Analysis of Vibration Energy Harvesting Performance of Thermo-Electro-Elastic Microscale Devices Based on Generalized Thermoelasticity

Abstract

Piezoelectric material structures with an excellent mechatronic coupling property effectively promote self-power energy harvesting in micro-/nano-electro-mechanical systems (MEMS/NEMS). Therein, the characteristics of the microscale and multi-physical aspects effect significant influence on performance, such as attaining a fast response and high power density. It is difficult to use the classical mechanical and heat conduction models to effectively explain and analyze microscale physical field coupling behaviors. The purpose of this study is to develop the piezoelectric thermoelastic theoretical model, firstly considering the non-uniform physical field. The generalized equations governing thermo-electro-elastic vibration energy harvesting in a microbeam model were obtained based on Hamilton’s principle and the generalized thermoelastic theory was developed by considering thermopolarization and thermal hysteresis behavior. After that, the explicit expressions for voltage and output power were derived using the assumed-modes method; meanwhile, effects such as the piezo-flexoelectric aspect, size dependence, etc. are discussed in detail. It was found that thermal and microscale effects significantly promote the voltage and output power. The research is also helpful for the design and optimization of self-powered and high-performance micro/nano devices and systems.

1. Introduction

Since the requirements of intelligence and rapid response were introduced, against the backdrop of the low-carbon and advanced manufacturing technology industries, harvesting renewable energy or waste energy into electrical energy from ambient environments for low- and self-powered devices such as actuators and sensors has been attracting enormous attention [1,2]. At present, piezoelectric material structures with an excellent mechatronic coupling property have been most widely adopted on microscale devices for energy harvesting, such as the galloping energy harvester [3]. In practice, there are several critical issues surrounding piezoelectric energy harvesting when applied in micro/nano systems and devices, in which the significant microscale effects and multi-physics field-coupling characteristics are exhibited in all mechanical mechanism and dynamical behaviors [4,5,6].

Among them, the piezoelectric materials show novel mechanical-electrical coupling characteristics at micro/nanoscale, in which the inhomogeneous strain field can be induced in addition to polarization and vice versa, and the inhomogeneous strain field or strain gradient generate a flexoelectric effect [7,8]. In addition, the flexoelectric effect appears in all dielectrics including centrosymmetric materials and shows a significant size-dependent property. In view of this, the outstanding properties and great application potential in the field of flexoelectricity have rekindled considerable research interest [9,10], in which the dielectric materials with the special electromechanical coupling property are further opening up new avenues in microscale device energy harvesting. Regarding piezo-flexoelectric energy harvesting, Yan et al. investigated flexoelectric nanobeam energy harvesting by considering a non-uniform geometric model via a finite element approach [11], and Shen and Deng [12] proposed a computational approach for flexodynamic effect analysis of nanoscale flexoelectric energy harvesting and found that the flexodynamic effect had a negative effect on energy harvesting efficiency. Considering the size dependence influence, the effects of strain gradient elasticity and structure size on the electromechanical response of flexoelectric energy harvesting were discussed in detail [13,14]. For the influence of loads and constraints on flexoelectric energy harvesting, Najar et al. took into account large displacements and restrictive boundary conditions and indicated that the better power density is obtained by larger scaling factors [15]. Zhang et al. studied electromechanical coupling responses by considering harmonic excitations with a proof mass on a cantilevered energy harvesting model [16]. Yvonnet et al. [17] evaluated the sensitivity of different parameters on energy harvesting by considering dynamic loads. From the above research results, it is indicated that the microscale effects have an essential influence on flexoelectric energy harvesting.

Moreover, multi-physics field coupling behaviors play a crucial role, especially in extreme or complex working environments in systems and devices in which thermal effects are a key challenge [18]. Generally, the involved thermoelastic coupling response is only approximately described by the classical Fourier heat conduction law, which predicts the temperature via infinite velocity propagation, which is in contradiction with real physical phenomena. Actually, heat has a finite thermal wave spread for fast-transient and laser processing etc. [19,20], especially under extreme heating conditions and at micro-and nanoscale dimensions [21]. To overcome this shortcoming, the generalized heat conduction models have been established successively, i.e., the C-V thermal wave model [19,20], dual-phase-lagging (DPL) model [22] and so on [23,24], in which the thermal hysteresis behavior has been depicted. Meanwhile, regarding thermo-electro-elastic coupling behaviors, the coupled piezo-thermoelastic theory was first proposed by Mindlin [25] based on the Fourier model, and then Chandrasekharaiah generalized and developed the piezo-thermoelastic theory by involving thermal relaxion time [26]. After that, He and Ma [27] formulated the piezoelectric-thermoelastic dynamic model based on generalized thermoelastic theory and discussed the multi-field dynamic response. Li et al. [28] developed piezoelectric thermoelastic theory within the extended thermodynamic framework and considered the memory-dependence feature. Beni et al. attempted to develop a generalized theory of flexothermoelasticity based on LS thermoelasticity and extracted constitutive relations [3]. Thermoelastic damping, considering piezoelectro-magneto-thermoelasticity in a composite microbeam model, was investigated by Singh et al. [29]. Srivastava and Mukhopadhyay investigated the thermoelastic vibration responses on a piezo-thermoelastic microbeam resonator by considering a DPL heat conduction model [30]. In addition, thermopolarization by depicting temperature polarization and gradient further reflects the thermodynamic properties in dielectrics [31,32] and has attracted attention. Among such studies, Yu et al. [33] considered the thermopolarization effect and constructed a thermo-electro-elastic coupling model based on a piezoelectric hollow cylinder, in which they found the thermopolarization coefficient can significantly regulate the stress distribution. Hrytsyna et al. developed a higher-grade theory and discussed the influence of the dynamic flexoelectric effect on a dielectric solids [34]. However, according to the current research, most investigations are initially devoted to the piezoelectric-thermoelastic theory, and there are few studies which focus on the thermo-electro-elastic coupling theory in the microscale multi-physics field.

Thus, with the advancement of piezoelectric-thermoelastic theory, the coupled thermo-electro-elastic vibration energy harvesting in micro/nano systems and devices has attracted widespread attention in various fields. Among others, Liu et al. [35] investigated the influence of thermoelastic dissipation on the piezoelectric vibration harvesting numerical calculation and experiment and discussed the temperature field influence on the piezoelectric structure. Zhang et al. [36] studied thermo-electric-elastic coupling in piezoelectric vibration energy harvesting by considering external heat flux and mechanical force load, and a model was developed using Green’s method. Considering microscale and thermoelastic coupling effects, Gu et al. [37] formulated the thermo-electric-elastic constitutive model firstly, in which the size-dependent and thermoelastic coupling effects on voltage and output power were systematically analyzed and discussed. The above research results reflect that the size dependence and multi-physics characteristics have significant influences on the efficiency of piezoelectric materials in vibration energy harvesting in microscale systems and devices.

Actually, understanding and interpreting microscale piezoelectric material structures vibration energy harvesting under complex physical circumstances is still imperfect based on existing theoretical frameworks. Besides that, to the best of the authors’ knowledge, the use of the existing piezoelectric-thermoelastic theory analysis model, considering size dependence and thermal relaxation effects, is extremely rare. The current work aims to generalize the thermo-electro-elastic theory to reveal the flexoelectric vibration energy harvesting performances in microscale systems and devices. Among them, the generalized piezoelectric thermoelastic theoretical model considering piezo-flexoelectricity and thermopolarization effects will be developed based on size dependence and non-Fourier heat conduction theories firstly. Furthermore, the governing equations of vibration energy harvesting in the simplified microbeam model in microscale devices will be derived by Hamilton’s principle and generalized thermoelastic theory. The effects of flexoelectricity, thermopolarization and microscale behaviors on the influences of the voltage, output power and deflection will be discussed. It is hoped that the present developed theoretical model and analysis results can help functional design and efficiency optimization in energy harvesting from microscale systems and devices.

2. The Establishment of Theory

2.1. Thermo-Electro-Elastic Constitutive Model

This section is devoted to developing the generalized piezoelectric thermoelastic theoretical model of microscale energy harvesting considering the thermo-electro-elastic multi-physical field coupling effect. For isotropic centrosymmetric crystalline flexoelectric dielectrics, the generalized internal energy density function $U$ based on the extended liner piezo-thermoelastic theory can be defined as

$$\begin{gathered} U={\displaystyle \frac{1}{2}}{c}_{ijkl}{\epsilon }_{ij}{\epsilon }_{kl}+{\displaystyle \frac{1}{2}}{a}_{ij}{P}_i{P}_j-{\displaystyle \frac{\rho c_E}{{2T}_0}}{\theta }^2+d_{ijk}{\epsilon }_{ij}{P}_k+f_{ijkl}{\epsilon }_{ij,k}{P}_l \\ -{\beta }_{ij}{\epsilon }_{ij}\theta -p_i\theta P_i-b_{ij}{\theta }_{,j}{P}_i+{\displaystyle \frac{1}{2}}{g}_{ijklmn}{\epsilon }_{ij,k}{\epsilon }_{lm,n} \end{gathered}$$

(1)

where ${\epsilon }_{ij}$ and ${\epsilon }_{ij,k}$ are the components for strain and strain gradient tensor, $P_i$ is the component of polarization vector, $\theta =T-T_0$ denotes temperature increment, and $T$ and $T_0$, respectively, represent absolute temperature and initial temperature. ${\theta }_{,j}$. $c_{ijkl}$, $d_{ijk}$ and $a_{ij}$ are the components of higher-order temperature, elastic constant, piezoelectric coefficient and reciprocal dielectric susceptibility. $c_E$, ${\beta }_{ij}$ and $p_i$ are the heat capacity, thermal expansion coefficient and pyroelectric coefficient, respectively. $\rho$ is density. In addition, $f_{ijkl}$, $b_{ij}$ and $g_{ijklmn}$, respectively, represent the flexoelectric coefficient, thermopolarization coefficient and strain gradient coupling coefficient. Correspondingly, the strain and strain gradient components satisfy the following relations:

$${\epsilon }_{ij}={\displaystyle \frac{1}{2}}\left(u_{i,j}+u_{j,i}\right), {\epsilon }_{ij,k}={\displaystyle \frac{1}{2}}\left(u_{i,jk}+u_{j,ik}\right)$$

(2)

where $u_i$ denotes the displacement component.

Through the free energy density function $\left(1\right)$, according to Toupin’s variational principle [38,39], the constitutive equations of the microscale piezoelectric material structure can be derived as

$$\begin{gathered} {\sigma }_{ij}={\displaystyle \frac{\partial U}{\partial {\epsilon }_{ij}}}=c_{ijkl}{\epsilon }_{kl}+d_{ijk}{P}_k-{\beta }_{ij}\theta \\ {\sigma }_{ij,k}^{\left(1\right)}={\displaystyle \frac{\partial U}{\partial {\epsilon }_{ij,k}}}=f_{ijkl}{P}_l+g_{ijklmn}{\epsilon }_{lm,n} \\ -E_i={\displaystyle \frac{\partial U}{\partial P_i}}=a_{ij}{P}_j+d_{ijk}{\epsilon }_{jk}+f_{ijkl}{\epsilon }_{jk,l}-p_i\theta -b_{ij}{\theta }_{,j} \end{gathered}$$

(3)

where ${\sigma }_{ij}$, ${\sigma }_{ij,k}^{\left(1\right)}$ and $E_i$, respectively, refer to the classical Cauchy stress, the high-order local stress and the local electric field components. Additionally, according to the nonlocal strain gradient theory [40], the total stress $t_{ij}$ on the microscale material structure is caused by classical stress ${\sigma }_{ij}$ and higher-order stress, and these satisfy the linear relationship

$$t_{ij}={\sigma }_{ij}-{\sigma }_{ij,k}^{\left(1\right)}=c_{ijkl}\left(1-l^2{\nabla }^2\right){\epsilon }_{kl}-f_{ijkl}{P}_l+d_{ijk}{P}_k-{\beta }_{ij}\theta$$

(4)

in which $l$ is the strain gradient length scale parameter and $l=\sqrt{g_{ijklmn}/c_{ijkl}}$ [13,15].

2.2. Generalized Thermoelastic Theory Model

For microscale material structures and some extreme situations in heat transport, the traditional Fourier’s law does not apply and a non-Fourier heat conduction model has been proposed: the dual phase-lag (DPL) theory in which two different delayed responses (time-delay parameters) [22] are present, i.e.,

$$\overline{q}\left(x,t+{\tau }_q\right)=-\kappa \nabla \theta \left(x,t+{\tau }_T\right)$$

(5)

where $\overline{q}$ and $\kappa$, respectively, are heat flux and thermal conductivity; ${\tau }_q$ and ${\tau }_T$, respectively, refer to the phase lag of heat flux and the phase lag of temperature associated with the time microscale effect.

Subsequently, considering thermoelastic behavior and combining the piezoelectric materials [26,41], the entropy equation is given as follows:

$$\rho s={\displaystyle \frac{\rho C_E}{T_0}}\theta +{\beta }_{ij}{\epsilon }_{ij}+p_i{E}_i$$

(6)

where $s$ is entropy per unit mass. In addition, considering the energy equation

$$-\nabla ·\overline{q}=\rho T_0{\displaystyle \frac{\partial s}{\partial t}}-\rho Q$$

(7)

${\epsilon }_{mm}$ denotes the volumetric strain and $Q$ is the heat source. In Simultaneous Equations (5)–(7), the generalized piezoelectric heat conduction model can be expressed as:

$$\kappa \left(1+{\tau }_T{\displaystyle \frac{\partial }{\partial t}}\right){\nabla }^2\theta =\left(1+{\tau }_q{\displaystyle \frac{\partial }{\partial t}}\right)\left(\rho C_E{\displaystyle \frac{\partial \theta }{\partial t}}+T_0{\beta }_{ij}{\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial t}}+T_0{p}_i{\displaystyle \frac{\partial E_i}{\partial t}}-\rho Q\right)$$

(8)

3. Modeling of Microbeam Structures

To clarify the microscale piezoelectric material structure vibration energy harvesting mechanism, the rectangular cantilever microbeam model is considered as shown in Figure 1. The construction of this model considers a small flexural deflection, and the length, thickness and width, respectively, are $L \left(0\le x\le L\right)$, $h$ $\left(-h/2\le z\le h/2\right)$ and $b \left(0\le y\le b\right)$. In addition, the $x$-axis and $z$-axis, respectively, are positioned along the axis directions of the micro-beam and thickness direction. It is assumed that the microbeam model is initially at the initial temperature $T_0$ and has an axisymmetric structure.

For the current problem, we consider the slender beam model, i.e., $h\ll L$, and assume that the displacement, strain, stress, heat flux, electric field and electric displacement components depend just on the length $x$ and time $t$. At this point, following the Euler–Bernoulli beam model hypothesis, the displacement fields can be expressed as:

$$u_z\left(x,y,z,t\right)=w\left(x,t\right), u_x\left(x,y,z,t\right)=-z{\displaystyle \frac{\partial w}{\partial x}}, u_y\left(x,y,z,t\right)=0$$

(9)

Accordingly, the strain and strain gradient components, ignoring third-order and higher-order differential terms of small deflection, can be written as:

$${\epsilon }_{xx}=-z{\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}, {\epsilon }_{xx,z}=-{\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}$$

(10)

Additionally, considering the thickness is smaller than the length on the model, it is assumed that the electric field correspondingly exists only in the thickness direction, i.e., $E_x=E_y=0$ [42]. Accordingly, the constitutive Equation (3) can be re-expressed as:

$$\begin{gathered} {\sigma }_{xx}=c_{1111}{\epsilon }_{xx}+d_{113}{P}_z-{\beta }_{11}\theta \\ {\sigma }_{xx,z}=f_{1133}{P}_z+g_{113113}{\epsilon }_{xx,z} \\ -E_z=a_{33}{P}_z+d_{113}{\epsilon }_{xx}+f_{1133}{\epsilon }_{xx,z}-p_1\theta -b_{11}{\theta }_{,z} \end{gathered}$$

(11)

Additionally, the total stress can be rewritten as:

$$t_{xx}=c_{1111}\left(1-l^2{\nabla }^2\right){\epsilon }_{xx}-f_{1133}{P}_z+d_{113}{P}_z-{\beta }_{11}\theta$$

(12)

Thereupon, the internal energy density $U$ can be simplified as:

$$\begin{gathered} U={\displaystyle \frac{1}{2}}{c}_{1111}{\epsilon }_{xx}^2+{\displaystyle \frac{1}{2}}{a}_{33}{P}_z^2-{\displaystyle \frac{\rho c_E}{{2T}_0}}{\theta }^2+d_{113}{\epsilon }_{xx}{P}_z+f_{1133}{\epsilon }_{xx,z}{P}_z \\ -{\beta }_{11}{\epsilon }_{xx}\theta -p_1\theta P_z-b_{11}{\theta }_{,z}{P}_z+{\displaystyle \frac{1}{2}}{g}_{113113}{\epsilon }_{xx,z}^2 \end{gathered}$$

(13)

In the absence of free electric charge, Gauss’s law can be expressed as:

$$-{\epsilon }_0{\displaystyle \frac{{\partial }^2\varnothing }{\partial z^2}}+{\displaystyle \frac{\partial P_z}{\partial z}}=0$$

(14)

where ${\epsilon }_0=8.854\times {10}^{-12} \mathrm{C}{\mathrm{V}}^{-1}{\mathrm{m}}^{-1}$ refers to permittivity of the medium and ∅ denotes the electric potential of the electric field. In addition, the electric field is assumed to exist only in the $z$ direction, so it can be obtained

$$E_z=-{\displaystyle \frac{\partial \varnothing }{\partial z}}$$

(15)

considering the electric boundary conditions $E_z=-{\displaystyle \frac{v\left(t\right)}{h}}\left(H\left(x\right)-H\left(x-L\right)\right)$ with the top-to-bottom surfaces of the electric field, in which $v\left(t\right)$ is the generated voltage and $H\left(x\right)$ is the Heaviside function. Therefore, the electric polarization $P_z$ and the electric field $E_z$ can be obtained

$$P_z={\displaystyle \frac{d_{113}}{a_{33}}}z{\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}+{\displaystyle \frac{p_1}{a_{33}}}\theta +{\displaystyle \frac{b_{11}}{a_{33}}}{\theta }_{,z}+{\displaystyle \frac{f_{1133}}{a_{33}}}{\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}-{\displaystyle \frac{1}{a_{33}}}{\displaystyle \frac{v\left(t\right)}{h}}$$

(16)

$$E_z=-{\displaystyle \frac{d_{113}}{a_{33}}}z{\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}-{\displaystyle \frac{v\left(t\right)}{h}}\left(H\left(x\right)-H\left(x-L\right)\right)$$

(17)

Subsequently, the stress and the higher-order stress term of Equation (13) based on Equations (16) and (17) can be obtained.

Based on this process, to derive the corresponding motion equation and the associated boundary conditions of the flexural vibration of the piezoelectric microbeam model, the extended Hamilton’s principle can be adopted, i.e.,

$$\delta {\int }_{t_1}^{t_2}\left(K-{\int }_{\mathsf{\Omega }}H\mathrm{d}\mathsf{\Omega }+W_{nc}\right)\mathrm{d}t=0$$

(18)

in which $\mathsf{\Omega }$ refers to the entire volume occupied. It should be pointed out that $H=U-{\displaystyle \frac{1}{2}}{\epsilon }_0{\varnothing }_{,z}^2+P_z{\varnothing }_{,z}$ is the electric enthalpy density. Meanwhile, $K=-{\displaystyle \frac{1}{2}}{\int }_0^L\rho A\left(\ddot{w}+{\ddot{w}}_b\right)\mathrm{d}x$ is the kinetic energy, $A$ is cross-sectional area and ${\ddot{w}}_b$ refers to the based acceleration. In addition, $W_{nc}={\int }_0^L{F}_{\mathrm{N}}\delta \left(x-L_F\right)\delta w\mathrm{d}x$ is the work done by nonconservative forces, in which $F_{\mathrm{N}}$ is the nonconservative forces at position $x=L_F$.

Just to make it clear, the electric enthalpy density on the variation can be expressed as:

$$\begin{gathered} \delta H=\left(a_{33}{P}_z-d_{113}z{\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}-p_1\theta -b_{11}{\theta }_{,z}-f_{1133}{\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}+{\displaystyle \frac{v\left(t\right)}{h}}\right)\delta P_z \\ -\left(-{\beta }_{11}z{\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}+p_1{P}_z+{\displaystyle \frac{\rho c_E}{T_0}}\theta \right)\delta \theta -b_{11}{P_z\delta \theta }_{,z}+\left(P_z-{\epsilon }_0{\displaystyle \frac{v\left(t\right)}{h}}\right)\delta \left({\displaystyle \frac{\partial \varnothing }{\partial z}}\right) \\ +\left(z^2{c}_{1111}{\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}-z{d}_{113}{P}_z+z{\beta }_{11}\theta -f_{13}{P}_z+l^2{c}_{1111}{\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}\right)\delta \left({\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}\right) \end{gathered}$$

(19)

As a result, the potential energy related term can be obtained as

$$\begin{gathered} {\int }_V\delta H\mathrm{d}V={\int }_V\left(\left(a_{33}{P}_z-d_{113}z{\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}-p_1\theta -b_{11}{\theta }_{,z}-f_{1133}{\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}+{\displaystyle \frac{v\left(t\right)}{h}}\right)\delta P_z\right. \\ -\left(-{\beta }_{11}z{\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}+p_1{P}_z+{\displaystyle \frac{\rho c_E}{T_0}}\theta \right)\delta \theta -b_{11}{P}_z\delta {\theta }_{,z}-{\displaystyle \frac{\partial }{\partial z}}\left(P_z-{\epsilon }_0{\displaystyle \frac{v\left(t\right)}{h}}\right)\delta \varnothing \\ \left.+{\displaystyle \frac{{\partial }^2}{\partial x^2}}\left(z^2{c}_{1111}{\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}-z{d}_{113}{P}_z+z{\beta }_{11}\theta -f_{1133}{P}_z+l^2{c}_{1111}{\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}\right)\delta w\right)\mathrm{d}V \\ +\left[{\int }_A\left(\left(z^2{c}_{1111}{\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}-z{d}_{113}{P}_z+z{\beta }_{11}\theta -f_{1133}{P}_z+l^2{c}_{1111}{\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}\right)\right.\delta \left({\displaystyle \frac{\partial w}{\partial x}}\right)\right. \\ {\left.\left.-{\displaystyle \frac{\partial }{\partial x}}\left(z^2{c}_{1111}{\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}-z{d}_{113}{P}_z+{\beta }_{11}\theta -f_{1133}{P}_z+l^2{c}_{1111}{\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}\right)\delta w\right)\mathrm{d}A\right]}_0^L \\ +{\left[{\int }_A\left(P_z-{\epsilon }_0{\displaystyle \frac{v\left(t\right)}{h}}\right)\delta \varnothing \mathrm{d}A\right]}_{-{\displaystyle \frac{h}{2}}}^{{\displaystyle \frac{h}{2}}} \end{gathered}$$

(20)

According to the above extended Hamilton’s principle, the relationship of electric polarization $P_z$ considering the thermos polarization effect based on the arbitrary value of $\delta P_z$ can also be obtained, which is consistent with Equation (16). Meanwhile, the motion equation obtained by considering $\delta w$ is arbitrary and in combination with Equation (16) can be derived as

$$\begin{gathered} {\int }_0^L\left(\rho A{\displaystyle \frac{{\partial }^2}{\partial t^2}}\left(w+w_b\right)+{\displaystyle \frac{{\partial }^2}{\partial x^2}}\left(\left(c_{11}-{\displaystyle \frac{d_{13}^2}{a_{33}}}\right)I{\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}-\left({\displaystyle \frac{d_{13}{p}_1}{a_{33}}}-{\beta }_{11}\right)M_T-2S_y\right.\right. \\ {\displaystyle \frac{f_{13}{d}_{113}}{a_{33}}}{\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}-{\displaystyle \frac{f_{13}{p}_1}{a_{33}}}{\int }_A\theta \mathrm{d}A-{\displaystyle \frac{f_{13}{b}_{11}}{a_{33}}}{\int }_A{\theta }_{,z}\mathrm{d}A-S_y{\displaystyle \frac{f_{13}{b}_{11}}{a_{33}}}{\theta }_{,z}+\left(l^2{c}_{11}-{\displaystyle \frac{f_{13}^2}{a_{33}}}\right) \\ A{\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}-\left(S_y{\displaystyle \frac{d_{13}}{a_{33}}}+A{\displaystyle \frac{f_{13}}{a_{33}}}\right)\left.{\displaystyle \frac{v\left(t\right)}{h}}\left(H\left(x\right)-H\left(x-L\right)\right)\right)-F_{\mathrm{N}}\delta \left(x-L_F\right))\mathrm{d}x=0 \end{gathered}$$

(21)

where

$$\begin{gathered} I={\int }_A{z}^2\mathrm{d}A \\ {S}_y={\int }_{A}z\mathrm{d}A \\ {M}_T={\int }_{A}z\theta \mathrm{d}A \end{gathered}$$

(22)

In addition, considering the previous assumptions on the neutral symmetry microbeam model, it can be set $c_{1111}=c_{11}$, $d_{113}=d_{13}$, $f_{1133}=f_{13}$ and result in $S_y=0$. Therefore, the motion governing Equation (21) can be simplified as

$$\begin{gathered} {\int }_0^L\left(m{\displaystyle \frac{{\partial }^{2}w}{\partial t^2}}+E{I}^{*}{\displaystyle \frac{{\partial }^{4}w}{\partial x^4}}\right.-{\mathsf{{\rm Y}}}_1{\displaystyle \frac{{\partial }^2{M}_T}{\partial x^2}}-{\mathsf{{\rm Y}}}_2{\int }_A{\displaystyle \frac{{\partial }^2\theta }{\partial x^2}}\mathrm{d}A-{\mathsf{{\rm Y}}}_3{\int }_A{\displaystyle \frac{{\partial }^2{\theta }_{,z}}{\partial x^2}}\mathrm{d}A \\ -{\mathsf{{\rm Y}}}_4{\displaystyle \frac{v\left(t\right)}{h}}\left({\delta }^{\prime }\left(x\right)-{\delta }^{\prime }\left(x-L\right)\right)-F_{\mathrm{N}}\delta \left(x-L_F\right))\mathrm{d}x=-m{\int }_0^L{\displaystyle \frac{{\partial }^2{w}_b}{\partial t^2}}\mathrm{d}x \end{gathered}$$

(23)

in which the volume mass $m=\rho A$ and the relevant parameter $E{I}^{\mathrm{*}}=\left(c_{11}-{\displaystyle \frac{d_{13}^2}{a_{33}}}\right)I+\left(l^2{c}_{11}-{\displaystyle \frac{f_{13}^2}{a_{33}}}\right)A$, ${\mathsf{{\rm Y}}}_1=\left({\displaystyle \frac{d_{13}{p}_1}{a_{33}}}-{\beta }_{11}\right)$, ${\mathsf{{\rm Y}}}_2={\displaystyle \frac{f_{13}{p}_1}{a_{33}}}$, ${\mathsf{{\rm Y}}}_3={\displaystyle \frac{f_{13}{b}_{11}}{a_{33}}}$, ${\mathsf{{\rm Y}}}_4=A{\displaystyle \frac{f_{13}}{a_{33}}}$.

For energy harvesting applications, the circuit governing equation should be considered; according to the electric displacement along the $z$-direction, the average electric displacement ${\overline{D}}_z$ can be expressed as

$${\overline{D}}_z={\displaystyle \frac{1}{h}}{\int }_V{D}_z\mathrm{d}V$$

(24)

in which $D_z=P_z-{\epsilon }_0{\varnothing }_{,z}$ denotes electric displacement for the microbeam; in addition, the electric current equation of electric displacement can be obtained

$$i\left(t\right)={\displaystyle \frac{v\left(t\right)}{R}}\begin{array}{c}={\displaystyle \frac{1}{h}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left({\int }_V\left(P_z-{\epsilon }_0{\varnothing }_{,z}\right)\mathrm{d}V\right)\end{array}$$

(25)

where $R$ is the external electrical load resistance. Therefore, using Equation (25), and considering Equations (15)–(17), the electric current governing equation can be developed

$${\displaystyle \frac{v\left(t\right)}{R}}=-{\mathsf{\Pi }}_1{\displaystyle \frac{\mathrm{d}v\left(t\right)}{\mathrm{d}t}}+{\mathsf{\Pi }}_2{\int }_0^L{\displaystyle \frac{{\partial }^{3}w}{\partial x^2\partial t}}\mathrm{d}x+{\mathsf{\Pi }}_3{\int }_V{\displaystyle \frac{\mathrm{d}\theta }{\mathrm{d}t}}\mathrm{d}V+{\mathsf{\Pi }}_4{\int }_V{\displaystyle \frac{\mathrm{d}{\theta }_{,z}}{\mathrm{d}t}}\mathrm{d}V$$

(26)

where the relevant parameter ${\mathsf{\Pi }}_1={\displaystyle \frac{bL}{h}}\left({\displaystyle \frac{1}{a_{33}}}+{\epsilon }_0\right)$, ${\mathsf{\Pi }}_2={\displaystyle \frac{{bf}_{13}}{a_{33}}}$, ${\mathsf{\Pi }}_3={\displaystyle \frac{p_1}{h{a}_{33}}}$, ${\mathsf{\Pi }}_4={\displaystyle \frac{b_{11}}{h{a}_{33}}}$.

Additionally, according to Equation (8), the generalized governing equation for piezoelectric heat conduction based on the microbeam model assumption can be derived

$$\begin{gathered} \kappa \left(1+{\tau }_T{\displaystyle \frac{\partial }{\partial t}}\right){\displaystyle \frac{{\partial }^2\theta }{\partial z^2}}=\left(1+{\tau }_q{\displaystyle \frac{\partial }{\partial t}}\right)\left({\mathsf{\Lambda }}_1{\displaystyle \frac{\partial \theta }{\partial t}}-\right.{\mathsf{\Lambda }}_{2}z{\displaystyle \frac{{\partial }^{3}w}{\partial x^2\partial t}}+{\mathsf{\Lambda }}_3{\displaystyle \frac{\partial {\theta }_{,z}}{\partial t}} \\ \left.-{\mathsf{\Lambda }}_5{\displaystyle \frac{\partial v\left(t\right)}{\partial t}}\left(H\left(x\right)-H\left(x-L\right)\right)-\rho Q\right) \end{gathered}$$

(27)

where the relevant parameter ${\mathsf{\Lambda }}_1=\rho C_E$, ${\mathsf{\Lambda }}_2=T_0\left({\beta }_{11}-p_1{\displaystyle \frac{d_{13}}{a_{33}}}\right)$, ${\mathsf{\Lambda }}_3={\displaystyle \frac{{T_0{p}_{1}b}_{11}}{a_{33}}}$, ${\mathsf{\Lambda }}_5={\displaystyle \frac{T_0{p}_1}{a_{33}h}}$.

Considering the microscale devices’ internal multi-field coupling processes, the temperature variation is assumed to be in terms of a $\mathrm{s}\mathrm{i}\mathrm{n}(z\pi /h)$ function along the thickness direction [43]; it can be given by

$$\theta \left(x,z,t\right)=\phi \left(x,t\right)\sin \left({\displaystyle \frac{z\pi }{h}}\right)$$

(28)

Naturally, ignoring the heat source effect, the governing equations involving flexoelectric and generalized thermoelasticity on energy harvesting from dielectric material structures can be further expressed as

$$\begin{array}{c}{\int }_0^L\left(m{\displaystyle \frac{{\partial }^{2}w}{\partial t^2}}+E{I}^{*}{\displaystyle \frac{{\partial }^{4}w}{\partial x^4}}\right.-{\mathsf{{\rm Y}}}_5{\displaystyle \frac{{\partial }^2\phi }{\partial x^2}}-{\mathsf{{\rm Y}}}_4{\displaystyle \frac{v\left(t\right)}{h}}\left({\delta }^{\prime }\left(x\right)-{\delta }^{\prime }\left(x-L\right)\right)\\ -F_{\mathrm{N}}\delta \left(x-L_F\right))\mathrm{d}x=-m{\int }_0^L{\displaystyle \frac{{\partial }^2{w}_b}{\partial t^2}}\mathrm{d}x\end{array}$$

(29)

$${\displaystyle \frac{v\left(t\right)}{R}}=-{\mathsf{\Pi }}_1{\displaystyle \frac{\mathrm{d}v\left(t\right)}{\mathrm{d}t}}+{\mathsf{\Pi }}_2{\int }_0^L{\displaystyle \frac{{\partial }^{3}w}{\partial x^2\partial t}}\mathrm{d}x+{\mathsf{\Pi }}_5{\int }_0^L{\displaystyle \frac{\partial \phi }{\partial t}}\mathrm{d}x$$

(30)

$$-\begin{array}{c}\kappa \left(1+{\tau }_T{\displaystyle \frac{\partial }{\partial t}}\right)\phi =\left(1+{\tau }_q{\displaystyle \frac{\partial }{\partial t}}\right)\left({\mathsf{\Lambda }}_6{\displaystyle \frac{\partial \phi }{\partial t}}-\right.\left.{\mathsf{\Lambda }}_7{\displaystyle \frac{{\partial }^{3}w}{\partial x^2\partial t}}\right)\end{array}$$

(31)

where the relevant parameter ${\mathsf{{\rm Y}}}_5={\displaystyle \frac{2Ah}{{\pi }^2}}{\mathsf{{\rm Y}}}_1-2b{\mathsf{{\rm Y}}}_3$, ${\mathsf{\Pi }}_5={\displaystyle \frac{2b\pi }{h}}{\mathsf{\Pi }}_4$, ${\mathsf{\Lambda }}_6={\displaystyle \frac{h^2{\mathsf{\Lambda }}_1}{{\pi }^2}}$, ${\mathsf{\Lambda }}_7={\displaystyle \frac{{\mathsf{\Lambda }}_2{h}^3}{24}}$.

4. Analytical Solution

To analyze and solve the vibration energy harvesting mechanism in a piezoelectric material microbeam structure, the assumed-modes method [44] is adopted. Thus, the related displacements and the temperature can be expressed as follows [45]:

$$w\left(x,t\right)=\sum _{i=1}^N{\mathsf{\varpi }}_i\left(t\right){\varphi }_i\left(x\right)={\mathsf{\varpi }}^{\mathrm{T}}\mathsf{\varphi }, \phi \left(x,t\right)=\sum _{i=1}^N{\vartheta }_i\left(t\right){\phi }_i\left(x\right)={\mathsf{\vartheta }}^{\mathrm{T}}\mathsf{\phi }$$

(32)

where ${\varphi }_i\left(x\right)$ and ${\phi }_i\left(x\right)$ are admissible trial functions, and ${\mathsf{\varpi }}_i\left(t\right)$ and ${\vartheta }_i\left(t\right)$ are unknown generalized coordinates. In addition, the trail functions on the symmetric cantilever Euler–Bernoulli beam model can be given by [45,46]

$$\begin{array}{c}{\varphi }_i\left(x\right)=\cos \left({\displaystyle \frac{{\lambda }_i}{L}}x\right)-\cosh \left({\displaystyle \frac{{\lambda }_i}{L}}x\right)+{\displaystyle \frac{\sin \left({\lambda }_i\right)-\sinh \left({\lambda }_i\right)}{\cos \left({\lambda }_i\right)+\cosh \left({\lambda }_i\right)}}\left[\sin \left({\displaystyle \frac{{\lambda }_i}{L}}x\right)-\sinh \left({\displaystyle \frac{{\lambda }_i}{L}}x\right)\right]\\ {\phi }_i\left(x\right)=\cos \left({\mu }_{i}x\right)\end{array}$$

(33)

in which ${\lambda }_i$ are the root of the relevant characteristic equations, and satisfy the following relationship [45,46]

$$1+\cos \left({\lambda }_{i}L\right)\cosh \left({\lambda }_{i}L\right)=0 \mathrm{a}\mathrm{n}\mathrm{d} {\mu }_i={\displaystyle \frac{\left(1+2i\right)\pi }{2}}$$

(34)

Furthermore, substituting Equation (33) into Equations (29)–(31) and ignoring additional external forces, the discrete governing equations for the structurally undamped Euler–Bernoulli beam model can be obtained as

$$\mathbf{M}\ddot{\mathsf{\varpi }}\left(t\right)+\mathbf{N}\mathsf{\varpi }\left(t\right)+\mathbf{K}\mathit{\vartheta }\left(t\right)-\mathsf{\Xi }v\left(t\right)=\mathbf{f}$$

(35)

$${\mathsf{\Xi }}^{\mathbf{T}}\dot{\mathsf{\varpi }}\left(t\right)+{\mathsf{\Theta }}^{\mathbf{T}}\dot{\mathit{\vartheta }}\left(t\right)+{\mathsf{\Pi }}_1\dot{v}\left(t\right)+{\displaystyle \frac{v\left(t\right)}{R}}=0$$

(36)

$$\begin{array}{c}{\tau }_q\mathbf{P}\ddot{\mathsf{\varpi }}\left(t\right)+\mathbf{P}\dot{\mathsf{\varpi }}\left(t\right)+{\tau }_q{\mathsf{\Lambda }}_6\mathbf{U}\ddot{\mathit{\vartheta }}\left(t\right)+\left(\kappa {\tau }_T+{\mathsf{\Lambda }}_6\right)\mathbf{U}\dot{\mathit{\vartheta }}\left(t\right)+\kappa \mathbf{U}\mathit{\vartheta }\left(t\right)=0\end{array}$$

(37)

where the matrix can be written as follows:

$$\begin{array}{c}\begin{array}{c}\begin{array}{c}{M}_{ij}=m{\int }_0^L{\varphi }_i\left(x\right){\varphi }_j\left(x\right)\mathrm{d}x\\ N_{ij}={\left(EI\right)}^{\star }{\int }_0^L{\varphi }_i^{\prime \prime }\left(x\right){\varphi }_j^{\prime \prime }\left(x\right)\mathrm{d}x\end{array}\\ K_{ij}=-{\mathsf{{\rm Y}}}_5{\int }_0^L{{\phi }_i^{\prime }\left(x\right)\phi }_j^{\prime }\left(x\right)\mathrm{d}x\\ {\mathsf{\Xi }}_i={\mathsf{\Pi }}_2{\int }_0^L{\varphi }_i^{\prime \prime }\left(x\right)\mathrm{d}x\\ {\mathsf{\Theta }}_{\mathit{i}}=-{\mathsf{\Pi }}_5{\int }_0^L{\phi }_i\left(x\right)\mathrm{d}x\end{array}\\ f_i=-m{\displaystyle \frac{{\partial }^2{w}_b}{\partial t^2}}{\int }_0^L{\varphi }_i\left(x\right)\mathrm{d}x\\ \begin{array}{c}{P}_{ij}={\mathsf{\Lambda }}_7{\int }_0^L{\varphi }_i^{\prime }\left(x\right){\varphi }_j^{\prime }\left(x\right)\mathrm{d}x\\ U_{ij}=-{\int }_0^L{\phi }_i\left(x\right){\phi }_j\left(x\right)\mathrm{d}x\end{array}\end{array}$$

(38)

Based on this, according to practical situations, the structural dissipation of the damping-controlled region is necessary to account for the microscale energy harvesting systems and devices in the resonant condition. In this work, the damping matrix will be introduced based on the Rayleigh damping as

$$\mathbf{D}=\chi \mathbf{M}+\gamma \mathbf{N}$$

(39)

in which $\chi$ and $\gamma$ are the constants of proportionality which can be developed by using two modal damping ratios, i.e.,

$$\left[\begin{array}{c}\gamma \\ \chi \end{array}\right]={\displaystyle \frac{2{\omega }_1{\omega }_2}{{\omega }_2^2-{\omega }_1^2}}\left[\begin{array}{cc}-1/{\omega }_2& 1/{\omega }_1\\ {\omega }_2& -{\omega }_1\end{array}\right]\left[\begin{array}{c}{\xi }_1\\ {\xi }_2\end{array}\right]$$

(40)

where ${\omega }_1$ and ${\omega }_2$ denote the first two natural frequencies of the microbeam model, and ${\xi }_1$ and ${\xi }_2$ are the damping ratios. Generally, ${\omega }_1={\left({\displaystyle \frac{1.875}{L}}\right)}^2\sqrt{{\displaystyle \frac{{\left(EI\right)}^{\star }}{\rho A}}}, {\omega }_2={\left({\displaystyle \frac{4.694}{L}}\right)}^2\sqrt{{\displaystyle \frac{{\left(EI\right)}^{\star }}{\rho A}}}$ based on the present model [47], and ${\zeta }_1={\zeta }_2=0.05$. Therefore, the corresponding motion in Equation (35) can be rewritten as

$$\mathbf{M}\ddot{\mathsf{\varpi }}\left(t\right)+\mathbf{D}\dot{\mathsf{\varpi }}\left(t\right)+\mathbf{N}\mathsf{\varpi }\left(t\right)+\mathbf{K}\mathit{\vartheta }\left(t\right)-{\mathsf{{\rm Y}}}_4\mathsf{\Xi }v\left(t\right)=\mathbf{f}$$

(41)

Meanwhile, considering the foundation vibration $\mathbf{f}$, it is reasonable to assume that the harmonic of the complex form is

$$\mathbf{f}=\mathbf{F}{\mathrm{e}}^{j\omega t}$$

(42)

in which the component $\mathbf{F}$ can be expressed as $F_i=m{W}_0{\omega }^2{\int }_0^L{\varphi }_i\left(x\right)\mathrm{d}x$; $W_0$ and $\omega$, respectively, represent amplitude of base excitation and harmonic frequency; and $j=\sqrt{-1}$. In addition, considering the existence of base vibration and the fact that the system is assumed to be linear, the generalized coordinates $\mathsf{\varpi }\left(t\right)$, $\vartheta \left(t\right)$ and voltage $v\left(t\right)$ can be expressed as

$$\mathsf{\varpi }\left(\mathit{t}\right)=\mathit{\Omega }{\mathrm{e}}^{j\omega t},\mathit{\vartheta }\left(\mathit{t}\right)=\mathit{\Gamma }{\mathrm{e}}^{j\omega t},v\left(t\right)=V{\mathrm{e}}^{j\omega t}$$

(43)

As a result, according to Equations (36), (37) and (41), and substituting Equations (42) and (43) into them, the complex-valued unknowns $\Omega$ and $V$ can be obtained as

$$V={\displaystyle \frac{\frac{-\left({\mathsf{\Xi }}^{\mathbf{T}}j\omega -\frac{{\mathsf{\Theta }}^{\mathbf{T}}j\omega \left(-{\tau }_q{\omega }^2+j\omega \right)\mathbf{P}}{\left(-{\tau }_q{\mathsf{\Lambda }}_6{\omega }^2+\left(\kappa {\tau }_T+{\mathsf{\Lambda }}_6\right)j\omega +\kappa \right)\mathbf{U}}\right)}{\left({-\omega }^2\mathbf{M}+j\omega \mathbf{D}+\mathbf{N}\right)-\frac{\mathbf{K}\left(-{\tau }_q{\omega }^2+j\omega \right)\mathbf{P}}{\left(-{\tau }_q{\mathsf{\Lambda }}_6{\omega }^2+j\omega \left(\kappa {\tau }_T+{\mathsf{\Lambda }}_6\right)+\kappa \right)\mathbf{U}}}\mathbf{F}}{{\frac{1}{R}+\mathsf{\Pi }}_{1}j\omega +\frac{{\mathsf{\Xi }}^{\mathbf{T}}j\omega -\frac{{\mathsf{\Theta }}^{\mathbf{T}}j\omega \left(-{\tau }_q{\omega }^2+j\omega \right)\mathbf{P}}{\left(-{\tau }_q{\mathsf{\Lambda }}_6{\omega }^2+\left(\kappa {\tau }_T+{\mathsf{\Lambda }}_6\right)j\omega +\kappa \right)\mathbf{U}}}{\left({-\omega }^2\mathbf{M}+j\omega \mathbf{D}+\mathbf{N}\right)-\frac{\mathbf{K}\left(-{\tau }_q{\omega }^2+j\omega \right)\mathbf{P}}{\left(-{\tau }_q{\mathsf{\Lambda }}_6{\omega }^2+j\omega \left(\kappa {\tau }_T+{\mathsf{\Lambda }}_6\right)+\kappa \right)\mathbf{U}}}\mathsf{\Xi }}}$$

(44)

$$\mathit{\Omega }={\displaystyle \frac{1}{\left(\left({-\omega }^2\mathbf{M}+j\omega \mathbf{D}+\mathbf{N}\right)-\frac{\mathbf{K}\left(-{\tau }_q{\omega }^2+j\omega \right)\mathbf{P}}{\left(-{\tau }_q{\mathsf{\Lambda }}_6{\omega }^2+j\omega \left(\kappa {\tau }_T+{\mathsf{\Lambda }}_6\right)+\kappa \right)\mathbf{U}}\right)}}\left(\mathbf{F}+\mathsf{\Xi }\mathit{V}\right)$$

(45)

Moreover, the output power $P$ and the displacements $w\left(x,t\right)$ of the vibration response can be obtained as

$$P=\left|{\displaystyle \frac{v^2\left(t\right)}{R}}\right|=\left|{\displaystyle \frac{V^2}{R}}\right|$$

(46)

It should be emphasized that if the thermal effect and polarization gradient effect are ignored, respectively, the analytical model is applicable to macroscopic and single physical field problems, and the analytical results obtained can be reduced to the results presented in [37,48].

5. Numerical Results and Discussion

In order to investigate the influence of the flexoelectric and thermal polarization effect on the microscale structural energy harvesting performance, the numerical results of the theoretical analysis model are discussed. In this section, the output power and voltage are studied in detail, in which the theory results of the microscale beam model based on the current established theory are also validated. This section considers the microbeam model with the following geometric and material parameters: the geometric dimensions are set as $h=5 \mathrm{n}\mathrm{m}$, $b=5h$ and $L=20h$ based on the Euler–Bernoulli beam model hypothesis, and PVDF is the substance utilized in the numerical computation. The material properties are displayed in

Table 1. Material-related performance parameters at $T_0=293 \mathrm{K}$ [14,49,50].

	Parameters	Unit	PVDF
Elastic modulus	$c_{11}$	GPa	3.7
Density	$\rho$	$\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$	1780
Poisson’s ratio	$\nu$	1	0.33
Thermal conductivity	$\kappa$	$\mathrm{W}/\mathrm{m}{\mathrm{K}}^{-1}$	$0.19$
Specific heat	$C_E$	$\mathrm{J}/\left(\mathrm{K}\mathrm{g}·\mathrm{K}\right)$	$2.62\times {10}^6$
Pyroelectric constants	$p_1$	$\mathrm{C}/\left({\mathrm{m}}^2\mathrm{K}\right)$	$-0.27\times {10}^{-4}$
Dielectric constants	$a_{33}$	$\mathrm{N}{\mathrm{m}}^2/{\mathrm{C}}^2$	$1.38\times {10}^{10}$
Piezoelectric coefficient	$d_{13}$	$\mathrm{C}/{\mathrm{m}}^2$	$-1.02\times {10}^9$
Flexoelectric coefficient	$f_{13}$	$\mathrm{C}/\mathrm{m}$	$179$
Vacuum permittivity	${ϵ}_0$	${\mathrm{C}}^2/\left(\mathrm{N}{\mathrm{m}}^2\right)$	$8.854\times {10}^{-12}$
Thermopolarization coefficient	$b_{11}$	$\mathrm{C}/\left(\mathrm{m}\mathrm{K}\right)$	$1\times {10}^{-7}$

.

During the course of analysis, the thermal relaxation time ${\tau }_q$ can be given by [51]

$${\tau }_q={\displaystyle \frac{3\kappa }{\rho C_E{v}^2}}$$

(47)

where $v$ is approximately replaced by elastic wave velocity, i.e., $v=\sqrt{c_{11}/\rho }$, and in that case, the temperature relaxation time ${\tau }_T=0.55{\tau }_q$ [52]. In addition, it should be mentioned that $W_0=9.81/{\omega }^2$ [13], and $\omega ={\omega }_1$ for the reduced-order modeling.

5.1. The Comparison of the Current Theoretical Analysis Model

In order to compare this model with the previous theoretical model, Figure 2 gives the voltage and output power with vibration frequency in the piezoelectric microbeam model, which the present model reduces to the theoretical analysis of Faroughi et al. [48] by neglecting thermal effect. According to Figure 2, when the thermal effect is ignored, the microelectromechanical energy harvesting performance of the piezoelectric material is significantly reduced in terms of both voltage and output power, particularly in regard to the maximum voltage and output power. In addition, considering the generalized thermoelastic effects, the vibrational frequency range corresponding to the maximum voltage and output power values undergoes notable variations. This analysis may contribute to the performance improvement and optimized design of piezoelectric self-powered energy harvesting systems in multi-physical environments, particularly in extreme thermal conditions.

To further explore the thermopolarization effect due to the temperature gradient changes in microscale structural materials on the basis of thermal effect, Figure 3 shows the influences of the thermopolarization effect on the voltage and output power. From Figure 3, it can be seen that the thermopolarization effect promotes the piezoelectric vibration energy harvesting performance, especially in the case of larger frequency ranges and at the maximum voltage and output power levels. Therefore, this takes into account the impact of temperature gradients, especially in nanoscale objects where heat propagation occurs at a finite speed such as in space–time microscale domains or under extreme thermal conditions [53].

5.2. The Influence of Microscale Effect on the Performance of Energy Harvesting

In what follows, we will analyze and discuss the piezoelectric material structure vibration energy harvesting performance on the influence of microscale characteristics. Through numerical simulation, it is found that the voltage and output power response increase significantly with the thickness $h$ on microbeam model, as shown in Figure 4. The voltage exhibits a linear relationship, while the output power has a nonlinear relationship. Furthermore, it is observed that the strain gradient coefficient significantly influences both voltage and output power, with both effects intensifying as nonlocal parameters increase, especially in the case of larger frequency ranges. In addition, as illustrated in Figure 5a,b, both the maximum voltage and maximum output power exhibit a notable increase with the increasing strain gradient coefficient. Notably, the results obtained from the classical continuous theory model (CL model) are smaller in comparison.

It should be mentioned that the above analysis reveals that the strain gradient coefficient exhibits a typical stiffness hardening behavior on the vibration energy harvesting [54]. The coupled electromechanical characteristics are instrumental in exploring the flexoelectricity effects induced by polarization and strain gradients on the microscale energy harvesting performance.

5.3. The Influence of Load Resistances on the Energy Harvesting Performance of Piezo-Flexoelectric Thermoelastic Microbeam

To explore the influence of load resistances on the improvement of the voltage and output power of the piezo-flexoelectric thermoelastic vibration energy harvesting, Figure 6a,b show the variations of the voltage and output power with the vibration frequency for given load resistances $R$. From Figure 3, it is evident that the voltage and output power decrease significantly as the load resistance increases. However, it is worth noting that the load resistance has virtually no effect on the frequency range at which the maximum voltage and output power occur, which means that the optimal load resistances have a strong influence on the thermo-electro-elastic vibration energy harvesting performance.

6. Conclusions

In this paper, to characterize the vibration energy harvesting performance of micro/nano piezoelectric material devices and systems, a size-dependent theoretical analytical model incorporating flexoelectricity and considering thermo-electro-elastic coupling behaviors on microscale energy harvesting was developed. Combining Hamilton’s principle and generalized thermoelastic theory, the thermo-electro-elastic governing equations were derived in a microbeam model. Then, approximated closed-form solutions for energy harvesting performance were obtained, and the thermal effect, thermopolarization and size dependence effects were discussed. The main conclusions can be summarized as follows:

(1) The thermal effect, considering relaxation time, has an obvious influence on energy harvesting performance, and significantly promotes the voltage and output power.

(2) The influences of the deformation gradient and polarization in nanoscale objects are obtained and discussed. If thermopolarization is taken into consideration, these influences can efficiently reflect the energy harvesting mechanism.

(3) The voltage and output power in the microbeam model of energy harvesting exhibit strong characteristics of the microscale effect, and the strain gradient shows typical stiffness hardening behavior within the voltage and output power.

As it is based on micro/nano electromechanical systems (MEMS/NEMS), the current work is only devoted to microscale thermo-electro-elastic theory, which will facilitate the enhancement of energy harvesting performance and efficiency on micro/nano systems or devices. Future endeavors grounded in both macroscopic and microscopic theories will explore cross-scale multi-physics theories.