Sensing Analysis of Piezoelectric Composite Beam
EVA KORMANIKOVA, KAMILA KOTRASOVA
Institute of Structural Engineering,
Civil Engineering Faculty, Technical University of Kosice,
Vysokoskolska 4, 04200 Kosice,
SLOVAKIA
Abstract: - The objective of this paper is to analyze the smart composite beam integrated with a piezoelectric
layer to use in sensing simulation. The beam consists of an 8-layered [0/45/-45/90]S bottom CFRP composite
layer and an upper PZT layer as a piezoelectric sensor. Linear 3D piezo-elasticity theory for static analysis has
been utilized. A prescribed downward displacement is applied at the free end of the cantilever beam to obtain
the voltage and electric field results of the outer surface of the piezoelectric layer. The analysis of the
piezoelectric composite beam is performed by ADINA FEM analysis.
Key-Words: - Sensing static analysis, FEM, 3D modeling, piezoelectricity, composite, cantilever beam.
Received: July 11, 2023. Revised: March 4, 2024. Accepted: May 11, 2024. Published: June 26, 2024.
1 Introduction
Piezoelectric materials exhibit certain special
characteristics that make them important
engineering materials. The materials belong to the
class of smart materials because they exhibit
inherent transducer characteristics. The discovery of
ferroelectric ceramics barium titanate, and lead
zirconate titanate (PZT) during the 1940s and 1950s
led to a spate of research activities on these
materials. Over the years, a wide variety of
transducers, sensors, and actuators have been
developed using the ceramic PZT, which is one of
the most sensitive piezoelectric materials. Another
important piezoelectric material that has created a
lot of interest is the polymer polyvinylidene fluoride
(PVDF), which was discovered in 1969. A polymer
exhibiting transducer characteristics has special
advantages over ceramic because of the more
flexible and less brittle nature of polymers. Zinc
oxide is a relatively newfound piezoelectric material
that has been used in nano-crystalline form for
piezoelectric applications such as micro-actuators
and sensing devices, [1], [2]. Piezoelectric sensors
and actuators are widely used in smart systems, [3],
[4], [5].
2 Problem Formulation
When a dielectric material belonging to a
noncentrosymmetric class (except the octahedral
class) is subjected to an external electric eld, there
will be asymmetric movement of the neighboring
ions, resulting in significant deformation of the
crystal and the deformation is directly proportional
to the applied electric eld. These materials exhibit
an electrostrictive effect due to the anharmonicity of
the bonds, but it is masked by the more significant
asymmetric displacement. The materials are called
piezoelectric materials. Piezoelectric materials
exhibit certain special characteristics that make
them important engineering materials. The materials
belong to the class of smart materials because they
exhibit inherent transducer characteristics, [6], [7],
[8].
2.1 Constitutive Equation
A piezoelectric material that is exposed to the small
electric field and low levels of mechanical stress
exhibits characteristics of linear behavior, [1]. The
constitutive equations describing the linear behavior
of the piezoelectric material are based on the
assumption that the resulting deformation is
composed of the deformation caused by the
mechanical stress and the deformation caused by the
electric voltage applied to the electrodes of the
piezoelectric material.
2.1.1 Expression using σ and E
Using the symmetrical properties of tensor
quantities, we can express the constitutive equations
using the tensor of mechanical stress σ and vector of
electric field intensity E as [6]
(1)
(2)
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2024.19.9
Eva Kormanikova, Kamila Kotrasova
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2.1.2 Expression using є and E
The second variant of the constitutive law is
expressed using the overall deformation field є and
intensity of the electric field E as
(3)
(4)
where:
and are the components of the vector of
electric induction (electric displacement field) and
electric intensity, respectively.
- permittivity at constant stress
- permittivity at constant strain
- coefficients of the stress vector
- coefficients of strain vector
- coefficients of the tensor of elasticity
constants for the constant electrical field
, , , are piezoelectric coefficients.
For PZT material, that is polled in the z-direction,
the equations (1), (2) are written in the form
(5)
(6)
Constitutive equations (3-4) can be written in the
form
(7)
(8)
where
(9)
3 Finite Element Method
In FE analysis, the piezoelectric medium is divided
into several small discrete elements called finite
elements. Each element has a set of interconnecting
points on its edges called the nodes. The variables,
called the degrees of freedom (DOF), such as
displacement, potential, etc., at any arbitrary point
within an element are expressed in terms of their
values at the nodal points, using a suitable
polynomial interpolation function Ni defined for
each variable, [9], [10].
For an element with n nodes, the displacement
of a point (x, y, z), denoted by u(x,y,z), is expressed
in terms of the nodal displacement values ui(x,y,z)
by
(10)
(11)
(12)
Then for the strain vector can be written:
(13)
with the simplest notation
(14)
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where matrix contains submatrices Nx, Ny and Nz
that are derivations of shape function over x,y,z,
respectively
(15)
and
(16)
The electric field in terms of electric potential φ is
given by
(17)
or in the form
(18)
where is the electric potential.
The potential at the point (x,y,z), denoted by
φ(x,y,z), is expressed in terms of the nodal potential
values φi (x,y,z) by
(19)
and the Eq. 18 can be written
(20)
where matrix contains submatrices Nx, Ny and Nz
that are derivations of shape function over x,y,z,
respectively
(21)
and
(22)
For static analysis of virtual work of internal
forces and virtual work of external mechanical
forces can be written
(23)
where represents vector of external nodal forces
(24)
Similarly, for virtual internal work of the
electric field and external work of electric charge
can be written
(25)
where represents vector of external nodal
charges
By substituting the Eqs. 13 and 20 into Eqs. 7
and 8 we can write constitutive equations in the
form
(26)
(27)
For virtual strain and electric fields
(28)
E (29)
By substituting the Eqs. 26 - 29 into Eqs. 23 and 25
we get
(30)
where individual submatrices have the form
(31)
(32)
(33)
(34)
Solving Eq.30 yields the value of the
displacement u and the electrical potential φ at
various points in the piezoelectric medium. Using
the u and φ values, the strain and the electric field in
the medium can be computed from Eqs. 14 and 20.
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DOI: 10.37394/232011.2024.19.9
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4 Problem Solution
This paper deals with piezoelectric unimorph. The
piezoelectric unimorph is a cantilever beam
consisting of a single strip of piezoelectric material
clamped at one end and free at the other end (Figure
1). The bending of the cantilever on the application
of displacement is used for sensing analysis. The
unimorph is made of a thin film of piezoelectric
material formed on a non-piezoelectric substrate
material. The system is clamped at one end, and a
displacement is applied at the free end, and we
investigate the voltage and electric field results. The
unimorph is used in d33 mode (Figure 1). In d31
mode, the piezoelectric layer is poled in a direction
perpendicular to the plane (direction 3) as shown in
Figure 1. Due to the transverse piezoelectric effect,
the piezoelectric film expands in the transverse
direction, which is direction 1. This results in the
bending of the unimorph. Using
microelectromechanical systems (MEMS)
micromachining techniques, micron-size
piezoelectric unimorph sensors have been
fabricated. The piezoelectric films are deposited on
silicon oxide or silicon nitride substrates. The
piezoelectric material used is PZT. The surface
micromachining technique is used for the
fabrication of the cantilever structure.
Fig. 1: Piezoelectric unimorph cantilever beam in
d33 mode: the piezoelectric layer is poled in
direction 3, and the material is strained in the same
direction
The cantilever shown below in Figure 2 is
composed of two sublayers: the top sublayer is a
piezoelectric material, and the bottom sublayer is an
elastic CFRP material. The interface between the
two layers is grounded. The 3-D FE analysis of an
unimorph cantilever is described.
The dimensions of the cantilever are (Figure 2):
L 0.1m, W 0.01m, h1 0.001m, h2 0.004m. The
material properties of the bottom sublayer made of
8-layered [0/45/-45/90]S CFRP composite are: E1
140 GPa, E2 E3 12.3 GPa, G12 G13 = G23 = 6.5
GPa,
12
13
23 0.38. After homogenization,
[11], [12] of the sublayer: E = 56,352 GPa,
23
0.315.
The piezoelectric material properties are:
Elastic constants: E1 E2 = 61 GPa, E3 53.2 GPa,
12 0.35,
13
23 0.38, G12 22.593 GPa, G13 =
G23 = 21.1 GPa.
Coupling constants: e13 e23 7.209 N/Vm, e33
15.118 N/Vm, e51 e62 12.332 N/Vm (where 1=x,
2=y, 3=z, 4=xy, 5=xz, 6=yz)
Dielectric constants:
11
22 1.53x10-8 C/Vm,
33
1.5x10-8 C/Vm.
The polarization (P) of the piezoelectric material
is along the z-direction (Figure 2). All the nodes on
the grounded face are made to have zero potential.
A prescribed downward displacement of 0.005 m is
applied at the free end of the cantilever beam.
Fig. 2: Dimensions of the cantilever beam
The way of modeling the example is following:
Model definition by 3D solid element with
piezoelectric option,
Defining the piezoelectric material polarization
direction,
Defining the piezoelectric element group is a
mesh generation of elements in an element group,
Defining the material axis as the same with
global axes direction,
Defining the structural and piezoelectric
boundary conditions and loading in the form of
displacement,
Plotting voltage and electric field results.
In Figure 3 can be seen the original and
deformed cantilever beam under displacement
loading and cutting line ‘MIDLINE’ for the
variation of the electric field z along the line in
meshing form.
Fig. 3: Original and deformed cantilever beam under
displacement loading
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2024.19.9
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In Figure 4 can be seen the variation of the
electric field z along the line ‘MIDLINE’ through
the length of the beam. The maximum value of
electric field z is 2.20652.106 V/m at a distance of
2.08333 mm.
Fig. 4: Electric field z along the middle line of a
cantilever beam
Fig. 5: Voltage along the beam length
We used the loading in the form of
displacement and performed the sensing analysis.
Material properties of interest in the analysis of
piezoelectric materials are: i. Components of the
piezoelectric coefficient matrix, ii. Components of
dielectric constant matrix, iii. Components of elastic
stiffness or compliance constant matrix. The
homogenization of an 8-layered [0/45/-45/90]S
CFRP composite is done to obtain the effective
elastic properties. The piezoelectric medium
requires both mechanical and electrical constraints
to be specified: i. Mechanical constraints: The
mechanical DOF displacement components of all
the nodes on the clamped parts of the structure are
made zero. ii. Electrical constraints: The electrical
DOF potential on all the nodes on the electrode
faces of the piezoelectric material that need to be
earthed are made zero.
5 Conclusion
The linear 3D piezo-elasticity theory for static
ADINA FEM analysis of the piezoelectric
composite beam has been utilized for obtaining the
voltage and electric field z under vertical
displacement 0.005 m. The maximum value of
electric field z is 2.20652.106 V/m at a distance of
2.08333 mm (Figure 4). The maximum voltage -
2238 V (Figure 5) is at the clamped end of the
unimorph.
By changing the dimensions of the unimorph
and the clamping conditions, higher voltage can be
achieved at lower displacement, [13], [14], [15],
[16]. It would be interesting to apply the choice of
the numerical mesh size for instance by reporting a
convergence analysis on the number of elements
and the error analysis of the numerical method, that
we will investigate in the further research study.
Acknowledgments:
We thank the Scientific Grant Agency of the
Ministry of Education of the Slovak Republic and
the Slovak Academy of Sciences under Projects
VEGA1/0363/21, and VEGA 1/0307/23.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
- Eva Kormanikova carried out the theoretical
analysis, conception and writing the paper.
- Kamila Kotrasova carried out the numerical
analysis and numerical example by FEM.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
VEGA1/0363/21, VEGA 1/0307/23
Conflict of Interest
Eva Kormanikova reports financial support was
provided by Scientific Grant Agency of the
MSVVaS and the SAV.
Alternatively, in case of no conflicts of interest
the following text will be published:
The authors have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
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Creative Commons Attribution License 4.0
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