Size Dependent Nonlinear Bending Analysis of a Flexoelectric Functionally Graded Nano-Plate Under Thermo-Electro-Mechanical Loads

Document Type: Research Paper

Authors

1 Mechanical Engineering Department, Shahrekord University, Shahrekord, Iran

2 Faculty of Engineering, Shahrekord University, Shahrekord, Iran

10.22034/jsm.2019.569280.1296

Abstract

The effects of flexoelectricity on thermo-electro-mechanical behavior of a functionally graded electro-piezo-flexoelectric nano-plate are investigated in this paper using flexoelectric modified and the Kirchhoff classic theories. Moreover, using the variation method and the principle of minimum potential energy for the first time, the coupled governing nonlinear differential equations of the nano-plate and their associated boundary conditions are obtained.  The functionally graded nano-plate is modeled using a power law equation along the plate thickness direction. The nano-plate behavior is analyzed under mechanical, electrical, and thermal loadings with different boundary conditions. It should be noted that the direct and reverse flexoelectric effects under different loading conditions were investigated.  Finally, the important quantities such as: the nano-plate deflection, the induced electrical voltage for different values of the length parameter, the power index related to the functionally graded behavior model and the geometric ratio parameter are determined. The results indicate that in the presence of flexoelectricity, the rigidity of the nano-plate increases. Also, the deflection and the generated electric potential along nano-plate thickness decreases. Finally, induced polarization decreases as a linear temperature variation is applied on the nano-plate.

Keywords


[1] Lao C.S., Kuang Q., Wang Z. L., Park M.C., Deng Y.L., 2007, Polymer functionalized piezoelectric-FET as humidity/chemical nanosensors, Applied Physics Letters 90: 262107.
[2] Tanner S.M., Gray J.M., Rogers C.T., Bertness K.A., Sanford N.A., 2007, High-Q GaN nanowire resonators and oscillator, Applied Physics Letters 91: 203117.
[3] Kogan S.M., 1996, Piezoelectric effect during inhomogeneous deformation and acoustic scattering of carriers in crystals, Soviet Physics, Solid State 5: 2069-2070.
[4] Majdoub M.S., Sharma P., Cagin T., 2008, Enhanced size-dependent piezoelectricity and elasticity in nanostructures due to the flexoelectric effect, Physical Review B 77: 125424.
[5] Kogan S.M., 1963, Piezoelectric effect under an inhomogeneous strain and acoustic scattering of carriers in crystals, Fizika Tverdogo Tela 5: 2829-2831.
[6] Maranganti R., Sharma N.D., Sharma P., 2006, Electromechanical coupling in nonopiezoelectric materials due to nanoscale nonlocal size effects: Green’s function solutions and embedded inclusions, Physical Review B 74: 014110.
[7] Shen S.P., Hu S.L., 2010, A theory of flexoelectricity with surface effect for elastic dielectrics, Journal of the Mechanics and Physics of Solids 58: 665-677.
[8] Yan Z., Jiang L.Y., 2013, Flexoelectric effect on the electroelastic responses of bending piezoelectric nano-beams, Journal of Applied Physics 113: 194102.
[9] Yan Z., Jiang L.Y., 2013, Size-dependent bending and vibration behavior of piezoelectric nano-beams due to flexoelectricity, Journal of Physics D: Applied Physics 46: 355502.
[10] Li A., Zhou S., Qi L., Chen X., 2015, A reformulated flexoelectric theory for isotropic dielectrics, Journal of Physics D: Applied Physics 48(46): 465202.
[11] Xu Y.M., Shen H.S., Zhang C.L., 2013, Nonlocal plate model for nonlinear bending of bilayer grapheme sheets subjected to transverse loads in thermal environment, Composite Structures 98: 294-302.
[12] Chen Y., Lee J.D., Eskandarian A., 2004, Atomistic viewpoint of the applicability of microcontinuum theories, International Journal of Solids Structures 41(8): 2085-2097.
[13] Li A.Q., Zhou S.J., Zhou S.S., Wang B.L., 2014, Size-dependent analysis of a three-layer micro-beam including electromechanical coupling, Composite Structures 116: 120-127.
[14] Hadjesfandiari A.R., 2013, Size-dependent piezoelectricity, International Journal of Solids Structures 50: 2781-2791.
[15] Liang X., Shen S.P., 2013, Size-dependent piezoelectricity and elasticity due to the electric field-strain gradient coupling and strain gradient elasticity, International Applied Mechanics 5: 1350015.
[16] Hu S.L., Shen S.P., 2009, Electric field gradient theory with surface effect for nano-dielectrics, Computers, Materials and Continua 13: 63-87.
[17] Yang J.S., 1999, Equations for the extension and flexure of electroelastic plate under strong electric fields, International Journal of Solids Structures 36: 3171-3192.
[18] Liu C., Ke L.L., Wang Y.S., Yang J., Kitipornchai S., 2013, Thermo-electro mechanical vibration of piezoelectric nano-plate based on the nonlocal theory, Composite Structures 106: 167-174.
[19] Yan Z., Jiang L.Y., 2011, Electromechanical response of a curved piezoelectric nano-beam with the consideration of surface effects, Journal of Physics D: Applied Physics 44: 365301.
[20] Ke L.L., Wang Y.S., 2012, Thermoelectric-mechanical vibration of piezoelectric nano-beams based on the nonlocal theory, Smart Materials and Structures 21(2): 025018.
[21] Zhang Z., Yan Z., Jiang L., 2014, Flexoelectric effect on the electroelastic responses and vibrational behaviors of a piezoelectric nano-plate, Journal of Applied Physics 116(1):014307.
[22] Zhang Z., Jiang L., 2014, Size effects on electromechanical coupling fields of a bending piezoelectric nano-plate due to surface effects and flexoelectricity, Journal of Applied Physics 116(13): 4308.
[23] Yang W., Liang X., Shen S., 2015, Electromechanical responses of piezoelectric nano-plate with flexoelectricity, Acta Mechanica 226: 3097-3110.
[24] Yan Z., Jiang L.Y., 2012, Vibration and buckling analysis of a piezoelectric nano-plate considering surface effects and in-plane constraint, Proceedings of the Royal Society of Series A 468: 3458-3475.
[25] Yan Z., Jiang L.Y, 2013, Size-dependent bending and vibration behavior of piezoelectric nano-beams due to flexoelectricity, Journal of Physics D: Applied Physics 46: 355502.
[26] Murmua T., Sienz J., Adhikari S., Arnold C., 2013, Nonlocal buckling of double-nano-plate-systems under biaxial compression, Composites Part B 44: 84-94.
[27] Li Y.S., Cai Z.Y., Shi S.Y., 2014, Buckling and free vibration of magneto electroelastic nano-plate based on nonlocal theory, Composite Structures 111: 522-529.
[28] Liang X., Hu S., Shen S., 2014, Effects of surface and flexoelectricity on a piezoelectric nano-beam, Smart Materials and Structures 23: 035020.
[29] Liang X., Shuling H., Shengping S., 2015, Size-dependent buckling and vibration behaviors of piezoelectric nanostructures due to flexoelectricity, Smart Materials and Structures 24: 105012.
[30] Yan Z., Jiang L.Y., 2015, Effect of flexoelectricity on the electroelastic fields of a hollow piezoelectric nanocylinder, Smart Materials and Structures 24: 065003.
[31] Ke L., Liu C., Wang Y.S., 2015, Free vibration of nonlocal piezoelectric nano-plate under various boundary conditions, Physica E 66: 93-106.
[32] Liu C., Ke L.L., Wang Y.S., Yang J., Kitipornchai S., 2013, Thermo-electro-mechanical vibration of piezoelectric nano-plate based on the nonlocal theory, Composite Structures 106: 167-174.
[33] Liang X., Yang W., Hu S., Shen S., 2016, Buckling and vibration of flexoelectric nanofilms subjected to mechanical loads, Journal of Physics D: Applied Physics 49: 115307.
[34] Alibeigi E., Tadi Beni Y., Mehralian F. 2018, Thermal buckling of magneto-electro- elastic piezoelectric nano-beams based on the modified couple stress theory, The European Physical Journal Plus 133: 133.
[35] Komijani M., Kiani Y., Esfahani S., Eslami M.,2 013, Vibration of thermo-electrically post-buckled rectangular functionally graded piezoelectric beams, Composite Structures 98: 143-152.
[36] Komijani M., Reddy J., Eslami M., 2014, Nonlinear analysis of microstructure-dependent functionally graded piezoelectric material actuators, Journal of the Mechanics and Physics of Solids 63: 214-227.
[37] Xiang H.J., Shi Z., 2009, Static analysis for functionally graded piezoelectric actuators or sensors under a combined electrothermal load, European Journal of Mechanics A 28: 338-346.
[38] Yang J., Xiang H., 2007, Thermo-electro-mechanical characteristics of functionally graded piezoelectric actuators, Smart Materials and Structures 16: 784.
[39] Tadi Beni Y., 2016, A nonlinear electro-mechanical analysis of nano-beams based on the size-dependent piezoelectricity, Journal of Mechanics 33: 289-301.
[40] Ke L.L., Wang Y.S., Yang J., Kitipornchai S., 2014, Free vibration of size-dependent magneto-electro-elastic nano-plate based on the nonlocal theory, Acta Mechanica Sinica 30: 516-525.
[41] Ke L.L, Wang Y.S., Reddy J.N., 2014, Thermo-electro-mechanical vibration of size-dependent piezoelectric cylindrical nanoshells under various boundary conditions, Composite Structures 116: 626-636.
[42] Kiani Y., Taheri S., Eslami M.R., 2011, Thermal buckling of piezoelectric functionally graded material beams, Journal of Thermal Stresses 34: 835-850.
[43] Kiani Y., Rezaei M., Taheri S., Eslami M.R., 2011, Thermo-electrical buckling of piezoelectric functionally graded material Timoshenko beams, International Journal of Mechanics and Materials in Design 7: 185-197.
[44] Ebrahimi F., Barati M.R., 2017, Vibration analysis of size-dependent flexoelectricnano-plates incorporating surface and thermal effects, Journal of Mechanics of Advanced Materials and Structures 25: 611-621.
[45] Ebrahimi F., Barati M.R., 2017, Modeling of smart magnetically affected flexoelectric/piezoelectric nanostructures incorporating surface effects, Nanomaterials and Nanotechnology 7(2): 1-11.
[46] Ebrahimi F., Ehyaei J., Babaei R., 2016, Thermal buckling of FGM nano-plates subjected to linear and nonlinear varing loads on Pasternak foundation, Advanced in Materials Research 5: 245-261.
[47] Ebrahimi F., Salari E., 2015, Size-dependent thermo-electrical buckling analysis of functionally graded piezoelectric nano-beams, Smart Materials and Structures 24: 125007.
[48] Ebrahimi F., Barati M.R., 2016, An exact solution for buckling analysis of embedded piezo- electro-magnetically actuated nanoscale beams, Advances in Nano Research 4: 65-84.
[49] Ebrahimi F., Barati M.R., 2017, Surface effects on the vibration behavior of flexoelectric nanobeams based on nonlocal elasticity theory, The European Physical Journal Plus 132: 11320.
[50] Ebrahimi F., Barati M.R., 2016, Electromechanical buckling behavior of smart piezoelectrically actuated higher-order size dependent graded nanoscale beams in thermal environment, International Journal of Smart and Nano Materials 7: 69-90.
[51] Tadi Beni, Y., 2016, Size-dependent analysis of piezoelectric nano-beams including electro-mechanical coupling, Mechanics Research Communications 75: 67-80.
[52] Shen S.P, Hu S.L., 2010, A theory of flexoelectricity with surface effect for elastic dielectrics, Journal of the Mechanics and Physics of Solids 58: 665-677.
[53] Eliseev E.A., Morozovska A.N., Glinchuk M.D., Blinc R., 2009, Spontaneous flexoelectric-flexomagnetic effect in nanoferroics, Physical Review B 79: 165433.
[54] Li J.U., 2004, The effective pyroelectric and thermal expansion coefficients of ferroelectric ceramics, Mechanics of Material 36: 949-958.
[55] Toupin R.A., 1956, The elastic dielectric, Rational Mechanics and Analysis 5: 849-915.
[56] Tadi Beni Y., Mehralian F., Razavi H., 2015, Free vibration analysis of size-dependent shear deformable functionally graded cylindrical shell on the basis of modified couple stress theory, Composite Structures 120: 65-78.
[57] Mehralian F., TadiBeni Y., 2016, Ansari R., Size dependent buckling analysis of functionally graded piezoelectric cylindrical nanoshell, Journal of Composite Structures 152: 45-61.
[58] Fei L., Zhuo X., Xiaoyong W., Xi Y., 2009, Determination of temperature dependence of piezoelectric coefficients matrix of lead zirconate titanate ceramics by quasi-static and resonance method, Journal of Physics D: Applied Physics 42: 095417.