本文基於協合應力偶理論(Consistent couple stress theory , CCST)，結合多重時間尺度方法，發展一套三維漸近理論，對嵌入彈性介質之簡支撐功能性微米板進行三維自由振動分析。文中假設材料組成之體積分率沿板厚度方向作冪次分佈，材料特性遵守二相複合材料混合定律，以預測複合材料中有效楊氏模數及有效質量密度等數值。此外，利用Winkler或Pasternak之基礎模型模擬微米板與基礎介質間的交互作用關係。文中結合多種數學方法，如多重尺度法、無因次化、漸近展開與連續積分等，獲得多階遞歸運動方程組。在首階問題中其控制方程式之微分運算子與古典板理論(Classical plate theory, CPT)相同，然發展至高階問題將變得越加複雜，出現許多不同的非齊次項，但階數間仍共用相同微分運算子，使公式易於歸納。文章最後依不同材料參數、微米板之幾何參數及不同模態頻率等，將結果繪製成圖表後，與其他文獻作比較及討論，結果顯示微小長度因子對微米板之振動頻率影響顯著。

Based on the consistent couple stress theory (CCST) incorporated with the multiple time scale method, the authors develop an asymptotic theory for a three-dimensional (3D) free vibration analysis of simply-supported, functionally graded microscale plates resting on an elastic medium. The material properties of the microplate are assumed to obey a power-law distribution of the volume fractions of the constituents through the thickness direction of the microplate, for which the effective material properties are estimated using the rule of mixtures. The interactions between the microplate and its foundation medium are simulated using either a Winkler or a Pasternak foundation model. Performing nondimensionalization, asymptotic expansion, and successive integration, the authors obtain recursive sets of motion equations for various order problems. The CCST-based classical plate theory (CPT) is derived as a first-order approximation of the 3D CCST. The motion equations for higher-order problems retain the same differential operators as those of the leading order problem although with different nonhomogeneous terms. Some 3D asymptotic solutions for the lowest natural frequency parameters of the microplate resting on an elastic medium at different vibration modes are carried out, for which they are shown to converge rapidly and to be in excellent agreement with the exact microplate solutions available in the literature when the material length scale parameter is taken to be zero.

[1] A.C.M. Chong, F. Yang, D.C.C. Lam, P. Tong, Torsion and bending of micron-scaled structures. J. Mater. Res. 14 (2001) 1052-1058.
[2] D.C.C. Lam, A.C.M. Chong, Indentation model and strain gradient plasticity law for glassy polymers. J. Mater. Res. 14 (1999) 3784-3788.
[3] W. Voigt Theoretische Studien fiber die Elastizitatsverhiltnisse der Kristalle (Theoretical Studies on the Elasticity Relationships of Crystals) Abh. Gesch. Wissenschaften (1887) 34.
[4] E. Cosserat, F. Cosserat Théorie des corps déformables (Theory of Deformable Bodies) A. Hermann et Fils, Paris (1909)
[5] R.D. Mindlin, H.F. Tiersten, Effects of couple-stresses in linear elasticity. Arch. Rational Mech. Anal. 11 (1962) 415-488.
[6] W.T. Koiter, Couple stresses in the theory of elasticity. I and II. Proc. Ned. Akad. Wet. (b) 67 (1964) 17-44.
[7] A.C. Eringen, Theory of micropolar elasticity, Fracture, 2, ed. H. Liebowitz, Academic Press, New York, 2 (1968) 662-729.
[8] F. Yang, A.C.M. Chong, D.C.C. Lam, P. Tong, Couple stress based strain gradient theory for elasticity. Int. J. Solids Sturct. 39 (2002) 2731-2743.
[9] A.R. Hadjesfandiari, G.F. Dargush, Couple stress theory for solids. Int. J. Solids Struct. 48 (2011) 2496-2510.
[10] A.R. Hadjesfandiari, Size-dependent thermoelasticity. Latin Amer. J. Solids Struct. 11 (2014) 1679-1708.
[11] A.R. Hadjesfandiari, Size-dependent piezoelectricity. J. Solids Struct. 50 (2013) 2781-2791.
[12] S.E. Alavi, M. Sadighi, M.D. Pazhooh, J.F. Ganghoffer, Development of size-dependent consistent couple stress theory of Timoshenko beams. Appl. Math. Modell. 79 (2020) 685-712.
[13] B.N. Patel, D. Pandit, S.M. Srinivasan, A simplified moment-curvature based approach for large deflection analysis of micro-beams using the consistent couple stress theory. Eur. J. Mech. A/Solids 66 (2017) 45-54.
[14] M. Ajri, M.M.S. Fakhrabadi, A. Rastgoo, Analytical solution for nonlinear dynamic behavior of viscoelastic nano-plates modelled by consistent couple stress theory. Latin Amer. J. Solids Struct. 15 (2018) e113.
[15] S.F. Dehkordi, Y.T. Beni, Electro-mechanical free vibration of single-walled piezoelectric/flexoelectric nano cones using consistent couple stress theory. Int. J. Mech. Sci. 128-129 (2017) 125-139.
[16] H. Razavi, A.F. Babadi, Y.T. Beni, Free vibration analysis of functionally graded piezoelectric cylindrical nanoshell based on consistent couple stress theory. Compos. Struct. 160 (2017) 1299-1309.
[17] A.H. Nayfeh, Introduction to Perturbation Techniques. John Wiley & Sons, Inc, New York, 1993.
[18] S. Srinivas, C.V. Joga Rao, A.K. Rao, An exact analysis for vibration of simply-supported homogeneous and laminated thick rectangular plates. J. Sound Vib. 12 (1970) 187-199.
[19] D.K. Jha, T. Kant, R.K. Singh, Free vibration of functionally graded plates with a higher-order shear and normal deformation theory. Int. J. Struct. Stab. Dyn. 13 (2013) 1350004.
[20] A. Nosier, J.N. Reddy, On vibration and buckling of symmetric laminated plates according to shear deformation theories. Part II, Acta Mech. 94 (1992) 145-169.
[21] M. Levinson, An accurate simply-supported theory of the statics and dynamics of elastic plates. Mech. Res. Commun. 7 (1980) 343-350.
[22] A.M. Zenkour, On vibration of functionally graded plates according to a refined trigonometric plate theory. Int. J. Struct. Stab. Dyn. 5 (2005) 279-297.
[23] T. Kant, K. Swaminathan, Free vibration of isotropic, orthotropic and multilayer plate based on higher-order refined theories. J. Sound Vib. 241 (2001) 319-327.
[24] R.P. Shimpi, H.G. Patel, A two variable refined plate theory for orthotropic plate analysis. Int. J. Solids Struct. 43 (2006) 6783-6799.
[25] H.T. Thai, S.E. Kim, A size-dependent functionally graded Reddy plate model based on a modified couple stress theory. Compos. Part B 45 (2013) 1636-1645.