# Modified Continuum Mechanics Modeling on Size-Dependent Properties of Piezoelectric Nanomaterials: A Review

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## Abstract

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## 1. Introduction

## 2. Novel Modified Continuum Theories for PNs

#### 2.1. Surface Piezoelectricity Theory

#### 2.2. Theory of Flexoelectricity

#### 2.3. Non-Local Piezoelectricity Theory

## 3. Size-Dependent Properties of PNs

#### 3.1. Electromechanical Coupling Behaviors of PNs

#### 3.2. Bending, Vibration and Buckling Behaviors of PNs

#### 3.3. Size-Dependent Dynamic Performance of PNs

## 4. Modeling of Piezoelectric Nanodevices

## 5. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic of (

**a**) a piezoelectric nanobeam; and (

**b**) a piezoelectric nanoplate with surface effects.

**Table 1.**Summary of the size-dependent mechanical and electromechanical coupling properties of piezoelectric nanomaterials (PNs) based on different theories. Acronyms: SPT (Surface piezoelectricity theory), TF (Theory of flexoelectricity), NPT (Non-local piezoelectricity theory), EMC (Electromechanical coupling).

Theories | Size-Dependent Properties | Materials | References |
---|---|---|---|

SPT | EMC fields | PZT-5H | [51,52,53,87] |

SPT | bending | PZT-5H | [52,53,97] |

SPT | vibration | PZT-5H | [54,55,90,92,93,94,97] |

SPT | buckling | PZT-5H | [53,90,92,93,95,96] |

SPT | wrinkling | PZT-5H | [91] |

SPT | wave propagation | PZT-4 | [115] |

SPT | wave propagation | PZT-5 | [116] |

SPT | dynamic characteristics | PZT-4 | [118] |

SPT | dynamic characteristics | CoFe${}_{2}$O${}_{4}$/BaTiO${}_{3}$ | [119] |

TF | EMC fields | ZnO | [88] |

TF | EMC fields | BaTiO${}_{3}$ | [72,74,76] |

TF | bending | BaTiO${}_{3}$ | [72,73,74] |

TF | vibration | BaTiO${}_{3}$ | [73,74] |

TF | bending and vibration | PZT-5H | [98] |

TF | buckling and vibration | Pb(Mg${}_{1/3}$Nb${}_{2/3}$O${}_{3}$) | [99] |

NPT | vibration | PZT-4 | [100,101,102] |

SPT and NPT | EMC fields | PZT-5H | [84] |

SPT and NPT | wave propagation | PZT-5H | [117] |

SPT and TF | bending | PZT-5H | [85,104] |

SPT and TF | vibration | PZT-5H | [104] |

SPT and TF | bending and vibration | BaTiO${}_{3}$ | [103] |

© 2017 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 ( http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Yan, Z.; Jiang, L.
Modified Continuum Mechanics Modeling on Size-Dependent Properties of Piezoelectric Nanomaterials: A Review. *Nanomaterials* **2017**, *7*, 27.
https://doi.org/10.3390/nano7020027

**AMA Style**

Yan Z, Jiang L.
Modified Continuum Mechanics Modeling on Size-Dependent Properties of Piezoelectric Nanomaterials: A Review. *Nanomaterials*. 2017; 7(2):27.
https://doi.org/10.3390/nano7020027

**Chicago/Turabian Style**

Yan, Zhi, and Liying Jiang.
2017. "Modified Continuum Mechanics Modeling on Size-Dependent Properties of Piezoelectric Nanomaterials: A Review" *Nanomaterials* 7, no. 2: 27.
https://doi.org/10.3390/nano7020027