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

^{1}

^{2}

^{*}

## Abstract

**:**

^{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

- Nguyen, T.D.; Mao, S.; Yeh, Y.W.; Purohit, P.K.; McAlpine, M.C. Nanoscale Flexoelectricity. Adv. Mater.
**2013**, 25, 946–974. [Google Scholar] [CrossRef] - Yudin, P.V.; Tagantsev, A.K. Fundamentals of flexoelectricity in solids. Nanotechnology
**2013**, 24, 1361–6528. [Google Scholar] [CrossRef] [PubMed] - Yuan, H.; Liu, J.; Gu, M.; Feng, S.; Zhou, M.; Luo, Y. Low-temperature sintering and electrical properties of Ba
_{0.68}Sr_{0.32}TiO_{3}thick films. Ceram. Int.**2021**, 47, 16641–16651. [Google Scholar] [CrossRef] - Li, K.; Luo, Y.; Xu, C.G. Measurement of the Transverse Flexoelectric Coefficient Based on BST Micro-cantilever Beam. Piezoelectrics Acoustooptics
**2014**, 10, 798–800. (In Chinese) [Google Scholar] - Ghobadi, A.; Golestanian, H.; Beni, Y.T.; Żur, K.K. On the size-dependent nonlinear thermo-electro-mechanical free vibration analysis of functionally graded flexoelectric nano-plate. Commun. Nonlinear Sci. Numer. Simul.
**2021**, 95, 105585. [Google Scholar] [CrossRef] - Kumar, A.; Kiran, R.; Kumar, R.; Jain, S.C.; Vaish, R. Flexoelectric effect in functionally graded materials: A numerical study. Eur. Phys. J. Plus
**2018**, 133, 141. [Google Scholar] [CrossRef] - Caiqin, C.; Jingbo, C.; Dongbo, L. Bending performance analysis of flexoelectric nanoplate considering electric field gradients. Chin. J. Theor. Appl. Mech.
**2022**, 54, 3088–3098. [Google Scholar] - Zhang, H.; Wang, C.M. Recent Development on Flexoelectric Effect and Its Application. J. Chin. Ceram. Soc.
**2023**, 51, 812–830. [Google Scholar] - Shu, J.Y.; King, W.E.; Fleck, N.A. Finite Elements for Materials With Strain Gradient Effects. Numer. Methods Eng.
**1999**, 44, 373–391. [Google Scholar] [CrossRef] - Amanatidou, E.; Aravas, N. Mixed Finite Element Formulations of Strain-Gradient Elasticity Problems. Comput. Methods Appl. Mech. Eng.
**2002**, 191, 1723–1751. [Google Scholar] [CrossRef] [Green Version] - Toupin, R.A. Elastic materials with couple stresses. Arch. Ration. Mech. Anal.
**1962**, 11, 385–414. [Google Scholar] [CrossRef] [Green Version] - Mindlin, R.D. Micro-Structure in Linear Elasticity. Arch. Ration. Mech. Anal.
**1964**, 16, 51–78. [Google Scholar] [CrossRef] - Xia, Z.C.; Hutchinson, J.W. Crack Tip Fields in Strain Gradient Plasticity. Mech. Phys. Solids
**1996**, 44, 1621–1648. [Google Scholar] [CrossRef] - Shu, J.; Fleck, N. The Prediction of a Size Effect in Micro-indentation. Solids Struct.
**1998**, 35, 1363–1383. [Google Scholar] [CrossRef] - Abdollahi, A.; Peco, C.; Millan, D.; Arroyo, M. Computational evaluation of the flexoelectric effect in dielectric solids. Appl. Phys.
**2014**, 116, 093502. [Google Scholar] [CrossRef] [Green Version] - Mao, S.; Purohit, P.K.; Aravas, N. Mixed Finite-Element Formulations in Piezoelectricity and Flexoelectricity. Proc. R. Soc. A
**2016**, 472, 20150879. [Google Scholar] [CrossRef] [Green Version] - Deng, F.; Deng, Q.; Yu, W.; Shen, S. Mixed finite elements for flexoelectric solids. J. Appl. Mech.
**2017**, 84, 081004. [Google Scholar] [CrossRef] - Nguyen, B.; Zhuang, X.; Rabczuk, T. NURBS-based formulation for nonlinear electro-gradient elasticity in semiconductors. Comput. Methods Appl. Mech. Eng.
**2019**, 346, 1074–1095. [Google Scholar] [CrossRef] - Shen, S.; Hu, S. A theory of flexoelectricity with surface effect for elastic dielectrics. J. Mech. Phys. Solids
**2010**, 58, 665–677. [Google Scholar] [CrossRef] - Hu, S.L.; Shen, S.P. Variational principles and governing equations in nano-dielectrics with the flexoelectric effect. Sci. China
**2010**, 53, 1497–1504. [Google Scholar] [CrossRef] - Shu, L.; Li, F.; Huang, W.; Wei, X.; Yao, X.; Jiang, X. Relationship between direct and converse flexoelectric coefficients. J. Appl. Phys.
**2014**, 116, 1788. [Google Scholar] [CrossRef] [Green Version]

**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} |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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