基于非局部弹性理论的分数阶粘弹性纳米板的振动
Vibration of Fractional-Order Viscoelastic Nanoplates Based on the Theory of Non-Local Elasticity
摘要: 本文基于Kirchhoff板理论，同时考虑了非局部弹性理论和分数阶Kelvin-Voigt粘弹性本构关系，利用Hamilton原理建立了粘弹性纳米板的控制方程。通过给出解的形式，利用拉普拉斯变换及其逆变换对问题就进行求解，在得到数值解后并分析了分数阶导数的阶数、非局部参数以及粘弹性系数对纳米板的振动影响。
Abstract: Based on Kirchhoff plate theory, this paper takes into account the nonlocal elasticity theory and the fractional Kelvin-Voigt viscoelastic constitutive relation. It establishes the governing equations of the viscoelastic nano-plate using the Hamilton principle. By giving the form of the solution and using the Laplace transform and its inverse to solve the problem, the influence of the order of fractional derivative, nonlocal parameter, and viscoelastic coefficient on the vibration of the nano-plate is analyzed after obtaining the numerical solution.
文章引用：薛晓路, 雷东侠. 基于非局部弹性理论的分数阶粘弹性纳米板的振动[J]. 力学研究, 2024, 13(2): 94-102. https://doi.org/10.12677/ijm.2024.132010

参考文献

[1] Gao, X.-L. and Zhang, G.Y. (2015) A Non-Classical Kirchhoff Plate Model Incorporating Microstructure, Surface Energy and Foundation Effects. Continuum Mechanics and Thermodynamics, 28, 195-213. [Google Scholar] [CrossRef] 
[2] Eringen, A.C. (1972) Nonlocal Polar Elastic Continua. International Journal of Engineering Science, 10, 1-16. [Google Scholar] [CrossRef] 
[3] Mindlin, R.D. (1964) Micro-Structure in Linear Elasticity. Archive for Rational Mechanics and Analysis, 16, 51-78. [Google Scholar] [CrossRef] 
[4] Lim, C.W., Zhang, G. and Reddy, J.N. (2015) A Higher-Order Nonlocal Elasticity and Strain Gradient Theory and Its Applications in Wave Propagation. Journal of the Mechanics and Physics of Solids, 78, 298-313. [Google Scholar] [CrossRef] 
[5] 周光泉, 刘孝敏. 粘弹性理论[M]. 合肥: 中国科学技术大学出版社, 1996.
[6] 陈文. 力学与工程问题的分数阶导数建模[M]. 北京: 科学出版社, 2010.
[7] Li, G., Zhu, Z. and Cheng, C. (2003) Application of Galerkin Method to Dynamical Behavior of Viscoelastic Timoshenko Beam with Finite Deformation. Mechanics of Time-Dependent Materials, 7, 175-188. [Google Scholar] [CrossRef] 
[8] Freundlich, J. (2019) Transient Vibrations of a Fractional Kelvin-Voigt Viscoelastic Cantilever Beam with a Tip Mass and Subjected to a Base Excitation. Journal of Sound and Vibration, 438, 99-115. [Google Scholar] [CrossRef] 
[9] Liu, X. and Li, D. (2020) A Link between a Variable-Order Fractional Zener Model and Non-Newtonian Time-Varying Viscosity for Viscoelastic Material: Relaxation Time. Acta Mechanica, 232, 1-13. [Google Scholar] [CrossRef] 
[10] Ansari, R., Faraji Oskouie, M. and Gholami, R. (2016) Size-Dependent Geometrically Nonlinear Free Vibration Analysis of Fractional Viscoelastic Nanobeams Based on the Nonlocal Elasticity Theory. Physica E: Low-Dimensional Systems and Nanostructures, 75, 266-271. [Google Scholar] [CrossRef] 
[11] Oskouie, M.F. and Ansari, R. (2017) Linear and Nonlinear Vibrations of Fractional Viscoelastic Timoshenko Nanobeams Considering Surface Energy Effects. Applied Mathematical Modelling, 43, 337-350. [Google Scholar] [CrossRef] 

为你推荐