Effect of surface stress and surface-induced stress on behavior of piezoelectric nanobeam

doi:10.1007/s10483-018-2346-8

Applied Mathematics and Mechanics (English Edition) ›› 2018, Vol. 39 ›› Issue (7): 953-966.doi: https://doi.org/10.1007/s10483-018-2346-8

Effect of surface stress and surface-induced stress on behavior of piezoelectric nanobeam

Yanmei YUE¹, Kaiyu XU^2,3, Xudong ZHANG⁴, Wenjing WANG⁵

1. Department of Engineering Mechanics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China;
2. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, China;
3. Department of Mechanics, College of Science, Shanghai University, Shanghai 200444, China;
4. State Grid Cangzhou Electric Power Supply Company, Cangzhou 061000, Hebei Province, China;
5. Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta T6G 3G8, Canada

收稿日期:2017-11-21 修回日期:2018-01-16 出版日期:2018-07-01 发布日期:2018-07-01
通讯作者: Kaiyu XU E-mail:kyxu@shu.edu.cn
基金资助:
Project supported by the National Natural Science Foundation of China (Nos. 10772106 and 11072138), the Natural Science Foundation of Shanghai (No. 15ZR1416100), and the Shanghai Leading Academic Discipline Project (No. S30106)

Effect of surface stress and surface-induced stress on behavior of piezoelectric nanobeam

Yanmei YUE¹, Kaiyu XU^2,3, Xudong ZHANG⁴, Wenjing WANG⁵

1. Department of Engineering Mechanics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China;
2. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, China;
3. Department of Mechanics, College of Science, Shanghai University, Shanghai 200444, China;
4. State Grid Cangzhou Electric Power Supply Company, Cangzhou 061000, Hebei Province, China;
5. Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta T6G 3G8, Canada

Received:2017-11-21 Revised:2018-01-16 Online:2018-07-01 Published:2018-07-01
Contact: Kaiyu XU E-mail:kyxu@shu.edu.cn
Supported by:
Project supported by the National Natural Science Foundation of China (Nos. 10772106 and 11072138), the Natural Science Foundation of Shanghai (No. 15ZR1416100), and the Shanghai Leading Academic Discipline Project (No. S30106)

摘要/Abstract

摘要： A new continuum model is developed to study the influence of surface stress on the behaviors of piezoelectric nanobeams. Different from existing piezoelectric surface models which only consider the surface properties, the proposed model takes surfaceinduced initial fields into consideration. Due to the fact that the surface-induced initial fields are totally different under various boundary conditions, two kinds of beams, the doubly-clamped beam and the cantilever beam, are analyzed. Furthermore, boundary conditions can affect not only the initial state of the piezoelectric nanobeam but also the forms of the governing equations. Based on the Euler-Bernoulli beam theory, the nonlinear Green-Lagrangian strain-displacement relationship is applied. In addition, the surface area change is also considered in the proposed model. The governing equations of the doubly-clamped and cantilever beams are derived by the energy variation principle. Compared with existing Young-Laplace models, the proposed model for the doubly-clamped beam is similar to the Young-Laplace models. However, the governing equation of the cantilever beam derived by the proposed model is very different from that derived by the Young-Laplace models. The behaviors of piezoelectric nanobeams predicted by these two models also have significant discrepancies, which is owing to the surface-induced initial fields in the bulk beam.

关键词: piezoelectric nanobeam, surface residual stress, surface effect, fourth-order nonlinear system, periodic solution, existence and uniqueness, asymptotic stability, nonlinear strain

Abstract: A new continuum model is developed to study the influence of surface stress on the behaviors of piezoelectric nanobeams. Different from existing piezoelectric surface models which only consider the surface properties, the proposed model takes surfaceinduced initial fields into consideration. Due to the fact that the surface-induced initial fields are totally different under various boundary conditions, two kinds of beams, the doubly-clamped beam and the cantilever beam, are analyzed. Furthermore, boundary conditions can affect not only the initial state of the piezoelectric nanobeam but also the forms of the governing equations. Based on the Euler-Bernoulli beam theory, the nonlinear Green-Lagrangian strain-displacement relationship is applied. In addition, the surface area change is also considered in the proposed model. The governing equations of the doubly-clamped and cantilever beams are derived by the energy variation principle. Compared with existing Young-Laplace models, the proposed model for the doubly-clamped beam is similar to the Young-Laplace models. However, the governing equation of the cantilever beam derived by the proposed model is very different from that derived by the Young-Laplace models. The behaviors of piezoelectric nanobeams predicted by these two models also have significant discrepancies, which is owing to the surface-induced initial fields in the bulk beam.

Key words: surface effect, nonlinear strain, surface residual stress, fourth-order nonlinear system, periodic solution, existence and uniqueness, asymptotic stability, piezoelectric nanobeam

中图分类号:

74B10

Yanmei YUE, Kaiyu XU, Xudong ZHANG, Wenjing WANG. Effect of surface stress and surface-induced stress on behavior of piezoelectric nanobeam[J]. Applied Mathematics and Mechanics (English Edition), 2018, 39(7): 953-966.

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Effect of surface stress and surface-induced stress on behavior of piezoelectric nanobeam

Effect of surface stress and surface-induced stress on behavior of piezoelectric nanobeam

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