# Research on Output Characteristics of Microscale BST Laminate Structure Based on Mixed Finite Element Method

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## Abstract

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^{1}continuous quadrilateral 8-node (displacement and potential) and 4-node (displacement gradient and Lagrange multipliers) flexoelectric mixed element. By comparing the numerical calculation results and analytical solutions of the electrical output characteristics of the microscale BST/PDMS laminated cantilever structure, it is proved that the mixed finite element method designed in this paper is an effective tool for studying the electromechanical coupling behavior of flexoelectric materials.

## 1. Introduction

^{0}continuous shape. The function achieves the same convergence as the C

^{1}continuous function. Amanatidou and Aravas [10] developed some mixed elements, which can obtain relatively accurate solutions in some classical problems (such as single-crystal cantilever beams), but the theory requires additional material intrinsic parameters that are difficult to determine experimentally. Couple stress theory [11] is a method for studying microscale effects, which has the advantage of having few additional material intrinsic constants and is relatively easy to determine experimentally. Regarding the general strain gradient elasticity [12] problem, Hutchinson [13] and Shu [14] took the couple-stress rotation angle as an additional nodal degree of freedom and solved the flexoelectric higher-order partial differential equation by the mixed finite element method, which verifies that the couple-stress Feasibility of theoretically dealing with flexoelectric problems.

## 2. Electrical Enthalpy Density Theoretical Model of Microscale Flexoelectric Effect

## 3. Constrained Variation Principle

## 4. Microscale Flexoelectric Mixed Cell Construction

## 5. Numerical Simulation and Performance Analysis of Microscale Flexoelectric Laminated Beams

#### 5.1. Output Voltage Analysis of Single Functional Layer BST/PDMS Laminated Structure

#### 5.2. Analysis of Output Voltage of Multilayer BST Laminated Structure

## 6. Conclusions

^{1}continuity of the element boundary is taken into account, a new mixed element is constructed. According to the parameter transformation method, a two-dimensional mixed finite element method related to the size effect and flexoelectric effect is developed.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Voltage of the flexoelectric beam under the concentrated load F, distributed load q, and couple load M (Single layer BST) (Ns = Numerical solution, As = Analytical solution).

**Figure 6.**Optimal output voltage of multi-layer micro scale flexoelectric laminated beam energy collection device.

**Figure 7.**Output voltage of BST sensitive layer with different number of layers varying with the thickness of BST layer.

**Figure 8.**Output voltage of micro scale flexoelectric laminated beam energy collection device with fixed thickness.

E | $\mathit{\nu}$ | $\mathit{\rho}$ | ${\mathit{\mu}}_{14}$ | ${\mathit{k}}_{33}$ | |
---|---|---|---|---|---|

BST | 152 GPa | 0.33 | 8.2 × 103 kg/m^{3} | 50 μC/m | 13,200 |

PDMS | 540 MPa | 0.49 | 970 kg/m^{3} |

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**MDPI and ACS Style**

Luo, Y.; Pu, T.; Liu, H. Research on Output Characteristics of Microscale BST Laminate Structure Based on Mixed Finite Element Method. *Micromachines* **2023**, *14*, 755.
https://doi.org/10.3390/mi14040755

**AMA Style**

Luo Y, Pu T, Liu H. Research on Output Characteristics of Microscale BST Laminate Structure Based on Mixed Finite Element Method. *Micromachines*. 2023; 14(4):755.
https://doi.org/10.3390/mi14040755

**Chicago/Turabian Style**

Luo, Ying, Tian Pu, and Hongguang Liu. 2023. "Research on Output Characteristics of Microscale BST Laminate Structure Based on Mixed Finite Element Method" *Micromachines* 14, no. 4: 755.
https://doi.org/10.3390/mi14040755