Transverse shear and normal deformation effects on vibration behaviors of functionally graded micro-beams

doi:10.1007/s10483-020-2662-6

Applied Mathematics and Mechanics (English Edition) ›› 2020, Vol. 41 ›› Issue (9): 1303-1320.doi: https://doi.org/10.1007/s10483-020-2662-6

Transverse shear and normal deformation effects on vibration behaviors of functionally graded micro-beams

Zhu SU¹, Lifeng WANG¹, Kaipeng SUN², Jie SUN²

1. State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China;
2. Shanghai Institute of Satellite Engineering, Shanghai 201109, China

收稿日期:2020-05-17 修回日期:2020-07-13 发布日期:2020-08-27
通讯作者: Zhu SU E-mail:suzhu@nuaa.edu.cn
基金资助:
Project supported by the National Natural Science Foundation of China (Nos. 51805250 and 11602145), the Natural Science Foundation of Jiangsu Province of China (No. BK20180429), the China Postdoctoral Science Foundation (No. 2019M660114), and the Jiangsu Planned Projects for Postdoctoral Research Funds of China (No. 2019K054)

Transverse shear and normal deformation effects on vibration behaviors of functionally graded micro-beams

Zhu SU¹, Lifeng WANG¹, Kaipeng SUN², Jie SUN²

1. State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China;
2. Shanghai Institute of Satellite Engineering, Shanghai 201109, China

Received:2020-05-17 Revised:2020-07-13 Published:2020-08-27
Contact: Zhu SU E-mail:suzhu@nuaa.edu.cn
Supported by:
Project supported by the National Natural Science Foundation of China (Nos. 51805250 and 11602145), the Natural Science Foundation of Jiangsu Province of China (No. BK20180429), the China Postdoctoral Science Foundation (No. 2019M660114), and the Jiangsu Planned Projects for Postdoctoral Research Funds of China (No. 2019K054)

摘要/Abstract

摘要： A quasi-three dimensional model is proposed for the vibration analysis of functionally graded (FG) micro-beams with general boundary conditions based on the modified strain gradient theory. To consider the effects of transverse shear and normal deformations, a general displacement field is achieved by relaxing the assumption of the constant transverse displacement through the thickness. The conventional beam theories including the classical beam theory, the first-order beam theory, and the higherorder beam theory are regarded as the special cases of this model. The material properties changing gradually along the thickness direction are calculated by the Mori-Tanaka scheme. The energy-based formulation is derived by a variational method integrated with the penalty function method, where the Chebyshev orthogonal polynomials are used as the basis function of the displacement variables. The formulation is validated by some comparative examples, and then the parametric studies are conducted to investigate the effects of transverse shear and normal deformations on vibration behaviors.

关键词: quasi-three dimensional theory, modified strain gradient theory, functionally graded (FG) micro-beam, size effect, vibration, general boundary condition

Abstract: A quasi-three dimensional model is proposed for the vibration analysis of functionally graded (FG) micro-beams with general boundary conditions based on the modified strain gradient theory. To consider the effects of transverse shear and normal deformations, a general displacement field is achieved by relaxing the assumption of the constant transverse displacement through the thickness. The conventional beam theories including the classical beam theory, the first-order beam theory, and the higherorder beam theory are regarded as the special cases of this model. The material properties changing gradually along the thickness direction are calculated by the Mori-Tanaka scheme. The energy-based formulation is derived by a variational method integrated with the penalty function method, where the Chebyshev orthogonal polynomials are used as the basis function of the displacement variables. The formulation is validated by some comparative examples, and then the parametric studies are conducted to investigate the effects of transverse shear and normal deformations on vibration behaviors.

Key words: quasi-three dimensional theory, modified strain gradient theory, functionally graded (FG) micro-beam, size effect, vibration, general boundary condition

中图分类号:

74H45
70J10

Zhu SU, Lifeng WANG, Kaipeng SUN, Jie SUN. Transverse shear and normal deformation effects on vibration behaviors of functionally graded micro-beams[J]. Applied Mathematics and Mechanics (English Edition), 2020, 41(9): 1303-1320.

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Transverse shear and normal deformation effects on vibration behaviors of functionally graded micro-beams

Transverse shear and normal deformation effects on vibration behaviors of functionally graded micro-beams

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