热环境中多孔功能梯度材料微圆板的自由振动分析
Free Vibration Analysis of Porous Functionally Graded Materials Circular Micro-Plates in Thermal Environment
摘要: 基于经典板理论和修正偶应力理论，研究热环境中多孔功能梯度材料(FGM)微圆板的自由振动问题。首先利用Hamilton原理推导热环境中含有材料尺度参数的多孔FGM微圆板自由振动的控制微分方程并进行无量纲化。然后应用微分变换法(DTM)对多孔FGM微圆板自由振动的无量纲控制微分方程和边界条件进行变换，得到计算自由振动无量纲固有频率的代数特征方程。最后通过MATLAB编程计算并分析了梯度指数、孔隙率、不同升温方式和材料尺度参数对多孔FGM微圆板无量纲固有频率的影响。结果表明：材料尺度参数的增加，无量纲固有频率会增大；梯度指数影响频率，反映了材料从陶瓷到金属的转变特性；孔隙率会削弱刚度，进而影响固有频率。研究结果可为今后热环境中多孔FGM微圆板的设计与应用提供数据支持。
Abstract: The free vibration problem of porous functional graded material (FGM) circular micro-plates in a thermal environment is investigated based on the classical plate theory and the modified coupled stress theory. Firstly, Hamilton’s principle is used to derive the governing differential equations for the free vibration of porous FGM circular micro-plates in thermal environment containing material scale parameters and to carry out the dimensionless quantization. Then the differential transformation method (DTM) is applied to transform the dimensionless governing differential equations and boundary conditions of the free vibration of the porous FGM circular micro-plate to obtain the algebraic characteristic equations for calculating the dimensionless natural frequency of the free vibration. Finally, the effects of gradient index, porosity, different heating methods and material scale parameters on the dimensionless natural frequency of porous FGM circular micro-plates are calculated and analyzed by MATLAB programming. The results show that the dimensionless natural frequencies increase with the increase of the material scale parameter; the gradient index affects the frequency, reflecting the transition characteristics of the material from ceramic to metal; and the porosity weakens the stiffness, which in turn affects the natural frequencies. The results can provide data support for the design and application of porous FGM circular micro-plates in thermal environments in the future.
文章引用：滕兆春, 薛刚, 雷飞鹏. 热环境中多孔功能梯度材料微圆板的自由振动分析[J]. 力学研究, 2024, 13(2): 39-51. https://doi.org/10.12677/ijm.2024.132005

参考文献

[1] 马涛, 赵忠民, 刘良祥, 等. 功能梯度材料的研究进展及应用前景[J]. 化工科技, 2012, 20(1): 71-75.
[2] Bever, M.B. and Shen, M. (1974) The Morphology of Polymeric Alloys. Materials Science Engineering, 15, 145-157. [Google Scholar] [CrossRef] 
[3] Jun, T. (2012) Development and Application of Functionally Gradient Materials. Proceedings of the 2012 International Conference on Industrial Control and Electronics Engineering, Xi’an, 23-25 August 2012, 1022-1025.
[4] Narayan, R.J., Hobbs, L.W., Jin, C. and Rabiei, A. (2006) The Use of Functionally Gradient Materials in Medicine. Journal of Materials Science: Materials in Medicine, 58, 52-56. [Google Scholar] [CrossRef] 
[5] Reddy, J.N. and Chin, C.D. (1998) Thermomechanical Analysis of Functionally Graded Cylinders and Plates. Journal of Thermal Stresses, 21, 593-626. [Google Scholar] [CrossRef] 
[6] Pandey, P.M., Rathee, S., Srivastava, M. and Jain, P.K., Eds. (2021) Functionally Graded Materials (FGMs): Fabrication, Properties, Applications, and Advancements. CRC Press, Boca Raton. [Google Scholar] [CrossRef] 
[7] Shen, H.S. (2009) Functionally Graded Materials: Nonlinear Analysis of Plates and Shells. CRC Press, Boca Raton.
[8] Ebrahimi, F. and Jafari, A. (2017) A Four-Variable Refined Shear-Deformation Beam Theory for Thermo-Mechanical Vibration Analysis of Temperature-Dependent FGM Beams with Porosities. Mechanics of Advanced Materials and Structures, 25, 212-224. [Google Scholar] [CrossRef] 
[9] 苏盛开, 黄怀纬. 多孔功能梯度梁的热-力耦合屈曲行为[J]. 复合材料学报, 2017, 37(12): 2794-2799.
[10] Daikh, A.A. and Zenkour, A.M. (2019) Free Vibration and Buckling of Porous Power-Law and Sigmoid Functionally Graded Sandwich Plates Using a Simple Higher-Order Shear Deformation Theory. Materials Research Express, 6, Article 115707. [Google Scholar] [CrossRef] 
[11] Nguyen, L.B., Nguyen-Xuan, H., Thai, C.H. and Phung-Van, P. (2023) A Size-Dependent Effect of Smart Functionally Graded Piezoelectric Porous Nanoscale Plates. International Journal of Mechanics and Materials in Design, 19, 817-830. [Google Scholar] [CrossRef] 
[12] Mirzaei, S., Hejazi, M. and Ansari, R. (2023) Isogeometric Analysis of Small-Scale Effects on the Vibration of Functionally Graded Porous Curved Microbeams Based on the Modified Strain Gradient Elasticity Theory. Acta Mechanica, 234, 4535-4557. [Google Scholar] [CrossRef] 
[13] Teng, Z.C., Wang, W.B. and Gu, C.L. (2022) Free Vibration and Buckling Characteristics of Porous Functionally Graded Materials (FGMs) Micro-Beams Based on the Modified Couple Stress Theory. Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 102, e202100219. [Google Scholar] [CrossRef] 
[14] Akbaş, S.D., Doğuşcan, Ş., Akgöz, B., et al. (2022) Dynamic Analysis of Functionally Graded Porous Microbeams under Moving Load. Transport in Porous Media, 142, 209-227. [Google Scholar] [CrossRef] 
[15] Rezaei, M., Khorshidvand, R.A., Khorsandijou, M.S. and Jabbari, M. (2023) Parametric Study for Modified Couple Stress Theory on Postbuckling of Size-Dependent FG Saturated Porous Mindlin Microplates. Physica Scripta, 98, Article 065958. [Google Scholar] [CrossRef] 
[16] Rahmani, A., Faroughi, S. and Friswell, M.I. (2020) The Vibration of Two-Dimensional Imperfect Functionally Graded (2D-FG) Porous Rotating Nanobeams Based on General Nonlocal Theory. Mechanical Systems and Signal Processing, 144, Article 106854. [Google Scholar] [CrossRef] 
[17] Aria, A.I., Rabczuk, T. and Friswell, M.I. (2019) A Finite Element Model for the Thermo-Elastic Analysis of Functionally Graded Porous Nanobeams. European Journal of Mechanics-A/Solids, 77, Article 103767. [Google Scholar] [CrossRef] 
[18] Reddy, J.N. and Berry, J. (2012) Nonlinear Theories of Axisymmetric Bending of Functionally Graded Circular Plates with Modified Couple Stress. Composite Structures, 94, 3664-3668. [Google Scholar] [CrossRef] 
[19] Wu, T.Y., Wang, Y.Y. and Liu, G.R. (2002) Free Vibration Analysis of Circular Plates Using Generalized Differential Quadrature Rule. Computer Methods in Applied Mechanics and Engineering, 191, 5365-5380. [Google Scholar] [CrossRef] 
[20] Pradhan, K.K. and Chakraverty, S. (2015) Free Vibration of Functionally Graded Thin Elliptic Plates with Various Edge Supports. Structural Engineering and Mechanics, 53, 337-354. [Google Scholar] [CrossRef] 

为你推荐