The Effect of Modified Couple Stress Theory on Buckling and Vibration Analysis of Functionally Graded Double-Layer Boron Nitride Piezoelectric Plate Based on CPT

Document Type : Research Paper


Department of Solid Mechanics ,Faculty of Mechanical Engineering, University of Kashan


In this article, the effect of size-dependent on the buckling and vibration analysis of functionally graded (FG) double-layer boron nitride plate based on classical plate theory (CPT) under electro-thermo-mechanical loadings which is surrounded by elastic foundation is examined. This subject is developed using modified couple stress theory. Using Hamilton's principle, the governing equations of motion are obtained by applying a modified couple stress and von Karman nonlinear strain for piezoelectric material and Kirchhoff plate. These equations are coupled for the FG double-layer plate using Pasternak foundation and solved using Navier’s type solution. Then, the dimensionless natural frequencies and critical buckling load for simply supported boundary condition are obtained. Also, the effects of material length scale parameter, elastic foundation coefficients and power law index on the dimensionless natural frequency and critical buckling load are investigated. The results demonstrate that the dimensionless natural frequency of the piezoelectric plate increases steadily by growing the power law index. ‌‌Also, the effect of the power law index on the dimensionless critical buckling load of double layer boron nitride piezoelectric for higher dimensionless material length scale parameter is the most.


[1] Yang F., Chong A.C.M, Lam D.C.C., Tong P., 2002, Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures 39: 2731-2743.
[2] Wang L., 2011, A modified nonlocal beam model for vibration and stability of nanotubes conveying fluid, Physica E: Low-dimensional Systems and Nanostructures 44: 25-28.
[3] Akgoz B., Civalek O., 2011, Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams, International Journal of Engineering Science 49: 1268-1280.
[4] Chen W., Li L., Xua M., 2011, A modified couple stress model for bending analysis of composite laminated beams with first order shear deformation, Composite Structures 93: 2723-2732.
[5] Asghari M., 2012, Geometrically nonlinear micro-plate formulation based on the modified couple stress theory, International Journal of Engineering Science 51: 292-309.
[6] Reddy J.N., Kim J., 2012, A nonlinear modified couple stress-based third-order theory of functionally graded plates, Composite Structures 94: 1128-1143.
[7] Chen W., Wei C, Sze K.Y., 2012, A model of composite laminated Reddy beam based on a modified couple-stress theory, Composite Structures 94: 2599-2609.
[8] Chen W., XubM., Li L., 2012, A model of composite laminated Reddy plate based on new modified couple stress theory, Composite Structures 94: 2143-2156.
[9] Ke L.L., Wang Y.S., Yang J., Kitipornchai S., 2012, Nonlinear free vibration of size-dependent functionally graded microbeams, International Journal of Engineering Science 50: 256-267.
[10] Wang L., Xu Y.Y., Ni Q., 2013, Size-dependent vibration analysis of three-dimensional cylindrical microbeams based on modified couple stress theory: A unified treatment, International Journal of Engineering Science 68: 1-10.
[11] Farokhi H., Ghayesh M.H., Amabili M., 2013, Nonlinear dynamics of a geometrically imperfect microbeam based on the modified couple stress theory, International Journal of Engineering Science 68: 11-23.
[12] Simsek M., Reddy J.N., 2013, A unified higher order beam theory for buckling of a functionally graded microbeam embedded in elastic medium using modified couple stress theory, Composite Structures 101: 47-58.
[13] Ghorbanpour Arani A., Rahnama Mobarakeh M., Shams Sh., Mohammadimehr M., 2012, The effect of CNT volume fraction on the magneto-thermo-electro-mechanical behavior of smart nanocomposite cylinder, Journal of Mechanical Science and Technology 26 (8): 2565-2572.
[14] Mohammadimehr M., Rousta Navi B., Ghorbanpour Arani A., 2015, Surface stress effect on the nonlocal biaxial buckling and bending analysis of polymeric piezoelectric nanoplate reinforced by CNT using Eshelby-Mori-Tanaka approach, Journal of Solid Mechanics 7(2) :173-190
[15] Mohammadimehr M., Golzari E., 2014, The elliptic phenomenon effect of cross section on the torsional buckling of a nanocomposite beam reinforced by a single-walled carbon nanotube, Proceedings of the Institution of Mechanical Engineers, Part N: Journal of Nanoengineering and Nanosystems, doi:10.1177/1740349914552307.
[16] Mohammadimehr M., Mohandes M., Moradi M., 2014, Size dependent effect on the buckling and vibration analysis of double bonded nanocomposite piezoelectric plate reinforced by boron nitride nanotube based on modified couple stress theory, Journal of Vibration and Control, doi:10.1177/1077546314544513.
[17] Asghari M., Ahmadian M.T., Kahrobaiyan M.H., Rahaeifard M., 2010, On the size-dependent behavior of functionally graded micro-beams, Materials and Design 31: 2324-2329.
[18] Liew K.M., Yang J., Kitipornchai S., 2003, Postbuckling of piezoelectric FGM plates subject to thermo-electro-mechanical loading, International Journal of Solids and Structures 40: 3869-3892.
[19] Rao B.N., Kuna M., 2008, Interaction integrals for fracture analysis of functionally graded piezoelectric materials, International Journal of Solids and Structures 45: 5237-5257.
[20] Golmakani M.E., Kadkhodayan M., 2011, Nonlinear bending analysis of annular FGM plates using higher-order shear deformation plate theories, Composite Structures 93: 973-982.
[21] Kim J., Reddy J.N., 2013, Analytical solutions for bending, vibration, and buckling of FGM plates using a couple stress-based third-order theory, Composite Structures 103: 86-98.
[22] Liu C., Ke L.L., Wanga Y.S., Yang J., Kitipornchai S., 2013, Thermo-electro-mechanical vibration of piezoelectric nanoplates based on the nonlocal theory, Composite Structures 106: 167-174.
[23] Ozgan K., Daloglu A.T., 2008, Effect of transverse shear strains on plates resting on elastic foundation using modified Vlasov model, Thin-Walled Structures 46: 1236-1250.
[24] Fallah A., Aghdam M.M., 2011, Nonlinear free vibration and post-buckling analysis of functionally graded beams on nonlinear elastic foundation, European Journal of Mechanics A/Solids 30: 571-583.
[25] Ghorbanpour Arani A., Hashemian M., Loghman A., Mohammadimehr M., 2011, Study of dynamic stability of the double-walled carbon nanotube under axial loading embedded in an elastic medium by the energy method, Journal of applied mechanics and technical physics 52 (5): 815-824.
[26] Zenkour A.M., 2010, Hygro-thermo-mechanical effects on FGM plates resting on elastic foundations, Composite Structures 93: 234-238.
[27] Kiani Y., Akbarzadeh A.H., Chen Z.T., Eslami M.R., 2012, Static and dynamic analysis of an FGM doubly curved panel resting on the Pasternak-type elastic foundation, Composite Structures 94: 2474-2484.
[28] Khalili S.M.R., Abbaspour P., Malekzadeh Fard K., 2013, Buckling of non-ideal simply supported laminated plate on Pasternak foundation, Applied Mathematics and Computation 219: 6420-6430.
[29] Rahmati A. H., Mohammadimehr M., 2014, Vibration analysis of non-uniform and non-homogeneous boron nitride nanorods embedded in an elastic medium under combined loadings using DQM, Physica B: Condensed Matter 440: 88-98.
[30] Brush D., Almroth B., 1975, Buckling of Bars, Plates and Shells, McGraw-Hill, New York.
[31] Thai H.T., Vo T.H.P., 2013, A size-dependent functionally graded sinusoidal plate model based on a modified couple stress theory, Composite Structures 96: 376-383.
[32] Thai H.T., Kim S.E., 2013 A size-dependent functionally graded Reddy plate model based on a modified couple stress theory, Composites: Part B 45: 1636-1645.
[33] Reddy J.N., 2003, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, CRC Press, Boca Raton.
[34] Yang J., Ke L.L., Kitipornchai S., 2010, Nonlinear free vibration of single-walled carbon nanotubes using nonlocal Timoshenko beam theory, Physica E 42: 1727-1735.
[35] Thai H.T., Choi D.H., 2013, Size-dependent functionally graded Kirchhoff and Mindlin plate models based on a modified couple stress theory, Composite Structures 95: 142-153.
[36] Mohammadimehr M., Saidi A.R., Ghorbanpour Arani A., Arefmanesh A., Han Q., 2010, Torsional buckling of a DWCNT embedded on winkler and pasternak foundations using nonlocal theory, Journal of Mechanical Science and Technology 24(6) : 1289-1299.
[37] Mosallaie Barzoki A.A., Ghorbanpour Arani A., Kolahchi R., Mozdianfard M.R., 2012, Electro-thermo-mechanical torsional buckling of a piezoelectric polymeric cylindrical shell reinforced by DWBNNTs with an elastic core, Applied Mathematical Modelling 36: 2983-2995.
[38] Ghorbanpour Arani A., Hashemian M., 2012, Electro-Thermo-Dynamic Buckling of Embedded DWBNNT Conveying Viscous Fluid, Journal of Solid Mechanics 4: 15-32.
[39] Mohammadimehr M., Rahmati A. H., 2013, Small scale effect on electro-thermo-mechanical vibration analysis of single-walled boron nitride nanorods under electric excitation, Turkish Journal of Engineering & Environmental Sciences 37 :1-15.