# Application of Multi-Parameter Perturbation Method to Functionally-Graded, Thin, Circular Piezoelectric Plates

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## Abstract

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## 1. Introduction

## 2. Mechanical Model and Basic Equations

## 3. Application of Multi-Parameter Perturbation Method

#### 3.1. Nondimensionalization and Perturbation Expansions

#### 3.2. Zero-Order Perturbation Solution

#### 3.3. First-Order Perturbation Solution

_{31}, substituting Equation (19) into Equations (16)–(18), we have the first-order perturbation equations with respect to ${D}_{31}$ as follows:

## 4. Numerical Simulation and Comparison with Perturbation Solution

#### 4.1. Numerical Simulation

#### 4.2. Comparison with Perturbation Solution

_{3}-4 (Generally abbreviated as PZT-4) in a dimensionless form are given in Table 2, according to Equation (15a)), thus comparing the theoretical solution with the numerical example in a dimensionless form, as shown in Figure 3. Note that all stress components in Figure 3 are given in dimensionless values (please refer to Equation (15b)).

## 5. Results and Discussions

#### 5.1. Effect of Gradient Index on Solution

#### 5.2. Deflection Comparison Between FGPM and FGM Plates

## 6. Conclusions

- (i)
- Adopting the three piezoelectric coefficients as perturbation parameters follows the basic idea of perturbation theory—i.e., if pure FGM plates without piezoelectric effects are taken as the undisturbed system, and the piezoelectric effect may be introduced as a kind of disturbance, then FGPM plates may be regarded as a disturbed system; thus, the solution for pure FGM plates may be easily obtained as a zero-order solution of the disturbed system.
- (ii)
- In our perturbation, two stress functions and one electrical potential function were selected as basic functions. It was found that in the zero-order perturbation solution, only stress functions were not zero; thus, the zero-order solution actually corresponded to the elastic solution concerning elastic stress, elastic strain, and elastic displacement, this conclusion is consistent with the former conclusion: while in the first-order perturbation solution, only the electrical potential function was not zero; thus, the first-order solution actually corresponded to the electrical solution, concerning electrical potential, electrical field intensity, and electrical displacement.
- (iii)
- The deformation magnitude of FGPM plates is generally smaller than that of corresponding FGM plates, showing the well-known piezoelectric stiffening effect. From the point of view of energy conservation and transformation, a portion of the work done by applied external loads is transformed into electrical energy, due to the piezoelectric effect, thus decreasing elastic strain energy stored in FGPM plates and resulting in the corresponding decrease in deformation magnitude.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Zero-Order Perturbation Solution

Gradient Index | $\mathit{\alpha}=-2$ | $\mathit{\alpha}=0$ | $\mathit{\alpha}=2$ | $\mathit{\alpha}=5$ |
---|---|---|---|---|

${\eta}_{n}$ | 0.157 | 0 | −0.157 | −0.307 |

Gradient Index | $\mathit{\alpha}=-2$ | $\mathit{\alpha}=1$ | $\mathit{\alpha}=2$ | $\mathit{\alpha}=5$ |
---|---|---|---|---|

${A}_{5}^{(0)}$(×10^{10}) | −2.517 | −2.070 | 2.517 | −1.252 |

${A}_{6}^{(0)}$(×10^{10}) | 1.830 | 1.647 | 1.830 | 1.879 |

## Appendix B. First-Order Perturbation Solution

_{31}, we may have three equations of ${f}_{11}^{\left(1\right)}\left(\eta \right),{f}_{12}^{\left(1\right)}\left(\eta \right),{f}_{13}^{\left(1\right)}\left(\eta \right),{g}_{11}^{\left(1\right)}\left(\eta \right),{g}_{12}^{\left(1\right)}\left(\eta \right),{h}_{11}^{\left(1\right)}\left(\eta \right),{h}_{12}^{\left(1\right)}\left(\eta \right),\mathrm{and}\text{}{h}_{13}^{\left(1\right)}\left(\eta \right)$. The satisfaction for these equations will give the conditions that the same power terms of $\beta $ are uniformly zero; thus, we obtain

_{31}. By using boundary conditions (38) and (39), these constants are determined as

_{33,}substituting Equation (20) into Equations (40)–(42), we may have three equations of ${f}_{21}^{\left(1\right)}\left(\eta \right),{f}_{22}^{\left(1\right)}\left(\eta \right),{f}_{23}^{\left(1\right)}\left(\eta \right),{g}_{21}^{\left(1\right)}\left(\eta \right),{g}_{22}^{\left(1\right)}\left(\eta \right),{h}_{21}^{\left(1\right)}\left(\eta \right),{h}_{22}^{\left(1\right)}\left(\eta \right),\mathrm{and}{h}_{23}^{\left(1\right)}\left(\eta \right)$. The satisfaction for these equations will give the conditions that the same power terms of $\beta $ are uniformly zero; thus, we obtain

_{33}. By using boundary conditions (43) and (44), these constants are determined as

_{15}. By using boundary conditions (48) and (49), these constants are determined as

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