20) determine the resonance frequencies and modes. For
symmetric modes, A2 ¼ 0.
Next consider forced vibrations. From Eq. (5.14)1,3 we obtain
0 e33 V 2h e33 V
¼
A1 ¼ ,
e233
c
ξ
cos
ξh
e
33 ξh
cos
ξh
2
sin
ξh
2
c
33
33
e
ε
33
2 33 sin ξh 2h ε
33
ð5:24Þ
161
5.2 Thickness Shear of a Ceramic Plate Through e15
c33 ξ cos ξh 0 e33
2 sin ξh V ε
V c33 ξ cos ξh
¼
:
B1 ¼ 33
e2
c
ξ
cos
ξh
e
33
33
c33 ξh cos ξh 2ε3333 sin ξh
2
e33
2 sin ξh 2h ε
33
ð5:25Þ
Hence
D3 ¼ ε33 B1 ¼ ε33
V
ξh
¼ σ e ,
2h ξh k233 tan ξh
ð5:26Þ
where σ e is the surface free charge per unit area on the electrode at x3 ¼ h. The
frequency dependent capacitance per unit area is
C¼
σ e ε33
ξh
¼
:
V
2h ξh k 233 tan ξh
ð5:27Þ
The denominator of Eq. (5.27), if set to zero, gives Eq. (5.22). We note the following
limits:
ε33
,
2h
ε33
1
ε33 lim C ¼
1 þ k 233 ¼ C 0 ,
¼
2
ω!0
2h
2h
k33
1
2
1 þ k33
lim C ¼
e33 !0
ð5:28Þ
where C0 is the static capacitance.
5.2
Thickness Shear of a Ceramic Plate Through e15
Consider an unbounded plate of ceramics poled along x3 (see Fig. 5.2). The plate
surfaces at x2 ¼ h are traction-free and are electroded. A time-harmonic voltage is
applied across the plate thickness.
For the specific material orientation in Fig. 5.2, the plate is driven into antiplane
motion. From Sect. 2.9, the governing equations and boundary conditions are:
x2
Fig. 5.2 An electroded
ceramic plate with in-plane
poling
2h
x1
162
5 Vibrations of Finite Bodies
c∇2 u þ e∇2 φ ¼ ρ€u,
e∇2 u ε∇2 φ ¼ 0,
T 23 ¼ 0, x2 ¼ h,
φðx2 ¼ hÞ φðx2 ¼ hÞ ¼ Vexpðiω t Þ:
ð5:29Þ
By thickness vibration we mean fields in the following form:
u ¼ uðx2 Þ,
φ ¼ φðx2 Þ,
ð5:30Þ
where the exp(iωt) factor has been dropped. The nontrivial components of the strain
and electric fields are
S4 ¼ 2S23 ¼ u, 2 ,
E2 ¼ φ, 2 :
ð5:31Þ
The nontrivial stress and electric displacement components are
T 4 ¼ cu, 2 þ eφ, 2 ,
D2 ¼ eu, 2 εφ, 2 :
ð5:32Þ
The equations of motion and charge reduce to
cu, 22 þ eφ, 22 ¼ ρω2 u,
eu, 22 εφ, 22 ¼ 0:
ð5:33Þ
Equation (5.33)2 can be integrated to yield
e
φ ¼ u þ B1 x2 þ B2 ,
ε
ð5:34Þ
where B1 and B2 are integration constants, and B2 is immaterial. Substituting
Eq. (5.34) into the expressions for T23, D2, and Eq. (5.33)1, we obtain
T 23 ¼ cu, 2 þ eB1 ,
D2 ¼ εB1 ,
ð5:35Þ
cu, 22 ¼ ρ ω2 u,
ð5:36Þ
e2
c ¼ c 1 þ k2 , k2 ¼ :
εc
ð5:37Þ
where
5.2 Thickness Shear of a Ceramic Plate Through e15
163
The general solution to Eq. (5.36) and the corresponding expression for the electric
potential are
u ¼ A1 sin ξx2 þ A2 cos ξx2 ,
e
φ ¼ ðA1 sin ξx2 þ A2 cos ξx2 Þ þ B1 x2 þ B2 ,
ε
ð5:38Þ
where A1 and A2 are integration constants, and
ρ
ξ2 ¼ ω 2 :
c
ð5:39Þ
T 23 ¼ cðA1 ξ cos ξx2 A2 ξ sin ξx2 Þ þ eB1 :
ð5:40Þ
The expression for stress is then
The boundary conditions require that
cA1 ξ cos ξh cA2 ξ sin ξh þ eB1 ¼ 0,
cA1 ξ cos ξh þ cA2 ξ sin ξh þ eB1 ¼ 0,
e
2 A1 sin ξh þ 2B1 h ¼ V,
ε
ð5:41Þ
or, add the first two, and subtract the first two from each other:
cA1 ξ cos ξh þ eB1 ¼ 0,
cA2 ξ sin ξh ¼ 0,
e
2 A1 sin ξh þ 2B1 h ¼ V:
ε
ð5:42Þ
Equation (5.42) decouples into two sets of equations. One is Eq. (5.42)2 alone. The
other consists of Eq. (5.42)1,3.
For free vibrations we have V ¼ 0. Equation (5.42)2 determines symmetric modes
for which
cA2 ξ sin ξh ¼ 0:
ð5:43Þ
sin ξh ¼ 0,
ð5:44Þ
Nontrivial solutions may exist if
or
ξðnÞ h ¼
nπ
,
2
n ¼ 0, 2, 4, 6, . . . ,
ð5:45Þ
164
5 Vibrations of Finite Bodies
which determines the resonance frequencies as
ω
ðnÞ
2π
¼
2h
rffiffiffi
c
,
ρ
n ¼ 0, 2, 4, 6, . . . :
ð5:46Þ
Equation (5.44) implies that B1¼0 and A1¼0. The corresponding modes are
uðnÞ ¼ cos ξðnÞ x2 ,
φðnÞ ¼
e
cos ξðnÞ x2 ,
ε
ð5:47Þ
where n ¼ 0 is a rigid-body mode. For antisymmetric modes
cA1 ξ cos ξh þ eB1 ¼ 0,
e
2 A1 sin ξh þ 2B1 h ¼ 0:
ε
ð5:48Þ
The resonance frequencies are determined by
cξ cos ξh
e
sin ξh
ε
e e2
h ¼ cξh cos ξh ε sin ξh ¼ 0,
ð5:49Þ
or
tan ξh ¼
ξh
,
k2
ð5:50Þ
where
e2
e2
k2
¼
:
k2 ¼ ¼ 2
ε
c εc 1 þ k
1 þ k2
ð5:51Þ
Equations (5.50) and (5.48) determine the resonance frequencies and antisymmetric
modes. For antisymmetric modes, A2 ¼ 0.
For forced vibrations, from Eq. (5.42)1,3, we have
0 e V 2h eV
¼
A1 ¼ ,
2
2
c
ξ
cos
ξh
e
cξh cos ξh 2eε sin ξh
e
2 sin ξh 2h ε
ð5:52Þ
165
5.3 Thickness Shear of a Quartz Plate Through e26
cξ cos ξh 0 e
2 sin ξh V V cξ cos ξh
¼
B1 ¼ ε
:
2
2
c
ξ
cos
ξh
e
c
ξh
cos
ξh 2eε sin ξh
e
2 sin ξh 2h ε
ð5:53Þ
Hence
D2 ¼ εB1 ¼ ε
V
ξh
¼ σ e ,
2h ξh k2 tan ξh
ð5:54Þ
where σ e is the surface free charge per unit area on the electrode at x2 ¼ h. The
capacitance per unit area is
C¼
σe
ε
ξh
¼
:
2h ξh k2 tan ξh
V
ð5:55Þ
The denominator of Eq. (5.55), if set to zero, gives the frequency equation for the
antisymmetric modes (see Eq. (5.50)). We note the following limits:
ε
,
e!0
2h
ε
1
ε
1 þ k2 :
lim C ¼
¼
2
ω!0
2h
2h
k
1
2
1þk
lim C ¼
ð5:56Þ
Obviously, the problem treated in the previous section and this section are the same
mathematically. However, separate and independent treatments are presented
because of their importance in applications.
5.3
Thickness Shear of a Quartz Plate Through e26
Consider an unbounded plate of rotated Y-cut quartz (see Fig. 5.3). The plate
surfaces at x2 ¼ h are traction-free and are electroded. A time-harmonic voltage
is applied across the plate thickness.
x2
Fig. 5.3 An electroded
plate of rotated Y-cut quartz
2h
x1
166
5 Vibrations of Finite Bodies
The governing equations for rotated Y-cut quartz can be found in Sect. 2.10. The
boundary conditions are
T 2 j ¼ 0, x2 ¼ h,
φðx2 ¼ hÞ φðx2 ¼ hÞ ¼ Vexpðiωt Þ:
ð5:57Þ
For the specific material orientation of rotated Y-cut quartz, the plate is driven into
thickness-shear vibration described by the following fields:
u1 ¼ u1 ðx2 Þexpðiω t Þ,
φ ¼ φðx2 Þexpðiω t Þ:
u2 ¼ u3 ¼ 0,
ð5:58Þ
The nontrivial components of strain, electric field, stress, and electric displacement
are
2S12 ¼ u1, 2 ,
E2 ¼ φ, 2 ,
ð5:59Þ
and
T 31 ¼ c56 u1, 2 þ e25 φ, 2 , T 12 ¼ c66 u1, 2 þ e26 φ, 2 ,
D2 ¼ e26 u1, 2 ε22 φ, 2 , D3 ¼ e36 u1, 2 ε23 φ, 2 ,
ð5:60Þ
where the time-harmonic factor has been dropped. The equations of motion and the
charge equation reduce to
T 21, 2 ¼ c66 u1, 22 þ e26 φ, 22 ¼ ρ€u1 ¼ ρω2 u1 ,
D2, 2 ¼ e26 u1, 22 ε22 φ, 22 ¼ 0:
ð5:61Þ
Equation (5.61)2 can be integrated to yield
φ¼
e26
u1 þ B1 x2 þ B2 ,
ε22
ð5:62Þ
where B1 and B2 are integration constants, and B2 is immaterial. Substituting
Eq. (5.62) into the expressions for T21, D2, and Eq. (5.61)1, we obtain
T 21 ¼ c66 u1, 2 þ e26 B1 ,
D2 ¼ ε22 B1 ,
ð5:63Þ
c66 u1, 22 ¼ ρω2 u1 ,
ð5:64Þ
and
5.3 Thickness Shear of a Quartz Plate Through e26
167
where
c66 ¼ c66 1 þ k226 ,
k226 ¼
e226
:
ε22 c66
ð5:65Þ
The general solution to Eq. (5.64) and the corresponding expression for the electric
potential are
u1 ¼ A1 sin ξx2 þ A2 cos ξx2 ,
e26
φ ¼ ðA1 sin ξx2 þ A2 cos ξx2 Þ þ B1 x2 þ B2 ,
ε22
ð5:66Þ
where A1 and A2 are integration constants, and
ξ2 ¼
ρ 2
ω:
c66
ð5:67Þ
Then the expression for the stress component relevant to boundary conditions
becomes
T 21 ¼ c66 ðA1 ξ cos ξx2 A2 ξ sin ξx2 Þ þ e26 B1 :
ð5:68Þ
The boundary conditions in Eq. (5.57) require that
c66 A1 ξ cos ξh c66 A2 ξ sin ξh þ e26 B1 ¼ 0,
c66 A1 ξ cos ξh þ c66 A2 ξ sin ξh þ e26 B1 ¼ 0,
e26
2 A1 sin ξh þ 2B1 h ¼ V:
ε22
ð5:69Þ
We add Eq. (5.69)1,2, and also subtract them from each other. Then Eq. (5.69)
becomes:
c66 A1 ξ cos ξh þ e26 B1 ¼ 0,
c66 A2 ξ sin ξh ¼ 0,
e26
2 A1 sin ξh þ 2B1 h ¼ V:
ε22
ð5:70Þ
Equation (5.70) decouples into two sets of equations. We treat free vibration and
forced vibration separately below.
First, consider free vibration with V ¼ 0. For symmetric modes, A1 ¼ B1 ¼ 0.
Equation (5.70) reduces to
c66 A2 ξ sin ξh ¼ 0:
ð5:71Þ
168
5 Vibrations of Finite Bodies
Nontrivial solutions may exist if
sin ξh ¼ 0,
ð5:72Þ
or
ξðnÞ h ¼
nπ
,
2
n ¼ 2, 4, 6, . . . ,
ð5:73Þ
which determines the following resonance frequencies:
ω
ðnÞ
nπ
¼
2h
rffiffiffiffiffiffi
c66
,
ρ
n ¼ 2, 4, 6, . . . :
ð5:74Þ
The corresponding modes are
ðnÞ
u1 ¼ cos ξðnÞ x2 ,
φðnÞ ¼
e26
cos ξðnÞ x2 :
ε22
ð5:75Þ
For antisymmetric modes, A2¼0. Equation (5.70) reduces to
c66 A1 ξ cos ξh þ e26 B1 ¼ 0,
e26
2 A1 sin ξh þ 2B1 h ¼ 0:
ε22
ð5:76Þ
The resonance frequencies are determined by
c66 ξ cos ξh
e26
ε sin ξh
22
e26 e226
¼
c
ξh
cos
ξh
sin ξh ¼ 0,
66
h ε22
ð5:77Þ
or
cot ξh ¼
k226
,
ξh
ð5:78Þ
where
e2
e2
k226
26 2 ¼
k226 ¼ 26 ¼
:
ε22 c66 ε22 c66 1 þ k26
1 þ k226
ð5:79Þ
Equations (5.78) and (5.76) determine the resonance frequencies and modes.
For rotated Y-cut quartz, k226 is very small. If k226 is taken to be zero, the resonant
frequencies determined by Eq. (5.78) are the same as those in Eq. (5.73) but
n assumes 1, 3, 5, and so on. For a small and nonzero k226 , Eq. (5.78) can be solved
169
5.3 Thickness Shear of a Quartz Plate Through e26
approximately. For example, for the most widely used fundamental mode with n ¼ 1,
we can write [2]
ξh ¼
π
Δ,
2
ð5:80Þ
where Δ is a small number. Substituting Eq. (5.80) into Eq. (5.78), for small k226 and
small Δ, we obtain
2
Δ ffi k226 ,
π
ð5:81Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi rffiffiffiffiffiffi π c66
4 2
π c66 1 þ k226
4
1 2 k26 ¼
1 2 k226
ffi
2h ρ π 2h
π
ρ
rffiffiffiffiffiffi
π c66
1 2
4 2
ffi
1 þ k 26
1 2 k26
2hrffiffiffiffiffiffi
2
π
ρ
rffiffiffiffiffiffi
π c66
1 2
4 2
π c66
ffi
1 þ k 26 2 k26 >
:
2h ρ
2
π
2h ρ
ð5:82Þ
or
ω
ð1Þ
Static thickness-shear deformation and the first few thickness-shear modes in a
rotated Y-cut quartz plate are shown in Fig. 5.4.
For forced vibration, from Eq. (5.70), we have A2 ¼ 0 and
0 e26 V 2h e26 V
¼
A1 ¼ ,
e226
ξ cos ξh e26 2
ξh
cos
ξh
2
sin
ξh
c
c66
66
e
ε
22
2 26 sin ξh 2h ε
22
ð5:83Þ
Fig. 5.4 Displacement
distribution of thicknessshear deformation and
modes along plate thickness
Static
n=1
n=2
n=3
170
5 Vibrations of Finite Bodies
c66 ξ cos ξh 0 e26
2 sin ξh V ε
V c66 ξ cos ξh
¼
:
B1 ¼ 22
e2
c
ξ
cos
ξh
e
66
26
c66 ξh cos ξh 2ε2622 sin ξh
2
e26
2 sin ξh 2h ε
22
ð5:84Þ
Hence
D2 ¼ ε22 B1 ¼ ε22
V
ξh
¼ σ e ,
2h ξh k226 tan ξh
ð5:85Þ
where σ e is the surface free charge per unit area on the electrode at x2 ¼ h. The
frequency dependent capacitance per unit area is then
C¼
σ ε22
ξh
¼
:
V
2h ξh k226 tan ξh
ð5:86Þ
The denominator of Eq. (5.86), if set to zero, gives Eq. (5.78). Note the following
limits:
ε22
,
2h
ε22
1
ε22 1 þ k 226 :
lim C ¼
¼
2
ω!0
2h
2h
k26
1
2
1 þ k26
lim C ¼
e26 !0
5.4
ð5:87Þ
Thickness Shear of a Quartz Plate Through e36
Consider an unelectroded plate of rotated Y-cut quartz (see Fig. 5.5). The plate
surfaces at x2 ¼ h are traction-free and are unelectroded. The free space electric
field is neglected.
The plate in Fig. 5.5 can be excited into thickness-shear vibration by a lateral
electric field [3] E3 through the piezoelectric constant e36. In the analysis below [4],
we assume the presence of a spatially uniform driving electric field E3 ¼ E and its
time-harmonic factor is dropped as usual. We begin with the following trial fields.
x2
Fig. 5.5 An unelectroded
plate of rotated Y-cut quartz
2h
x1
5.4 Thickness Shear of a Quartz Plate Through e36
171
They will be shown to satisfy all governing equations and boundary conditions of
the theory of piezoelectricity:
u1 ¼ u1 ðx2 Þ, u2 ¼ u3 ¼ 0,
φ ¼ ϕðx2 Þ Ex3 :
ð5:88Þ
The nontrivial components of the strain, electric field, stress, and electric displacement components are, correspondingly,
2S12 ¼ u1, 2 ,
E 2 ¼ ϕ, 2
E 3 ¼ E,
T 31 ¼ c56 u1, 2 þ e25 ϕ, 2 e35 E,
T 21 ¼ c66 u1, 2 þ e26 ϕ, 2 e36 E,
D2 ¼ e26 u1, 2 ε22 ϕ, 2 þ ε23 E,
D3 ¼ e36 u1, 2 ε32 ϕ, 2 þ ε33 E:
ð5:89Þ
ð5:90Þ
The equation of motion and the charge equation of electrostatics left to be satisfied
by u1 and ϕ are
T 21, 2 ¼ c66 u1, 22 þ e26 ϕ, 22 ¼ ρω2 u1 ,
D2, 2 ¼ e26 u1, 22 ε22 ϕ, 22 ¼ 0:
ð5:91Þ
The displacement and potential fields determined from Eq. (5.91) are
u1 ¼ A1 sin ξx2 þ A2 cos ξx2 ,
e26
ϕ ¼ ðA1 sin ξx2 þ A2 cos ξx2 Þ þ B1 x2 þ B2 ,
ε22
ð5:92Þ
c66 . Then the stress and electric displacement components needed
where ξ2 ¼ ρω2 =
for boundary conditions are
T 21 ¼ c66 ðA1 ξ cos ξx2 A2 ξ sin ξx2 Þ þ e26 B1 e36 E,
ð5:93Þ
D2 ¼ ε22 B1 þ ε23 E:
ð5:94Þ
For traction-free and unelectroded plate surfaces, we have
T 21 ðx2 ¼ hÞ ¼ c66 ðA1 ξ cos ξh A2 ξ sin ξhÞ þ e26 B1 e36 E ¼ 0,
ð5:95Þ
T 21 ðx2 ¼ hÞ ¼ c66 ðA1 ξ cos ξh þ A2 ξ sin ξhÞ þ e26 B1 e36 E ¼ 0,
ð5:96Þ
D2 ðx2 ¼ hÞ ¼ ε22 B1 þ ε23 E ¼ 0:
ð5:97Þ
Equations (5.95), (5.96), and (5.97) can be solved for A1, A2, and B1. From these
equations it can be seen that the lateral electric field E drives the plate as effective
surface traction and charge.
172
5.5
5 Vibrations of Finite Bodies
Mass Sensitivity of Thickness-Shear Frequency
The electrodes on the crystal plate surfaces in Sects. 5.1, 5.2, and 5.3 were assumed
to be very thin. Their mechanical effects such as inertia and stiffness were neglected.
In this section we study the inertial effect of the electrodes only, which can also be
viewed as the effect of additional mass layers on the surfaces of a plate with very thin
electrodes with negligible mass (see Fig. 5.6).
From Eq. (5.61), the governing equations are still
T 21, 2 ¼ c66 u1, 22 þ e26 φ, 22 ¼ ρ€u1 ¼ ρω2 u1 ,
D2, 2 ¼ e26 u1, 22 ε22 φ, 22 ¼ 0:
ð5:98Þ
With consideration of the mass layer inertia, the boundary conditions are
T 2 j ¼ 2ρ0 h0 €uj , x2 ¼ h,
T 2 j ¼ 2ρ0 h00 €uj , x2 ¼ h,
φðx2 ¼ hÞ ¼ φðx2 ¼ hÞ,
ð5:99Þ
where ρ0 is the mass layer density. The general solution to Eq. (5.98) is still given by
Eq. (5.66):
u1 ¼ A1 sin ξx2 þ A2 cos ξx2 ,
e26
φ ¼ ðA1 sin ξx2 þ A2 cos ξx2 Þ þ B1 x2 þ B2 ,
ε22
ð5:100Þ
c66 . For boundary conditions we also need the following field from
where ξ2 ¼ ρω2 =
Eq. (5.68):
T 21 ¼ c66 ðA1 ξ cos ξx2 A2 ξ sin ξx2 Þ þ e26 B1 :
ð5:101Þ
Substituting Eqs. (5.100) and (5.101) into the boundary conditions in Eq. (5.99), we
obtain
x2
Fig. 5.6 An electroded
plate of rotated Y-cut quartz
with mass layers
2h⬘
Mass layers
2h⬘
Crystal
plate
h
h
x1
5.5 Mass Sensitivity of Thickness-Shear Frequency
173
c66 ξðA1 cos ξh A2 sin ξhÞ e26 B1
¼ ω2 2ρ0 h0 ðA1 sin ξh þ A2 cos ξhÞ,
c66 ξðA1 cos ξh þ A2 sin ξhÞ þ e26 B1
¼ ω2 2ρ0 h00 ðA1 sin ξh þ A2 cos ξhÞ,
e26
A1 sin ξh þ B1 h ¼ 0:
ε22
ð5:102Þ
For nontrivial solutions of the undetermined constants, the determinant of the
coefficient matrix of Eq. (5.102) has to vanish. This results in the following
frequency equation:
2
0 0
c66 ξ cos ξh þ ω2 2ρ0 h00 sin ξh
c66 ξ cos ξh ω 2ρ h sin ξh
e26
sin ξh
ε22
c66 ξ sin ξh þ ω2 2ρ0 h0 cos ξh
c66 ξ sin ξh þ ω2 2ρ0 h00 cos ξh
0
e26 e26 ¼ 0:
h ð5:103Þ
00
0
We examine the special case of symmetric mass layers when h ¼ h . In this case
the modes can be separated into symmetric and antisymmetric ones and Eq. (5.103)
splits into two factors. For symmetric modes A1 ¼ 0 and B1 ¼ 0. The frequency
equation is
tan ξh ¼ Rξh,
ð5:104Þ
where R is the mass ratio defined by
R¼
2ρ0 h0
:
ρh
ð5:105Þ
For antisymmetric modes A2 ¼ 0. The frequency equation is
cot ξh k226
¼ Rξh:
ξh
ð5:106Þ
For the fundamental antisymmetric mode, if we write [2]
ξh ¼
π
Δ,
2
ð5:107Þ
we can obtain from Eq. (5.106), for small Δ, small k226 , and small R,
2
π
Δ ffi k226 þ R,
π
2
ð5:108Þ
174
5 Vibrations of Finite Bodies
which implies that
ω
ð1Þ
π
ffi
2h
rffiffiffiffiffiffi 4 2
c66
1 2 k26 R :
π
ρ
ð5:109Þ
Obviously, the mass layer inertia represented by R lowers the frequency as expected.
When R ¼ 0, Eq. (5.109) reduces to Eq. (5.82). The frequency effect of the inertia of
the mass layer can be used to design sensors for measuring the mass layer density or
thickness [5]. These mass sensors are called quartz crystal microbalances (QCMs).
5.6
Azimuthal Shear of a Ceramic Cylinder
Consider an infinite circular ceramic cylinder of inner radius a and outer radius b (see
Fig. 5.7). It is made of ceramics with azimuthal poling. We arrange (2,3,1) along
(r,θ,z) so that the poling direction is 3. The inner and outer surfaces are electroded
and the electrodes may be open or shorted. There is no mechanical load applied.
We study azimuthal or circumferential shear vibrations described by
uθ ¼ uθ ðr Þexpðiω t Þ,
φ ¼ φðr Þexpðiω t Þ:
ur ¼ uz ¼ 0,
ð5:110Þ
For free vibrations, the boundary conditions are
n j T ji ¼ 0, r ¼ a, b,
φðr ¼ aÞ ¼ φðr ¼ bÞ, if the electrodes are shorted,
or Dr ¼ 0, r ¼ a, b, if the electrodes are open:
ð5:111Þ
Corresponding to Eq. (5.110), the nontrivial components of strain, electric field,
stress, and electric displacement are
Fig. 5.7 A ceramic cylinder
with azimuthal poling
b
2
a
3
P
5.6 Azimuthal Shear of a Ceramic Cylinder
175
duθ uθ
,
dr
r
dφ
E2 ¼ Er ¼ ,
dr
duθ uθ
dφ
T 4 ¼ T rθ ¼ c44
þ e15 ,
dr
r
dr
duθ uθ
dφ
D2 ¼ Dr ¼ e15
ε11 :
dr
dr
r
S4 ¼ 2Srθ ¼
ð5:112Þ
ð5:113Þ
Thus on the boundary surfaces at r ¼ a and b there are no circumferential electric
fields. The electric potential assumes constant values on the electrodes as required.
The stress components Trr and Trz vanish everywhere, particularly on the lateral
surfaces of the cylinder. The equation of motion and the charge equation to be
satisfied are
dT rθ 2
þ T rθ ¼ ρω2 uθ ,
r
dr
1
ðrDr Þ, r ¼ 0:
r
ð5:114Þ
Equation (5.114)2 can be integrated as
Dr ¼ e15
C3
,
r
ð5:115Þ
where C3 is an integration constant. Then, from Eq. (5.113)2 we have
φ, r ¼
e15
C3
2Srθ :
ε11
r
ð5:116Þ
The substitution of Eq. (5.116) into Eq. (5.113)1 gives
duθ uθ
e15
C3
2Srθ T rθ ¼ c44
þ e15
r
ε11
r
dr
2
duθ uθ
e15 C 3
,
¼ c44
dr
r
ε11 r
ð5:117Þ
where
c44 ¼ c44 1 þ k215 ,
k215 ¼
e215
:
ε11 c44
ð5:118Þ
176
5 Vibrations of Finite Bodies
Substituting Eq. (5.117) into Eq. (5.114)1, we obtain
2
d uθ 1duθ 1
e215 C3
þ
u
c44
þ
θ
ε11 r 2
dr 2 r dr r2
2
duθ uθ
e215 C3
þ c44
¼ ρω2 uθ ,
2
r
dr
r
ε11 r 2
ð5:119Þ
which can be written as
2
d uθ 1duθ 1
e215 C3
2
c44
þ
u
u
¼
,
þ
ρω
θ
θ
r dr r 2
ε11 r 2
dr 2
ð5:120Þ
d 2 uθ 1 duθ 1
C3
uθ þ ξ2 uθ ¼ k215 2 ,
þ
r dr r 2
r
dr 2
ð5:121Þ
e2
e2
k215
15 2 ¼
k215 ¼ 15 ¼
:
ε11 c44 ε11 c44 1 þ k15
1 þ k215
ð5:122Þ
or
where
ξ2 ¼
ρω2
,
c44
We introduce a dimensionless radial coordinate R ¼ ξ r. Equation (5.121)
becomes
d2 uθ 1 duθ
1
C3
þ 1 2 uθ ¼ k215 2 ,
þ
2
R
dR
R
R
dR
ð5:123Þ
which is Bessel’s equation of order one.
In the following we consider the case when the electrodes at r ¼ a and b are
open. The corresponding electrical boundary conditions in Eq. (5.111) imply that
C3 ¼ 0. Then the general solution to Eq. (5.123) is
uθ ¼ C 1 J 1 ðRÞ þ C2 Y 1 ðRÞ
¼ C 1 J 1 ðξr Þ þ C 2 Y 1 ðξr Þ,
ð5:124Þ
where J1 and Y1 are the first-order Bessel functions of the first and second kind,
respectively. From Eq. (5.117) the shear stress is
5.7 Axial Shear of a Ceramic Cylinder
T rθ
177
duθ uθ
¼ c44
dr 0 r
¼ c44 C 1 J 1 ðξr Þξ þ C 2 Y 01 ðξr Þξ
1
c44 ½C 1 J 1 ðξr Þ þ C 2 Y 1 ðξ r Þ
r
J 1 ðξ r Þ
Y 1 ðξ r Þ
¼ C1 c44 ξ J 01 ðξr Þ þ C 2 c44 ξ Y 01 ðξr Þ :
ξr
ξr
ð5:125Þ
The traction-free boundary conditions require that
J 1 ð ξ aÞ
Y 1 ð ξ aÞ
þ C 2 c44 ξ Y 01 ðξaÞ ¼ 0,
ξa
ξa
J 1 ð ξ bÞ
Y 1 ð ξ bÞ
þ C 2 c44 ξ Y 01 ðξbÞ ¼ 0:
C 1 c44 ξ J 01 ðξbÞ ξb
ξb
C 1 c44 ξ J 01 ðξaÞ ð5:126Þ
The frequency equation is given by that the determinant of the coefficient matrix of
Eq. (5.126) has to vanish, that is,
J 0 ðξaÞ J 1 ðξ aÞ
1
ξa
J
1
0
J ðξbÞ ðξ bÞ
1
ξb
5.7
Y 1 ðξ aÞ ξ a ¼ 0:
Y 1 ðξ bÞ Y 01 ðξbÞ ξb Y 01 ðξaÞ ð5:127Þ
Axial Shear of a Ceramic Cylinder
Consider an infinite ceramic circular cylinder of inner radius a and outer radius b (see
Fig. 5.8). We orient (1,2,3) along (r,θ,z) and the poling direction is 3. The inner and
outer surfaces are electroded.
We study axial-shear vibrations which are antiplane motions. The notation for
antiplane motions in Sect. 2.9 is employed. The governing equations are:
c∇2 u ¼ ρ€u, a < r < b,
∇2 ψ ¼ 0, a < r < b,
e
φ ¼ ψ þ u, a < r < b:
ε
For boundary conditions we consider the following possibilities:
ð5:128Þ
178
5 Vibrations of Finite Bodies
x2
Fig. 5.8 A ceramic cylinder
with axial poling
b
x1
a
Ceramic
u ¼ 0, r ¼ a,
or T rz ¼ 0,
φðr ¼ aÞ ¼ φðr
or Dr ¼ 0,
b, if the cylindical surfaces are fixed,
r ¼ a, b, if the cylindical surfaces are free;
¼ bÞ, if the electrodes are shorted,
r ¼ a, b, if the electrodes are open:
ð5:129Þ
For axisymmetric motions [6] we look for solutions in the following form:
uðr; t Þ ¼ uðr Þexpðiωt Þ,
ψ ðr; t Þ ¼ ψ ðr Þexpðiωt Þ,
ð5:130Þ
and drop the exponential time factor. The equations for u and ψ reduce to
2
∂ u 1 ∂u
ρω2
¼
u,
þ
∂r 2 r ∂r
c
2
∂ ψ 1 ∂ψ
¼ 0:
∇2 ψ ¼ 2 þ
∂r
r ∂r
∇2 u ¼
ð5:131Þ
The general solution to Eq. (5.131) is
u ¼ A1 J 0 ðξr Þ þ A2 Y 0 ðξr Þ,
r
ψ ¼ A3 ln þ A4 ,
b
ð5:132Þ
where A1, A2, A3, and A4 are undetermined constants, J0 and Y0 are zero-order Bessel
functions of the first and second kind, and
ξ2 ¼
ρω2
:
c
The stress and electric displacement components are
ð5:133Þ
5.7 Axial Shear of a Ceramic Cylinder
179
e
r
φ ¼ ½A1 J 0 ðξr Þ þ A2 Y 0 ðξr Þ þ A3 ln þ A4 ,
ε
b
A3
T rz ¼ cuz, r þ eψ , r ¼ cξ½A1 J 1 ðξr Þ þ A2 Y 1 ðξr Þ þ e ,
r
A3
Dr ¼ εψ , r ¼ ε ,
r
ð5:134Þ
where J 00 ¼ J 1 and Y 00 ¼ Y 1 have been used.
In the case when the two cylindrical surfaces are fixed and the two electrodes are
shorted and grounded, we have
u ¼ 0,
φ ¼ 0,
r ¼ a, b,
r ¼ a, b,
ð5:135Þ
ψ ¼ 0,
r ¼ a, b:
ð5:136Þ
A3 ¼ 0,
A4 ¼ 0,
ð5:137Þ
which implies that
Hence
and the frequency equation is
J 0 ðξaÞ Y 0 ðξaÞ J 0 ðξbÞ Y 0 ðξbÞ ¼ 0:
ð5:138Þ
In the case when the cylindrical surfaces are traction-free and electrically open,
we have
T rz ¼ 0,
Dr ¼ 0,
r ¼ a, b,
r ¼ a, b:
ð5:139Þ
Then A3 ¼ 0 and the frequency equation is
J 1 ðξaÞ Y 1 ðξaÞ J 1 ðξbÞ Y 1 ðξbÞ ¼ 0:
ð5:140Þ
If the cylindrical surfaces are traction-free with shorted and grounded electrodes,
we have
180
5 Vibrations of Finite Bodies
T rz ¼ 0, r ¼ a, b,
φ ¼ 0, r ¼ a, b:
ð5:141Þ
In this case the frequency equation is
J 1 ðξaÞ
J 1 ðξbÞ
Y 1 ðξaÞ Y 1ðξbÞ b
k2 J 0 ðξaÞ J 0 ðξbÞ J 1 ðξaÞ aJ 1 ðξbÞ ¼
:
a
b
ξb ln Y 0 ðξaÞ Y 0 ðξbÞ Y 1 ðξaÞ Y 1 ðξbÞ b
a
ð5:142Þ
For large x, Bessel functions can be approximated by
rffiffiffiffiffi
2
νπ π
J v ð xÞ ffi
cos x ,
πx
2 4
rffiffiffiffiffi
2
νπ π
Y v ð xÞ ffi
sin x :
πx
2 4
ð5:143Þ
Then it can be shown that, for large a and b, Eq. (5.142) simplifies to
"
rffiffiffi#
b
b
k2
sin ξðb aÞ ¼
1þ
cos ξðb aÞ 2
:
a
a
ξb ln ab
ð5:144Þ
Setting 2h ¼ ba and letting a, b!1, we have
a
b 2h
2h
2h
¼ ξb ln 1 ffi ξb ¼ ξ2h,
ξb ln ¼ ξb ln
b
b
b
b
sin ξðb aÞ ¼ sin ðξ2hÞ, rffiffiffi
b
b
1þ
cos ξðb aÞ 2
a
a
ffi 2 cos ðξ2hÞ 2 ¼ 2½1 cos ðξ2hÞ ¼ 4 sin 2 ξh:
ð5:145Þ
Then Eq. (5.144) reduces to
ξh
tan ξh ¼ 2 ,
k
ð5:146Þ
which is the frequency equation for thickness-shear vibrations of an electroded
ceramic plate given by Eq. (5.50).
181
5.8 Extension of a Ceramic Rod Through e33
5.8
Extension of a Ceramic Rod Through e33
Consider a cylindrical rod of length l made from polarized ceramics with axial poling
(see Fig. 5.9). The shape of the cross section of the rod can be arbitrary but it has to
be small compared to the length of the rod. The entire surface of the rod is tractionfree. The cylindrical surface is unelectroded. The electric field in the surrounding
free space is neglected. The two end faces are electroded. The rod can be excited into
extensional vibration through e33 if a voltage is applied across the two end electrodes. In this section we consider free vibrations with open electrodes. The wavelength of the vibration modes is much larger than the width and thickness of the rod.
We consider a thin rod and make the following approximations of uniaxial stress
and electric displacement:
T 33 ¼ T 33 ðx3 ; t Þ, all other T ij ¼ 0,
D3 ¼ D3 ðx3 ; t Þ, D1 ¼ D2 ¼ 0:
ð5:147Þ
Thus the boundary conditions on the lateral surface are satisfied. The relevant
equations of motion and charge reduce to
T 33, 3 ¼ ρ€u3 ,
D3, 3 ¼ 0:
ð5:148Þ
From Eq. (2.19), the relevant constitutive relations take the following form:
S33 ¼ s33 T 33 þ d33 E3 ,
D3 ¼ d33 T 33 þ ε33 E3 ,
ð5:149Þ
S33 ¼ u3, 3 , E3 ¼ φ, 3 ,
E
T
s33 ¼ s33
, ε33 ¼ ε33
:
ð5:150Þ
where
From Eq. (5.149) we obtain
Fig. 5.9 An axially poled
ceramic rod
x1
Electrode
Electrode
P
x2
l
Traction-free, unelectroded
x3
182
5 Vibrations of Finite Bodies
1
1
ðS33 d33 E 3 Þ ¼
u3, 3 þ d 33 φ, 3 ,
s33
s33
h
l 2 i
d 33
d233
d33
S33 þ ε33 u3, 3 ε33 1 k33
E3 ¼
φ, 3 ,
D3 ¼
s33
s33
s33
T 33 ¼
ð5:151Þ
where
l
k33
2
¼
d233
:
ε33 s33
ð5:152Þ
Substituting Eq. (5.151) into Eq. (5.148), we obtain
1
u3, 33 þ d33 φ, 33 ¼ ρ€u3 ,
s33
h
l 2 i
d 33
u3, 33 ε33 1 k33
φ, 33 ¼ 0:
s33
ð5:153Þ
Consider the case of open end electrodes. The electrical end conditions are
D3 ¼ 0,
x3 ¼ l=2:
ð5:154Þ
Equation (5.148)2 implies that D3 is independent of x3. Hence, from Eq. (5.154), D3
is identically zero, which implies that
φ, 3 ¼
d33
h
l 2 i u3 , 3
ε33 s33 1 k 33
l 2
k33
1
¼
u3 , 3 :
d33 1 k l 2
ð5:155Þ
33
Then
1
l 2 i u3 , 3 ,
1 k 33
ð5:156Þ
1
h
l 2 i u3, 33 ¼ ρ€u3 :
s33 1 k33
ð5:157Þ
h
T 33 ¼
s33
and Eq. (5.153)1 becomes
For traction-free mechanical end conditions, in harmonic motions, the eigenvalue
problem is
5.9 Extension of a Ceramic Rod Through e31
h
l 2 i 2
u3, 33 þ ρs33 1 k33
ω u3 ¼ 0,
u3, 3 ¼ 0, x3 ¼ l=2:
183
l=2 < x3 < l=2,
ð5:158Þ
The solution of ω ¼ 0 and u3 ¼ constant represents a rigid-body mode. For the rest of
the modes we try u3 ¼ sinkx1. Then, from Eq. (5.158)1,
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h
l 2 i
k ¼ ω ρs33 1 k33
:
ð5:159Þ
To satisfy Eq. (5.158)2 we must have
l
cos k ¼ 0,
2
)
l nπ
k ð nÞ ¼ ,
2
2
n ¼ 1, 3, 5, . . . ,
ð5:160Þ
or
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h
l 2 i l nπ
ω
¼ ,
ρs33 1 k 33
2
2
nπ
ðnÞ
ω ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
, n ¼ 1, 3, 5, . . . :
h
l 2 i
l ρs33 1 k 33
ðnÞ
ð5:161Þ
Similarly, by considering u3 ¼ coskx1, the following frequencies can be determined:
nπ
ωðnÞ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
,
h
l 2 i
l ρs33 1 k33
n ¼ 2, 4, 6, . . . :
ð5:162Þ
l
raises the resonant frequencies, which is
Equations (5.161) and (5.162) show that k33
the so-called piezoelectric stiffening effect. The frequencies in Eqs. (5.161) and
(5.162) are integral multiples of ω(1) and are called harmonics. ω(1) is called the
fundamental and the rest are called the overtones.
5.9
Extension of a Ceramic Rod Through e31
Consider a rectangular ceramic rod of length l, width w, and thickness t as shown in
Fig. 5.10, where l > > w > > t. The poling direction is along x3. We are interested in
the extensional vibration of the rod [3].
As an approximation, it is appropriate to take the vanishing boundary stresses on
the surfaces bounding the two small dimensions to vanish everywhere.
Consequently
184
5 Vibrations of Finite Bodies
x3
Fig. 5.10 A ceramic rod
with rectangular cross
section
x2
x1
t
l
T 11 ¼ T 1 ðx1 ; t Þ, and all other T ij ¼ 0:
w
ð5:163Þ
If the surfaces of the area lw are fully electroded with a driving voltage V across
the electrodes, the appropriate electrical conditions are
E1 ¼ E 2 ¼ 0,
V
E3 ¼ :
t
ð5:164Þ
The pertinent constitutive relations are from Eq. (2.19):
S1 ¼ s11 T 1 þ d31 E 3 ,
D3 ¼ d 31 T 1 þ ε33 E3 ,
ð5:165Þ
E
s33 ¼ s33
,
ð5:166Þ
where
T
ε33 ¼ ε33
:
Equation (5.165)1 can be inverted to give
T1 ¼
1
d 31
S1 E3 :
s11
s11
ð5:167Þ
Then the differential equation of motion and boundary conditions are
1
u1, 11 ¼ ρ€u1 , l=2 < x1 < l=2,
s11
1
d31 V
¼ 0, x1 ¼ l=2:
T 1 ¼ u1 , 1 þ
s11
s11 t
ð5:168Þ
Equation (5.168) shows that the applied voltage effectively acts like two extensional
end forces on the rod. For free vibrations, the electrodes are shorted and V ¼ 0. We
look for a solution in the form
u1 ðx1 ; t Þ ¼ u1 ðx1 Þexpðiω t Þ:
ð5:169Þ
5.9 Extension of a Ceramic Rod Through e31
185
Then the eigenvalue problem is
u1, 11 þ ρs11 ω2 u1 , l=2 < x1 < l=2,
u1, 1 ¼ 0, x1 ¼ l=2:
ð5:170Þ
The solution of ω ¼ 0 and u1 ¼ constant represents a rigid-body mode. For the rest of
the modes we try u1 ¼ sinkx1. Then, from Eq. (5.170)1,
pffiffiffiffiffiffiffiffi
k ¼ ω ρs11 :
ð5:171Þ
To satisfy (5.170)2 we must have
l
cos k ¼ 0,
2
)
l nπ
k ð nÞ ¼ ,
2
2
n ¼ 1, 3, 5, . . . ,
ð5:172Þ
or
pffiffiffiffiffiffiffiffi l nπ
ωðnÞ ρs11 ¼ ,
2
2
nπ
ωðnÞ ¼ pffiffiffiffiffiffiffiffi ,
l ρs11
n ¼ 1, 3, 5, . . . :
ð5:173Þ
Similarly, by considering u1 ¼ coskx1, the following frequencies can be determined:
nπ
ωðnÞ ¼ pffiffiffiffiffiffiffiffi ,
l ρs11
n ¼ 2, 4, 6, . . . :
ð5:174Þ
The frequencies in Eqs. (5.173) and (5.174) do not show any piezoelectric stiffening
effect because the electric field is zero in this case when the two electrodes are
shorted.
If the surfaces of the area lt are fully electroded with a driving voltage V across the
electrodes, the appropriate electrical conditions are
E1 ¼ 0,
D3 ¼ 0,
V
E2 ¼ :
w
ð5:175Þ
The pertinent constitutive relations are
S1 ¼ s11 T 1 þ d21 E 2 þ d31 E 3 ,
D2 ¼ d21 T 1 þ ε22 E 2 þ ε23 E 3 :
ð5:176Þ
From the boundary conditions on the areas of lw, we take the following to be
approximately true everywhere:
D3 ¼ d31 T 1 þ ε32 E 2 þ ε33 E 3 ¼ 0:
ð5:177Þ
186
5 Vibrations of Finite Bodies
With Eq. (5.177), we can write Eq. (5.176) as
S1 ¼ ~s 11 T 1 þ d~ 21 E2 ,
D2 ¼ d~ 21 T 1 þ e
ε 22 E 2 ,
ð5:178Þ
~s 11 ¼ s11 d231 =ε33 ,
d~ 21 ¼ d21 d31 ε23 =ε33 ,
e
ε 22 ¼ ε22 ε223 =ε33 :
ð5:179Þ
where
If the surfaces of areas lw and lt are not electroded, the appropriate electrical
conditions are
D2 ¼ D3 ¼ 0:
ð5:180Þ
In this case, from Eq. (2.20), the pertinent constitutive relations are
S1 ¼ s11 T 1 þ g11 D1 ,
E 1 ¼ g11 T 1 þ β11 D1 :
5.10
ð5:181Þ
Radial Vibration of a Circular Ceramic Plate
A circular disk of ceramics poled in the thickness direction is positioned in a
coordinate system as shown in Fig. 5.11. The top and bottom faces of the disk are
traction-free and are completely coated with electrodes. The electrodes are
connected to a voltage source of potential V exp (iω t). We consider axisymmetric
radial modes [3].
Under these circumstances, the boundary conditions at x3 ¼ b are
T 3 j ¼ 0, x3 ¼ b,
V
φ ¼ expðiω t Þ, x3 ¼ b:
2
ð5:182Þ
x3
Fig. 5.11 A circular
ceramic plate with thickness
poling
a
x2
2b
P
x1
5.10
Radial Vibration of a Circular Ceramic Plate
187
Since T3, T4, and T5 vanish on both major surfaces of the plate and the plate is thin,
these stresses cannot depart much from zero. Consequently they are assumed to
vanish throughout. Thus we assume that
T 3 ¼ T 4 ¼ T 5 ¼ 0:
ð5:183Þ
Furthermore, since the plate is thin and has conducting surfaces,
E 1 ¼ 0,
E2 ¼ 0,
V
E 3 ¼ expðiω t Þ:
2b
ð5:184Þ
∂
¼ 0:
∂θ
ð5:185Þ
We consider radial modes with
uθ ¼ 0,
The constitutive relations are
p
p
p
T rr ¼ c11
ur, r þ c12
ur =r þ e31
φ, 3 ,
p
p
p
T θθ ¼ c11 ur =r þ c12 ur, r þ e31
φ, 3 ,
T rθ ¼ 0,
p
p
D3 ¼ e31
ður, r þ ur =r Þ ε33
φ, 3 ,
ð5:186Þ
p
c11
¼ c11 c213 =c33 ,
p
c12
¼ c12 c213 =c33 ,
p
¼ e31 e33 c13 =c33 ,
e31
p
ε33
¼ ε33 þ e233 =c33 ,
ð5:187Þ
where
are the effective material constants for a thin plate after the relaxation of the normal
stress in the thickness direction. The one remaining equation of motion in cylindrical
coordinates is
∂T rr T rr T θθ
þ
¼ ρ€ur :
∂r
r
ð5:188Þ
Substituting from Eq. (5.186) for the stress components, we obtain
p c11
u, rr þ ur, r =r ur =r 2 ¼ ρ€ur ,
ð5:189Þ
188
5 Vibrations of Finite Bodies
which, since we are assuming a steady-state problem with frequency ω, becomes
ur, rr þ
ur , r
1
þ ξ2 2 ur ¼ 0,
r
r
ð5:190Þ
where
ξ2 ¼
ω2
ðvp Þ2
,
p
ðvp Þ2 ¼ c11
=ρ:
ð5:191Þ
Equation (5.190) can be written as Bessel’s equation of order one. For a solid disk,
the motion at the origin is zero and the general solution is
ur ¼ BJ 1 ðξ r Þexpðiω t Þ,
ð5:192Þ
where J1 is the first kind Bessel function of the first order. Equation (5.192) is subject
to the boundary condition
T rr ¼ 0,
r ¼ a,
ð5:193Þ
J1
p V
,
¼ e31
2b
a
ð5:194Þ
hence Eq. (5.193) requires that
p dJ 1 c11
B dr
p
þ c12
B
r¼a
where, for convenience, the argument of the Bessel function is not written. From
Eq. (5.194), B can be expressed in terms of V as follows:
B ¼ ð1 σ p Þ
J 1 ð ξ aÞ
ξJ 0 ðξ aÞ
a
1 p
e31
p
c11
V
,
2b
ð5:195Þ
where
dJ 1 ðxÞ
J 1 ð xÞ
¼ J 0 ð xÞ ,
dx
x
ð5:196Þ
p
p
σ p ¼ c12
=c11
,
ð5:197Þ
has been used and
which may be interpreted as a planar Poisson’s ratio, since the material is isotropic in
the plane normal to x3. The total charge on the electrode at the bottom of the plate is
given by
5.11
Orthogonality of Modes
189
Z
Q ¼
Z
D3 dA ¼ 2π
e
a
D3 rdr:
ð5:198Þ
0
A
The substitution of Eq. (5.192) into Eq. (5.186)4 and then into Eq. (5.198) yields
p
p
Qe ¼ 2πe31
aBJ 1 ðξ aÞ πε33
Va2 =2b:
ð5:199Þ
Hence we obtain for the current that flows to the bottom electrode of the plate as
I¼
dQe
dt"
#
p 2
p
2 k 31
J 1 ð ξ aÞ
ε33
πa2 V
,
1
¼ iω
2b
ð1 σ p ÞJ 1 ðξ aÞ ξ aJ 0 ðξ aÞ
ð5:200Þ
where
p 2
k31
p 2
e31
¼ p p :
ε33 c11
ð5:201Þ
For free vibrations, the applied voltage is zero. From Eq. (5.194),
dJ 1 J1
þ σ p ¼ 0,
dr r¼a
a
ð5:202Þ
which determines resonance frequencies. Or, at resonances, the current goes to
infinity. This condition is determined by setting the square bracketed factor in the
denominator of Eq. (5.200) equal to zero. The resulting equation is
ξ aJ 0 ðξ aÞ
¼ 1 σp ,
J 1 ð ξ aÞ
ð5:203Þ
which can be brought into the same form as Eq. (5.202). The antiresonance frequencies result when the current goes to zero. The resulting equation is
p 2
ξ aJ 0 ðξ aÞ
¼ 1 σ p 2 k31
:
J 1 ð ξ aÞ
5.11
ð5:204Þ
Orthogonality of Modes
The free vibration of a piezoelectric body with frequency ω is governed by the
differential equations
190
5 Vibrations of Finite Bodies
T ji, j ðu; φÞ ¼ ρω2 ui ,
Di, i ðu; φÞ ¼ 0,
ð5:205Þ
T ji ðu; φÞ ¼ cjikl Skl ekji E k ¼ cjikl uk, l þ ekji φ, k ,
Di ðu; φÞ ¼ eikl Skl þ εik E k ¼ eikl uk, l εik φ, k :
ð5:206Þ
where
From Eqs. (2.32), (2.34), and (2.35), the homogeneous boundary conditions are
ui ¼ 0 on Su ,
n j T ji ¼ 0 on ST ,
φ ¼ 0 on Sφ ,
ni Di ¼ 0 on SD :
ð5:207Þ
For any two vectors u and v as well as two scalars φ and ψ, with the divergence
theorem, we have
R V T jiR, j ðu;
φÞvi þ Di, i u; φ ψ dV
¼ V f T ji ðu; φÞvi , j T ji ðu; φÞvi, j
þ ½Di ðu; φÞψ , i Di ðu; φÞψ , i gdV
R ¼ R S n j T ji ðu; φÞvi þ ni Di u; φ ψ dS
V cjikl uk, l vi, j þ ekji φ, k vi, j þ eikl uk, l ψ , i εik φ, k ψ , i dV:
ð5:208Þ
Similarly,
R V T jiR, j ðv; ψ Þui þ Di, i v; ψ φ dV ¼ R S n j T ji ðv; ψ Þui þ ni Di v; ψ φ dS
V cjikl vk, l ui, j þ ekji ψ , k ui, j þ eikl vk, l φ, i εik ψ , k φ, i dV:
ð5:209Þ
Subtracting Eqs. (5.208) and (5.209), using the symmetries of the material constants,
we obtain the following identity [7] which corresponds to Green’s identity for the
Laplacian operator and the reciprocal theorem in elasticity:
R V T jiR, j ðu; φÞvi þ Di, i u; φ ψ dV
RV T ji, j ðv; ψ Þui þ Di, i v;
ψ φ dV
¼ R S n j T ji ðu; φÞvi þ ni Di u; φ ψ dS
S n j T ji ðv; ψ Þui þ ni Di v; ψ φ dS:
ð5:210Þ
Now consider two vibration modes corresponding to two different resonance
frequencies ω(m) and ω(n), that is,
5.11
Orthogonality of Modes
191
2 ðmÞ
ðmÞ
T ij, i þ ρ ωðmÞ u j ¼ 0,
ð5:211Þ
ðmÞ
Di, i ¼ 0,
and
2 ðnÞ
ðnÞ
T ij, i þ ρ ωðnÞ u j ¼ 0,
ð5:212Þ
ðnÞ
Di, i ¼ 0:
The modes in Eqs. (5.211) and (5.212) also satisfy the boundary conditions in
Eq. (5.207). In Eq. (5.210), letting
u ¼ uðmÞ , φ ¼ φðmÞ ,
v ¼ uðnÞ , ψ ¼ φðnÞ ,
ð5:213Þ
using Eqs. (5.207), (5.211) and (5.212), we obtain
ω
ðnÞ
2
ω
ðmÞ
2
Z
ðmÞ ðnÞ
ρu j u j dV ¼ 0:
ð5:214Þ
V
Since the two resonance frequencies are different, Eq. (5.214) implies that
Z
ðmÞ ðnÞ
ρu j u j dV ¼ 0,
ð5:215Þ
V
which is called the orthogonality of modes [2].
In Eq. (5.208), if we choose
u ¼ uðnÞ ,
v ¼ uðnÞ ,
φ ¼ φðnÞ ,
ψ ¼ φð n Þ ,
ð5:216Þ
we obtain
2 R
ðnÞ ðnÞ
ωðnÞ V ρu j u j dV
R
ðnÞ ðnÞ
ðnÞ ðnÞ
ðnÞ ðnÞ
ðnÞ ðnÞ
¼ V cjikl uk, l ui, j þ ekji φ, k ui, j þ eikl uk, l φ, i εik φ, k φ, i dV,
ð5:217Þ
which can be further written as
R
ω
ðnÞ
2
¼
ðnÞ ðnÞ
V
ðnÞ ðnÞ
ðnÞ ðnÞ
cjikl uk, l ui, j þ 2ekji φ, k ui, j εik φ, k φ, i
R
ðnÞ ðnÞ
V ρu j u j dV
dV
:
ð5:218Þ
192
5 Vibrations of Finite Bodies
Equation (5.218) can be viewed as the ratio between the electric enthalpy and the
kinetic energy corresponding to the mode. More generally, we can define the
following functional for the variational formulation of the free vibration
eigenvalue problem:
R Rðu; φÞ ¼
V
cjikl uk, l ui, j þ 2ekji φ, k ui, j εik φ, k φ, i dV
R
,
V ρu j u j dV
ð5:219Þ
where the admissible u and φ satisfy the boundary conditions in Eq. (5.207). Then it
can be shown [7] that R assumes stationary values when u ¼ u(n) and φ ¼ φ(n), and
the stationary value of R is (ω(n))2. R is called the Rayleigh quotient.
References
1. H.F. Tiersten, Thickness vibrations of piezoelectric plates. J. Acoust. Soc. Am. 35, 53–58 (1963)
2. H.F. Tiersten, Linear Piezoelectric Plate Vibrations (Plenum, New York, 1969)
3. A.H. Meitzler, H.F. Tiersten, A.W. Warner, D. Berlincourt, G.A. Couqin, F.S. Welsh III, IEEE
Standard on Piezoelectricity (IEEE, New York, 1988)
4. B. Liu, Q. Jiang, J.S. Yang, Fluid-induced frequency shift in a piezoelectric plate driven by lateral
electric fields. Int. J. Appl. Electromagn. Mech. 34, 171–180 (2010)
5. G. Sauerbrey, Verwendung von schwingquarzen zur wägung dünner schichten und zur
mikrowägung. Z. Phys. 155, 206–222 (1959)
6. N.T. Adelman, Y. Stavsky, Radial vibrations of axially polarized piezoelectric ceramic cylinders.
J. Acoust. Soc. Am. 57, 356–360 (1975)
7. J.S. Yang, R.C. Batra, Free vibrations of a piezoelectric body. J. Elast. 34, 239–254 (1994)
Chapter 6
Linear Theory for Small Fields
on a Finite Bias
The theory of linear piezoelectricity assumes infinitesimal deviations from an ideal
reference state of the material in which there are no pre-existing mechanical and/or
electrical fields (initial or biasing fields). The presence of biasing fields makes a
material apparently behave like a different one, and renders the linear theory of
piezoelectricity invalid. The behavior of electroelastic bodies under biasing fields
can be described by the theory for infinitesimal incremental fields superposed on
finite biasing fields, which is a consequence of the nonlinear theory of
electroelasticity when it is linearized around the bias. Knowledge of the behavior
of electroelastic bodies under biasing fields is important in many applications
including the buckling of thin electroelastic structures, frequency stability of
piezoelectric resonators, acoustic wave sensors based on frequency shifts due to
biasing fields, characterization of nonlinear electroelastic materials by propagation
of small-amplitude waves in electroelastic bodies under biasing fields, and
electrostrictive ceramics which operate under a biasing electric field. This chapter
presents the linear theory for small fields superposed on finite biasing fields in an
electroelastic body.
6.1
Nonlinear Spring
The basic concept of small fields superposed on finite biasing fields can be well
explained by a simple nonlinear spring. Consider the following spring-mass system
0
(see Fig. 6.1). When the spring is stretched by x, the force in the spring is kx + k x2,
0
where k and k are linear and nonlinear spring constants, respectively.
The reference state in Fig. 6.1a is the natural state of the spring when there is no
force or deformation in it. Under an initial, constant force f0, the mass m is in
equilibrium with an initial stretch x0 in the spring (see Fig. 6.1b) such that
© Springer Nature Switzerland AG 2018
J. Yang, An Introduction to the Theory of Piezoelectricity, Advances in Mechanics
and Mathematics 9, https://doi.org/10.1007/978-3-030-03137-4_6
193
194
6 Linear Theory for Small Fields on a Finite Bias
Fig. 6.1 Reference, initial,
and present states of a
nonlinear spring-mass
system
m
x
O
(a) Reference state.
f0
kx0+k⬘(x0)2
O
x
x0
(b) Initial state (static).
k(x0 +D x)+k⬘( x0 +Dx)2
O
f0 +Df
x0 x0 +Dx
x
(c) Present state (dynamic).
f 0 ¼ kx0 þ k0 x20 :
ð6:1Þ
Then a small and dynamic incremental force Δf is applied. The mass is in small
amplitude motion around x0 with current position x0 + Δx (see Fig. 6.1c). Since both
Δf and Δx are small, we want to derive a linear relationship between them. In the
current state in Fig. 6.1c the equation of motion for the mass is
h
i
d2
m 2 ðx0 þ ΔxÞ ¼ ðf 0 þ Δf Þ kðx0 þ ΔxÞ þ k0 ðx0 þ ΔxÞ2
dt
h
i
ð6:2Þ
¼ ðf 0 þ Δf Þ kx0 þ kΔx þ k0 x20 þ k0 2x0 Δx þ k 0 ðΔxÞ2
h
i
¼ f 0 kx0 k0 x20 þ Δ f kΔx k0 2x0 Δx k 0 ðΔxÞ2 :
Using Eq. (6.1) and the smallness of Δx, we obtain
m
d2
ðΔxÞ ¼ Δf kΔx k 0 2x0 Δx k0 ðΔxÞ2
dt 2
ffi Δf ðk þ k0 2x0 ÞΔx
¼ Δf ke Δx,
ð6:3Þ
where
ke ¼ k þ 2k 0 x0
ð6:4Þ
6.2 Formulation of the Problem
195
is an effective linear spring constant for small-amplitude motions around the initial
state x0. Thus at the state with an initial stretch, the nonlinear spring responds to
small, incremental forces like a linear spring with an effective linear spring constant
ke. It is important to notice that ke depends on x0 and the nonlinear spring constant k0 .
6.2
Formulation of the Problem
The concept in the previous section can be generalized to an electroelastic body
[1–3]. Consider the following three states of an electroelastic body (see Fig. 6.2).
There are three coincident Cartesian coordinate systems. XK are for the reference
state, Greek coordinates and indices ξα are for the initial state, and yi are for the
current state. Some of the associated tensors may depend on all three coordinates
and are called three-point tensors.
1. The reference state. In this state the body is undeformed and free of electric fields.
A generic point at this state is denoted by X with Cartesian coordinates XK. The
mass density is ρ0.
2. The initial state is indicated by a superscript “1”. In this state the body is deformed
finitely and statically under the action of body force f1, body free charge ρE1,
prescribed surface position ξ, surface traction T1α , surface potential φ1 and surface
free charge σ E1 . The displacement field is denoted by w(X). The position of
the material point associated with X is given by ξ ¼ ξ(X) or ξα ¼ ξα(X) with
strain E 1KL . The electric potential in this state is denoted by φ1 which is responsible
for a finite static electric field E1α . φ1 is a function of ξ, but ξ ¼ ξ(X) allows us to
view φ1 as a function of X. The deformation and fields in this state are the initial
or biasing fields. ξ(X) and φ1(X) satisfy the following static equations of
nonlinear electroelasticity:
ξα, K ,
J 1 ¼ det
E 1KL ¼ ξα, K ξα, L δKL =2,
W 1K ¼ φ1, K ,
E 1α ¼ φ1, α ,
0 ∂χ ,
T S1
KL ¼ ρ ∂ E KL 1
1
EKL , W K
P 1K ¼ ρ0 ∂∂χ
,
WK 1
1
ð6:5Þ
ð6:6Þ
E KL , W K
1
K 1Kα ¼ ξα, L T S1
KL þ M Kα ,
1
1
1
D K ¼ ε0 E K þ P K ,
ð6:7Þ
196
6 Linear Theory for Small Fields on a Finite Bias
Reference
state
Initial
state
w (Bias)
3
X
u (Increment)
x=X+w,ϕ 1
y=x+u
ϕ =ϕ1+ ϕ~
Present
state
2
1
Fig. 6.2 Reference, initial, and present states of an electroelastic body
1
M 1Kα ¼ J 1 X K , β ε0 E 1β E 1α E 1γ E 1γ δβα ,
2
E 1K ¼ J 1 X K , α X L, α W 1L ,
K 1Kα, K þ ρ0 f 1α ¼ 0,
D 1K , K ¼ ρE1 :
ð6:8Þ
ð6:9Þ
3. The present state. In this state, the body is under the action of time-dependent
electromechanical loads fi, ρE, yi , Ti , φ, and σ E inside the body and on its surface.
The current or final position of X is given by y ¼ ξ + u. The final electric potential
e . We consider the case when u and φ
e are infinitesimal and dynamic.
is φ ¼ φ1 þ φ
u is a function of ξ and time but ξ ¼ ξ(X) allows us to view it as a function of
X and time. Hence y may also be viewed as a function of X and time. Similarly, φ
e are functions of y and time but they can be viewed as functions of X and
and φ
time. y(X,t) and φ(X,t) satisfy the following dynamic equations of nonlinear
electroelasticity:
J ¼ det yi, K ,
E KL ¼ yi, K yi, L δKL =2,
W K ¼ φ, K ,
E i ¼ φ, i ,
S
T KL
¼ ρ0 ∂∂χ
,
E KL E , W
KL K
P K ¼ ρ0 ∂∂χ
,
WK E KL , W K
ð6:10Þ
ð6:11Þ
6.3 Linearization About a Bias
197
S
K Lj ¼ y j, K T KL
þ M Lj ,
D K ¼ ε0 E K þ P K ,
1
M Lj ¼ JX L, i ε0 E i E j Ek Ek δij ,
2
E K ¼ JX K , i X L, i W L ,
K Lj, L þ ρ0 f j ¼ ρ0€yj ,
D K , K ¼ ρE :
ð6:12Þ
ð6:13Þ
ð6:14Þ
Our goal is to derive linear equations governing the small incremental fields u(X,t)
e (X,t).
and φ
6.3
Linearization About a Bias
e are assumed to be infinitesimal. To show this more
The incremental fields u and φ
specifically, we write y and φ as
yi ðX; t Þ ¼ δiα ½ξα ðX; t Þ þ λuα ðX; t Þ,
φðX; t Þ ¼ φ1 ðX; t Þ þ λe
φ ðX; t Þ,
ð6:15Þ
where a small and dimensionless parameter λ is introduced to indicate the smallness
of the incremental fields. In the following, terms quadratic in or of higher order of λ
will be dropped. At the end λ can be set to 1. The substitution of Eq. (6.15) into
Eq. (6.10)2,3 yields
~ KL ,
EKL ffi E1KL þ λ E
1
~ K,
WK ¼ WK þ λ W
ð6:16Þ
where the incremental fields are indicated by a superimposed tilde “~”. In Eq. (6.16),
~ KL ¼ ξα, K uα, L þ ξα, L uα, K =2 ¼ E
~ LK ,
E
~
φ, K :
W K ¼ e
ð6:17Þ
Further substitution of Eq. (6.16) into Eq. (6.11) gives
S
~S
T KL
ffi T S1
KL þ λ T KL ,
1
P K ffi P K þ λ P~ K ,
where
ð6:18Þ
198
6 Linear Theory for Small Fields on a Finite Bias
2
2
S
S
~ MN þρ0 ∂ χ ~ M ¼ T~ LK
T~ KL
W
¼ ρ0 ∂ EKL∂∂χEMN 1 1 E
,
∂ E KL ∂ W M E 1 , W 1
E KL , W K
KL
K
2
2
~ MN ρ0 ∂ χ ~ M:
W
P~ K ¼ ρ0 ∂ W∂K ∂χEMN 1 1 E
∂ WK ∂ WM 1
1
E KL , W K
ð6:19Þ
E KL , W K
To the first order of λ, using Eq. (6.15)1, we have
X K , j ¼ X K , α ξα, j ¼ X K , α ðδiα yi λuα Þ, j
¼ X K , α δiα δij λuα, j ¼ X K, α δ jα λuα, L X L, j
¼ X K , α δ jα λuα, L X L, βξβ, j
¼ X K , α δ jα λuα, L X L, β δ jβ λuβ, M X M , j
ffi X K , α δ jα λuα, L X L, β δ jβ
¼ δ jα X K , α λX K , α uα, L X L, β δβj :
ð6:20Þ
From Eq. (1.22), using Eq. (6.15)1, we obtain
1
JX K , j ¼ εKLM ε jlm yl, L ym, M
2
ffi J 1 δ jα X K , α þ λεKLM ε jβγ ξβ, L uγ, M
¼ J 1 δ jα X K , α þ λεKLM δ jα εαβγ ξβ, L uγ, M
¼ J 1 δ jα X K , α þ λεKLM δ jα εγαβ ξβ, L uγ, M :
ð6:21Þ
With Eq. (1.18)2, Eq. (6.21) becomes
JX K , j ¼ J 1 δ jα X K , α þ λεKLM δ jα J 1 εPQL X P, γ X Q, α uγ, M
¼ J 1 δ jα X K , α þ λδ jα J 1 εKLM εPQL X P, γ X Q, α uγ, M
¼ J 1 δ jα X K , α þ λδ jα J 1 εLMK εLPQ X P, γ X Q, α uγ, M
¼ J 1 δ jα X K , α þ λδ jα J 1 ðδMP δKQ δMQ δKP ÞX P,γ X Q, α uγ, M
¼ J 1 δ jα X K , α þ λδ jα J 1 X M , γ X K , α X K , γ X M , α uγ, M
¼ J 1 δ jα X K , α þ λJ 1 δ jα X K , α X L, γ X K , γ X L, α uγ, L ,
ð6:22Þ
where the ε-δ identity in Eq. (1.15) has been used. For the electric field, using
Eqs. (6.16)2 and (6.20), we have
Ei ¼ φ, i ¼ φ, K X K , i ¼ W K X K , i
~ M X M , α E1β uβ, M X M , α :
ffi δiα E 1α þ λδiα W
ð6:23Þ
Then, from Eq. (6.13),
~ Kα ,
M Ki ¼ δiα M 1Kα þ λM
~K,
E K ¼ E 1K þ λE
where
ð6:24Þ
6.3 Linearization About a Bias
199
~ Kα ¼ gKαLγ uγ, L r LKα W
~ L,
M
~
~
E K ¼ r KLγ uγ, L þ lKL W L ,
ð6:25Þ
and
gKαLγ ¼ ε0 J 1 E 1α E 1β X K , β X L, γ X K , γ X L, β E 1α E1γ X K , β X L, β
1
þ E1β E 1γ X K , α X L, β X K , β X L, α þ E 1β E1β X K , γ X L, α X K , α X L, γ ,
2
r KLγ ¼ ε0 J 1 E1α X K , α X L, γ E 1α X K , γ X L, α E1γ X K , α X L, α ,
ð6:26Þ
lKL ¼ ε0 J 1 X K , α X L, α :
We note that the expression in Eq. (6.26)1 has differences from the corresponding
expression in Eq. (3.12) of [3]. The author of [3] agreed with that Eq. (3.12) of [3]
has errors during a phone conversation without written communication. Therefore
the expression in Eq. (6.26)1 in the above should be verified before it is used. Then,
from Eq. (6.12), we have
~ Kα ,
K Ki ffi δiα K 1Kα þ λK
~ K,
D K ffi D 1K þ λ D
ð6:27Þ
~
~
~ Kα ¼ uα, L T S1
K
KL þ ξα, L T KL þ M Kα ,
~ K þ P~ K :
~ K ¼ ε0 E
D
ð6:28Þ
where
With Eqs. (6.19) and (6.25), we can write Eq. (6.28) as
~ M,
~ Lγ ¼ GLγMα uα, M RMLγ W
K
~ L,
~ K ¼ RKLγ uγ, L þ LKL W
D
ð6:29Þ
where
2
∂ χ
GKαLγ ¼ ξα, M ρ0 ∂ EKM
ξ þ T S1
KL δαγ þ gKαLγ ¼ GLγKα ,
∂ E LN E 1 , W 1 γ , N
KL
K
2
RKLγ ¼ ρ0 ∂ W∂K ∂χEML 1 1 ξγ, M þ r KLγ ,
EKL , W K
2
∂ χ
0
LKL ¼ ρ ∂ W K ∂ W L 1 1 þ lKL ¼ LLK :
ð6:30Þ
E KL , W K
Equation (6.29) shows that the incremental stress tensor and electric displacement
vector depend linearly on the incremental displacement gradient and potential
gradient. In Eq. (6.29), GKαLγ , RKLγ , and LKL are called the effective or apparent
200
6 Linear Theory for Small Fields on a Finite Bias
elastic, piezoelectric, and dielectric constants of the material under biasing fields.
They depend on the initial deformation ξα(X) and electric potential φ1(X). Since
the fields in the present state satisfy Eq. (6.14) and the biasing fields satisfy
Eq. (6.9), we have
~ Kα, K þ ρ0 ~f α ¼ ρ0 €uα ,
K
~ K, K ¼ e
D
ρE ,
ð6:31Þ
where ~f α and e
ρ E are determined from
f i ¼ δiα f 1α þ λ~f α ,
ρE ¼ ρE1 þ λe
ρE :
6.4
ð6:32Þ
Boundary-Value Problem
Consider an electroelastic body which occupies a region V in the reference state and
whose boundary surface S is partitioned according to Eq. (1.253) as
Sy [ ST ¼ Sφ [ SD ¼ S,
Sy \ ST ¼ Sφ \ SD ¼ 0:
ð6:33Þ
e we have the following equations and
In summary, for the incremental fields u and φ
boundary conditions. From Eq. (6.31), the equations of motion and charge are
~ Kα, K þ ρ0 ~f α ¼ ρ0 €uα
K
~ K, K ¼ e
D
ρ E in V:
in V,
ð6:34Þ
The constitutive relations are from Eq. (6.29)
~ Lγ ¼ GLγMα uα, M þ RMLγ φ
e, M
K
~ K ¼ RKLγ uγ, L LKL φ
e , L in
D
in V,
V,
ð6:35Þ
where Eq. (6.17)2 has been used. The boundary conditions are
uα ¼ uα on Sy ,
e¼φ
e on Sφ ,
φ
and
ð6:36Þ
6.5 Effects of a Small Bias
201
~ Lα ¼ T~ α on ST ,
NLK
~ K ¼ e
NK D
σ E on SD ,
ð6:37Þ
where the incremental surface loads are defined by
T~ α ¼ δαi Ti T1α ,
σe E ¼ σ E σ E1 :
ð6:38Þ
For dynamic problems proper initial conditions need to be imposed.
It can be verified that the stationary condition of the following variational
functional under the constraints of the essential boundary conditions in Eq. (6.36)
yields Eq. (6.34) and the natural boundary conditions in Eq. (6.37):
R t R 1
1
e ¼ t0 dt V ρ0 u_ α u_ α GKαLγ uK , α uL, γ
Π u; φ
2
2
1
e
e
e
e dV
RKLγ φ , K uL, γ þ LKL φ , K φ , L þ ρ0 ~f α uα e
ρE φ
2
Rt R
Rt R
e dS:
þ t0 dt ST T~ α uα dS t0 dt SD σe E φ
6.5
ð6:39Þ
Effects of a Small Bias
In some applications, the biasing deformations and fields are also infinitesimal and
are governed by the linear theory of piezoelectricity:
T 1KL, K þ ρ0 f 1L ¼ 0,
D1K , K ¼ ρE1 ,
T 1KL ¼ cKLMN E1MN eMKL W 1M ,
D1K ¼ eKMN E 1MN þ εKM W 1M ,
1
E1KL ¼ ðwK , L þ wL, K Þ ¼ E 1LK ,
2
W 1K ¼ φ1, K :
ð6:40Þ
ð6:41Þ
ð6:42Þ
Since the biasing fields are infinitesimal, we consider their first-order effects on the
incremental fields only. As to be seen below, in this case the following free energy
density is sufficient:
202
6 Linear Theory for Small Fields on a Finite Bias
1
1
ρ0 χ ¼ cABCD EAB ECD eABC W A EBC χ AB W A W B
2
2
1
1
þ cABCDEF E AB E CD EEF þ dABCDE W A E BC EDE
6
2
1
1
bABCD W A W B E CD χ ABC W A W B W C ,
2
6
ð6:43Þ
where we have simplified the notation of the third-order material constants in
Eq. (1.271) and denoted
cABCDEF ¼ c
,
d ABCDEF ¼ d
,
3 ABCDEF
1 ABCDE
χ ABC ¼ χ
ð6:44Þ
:
3 ABC
Then, neglecting terms quadratic in the gradients of the small w and φ1, the effective
material constants in Eq. (6.30) take the following form:
GKαLγ ¼ cKαLγ þ b
c KαLγ ,
RKLγ ¼ eKLγ þ b
e KLγ ,
ð6:45Þ
LKL ¼ εKL þ b
ε KL ,
where
b
c KαLγ ¼ T 1KL δαγ þ cKαLN wγ, N þ cKNLγ wα, N
þ cKαLγAB E 1AB þ d AKαLγ W 1A ,
b
e KLγ ¼ eKLM wγ, M d KLγAB E1AB þ bAKLγ W 1A
þ ε0 W 1K δLγ W 1L δKγ W 1M δMγ δKL ,
b
ε KL ¼ bKLAB E 1AB þ χ KLA W 1A þ ε0 E 1MM δKL 2E 1KL :
ð6:46Þ
Equations (6.45) and (6.46) show explicitly that GKαLγ , RKLγ , and LKL depend on
the small initial deformations and fields linearly. Therefore, when these initial
fields are not uniform, the equations for the incremental fields are with variable
coefficients. The effective material constants GKαLγ , RKLγ , and LKL in general have
lower symmetry than the fundamental linear elastic, piezoelectric, and dielectric
constants cKαLγ , eKLγ, and εKL. This is called the induced anisotropy or symmetry
breaking due to the biasing fields. There can be as many as 45 apparently
independent components for GKαLγ , 27 components for RKLγ , and 6 components
for LKL. It is important to notice that the third-order material constants in
Eq. (6.44) are necessary for a complete description of the first-order effects of
the biasing fields.
6.6 Frequency Perturbation
6.6
203
Frequency Perturbation
Many piezoelectric devices are resonant devices for which frequency consideration
is of fundamental importance. An analysis of these devices based on linear piezoelectricity can provide the basic vibration characteristics such as resonance frequencies and modes. However, devices designed based on linear piezoelectricity are
deficient in many applications. Knowledge of the frequency stability due to environmental effects, for example, temperature change, force, and acceleration, which
cause biasing fields and frequency shifts is often required for a successful design. For
the first-order effects of small biasing fields on resonance frequencies, we need to
study the eigenvalue problem of the free vibration of an electroelastic body with the
presence of a small bias. The governing equations are, from the homogeneous form
of Eqs. (6.34) and (6.35):
e , ML ¼ ρ0 ω2 uγ
GLγMα uα, ML þ RMLγ φ
e , LK ¼ 0 in V,
RKLγ uγ, LK LKL φ
in
V,
ð6:47Þ
along with the homogeneous form of the boundary conditions in Eqs. (6.36) and
(6.37):
uα ¼ 0 on Sy ,
e ¼ 0 on Sφ ,
φ
ð6:48Þ
~ Lα ¼ 0 on ST ,
NLK
~ K ¼ 0 on SD :
NK D
ð6:49Þ
In a way similar to the derivation of Eq. (5.218), we have the following Rayleigh
quotient [4] for the resonance frequency with the present of the biasing fields:
R ω ¼
2
V
e , M uγ, L LKL φ
e, K φ
e , L dV
GLγMα uα, M uγ, L þ RKLγ φ
R
:
0
V ρ uα uα dV
ð6:50Þ
Equation (6.50) can be interpreted as the variational formulation [4] of the eigenvalue problem of the free vibration of an electroelastic body under biasing fields.
Let the resonance frequencies and modes of the body without the biasing fields be
ω(n), u(n), and φ(n). With the presence of small biasing fields, we denote the
frequency by ω ¼ ω(n) + Δω where Δω is small and (Δω)2 is negligible. We use
the Ritz method and perform an approximate variational analysis on ω based on
Eq. (6.50). The trial functions of the Ritz method are simply the unperturbed modes
u(n) and φ(n) [4]. Then, from Eq. (6.50),
204
6 Linear Theory for Small Fields on a Finite Bias
ωðnÞ
2
ffi
R
þ 2ωðnÞ Δω
ðnÞ
V
ðnÞ
ðnÞ ðnÞ
ðnÞ ðnÞ
GLγMα uα, M uγ, L þ RKLγ φ, M uγ, L LKL φ, K φ, L dV
:
R
0 ðnÞ ðnÞ
V ρ uα uα dV
ð6:51Þ
With Eqs. (5.218) and (6.45), Eq. (6.51) reduces to
R
Δω ffi
ðnÞ
V
ðnÞ
ðnÞ ðnÞ
ðnÞ ðnÞ
b
e KLλ φ, M uγ, L b
ε KL φ, K φ, L dV
c LγMα uα, M uγ, L þ 2b
,
R
ð
n
Þ
ð
n
Þ
2ωðnÞ V ρ0 uα uα dV
ð6:52Þ
which is the perturbation integral in [5].
6.7
Linearization Using Total Stress
With the total free energy ψ in Eq. (1.264), the derivation for the equations of the
incremental fields can be written in a more compact form. Under Eq. (6.15), we can
write Eqs. (1.265) and (1.266) as:
∂ψ
ffi T 1KL þ λT~ KL ,
∂ EKL
∂ψ
~ K,
D K ¼ ρ0
ffi D 1K þ λ D
∂ WK
T KL ¼ ρ0
ð6:53Þ
where
2
T~ KL ¼ ρ0 ∂ EKL∂ ∂ψEMN 2
~ MN þ ρ0 ∂ ψ ~ M,
E
W
∂
W
∂
E
KL
M E1 , W 1
E1KL , W 1K
KL
K
2
2
~ MN ρ0 ∂ ψ ~ M:
~ K ¼ ρ0 ∂ ψ E
W
D
∂ W K ∂ E MN 1
∂ WK∂ WM 1
1
1
E KL , W K
ð6:54Þ
E KL , W K
From Eq. (1.267),
~ Kα ,
K Ki ¼ yi, L T KL ffi δiα K 1Kα þ λK
ð6:55Þ
~ Kα ¼ uα, L T 1KL þ ξα, L T~ KL :
K
ð6:56Þ
where
References
205
Then
~ M,
~ Lγ ¼ GLγMα uα, M RMLγ W
K
~ L,
~ K ¼ RKLγ uγ, L þ LKL W
D
ð6:57Þ
where
2
∂ ψ
GKαLγ ¼ ξα, M ρ0 ∂ EKM
ξ þ T 1KL δαγ ¼ GLγKα ,
∂ E LN E 1 , W 1 γ , N
KL
K
2
RKLγ ¼ ρ0 ∂ W∂K ∂ψEML 1 1 ξγ, M ,
EKL , W K
2
∂ ψ 0
LKL ¼ ρ ∂ W K ∂ W L 1 1 ¼ LLK :
ð6:58Þ
E KL , W K
References
1. J.C. Baumhauer, H.F. Tiersten, Nonlinear electroelastic equations for small fields superposed on
a bias. J. Acoust. Soc. Am. 54, 1017–1034 (1973)
2. H.F. Tiersten, Electroelastic interactions and the piezoelectric equations. J. Acoust. Soc. Am. 70,
1567–1576 (1981)
3. H.F. Tiersten, On the accurate description of piezoelectric resonators subject to biasing deformations. Int. J. Eng. Sci. 33, 2239–2259 (1995)
4. J.S. Yang, Free vibrations of an electroelastic body under biasing fields. IEEE Trans. Ultrason.
Ferroelect. Freq. Contr. 52, 358–364 (2005)
5. H.F. Tiersten, Perturbation theory for linear electroelastic equations for small fields superposed
on a bias. J. Acoust. Soc. Am. 64, 832–837 (1978)
Chapter 7
Other Effects
In this chapter we discuss a few effects that are external to the quasistatic theory of
electroelasticity developed in Chap. 1 which is valid through Chap. 6. These include
the effects of heat conduction; mechanical and electrical dissipations due to viscosity, dielectric loss and semiconduction; nonlocal and gradients effects; as well as
electromagnetic effects.
7.1
Nonlinear Thermal and Dissipative Effects
Thermal and viscous effects often appear together and are treated in this section
including dielectric loss. For this purpose the energy equation in the integral form in
Eq. (1.128) needs to be extended to include thermal effects [1–3]:
d
dt
1
ρ v v þ ε dv
v R 2
R
¼ v ρ f þ f E v þ wE þ ργ dv þ s ðt v n qÞds,
Z
ð7:1Þ
where q is the heat flux vector and γ is the body heat source per unit mass. The
second law of thermodynamics needs to be added as follows [1–3]:
d
dt
Z
Z
ρηdv v
ργ
dv v θ
Z
s
qn
ds,
θ
ð7:2Þ
where η is the entropy per unit mass and θ is the absolute temperature. Equations
(7.1) and (7.2) can be converted to differential form using the divergence theorem.
We have (compare to Eq. (1.142))
© Springer Nature Switzerland AG 2018
J. Yang, An Introduction to the Theory of Piezoelectricity, Advances in Mechanics
and Mathematics 9, https://doi.org/10.1007/978-3-030-03137-4_7
207
208
7
ρε_ ¼ τij v j, i þ wE þ ργ qi, i ,
ργ q
ρη_ i :
θ
θ ,i
Other Effects
ð7:3Þ
ð7:4Þ
Eliminating γ from Eqs. (7.3) and (7.4), we obtain the C-D (Clausius–Duhem)
inequality as:
q θ, i
ρ θη_ ε_ þ τij v j, i þ ρE i π_ i i 0:
θ
ð7:5Þ
Corresponding to Eq. (1.170), a free energy χ can be introduced through the
following Legendre transform:
χ ¼ ε θη Ei π i :
ð7:6Þ
Then the energy equation in Eq. (7.3) and the C-D inequality in Eq. (7.5) become
(compare to Eq. (1.172))
ρ χ_ þ ηθ_ þ η_ θ ¼ τij v j, i Pi E_ i þ ργ qi, i ,
ð7:7Þ
q θ, i
ρ χ_ þ ηθ_ þ τij v j, i Pi E_ i i 0:
θ
ð7:8Þ
and
We introduce the following material heat flux and material temperature gradient:
QK ¼ JX K , k qk ,
Θ K ¼ θ , K ¼ θ , k yk , K :
ð7:9Þ
Then Eqs. (7.7) and (7.8) can be written in material form as (compare to Eq. (1.208))
S _
ρ0 χ_ þ ηθ_ þ η_ θ ¼ T KL
E KL P K W_ K þ ρ0 γ QK , K ,
ð7:10Þ
Q ΘK
S _
E KL P K W_ K K
0:
ρ0 χ_ þ ηθ_ þ T KL
θ
ð7:11Þ
For constitutive relations we start with the following expressions:
χ ¼ χ ðE KL ; W K ; θ; ΘK Þ,
S
S
E KL ; W K ; θ; ΘK ; E_ KL ; W_ K ,
T KL
¼ T KL
P K ¼ P K EKL ; W K ; θ; ΘK ; E_ KL ; W_ K ,
QK ¼ QK E KL ; W K ; θ; ΘK ; E_ KL ; W_ K :
ð7:12Þ
7.1 Nonlinear Thermal and Dissipative Effects
209
The substitution of Eq. (7.12) into the C-D inequality in Eq. (7.11) yields
∂χ _
∂χ _
0
ρ
ΘK ρ ηþ
θ
∂θ ∂ ΘK
1
S
0 ∂χ
0 ∂χ
_
E KL P K þ ρ
W_ K QK ΘK 0:
þ T KL ρ
∂ EKL
∂W K
θ
0
ð7:13Þ
Since Eq. (7.13) is linear in Θ_ K and θ_ , for the inequality to hold χ cannot depend on
ΘK, and η needs to be related to χ by
η¼
∂χ
:
∂θ
ð7:14Þ
S
We break T KL
and P K into reversible and dissipative parts as follows:
S
R
D
T KL
¼ T KL
þ T KL
,
P K ¼ P KR þ P KD ,
ð7:15Þ
where
∂χ
∂χ
, P KR ¼ ρ0
,
∂EKL
∂W K
χ ¼ χ ðEKL ; W K ; θÞ,
D
D
T KL ¼ T KL
E KL ; W K ; θ; ΘK ; E_ KL ; W_ K ,
P KD ¼ P KD EKL ; W K ; θ; ΘK ; E_ KL ; W_ K :
R
T KL
¼ ρ0
ð7:16Þ
ð7:17Þ
Then what is left from the C-D inequality in Eq. (7.13) is
1
D _
T KL
E KL P KD W_ K QK ΘK 0:
θ
ð7:18Þ
From Eqs. (7.10) and (7.14) we obtain the heat equation as
D _
E KL P KD W_ K þ ρ0 γ QK , K ,
ρ0 θη_ ¼ T KL
ð7:19Þ
where Eq. (7.16) has been used.
In summary, the nonlinear equations for thermoviscoelectroelasticity are
ρ0 ¼ ρJ,
K Lk, L þ ρ0 f k ¼ ρ0€yk ,
D K , K ¼ ρE ,
D _
E KL P KD W_ K þ ρ0 γ QK , K ,
ρ0 θη_ ¼ T KL
ð7:20Þ
210
7
Other Effects
with constitutive relations given by Eqs. (7.14), (7.15), (7.16), and (7.17) which are
restricted by Eq. (7.18). The equation for the conservation of mass in Eq. (7.20)1 can
be used to determine ρ separately from the other equations. Equations (7.20)2,3,4 can
be written as five equations for yi(X, t), φ(X, t) and θ(X, t). On the boundary surface
S, the thermal boundary condition may be either prescribed temperature or heat flux
N L QL ¼ Q:
7.2
ð7:21Þ
Linear Thermal and Dissipative Effects
For small deformations and weak electric fields, corresponding to the linear theory of
piezoelectricity, we have
Di, i ¼ ρe ,
T ji, j þ ρ0 f i ¼ ρ0 €ui ,
ρ0 θη_ ¼ T ijD S_ ij PiD E_ i þ ρ0 γ qi, i :
ð7:22Þ
The reversible part of the constitutive equations for small deformations and weak
electric fields are determined by χ ¼ χ(Sij, Ei, θ) and
∂χ
,
∂θ
∂χ
T ijR ¼ ρ0
,
∂Sij
∂χ
PiR ¼ ρ0
:
∂E i
η¼
ð7:23Þ
In order to linearize the constitutive relations thermally we expand χ into a power
series about θ ¼ T0, Sij ¼ 0, and Ei ¼ 0, where T0 is a uniform reference temperature.
Denoting T ¼ θT0, assuming |T/T0|<<1, and keeping quadratic terms only, we can
write
1
1
ρ0 χ ¼ cijkl Sij Skl ekij Ek Sij χ ij E i E j
2
2
1α 2
T βkl Skl T λk E k T,
2T 0
ð7:24Þ
where βij are the thermoelastic constants, λk are the pyroelectric constants and α is
related to the specific heat. Equations (7.23) and (7.24) yield
7.2 Linear Thermal and Dissipative Effects
211
T ijR ¼ cijkl Skl ekij E k βij T,
DiR ¼ ε0 E i þ PiR ¼ eijk S jk þ εij E j þ λi T,
α
ρ0 η ¼ T þ βkl Skl þ λk E k ,
T0
ð7:25Þ
which are the equations for linear thermopiezoelectricity given by Mindlin [4]. For
the dissipative part of the constitutive relations we choose the linear relations
T ijD ¼ μijkl S_ kl αkij E_ k ,
DiD ¼ PiD ¼ βijk S_ jk þ ζ ij E_ j ,
qk ¼ κ kl T , l :
ð7:26Þ
In the following we will assume βijk ¼ αijk. Equation (7.26) is restricted by
q T,i
T ijD S_ ij PiD E_ i i 0:
T0
ð7:27Þ
A dissipation function can be introduced as follows:
1
1
Ψ S_ kl ; E_ j ¼ μijkl S_ kl S_ ij αkij E_ k S_ ij ζ ij E_ i E_ j ,
2
2
ð7:28Þ
where by Eqs. (7.22)3, (7.26)1,2, and (7.27) can be written as
ρ0 T 0 η_ ¼ 2Ψ S_ ij ; E_ i þ ρ0 γ qi, i ,
∂Ψ
∂Ψ
T ijD ¼
, PiD ¼ ,
_
∂E_ i
∂S ij
κ ij T , i T , j
2Ψ S_ ij ; E_ i þ
0:
T0
ð7:29Þ
To ensure that the last one in Eq. (7.29) is satisfied, we can choose
Ψ S_ kl ; E_ j 0,
κij θ, i θ, j 0,
ð7:30Þ
which implies that μijkl, ζ ij, and κij are positive definite. The formal similarity
between Eq. (7.28) and the first three terms on the right-hand side of Eq. (7.24)
suggests that the structures of μijkl, ζ ij, and αijk are the same as those of cijkl, χ ij, and
eijk, which are known for various crystal classes.
When the thermoelastic and pyroelectric effects are small, they can be neglected.
Then the above equations for the linear theory reduce to two one-way coupled
systems of equations. One represents the theory of viscopiezoelectricity with
212
7
Other Effects
Di, i ¼ ρe ,
T ji, j þ ρ0 f i ¼ ρ0 €ui ,
ð7:31Þ
T ijR ¼ cijkl Skl ekij E k ,
DiR ¼ eijk S jk þ εij E j ,
ð7:32Þ
T ijD ¼ μijkl S_ kl αkij E_ k ,
DiD ¼ αijk S_ jk þ ζ ij E_ j :
ð7:33Þ
The other governs the temperature field with
ρ0 T 0 η_ ¼ T ijD S_ ij DiD E_ i þ ρ0 γ qi, i ,
α
ρ0 η ¼ T:
T0
ð7:34Þ
Once the mechanical and electric fields are obtained from Eqs. (7.31), (7.32), and
(7.33), they can be substituted into Eq. (7.34) to solve for the temperature field T.
For harmonic motions with a factor of exp(iω t) in the complex notation, the
linear constitutive relations in Eqs. (7.32) and (7.33) can be written as
T ij ¼ cijkl Skl þ μijkl S_ kl ekij E k αkij E_ k
¼ cijkl þ iω μijkl Skl ekij þ iω αkij Ek ,
Di ¼ eijk S jk þ αijk S_jk þ εijE j þ ζ ij E_ j
¼ eijk þ iωαijk S jk þ εij þ iω ζ ij E j :
ð7:35Þ
Formally, Eq. (7.35) is like the constative relations of piezoelectricity where the
apparent material constants are complex [5] and frequency-dependent.
7.3
Semiconduction
Piezoelectric materials may be dielectrics or semiconductors. Mechanical fields
and mobile charges in piezoelectric semiconductors can interact through electric
fields, which is called the acoustoelectric effect. The basic behaviors of piezoelectric semiconductors can be described by a simple extension of the theory of
piezoelectricity. The three-dimensional phenomenological theory consists of the
equation of motion, the charge equation of electrostatics, and the conservation of
charge for electrons and holes (continuity equations) [6, 7]:
7.3 Semiconduction
213
T ji, j ¼ ρ€
uj ,
Di, i ¼ q p n þ N þ
D NA ,
J in, i ¼ qn_ ,
J ip, i ¼ qp_ ,
ð7:36Þ
where q ¼ 1.6 1019coul is the electronic charge. p and n are the concentrations
of holes and electrons. N þ
D and N A are the concentrations of impurities of donors
p
n
and accepters. J i and J i are the hole and electron current densities. In Eq. (7.36),
we have neglected carrier recombination and generation [6]. Constitutive relations
accompanying Eq. (7.36) describing material behaviors can be written in the
following form:
E
T kl þ dkij E k ,
Sij ¼ sijkl
Di ¼ dikl T kl þ εikT E k ,
J in ¼ qnμijn E j þ qDijn n, j ,
J ip ¼ qpμijp E j qDijp p, j ,
ð7:37Þ
where μijn and μijp are the carrier mobilities. Dijn and Dijp are the carrier diffusion
constants.
Let
n ¼ n0 þ Δn,
p ¼ p0 þ Δp,
ð7:38Þ
where
n0 ¼ N þ
D,
p0 ¼ N A:
ð7:39Þ
They are constants for uniform impurities which we assume. In the reference state
n ¼ n0, p ¼ p0, and other fields are zero. Then Eq. (7.36)2–4 become
Di, i ¼ qðΔp ΔnÞ,
∂ðΔnÞ
,
J in, i ¼ q
∂t
∂ðΔpÞ
:
J ip, i ¼ q
∂t
ð7:40Þ
Consider the case of small Δn and Δp. We linearize Eq. (7.37)3,4 as
J in ¼ qn0 μijn E j þ qDijn ðΔnÞ, j ,
J ip ¼ qp0 μijp E j qDijp ðΔpÞ, j :
ð7:41Þ
214
7.4
7
Other Effects
Extension of a Semiconducting Fiber
As an example consider the static extension of the piezoelectric semiconductor fiber
shown in Fig. 7.1 [8]. The shape of the cross section A may be arbitrary. The fiber is
within |x3| < L and is assumed to be long or thin, that is, its length is much larger than
the characteristic dimension of the cross section. It is made from a piezoelectric
semiconductor crystal of class 6mm which includes ZnO as a special case. The
c-axis of the crystal is along the axis of the fiber, that is, the x3 axis. The lateral
surface of the fiber is traction-free. It is under the action of a pair of equal and
opposite axial forces f which produces an axial stress T per unit cross-sectional area.
We are interested in the effects of T on the carrier distributions and electromechanical fields in the fiber.
The problem is of uniaxial stress and electric displacement (see Sect. 3.1). The
currents are also approximately uniaxial. In this case Eqs. (7.36)1 and (7.40) reduce to
T 3, 3 ¼ 0,
D3, 3 ¼ qðΔp ΔnÞ,
J 3n, 3 ¼ 0,
J 3p, 3 ¼ 0,
ð7:42Þ
T 3 ¼ c33 S3 e33 E3 ,
D3 ¼ e33 S3 þ ε33 E3 ,
ð7:43Þ
∂ðΔnÞ
,
∂x3
p
p ∂ðΔpÞ
J 3p ffi qp0 μ33
E3 qD33
,
∂x3
ð7:44Þ
where
n
n
J 3n ffi qn0 μ33
E3 þ qD33
S3 ¼ u3 , 3 ,
E 3 ¼ φ, 3 ,
ð7:45Þ
E
,
c33 ¼ 1=s33
E
e33 ¼ d 33 =s33
,
T
E
ε33 ¼ ε33 d233 =s33
:
ð7:46Þ
c33 , e33 , and ε33 are the one-dimensional effective elastic, piezoelectric, and dielectric
constants because of the introduction of the one-dimensional conditions for thin
x1
Fig. 7.1 A piezoelectric
semiconductor fiber of
crystals of class 6mm
T=f/A
T=f/A
c
x3
x2
2L
7.4 Extension of a Semiconducting Fiber
215
fibers, that is, setting T1 ¼ T2 ¼ 0, D1 ¼ D2 ¼ 0, and J1 ¼ J2 ¼ 0 for both electrons
and holes in the three-dimensional constitutive relations. We consider an electrically
isolated fiber. There are no concentrated free charges at the fiber ends and there are
no currents flowing in or out of the fiber at its ends. In this case the boundary
conditions are
T 3 ðLÞ ¼ T,
D3 ðLÞ ¼ 0,
ð7:47Þ
J 3n ðLÞ ¼ 0,
J 3p ðLÞ ¼ 0:
ð7:48Þ
In addition, the fiber is assumed to be electrically neutral at the reference state.
Therefore Δn and Δp must satisfy the following charge neutrality conditions:
Z
L
L
Z
Δndx ¼ 0,
L
L
Δpdx ¼ 0:
ð7:49Þ
Equation (7.42) consists of linear ordinary differential equations with constant
coefficients. Its solution satisfying Eqs. (7.47), (7.48), and (7.49) can be obtained in a
systematic and straightforward manner. Since there are no boundary conditions
prescribed directly on the mechanical displacement and the electric potential, the
mechanical displacement may have an arbitrary constant representing a rigid-body
translation of the fiber along x3. At the same time the electric potential may have an
arbitrary constant which does not make any difference in the electric field it produces. To uniquely determine the arbitrary constants in the mechanical displacement
and the electric potential, we set
u3 ð0Þ ¼ 0,
φð0Þ ¼ 0:
ð7:50Þ
Then the electromechanical fields in the fiber are found to be
e33 T
sinhκx3 ,
T
ε33 c33 κcoshκL
e33 T
coshκx3 ,
E3 ¼ T
ε33 c33 coshκL
T
e33 T
coshκx3 þ e33 ,
D3 ¼ c33 coshκL
c33
ð7:51Þ
n
μ33
e33 T
sinhκx3 ,
n T
ε33 c33 κcoshκL
D33
μp
e33 T
sinhκx3 ,
Δp ¼ p0 33
p T
ε33 c33 κcoshκL
D33
ð7:52Þ
φ¼
Δn ¼ n0
216
7
Other Effects
n
μ33
e33 T
sinhκx3 ,
n T
ε33 c33 κcoshκL
D33
μp
e33 T
sinhκx3 ,
p ¼ p0 p0 33
p T
D33 ε33 c33 κcoshκL
ð7:53Þ
T
e233 T
sinhκx3 þ x3 ,
T
2
c33
ε33 c33 κcoshκL
T
e233 T
S3 ¼ T 2
coshκx3 þ ,
c33
ε33 c33 coshκL
T 3 ¼ T,
ð7:54Þ
n
p
q μ33
μ33
κ ¼ T
n n0 þ p p0 ,
ε33 D33
D33
ð7:55Þ
n ¼ n0 þ n0
u3 ¼ where
2
n
D33
kθ
¼ ,
n
q
μ33
p
D33
kθ
:
p ¼
q
μ33
ð7:56Þ
θ is the absolute temperature. Equation (7.56) is called the Einstein relation. The
fields in Eqs. (7.51), (7.52), (7.53), and (7.54) are either symmetric or antisymmetric
about the center of the fiber. It can be verified that when n0, p0 ! 0 and hence
κ ! 0, the fields in Eqs. (7.51), (7.52), (7.53), and (7.54) reduce to the fields in Sect.
3.1 for a nonconducting piezoelectric fiber under an axial force.
For numerical results consider an n-type ( p ¼ 0) ZnO fiber with 2 L ¼ 1.2 μm, n0
21 3
¼ Nþ
D ¼ 10 m , and f ¼ 8.5 nN or lower which produces and axial stress
T ¼ 3.272 105 N/m2 or lower. For these data, the electric potential, axial electric
field, electron concentration and axial displacement are shown in Fig. 7.2. The fields
in Fig. 7.2a–c vary more rapidly near the ends of the fiber than in the central part. The
axial displacement in Fig. 7.2d is dominated by the linear term in Eq. (7.54)1. When
f increases, the fields become stronger as expected. Numerical calculations also show
that when n0 increases, κ increases according to Eq. (7.55), the hyperbolic functions
have larger values and they change more rapidly near the ends of the fiber. Accordingly the fields are stronger near the ends.
7.5
Nonlocal Effects
Nonlocality arises from the consideration of finite-distance or long-range interactions. In one-dimensional lattice dynamics it has been shown that a nonlocal theory
includes, besides the interactions between neighboring atoms, interactions among
non-neighboring atoms as well [9]. Nonlocality in constitutive relations is needed in
7.5 Nonlocal Effects
217
Fig. 7.2 Axial distributions of electromechanical fields for different values of f. (a) Electric
potential. (b) Axial electric field. (c) Electron concentration. (d) Axial mechanical displacement
modeling phenomena whose macroscopic characteristic length is not much larger
than the microscopic characteristic length. Consider a piezoelectric body V. Within
the linear theory, the nonlocal constitutive relations are given by [10]
R T ij ðxÞ ¼ R V cijkl ðx; x0 ÞSkl ðx0 Þ ekij ðx; x0 ÞEk ðx0 Þ dV ðx0 Þ,
Di ðxÞ ¼ V ½eikl ðx; x0 ÞSkl ðx0 Þ þ εik ðx; x0 ÞE k ðx0 ÞdV ðx0 Þ:
ð7:57Þ
As a special case, when the nonlocal material moduli in Eq. (7.57) are Dirac delta
functions, Eq. (7.57) reduces to the local constitutive relations of the theory of
piezoelectricity. The substitution of Eq. (7.57) into the equation of motion and the
charge equation results in integral-differential equations which are usually
complicated.
In the following we present an example of nonlocal electrostatics [11] which is
simple and is sufficient to show the basic effects of nonlocality. Consider an
unbounded dielectric plate as shown in Fig. 7.3. The plate is electroded and a voltage
is applied. We want to obtain its capacitance from the nonlocal theory.
218
7
x
Fig. 7.3 A thin dielectric
plate
Other Effects
Electrode
h
j =V
Dielectric
j =0
0
Electrode
The problem is one-dimensional. The boundary-value problem consists of:
dD
¼ 0, 0 < x < h,
dx
D ¼ ε0 ERþ P, 0 < x < h,
h
P ¼ ε0 χ 0 K ðx0 ; xÞEðx0 Þdx0 ,
dφ
E ¼ , 0 < x < h,
dx
φð0Þ ¼ 0, φðhÞ ¼ V,
0 < x < h,
ð7:58Þ
where, in this section, χ is the dimensionless relative electric susceptibility, differing
0
from the one in Eq. (2.10) by a factor ε0. When the kernel function K(x , x) has the
following special form:
K ðx0 ; xÞ ¼ δðx0 xÞ,
ð7:59Þ
Eq. (7.58) reduces to classical electrostatics. The dielectric material is assumed to be
0
homogeneous and isotropic. Hence K(x , x) must be invariant under translation and
inversion. We have
K ðx0 ; xÞ ¼ K ðx0 xÞ ¼ K ðx x0 Þ:
ð7:60Þ
0
Qualitatively, K(x0 , x) should be large near x ¼ x and decaying away from there. We
choose the following kernel function
K ð x0 xÞ ¼
1
expðjx x0 j=αÞ,
2α
α > 0,
ð7:61Þ
where α is a microscopic parameter with the dimension of length. It is a characteristic
0
length of microscopic interaction. It is easy to verify that the K(x , x) in Eq. (7.61) has
the following properties:
Z
lim K ¼ 0,
α ! 0þ
0
x 6¼ x
þ1
1
Kdx ¼ 1:
ð7:62Þ
7.5 Nonlocal Effects
219
Hence
lim K ¼ δðx0 xÞ,
ð7:63Þ
α!0þ
0
which shows that K(x , x) does include the local form as a limit case. We also note
0
that the above K(x , x) is the fundamental solution of the following differential
operator
2
α2
∂ K
þ K ¼ δðx0 xÞ:
∂x2
ð7:64Þ
Integrating Eq. (7.58)1 once, with the use of Eq. (7.58)2,3 we obtain
Z
h
D ¼ ε0 E ð xÞ þ ε0 χ
K ðx0 ; xÞE ðx0 Þdx0 ¼ σ e ,
ð7:65Þ
0
where σ e is an integration constant which physically represents the surface free
charge density on the electrode at x ¼ h. Equation (7.65) can be written as
Z
h
E ðxÞ ¼ χ
K ðx0 ; xÞE ðx0 Þdx0 0
σe
,
ε0
ð7:66Þ
which is a Fredholm integral equation of the second kind for the electric field E.
Instead of solving Eq. (7.66) directly, we proceed as follows. With Eq. (7.64), we
differentiate Eq. (7.66) with respect to x twice and obtain
R h ∂ K ð x0 ; xÞ
d 2 E ð xÞ
¼
χ
Eðx0 Þdx0
0
∂ x2
dx2
Rh 1
¼ χ 0 2 ½K ðx0 ; xÞ δðx0 xÞEðx0 Þdx0
αZ
h
1
σe
¼ 2 χ
K ðx0 ; xÞEðx0 Þdx0 α
ε0
Z 0h
1
1 σe
þ 2χ
δðx0 xÞEðx0 Þdx0 þ 2
α
α ε0
0
1
σe
¼ 2 EðxÞ þ χEðxÞ þ
:
α
ε0
2
ð7:67Þ
Hence a solution E of the integral equation in Eq. (7.66) also satisfies the following
differential equation
α2
d2 E
σe
ð
1
þ
χ
ÞE
¼
:
ε0
dx2
ð7:68Þ
220
7
Other Effects
The general solution to Eq. (7.68) can be obtained easily. It has two exponential
terms from the corresponding homogeneous equation, and a constant term which is
the particular solution. The general solution has two new integration constants.
These two integration constants result from the differentiation in obtaining the
differential equation in Eq. (7.68) from the integral equation in Eq. (7.66). Hence
the solution to Eq. (7.68) may not satisfy Eq. (7.66). We substitute the general
solution of Eq. (7.68) back into Eq. (7.66), which determines the two new
integration constants. Then, with the two boundary conditions in Eq. (7.58), we
can determine σ e and another integration constant resulting from integrating E for
φ, and thus obtain the nonlocal electric potential distribution φ as
3
h
kh
sinhk x þ sinh
6x χ
2
27
6
7
1 þ 2χ
φ¼4 þ
kh
kh
kh 5
h kh
cosh þ kαsinh
2
2
x
φ0 ¼ V,
h
2
tanhkh
2
1
V,
kh
1þkαtanh 2
ð7:69Þ
where φ0 is the local solution for comparison, and
pffiffiffiffiffiffiffiffiffiffiffi
1þχ
:
k¼
α
ð7:70Þ
The nonlocal electric field distribution E and the local solution E0 are
3
h
coshk x 6
7
2
6
7
E ¼ 41 þ χ
1 þ 2χ
kh
kh
kh5
cosh þ kαsinh
2
2
V
E0 ¼ :
h
2
tanhkh
2
kh
1þkαtanh 2
1
E0 ,
ð7:71Þ
Denoting the capacitance per unit electrode area from the local theory by C0 and the
one from the nonlocal theory by C, we have
C¼
C0 ¼
1 þ 2χ
kh
tanhkh
2
1
kh
1þkαtanh 2
ε0 ð 1 þ χ Þ
:
h
C0 ,
ð7:72Þ
With the expression of k in Eq. (7.70), we write Eq. (7.72)1 in the following form:
7.5 Nonlocal Effects
C
¼
C0
221
!1
pffiffiffiffiffiffiffiffiffiffiffi h tanh 1 þ χ 2α
χ
pffiffiffiffiffiffiffiffiffiffiffi h pffiffiffiffiffiffiffiffiffiffiffi
1 þ pffiffiffiffiffiffiffiffiffiffiffi h
:
1 þ χ 2α 1 þ 1 þ χ tanh 1 þ χ 2α
ð7:73Þ
It is seen that the thin film capacitance from the nonlocal theory depends on the ratio
h/2α of the film thickness to the microscopic characteristic length. From Eq. (7.73)
have
C=C 0 < 1,
lim
h
α!1
C
¼ 1,
C0
ð7:74Þ
ð7:75Þ
which shows that when the film thickness is large compared to the microscopic
characteristic length, the nonlocal solution approaches the local solution from below.
We also have the limit
lim
h
α!0
C
1
< 1,
¼
C0 1 þ χ
ð7:76Þ
which shows that the nonlocal and local solutions differ more for materials with a
larger value of χ. We plot C/C0 from Eq. (7.73) as a function of h/2α for χ ¼ 1,
10, and 100 in Fig. 7.4. It is seen that for a film with a moderate value of χ ¼ 100,
when the thickness h/2α ffi 10, there is a deviation of about 10% from the local
theory which has a fixed value of 1. The figure shows that C/C0 < 1, the deviation
from 1 increases as h decreases, and it disappears when h is large. This is the
so-called size effect of thin film capacitance.
The spatial distributions of the electric field for χ ¼ 10 and for two values of h/
2α ¼ 1 and 5, respectively, are shown in Fig. 7.5. The field is large near the
electrodes compared to the local solution with the fixed value of 1. The curve with
h/2α ¼ 5 has a larger electric field near the electrodes than the curve with h/2α ¼ 1.
This is a boundary effect exhibited by the nonlocal theory. Even for a thick capacitor,
Eq. (7.71) still yields
lim
h
α!1
E ð hÞ
χ
pffiffiffiffiffiffiffiffiffiffiffi > 1:
¼1þ
E0
1þ 1þχ
ð7:77Þ
When χ ¼ 10, Eq. (7.77) yields a value of 3.32. For materials with a large χ the value
of Eq. (7.7) can be large. Since E is large near the electrodes and D is a constant,
P must be smaller near the electrodes than near the center of the plate.
The spatial distributions of the normalized deviation of the nonlocal electric
potential from the local solution when χ ¼ 10 and h/2α ¼ 1 and 5, respectively,
are shown in Fig. 7.6. The curve with h/2α ¼ 5 shows a smaller deviation.
222
7
Other Effects
1
Fig. 7.4 Capacitance for
χ ¼ 1, 10, and 100
0.8
C/C0
c=1
0.6
c=10
0.4
c=100
0.2
0
0
Fig. 7.5 Electric field
distribution for χ ¼ 10, h/
2α ¼ 1 and 5
5
3.5
c=10, h=2a
3
2.5
E/E0
10
h/2a
c=10, h=10a
2
1.5
1
0.5
0
0
1
0.5
x/h
Fig. 7.6 Electric potential
deviation for χ ¼ 10, h/
2α ¼ 1 and 5
c=10, h=2a
0.08
c=10, h=10a
0.04
(f−f 0)/V 0
0
0.2
0.4
0.6
-0.04
-0.08
x/h
0.8
1
7.6 Gradient Effect
223
Finally, we note that in Eq. (7.68) the small parameter α appears as the coefficient
of the term with the highest derivative. Hence when α tends to zero we have a
singular perturbation problem of boundary layer type of a differential equation. For
this type of problems, when the small parameter is set to zero, certain boundary
conditions have to be dropped because the order of the differential equation is
lowered.
7.6
Gradient Effect
Gradient effects of strain and polarization (or electric field) can also be included in
constitutive relations. These effects can be shown to be related to weak nonlocal
effects. For example, consider a one-dimensional nonlocal constitutive relation
between Y and X in a homogeneous and unbounded medium. We have
R þ1
Y ðxÞ ¼ 1 K ðx0 xÞX ðx0 Þdx0
R þ1
¼ 1 K ðx0 xÞX ½x þ ðx0 xÞdx0
R þ1
¼ 1 K ðx0 xÞ½X ðxÞ þ X 0 ðxÞðx0 xÞ þ dx0
R þ1
ffi 1 K ðx0 xÞ½X ðxÞ þ X 0 ðxÞðx0 xÞdðx0 xÞ
R þ1
¼ 1 K ðx0 xÞX ðxÞdðx0 xÞ
R þ1
þ 1 K ðx0 xÞX 0 ðxÞðx0 xÞdðx0 xÞ
R þ1
¼ X ðxÞ 1 K ðx0 xÞd ðx0 xÞ
R þ1
þX 0 ðxÞ 1 K ðx0 xÞðx0 xÞdðx0 xÞ
¼ aX ðxÞ þ bX 0 ðxÞ,
ð7:78Þ
R þ1
a ¼ 1 K ðx0 xÞd ðx0 xÞ,
R þ1
b ¼ 1 K ðx0 xÞðx0 xÞd ðx0 xÞ:
ð7:79Þ
where
Therefore, to the lowest order of approximation, the nonlocal constitutive relation
reduces to a local one, and to the next order a gradient term arises.
Gradient terms in constitutive relations can also be introduced in the following
procedure from lattice dynamics with a finite microscopic interaction distance. Consider the extensional motion of a one-dimensional spring-mass system (see Fig. 7.7).
The motion of the ith particle is governed by the finite difference equation
m€uðiÞ ¼ k½uði þ 1Þ uðiÞ k½uðiÞ uði 1Þ,
ð7:80Þ
224
7
Fig. 7.7 A spring-mass
system
k
m
k
m
k
Other Effects
m
i-1
i
i+1
x-a
x
x+a
k
or, with the introduction of x
m€
uðxÞ ¼ k ½uðx þ aÞ þ uðx aÞ 2uðxÞ
1
1 0
1
¼ k uðxÞ þ u0 ðxÞa þ u00 ðxÞa2 þ u00 ðxÞa3 þ u0000 ðxÞa4 þ 2
6
24
1
1 0
þ uðxÞ u0 ðxÞa þ u00 ðxÞa2 u00 ðxÞa3
2
6
1 0000
4
þ u ðxÞa þ 2uðxÞ
24
1
ffi k u00 ðxÞa2 þ u0000 ðxÞa4 ¼ T 0 ðxÞa,
12
ð7:81Þ
where an extensional “force” T is introduced by the following constitutive relation
T ¼ kau0 ðxÞ þ
ka3 000
u ðxÞ,
12
ð7:82Þ
which formally can be viewed as dependent on the “strain” u0 and its second
gradient. It should be noted that, for strain gradient elasticity, according to Mindlin
[12], a continuum theory with the first strain gradient is fundamentally flawed in that
it is qualitatively inconsistent with lattice dynamics and the second strain gradient
needs to be included to correct the inconsistency. In the rest of this section, we focus
on the gradients of the electric variables rather than the strain.
Mindlin generalized the theory of piezoelectricity by allowing the stored energy
density to depend on the polarization gradient Pj,i [13]:
Z
Πðui ; Pi ; φÞ ¼
V
1
W Sij ; Pi ; P j, i ε0 φ, i φ, i þ φ, i Pi dV,
2
ð7:83Þ
where boundary terms are dropped for simplicity. The stationary conditions of the
above functional for independent variations of ui, φ, and Pi are
∂W
∂Sij , i
ε0 φ, ii
¼ 0,
þPi, i ¼0,
∂W
∂W
þ
φ, i ¼ 0:
∂Pi
∂P j, i , j
ð7:84Þ
7.6 Gradient Effect
225
Equation (7.84) represents seven equations for ui, Pi, and φ. If the dependence of
W on the polarization gradient is dropped, Eq. (7.84) reduces to the theory of linear
piezoelectricity. The inclusion of polarization gradient is supported by lattice
dynamics [14, 15]. The polarization gradient theory and lattice dynamics both
predict the thin film capacitance to be smaller than the classical result [15], as
shown in Fig. 7.4.
Instead of the polarization gradient, the electric field gradient can also be included
in constitutive relations [16]. The electric field gradient theory is equivalent to the
theory of dielectrics with electric quadrupoles [17] because the electric quadrupole is
the thermodynamic conjugate of the electric field gradient. Consider the following
functional [18]:
1
R
Πðui ; φÞ ¼ V W Sij ; E i ; E i, j ε0 Ei Ei f i ui þ ρe φ dV
2
R
∂φ
þπ
S t i ui þ dφ
dS,
∂n
ð7:85Þ
where d is related to surface free charge. The presence of the π term is variationally
consistent. Consider, for example,
1
W Sij ; E i ; E i, j ε0 Ei Ei
2
1
¼ H Sij ; Ei ε0 αijkl Ei, j E k, l ,
2
ð7:86Þ
where H is the usual electric enthalpy function of piezoelectric materials given in
Eq. (2.9), which is repeated below:
1
1
H Sij ; E i ¼ cijkl Sij Skl eikl Ei Skl εij E i E j :
2
2
ð7:87Þ
αijkl are new material constants due to the introduction of the electric field gradient
into the energy density function. They have the dimension of (length)2. Physically,
they may be related to characteristic lengths of microstructural interactions. Since
Ei,j ¼ Ej,i, αijkl have the same structure as cijkl as required by crystal symmetry. For
W to be negative definite in the case of pure electric phenomena without
mechanical fields, we require αijkl to be positive definite like εij. With the following
constraints
Sij ¼ ui, j þ u j, i =2,
E i ¼ φ, i ,
ð7:88Þ
from the variational functional in Eq. (7.85), for independent variations of ui and φ in
V, we have
226
7
Other Effects
T ji, j þ f i ¼ 0,
Di, i ¼ ρe ,
ð7:89Þ
∂W
¼ cijkl Skl ekij E k ,
∂Sij
Di ¼ ε0 E i þ Pi ¼ εij E j þ eikl Skl ε0 αijkl E k, lj ,
Pi ¼ Πi Πij, j ¼ ε0 χ ij E j þ eikl Skl ε0 αijkl Ek, lj ,
ð7:90Þ
∂W
¼ eikl Skl þ ε0 χ ij E j ,
∂Ei
∂W
Πij ¼ ¼ ε0 αijkl Ek, l ,
∂E i, j
ð7:91Þ
where we have denoted
T ij ¼
Πi ¼ and εij ¼ ε0(δij + χ ij). In this section χ ij is the relative electric susceptibility, differing
from the one in Eq. (2.10) by a factor ε0. When the energy density does not depend
on the electric field gradient, the equations reduce to the linear theory of piezoelectricity. The first variation of the functional also implies the following as possible
forms of boundary conditions on S:
δui ¼ 0,
RT jin j ¼ t i or
n
d
δφ þ Π
D
i i
S
ij n j ð∇s δφÞi dS ¼ 0,
∂φ
Πij n j ni ¼ π or δ
¼ 0,
∂n
ð7:92Þ
where ∇s is the surface gradient operator. One obvious possibility of Eq. (7.92)2 is
δφ ¼ 0 on S. With successive substitutions from Eqs. (7.90), (7.91), and (7.88),
Eq. (7.89) can be written as four equations in terms of ui and φ:
cijkl uk, lj þ ekij φ, kj þ f i ¼ 0,
eikl uk, li εij φ, ij þ ε0 αijkl φ, ijkl ¼ ρe :
ð7:93Þ
To see the most basic effects of the electric field gradient, consider the infinite
plate capacitor shown in Fig. 7.8 [19].
Fig. 7.8 A thin dielectric
plate
x
h
0
Electrode
f= V
Dielectric
f = -V
-h
Electrode
7.6 Gradient Effect
227
The problem is one-dimensional. We assume that the material is isotropic so that
there is no piezoelectric coupling and the problem is electrostatic. The equations and
boundary conditions from the electric field gradient theory are
dD
¼ 0, h < x < h,
dx
D ¼ ε0 E þ P, h < x < h,
d2 E
P ¼ ε0 χE ε0 α 2 , h < x < h,
dx
dφ
E ¼ , h < x < h,
dx
φðhÞ ¼ V, φðhÞ ¼ V,
ð7:94Þ
where χ is the relative electric susceptibility. From Eq. (7.94) the following equation
for φ can be obtained:
2
d4 φ
2d φ
k
¼ 0,
dx4
dx2
ð7:95Þ
where
k2 ¼
1þχ
:
α
ð7:96Þ
The general solution to Eq. (7.95) can be obtained in a straightforward manner. The
antisymmetric solution for φ is
φ ¼ C1 x þ C2 sinhkx,
ð7:97Þ
where C1 and C2 are integration constants. Due to the introduction of the electric
field gradient, the order of the equation for φ is now higher than the Laplace equation
in the classical theory of electrostatics. Therefore more boundary conditions are
needed. Following Mindlin, we prescribe the following additional boundary condition [15]:
V
Pðx ¼ hÞ ¼ λε0 χ ,
h
ð7:98Þ
where 0 λ 1 is a parameter. λ ¼ 1 represents the classical solution. Equation
(7.98) is for Mindlin’s polarization theory. For the electric field gradient theory here,
its equivalent form is
V
E ðx ¼ hÞ ¼ λ :
h
ð7:99Þ
228
7
Other Effects
With the solution in Eq. (7.97), the boundary conditions in Eqs. (7.94) and (7.99),
and the identification of the relation between an integration constant with the surface
charge on the electrode at x ¼ h, we obtain the capacitance C per unit area, the
potential φ and the electric field E as
1 þ λχ tanhkh
C
kh
¼
,
C0
1 þ χ tanhkh
kh
!
1 þ λχ tanhkh
sinhkx
kh
1
,
tanhkh
sinhkh
1 þ χ kh
!
1 þ λχ tanhkh
coshkx
kh
kh,
1
sinhkh
1 þ χ tanhkh
kh
φ 1 þ λχ tanhkh
x
kh
¼
þ
tanhkh h
V
1 þ χ kh
1 þ λχ tanhkh
E
kh
¼
þ
E0
1 þ χ tanhkh
kh
ð7:100Þ
ð7:101Þ
ð7:102Þ
where
C0 ¼
ε
,
2h
V
E0 ¼ ,
h
ð7:103Þ
are the capacitance and electric field from the classical theory, and ε ¼ ε0(1 + χ) is
the electric permittivity. Equation (7.100) is exactly the same as the result of the
polarization gradient theory in [15], and its behavior is qualitatively the same as what
is shown in Fig. 7.4.
7.7
Electromagnetic Effects
The theory of linear piezoelectricity is based on the quasistatic approximation in
[20]. In the theory, the mechanical equations are dynamic but the electric and
magnetic equations appear to be static. The electric field and the magnetic field are
not dynamically coupled. When the fully dynamic form of Maxwell’s equations is
used, the corresponding theory for coupled electromagnetomechanical fields is
called piezoelectromagnetism by some researchers [21]. For a piezoelectric but
nonmagnetizable dielectric body, the three-dimensional theory of linear
piezoelectromagnetism consists of the equations of motion and Maxwell’s equations
as shown by
T ji, j þ ρf i ¼ ρ€ui ,
εijk E k, j ¼ B_ i , εijk H k, j ¼ D_ i ,
Bi, i ¼ 0, Di, i ¼ 0,
ð7:104Þ
7.7 Electromagnetic Effects
229
and the following constitutive relations:
T ij ¼ cijkl Skl ekij E k ,
Di ¼ eijk S jk þ εij E j ,
B i ¼ μ0 H i ,
ð7:105Þ
where Bi is the magnetic induction, Hi is the magnetic field, and μ0 is the magnetic
permeability of free space. With Eq. (7.105), we can write Eq. (7.104) as
cijkl uk, li ekij E k, i ¼ ρ€uj ,
εijk Ek, j ¼ B_ i ,
1
εijk Bk, j ¼ eikl u_ k, l þ εik E_ k :
μ0
ð7:106Þ
Consider the special case of antiplane motions of polarized ceramics. We have [22]
u1 ¼ u2 ¼ 0, u3 ¼ u3 ðx1 ; x2 ; t Þ,
E 1 ¼ E 1 ðx1 ; x2 ; t Þ, E 2 ¼ E 2 ðx1 ; x2 ; t Þ,
H 1 ¼ H 2 ¼ 0, H 3 ¼ H 3 ðx1 ; x2 ; t Þ:
E 3 ¼ 0,
ð7:107Þ
The nonzero components of Sij, Tij, Di, and Bi are
S4 ¼ u3 , 2 , S5 ¼ u3 , 1 ,
T 4 ¼ c44 u3, 2 e15 E 2 , T 5 ¼ c44 u3, 1 e15 E1 ,
D1 ¼ e15 u3, 1 þ ε11 E1 , D2 ¼ e15 u3, 2 þ ε11 E2 ,
B3 ¼ μ0 H 3 :
ð7:108Þ
The nontrivial ones of the equations of motion and Maxwell’s equations in
Eq. (7.104) take the following form:
c44 ðu3, 11 þ u3, 22 Þ e15 ðE 1, 1 þ E 2, 2 Þ ¼ ρ€u3 ,
e15 ðu3, 11 þ u3, 22 Þ þ ε11 ðE 1, 1 þ E 2, 2 Þ ¼ 0,
E2, 1 E1, 2 ¼ μ0 H_ 3 ,
H 3, 2 ¼ e15 u_ 3, 1 þ ε11 E_ 1 ,
H 3, 1 ¼ e15 u_ 3, 2 þ ε11 E_ 2 :
ð7:109Þ
Eliminating the electric field components from Eq., (7.109)1,2, we obtain
c44 ðu3, 11 þ u3, 22 Þ ¼ ρ€u3 ,
ð7:110Þ
where c44 ¼ c44 þ e215 =ε11 . Differentiating Eq. (7.109)3 with respect to time once
and substituting from Eq. (7.109)4,5, we have
230
7
€ 3:
H 3, 11 þ H 3, 22 ¼ ε11 μ0 H
Other Effects
ð7:111Þ
The above equations can be written as
c44 ∇2 u3 ¼ ρ€u3 ,
€ 3,
∇2 H 3 ¼ ε0 μ0 H
_
D ¼ i3 ∇H 3 ,
ð7:112Þ
where ∇ and ∇2 are the two-dimensional gradient operator and Laplacian, respectively. D is the electric displacement in the (x1,x2) plane. i3 is the unit vector in the x3
direction. Equations (7.112)1,2 govern the displacement and magnetic fields. Once u3
and H3 are determined, D1 and D2 can be obtained from Eq. (7.112)3. Then the
electric field and the stress components can be obtained from constitutive relations.
From Eqs. (7.112)3 and (2.98)3,4, it can be seen that physically the ψ introduced by
Bleustein [23] is related to H3.
To see some of the electromagnetic effects specifically, we study the propagation
of surfaces waves in a ceramic half space [22]. The corresponding quasistatic
problem was analyzed in Sect. 4.3. Consider a ceramic half space poled in the x3
direction (see Fig. 7.9).
For surface waves propagating in the x1 direction, we have
u3 ¼ Uexpðξ2 x2 Þ cos ðξ1 x1 ω t Þ,
H 3 ¼ Hexpðη2 x2 Þ cos ðξ1 x1 ω t Þ,
ð7:113Þ
where U, H, ξ1, ξ2, η2, and ω are constants. The substitution of Eq. (7.113) into
Eqs. (7.110) and (7.111) results in
ξ22 ¼ ξ21 ρω2 =
c44 > 0,
η22 ¼ ξ21 ε11 μ0 ω2 > 0,
ð7:114Þ
where the inequalities are for decaying behavior from the surface. From
Eqs. (7.109)4,5 and (7.113) we obtain
Fig. 7.9 A ceramic half
space
Free space
x1
Ceramic
x2
Propagation
direction
7.7 Electromagnetic Effects
231
1 e15 ω ξ1 Uexpðξ2 x2 Þ
ε11 ω
þ η2 Hexpðη2 x2 Þ sin ðξ1 x1 ω t Þ,
1 E2 ¼
e15 ω ξ2 Uexpðξ2 x2 Þ
ε11 ω
þ ξ1 Hexpðη2 x2 Þ cos ðξ1 x1 ωt Þ:
E1 ¼
ð7:115Þ
First consider the case when the surface at x2 ¼ 0 is electroded with a perfect
conductor for which we have E1 ¼ 0. The electrode is assumed to be very thin with
negligible mass. Hence we have the traction-free boundary condition that T4 ¼ 0 on
the surface. Then from the T4 in Eqs. (7.108) and (7.115)1 we obtain
e15 ω ξ1 U þ η2 H ¼ 0,
ε11 c44 ω ξ2 U þ e15 ξ1 H ¼ 0:
ð7:116Þ
For nontrivial solutions of U and H, the determinant of the coefficient matrix of
Eq. (7.116) has to vanish which, with Eq. (7.114), leads to
sffiffiffiffiffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
v2
v2
1 2 1 α n2 2 ¼ k215 ,
vT
vT
ð7:117Þ
where
ω2
v2T
c44
2
,
α
¼
,
v
¼
,
T
ρ
c2
ξ21
1
ε11
e2
c2 ¼
, n2 ¼ , k215 ¼ 15 :
ε0 μ 0
ε0
ε11 c44
v2 ¼
ð7:118Þ
In Eq. (7.118), v is the surface wave speed, vT is the speed of plane shear waves
propagating in the x1 direction from quasistatic piezoelectricity, α is the ratio of
acoustic and light wave speeds which is normally a very small number, c is the speed
of light in a vacuum, and n is the refractive index in the x1 direction. Equation
(7.117) is an equation for the surface wave speed v. Waves with speed determined by
Eq. (7.117) are clearly nondispersive. Since α is very small, it is simpler and more
revealing to examine the following perturbation solution of Eq. (7.117) relevant to
Bleustein–Gulyaev waves for small α:
v2 ffi v2T 1 k415 1 αn2 k415 :
ð7:119Þ
It is seen that the effect of electromagnetic coupling on the speed of Bleustein–
Gulyaev waves is of the order of αn2 k415 . As a numerical example we consider
PZT-7A. Calculation shows that
232
7
k15 ¼ 0:671, n2 ¼ 460,
α ¼ 6:85 109 , αn2 k415 ¼ 6:38 107 :
Other Effects
ð7:120Þ
Hence the modification on the acoustic wave speed due to electromagnetic coupling
is very small and is negligible in most applications. When α is set to zero, or when
the speed of light approaches infinity, Eq. (7.119) reduces to the speed of the
Bleustein–Gulyaev waves in Eq. (4.64).
Next consider the case when the surface of the half space at x2 ¼ 0 is
unelectroded. In this case electromagnetic waves also exist in the free space of
x2 < 0. The solution for the free space x2 < 0 can be written as:
H 3 ¼ Hexp
η2 x2 cos ðξ1 x1 ω t Þ,
ð7:121Þ
and η2 are undetermined constants. Substituting Eq. (7.121) into
where H
Eq. (7.111), with ε11 replaced by ε0 for free space, we obtain
η22 ¼ ξ21 ε0 μ0 ω2 > 0:
ð7:122Þ
The electric field generated by the H3 in Eq. (7.121) through Eq. (7.109) with u3
dropped and ε11 replaced by ε0 for fee space is given by
1
η2 Hexp
η2 x2 sin ðξ1 x1 ω t Þ,
ε0 ω
1
ξ1 Hexp
E2 ¼
η2 x2 cos ðξ1 x1 ω t Þ:
ε0 ω
E1 ¼ ð7:123Þ
We impose the continuity of E1 and H3 at x2 ¼ 0 as well as the traction-free boundary
condition that T4 ¼ 0. This implies that
1
ðe15 ω ξ1 U þ η2 H Þ þ
1
¼ 0,
η H
ε0 ω 2
ε11 ω
¼ 0,
HH
ε11 c44 ω ξ2 U þ e15 ξ1 H ¼ 0:
ð7:124Þ
For nontrivial solutions the determinant of the coefficient matrix has to vanish,
which leads to
sffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!
v2
v2
v2
2
2
1 2
1 αn 2 þ n 1 α 2 ¼ k215 :
vT
vT
vT
ð7:125Þ
Equation (7.125) is an equation for v. Again, the waves are nondispersive. A
perturbation solution of Eq. (7.125) relevant to Bleustein–Gulyaev waves to the
first order in α is
7.8 Quasistatic Approximation
233
"
v ffi
2
v2T
1
#"
k415
ð 1 þ n2 Þ 2
1 α2n
k415
2
#
ð 1 þ n2 Þ 3
:
ð7:126Þ
Numerical calculation shows that, for PZT-7A,
α2n2
k415
ð 1 þ n2 Þ 3
¼ 1:30 1016 :
ð7:127Þ
When α is set to zero, Eq. (7.127) reduces to Eq. (4.72).
7.8
Quasistatic Approximation
The quasistatic approximation made in the theory of piezoelectricity can be considered as the lowest order approximation of the fully dynamic theory given by
Eq. (7.106) through the following perturbation procedure [20]. Consider an acoustic
wave with frequency ω in a piezoelectric crystal of characteristic dimension L. We
scale the various independent and dependent variables with respect to characteristic
quantities:
xi
ξi ¼ , τ ¼ ωt,
L
ui
U i ¼ , bi ¼ cBi ,
L
ð7:128Þ
1
c ¼ pffiffiffiffiffiffiffiffiffi
ε0 μ 0
ð7:129Þ
where
is the speed of light in free space and the scaling yields a b in the same unit as E.
Then Eq. (7.106) takes the following form:
2
2
1
∂ Uk
1 ∂E k
∂ Uj
cijkl
ekij
¼ ρω2 L
,
L
∂ξl ∂ξi L
∂ξi
∂τ2
1 ∂E k
ω ∂bi
εijk
,
¼
L ∂ξ j
c ∂τ
2
1
∂bk
eikl ∂ U k
εik ∂E k
þ ωε0
,
εijk
¼ ωε0
cLμ0 ∂ξ j
ε0 ∂ξl ∂τ
ε0 ∂τ
ð7:130Þ
234
7
Other Effects
or
2
2
∂ Uk
∂Ek
∂ Uj
ekij
¼ ρω2 L2
,
∂ξl ∂ξi
∂ξi
∂τ2
∂E k
∂bi
,
εijk
¼ η
∂ξ j
∂τ
!
cijkl
ð7:131Þ
2
εijk
∂bk
eikl ∂ U k εik ∂E k
þ
¼η
,
∂ξ j
ε0 ∂ξl ∂τ ε0 ∂τ
where
η¼
ωL
<< 1:
c
ð7:132Þ
To the lowest order,
2
2
∂ Uk
∂Ek
∂ Uj
ekij
¼ ρω2 L2
,
∂ξl ∂ξi
∂ξi
∂τ2
∂E k
εijk
¼ 0,
∂ξ j
∂bk
εijk
¼ 0,
∂ξ j
ð7:133Þ
cijkl uk, li ekij Ek, i ¼ ρ€uj ,
εijk E k, j ¼ 0,
εijk H k, j ¼ 0,
ð7:134Þ
cijkl
or
which is the quasistatic form for the theory of piezoelectricity.
References
1. H.F. Tiersten, On the nonlinear equations of thermoelectroelasticity. Int. J. Eng. Sci. 9, 587–604
(1971)
2. H.F. Tiersten, On the influence of material objectivity on electroelastic dissipation in polarized
ferroelectric ceramics. Math. Mech. Solids 1, 45–55 (1996)
3. J.S. Yang, Nonlinear equations of thermoviscoelectroelasticity. Math. Mech. Solids 3, 113–124
(1998)
4. R. D. Mindlin, On the equations of motion of piezoelectric crystals, in: Problems of Continuum
Mechanics (N.I. Muskhelishvili 70th Birthday volume), 1961, pp. 282–290
5. R. Holland, E.P. EerNisse, Design of Resonant Piezoelectric Devices (MIT Press, Cambridge,
MA, 1969)
References
235
6. R.F. Pierret, Semiconductor Fundamentals, 2nd edn. (Addison-Wesley, Reading, 1988)
7. A.R. Hutson, D.L. White, Elastic wave propagation in piezoelectric semiconductors. J. Appl.
Phys. 33, 40–47 (1962)
8. C.L. Zhang, X.Y. Wang, W.Q. Chen, J.S. Yang, An analysis of the extension of a ZnO
piezoelectric semiconductor nanofiber under an axial force. Smart Mater. Struct. 26, 025030
(2017)
9. A.C. Eringen, B.S. Kim, Relation between non-local elasticity and lattice dynamics. Crystal
Lattice Defects 7, 51–57 (1977)
10. A.C. Eringen, Theory of nonlocal piezoelectricity. J. Math. Phys. 25, 717–727 (1984)
11. J.S. Yang, Thin film capacitance in case of a non-local polarization law. Int. J. Appl.
Electromagn. Mech. 8, 307–314 (1997)
12. R.D. Mindlin, Elasticity, piezoelectricity and crystal lattice dynamics. J. Elast. 2, 217–282
(1972)
13. R.D. Mindlin, Polarization gradient in elastic dielectrics. Int. J. Solids Struct. 4, 637–642 (1968)
14. A. Askar, P.C.Y. Lee, A.S. Cakmak, Lattice-dynamics approach to the theory of elastic
dielectrics with polarization gradient. Phys. Rev. B 1, 3525–3537 (1970)
15. R.D. Mindlin, Continuum and lattice theories of influence of electromechanical coupling on
capacitance of thin dielectric films. Int. J. Solids Struct. 5, 1197–1208 (1969)
16. L.D. Landau, E.M. Lifshitz, Electrodynamics of Continuous Media, 2nd edn. (ButterworthHeinemann, Oxford, 1984)
17. A.C. Eringen, G.A. Maugin, Electrodynamics of Continua, vol I (Springer-Verlag, New York,
1990)
18. X.M. Yang, Y.T. Hu, J.S. Yang, Electric field gradient effects in anti-plane problems of
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41–45 (2004)
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1785–1796 (1976)
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(1968)
Chapter 8
Piezoelectric Devices
This chapter presents theoretical analyses on a few piezoelectric devices. Sections
8.1 through 8.3 are based on the exact theory of linear piezoelectricity. In particular,
Sect. 8.2 is an antiplane problem for which the notation of Section 2.9 is followed.
Sections 8.4 and 8.5 employ approximate one-dimensional equations for thin rods.
Sections 8.6 and 8.7 are on elastic rods and beams with piezoelectric elements in
extension and bending, respectively.
8.1
Generator or Energy Harvester
In this section we analyze a simple problem for mechanical to electrical energy
conversion in piezoelectric materials. Devices for this purpose may be called
generators or energy harvesters in different applications. To be simple we study a
ceramic plate in thickness-stretch vibration under surface mechanical loads (see
Fig. 8.1) [1]. This is an idealized structure convenient for showing the energy
conversion process but it is not a practical device.
The surfaces of the plate are electroded and the electrodes are connected by a
circuit whose impedance is denoted by Z per unit area when the motion is timeharmonic. When the plate is driven by a time-harmonic surface normal stress, the
mechanical boundary conditions are
T 3 j ¼ δ3 j pexpðiωt Þ,
x3 ¼ h:
ð8:1Þ
On each of the two electrodes, the electric potential is spatially constant, varying
with time. We denote the potential difference between the two electrodes by a
voltage V
© Springer Nature Switzerland AG 2018
J. Yang, An Introduction to the Theory of Piezoelectricity, Advances in Mechanics
and Mathematics 9, https://doi.org/10.1007/978-3-030-03137-4_8
237
238
8 Piezoelectric Devices
Fig. 8.1 A ceramic plate in
thickness-stretch vibration
T33 = p exp(iωt), T31 = T32 = 0
P
x3
2h
x1
Z
T33 = pexp(iωt), T31 = T32 = 0
φðx3 ¼ hÞ φðx3 ¼ hÞ ¼ V:
ð8:2Þ
The free charge per unit area on the electrode at x3 ¼ h and the current per unit area
that flows out of the electrode at x3 ¼ h are
Qe ¼ D3 ,
I ¼ Q_ e :
ð8:3Þ
For time-harmonic motions, we use the following complex notation:
e ; Igexpðiω t Þg:
Q
fV; Qe ; I g ¼ ReffV;
ð8:4Þ
e :
I ¼ iω Q
ð8:5Þ
Then
We also have the following relationship for the output circuit:
I ¼ V=Z:
ð8:6Þ
For thickness-stretch motions the complex solutions are in the following form:
u3 ¼ u3 ðx3 Þexpðiωt Þ,
φ ¼ φðx3 Þexpðiωt Þ:
u1 ¼ u2 ¼ 0,
ð8:7Þ
The nontrivial components of strain and electric field are
S33 ¼ u3, 3 ,
E3 ¼ φ3 :
ð8:8Þ
In Eq. (8.8) and from hereon, we drop the time-harmonic factor for simplicity. The
nontrivial stress and electric displacement components are
T 11 ¼ T 22 ¼ c13 u3, 3 þ e31 φ, 3
T 33 ¼ c33 u3, 3 þ e33 φ, 3 ,
D3 ¼ e33 u3, 3 ε33 φ, 3 :
The equations to be satisfied are
ð8:9Þ
8.1 Generator or Energy Harvester
239
c33 u3, 33 þ e33 φ, 33 ¼ ρω2 u3 ,
e33 u3, 33 ε33 φ, 33 ¼ 0:
ð8:10Þ
Equation (8.10)2 can be integrated to yield
φ¼
e33
u3 þ B1 x3 þ B2 ,
ε33
ð8:11Þ
where B1 and B2 are integration constants, and B2 is immaterial. Substituting
Eq. (8.11) into the expressions for T33, D3, and Eq. (8.10)1, we obtain
T 33 ¼ c33 u3, 3 þ e33 B1 ,
D3 ¼ ε33 B1 ,
ð8:12Þ
c33 u3, 33 ¼ ρ ω2 u3 ,
ð8:13Þ
where
c33 ¼ c33 1 þ k233 ,
k233 ¼
e233
:
ε33 c33
ð8:14Þ
The general solution to Eq. (8.13) and the corresponding expression for the electric
potential are
u3 ¼ A1 sin ξx3 þ A2 cos ξx3 ,
e33
φ ¼ ðA1 sin ξx3 þ A2 cos ξx3 Þ þ B1 x3 þ B2 ,
ε33
ð8:15Þ
where A1 and A2 are integration constants, and
ξ2 ¼
ρ 2
ω:
c33
ð8:16Þ
The expression for the relevant stress is then
T 33 ¼ c33 ðA1 ξ cos ξx3 A2 ξ sin ξx3 Þ þ e33 B1 :
ð8:17Þ
The boundary conditions require that
c33 A1 ξ cos ξh c33 A2 ξ sin ξh þ e33 B1 ¼ p,
c33 A1 ξ cos ξh þ c33 A2 ξ sin ξh þ e33 B1 ¼ p:
ð8:18Þ
Adding and subtracting the two equations in Eq. (8.18), respectively, we obtain
240
8 Piezoelectric Devices
c33 A1 ξ cos ξh þ e33 B1 ¼ p,
c33 A2 ξ sin ξh ¼ 0,
ð8:19Þ
which implies that A2 ¼ 0. From Eqs. (8.3), (8.5) and (8.12)2,
e ¼ ε33 B1 ,
Q
I ¼ iωε33 B1 :
ð8:20Þ
Then the circuit condition in Eq. (8.6) can be written as
iωε33 B1 ¼ V=Z:
ð8:21Þ
From Eqs. (8.2) and (8.15)2, the voltage across the electrodes takes the following
form:
e33
A1 sin ξh þ B1 h ¼ V:
2
ε33
ð8:22Þ
Solving
Equations (8.19)1, (8.21), and (8.22) are three equations for A1, B1 and V.
these equations, we obtain
1
,
2
sin ξh
k33
ξh cot ξh 1 þ Z=Z 0
e33 p
1
B1 ¼ ,
ε33 c33 ð1 þ Z=Z 0 Þξh cot ξh k233
V
A1 ¼
p
h
c33
¼ iω e33
1
p
Z
,
c33 ð1 þ Z=Z 0 Þξh cot ξh k233
ð8:23Þ
where
e2
k233
,
k233 ¼ 33 ¼
ε33 c33 1 þ k233
Z0 ¼
1
,
iωC 0
C0 ¼
ε33
:
2h
ð8:24Þ
In the above, C0 is a reference capacitance per unit area of the plate. From Eq. (8.23)
the resonance frequencies are determined by setting the denominator of the expressions to zero:
ξh cot ξh k233
¼ 0:
1 þ Z=Z 0
ð8:25Þ
When Z ¼ 0 or the electrodes are shorted, Eq. (8.25) reduces to Eq. (5.22). From
Eqs. (8.20) and (8.23) we have
8.1 Generator or Energy Harvester
241
p
1
I ¼ iωe33
:
c33 ð1 þ Z=Z 0 Þξh cot ξh k233
ð8:26Þ
With the complex notation and the circuit condition, the average output electrical
power per unit plate area over a period is given by
1 ∗ ∗ 1∗
1 2
I V þ I V ¼ I I ðZ þ Z ∗ Þ ¼ jIj RefZ g
4
4
2
2
1 2 2 ε33 j k233 j
1
,
RefZ g
¼ pω
2
2
j c33 j
ð1 þ Z=Z 0 Þξh cot ξh k33 P2 ¼
ð8:27Þ
where an asterisk represents complex conjugate. Equation (8.27) shows that the
output power P2 depends on the input stress amplitude p quadratically, as expected.
The output power also explicitly depends on ω2, ε33, and j k233 j linearly, and is
inversely proportional to c33 . Therefore, a large dielectric constant ε33, a large
electromechanical coupling factor j k233 j, and a small stiffness c33 are helpful for
raising the output power. ω and j k233 j, and hence ε33, e33, c33 , and ρ are also present
implicitly in the last factor in Eq. (8.27). When the driving frequency is near a
resonance frequency, the denominator of this factor is very small, leading to a large
output power. We also see from Eq. (8.27) that the output power P2 depends on Re
{Z} linearly for a small load impedance Z, and it diminishes as Z becomes large. The
output power P2 depends only on the real part of the load impedance (resistance).
The imaginary part of Z is inductive/capacitive, taking energy away and giving it
back periodically with no net energy consumption when averaged in a period.
To calculate the mechanical input power we need the velocities at the plate
surfaces
v3 ðx3 ¼ hÞ ¼ iωA1 sin ξh
p
1
:
¼ iω h
c33
k233
ξh cot ξh 1 þ Z=Z 0
ð8:28Þ
Therefore the input mechanical power is
1 1
P1 ¼ ð2Þ pv∗
þ p∗ v 3 ¼ p v ∗
3 þ v3 ¼ pRefv3 g
4 3
2
¼ ωp2 h
Im c∗
33
k233
1þZ=Z 0
ξh cot ξh
∗
2
2
k33
ξh
cot
ξh
jc33 j2 1þZ=Z
0
The efficiency of the energy conversion is defined as
:
ð8:29Þ
242
8 Piezoelectric Devices
7.00E-05
p2
6.00E-05
5.00E-05
4.00E-05
Z=(3+i )Z0
Z=(1+i )Z0
3.00E-05
2.00E-05
Z=i Z0
Z=3i Z0
w (1/sec.)
1.00E-05
0.00E+00
0.0E+00 2.5E+05 5.0E+05 7.5E+05
1.0E+06 1.3E+06 1.5E+06
Fig. 8.2 Power density as a function of the driving frequency ω
η¼
P2
1
¼
P1 2h
ωε33 j k233 j RefZ g
2
∗ :
k 33
jð1þZ=Z 0 Þj2
∗
Im c33 1þZ=Z 0 ξh cot ξh
j
c33 j
ð8:30Þ
It depends on the load impedance Z linearly for a small load. It vanishes with a large
Z. It also becomes large at the resonance frequencies. Another quantity of practical
interest is the power density defined as the output power per unit volume. In our case,
it is given by
p2 ¼
P2
:
2h
ð8:31Þ
For a numerical example, consider PZT-5H. The mechanical damping of the
material is introduced through replacing c33 by c33(1 + iQ1), where Q ¼ 1 102 is
the material quality factor. The device thickness h is chosen, rather arbitrarily, as
h ¼ 1 cm. Figure 8.2 shows the output power density versus the driving frequency
for different values of the load impedance. The system has infinitely many resonance
frequencies. Only the behavior near the first resonance frequency of 715,000 1/sec.
is shown. Near resonance the output is maximal.
8.2
Transducer for Acoustic Wave Generation
Consider the generation of antiplane acoustic waves in an elastic half space by a
piezoelectric plate transducer (see Fig. 8.3) [2]. The piezoelectric layer is ceramics
poled along the x3 direction. It is driven by an alternating voltage. Waves are
generated and they propagate in the elastic half space away from the ceramic layer.
8.2 Transducer for Acoustic Wave Generation
Fig. 8.3 A ceramic
transducer on an elastic half
space
243
j =Vexp(-iwt)
Traction-free
Piezoelectric
j=0
h
x1
Elastic
x2
Propagation
direction
With the notation in Sect. 2.9, the governing equations and the boundary as well
as continuity conditions are
c∇2 u ¼ ρ€u, h < x2 < 0,
∇2 ψ ¼ 0, h < x2 < 0,
e
φ ¼ ψ þ u, h < x2 < 0,
ε
b
c ∇2 u ¼ b
ρ €u,
x2 > 0,
T 23 ðx2 ¼ hÞ ¼ 0,
φðx2 ¼ hÞ φðx2 ¼ 0Þ ¼ Vexpðiω t Þ,
uðx2 ¼ 0 Þ ¼ uðx2 ¼ 0þ Þ,
T 23 ðx2 ¼ 0 Þ ¼ T 23 ðx2 ¼ 0þ Þ,
u is outgoing, x2 ! þ1,
ð8:32Þ
ð8:33Þ
ð8:34Þ
where b
ρ and b
c are the mass density and shear modulus of the elastic half space.
For –h < x2 < 0, the solutions can be written as
u ¼ ðA1 cos ξx2 þ A2 sin ξx2 Þexpðiω t Þ,
ψ ¼ ðB1 x2 þ B2 Þexpðiω t Þ,
ð8:35Þ
where A1, A2, B1, and B2 are undetermined constants, and
ξ2 ¼
ρω2
:
c
ð8:36Þ
For boundary and continuity conditions, we need
T 23 ¼ cu, 2 þ eψ , 2
¼ cA1 ξ sin ξx2 þ cA2 ξ cos ξx2 þ eB1 expðiω t Þ,
e
φ ¼ψ þ u
e ε
e
¼ A1 cos ξx2 þ A2 sin ξx2 þ B1 x2 þ B2 expðiω t Þ:
ε
ε
For x2 > 0, the solution can be written as
ð8:37Þ
244
8 Piezoelectric Devices
b
b
u ¼ Aexpi
ξ x2 ω t ,
ð8:38Þ
b is an undetermined constant, and
where A
b
ρω
b
:
ξ2 ¼
b
c
2
ð8:39Þ
Equation (8.38) already satisfies the condition that the waves in the elastic medium
are propagating away from the ceramic layer (radiation condition for unidirectional
waves). For continuity conditions we need
b
b
T 23 ¼ b
c u, 2 ¼ b
c ib
ξ Aexpi
ξx2 ω t :
ð8:40Þ
Then the continuity and boundary conditions take the following form:
cA1 ξ sin ξh þ cA2 ξ cos ξh þ eB1 ¼ 0,
e
e
e
A1 cos ξh A2 sin ξh B1 h þ B2 A1 B2 ¼ V,
ε
ε
ε
b
A1 ¼ A,
b
cA2 ξ þ eB1 ¼ b
c ib
ξ A:
ð8:41Þ
Our main interest is to find the following:
1 cos ξh
b ¼ e
V,
A
c
Δ
ð8:42Þ
b
c
ξh
Δ ¼ ð1 cos ξhÞ k2 ð cos ξh 1Þ i b
c
!
b
cb
ξ þ sin ξh þ i
ξh k2 sin ξh ,
cξ
e2
k2 ¼ :
ε
c
ð8:43Þ
where
b depends strongly on the driving frequency. Numerical results
It can be seen that A
b is large at a series of discrete frequencies.
[2] show that, similar to Fig. 8.2, A
8.3
Power Delivery Through a Wall
In certain applications some energy needs to be delivered through a wall. One way of
doing it is using two piezoelectric transducers and acoustic waves [3]. One transducer is for generating acoustic waves which propagate through the wall. The other
8.3 Power Delivery Through a Wall
245
Area S
Electrode
x3=h0+h1
j =V1
x3=h0
j =0
x3=-h0
j =0
x3=-h0-h2
j =V2
x3
I1
Transducer
Metal plate
P
x1
O
Transducer
ZL
P
I2
Electrode
Fig. 8.4 An elastic metal plate between two piezoelectric transducers
transducer is on the other side of the wall for converting the acoustic wave energy to
electric energy. Specifically, consider the structure in Fig. 8.4 in which a metal plate
representing the wall is sandwiched by two piezoelectric layers. One is driven by a
prescribed electric voltage source for acoustic wave generation. The other converts
the acoustic energy into electric energy to power a load circuit with impedance ZL.
The transducers are from ceramics poled in the thickness direction. Hence the
plate is driven into thickness-stretch motion. We neglect edge effects. For the
piezoelectric layers, the governing equations are
c33 u3, 33 þ e33 φ, 33 ¼ ρ u€3 ,
e33 u3, 33 ε33 φ, 33 ¼ 0:
ð8:44Þ
The relevant stress component T33 and the electric displacement component D3 are
given by
T 33 ¼ c33 u3, 3 þ e33 φ, 3 ,
D3 ¼ e33 u3, 3 ε33 φ, 3 :
ð8:45Þ
Similarly, for the elastic layer with mass density ρ0 and the relevant elastic constant
c0 , the governing equations are
c0 u3, 33 ¼ ρ0 €u3 ,
ð8:46Þ
T 33 ¼ c0 u3, 3 :
ð8:47Þ
The boundary conditions at the traction-free top of the driving or input transducer
which is also the top of the whole structure are
T 33 ðh0 þ h1 Þ ¼ 0,
φ ¼ V 1,
ð8:48Þ
246
8 Piezoelectric Devices
where V1 is considered known. At the bottom of the driving transducer, the electromechanical continuity and boundary conditions are
u3 hþ
0 ¼ u3 h0 , T 33 hþ
0 ¼ T 33 h0 ,
þ
φ h0 ¼ 0:
ð8:49Þ
Similarly, at the top of the output transducer, the boundary and continuity conditions
are
þ
u3 h
0 þ ¼ u3 h
0 ,
T 33 h0 ¼ T 33 h
0 ,
φ h
¼
0:
0
ð8:50Þ
At the bottom of the output transducer, the boundary conditions are
T 33 ½ðh0 þ h2 Þ ¼ 0,
φ ½ðh0 þ h2 Þ ¼ V 2 ,
ð8:51Þ
where V2 is unknown. The current density per unit area flowing into the driving
electrode at the top of the input transducer where x3 ¼ h0 + h1 is given by:
I 1 ¼ Q_ 1e ¼ iω Q1e ,
Q1e ¼ D3 ðh0 þ h1 Þ,
ð8:52Þ
where Q1e is the charge per unit area on the driving electrode at x3 ¼ h0 + h1. The
current density flowing out of the bottom electrode of the output transducer and the
charge on it are
I 2 ¼ Q_ 2e ¼ iω Q2e ,
Q2e ¼ D3 ½ðh0 h2 Þ:
ð8:53Þ
In time-harmonic motions, for the output circuit, we have the following equation
I 2 S ¼ V 2 =Z L ,
ð8:54Þ
where S is the electrode area under consideration. Equations (8.48), (8.49), (8.50)
and (8.51) and Eq. (8.54) are eleven equations that will be used to determine the
eleven undetermined constants in the solutions of each layer.
For time-harmonic motions, we use the complex notation and write
fu3 ðx3 Þ; φðx3 Þ; V 1 ; V 2 ; I 1 ; I 2 g
¼ ReffU ðx3 Þ; Φðx3 Þ; V1 ; V2 ; I1 ; I2 gexpðiωt Þg:
ð8:55Þ
8.3 Power Delivery Through a Wall
247
Then the governing equations in each layer reduce to ordinary differential equations
of constant coefficients. The solutions for each layer can be obtained in a straightforward manner and are written as, from the top layer (the driving transducer) to the
bottom,
U ðx3 Þ ¼ A1 sin ηx3 þ B1 cos ηx3 , h0 x3 h0 þ h1 ,
Φðx3 Þ ¼ F 1 x3 þ G1
e33
þ ðA1 sin ηx3 þ B1 cos ηx3 Þ, h0 x3 h0 þ h1 ,
ε33
U ðx3 Þ ¼ A2 sin η0 x3 þ B2 cos η0 x3 ,
ð8:56Þ
h0 x 3 h0 ,
ð8:57Þ
U ðx3 Þ ¼ A3 sin ηx3 þ B3 cos ηx3 , h0 h2 x3 h0 ,
Φðx3 Þ ¼ F 2 x3 þ G2
e33
þ ðA3 sin ηx3 þ B3 cos ηx3 Þ, h0 h2 x3 h0 ,
ε33
ð8:58Þ
where
1=2
η ¼ ρω2 =
c33
,
1=2
η0 ¼ ρ0 ω2 =c0
,
c33 ¼ c33 þ e233 =ε33 :
ð8:59Þ
A1 through A3, B1 through B3, F1, F2, G1, and G4 are ten undetermined constants.
Substituting the above solutions into the boundary and continuity conditions in
Eqs. (8.48), (8.49), (8.50), and (8.51) and the circuit equation in Eq. (8.54), we
obtain eleven equations for the above ten undetermined constants plus the output
voltage V2 which is unknown.
For a numerical example, consider PZT-5H for the piezoelectric transducers. For
the elastic plate, consider steel with ρ0 ¼7850 kg/m3 and c0 ¼2.69 1011 N/m2. In our
numerical calculation, c33 and c0 are replaced by c33(1 + iQ1) and c0 (1 + iQ1),
where c33, c0 and Q are real numbers. Q ¼ 102. h0 ¼ 3 mm, h1 ¼ 1 mm, h2 ¼ 2 mm,
and S ¼ 0.01 m2. Figure 8.5 shows the normalized output voltage j V2 =V1 j versus
the driving frequency ω. The output voltage assumes maxima at the thickness-stretch
resonance frequencies. The highest peak in this example occurs at the fourth
resonance. At the lowest resonance, the output transducer is not severely stressed
because of the traction-free boundary condition at the bottom surface. Correspondingly, the electric field within the output transducer generated through piezoelectric
coupling is not strong. This leads to a moderate output voltage. At the next few
resonances, the stress and the corresponding electric field in the output transducer
become increasingly stronger, leading to an increasing output voltage. This trend,
however, reverses after the fourth resonance when there are stress nodal points with
zero stress present within the output transducer. Across the nodal points the stress
reverses its sign and so does the piezoelectrically generated electric field. This causes
voltage cancellation across the nodal points. Hence, the voltage across the output
transducer decreases.
248
8 Piezoelectric Devices
Fig. 8.5 Normalized output
voltage versus driving
frequency
8.4
Transformer
Piezoelectric materials can be used to make transformers for raising or lowering a
voltage. The basic behavior of a piezoelectric transformer can be described by the
linear theory of piezoelectricity. In this section, we perform a theoretical analysis
[4] on Rosen transformers operating with extensional modes of ceramic rods.
Consider the Rosen transformer of length a + b, width w and thickness h as
shown in Fig. 8.6. It is a slender rod with a, b >> w >> h. The driving or input
part in –a < x1 < 0 is poled in the x3 direction and is electroded at x3 ¼ 0 and x3 ¼ h.
The electrodes are represented by areas bounded by the thick lines in the figure.
The output part with 0 < x1 < b is poled in the x1 direction with one output electrode
at the end where x1 ¼ b. The other output electrode is shared with the driving part
(where φ ¼ 0). Under a time-harmonic driving voltage V1(t) with a proper
frequency, the transformer can be driven into length extensional vibration as a
rod and produce an output voltage V2(t). The output electrodes are connected by an
output or load circuit whose impedance is denoted by ZL.
Since the transformer is nonuniform with the poling directions in the two parts
oriented differently, we need to analyze each part separately and then apply
continuity as well as boundary conditions. For the extension of thin rods we
make the usual uniaxial stress approximation:
T 2 ¼ T 3 ¼ T 4 ¼ T 5 ¼ T 6 ¼ 0,
ð8:60Þ
which is valid for both the driving and output parts. For the electric field in the
driving part in –a < x1 < 0, based on the electrode configuration, we approximately
have
8.4 Transformer
249
x3
j = V1
j =V2
x2
I1
h
P
P
a
w
b
j=0
x1
I2
ZL
Fig. 8.6 Rosen piezoelectric transformer
φ¼
x3
V 1,
h
ð8:61Þ
which implies that
E1 ¼ E 2 ¼ 0,
1
E3 ¼ V 1 :
h
ð8:62Þ
The relevant equation of motion and constitutive relations are
T 1, 1 ¼ ρ€u1 ,
S1 ¼ s11 T 1 þ d 31 E3 ,
D3 ¼ d31 T 1 þ ε33 E 3 :
ð8:63Þ
From Eqs. (8.60), (8.61), (8.62), and (8.63) we can obtain the following equation for
u1 and expressions for T1 and D3:
1
u1, 11 ¼ ρ€u1 ,
s11
1
d31
V1 ,
u1, 1 þ
T1 ¼
s11
h
d 31
V1
u1, 1 ε33 ,
D3 ¼
s11
h
ð8:64Þ
where
ε33 ¼ ε33 1 k231 ,
k 231 ¼
d231
:
ε33 s11
At the left end, we have the following boundary condition:
ð8:65Þ
250
8 Piezoelectric Devices
1
d31
V 1 ¼ 0,
T1 ¼
u1, 1 þ
s11
h
x1 ¼ a:
ð8:66Þ
The current flowing out of the driving electrode at x3 ¼ h is given by
I 1 ¼ Q_ 1e ,
Z
Q1e ¼ w
0
a
D3 dx1 ,
ð8:67Þ
where Q1e is the charge on the driving electrode at x3 ¼ h.
For the output part in 0 < x1 < b, in addition to Eq. (8.60), we approximately have
D2 ¼ D3 ¼ 0,
ð8:68Þ
which implies, from the constitutive equations, that
E2 ¼ E 3 ¼ 0:
ð8:69Þ
Hence the dominating electric field component is
E 1 ¼ φ, 1 ,
φ ¼ φðx1 ; t Þ:
ð8:70Þ
The relevant equation of motion, the charge equation, and constitutive relations are
T 1, 1 ¼ ρ€u1 ,
D1, 1 ¼ 0,
S1 ¼ s33 T 1 þ d 33 E1 ,
D1 ¼ d33 T 1 þ ε33 E 1 :
ð8:71Þ
The equation for u1 and the expressions for T1, D1, and φ can be written as
1
u1, 11 ¼ ρ€u1 ,
s33
1
u1, 1 k 233 c1 ,
T1 ¼
s33
d33
D1 ¼
c1 ,
s33
1
φ ¼ ð c 1 x 1 u1 Þ þ c 2 ,
d 33
where
ð8:72Þ
8.4 Transformer
251
s33 ¼ s33 1 k233
,
d33 ¼ d33
!
1
1 2 ,
k33
k233 ¼
d 233
:
s33 ε33
ð8:73Þ
c1(t) and c2(t) are two integration constants which may still be functions of time. The
following boundary conditions need to be satisfied at the right end:
1
u1, 1 k233 c1 ¼ 0,
s33
φ ¼ V 2 ðt Þ, x1 ¼ b:
T1 ¼
x1 ¼ b,
ð8:74Þ
Physically, c1 is related to the electric charge Q2e and hence the current I2 on the
output electrode at x1 ¼ b:
D1 ¼
d33
Qe
c1 ¼ 2 ,
s33
wh
I 2 ¼ Q_ 2e :
ð8:75Þ
At the junction of the two parts where x1 ¼ 0, the following continuity and
boundary conditions need to be prescribed:
u1 ð0 Þ ¼ u1 ð0þ Þ,
T 1 ð0 Þ ¼ T 1 ð0þ Þ,
1
φð 0 þ Þ ¼ V 1 :
2
ð8:76Þ
Since piezoelectric transformers operate under a time-harmonic driving voltage, we
employ the complex notation and write
fu1 ; φ; V 1 ; V 2 I 1 ; I 2 ; c1 ; c2 g
¼ ReffU; Φ; V1 ; V2 I1 ; I2, C1 ; C2 gexpðiωt Þg:
ð8:77Þ
For the output electrodes, under the complex notation, we have the following circuit
condition:
I2 ¼ V2 =Z L :
ð8:78Þ
When ZL ¼ 0 or 1, we have shorted or open output electrodes.
For free vibrations we set V1 ¼ 0 for shorted driving electrodes and c1 ¼ 0 for
open output electrodes. The equations and boundary conditions all become homogeneous. Equations (8.64)1, (8.72)1, (8.66), (8.74)1, and (8.76)1,2 reduce to
252
8 Piezoelectric Devices
1
U , 11 ¼ ρω2 U, a < x1 < 0,
s11
1
U , 11 ¼ ρω2 U, 0 < x1 < b,
s33
1
1
U , 1 ðaÞ ¼ 0,
U , 1 ðbÞ ¼ 0,
s11
s33
1
1
U ð0 Þ ¼ U ð0þ Þ,
U , 1 ð0 Þ ¼ U , 1 ð0þ Þ,
s11
s33
ð8:79Þ
where ω is an unknown resonance frequency at which nontrivial solutions of U to the
above equations exist. The solution of Eq. (8.79) when ω ¼ 0 is a rigid-body
displacement with U being a constant. It is not of interest here. The other solutions
of Eq. (8.79) are given by
ω ¼ ωðnÞ ,
U ðnÞ ¼
8 1
2
s
>
ðnÞ
>
< s1133 sin k x1 þ
n ¼ 1, 2, 3, . . . ,
ð8:80Þ
1
cos k ðnÞ x1 , a < x1 < 0,
tan kðnÞ b
1
sin kðnÞ x1 þ
cos kðnÞ x1 , 0 < x1 < b,
tan kðnÞ b
>
>
:
ð8:81Þ
where
kðnÞ
2
2
¼ ρs11 ωðnÞ ,
2
2
kðnÞ ¼ ρ
s33 ωðnÞ :
ð8:82Þ
ω(n) is the nth root of the following frequency equation:
1=2
tan ka
s11
:
¼ s33
tan kb
ð8:83Þ
We note that the width w and thickness h do not appear in the frequency equation.
This is as expected from a one-dimensional model. Once the displacement U is
known, the potential Φ in the output part can be found from Eqs. (8.72)4 and (8.76)3.
We have
ΦðnÞ ¼
8
<
0, a < x1 < 0,
1
1 ðnÞ
ðnÞ x1 1 ,
cos
k
: sin k x1 þ
d33
tan kðnÞ b
0 < x1 < b:
ð8:84Þ
As a numerical example, consider a transformer made from polarized ceramics PZT5H. We consider a transformer with a ¼ b ¼ 22 mm, w ¼ 10 mm, and h ¼ 2 mm.
With the above data, the first root of the frequency equation in Eq. (8.83), which is
the resonance frequency of the operating mode, is found to be.
a
b
1 U(1)
0.8
0.6
0.4
0.2
0
-0.2 -1
-0.4
-0.6
-0.8
-1
1
F (1)
x1/a
0.5
-0.5
0
0.5
1
x1 /a
0
-1
-0.5
0
0.5
1
Fig. 8.7 (a) Normalized U(1). (b) Normalized Φ(1) of the first extensional mode
f ð1Þ ¼ ωð1Þ =2π ¼ 36:35 kHz:
ð8:85Þ
The mode shape of the operating mode U(1) is shown in Fig. 8.7a. It is normalized by
its maximum. Normalized Φ(1) as a function of x1 is shown in Fig. 8.7b. Φ(1) rises in
the output part. This is why the nonuniform ceramic rod in Fig. 8.6 can work as a
transformer.
For the forced vibration driven by V 1 ðt Þ ¼ V1 expðiωt Þ, from Eqs. (8.64)1, (8.72)1,
(8.66), (8.74)1, and (8.76)1,2, we have the following nonhomogeneous problem for U:
1
U , 11 ¼ ρω2 U, a < x1 < 0,
s11
1
U , 11 ¼ ρω2 U, 0 < x1 < b,
s
33
1
d 31 V 1 ¼ 0, x1 ¼ a,
U, 1 þ
s11
h
1
U , 1 k233 C 1 ¼ 0, x1 ¼ b,
s33
U ð0 Þ ¼ U ð0þ Þ,
1
d31 1
V1 ¼
U , 1 ð 0 Þ þ
U , 1 ð0þ Þ k 233 C1 :
s11
s33
h
ð8:86Þ
The solution to Eq. (8.86) can be written as
8
d 31 V1
>
2
>
>
α
þ
β
k
C
11
11 33 1 sin kx1
>
>
h
>
>
>
>
d31 V1
>
2
>
þ β12 k33 C 1 cos kx1 , a < x1 < 0,
þ α12
<
h
U¼
d31 V1
>
2
>
1
>
þ β22 k 33 C1 sin kx
α22
>
>
h
>
>
>
>
d 31 V1
>
1 , 0 < x1 < b,
>
þ β12 k233 C 1 cos kx
þ α12
:
h
where
ð8:87Þ
254
8 Piezoelectric Devices
pffiffiffiffiffiffiffiffi
k ¼ ω ρs11 ,
pffiffiffiffiffiffiffiffi
k ¼ ω ρ
s33 ,
1
1
sin kað cos ka 1Þ ,
s33 cos kb
α11 ¼ Δ cos ka
k cos ka
1
ð cos ka 1Þ,
α12 ¼ s33 cos kb
Δ
1
ð cos ka 1Þ,
α22 ¼ s33 sin kb
Δ
1
,
β11 ¼ s11 sin ka 1 cos kb
Δ
1
,
β12 ¼ s11 cos ka 1 cos kb
Δ
1
1
β22 ¼ s11 sin kb cos ka 1 cos kb þ k cos kb
,
Δ cos kb
cos ka þ s33 k sin ka cos kb:
Δ ¼ s11 k sin kb
ð8:88Þ
ð8:89Þ
ð8:90Þ
ð8:91Þ
It can be verified that Δ ¼ 0 implies the frequency equation in Eq. (8.83). With
Eq. (8.87), from Eqs. (8.72)4 and (8.76)3, we obtain the voltage distribution in the
output part as
1
1
d31 V1
1
þ β22 k233 C 1 sin kx
Φ ¼ V1 þ C1 x1 α22
2
d33
h
d31 V1
1 , 0 < x1 < b:
þ β12 k233 C 1 1 cos kx
þ α12
h
ð8:92Þ
V1 is considered given. Equations (8.87) and (8.92) still have an unknown constant
C1. From Eqs. (8.92) and (8.74)2 the output voltage is
V2 ¼ ΦðbÞ ¼ Γ1 V1 Z 2 I2 ,
ð8:93Þ
where
1 d31 1
α22 sin kb
,
α12 1 cos kb
Γ1 ¼ þ 2 d33 h
b
1 1
þ β12 k2 1 cos kb
b β22 k233 sin kb
,
Z2 ¼
33
2 b
iωε33 wh
1 k 33
ð8:94Þ
and Eq. (8.75) has been used to replace C1 by I2 . From Eqs. (8.67), (8.64)3, and
(8.87), the driving current can be written as
I1 ¼ V1 =Z 1 þ Γ2 I2 ,
where
ð8:95Þ
8.4 Transformer
Fig. 8.8 Transforming ratio
versus driving frequency
255
50
|V 2/V 1|
Z L =2000kW
40
30
20
Z L =1000kW
10
0
33
35
37
39
Frequency (kHz)
"
#
1
k231
iωε33 wa
,
1=Z 1 ¼ a ðα12 þ α11 sin ka α12 cos kaÞ
2
a
h
1 k31
s33 d31 1
Γ2 ¼ k233
ðβ þ β11 sin ka β12 cos kaÞ:
s11 d33 h 12
41
ð8:96Þ
Since V1 is given, solving Eqs. (8.93), (8.95), and the circuit condition in Eq. (8.78),
we obtain the transforming ratio, the output current, and the input admittance as
Γ1 Z L
V2
¼
,
V1 Z L þ Z 2
I2 ¼
Γ1
V1 ,
ZL þ Z2
1
Γ1 Γ2
I1
¼
:
V 1 Z1 ZL þ Z2
ð8:97Þ
For numerical results of the forced vibration analysis, some damping in the system is
necessary. This is achieved by allowing the elastic material constants s11 and s33 to
assume complex values. In our calculations s11 and s33 are replaced by s11(1 iQ1)
and s33(1 iQ1), where Q is a real number. We fix Q ¼ 1000. Figure 8.8 shows the
transforming ratio as a function of the driving frequency for two values of the load
impedance ZL. The transforming ratio assumes its maximum as expected near the
first resonance frequency ( f (1) ¼ 36.35 kHz). This shows that the transformer is a
resonant device effective at a particular frequency. The output voltage can be many
times higher than the driving voltage. When the ZL increases, the output voltage
increases. At the same time the output current decreases.
256
8.5
8 Piezoelectric Devices
Gyroscope
Piezoelectric gyroscopes are vibratory gyroscopes whose vibrations are excited and
detected piezoelectrically. Two vibration modes are involved in the operation of a
piezoelectric gyroscope. When the gyroscope is not rotating, the two modes have
material particles moving in perpendicular directions so that they are coupled by the
Coriolis force (or more precisely the Coriolis acceleration) when the gyroscope is
rotating. Furthermore, the resonance frequencies of the two modes can be made as
close as wanted in order that the gyroscope operates at the so-called double-resonant
condition with high sensitivity. When a gyroscope is excited into vibration by an
applied alternating voltage in one of the two modes (the primary mode) and is
attached to a body rotating with an angular rate Ω, the Coriolis force of the primary
mode excites the other mode (the secondary mode) through which Ω can be detected
from electrical signals accompanying the secondary mode. A Piezoelectric gyroscope is in small amplitude vibration in the reference frame rotating with it. The
relative equilibrium state in the rotating reference frame has initial deformation/
stress due to the centrifugal force of the order of Ω2. Therefore an accurate description of the motion of a piezoelectric gyroscope requires the theory for small dynamic
fields superposed on initial fields proportional to Ω2. Since the main interest is on a
linear effect of Ω, in the analysis of piezoelectric gyroscopes the effect of the initial
fields on the incremental vibrations are usually neglected. The equations for the
small-amplitude vibration in the rotating frame used by most researchers are
T ji, j ¼ ρ€ui 2ρεijk Ω j u_ k ρ Ωi Ω j u j Ω j Ω j ui ,
Di, i ¼ 0,
T ij ¼ cijkl Skl ekij E k , Di ¼ eijk S jk þ εij E j ,
Sij ¼ ui, j þ u j, i =2, E i ¼ φ, i ,
ð8:98Þ
which still has some second-order effect of Ω due to the centrifugal acceleration
associated with the vibration.
The basic behavior of piezoelectric gyroscopes can be shown by the simple
example below [5]. Consider a concentrated mass M connected to two thin rods of
polarized ceramics as shown in Fig. 8.9. The two rods are electroded at the side
surfaces, with electrodes shown by the thick lines. Under a time-harmonic driving
voltage V1(t), the rod along the x direction is driven into extensional vibrations. If the
system is rotating about the normal of the (x,y) plane at an angular rate Ω, it produces
a voltage output V2(t) across the width of the rod along the y direction. We show
below that V2(t) is approximately proportional to Ω when Ω is small and therefore it
can be used to detect Ω conveniently.
For long or thin rods (L >> h) the flexural rigidity is very small. The rods do not
resist bending but can still provide extensional forces. When the mass of M is much
larger than that of the rods, the inertial effect of the rods can be discounted. Then the
rods are effectively like two springs with piezoelectric coupling. Let the
8.5 Gyroscope
257
Fig. 8.9 A simple
piezoelectric gyroscope
y
j =0
j =V2
v: Secondary motion
P
u: Primary motion
L
h
j =V1
x
M
h
P
j =0
W
L
displacements of M in the x and y directions be u(t) and v(t). We consider small
amplitude vibrations of M in the rotating (x,y) coordinate system. For each rod we
also introduce a local coordinate system with the x1 axis along the axis of the rod and
the x3 axis along the poling direction.
Consider the rod along the x direction first. Neglecting the dynamic effect in the
rod due to inertia, the axial strain in the rod can be written as
S1 ¼ u=L:
ð8:99Þ
With respect to the local coordinate system, the electric field corresponding to the
configuration of the driving electrodes shown can be written as
E 1 ¼ E2 ¼ 0,
E 3 ¼ V 1 =h,
ð8:100Þ
where the driving voltage V1 is considered given and is time-harmonic. For thin rods
in extension, the dominating stress component is the axial stress component T1. All
other stress components can be treated as zero. Under the above stress and electric
field conditions, the constitutive relations take the form
S1 ¼ s11 T 1 þ d 31 E3 ,
D3 ¼ d 31 T 1 þ ε33 E3 ,
ð8:101Þ
where D3 is the component of the electric displacement vector in the local coordinate
system. s11, d31, and ε33 are the relevant elastic, piezoelectric, and dielectric constants. From Eq. (8.101) we can solve for T1 and D3 in terms of S1 and E3:
258
8 Piezoelectric Devices
1
d31
1 u d31 V 1
þ
,
S1 E3 ¼ s11
s11 L s11 h
s11
d 31
d 31 u
V1
ε33
D3 ¼
,
S1 þ ε33 E 3 ¼ s11
s11 L
h
T1 ¼
ð8:102Þ
where Eqs. (8.99) and (8.100) have been used, and
ε33 ¼ ε33 1 k 231 ,
k 231 ¼ d 231 =ðε33 s11 Þ:
ð8:103Þ
The axial force in the rod and the electric charge on the electrode at the upper surface
of the rod are given by
F 1 ¼ T 1 h ¼ Ku þ
Q1e ¼ D3 L ¼
d31
V 1,
s11
d 31
u þ C0 V 1 ,
s11
ð8:104Þ
where
K¼
h
,
s11 L
C0 ¼
ε33 L
:
h
ð8:105Þ
K and C0 represent the effective stiffness and the static capacitance of the rod. The
electric current on the electrode is related to the charge by
I 1 ¼ Q_ 1e ¼ d31
u_ C0 V_ 1 :
s11
ð8:106Þ
Similarly, for the rod along the y direction, the axial force and electric current are
given by
d 31
V 2,
s11
d 31
I 2 ¼ v_ C0 V_ 2 :
s11
F 2 ¼ Kv þ
ð8:107Þ
In piezoelectric gyroscope applications, neither V2 nor I2 is known. The output or
sensing electrodes across the rod along the y direction are connected by an electric
circuit with an impedance Z for time-harmonic motions. We have the following
circuit condition:
I 2 ¼ V 2 =Z:
The equations of motion for M are
ð8:108Þ
8.5 Gyroscope
259
F 1 ¼ M u€ 2Ωv_ Ω2 u ,
F 2 ¼ M €v þ 2Ωu_ Ω2 v ,
ð8:109Þ
where the Coriolis and centrifugal accelerations are included.
For time-harmonic motions we use the complex notation
ðu; v; V 1 ; V 2 ; I 1 ; I 2 Þ ¼ u; v; V1 ; V2 ; I1 ; I2 expðiω t Þ:
ð8:110Þ
The system is governed by the following three linear algebraic equations for u, v, and
V2 , with V1 as a driving term:
d31
M ω2 þ Ω2 K u þ 2iωΩM v ¼ V1 ,
s11
2
d 31 2
V 2 ¼ 0,
2iωΩM u þ M ω þ Ω K v þ
s11
d31
Z0 V 2 ¼ 0:
v þ C0 1 þ
s11
Z
ð8:111Þ
Some damping is introduced by the complex elastic constant s11(1 iQ1) where
Q is the material quality factor. Once Eq. (8.111) is solved, the currents can be
calculated. We have
2
V2 1
Z
2 k 31
2iω
Ωω
¼
,
0
2
V1 Δ
1 k31 Z þ
" Z0
!#
I1
1 k231
k 231
Z
2
2
2
2
¼1
ω ω þ Ω ω0 1 þ
,
V1 =Z 0
Δ1 k231 0
1 k 231 Z þ Z 0
2
I2
1
Z0
2 k 31
2iω
Ωω
¼
,
0
V1 =Z 0 Δ
1 k2 Z þ Z 0
ð8:112Þ
31
where
K
,
M
"
2
2
2
2
Δ ¼ ω þ Ω ω0 ω þ Ω2 ω20 1 þ
ω20 ¼
k 231
Z
2 Z þZ
1 k 31
0
!#
4ω2 Ω2 :
ð8:113Þ
The output voltage as a function of the driving frequency ω is plotted in Fig. 8.10
for the case of open sensing electrodes (Z ¼ 1) and two values of Ω. There are two
resonance frequencies with normalized values near 1. Near these two frequencies,
the voltage sensitivity assumes maxima. If smaller values of Q are used, the peaks
become narrower and higher.
260
8 Piezoelectric Devices
Fig. 8.10 Output voltage
versus driving frequency ω
0.5
V2 / V1
Z / Z0 = ¥
0.4
W/w 0 =0.002
0.3
0.2
W/w 0 =0.001
0.1
w /w
0.0
0.8
Fig. 8.11 Output voltage
versus angular rate Ω
0.4
0.9
1.0
1.1
1.2
0
1.3
V2 / V1 m ax
0.3
Z / Z 0 = 0 .1
0.2
Z / Z 0 = 0.05
0.1
0.0
0.000
W/w0
0.001
0.002
The dependence of the normalized maximal output voltage for one of the two
peaks shown in Fig. 8.10 on the angular rate Ω is shown in Fig. 8.11 for two values
of the load Z. When Ω is much smaller than ω0, the relation between the output
voltage and Ω is essentially linear, as shown in Eq. (8.112)1. Therefore these
gyroscopes are convenient for detecting an angular rate that is relatively slow
compared to the resonance frequency.
8.6
Elastic Rod with Piezoelectric Actuator/Sensor
Consider the longitudinal extension of a composite rod consisting of an elastic layer
and two identical piezoelectric layers of ceramics poled along x3 (see Fig. 8.12). We
assume L >> h and L >> b >> h0 .
8.6 Elastic Rod with Piezoelectric Actuator/Sensor
a
261
b
x3
x3
I
h⬘
P
x1
2h
x2
V
P
h⬘
b
L
Fig. 8.12 (a) A composite rod for extension. (b) Its cross section
For thin rods we make the following approximation:
T 1 ¼ T 1 ðx1 ; t Þ,
T 2 ¼ T 3 ¼ T 4 ¼ T 5 ¼ T 6 ¼ 0:
ð8:114Þ
The extensional displacement is approximated by
u1 ¼ u1 ðx1 ; t Þ:
ð8:115Þ
S1 ¼ u1, 1 :
ð8:116Þ
The corresponding axial strain is
The electric fields in the piezoelectric layers are approximated by
E 1 ¼ 0,
E 2 ¼ 0,
V
E3 ¼ 0 :
h
ð8:117Þ
The stress-strain relation of the elastic layer in the middle is
T 1 ¼ ES1 ,
ð8:118Þ
where E is Young’s modulus. The relevant constitutive relations for the ceramic
layers can be written as
S1 ¼ s11 T 1 þ d31 E 3 ,
D3 ¼ d 31 T 1 þ ε33 E3:
ð8:119Þ
From Eq. (8.119) we solve for the axial stress T1 and the transverse electric
displacement D3
262
8 Piezoelectric Devices
1
T 1 ¼ s1
11 S1 s11 d 31 E 3 ,
1
D3 ¼ s11 d31 S1 þ ε33 E3 ,
ð8:120Þ
where
ε33 ¼ ε33 1 k 231 ,
k 231 ¼ d 231 =ðε33 s11 Þ:
ð8:121Þ
The total extensional force over the entire cross section in Fig. 8.12b is obtained
by integrating the stress over the cross section:
Z
N¼
T 1 dx2 dx3 ¼ cð0Þ S1 eð0Þ E 3 ,
ð8:122Þ
where
cð0Þ ¼ 2hbE þ 2h0 b
eð0Þ ¼ 2h0 b
d31
:
s11
1
,
s11
ð8:123Þ
c(0) is the extensional rigidity of the composite rod. The equation of motion is
obtained by applying Newton’s second law to the differential element in Fig. 8.13.
We have
ðN þ dN Þ N ¼ ðρ2hb þ ρ0 2h0 bÞðdx1 Þ€u1 ,
ð8:124Þ
∂N
¼ 2bðρh þ ρ0 h0 Þ€
u1 ,
∂x1
ð8:125Þ
or
where ρ and ρ0 are the mass densities of the elastic and piezoelectric layers.
The electric charge on the top electrode at x3 ¼ h + h0 is given by
Z
Qe ¼ b
L
D3 dx1 :
ð8:126Þ
0
Fig. 8.13 A differential
element of the rod
N
N+dN
dx1
x1
8.7 Elastic Beam with Piezoelectric Actuator/Sensor
263
The current flowing out of the top electrodes of the two piezoelectric layers
together is
I ¼ 2Q_ e :
ð8:127Þ
When the motion is time-harmonic, under the complex notation, we write
ðiωt Þg and V ¼ RefVexp
ðiωt Þg. In general the electrodes may be
I ¼ RefIexp
connected to a circuit whose impedance is Z. Then we have the following circuit
equation:
I ¼ V=Z:
ð8:128Þ
As a simple example of the actuation of the rod by a known voltage V, consider a
static and free rod with N ¼ 0. From Eqs. (8.122), (8.123), and (8.117) we obtain the
axial strain as
S1 ¼
eð0Þ
h0 bd 31 =s11 V
E
¼
:
3
hbE þ h0 b=s11 h0
cð0Þ
ð8:129Þ
Conversely, for a simple example of the sensing of an axial strain S1 in the rod
produced by some mechanical load, consider a rod in static uniform extension with
an open circuit between the top and bottom electrodes (Z ¼ 1). Then Qe ¼ 0 and
D3 ¼ 0. From Eqs. (8.120)2 and (8.117) we obtain the output voltage as
V¼
8.7
d 31 0
h S1 :
ε33 s11
ð8:130Þ
Elastic Beam with Piezoelectric Actuator/Sensor
In this section we consider the bending of a composite beam consisting of an elastic
layer and two piezoelectric layers of ceramics poled along x3, respectively (see
Fig. 8.14). We assume L >> h and L >> b >> h0 . Figure 8.14 differs from Fig. 8.12
only in that the polarization of the lower piezoelectric layer is reversed. This causes a
sign difference between the piezoelectric constants of the two ceramic layers. Under
an applied voltage, one of the two ceramic layers contracts while the other extends or
vice versa and thus bending of the composite beam in the (x1, x3) plane is created.
For bending the dominating stress is the axial stress T1:
264
8 Piezoelectric Devices
a
b
x3
x3
I
h⬘
P
x1
2h
x2
V
P
h⬘
b
L
Fig. 8.14 (a) A composite beam for bending. (b) Its cross section
T 1 ¼ T 1 ðx1 ; x3 ; t Þ,
T 2 ¼ T 3 ¼ T 4 ¼ T 6 ¼ 0:
ð8:131Þ
u3 ffi u3 ðx1 ; t Þ:
ð8:132Þ
The flexural displacement is
The axial displacement is, accordingly,
u1 ffi x3 u3, 1 :
ð8:133Þ
Then the axial strain is approximately given by
S1 ¼ u1, 1 ffi x3 u3, 11 :
ð8:134Þ
The electric fields in the ceramic layers are
E 1 ¼ 0,
E2 ¼ 0,
V
E3 ¼ :
h
ð8:135Þ
The stress-strain relation for the elastic layer in the middle is
T 1 ¼ ES1 ,
ð8:136Þ
where E is Young’s modulus. The relevant constitutive relations for the ceramic
layers can be written as
S1 ¼ s11 T 1 d31 E3 ,
D3 ¼ d31 T 1 þ ε33 E 3,
ð8:137Þ
8.7 Elastic Beam with Piezoelectric Actuator/Sensor
265
where the upper signs are for the upper layer and the lower signs are for the lower
layer. We solve Eq. (8.137) for the axial stress T1 and the transverse electric
displacement D3 to obtain
1
T 1 ¼ s1
11 S1 s11 d 31 E 3 ,
1
D3 ¼ s11 d31 S1 þ ε33 E 3 ,
ð8:138Þ
where
ε33 ¼ ε33 1 k 231 ,
k 231 ¼ d 231 =ðε33 s11 Þ:
ð8:139Þ
The electric charge on the top electrode of the upper ceramic layer at x3 ¼ h + h0 is
given by
Z
L
Qe ¼ b
D3 ðx3 ¼ h þ h0 Þdx1 :
ð8:140Þ
0
Then the current in Fig. 8.14a is given by
I ¼ 2Q_ e :
ð8:141Þ
When the motion is time-harmonic, under the complex notation, we write
ðiωt Þg and V ¼ RefVexp
ðiωt Þg. In general the electrodes may be
I ¼ RefIexp
connected to a circuit whose impedance is Z. Then we have the following circuit
equation:
I ¼ V=Z:
ð8:142Þ
The bending moment M is defined by the following integral over the cross section
of the beam:
Z
M¼
V
x3 T 1 dx2 dx3 ¼ Du3, 11 þ s1
11 d 31 0 2G,
h
ð8:143Þ
where Eqs. (8.134), (8.135), (8.136), (8.137), and (8.138) have been used and we
have denoted
i
2 3 2 1 h
3
Eh þ s11 ðh þ h0 Þ h3 b,
3
3
h0 0
G¼ hþ
h b:
2
D¼
ð8:144Þ
D is the bending stiffness of the beam and G is the first moment of the cross-sectional
area of one of the ceramic layers about the x2 axis. The equation for flexural motion
266
8 Piezoelectric Devices
Fig. 8.15 A differential
element of the beam
M+dM
M
x1
Q
Q+dQ
dx1
is obtained by applying Newton’s second law to the differential element in Fig. 8.15
in the x3 direction. We have
ðQ þ dQÞ Q ¼ ðρ2hb þ ρ0 2h0 bÞðdx1 Þ€u3 ,
ð8:145Þ
∂Q
¼ 2bðρh þ ρ0 h0 Þ€
u3 ,
∂x1
ð8:146Þ
or
where Q is the transverse shear force for the notation in this section only. ρ and ρ0 are
the mass densities of the elastic and piezoelectric layers.
In addition, taking moment about a point on the right face of the element in
Fig. 8.15 gives
M þ Qdx1 ðM þ dM Þ ffi 0,
ð8:147Þ
where the rotatory inertia of the element has been neglected. Equation (8.147) can be
written as
Q¼
∂M
:
∂x1
ð8:148Þ
The substitution of Eq. (8.148) into Eq. (8.146) yields
∂M
¼ 2bðρh þ ρ0 h0 Þ€u3 ,
∂x21
ð8:149Þ
where M is given by Eq. (8.143).
As a simple example of the actuation of the beam by a known voltage V, consider
a static and free beam with M ¼ 0. From Eq. (8.143) we obtain the beam bending
curvature as
References
267
u3, 11 ¼
V
s1
11 d 31 0 2G
h
=D:
ð8:150Þ
References
1. J.S. Yang, H.G. Zhou, Y.T. Hu, Q. Jiang, Performance of a piezoelectric harvester in thicknessstretch mode of a plate. IEEE Trans. Ultrason. Ferroelect. Freq. Contr. 52, 1872–1876 (2005)
2. P. Li, F. Jin, W.Q. Chen, J.S. Yang, Effects of interface bonding on acoustic wave generation in
an elastic body by surface-mounted piezoelectric transducers. IEEE Trans. Ultrason. Ferroelect.
Freq. Contr. 60, 1957–1963 (2013)
3. Y.T. Hu, X.S. Zhang, J.S. Yang, Q. Jiang, Transmitting electric energy through a metal wall by
acoustic waves using piezoelectric transducers. IEEE Trans. Ultrason. Ferroelect. Freq. Contr.
50, 773–781 (2003)
4. J.S. Yang, X.S. Zhang, Extensional vibration of a nonuniform piezoceramic rod and high voltage
generation. Int. J. Appl. Electromagn. Mech. 16, 29–42 (2002)
5. J.S. Yang, H.Y. Fang, A piezoelectric gyroscope based on extensional vibrations of rods. Int.
J. Appl. Electromagn. Mech. 17, 289–300 (2003)
Appendices
Appendix 1: Derivation of fE, cE, and wE by a Polarized
Particle
This appendix presents a derivation of the electric body force, couple and power
using a polarized particle. Consider the current state of a particle of a deformable and
polarizable material. The mass center of the particle is at y. Its negative charge center
is at y, and its positive charge center is at y+. The mass density is ρ(y). The negative
charge density is μ(y), and the positive charge density is μ+(y+).
From Fig. A1.1, obviously,
y ¼ y þ Δy ,
yþ ¼ y þ Δyþ ,
Δy ¼ Δyþ Δy :
ðA1:1Þ
Accordingly, the polarization vector is defined by
Pi ¼ μþ ðyþ ÞΔyi :
ðA1:2Þ
We assume charge neutrality of the particle, that is,
μ ðy Þ þ μþ ðyþ Þ ¼ 0:
ðA1:3Þ
The material time derivative of the polarization vector can be calculated from
Eq. (A1.2) as
© Springer Nature Switzerland AG 2018
J. Yang, An Introduction to the Theory of Piezoelectricity, Advances in Mechanics
and Mathematics 9, https://doi.org/10.1007/978-3-030-03137-4
269
270
Appendices
Fig. A1.1 A polarized
material particle
Dy+
y+
3
Dy
Dy
y
y
-
2
1
d
P_ i ¼ μ_ þ ðyþ ÞΔyi þ μþ ðyþ Þ ðΔyi Þ
dt
μ_ þ ðyþ Þ þ þ
¼ þ þ μ ðy ÞΔyi þ μþ ðyþ ÞΔvi
μ ðy Þ
μ_ þ ðyþ Þ
¼ þ þ Pi þ μþ ðyþ Þvi, j Δy j
μ ðy Þ
μ_ þ ðyþ Þ
¼ þ þ Pi þ vi, j P j
μ ðy Þ
ρ_
¼ Pi þ vi, j P j :
ρ
ðA1:4Þ
Multiplying Eq. (A1.4) by Ei, we obtain
E i vi, j P j ¼ E i P_ i ρ_
Pi E i :
ρ
Then the electric body force fE can be calculated from
ðA1:5Þ
Appendices
271
μ ðy ÞE i ðy Þ þ μþ ðyþ ÞEi ðyþ Þ
þ þ
þ
¼ μ ðy ÞE
h i ðy þ Δy Þ þ μ ðyi ÞE i ðy þ Δy
h Þ
i
¼ μ ðy Þ E i ðyÞ þ E i, j ðyÞΔyj þ μþ ðyþ Þ E i ðyÞ þ E i, j ðyÞΔyþj
¼ h½μ ðy Þ þ μþ ðyþ ÞEi ðyÞ i
þ μ ðy ÞΔyj þ μþ ðyþ ÞΔyþj Ei, j ðyÞ
h
i
¼ 0 þ μþ ðyþ ÞΔyj þ μþ ðyþ ÞΔyþj Ei, j ðyÞ
h
i
¼ μþ ðyþ Þ Δyj þ Δyþj Ei, j ðyÞ
ðA1:6Þ
¼ μþ ðyþ ÞΔy j E i, j ðyÞ ¼ P j E i, j
¼ ðP ∇EÞi ¼ f iE :
The electric body couple cE can be calculated from
εijk yj μ ðy ÞE k ðy Þ þ εijk yþj μþ ðyþ ÞEk ðyþ Þ
¼ εijk y j þ Δyj μ ðy ÞE k ðy Þ þ εijk y j þ Δyþj μþ ðyþ ÞE k ðyþ Þ
¼ εijk y j ½μ ðy ÞE k ðy Þ þ μþ ðyþ ÞE k ðyþ Þ
h
i
þ εijk Δyj μ ðy ÞE k ðy Þ þ Δyþj μþ ðyþ ÞEk ðyþ Þ
h
i
ffi εijk y j f kE þ εijk Δyj μþ ðyþ ÞE k ðyÞ þ Δyþj μþ ðyþ ÞE k ðyÞ
¼ εijk y j f kE þ εijk Δyj þ Δyþj μþ ðyþ ÞEk ðyÞ
ðA1:7Þ
¼ εijk y j f kE þ εijk Δy j μþ ðyþ ÞE k ðyÞ
¼ εijk y j f kE þ εijk P j Ek ðyÞ
¼ εijk y j f kE þ ciE ,
where
ciE ¼ εijk P j E k :
The electric body power wE can be calculated from
ðA1:8Þ
272
Appendices
μ ðy ÞE i ðy Þvi ðy Þ þ μþ ðyþ ÞEi ðyþ Þvi ðyþ Þ
¼ μ ðy ÞE i ðy Þvi ðy þ Δy Þ
þ μþ ðyþ ÞE i ðyþ Þvhi ðy þ Δyþ Þ
i
¼ μ ðy ÞE i ðy Þ vi ðyÞ þ vi, j ðyÞΔyj
h
i
þ μþ ðyþ ÞE i ðyþ Þ vi ðyÞ þ vi, j ðyÞΔyþj
¼ ½μ ðy ÞEi ðy Þ þ μþ ðyþ ÞEi ðyþ Þvi ðyÞ
þ μ ðy ÞE i ðy Þvi, j ðyÞΔyj þ μþ ðyþ ÞEi ðyþ Þvi, j ðyÞΔyþj
ffi f iE vi þ Δyj þ Δyþj μþ ðyþ ÞEi ðyÞvi, j ðyÞ
ðA1:9Þ
¼ f iE vi þ Δy j μþ ðyþ ÞE i ðyÞvi, j ðyÞ
¼ f iE vi þ P j E i ðyÞvi, j ðyÞ
ρ_
¼ f iE vi þ E i P_ i Pi E i
ρ
¼ f iE vi þ ρEk π_ k ¼ f iE vi þ wE ,
where
wE ¼ ρEk π_ k ,
ðA1:10Þ
and Eqs. (A1.5) and (1.126) have been used.
Appendix 2: Derivation of fE, cE, and wE by Polarization
Charges
Using the effective volume and surface polarization charges in Eq. (1.68), the
electric body force fE can be calculated from
R
R
ERj dv þ s σ p E j ds R
¼ v ðhPi, i ÞE j dv þ s ni Pi Eij ds
R
¼ v ðPi, i ÞE j þ Pi E j , i dv
R
R
¼ v Pi E j, i dv ¼ v f Ej dv:
vρ
p
The electric body couple cE can be calculated from
ðA2:1Þ
Appendices
R
273
R
E k dv þ s εijk x j σ p E k ds
R
¼ v εhijk x j ðPm, m ÞEk dv þ s εijk x j nm Pm Eik ds
R
¼ v εijk x j ðPm, m ÞE k þ εijk x j Pm Ek , m dv
R
¼ v εijk x j ðPm, m ÞE k þεijk δ jm Pm E k þ x j Pm, m E k þ x j Pm E k, m dv
R
¼ v εijk P j E k þ εijkx j f kE dv
R
¼ v ciE þ εijk x j f kE dv,
v εijk xRj ρ
p
ðA2:2Þ
where
ciE ¼ εijk P j E k :
ðA2:3Þ
The electric body power wE can be calculated from
R
R
ERi vidv þ s σ p E i vi ds R
¼ v hP j, j E i vi dv þ s n j P j Eiivi ds
R ¼ v P j, j E i vi þ P j E i vi , j dv
R
¼ Rv P j, j Ei vi þ P j, j Eivi þ P j E i, j vi þ P j E i vi, j dv
¼ v P j E i, j vi þ P j E i vi, j dv
R
ρ_
¼ v f iE vi þ Ei P_ i Pi E i dv
ρ R
R
¼ v f iE vi þ ρE k π_ k dv ¼ v f iE vi þ wE dv,
vρ
p
ðA2:4Þ
where
wE ¼ ρEk π_ k ,
ðA2:5Þ
and Eqs. (A1.5) and (1.126) have been used.
Appendix 3: List of Notation
For the nonlinear theory of electroelasticity the notation and terminology vary
among researchers. In the present book we mainly follow Harry F. Tiersten’s papers
in [1–8]:
δij, δKL – Kronecker delta
δiK, δKi – Coordinate transformation coefficients
εijk, εIJK – Permutation tensor
XK – Reference coordinates of a material point
yi – Present coordinates of a material point
uK – Mechanical displacement vector
274
Appendices
J – Jacobian
CKL – Deformation tensor
EKL – Finite strain tensor
vi – Velocity vector
dij – Deformation rate tensor
ωij – Spin tensor
d/dt – Material time derivative
ρ – Present mass density
ρ0 – Reference mass density
ρe – Present volume free charge density
ρE – Reference volume free charge density
σ e – Present surface free charge density
σ E – Reference surface free charge density
ρ p – Present volume polarization charge density
σ p – Present surface polarization charge density
Qe – Total free charge
I – Electric current
ε0 – Electric permittivity of free space
μ0 – Magnetic permeability of free space
φ – Electrostatic potential
Ei – Electric field
Pi – Electric polarization per unit present volume
π i – Electric polarization per unit mass
Di – Electric displacement vector
WK – Reference electric potential gradient
E K – Reference electric field vector
PK – Reference electric polarization vector
D K – Reference electric displacement vector
f Ej – Electric body force per unit present volume
c Ej – Electric body couple per unit present volume
wE – Electric body power per unit present volume
fj – Mechanical body force per unit mass
τij – Cauchy stress tensor (asymmetric)
T ijE – Maxwell stress tensor (asymmetric)
S
– Symmetric stress tensor in spatial, two-point (asymmetric), and
τijS , FLj, T KL
material form
ES
T ijES , MLj, T KL
– Symmetric Maxwell stress tensor in spatial, two-point (asymmetric), and material form
tij, KLj, T KL – Symmetric total stress tensor in spatial, two-point (asymmetric), and
material form
Tk – Mechanical surface traction per unit reference area
tk – Mechanical surface traction per unit present area
ε – Internal energy per unit mass
Appendices
275
χ – Free energy per unit mass
ψ – Total free energy per unit mass
θ – Absolute temperature
η – Entropy per unit mass
γ – Body heat source per unit mass
qk – Present heat flux vector
QK – Reference heat flux vector
For the theory of linear theory we mainly follow the notation of the IEEE
Standard on Piezoelectricity [9]:
xk – Reference coordinates
ρ – Reference mass density
fj – Body force per unit mass
H – Electric enthalpy
Tkl – Linear stress tensor
Skl – Linear strain tensor
ψ – Dissipation function
Q – Material quality factor
* – Complex conjugate
Appendix 4: Material Constants
Electric permittivity of free space ε0 ¼ 8.854 1012 F/m.
Magnetic permeability of free space μ0 ¼ 12.57 107 H/m.
Electronic charge q ¼ 1.602 1019 C.
Boltzmann constant k ¼ 1.381 1023 J/K.
Material constants of a few widely used single crystals and ceramics, most of them
piezoelectric, are listed below in the order of aluminum nitride, langanite, langasite,
langatate, lithium niobate, lithium tantalate, poled ceramics, quartz, silicon, and zinc
E
oxide. The elastic and dielectric constants given are cpq
and εijS .
Aluminum Nitride (AlN) [10]
ρ ¼ 3260 kg/m3,
0
345
B 125
B
B 120
cpq ¼ B
B 0
B
@ 0
0
125
345
120
0
0
0
120 0
0
120 0
0
395 0
0
0 118 0
0
0
118
0
0
0
1
0
0 C
C
0 C
C 109 N=m2 ,
0 C
C
0 A
110
276
Appendices
eip
0
1
0
0
0
0:48 0
0
0
0:48
0
0 A C=m2 ,
0:58 1:55
0
0
0
0
1
8:0 0
0
εij ¼ @ 0 8:0 0 A 1011 F=m:
0
0 9:5
0
¼@ 0
0:58
Langanite, Langasite, and Langatate [11]
These crystals have the same symmetry as quartz. Their material constants are
ρ ðkg=m3 Þ c11 1010 N=m2
c66
c33
c44
c14
c13
e11 ðC=m2 Þ
e14
ε11 =ε0
ε33 =ε0
Langanite
6028:900
19:299
4:116
26:465
4:956
1:485
10:225
0:452
0:061
20:089
79:335
Langasite
5739:200
18:849
4:221
26:168
5:371
1:415
9:688
0:402
0:130
19:620
49:410
Langatate
6150:400
18:852
4:032
26:180
5:110
1:351
10:336
0:456
0:094
18:271
78:950
Lithium Niobate (LiNbO3)
The second-order material constants for lithium niobate are [12, 13]:
ρ ¼ 4700 kg=m3 ,
1
2:03 0:53 0:75 0:09
0
0
B 0:53 2:03 0:75 0:09
0
0 C
C
B
C
B 0:75 0:75 2:45
0
0
0
C 1011 N=m2 ,
B
cpq ¼ B
C
0:09
0:09
0
0:60
0
0
C
B
@ 0
0
0
0
0:60 0:09 A
0
0
0
0
0:09 0:75
0
1
0
0
0
0
3:70 -2:50
eip ¼ @ -2:50 2:50
0
3:70
0
0 A C=m2 ,
0:20 0:20 1:30
0
0
0
0
Appendices
277
0
38:9
εij ¼ @ 0
0
1
0
0
38:9
0 A 1011 F=m:
0
25:7
The temperature derivatives of the second-order material constants are given in
[14]. The third-order material constants can be found in [15].
Lithium Tantalate (LiTaO3)
The second-order material constants for lithium tantalate are [12, 13]:
ρ ¼ 7450 kg=m3 ,
1
2:33 0:47 0:80 0:11
0
0
B 0:47 2:33 0:80 0:11
0
0 C
C
B
B 0:80 0:80 2:75
0
0
0 C
C 1011 N=m2 ,
cpq ¼ B
B 0:11 0:11
0
0:94
0
0 C
C
B
@ 0
0
0
0
0:94 0:11 A
0
0
0
0
0:11 0:93
0
1
0
0
0
0 2:6 -1:6
0 A C=m2 ,
eip ¼ @ -1:6 1:6 0 2:6 0
0
0 1:9 0
0
0
0
1
36:3
0
0
36:3
0 A 1011 F=m:
εij ¼ @ 0
0
0
38:2
0
The temperature derivatives of the second-order material constants can be found
in [14].
Poled Ceramics
The material matrices for PZT-2 are [16]
0
13:5
B 6:79
B
B 6:81
cpq ¼ B
B 0
B
@ 0
0
ρ ¼ 7600 kg=m3 ,
6:79
13:5
6:81
0
0
0
6:81
6:81
11:3
0
0
0
0
0
0
2:22
0
0
0
0
0
0
2:22
0
1
0
0 C
C
0 C
C 1010 N=m2 ,
0 C
C
0 A
3:36
278
Appendices
eip
0
0
¼@ 0
1:9
0
0
0
0
0 9:8
1:9 9:0 0
0
0
504ε0
εij ¼ @ 0
504ε0
0
0
9:8
0
0
1
0
0 A C=m2 ,
0
1
0
0 A:
260ε0
The material matrices for PZT-5H are [16]
ρ ¼ 7500 kg=m3 ,
1
12:6 7:95 8:41
0
0
0
B 7:95 12:6 8:41
0
0
0 C
C
B
C
B 8:41 8:41 11:7
0
0
0
C 1010 N=m2 ,
B
cpq ¼ B
C
0
0
0
2:30
0
0
C
B
@ 0
0
0
0
2:30
0 A
0
0
0
0
0
2:33
0
1
0
0
0
0
17:0 0
eip ¼ @ 0
0
0
17:0
0
0 A C=m2 ,
6:5 6:5 23:3
0
0
0
0
1
0
0
1700ε0
εij ¼ @ 0
1700ε0
0 A:
0
0
1470ε0
0
Quartz (SiO2)
When referred to the crystallographic axes, the second-order material constants for
left-hand quartz have the following values [13, 17]:
ρ ¼ 2649 kg=m3 ,
0
86:74
B 6:99
B
B 11:91
cpq ¼ B
B 17:91
B
@ 0
0
6:99
86:74
11:91
17:91
0
0
11:91
11:91
107:2
0
0
0
17:91
17:91
0
57:94
0
0
1
0
0
0
0 C
C
0
0 C
C 109 N=m2 ,
0
0 C
C
57:94 17:91 A
17:91 39:88
Appendices
eip
279
0
1
0:171 0:171 0 0:0406
0
0
¼@ 0
0
0
0
0:0406 0:171 A C=m2 ,
0
0
0
0
0
0
0
1
39:21
0
0
εij ¼ @ 0
39:21
0 A 1012 F=m:
0
0
41:03
Temperature derivatives of the second-order elastic constants of quartz at 25 C
are [18].
pq
(1/cpq)(dcpq/dT)(106/ C)
pq
(1/cpq)(dcpq/dT)(106/ C)
11
18.16
33
66.60
44
89.72
12
1222
66
126.7
13
178.6
14
49.21
For quartz there are thirty one nonzero third-order elastic constants among which
fourteen are independent. They are given in the following table. These values, at
25 C, are all in 1011 N/m2 [19].
Constant
c111
c112
c113
c114
c123
c124
c133
c134
c144
c155
c222
c333
c344
c444
Value
2.10
3.45
+0.12
1.63
2.94
0.15
3.12
+0.02
1.34
2.00
3.32
8.15
1.10
2.76
Standard error
0.07
0.06
0.06
0.05
0.05
0.04
0.07
0.04
0.07
0.08
0.08
0.18
0.07
0.17
In addition, there are seventeen relationships among the third-order elastic constants
of quartz [20].
AT-cut quartz is a special case of rotated Y-cut quartz (θ ¼ 35.25 , see Fig. 2.3)
whose second-order material constants are [13]:
280
Appendices
1
86:74 8:25 27:15 3:66
0
0
B 8:25 129:77 7:42
5:7
0
0 C
C
B
C
B 27:15 7:42 102:83 9:92
0
0
C 109 N=m2 ,
B
cpq ¼ B
5:7
9:92
38:61
0
0 C
C
B 3:66
@ 0
0
0
0
68:81 2:53 A
0
0
0
0
2:53 29:01
0
1
0:171 0:152 0:0187 0:067
0
0
eip ¼ @ 0
0
0
0
0:108
0:095 A C=m2 ,
0
0
0
0
0:0761 0:067
0
1
39:21
0
0
εij ¼ @ 0
39:82 0:86 A 1012 F=m:
0
0:86 40:42
0
The fourth-order elastic constants are mostly unknown. A very useful one for AT-cut
quartz can be found in [21]:
E
c6666
¼ 77 1011 N=m2 :
Silicon (Si)
Silicon is a nonpiezoelectric semiconductor. It is a cubic crystal belonging to m3m
with 3 independent elastic constants and one independent dielectric constant [16]:
0
16:57
B 6:39
B
B 6:39
cpq ¼ B
B 0
B
@ 0
0
ρ ¼ 2332kg=m3
1
6:39
0
0
0
6:39
0
0
0 C
C
16:57
0
0
0 C
C 1010 N=m2 ,
0
7:956
0
0 C
C
0
0
7:956
0 A
0
0
0
7:956
0
1
11:7
0
0
εij ¼ @ 0
11:7
0 Aε0 :
0
0
11:7
6:39
16:57
6:39
0
0
0
Zinc Oxide (ZnO) [16]
ρ ¼ 5680kg/m3,
Appendices
281
1
20:97 12:11 10:51
0
0
0
B 12:11 20:97 10:51
0
0
0 C
C
B
B 10:51 10:51 21:09
0
0
0 C
C 1010 N=m2 ,
cpq ¼ B
C
B 0
0
0
4:247
0
0
C
B
@ 0
0
0
0
4:247
0 A
0
0
0
0
0
4:43
0
1
0
0
0
0
0:48 0
eip ¼ @ 0
0
0
0:48
0
0 A C=m2 ,
0:573 0:573 1:32
0
0
0
0
1
8:55
0
0
8:55
0 Aε0 :
εij ¼ @ 0
0
0
10:2
0
References
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Soc. Am. 57, 660–666 (1975)
2. H.F. Tiersten, On the nonlinear equations of thermoelectroelasticity. Int. J. Engng Sci. 9,
587–604 (1971)
3. H.G. de Lorenzi, H.F. Tiersten, On the interaction of the electromagnetic field with heat
conducting deformable semiconductors. J. Math. Phys. 16, 938–957 (1975)
4. H.F. Tiersten, A Development of the Equations of Electromagnetism in Material Continua
(Springer, New York, 1990)
5. J.C. Baumhauer, H.F. Tiersten, Nonlinear electroelastic equations for small fields superposed
on a bias. J. Acoust. Soc. Am. 54, 1017–1034 (1973)
6. H.F. Tiersten, On the accurate description of piezoelectric resonators subject to biasing deformations. Int. J. Eng. Sci. 33, 2239–2259 (1995)
7. H.F. Tiersten, Perturbation theory for linear electroelastic equations for small fields superposed
on a bias. J. Acoust. Soc. Am. 64, 832–837 (1978)
8. H.F. Tiersten, On the influence of material objectivity on electroelastic dissipation in polarized
ferroelectric ceramics. Math. Mech. Solids 1, 45–55 (1996)
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Standard on Piezoelectricity (IEEE, New York, 1988)
10. K. Tsubouchi, K. Sugai, N. Mikoshiba, AlN material constants evaluation and SAW properties
on AlN/Al2O3 and AlN/Si, in Proc. IEEE Ultrasonics Symp., (1981), pp. 375–380
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langanite and langasite. Proceedings of the IEEE/EIA International Frequency Control Symposium and Exhibition, (2000), pp. 200–205
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crystals in class (3m). J. Acoust. Soc. Am. 42, 1223–1231 (1967)
13. H.F. Tiersten, Linear Piezoelectric Plate Vibrations (Plenum, New York, 1969)
14. R.T. Smith, F.S. Welsh, Temperature dependence of the elastic, piezoelectric, and dielectric
constants of lithium tantalate and lithium niobate. J. Appl. Phys. 42, 2219–2230 (1971)
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17. R. Bechmann, Elastic and piezoelectric constants of alpha-quartz. Phys. Rev. 110, 1060–1061
(1958)
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and Mathematics 9, https://doi.org/10.1007/978-3-030-03137-4
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18. B.K. Sinha, H.F. Tiersten, First temperature derivatives of the fundamental elastic constants of
quartz. J. Appl. Phys. 50, 2732–2739 (1979)
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21. H.F. Tiersten, Analysis of intermodulation in thickness-shear and trapped energy resonators.
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Index
A
Acoustic
impedance, 122
tensor, 114
wave generation, 242
Acoustoelectric effect, 212
Antiplane, 70, 75
Antiresonance, 189
B
Beam, 263
Bleustein–Gulyaev wave, 122
C
Capacitance
frequency dependent, 161, 165, 170
static, 91, 93, 100
Cauchy stress, 17
Ceramics, 68, 277, 278
Christoffel tensor, 114
Clausius–Duhem inequality, 208
Compatibility, 57
Conservation
angular momentum, 19, 34
charge, 212
energy, 24, 34
linear momentum, 16, 34
mass 12, 33
Continuity condition, 38
Continuity equation, 212
Coulomb’s law, 39
Cylinder, 95, 97, 98, 105, 150, 152, 174
Cylindrical coordinates, 77
D
Deformation
gradient, 3
rate, 9
tensor, 5
Dislocation, 109
Dispersion relation, 135, 146, 148
Dissipation function, 211
Double resonance, 255
E
Effective
material constants, 187, 202
spring constant, 195
Eigenvalue problem, 114, 182, 185, 192, 203
Electric
body couple, 20, 271, 272
body force, 17, 270, 272
body power, 24, 271, 273
enthalpy, 55
Electromechanical coupling factor, 84, 86, 89,
91, 94
Electrostatic potential, 30
Energy integral, 61
ε–δ identity, 3
Essential boundary condition, 46, 201
F
Face-shear wave, 135
Faraday’s law, 27
Fiber, 214
Finite strain, 7
Four-vector, 65
© Springer Nature Switzerland AG 2018
J. Yang, An Introduction to the Theory of Piezoelectricity, Advances in Mechanics
and Mathematics 9, https://doi.org/10.1007/978-3-030-03137-4
285
286
Free energy, 30, 47, 54, 201, 208
Fundamental material constants, 48
Fundamental solution, 102, 219
G
Gap wave, 146
Gauss’s law, 27, 28
Gradient
electric field, 225
polarization, 224
strain, 224
Green’s identity, 190
Gyroscope, 255
H
Harmonics
fundamental, 183
overtone, 183
I
Induced anisotropy, 202
Interface wave, 130
Internal energy, 24, 55
Invariants, 68
J
Jacobian, 3
Jump condition, 38
K
Kronecker delta, 1
L
Lagrange multiplier, 63
Langanite, 276
Langasite, 276
Langatate, 276
Lateral electric field, 170
Lattice, 223
Legendre transform, 30, 55, 208
Lithium niobate, 76, 276
Lithium tantalate, 76, 277
Love wave, 143
M
Material constants
complex, 212
effective, 187, 202
Index
fundamental, 48
Matrix notation, 66
Maxwell’s equations, 228
Maxwell stress tensor, 30
Motional capacitance, 161, 165
Multiply-connected domain, 57
N
Natural boundary condition, 46, 201
Nonlocal, 216
O
Orthogonality, 114, 189
P
Permutation tensor, 3
Perturbation, 203
Piezoelectric effect
converse, 12
direct, 12
Piezoelectric stiffening, 83, 97, 114–117,
183, 185
Plate
lateral field excitation, 170
thickness
field excitation, 157, 161, 165
shear, 87, 89, 91, 161, 165, 170
stretch, 84, 157, 237, 245
twist, 135
Polarization
electronic, 11
ionic, 11
orientational, 11
per unit mass, 24
Polarization charge, 11
Polarization gradient, 224
Positive definiteness, 55
Poynting
theorem, 60
vector, 61
Q
Quadrupole, 225
Quality factor, 242, 259
Quartz, 72, 278
Quasistatic approximation, 233
R
Radiation condition, 153, 244
Rayleigh quotient, 192, 203
Index
Reciprocal theorem, 190
Rod, 81, 181, 183, 260
S
Second-order theory, 50
Semiconduction, 212
Simple-connected domain, 57
Snell’s law, 121
Spherical coordinates, 78
Spin tensor, 9
Spring-mass system, 194, 224
Strain
finite, 7
linear (infinitesimal), 54
rate, 9
Stress
Cauchy, 17
electrostatic, 30
Maxwell, 30
symmetric, 31
total, 31
Superposition, 59
Surface wave, 122, 125, 127, 230
Symmetry breaking, 202
T
Thermopiezoelectricity, 211
Thermoviscoelectroelasticity, 209
287
Thickness field excitation, 157, 161, 165
Thickness shear, 87, 89, 91, 161, 165, 170
Thickness stretch, 84, 157, 237, 245
Thickness twist, 135
Third-order theory, 47
Total
free energy, 47, 54
stress, 31
Transformer, 248
U
Uniqueness, 61
V
Variation
eigenvalue problem, 192, 203
generalized, 64
Hamiltonian, 62
mixed, 64
W
Wave
front, 113
normal incidence, 121
phase, 113
phase velocity, 113
plane, 113