1. Introduction
With the rapid development of modern science and technology, there is a tendency toward ultra-precision and miniaturization in the smart material and smart structure research. There is a large market for smart devices at the nano-scale, when the flexoelectric effect, which is neglected at the macro-scale, plays an increasingly vital role due to its high electromechanical coupling. The flexoelectric effect is a kind of electromechanical coupling caused by strain gradients or non-uniform deformations [
1,
2,
3,
4], and this electromechanical phenomenon is size-dependent at the nano-scale [
5]. Therefore, it is essential to understand and analyze the flexoelectric effect in nanoscale materials and structures.
Researchers have carried out many studies on the size-dependent static and vibration behaviors in micro-nanostructures considering the flexoelectric effect. Zhang et al. [
5] established a Timoshenko dielectric beam model considering the direct flexoelectric effect and found that the deflection of cantilever and simply supported beams decreased with increasing beam thickness. Their results showed that flexoelectricity plays a major role in the electromechanical coupling response of piezoelectric beams when the beam thickness is at the nano-scale. Liang et al. [
6,
7] resolved and discussed the role of flexoelectric and surface effects in cantilever beam structures. Their results showed that surface and flexoelectric effects can reduce the bending deformation of the structure. Zhou et al. [
8] investigated the flexoelectric effects in piezoelectric nanobeams with three different electrical boundary conditions based on the classical Euler–Bernoulli beam model. They gave the analytical expression of the induced electric potential of flexoelectric beams. Sladek et al. [
9] analyzed curved nanoscale Timoshenko beams with the flexoelectric effect. The deflection, rotation, and induced electric intensity have been presented for various flexoelectric coefficients and the beam curvature. Park et al. [
10] designed and analyzed a structural model based on piezoelectric polymer (PVDF) film with strain gradient response, and further extended the application of dielectric materials to flexoelectric sensors. Malikan and Eremeyev [
11] modeled the dynamics of a visco-piezo-flexoelectric nanobeam considering a converse flexoelectric effect. Their results showed that the viscoelastic coupling will have an influence on the flexoelectricity property of the material. Yang and Zu [
12] investigated the flexoelectric effect on the natural frequency of the conventional cantilever beam harvesting structure with an end mass block. Yan [
13] elaborated the flexoelectric effect in composite flat plate harvesters by the weighted residual method based on the Kirchhoff plate theory. Liang et al. [
14] studied the buckling and vibration of flexoelectric nanofilms under the mechanical loading. Chang [
15,
16] used the differential quadrature method and the finite element method to study the longitudinal vibration of nanobeams with variable cross-sections, respectively. Lin et al. [
17] investigated the effects of end mass blocks and beam dimensions on the natural frequency and the effective frequency shift of a flexoelectric beam. Therefore, the study of flexoelectric structure properties is of directional guidance for the development and application of nanoresonators and nanosensors, etc.
Recently, there have been some research reports on nanostructures considering flexible substrates and magnetic fields. Hong [
18] examined the static bending and free vibration of piezoelectric functionally graded plates on a two-parameter elastic foundation. Baradaran et al. [
19] studied the surface effect on the static bending of nanowires on an elastic foundation. Ebrahimi and Barati [
20] evaluated the buckling of flexoelectric nanobeams with an elastic foundation based on non-local and surface elasticity theories. They found that the nanostructures could tolerate higher buckling loads due to the flexoelectric and surface effects at the nano-scale. Yinusa et al. [
21] analyzed the transverse and longitudinal vibrations and stability of the carbon nanotube in a magnetic environment. They determined that the magnetic term has a 20% attenuation or damping effect on the system vibration. Based on the variation method and the principle of minimum potential energy, Gobadi et al. [
22] studied the thermo-electro-magnetic mechanical behavior of flexoelectric nanoplates, in which the analytical solutions have been presented. Akgoz and Civalek [
23] analyzed the size-dependent stability of single-walled carbon nanotubes surrounded by a two-parameter elastic substrate. They found that increasing the Winkler and Pasternak parameters can increase the buckling load of carbon nanotubes. Jalaei et al. [
24] studied the transient response of viscoelastic functionally graded nanobeams under dynamic loads and magnetic fields. The results showed that the oscillation amplitude decreases while the number of periods of nanobeams increases by increasing the magnetic field and the length scale parameter. Barati [
25] analyzed the vibration characteristics of flexoelectric beams attached to a nonlinear foundation under the short-circuit electrical condition based on the surface elasticity and non-local elasticity theories. Recently, Xu et al. [
26] established a rectangular piezoelectric cantilever beam energy harvester with a copper substrate. Employing the finite element method, the influence of the copper substrate size on the output performance of piezoelectric harvesters was analyzed, and the optimal size of the substrate was obtained to achieve the maximum voltage output at a low frequency. The substrates and external fields have significant effects on the electromechanical properties of flexoelectric or piezoelectric actuators, sensors or harvesters. It is not clear how the coupling of substrate parameters and magnetic fields acts on the electromechanical responses of sensors or harvesters. However, to our knowledge, none of the previous studies mentioned how the linear elastic substrate and magnetic fields affect the static bending and vibration behaviors of flexoelectric sensors under the open-circuit electrical condition. The induced electric potential and the natural frequency under the OCI condition are very important performances of the flexoelectric sensors or energy harvesters.
The purpose of the present paper was to study the bending behavior and vibration properties of a flexoelectric cantilever beam attached to a linear elastic substrate under the OCI condition. Based on the electrical Gibbs free energy density and the Hamilton’s variational principle, the dynamic governing equations and the corresponding general boundary conditions were derived. Then, the characteristic equations of the natural frequency and the static electromechanical responses were further obtained. The bending behavior, vibration response, and the effects of linear elastic parameters and magnetic field on structural performance are discussed in detail.
2. Basic Theory of Flexoelectric Materials
Based on the electrical Gibbs free energy density function, according to the traditional piezoelectric theory, the basic theoretical model of the flexoelectric material is constructed with the interaction between the electric field and the strain gradient. Thereby, electrical Gibbs free energy density function
of the material can be written as:
where
, and
are the material property parameters, respectively.
denotes the dielectric coefficient,
denotes the modulus of elasticity,
is the piezoelectric coefficient,
denotes the positive flexoelectric coefficient, and
denotes the inverse flexoelectric coefficient.
is the electric field,
is the electric field gradient,
denotes the strain, and
denotes the strain gradient. Sharma et al. [
27] investigated the flexoelectric effect equivalent to piezoelectricity in bending film and defined
as the effective flexoelectric coefficient. Considering the effective flexoelectric coefficient, Equation (1) can be rewritten as:
Under the assumption of linear deformation, the expression between the strain and strain gradient and its displacement
is:
Correspondingly, the constitutive equations for flexoelectric materials can be further obtained under linear small deformation conditions:
where,
are the Cauchy stress tensor, the higher-order stress tensor, and electric displacement vector, respectively. Substituting Equations (5)–(7) into Equation (2), an alternative expression for the electrical Gibbs free energy density function is obtained:
3. Analysis Model of Flexoelectric Beams Based on Linear Elastic Substrates
The present research object is the flexoelectric cantilever beam structure with an end mass based on linear elastic substrates. The cantilever beam model is shown in
Figure 1. The length, width, and thickness of the cantilever beam are
,
, and
, respectively. The mass of the end mass block is
, where the end mass block is set as a cube and the side length is
. The force
F is applied at the end of the beam. The top and bottom surfaces of the beam are covered with electrodes, in which the thickness and stiffness of electrode layers would be neglected. This cantilever beam structure is connected to a linear elastic substrate and subjected to an in-plane axial magnetic field
. The linear elastic substrate can be simulated by a two-parameter linear elastic foundation model consisting of linear and shear layers. Here,
is the Pasternak constant, which describes the shear effect, and
is the Winkler constant, which describes the tensile effect.
The Hamilton’s variational expression for a flexoelectric cantilever beam is [
6]:
where
, and
are the total kinetic energy of the system, the total electrical Gibbs free energy, and the work done by the external loads, respectively. Equation (9) can be written as:
where
is the flexoelectric material density,
is the primary derivative of the absolute displacement of the flexoelectric beam with respect to time, where
.
denotes the transverse displacement of the neutral layer of the beam along the
direction, which is the deflection of the beam.
is the moment of inertia corresponding to the end mass block.
denotes the free charge density of the electrodes on the top and bottom surfaces of the beam.
is the electric potential on the surfaces generated by the bending of the cantilever beam.
and
are the volume and top and bottom surface areas of the beam, respectively.
is the work performed on the system by the external conditions (the substrate and magnetic field). For a planar beam deformation, combined with the Maxwell relation, the form of the Lorentz force generated under the action of the magnetic field can be expressed as:
[
28], where
is the Lorentz force,
is magnetic parliamentary, and
is the cross-sectional area of the beam. Combined with the two-parameter foundation model, the work variation of the external fields can be written as:
[
20,
25,
28].
According to the Euler–Bernoulli theory, the displacement expression of the flexoelectric beam is:
where
are the displacements along
and
directions, respectively, and
is the deflection of the beam structure as bending. The expression of strain and strain gradient of the flexoelectric cantilever beam is:
Under the condition of linear small deformation, the constitutive equations of the flexoelectric material can be further obtained:
The electric field
inside the cantilever beam can also be expressed as a negative gradient of the internal electric potential
along the thickness direction:
No free charge exists in flexoelectric cantilever beams, and thus, the electric displacement should satisfy the Gauss’s law, i.e.,
. After substituting Equation (15) into Gauss’s law, combined with Equation (16), we obtained:
Under the OCI condition, it is assumed that the electric potential on the top surface of the cantilever beam is
and the electric potential on the bottom surface is
. Then the electric potential difference between the top and bottom surfaces of the beam is
. Solving Equation (17) and combining with the electrical boundary conditions, the expression for the internal electric potential of the flexoelectric cantilever beam can be obtained:
where
is the potential value of the neutral axis of the beam with respect to
. By the above equation, the electric field
, stress
, higher order stress
, and electric displacement
can be obtained:
Substituting Equations (19)–(22) into the Equation (8), we can obtain the expansion of the electrical Gibbs free energy density function
of the cantilever beam structure:
is a functional expression related to time
t in the vibration. Therefore, the electrical Gibbs free energy density function could be expanded by the generalized variational method, i.e.,
where
is the effective bending rigidity of the piezoelectric nanobeam.
By using the parallel axis theorem, the expression for the moment of inertia of the end mass block can be obtained as:
Then, by substituting Equations (24) and (25) into Equation (9), the generalized Hamilton’s variational equation of the flexoelectric cantilever beam structure is obtained:
where
is the second derivative of the deflection
with respect to time,
is the Diracdelta function. The following variational expansion is applied in the further derivation of Equation (26):
3.1. The Bending Response of Flexoelectric Beams Attached to Linear Elastic Substrates
The top and bottom surfaces of a flexoelectric cantilever beam are covered with electrodes, thus the top and bottom surfaces are electrical equipotential bodies and the electric potential difference
is a function independent of
. In the analysis of the beam structure statically, let
t = 0 and
= 0. For any
, Equation (26) is satisfied, in which the electromechanical coupling governing equation and boundary conditions of the flexoelectric cantilever beam under the OCI condition can be obtained:
Similarly, for Equation (26),
can be chosen arbitrarily, and there is the following relation:
Under OCI conditions, the charges on the bending surfaces of the flexoelectric beam are redistributed, however, the total surface free charge or the surface electric displacement of the flexoelectric beam is zero [
8,
29,
30]. Thus, the electrical boundary condition expression is:
So, the electric potential could be obtained for the flexoelectric cantilever beam.
Therefore, the expressions of the deflection and induced electric potential as only force
(case I) are:
The expressions of the deflection and induced electric potential as the force
and the magnetic field (case II) are:
where
.
Based on a linear elastic substrate, the expressions of the deflection and induced electric potential as only force
(case III) are:
where:
Based on a linear elastic substrate, the expressions of the deflection and induced electric potential acting on the force and the magnetic field (case IV) are the same as in the case III, with the difference of the parameter which should be changed to .
3.2. The Vibration Response of Flexoelectric Beams Attached to Linear Elastic Substrates
When the vibration characteristics of the beam structure are analyzed,
t ≠ 0 and
F = 0 have been set. Therefore, Equation (26) can be satisfied for any choice of
, so that the dynamic governing equation of the flexoelectric cantilever beam system under the OCI condition can be obtained:
where
denotes the mass per unit length of the flexoelectric beam.
Correspondingly, the dynamic induced electric potential could be expressed as
. From Equation (26), the dynamic mechanical boundary conditions of the flexoelectric beam based on the linear elastic substrate under the OCI condition can be obtained:
By the separated variables method, the characteristic equation for the natural frequency of the flexoelectric energy harvester based on a linear elastic substrate under the OCI condition can be solved. The solution of this equation can be set according to the form of Equation (38) as [
31,
32]:
where
denotes the modal vibration pattern,
are the eigenvalues of the structural vibration, the five parameters
,
,
, and
are independent of
and
,
denotes the generalized coordinate, and
is the imaginary root. After substituting Equations (40) and (41) into Equations (38) and (39), the dynamic governing equation and the corresponding boundary conditions of the cantilever beam structure can be obtained:
The circular frequency expression of the natural vibration of the system can be obtained from Equation (42):
. Then, the following relation can be obtained after substituting Equation (41) into Equation (43):
To ensure that the system of homogeneous Equation (44) has non-zero solutions, the coefficient determinant corresponding to the equations must be zero. By solving and simplifying the matrix determinant, the characteristic equation for the natural frequency of this flexoelectric energy harvester under the OCI condition can be obtained as:
where
Equation (46) is a transcendental equation with respect to the eigenvalues . The analytical solutions of cannot be written exactly by conventional methods, but it is possible to obtain a series of values of by the numerical method. Then, the natural frequency in the OCI condition could be obtained. Under the short-circuit electrical condition, the surface induced electric potential of the flexoelectric energy harvester is zero. Hence, the natural frequency of the beam structure can be obtained by the same method of setting in the corresponding boundary condition.