Goodarzi, M., Mohammadi, M., Khooran, M., Saadi, F. (2016). Thermo-Mechanical Vibration Analysis of FG Circular and Annular Nanoplate Based on the Visco-Pasternak Foundation. Journal of Solid Mechanics, 8(4), 788-805.

M Goodarzi; M Mohammadi; M Khooran; F Saadi. "Thermo-Mechanical Vibration Analysis of FG Circular and Annular Nanoplate Based on the Visco-Pasternak Foundation". Journal of Solid Mechanics, 8, 4, 2016, 788-805.

Goodarzi, M., Mohammadi, M., Khooran, M., Saadi, F. (2016). 'Thermo-Mechanical Vibration Analysis of FG Circular and Annular Nanoplate Based on the Visco-Pasternak Foundation', Journal of Solid Mechanics, 8(4), pp. 788-805.

Goodarzi, M., Mohammadi, M., Khooran, M., Saadi, F. Thermo-Mechanical Vibration Analysis of FG Circular and Annular Nanoplate Based on the Visco-Pasternak Foundation. Journal of Solid Mechanics, 2016; 8(4): 788-805.

Thermo-Mechanical Vibration Analysis of FG Circular and Annular Nanoplate Based on the Visco-Pasternak Foundation

^{1}Department of Engineering, College of Mechanical Engineering, Ahvaz Branch, Islamic Azad University, Ahvaz, Iran

^{2}Department of Mechanical Engineering, Shahid Chamran University of Ahvaz, Iran

Abstract

In this study, the vibration behavior of functional graded (FG) circular and annular nanoplate embedded in a Visco-Pasternak foundation and coupled with temperature change is studied. The effect of in-plane pre-load and temperature change are investigated on the vibration frequencies of FG circular and annular nanoplate. To obtain the vibration frequencies of the FG circular and annular nanoplate, two different size dependent theories are utilized. The material properties of the FGM nanoplates are assumed to vary in the thickness direction and are estimated through the Mori–Tanaka homogenization technique. The FG circular and annular nanoplate is coupled by an enclosing viscoelastic medium which is simulated as a Visco- Pasternak foundation. By using the modified strain gradient theory (MSGT) and the modified couple stress theory (MCST), the governing equation is derived for FG circular and annular nanoplate. The differential quadrature method (DQM) and the Galerkin method (GM) are utilized to solve the governing equation to obtain the frequency vibration of FG circular and annular nanoplate. The results are subsequently compared with valid result reported in the literature. The effects of the size dependent, the in-plane pre-load, the temperature change, the power index parameter, the elastic medium and the boundary conditions on the natural frequencies are investigated. The results show that the size dependent parameter has an increasing effect on the vibration response of circular and annular nanoplate. The temperature change also play an important role in the mechanical behavior of the FG circular and annular nanoplate. The present analysis results can be used for the design of the next generation of nanodevices that make use of the thermal vibration properties of the nanoplate

[1] Sallese J.M., Grabinski W., Meyer V., Bassin C., Fazan P., 2001, Electrical modeling of a pressure sensor MOSFET, Sensors and Actuators A: Physical 94: 53-58. [2] Nabian A., Rezazadeh G., Haddad-derafshi M., Tahmasebi A., 2008, Mechanical behavior of a circular microplate subjected to uniform hydrostatic and non-uniform electrostatic pressure, Microsystem Technologies 14: 235-240. [3] Bao M., Wang W., 1996, Future of microelectromechanical systems (MEMS), Sensors and Actuators A: Physical 56: 135-141. [4] Younis M.I., Abdel-Rahman E.M., Nayfeh A., 2003, A reduced-order model for electrically actuated microbeambased MEMS, Journal of Microelectromechanical Systems 12: 672-680. [5] Batra R.C., Porﬁri M., Spinello D, 2006, Electromechanical model of electrically actuated narrow microbeams, Journal of Microelectromechanical Systems 15: 1175-1189. [6] Nayfeh A.H., Younis M.I., Abdel-Rahman E.M., 2007, Dynamic pull-in phenomenon in MEMS resonators, Nonlinear Dynamics 48: 153-163. [7] Nayfeh A.H., Younis M.I., 2004, Modeling and simulations of thermoelastic damping in microplates, Journal of Micromechanics and Microengineering 14: 1711-1717. [8] Zhao X.P., Abdel-Rahman E.M., Nayfeh A.H., 2004, A reduced-order model for electrically actuated microplates, Journal of Micromechanics and Microengineering 14: 900-906. [9] Machauf A., Nemirovsky Y., Dinnar U., 2005, A membrane micropump electrostatically actuated across the working ﬂuid, Journal of Micromechanics and Microengineering 15: 2309-2316. [10] Wong E.W., Sheehan P.E., Lieber C.M., 1997, Nanobeam mechanics: elasticity, strength and toughness of nanorods and nanotubes, Science 277: 1971-1975. [11] Zhou S.J., Li Z.Q., 2001, Metabolic response of Platynota stultana pupae during and after extended exposure to elevated CO2 and reduced O2 atmospheres, Shandong University Technology 31: 401-409. [12] Mohammadi V., Ansari R., Faghih Shojaei M., Gholami R., Sahmani S., 2013, Size-dependent dynamic pull-in instability of hydrostatically and electrostatically actuated circular microplates, Nonlinear Dynamics 73: 1515-1526. [13] 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. [14] Eringen A.C., 1983, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal Applied Physics 54: 4703-4711. [15] Asemi H.R., Asemi S.R., Farajpour A., Mohammadi M., 2015, Nanoscale mass detection based on vibrating piezoelectric ultrathin films under thermo-electro-mechanical loads, Physica E 68: 112-122. [16] Danesh M., Farajpour A., Mohammadi M., 2012, Axial vibration analysis of a tapered nanorod based on nonlocal elasticity theory and differential quadrature method, Mechanics Research Communications 39: 23-27. [17] Aydogdu M., 2009, Axial vibration of the nanorods with the nonlocal continuum rod model, Physica E 41: 861-864. [18] Mohammadi M., Farajpour A., Goodarzi M., Heydarshenas R., 2013, Levy type solution for nonlocal thermo-mechanical vibration of orthotropic mono-layer graphene sheet embedded in an elastic medium, Journal of Solid Mechanics 5(2): 116-132. [19] Mohammadi M., Farajpour A., Goodarzi M., Dinari F., 2014, Thermo-mechanical vibration analysis of annular and circular graphene sheet embedded in an elastic medium, Latin American Journal of Solids and Structures 11 (4): 659-682. [20] Duan W. H., Wang C. M., 2007, Exact solutions for axisymmetric bending of micro/nanoscale circular plates based on nonlocal plate theory, Nanotechnology 18: 385704. [21] Mohammadi M., Moradi A., Ghayour M., Farajpour A., 2014, Exact solution for thermo-mechanical vibration of orthotropic mono-layer graphene sheet embedded in an elastic medium, Latin American Journal of Solids and Structures 11(3): 437-458. [22] Asemi S.R., Farajpour A., Asemi H.R., Mohammadi M., 2014, Influence of initial stress on the vibration of double-piezoelectric-nanoplate systems with various boundary conditions using DQM, Physica E 63: 169-179. [23] Mohammadi M., Goodarzi M., Ghayour M. Farajpour A., 2013, Inﬂuence of in-plane pre-load on the vibration frequency of circular graphene sheet via nonlocal continuum theory, Composites Part B 51: 121-129. [24] Goodarzi M., Mohammadi M., Farajpour A., Khooran M., 2014, Investigation of the effect of pre-stressed on vibration frequency of rectangular nanoplate based on a visco pasternak foundation, Journal of Solid Mechanics 6: 98-121. [25] Farajpour A., Shahidi A. R., Mohammadi M., Mahzoon M., 2012, Buckling of orthotropic micro/nanoscale plates under linearly varying in-plane load via nonlocal continuum mechanics, Composite Structures 94: 1605-1615. [26] Mohammadi M., Farajpour A., Goodarzi M., Mohammadi H., 2013, Temperature effect on vibration analysis of annular graphene sheet embedded on visco-Pasternak foundation, Journal of Solid Mechanics 5(3): 305-323. [27] Wang C.M., Duan W.H., 2008, Free vibration of nanorings/arches based on nonlocal elasticity, Journal of Applied Physics 104: 014303. [28] Moosavi H., Mohammadi M., Farajpour A., Shahidi S. H., 2011, Vibration analysis of nanorings using nonlocal continuum mechanics and shear deformable ring theory, Physica E 44: 135-140. [29] Mohammadi M., Goodarzi M., Ghayour M., Alivand S., 2012, Small scale effect on the vibration of orthotropic plates embedded in an elastic medium and under biaxial in-plane pre-load via nonlocal elasticity theory, Journal of Solid Mechanics 4(2) : 128-143. [30] Wang B., Zhao J., Zhou S., 2010, A micro scale Timoshenko beam model based on strain gradient elasticity theory, European Journal of Mechanics A/Solids 29: 591-599. [31] Ansari R., Gholami R., Sahmani S., 2011, Free vibration of size-dependent functionally graded microbeams based on a strain gradient theory, Composite Structures 94: 221-228. [32] Ansari R., Gholami R., Sahmani S., 2012, Study of small scale effects on the nonlinear vibration response of functionally graded Timoshenko microbeams based on the strain gradient theory, Journal of Computational and Nonlinear Dynamics 7: 031010 . [33] Sahmani S., Ansari R., 2013, On the free vibration response of functionally graded higher-order shear deformable microplates based on the strain gradient elasticity theory, Composite Structures 95: 430-442. [34] Ghayesh M.H., Amabili M., Farokhi H., 2013, Nonlinear forced vibrations of a microbeam based on the strain gradient elasticity theory, International Journal of Engineering Science 63: 52-60. [35] Mohammadi M., Farajpour A., Moradi M., Ghayour M., 2013, Shear buckling of orthotropic rectangular graphene sheet embedded in an elastic medium in thermal environment, Composites Part B: Engineering 56: 629-637. [36] Civalek Ö., Akgöz B., 2013, Vibration analysis of micro-scaled sector shaped graphene surrounded by an elastic matrix, Computational Materials Science 77: 295-303. [37] Murmu T., Pradhan S.C., 2009, Vibration analysis of nano-single-layered graphene sheets embedded in elastic medium based on nonlocal elasticity theory, Journal of Applied Physics 105: 064319. [38] Pradhan S.C., Phadikar J.K., 2009, Small scale effect on vibration of embedded multi layered graphene sheets based on nonlocal continuum models, Physics Letters A 373: 1062-1069. [39] Wang Y. Z., Li F. M., Kishimoto K., 2011, Thermal effects on vibration properties of doublelayered nanoplates at small scales, Composites Part B: Engineering 42: 1311-1317. [40] Reddy C.D., Rajendran S., Liew K.M., 2006, Equilibrium configuration and continuum elastic properties of finite sized graphene, Nanotechnology 17: 864-870. [41] Aksencer T., Aydogdu M., 2011, Levy type solution method for vibration and buckling of nanoplates using nonlocal elasticity theory, Physica E 43: 954-959. [42] Malekzadeh P., Setoodeh A.R., Alibeygi Beni A., 2011, Small scale effect on the thermal buckling of orthotropic arbitrary straight-sided quadrilateral nanoplates embedded in an elastic medium, Composite Structures 93: 2083-2089. [43] Satish N., Narendar S., Gopalakrishnan S., 2012, Thermal vibration analysis of orthotropic nanoplates based on nonlocal continuum mechanics, Physica E 44: 1950-1962. [44] Prasanna Kumar T.J., Narendar S., Gopalakrishnan S., 2013, Thermal vibration analysis of monolayer graphene embedded in elastic medium based on nonlocal continuum mechanics, Composite Structures 100: 332-342. [45] Farajpour A., Mohammadi M., Shahidi A.R., Mahzoon M., 2011, Axisymmetric buckling of the circular graphene sheets with the nonlocal continuum plate model, Physica E 43: 1820-1825. [46] Mohammadi M., Ghayour M., Farajpour A., 2013, Free transverse vibration analysis of circular and annular graphene sheets with various boundary conditions using the nonlocal continuum plate model, Composites Part B: Engineering 45: 32-42. [47] Lam D.C.C., Yang F., Chong A.C.M., Wang J., Tong P., 2003, Experiments and theory in strain gradient elasticity, Journal of the Mechanics and Physics of Solids 51: 1477-1508. [48] Timoshenko S.P., Goodier J.N., 1970, Theory of Elasticity, McGraw-Hill, New York. [49] Saadatpour M. M., Azhari M., 1998, The Galerkin method for static analysis of simply supported plates of general shape, Computers and Structures 69: 1-9. [50] Farajpour A., Danesh M., Mohammadi M., 2011, Buckling analysis of variable thickness nanoplates using nonlocal continuum mechanics, Physica E 44: 719-727. [51] Shu C., 2000, Differential Quadrature and its Application in Engineering, Berlin, Springer. [52] Bert C.W., Malik M., 1996, Differential quadrature method in computational mechanics: a review, Applied Mechanics Reviews 49: 1-27. [53] Malekzadeh P., Setoodeh A. R., Alibeygi Beni A., 2011, Small scale effect on the thermal buckling of orthotropic arbitrary straight-sided quadrilateral nanoplates embedded in an elastic medium, Composite Structures 93: 2083-2089. [54] Wang Y. Z., Li F. M., Kishimoto K., 2011, Thermal effects on vibration properties of double-layered nanoplates at small scales, Composites Part B: Engineering 42: 1311-1317. [55] Ke L. L., Yang J., Kitipornchai S., Bradford M. A., 2012, Bending, buckling and vibration of size-dependent functionally graded annular microplates, Composite Structures 94: 3250-3257. [56] Leissa A.W., Narita Y., 1980, Natural frequencies of simply supported circular plates, Journal of Sound and Vibration 70: 221-229. [57] Kim C.S., Dickinson S.M., 1989, On the free, transverse vibration of annular and circular, thin, sectorial plates subject to certain complicating effects, Journal of Sound and Vibration 134: 407-421. [58] Qiang L.Y., Jian L., 2007, Free vibration analysis of circular and annular sectorial thin plates using curve strip Fourier P-element, Journal of Sound and Vibration 305: 457-466. [59] Zhou Z.H., Wong K.W., Xu X.S., Leung A.Y.T., 2011, Natural vibration of circular and annular thin plates by Hamiltonian approach, Journal of Sound and Vibration 330: 1005-1017. [60] Carrington H., 1925, The frequencies of vibration of ﬂat circular plates ﬁxed at the circumference, Philosophical Magazine 6: 1261-1264. [61] Leissa A.W., 1969, Vibration of Plates, Ofﬁce of Technology Utilization, Washington. [62] Chakraverty S., Bhat R.B., Stiharu I., 2001, Free vibration of annular elliptic plates using boundary characteristic orthogonal polynomials as shape functions in the Rayleigh–Ritz method, Journal of Sound and Vibration 241: 524-539.