A finite element model for nonlinear behaviour of piezoceramics
under weak electric fields
M.K. Samala,∗ , P. Seshub , S. Parasharc , U. von Wagnerd , P. Hagedornc , B.K. Duttaa ,
H.S. Kushwahaa
a Reactor Safety Division, Bhabha Atomic Research Centre, Trombay, Mumbai-400085, India
b Department of Mechanical Engineering, Indian Institute of Technology Bombay, Mumbai-400076, India
c Department of Applied Mechanics, Technical University of Darmstadt, 64289 Darmstadt, Germany.
d Department of Applied Mechanics, Technical University of Berlin, 10587 Berlin, Germany.
Abstract
It has been experimentally observed that piezoceramic materials exhibit different types of nonlinearities under
different combinations of electric and mechanical fields. When excited near resonance in the presence of weak e to a
Duffinor such as jump phenomena and presence of superharmonics in the response spectra. There has not been much
work in the litrature to model these types of nonlinearities. Some authors have developed one-dimensional models
for the above phenomenon and derived closed-form solutions for the displacement response of piezo-actuators.
However, the generalized three-dimensional (3-D) formulation of electric enthalpy, the variational formulation
and the FEM implementation have not yet been addressed, which are the focus of this paper. In this work, these
nonlinearities have been modelled in a 3-D piezoelectric continuum using higher order quadratic and cubic terms in
the generalized electric enthalpy density function. The coupled nonlinear finite element equations have been derived
using variational formulation. A special linearization technique for assembling the nonlinear matrices and solution
of the resulting nonlinear equations has been developed. The method has been used for simulating the nonlinear
frequency response of a lead zirconate titanate plate excited near its first in-plane vibration resonance frequency
with sinusoidal excitations of different electric field strengths. The results have been compared with those of the
experiment.
Keywords: Finite element method; Piezoceramic; Nonlinear modelling; Jump phenomena
1465
1. Introduction
Smart materials, especially piezoceramics, find a wide range of applications in many industries for
active vibration control, shape control and health monitoring of structures [1,2]. These piezoceramics
have been extensively studied in recent years for actuator and sensor applications because of their high
electro-mechanical coupling coefficient. Some special classes of these actuators, known as ultrasonic
transducers, are driven at resonance frequency. Depending upon the direction of polarization and direction
of actuation, these can be known as axial, transverse or flexural transducers. High-power transverse
piezoelectric actuators are used for ultrasonic cleaners, piezo-vibrators and machine cutters, etc. It is
well known that the mechanical and electrical responses of a piezoelectric material are coupled. When
the applied electric field is low and the strains are also low, the behaviour of piezoceramics is almost
linear. But, it has been seen from experiments that the piezoelectric materials exhibit a wide range of
nonlinear effects under both high and low electric fields when operated in the presence of high and low
stress fields. The nonlinear behaviours under high and low electric fields are different in some aspects.
For example, the nonlinearities observed under high electric fields are hysteresis behaviour between the
electric field and strain, nonlinear relation between electric field and mechanical displacement etc. The
nonlinearities observed under weak electric fields are jump phenomena (Fig. 1), dependence of resonance
frequency on vibration amplitude, presence of superharmonics in the response spectra and nonlinear
relationship between applied electric voltage and mechanical displacement, etc. The significant change
of the displacement amplitude at a particular frequency is called a “jump phenomenon”. The frequency
at which this phenomenon occurs is also different for increasing (sweep-up) and decreasing (sweepdown) frequency modes. The problem associated with piezo-actuators showing the jump phenomena
can be excessive heat generation, mechanical breakdown and instability of vibration when operating
in the resonant mode, etc. The different nonlinear behaviours of piezo-actuators can also affect other
applications such as the active shape and vibration control of structures [3,4].
In this work, experiments have been conducted on lead zirconate titanate (PZT) plates under electric
fields of varying field strengths to study these nonlinearities. The first in-plane vibration mode of the plate
is being studied to observe the effect of piezoelectric nonlinearity. Based on the observed experimental
Fig. 1. Jump phenomena observed in the sweepup and sweepdown frequency response behaviour of a piezo-plate.
1466
behaviour (similar to a Diffing oscillator), theoretical models have been developed for the electric enthalpy
density function, which is subsequently used in Hamilton’s principle to derive governing equations for the
piezo-continuum. A generalized nonlinear finite element model has been developed from the governing
equations so that it can be applied to predict the actual behaviour of piezo-actuators or resonators operating
in resonance mode for which closed-form solutions cannot be derived.
There are many models for nonlinearities under strong electric fields (e.g., hysteresis), which are mainly
based on the classical Preisach model and its modified versions [5]. However, there are very few research
works existing in the literature which deal with nonlinear effects under weak electric fields. Beige and
Schmidt [6] first described these nonlinear effects while conducting experiments on longitudinal vibrations
of piezo-rods [6]. They modelled these nonlinearities using higher order quadratic and cubic terms in the
energy expression contributed by elastic, piezoelectric and dielectric continuum.
Von Wagner and Hagedorn [7] and von Wagner [8] modelled both the nonlinear d31 and d33 effects of
typical PZT-based piezoceramics (i.e., PIC 181 and PIC 255) by assuming that the softening behaviour
of the material is due to the nonlinear dependence of Young’s modulus and the piezoelectric coefficient
on the mechanical and electrical field variables [7,8]. They derived the electric enthalpy density functions
for the one-dimensional (1-D) continuum. The governing 1-D nonlinear equation was derived using
variational formulation (Hamilton’s principle) and was solved using Rayleigh–Ritz method. The shape
function of only one mode of vibration of the structure was used as the weighting function in this method.
The discretized equation was solved for the amplitude of vibration using perturbation techniques and
harmonic balance methods. The unknowns in the model, i.e., the nonlinear parameters in the constitutive
relations were determined by minimizing the error between experimental and theoretical results. This was
one of the first attempts to obtain closed-form solutions for simplified 1-D problems. However, when the
structures to be modelled have complicated geometry, boundary condition and loading types (as in the
case of machine cutters, piezo-resonators, etc.), it is no longer possible to obtain closed-form solutions
for these classes of nonlinear problems. Our aim here is to formulate the generalized 3-D electric enthalpy
density function. Once the electric enthalpy density function is formulated, the governing equations can
be derived using variational formulation. However, one has to resort to some numerical method (e.g.,
FEM) to solve the system of equations.
Several authors have used FEM to model various piezoelectric material systems [9–15]. However,
their work is mainly directed towards modelling of the linear piezoelectric behaviour. Saravanos et al.
[16] have developed a layer-wise FE model for the dynamic analysis of piezoelectric composite plates.
Recently, Wang [17] has developed an FE model for static and dynamic analysis of piezoelectric bimorphs.
Wang et al. [18] have studied the dynamic stability behaviour of FE models for piezo-composite plates.
A generalized FE model taking into consideration of the nonlinearity of a piezo-continuum similar to a
Duffing oscillator is not available in the literature. In this work, the generalized 3-D nonlinear constitutive
equations for the nonlinear problem have been derived after the electric enthalpy density function had
been conceived. Based on this, the FEM-based solution technique has been developed. The method has
been applied to calculate the nonlinear frequency response behaviour of a PZT plate excited near its first
in-plane resonance frequency for which experimental results are available.
2. Constitutive equations of the piezoelectric continuum
The electric enthalpy density function H is generally used to derive the governing equations
of the coupled piezoelectric continuum and for the linear piezoelectric behaviour. It is given as (IEEE
1467
standard [19])
H = 21 CijE kl Sij Skl − ekij Ek Sij − 21 Sij Ei Ej ,
(1)
where CijE kl is the fourth-order elastic tensor under a constant electric field, ekij is the third order piezoelectric tensor, Sij is the second-order dielectric tensor, Sij is the strain tensor and Ei is the electric field
vector. The stress tensor Tij and the electric displacement vector Di can be derived from H using the
expressions Tij = jH /jSij and Di = −jH /jEi , respectively. It can be easily seen that the expressions for
Tij and Di are coupled with the secondary field variables Sij (strain tensor) and Ei (electric field vector),
respectively. Sij is expressed in terms of the mechanical displacement vector ui as Sij = (ui,j + uj,i )/2
and Ei is expressed in terms of electric potential as Ei = −,i . Keeping in mind the fact that stress and
strain tensor are symmetric, H can be simplified and written in terms of second-order tensors as
H = 21 CijE Si Sj − eij Ei Sj − 21 Sij Ei Ej .
(2)
In order to model the nonlinearities in a coupled piezoelectric medium, we propose to modify the linear
electric enthalpy density function in the following way. It was experimentally observed that second- and
third-order harmonics are present in the response spectra of PZT piezoceramic plates and rod geometries.
The observed behaviour can be modelled using quadratic and cubic order terms in the energy terms of the
two coupled mediums. Hence, the generalized expression for nonlinear electric enthalpy density function
can be written as
Hnonl = 21 CijE Si Sj − emi Em Si − 21 Smn Em En + 13 CijE k Si Sj Sk + 41 CijE kl Si Sj Sk Sl − 13 Smno Em En Eo
∗
∗∗
− 41 Smnop Em En Eo Ep − 21 emij Em Si Sj − 21 emni
Em En Si − 13 emij
k Em Si Sj Sk
∗∗∗
∗∗∗
− 41 emnij
Em En Si Sj − 13 emnoi
Em En Eo Si ,
(3)
where the coefficients with 3 and 4 number of indices are higher order quadratic and cubic coefficients,
respectively. However, this form of Hnonl is not suitable for derivation of nonlinear FEM matrices using
variational formulation. Hence, the authors have envisaged a suitable matrix expression for Hnonl retaining
the nonlinear terms and it is given as follows:
Hnonl = 21 {S}T [C]{S} − {E}T [d][C]{S} − 21 {E}T [[T ] − [d][C][d]T ]{E} + 13 {S}T [C1 ]{S 2 }
+ 41 {S 2 }T [C21 ]{S 2 } + 41 {S}T [C22 ]{S 3 } − 21 {E}T [11 ]{S 2 } − 21 {E 2 }T [12 ]{S}
− 13 {E}T [1 ]{E 2 } − 13 {E}T [21 ]{S 3 } − 21 {E 2 }T [22 ]{S 2 } − 13 {E 3 }T [23 ]{S}
− 41 {E}T [21 ]{E 3 } − 41 {E 2 }T [22 ]{E 2 },
(4)
where {S} is the strain vector, [C] is the linear elasticity matrix, [d] is the linear piezoelectric coefficient
matrix, {E} is the electric field vector, [T ] is the dielectric coefficient matrix, [C1 ] is the quadratic
elasticity matrix, and [C21 ] and [C22 ] are cubic elasticity matrices at a constant electric field, respectively.
1468
The superscript E (matrices at a constant electric field) for these matrices has been omitted for clarity.
[11 ] and [12 ] are quadratic piezoelectric matrices, [21 ], [22 ] and [23 ] are cubic piezoelectric matrices,
and [0 ] = [[T ] − [d][C][d]T ], [1 ], [21 ] and [22 ] are linear, quadratic and cubic dielectric matrices
at constant strain field S, respectively. The superscript S (matrices at a constant strain field) for these
matrices has been omitted for clarity. The vectors with superscripts 2 and 3 are defined as the vectors with
square and cube of individual terms of the vectors. The first three terms of Hnonl correspond to the linear
electric enthalpy density function. Using this expression of Hnonl , the FE matrices have been derived in
the next section using variational formulation.
3. Variational formulation
The dynamic equations of a piezoelectric continuum can be derived from the Hamilton principle, in
which the Lagrangian and the virtual work are properly adapted to include the electrical, mechanical as
well as the coupled electro-mechanical terms. The potential energy density of a piezoelectric material
includes contributions from the strain energy and from the electrostatic energy. The Hamilton principle
can be written as
t1
t0
V
L dV dt +
t1
t0
W dt = 0,
(5)
where t0 and t1 define the time interval, L is the Lagrangian and is defined by the sum of kinetic energy
Tke and electrical enthalpy density H and given as L = (Tke − H ), and W is the virtual work done by
external mechanical and electrical forces. The kinetic energy is given as
Tke = 21 {u̇}T {u̇},
(6)
where is the mass density, {u} is the generalized displacement field and {u̇} is the generalized velocity
field. The nonlinear electric enthalpy density function Hnonl is given in Eq. (4). The virtual work done
by the external mechanical forces F and the applied electric charges Q for an arbitrary variation of the
displacement field {u} and electrical potential {}, both satisfying the essential boundary conditions
(i.e., {u} = 0 on surface As3 and {} = 0 on surface As4 ), can be written as
W =
{u} {FV } dV +
T
V
{u} {FAs } dAs + {u} {FP } −
T
As1
T
As2
dAs − Q + WD , (7)
where {FV } is the applied body force vector, {FAs } is the applied surface force vector (defined on the
surface As1 ), {FP } is the applied point load vector, is the electric potential, is the surface charge
conferred on surface As2 , Q is the applied concentrated electric charge and WD is the work done by
damping forces (we assume proportional viscous damping here). In this problem, there are no external
mechanical or electrical forces except the applied electric potential at the electrodes coated to the
surfaces of the piezoceramic. Now, putting the nonlinear electric enthalpy function Hnonl and kinetic
1469
energy Tke from Eqs. (4) and (6), the Hamilton’s principle becomes
t1 [{u}T {ü}] dV dt
−
V
t0


[C]{S} − [C]T [d]T {E} − [diag({S})][11 ]T {E} − 21 [12 ]T {E 2 }
t1  +[diag({S})][C21 ][diag({S})]{S} + 1 [C22 ]{S 3 } + 3 [diag({S 2 })][C22 ]T {S} 
4
4

{S}T 
−
 + 1 [C1 ]{S 2 } + 2 [diag({S})][C1 ]T {S} − [diag({S 2 })][ ]T {E}

t0
V
21
3
3
−[diag({S})][22 ]T {E 2 } − 13 [23 ]T {E 3 }
× dV dt


[d][C]{S} + [0 ]{E} + 21 [11 ]{S 2 }
t1  +[diag({E})][12 ]{S} + 1 [13 ]{S 2 } + 2 [diag({E})][13 ]T {E}

3
3

{E}T 
−
1
3
2
2
 + [ ]{S } + [diag({E})][ ]{S } + [diag({E })][ ]{S}

t0
V
3
21
22
23
+ 41 [1 ]{E 3 } + 43 [diag({E 2 })][1 ]T {E} + [diag({E})][2 ][diag({E})]{E}
×
dt
t1 t1
tdV
1
T
T
[{u} {FV }] dV dt +
[{u} {FAs }] dAs dt +
{u}T {FP } dt
+
t
V
t0
As1
t0
t1
t1
0t1 dAs dt −
Q dt −
[WD ] dt = 0,
−
t0
As2
t0
t0
(8)
where the matrix with ‘diag’ refers to a diagonal matrix with the main diagonal terms being the terms of
the vector that it contains. The virtual work done by viscous damping forces is given as
WD = {u}T [Cdamp ]{u̇} dV ,
(9)
V
where [Cdamp ] is the proportional damping coefficient matrix and is expressed in terms of mass [M] and
stiffness [K] matrices with the help of constants and as
[Cdamp ] = [M] + [K].
(10)
4. Derivation of nonlinear FE equations
The displacement field {u} and the electric potential at any point in a finite element are related to
the corresponding nodal values {ui } and {i } by means of the element shape function matrices [Nu ] and
[N ] as {u} = [Nu ]{ui } and = [N ]{i }, respectively. The strain field vector {S} and the electric field
vector {E} can be expressed in terms of {ui } and {i } with the help of [Bu ] and [B ] as {S} = [Bu ]{ui }
and {E} = [B ]{i }, where [Bu ] and [B ] are derivatives of shape function matrices. Incorporating these
expressions of {S} and {E} into the variational equation (8), collecting the terms with {ui }T and {i }T
and equating them to zero separately, we obtain the following coupled FE equations.
[M]{ü} + [Cdamp ]{u̇} + [Kuu ]{ui } + [Ku ]{i } = {fi },
(11)
[Ku ]{ui } + [K ]{i } = {gi },
(12)
where [M] is the mass matrix, [Kuu ] is the nonlinear mechanical stiffness matrix, [Ku ] and [Ku ]
are nonlinear piezoelectric stiffness matrices, [K ] is the nonlinear dielectric stiffness matrix, {fi } is
1470
the external mechanical force vector, and {gi } is the external electrical force vector respectively. The
expressions for these matrices are given in Appendix-A1.
It may be noted that the stiffness matrices are not constants. They depend implicitly upon the solution,
i.e., {ui } and {i }. In order to assemble the element stiffness matrices, a special linearization technique
(named Classical linearization after Salinas et al. [20]) has been adopted. In this technique, the initial
solutions for {ui } and {i } are taken as the solutions of the corresponding linear equations. The matrices
are then assembled and the global dynamic-coupled equations are solved using the Newmark- method
after applying proper mechanical and electrical boundary conditions. In this way, the next approximate
solution for the current time step is obtained. Again, the matrices are updated with the recent approximate
solution and the method is repeated till the two consecutive solution vectors converge within a pre-specified
tolerance limit.
5. Experimental work
Experiments were conducted for studying the nonlinear behaviour of piezoceramic plates for the inplane mode of vibration. The piezoceramics were excited close to their resonance frequencies in two
ways, i.e., sweep-up (SU) and sweep-down (SD). In SU the frequency was increased from a low to a high
value past the natural frequency and in SD, the process was reversed. This was done to observe the jump
phenomenon. The line diagram of the experimental set-up is shown in Fig. 2.
The excitation and measurements have been done with the help of a gain-phase analyser (HewlettPackard HP 4194A). It sends the excitation signal to the piezoceramic through a power amplifier (Bruel
and kjaer 2713). The dynamic response (velocity) signal is captured with the help of a laser vibrometer
(Polytec). The laser vibrometer has two units (i.e., optic unit and electronic unit). The optic unit (OFV
508) supplies the laser signal to the piezoceramic as well as senses the reflected signal. The electronic unit
Fig. 2. Experimental set-up for measuring the nonlinear dynamic response of the piezo-plate.
1471
Fig. 3. FFT of the time domain signal when the plate is excited at 1/3rd of the first natural frequency.
then processes the signal from the optic unit and it is fed to the test channel of the gain phase analyzer.
The input signal to the piezoceramic from the power amplifier is also supplied to the reference channel of
the gain phase analyser which then compares the signal from the test channel with respect to the reference
channel to determine the gain and the phase difference between the two signals. The two signals are also
viewed in the digital oscilloscope (Yokogawa DL708E). The applied voltages are measured from the
oscilloscope display. The displacement responses for various values of applied voltages are discussed in
the next section on FE analysis.
The Fast Fourier Transform (FFT) of the time domain response signal of piezoceramic has also been
conducted. For this purpose, the piezoceramic is excited with the help of a signal generator (HewlettPackard HP 33120A) instead of the gain-phase analyser. Sinusoidal excitation with a particular frequency
is set in the signal generator. The response signal is measured with the help of the laser vibrometer. The
time domain signal is then picked in the digital oscilloscope with a predefined sampling rate, sampling
time and record length of the data (which have been initially chosen in the oscilloscope settings). Then FFT
analysis is performed on the time domain data to know the presence of signals of frequencies other than
the excitation frequency. Fig. 3 shows the FFT analysis of the time domain signal of the laser vibrometer
from the piezo-plate when it is excited at one-third of the first in-plane natural frequency. It can be seen
from Fig. 3 that there are two distinct peaks observed in the frequency domain, i.e., one at the excitation
frequency and the other at a frequency which is three times the excitation frequency. This gives evidence
that cubic-type nonlinearity is dominant in the response of the piezo-plate.
6. FE analysis of in-pane vibration of a rectangular piezoelectric plate
The procedure developed in this work has been implemented in an in-house FE program “NLPIEZO”.
The program has been thoroughly tested using several case studies from the literature [21]. We present
here the FE analysis of the nonlinear response behaviour of a piezoceramic plate. The plate analysed in
this case is a rectangular plate of 70 mm length, 20 mm width and 4 mm thickness and is made up of PIC
181 supplied by “PI Ceramics”, Lederhose, Germany (www.piceramic.de) (Fig. 4a).
1472
Fig. 4. (a) Figure of the piezo-plate used in the investigation. (b) 2-D finite element mesh of one-quarter of the piezo-plate.
Fig. 5. Variation of x-direction displacement in the first in-plane mode of oscillation of the plate.
Fig. 6. Variation of y-direction displacement in the first in-plane mode of oscillation of the plate.
The plate is polarized along the thickness direction. Sinusoidal electric voltages of different amplitudes
are applied across the thickness. The first in-plane mode of the free–free vibration of the plate is along
its length direction. The variation of displacements along x (length) and y (width) directions for the first
mode are shown in Figs. 5 and 6, respectively. The amplitude of vibration is measured with the help of a
1473
W
L
Fig. 7. Symmetric boundary conditions used for the first mode of vibration.
laser vibrometer, which is focused along the length onto one end of the plate. The vibrometer measures
the velocity amplitude, which is converted into the displacement amplitude by dividing it with the applied
circular frequency of excitation . A frequency sweep is applied near the first resonance frequency of
the plate. It was observed that the frequency response path was not the same for the up-sweep (frequency
is increased) and down-sweep (frequency is decreased) method of excitation (Fig. 1). The nonlinear
softening behaviour and jump phenomena are clearly observed.
This response behaviour of the plate has been calculated using the nonlinear FE program NLPIEZO.
As the thickness of the plate is very small compared to other dimensions, a 2-D plane stress analysis
has been carried out. One-fourth of the plane of the plate has been analysed because of symmetry of
the first in-plane mode of vibration. The finite element mesh used for this study is shown in Fig. 4b. A
5 × 5 4-noded isoparametric FE mesh has been used. A mesh convergence study had been carried out
beforehand to ensure that further refining of mesh size is not required to obtain convergence in solution.
The mechanical boundary condition is applied at the symmetric planes of the plate (i.e., u1 = 0 at x = 0
and u2 = 0 at y = 0 where 1 and 2 refer to x and y directions, respectively (Fig. 7)).
The electric boundary condition is simple because we are not modelling the thickness of the plate.
Electric potential is applied across the thickness and the whole plane of the plate is coated with an electrode
(i.e., is imposed). Hence, the electric field can be assumed to be constant in the plane perpendicular
to the thickness of the plate and is given by electric potential per unit thickness with a negative sign.
Because of the above electrical boundary conditions, only the equation for the actuator [i.e., Eq. (11)]
needs to be solved. The electric potential applied serves as an external loading on the plate. The linear
and nonlinear material property matrices have been taken from reference [21]. These material property
matrices are usually obtained using optimization techniques. Analytical solutions for simple geometries
are obtained using perturbation method of solution of nonlinear differential equations and by comparing
analytical solutions with experimental ones, the parameters for a material are obtained. The details of
this method are given elsewhere [22]. The damping values (proportional damping constants) have been
calculated from the experimental results using the half-power method. The mass proportional damping
constant is assumed as zero and hence the stiffness proportional constant is used as the damping parameter.
The variation of the stiffness proportional damping constant with an electric field is shown in Fig. 11
(Appendix-A2). The material property matrices are given in Appendix-A2 where [S E ] is the compliance
matrix at a constant electric field.
1474
7. Results and discussion
Fig. 8 shows the comparison of the predicted and experimental nonlinear frequency response of the
plate for both up and down frequency sweep. Here we will briefly describe the method used to obtain
different responses by FEM in sweep-up and sweep-down conditions of excitation. For the sweep-up
condition, the system is excited at a frequency lower than the resonance frequency away from the jump
region. It may be noted that the nonlinear response has to be detained for each particular applied frequency
of excitation at a time.
Once the converged solution of the nonlinear system for a particular frequency is obtained, one needs to
proceed to the next frequency of excitation. An initial guess solution for a particular frequency is obtained
by solving the corresponding linear problem. This solution is used to calculate the nonlinear matrices and
then the linearized system is solved to obtain the next approximate solution. The process is repeated in
an iterative manner till convergence in the solution vector is achieved.
For the sweep-up frequency method, the converged response of the previous frequency (lower value)
is used as an initial guess for the next higher frequency of excitation instead of starting from the linear
problem. For the sweep-down method, the converged response of previous higher frequency is used
for the solution of the next lower frequency. In a nonlinear problem, there are multiple solutions for a
particular input (frequency of excitation) especially in the jump region. The convergence to a particular
solution depends upon the initial guess solution used and the solution attained is that which is closer to
the guess solution. This property has been exploited in this analysis to predict the different sweep-up and
sweep-down frequency response behaviour of the piezo-plate.
It can be observed from Fig. 8 that the model is able to accurately predict the nonlinear response
of the piezo-plate. In order to study the effect of different electric fields on the nonlinear response of
the piezo-plate, experiments have been conducted at different excitation voltages (i.e., from 3 to 35 V).
The FEM-calculated displacement responses per unit voltage near the first resonance frequency for both
sweep-up and sweep-down modes are shown in Fig. 9 for 5, 10 and 30 V, respectively.
Fig. 8. Comparison of FEM-predicted and experimental nonlinear response behaviour of the piezo- excited near the first resonance
frequency with 30 V.
1475
Fig. 9. Variation of displacement amplitude with electric field for the piezo-plate excited near the first resonance frequency
calculated by FEM.
Fig. 10. Variation of displacement amplitude with electric field for the piezo-plate excited near the first resonance frequency.
The results of FEM were very close to those of the experiments as evident in Fig. 8 for the applied
potential of 30 V. It can be seen from Fig. 9 that the displacement amplitude per unit voltage reduces
drastically as the electric field is increased. It can also be observed that there is a shift in the frequency of
maximum amplitude towards the lower frequency region for both sweep-up and sweep-down modes of
the experiment. It reflects a strong softening effect in the piezoceramic behaviour as the electric field is
increased. The change or decrease in the frequency corresponding to the maximum amplitude is higher
for higher electric fields. The variation of the calculated displacement amplitude from FEM for the whole
range of electric fields (i.e., 3 to 35 V divided by the thickness of the piezo-plate) is shown in Fig. 10. It
can be observed that a strong nonlinear relationship exists between the displacement amplitude and the
applied electric field.
1476
This nonlinear piezoelectric behaviour model has tremendous applications in the design of ultrasonic
piezomotors and other piezoelectric systems (such as machine cutters, resonators, etc.) where the piezo
is excited near its resonance frequency in order to obtain maximum response. In ring-type ultrasonic
piezomotors, the different modes of vibration of the piezo-ring are used to generate elliptical motion of
the stator and this in turn is used to move the rotor through a friction layer between the stator and rotor.
The torque–speed characteristics of the system depend upon the vibration behaviour of the piezo-ring
and the friction material. The nonlinear vibration behaviour modelled in this work may prove to be very
important and if not envisaged beforehand can affect the predicted performance of the system. Hence,
this behaviour should be considered carefully while modelling the performance of a piezoelectric system
for different applications.
Fig. 8 shows that the maximum amplitude of vibration of the plate is not at the natural frequency of
the plate, but at a frequency less than that. It means the material undergoes softening due to the applied
electric field and it occurs when the plate is excited near the natural frequency. The natural frequency of
a coupled electro-mechanical system not only depends upon the mechanical, piezoelectric and dielectric
material properties, but also upon the mechanical and electrical boundary conditions imposed. This
becomes clear when one considers the natural vibration of a piezo-plate with voltage-driven electrodes
(short-circuited) and charge-driven electrodes (open-circuited), respectively. In case of voltage-driven
electrodes, the electrical field has no effect on the mechanical stiffness of the system and the natural
frequency of vibration is the same as that of a pure mechanical system.
On the other hand, the natural frequency is increased for a charge-driven electrode. This is because
of the application of an external electric field boundary condition (i.e., applied electric field is zero
at the electrodes) for this problem. It may be noted that the application of natural electric boundary
conditions (electric potential) decreases the stiffness of the system (i.e., softening effect). Electric potential
is generated in a piezoelectric continuum due to mechanical strain. The distribution of mechanical strain
depends upon the frequency of excitation. When the system is vibrating at a particular mode, the strain
is maximum at the nodes and minimum at the antinodes. The formation of nodes and antinodes imposes
natural electrical boundary conditions on the problem and hence it decreases the natural frequency of
the system. This phenomenon is pronounced at high values of strain and hence is observed only when
the system is excited near the resonance frequencies. Also, it has been experimentally observed that this
phenomenon is more pronounced for lower modes. For higher modes, the effect of nonlinearity decreases.
The system also behaves like a nonlinear spring-damper system in addition to the observed softening.
This phenomenon can only be modelled using a nonlinear differential equation similar to Duffing’s
equation for the continuum. The equation has to be linearized in order to obtain a starting solution and
is then solved iteratively. In this way, the effective stiffness of the system becomes a function of field
variables. The nonlinear stiffness coefficient for the system is negative and hence it captures the softening
behaviour. The model developed in this work is also able to predict the jump phenomenon because of the
characteristics of Duffing’s equation.
8. Conclusion
In this work, a generalized nonlinear electric enthalpy density function incorporating higher order
nonlinear terms (quadratic and cubic) in the energy expression of the coupled piezoelectric medium
has been formulated. The scope of this formulation is to model the nonlinearities in a system whose
1477
governing equations are similar to a Duffing oscillator. These behaviours of a Duffing oscillator (i.e.,
jump phenomenon and dependence of resonance frequency on vibration amplitude) have been observed
through experiments on piezoceramic plates and rod geometries under weak electric fields when excited
near the resonance frequencies. This nonlinear electric enthalpy density function has been used to derive
the coupled FE equations through variational formulation. These equations have been solved using the
linearization technique. Experiments have been conducted on rectangular piezoceramic PIC 181 plates
at different external electric fields to study and quantify its nonlinear are given vibration behaviour with
electric potential applied across the thickness. The plate is excited with a sinusoidal voltage and then
frequency sweeps are given near the first resonance frequency and the frequency response is calculated
using the software NLPIEZO based on the FE formulation developed in this work. The calculated results
for the nonlinear response of the piezo-plate matched closely with those of the experiment. The following
conclusions can be drawn from the above study:
• The maximum amplitude of vibration of the plate for this case is not at the natural frequency of the
plate, but at a frequency less than that. It means that the material undergoes softening due to the applied
electric field and it occurs when the plate is excited near its natural frequency. The reason for this is
due to inherent imposition of natural electrical boundary conditions.
• The system also behaves like a nonlinear spring-damper system in addition to the observed softening.
This phenomenon can only be modelled using a nonlinear differential equation similar to Duffing’s
equation for the continuum.
• The model is able to predict the nonlinear response behaviour of the piezo-plate accurately. This
generalized 3-D formulation is incorporated into the FE software NLPIEZO and hence, it can be used
to model any piezoelectric system with complex geometry and boundary conditions. In future work,
it will be used to model nonlinearities of an ultrasonic piezomotor and a piezo-beam system.
Acknowledgements
The first author sincerely acknowledges the financial support provided by DAAD (German Academic
Exchange Service) for carrying out part of this research work at Technical University of Darmstadt, Germany. This work is supported by the Deutsche Forschungs Gemeinschaft vide research grant Ha1060/36.
The first two authors gratefully acknowledge the support received from Aeronautics Research and Development Board and Department of Science and Technology, India.
Appendix-A1 (Finite element matrices of the nonlinear piezoelectric continuum)
[M] = [Nu ]T [Nu ] dV
V
[Kuu ]


[Bu ]T [C][Bu ] + [Bu ]T [diag({S})][C21 ][diag({S})][Bu ]
 + 41 [Bu ]T [diag({S})][C22 ][diag({S 2 })][Bu ] + 43 [Bu ]T [diag({S 2 })][C22 ]T [Bu ]  dV ,
=
V
+ 13 [Bu ]T [C1 ][diag({S})][Bu ] + 23 [Bu ]T [diag({S})][C1 ]T [Bu ]
1478
[Ku ]


[Bu ]T [C]T [d]T [B ] + [Bu ]T [diag({S})][11 ]T [B ]
 dV ,
 + 1 [Bu ]T [12 ]T [diag({E})][B ] + [Bu ]T [diag({S 2 })][21 ]T [B ]
=
2
V
1
T
T
T
T
2
+[Bu ] [diag({S})][22 ] [diag({E})][B ] + 3 [Bu ] [23 ] [diag({E })][B ]
[Ku ] = [Ku ]T ,
[K ]


[B ]T [0 ][B ] + 13 [B ]T [1 ][diag({E})][B ] + 23 [B ]T [diag({E})][1 ]T [B ]
 + 1 [B ]T [21 ][diag({E 2 })][B ] + 3 [B ]T [diag({E 2 })][21 ]T [B ]

=−
4
4
V
T
+[B ] [diag({E})][22 ][diag({E})][B ]
× dV ,
[Cdamp ] = [M] + [Bu ]T [C][Bu ],
V
{fi } =
[Nu ] [FV ] dV +
T
V
{gi } = −
As2
As1
[Nu ]T [FAs ] dAs + [Nu ]T [FP ],
[N ]T dAs − [N ]T Q.
Appendix-A2 (Material properties of PIC 181)


1.1600 −0.4070 −0.4996
0
0
0
0
0
0 
 −0.4070 1.1750 −0.4996


0
0
0  −11 2
 −0.4996 −0.4996 1.4110
[S E ] = 
 10 m /N,
0
0
0
3.5330
0
0 


0
0
0
0
3.5330
0 
0
0
0
0
0
3.1640
0
0
0
0
3.89 0
[d] =
0
0
0
3.89
0
0 10−10 m/V,
−1.148 −1.148 2.53
0
0
0


4.7000 2.7914 2.6525
0
0
0
0
0
0 
 2.7914 4.7000 2.6525


2.6525
2.6525
3.9978
0
0
0  16

[C21 ] = − 
 10 N/m2 .
0
0
0.8465
0
0 
 0
 0
0
0
0
0.8465
0 
0
0
0
0
0
0.9452
The other nonlinear material properties have a negligible effect and are assumed to be zero. The mass
proportional damping constant is assumed to be zero. So, the stiffness proportional damping constant is
taken as the damping parameter. The variation of the damping parameter with applied electric field is
shown in Fig. 11.
1479
Fig. 11. Variation of stiffness-proportional damping constant with the applied electric field for the piezo-plate.
References
[1] E.F. Crawley, J. de Luis, Use of piezoelectric actuators as elements of intelligent structures, AIAA J. 25 (10) (1987)
1373–1385.
[2] S.S. Rao, M. Sunar, Piezoelectricity and its use in disturbance sensing and control of flexible structures: a survey, Appl.
Mech. Rev. 47 (4) (1994) 113–123.
[3] D. Sun, L. Tong, D. Wang, An incremental algorithm for static shape control of smart structures with nonlinear piezoelectric
actuators, Int. J. Solids Struct. 41 (2004) 2277–2292.
[4] Y.H. Zhou, H.S. Tzou, Active control of nonlinear piezoelectric circular shallow spherical shells, Int. J. Solids Struct. 37
(12) (2000) 1663–1677.
[5] X. Zhou, A. Chattopadhyay, Hysteresis behaviour and modelling of piezoceramic actuators, Trans. ASME—J. Appl. Mech.
68 (2001) 270–277.
[6] H. Beige, G. Schmidt, Electromechanical resonances for investigating linear and nonlinear properties of dielectrics,
Ferroelectrics 41 (1982) 39–49.
[7] U. von Wagner, P. Hagedorn, Piezo-beam-systems subjected to weak electric field: experiment and modelling of
nonlinearities, J. Sound Vib. 256 (5) (2002) 861–872.
[8] U. von Wagner, Non-linear longitudinal vibrations of piezoceramics excited by weak electric fields, Int. J. Non-Linear
Mech. 38 (2003) 565–574.
[9] H. Allik, T.J.R. Hughes, Finite element method for piezoelectric vibration, Int. J. Numer. Methods Eng. 2 (1970) 151–157.
[10] H. Allik, K.M. Webman, J.T. Hunt, Vibrational response of sonar transducers using piezoelectric finite elements, J. Acoust.
Soc. Am. 56 (1974) 1782–1791.
[11] H.S. Tzou, C.I. Tseng, Distributed piezoelectric sensor/actuator design for dynamic measurement/control of distributed
parameter systems: a piezoelectric finite element approach, J. Sound Vib. 138 (1) (1990) 17–34.
[12] H.S. Tzou, C.I. Tseng, Distributed modal identification and vibration control of continua: piezoelectric finite element
formulation and analysis, Trans. ASME—J. Dyn. Syst. Meas. Control 113 (1991) 500–505.
[13] K.H. TSung, K. Charles, Finite element analysis of composite structures containing distributed piezoceramic sensors and
actuators, AIAA J. 30 (3) (1993) 771–780.
[14] R. Simkovics, H. Landes, M. Kaltenbacher, R. Lerch, Nonlinear finite element analysis of piezoelectric transducers,
Technical Report 4.1, Johannes Kepler University of Linz, Austria, 1999a.
[15] R. Simkovics, H. Landes, M. Kaltenbacher, J. Hoffelner, R. Lerch, Finite element analysis of hysteresis effects in
piezoelectric transducers, Technical Report 4.2, Johannes Kepler University of Linz, Austria, 1999b.
[16] D.A. Saravanos, P.R. Heyliger, D.A. Hopkins, Layerwise mechanics and finite element for the dynamic analysis of
piezoelectric composite plates, Int. J. Solids Struct. 34 (1997) 359–378.
1480
[17] S.Y. Wang, A finite element model for the static and dynamic analysis of a piezoelectric bimorph, Int. J. Solids Struct. 41
(2004) 4075–4096.
[18] S.Y. Wang, S.T. Quek, K.K. Ang, Dynamic stability analysis of finite element modelling of piezoelectric composite plates,
Int. J. Solids Struct. 41 (2004) 745–764.
[19] IEEE standard, IEEE standards on piezoelectricity, ANSI/IEEE standard, The institute of Electrical and Electronic
Engineers, New York, 1988.
[20] D. Salinas, Y.W. Kwon, B.S. Ritter, A study of linearisation techniques in nonlinear FEM, Math. Modelling Sci. Comput.
1 (5) (1993) 480–493.
[21] M.K. Samal, Modelling of nonlinear behaviour of piezoelectric materials under weak electric fields, Masters Thesis,
Department of Mechanical Engineering, Indian Institute of Technology, Bombay, 2003.
[22] H. Dave, Longitudinal vibrations of piezoceramics subjected to weak electric fields: experiment and modelling, Masters
Thesis, Institut fuer mechanik, Technische Universitaet Darmstadt, Germany, 2002.