Moradi, A., Yaghootian, A., Jalalvand, M., Ghanbarzadeh, A. (2018). Magneto-Thermo mechanical vibration analysis of FG nanoplate embedded on Visco Pasternak foundation. Journal of Computational Applied Mechanics, 49(2), 395-407. doi: 10.22059/jcamech.2018.261764.300

Abbas Moradi; Amin Yaghootian; Mehdi Jalalvand; Afshin Ghanbarzadeh. "Magneto-Thermo mechanical vibration analysis of FG nanoplate embedded on Visco Pasternak foundation". Journal of Computational Applied Mechanics, 49, 2, 2018, 395-407. doi: 10.22059/jcamech.2018.261764.300

Moradi, A., Yaghootian, A., Jalalvand, M., Ghanbarzadeh, A. (2018). 'Magneto-Thermo mechanical vibration analysis of FG nanoplate embedded on Visco Pasternak foundation', Journal of Computational Applied Mechanics, 49(2), pp. 395-407. doi: 10.22059/jcamech.2018.261764.300

Moradi, A., Yaghootian, A., Jalalvand, M., Ghanbarzadeh, A. Magneto-Thermo mechanical vibration analysis of FG nanoplate embedded on Visco Pasternak foundation. Journal of Computational Applied Mechanics, 2018; 49(2): 395-407. doi: 10.22059/jcamech.2018.261764.300

Magneto-Thermo mechanical vibration analysis of FG nanoplate embedded on Visco Pasternak foundation

^{1}Department of Mechanical Engineering, Shahid Chamran University of Ahvaz, Ahvaz, Iran

^{2}Department of Mathematics and Computer Science, Shahid Chamran University of Ahvaz, Ahvaz, Iran

Receive Date: 12 July 2018,
Revise Date: 08 August 2018,
Accept Date: 01 September 2018

Abstract

In this study, the vibration behavior of functional graded (FG) circular and annular nanoplate embedded in Visco-Pasternak foundation and coupled with temperature change is studied. It is assumed that a uniform radial magnetic field acts on the top surface of the plate and the magnetic permeability coefficient of the plate along its thickness are assumed to vary according to the volume distribution function. The effect of in-plane pre-load and temperature change are investigated on the vibration frequencies of FG nanoplate. The FG 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 DQM and the Galerkin method are utilized to solve the governing equation to obtain the frequency vibration. The effects of the size dependent, the in-plane pre-load, the temperature change, the magnetic field, the power index parameter, the elastic medium and the boundary conditions on the natural frequencies are investigated. The temperature change plays an important role in the mechanical behavior of the FG circular and annular nanoplate. According to the results, the application of radial magnetic field to the top surface of the plate changes the state of stress in both tangential and radial direction. The present analysis results can be used for the design of the next generation of nano-devices that make use of the thermal vibration properties of the nanoplate.

[1] M. Bao, W. Wang, Future of microelectromechanical systems (MEMS), Sens. Actuators A, Phys. , Vol. 56, pp. 135-141, 1996.

[2] A. Nabian, G. Rezazadeh, M. Haddad-derafshi, A. Tahmasebi, Mechanical behavior of a circular microplate subjected to uniform hydrostatic and non-uniform electrostatic pressure., J. Microsystems Technol, Vol. 14, No. 235-240, 2008.

[3] J. M. Sallese, W. Grabinski, V. Meyer, C. Bassin, P. Fazan, Electrical modeling of a pressure sensor MOSFET, , Sens. Actuators A, Actuators A, Phys. , Vol. 94, pp. 53-58, 2001.

[4] R. C. Batra, M. Porﬁri, D. Spinello, Electromechanical model of electrically actuated narrow microbeams,, J. Microelectromech. Syst. , Vol. 15, pp. 1175-1189, 2006.

[5] A. H. Nayfeh, M. I. Younis, Modeling and simulations of thermoelastic damping in microplates,, J. Micromech. Microeng. , Vol. 14, pp. 1711-1717, 2004.

[6] A. H. Nayfeh, M. I. Younis, E. M. Abdel-Rahman, Dynamic pull-in phenomenon in MEMS resonators,, Nonlinear Dynamic., Vol. 48, pp. 153–163., 2006.

[7] M. I. Younis, E. M. Abdel-Rahman, A. Nayfeh, A reduced-order model for electrically actuated microbeambased MEMS,, J. Microelectromech. Syst., Vol. 12, pp. 672–680, 2003.

[8] X. P. Zhao, E. M. Abdel-Rahman, A. H. Nayfeh, A reduced-order model for electrically actuated microplates,, J. Micromech. Microeng. , Vol. 14, pp. 900-906, 2004.

[9] A. Machauf, Y. Nemirovsky, U. Dinnar, A membrane micropump electrostatically actuated across the working ﬂuid,, J. Micromech. Microeng, Vol. 15, pp. 2309–2316, 2005.

[10] Wong E.W., Sheehan P.E., Lieber C.M., Nanobeam mechanics: elasticity, strength and toughness of nanorods and nanotubes, Science, Vol. 277, pp. 1971–1975, 1997.

[11] Zhou S.J., Li Z.Q., Metabolic response of Platynota stultana pupae during and after extended exposure to elevated CO2 and reduced O2 atmospheres., Shandong University Technology Vol. 31, pp. 401-409, 2001.

[12] Mohammadi V., Ansari R., Faghih Shojaei M., Gholami R., Sahmani S., Size-dependent dynamic pull-in instability of hydrostatically and electrostatically actuated circular microplates, Nonlinear Dynamic Vol. 73, No. 1515-1526, 2013.

[13] Yang F., Chong A.C.M., Lam D.C.C., Tong P., Couple stress based strain gradient theory for elasticity., International Journal of Solids Structure Vol. 39, pp. 2731-2743, 2002.

[14] S. Adali, variation principles for transversely vibrating multi-walledcarbon nanotubes based on nonlocal Euler-Bernoulli beam model, International Nano Letters, Vol. 9, pp. 1737-1741, 2009.

[15] Aydogdu M., Axial vibration of the nanorods with the nonlocal continuum rod model, Physica Vol. 41, pp. 861-864, 2009.

[16] Babaei H., Shahidi A. R., Small-scale effects on the buckling of quadrilateral nanoplates based on nonlocal elasticity theory using the Galerkin method, Archive Applied Mechanics, Vol. 81, pp. 1051–1062, 2010.

[17] Danesh M., Farajpour A., Mohammadi M., Axial vibration analysis of a tapered nanorod based on nonlocal elasticity theory and differential quadrature method, Mechanics Research Communications, Vol. 39, pp. 23-27, 2012.

[18] Duan W. H., Wang C. M., , exact solutions for axisymmetric bending of micro/nanoscale circular plates based on nonlocal plate theory, Nanotechnology pp. 18: 385704, 2007.

[19] Eringen A.C., on differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal Applied Physics Vol. 54, pp. 4703-4711, 1983.

[20] Farajpour A, Danesh M, Mohammadi M., Buckling analysis of variable thickness nanoplates using nonlocal continuum mechanics, Physica Vol. E 44, pp. 719-727, 2011.

[21] Farajpour A., Shahidi A. R., Mohammadi M., Mahzoon M., Buckling of orthotropic micro/nanoscale plates under linearly varying in-plane load via nonlocal continuum mechanics, Composite Structures Vol. 94, pp. 1605-1615, 2012.

[22] Lu P., Lee H.P., Lu C., Zhang P.Q., Dynamic properties of flexural beams using a nonlocal elasticity model, Journal Applied Physics Vol. 99, pp. 073510, 2006.

[23] Mohammadi M., Goodarzi M., Ghayour M., Alivand S., 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 Vol. 4, pp. 128-143, 2012.

[24] Mohammadi M., Goodarzi M., Ghayour M., Farajpour A., Inﬂuence of in-plane pre-load on the vibration frequency of circular graphene sheet via nonlocal continuum theory, Composites Part B, Vol. 51, pp. 121–129, 2013.

[25] Mohammadi M., Moradi A., Ghayour M., Farajpour A., Exact solution for thermo-mechanical vibration of orthotropic mono-layer graphene sheet embedded in an elastic medium, Latin American Journal of Solids and Structures, Vol. 11(3), pp. 437-458, 2014.

[26] Moosavi H., Mohammadi M., Farajpour A., Shahidi S. H., Vibration analysis of nanorings using nonlocal continuum mechanics and shear deformable ring theory, Physica Vol. E 44, pp. 135-140, 2011.

[27] Pradhan S.C., Phadikar J.K., small scale effect on vibration of embedded multilayered graphene sheets based on nonlocal continuum models, Physic Letter A Vol. 373, pp. 1062-1069, 2009.

[28] Wang C.M., Duan W.H., , free vibration of nanorings/arches based on nonlocal elasticity, Journal Applied Physics, Vol. 104, pp. 014303, 2008.

[29] Wang Q., Wang C.M., the constitutive relation and small scale parameter of nonlocal continuum mechanics for modelling carbon nanotubes, Nanotechnology Vol. 18, pp. 7:075702, 2007.

[30] B. Wang, J. Zhao, S. Zhou, A micro scale Timoshenko beam model based on strain gradient elasticity theory, European Journal.Mech. A, Solids, Vol. 29, pp. 591–599, 2010.

[31] R. Ansari, R. Gholami, S. Sahmani, Free vibration of size-dependent functionally graded microbeams based on a strain gradient theory, Composite Structures, Vol. 94, pp. 221–228, 2011.

[32] R. Ansari, R. Gholami, S. Sahmani, Study of small scale effects on the nonlinear vibration response of functionally graded Timoshenko microbeams based on the strain gradient theory, ASME Journal Computational. Nonlinear Dynamics, Vol. 7, pp. 031010 2012.

[33] S. Sahmani, R. Ansari, On the free vibration response of functionally graded higher-order shear deformable microplates based on the strain gradient elasticity theory, Composite.Structures, Vol. 95, pp. 430–442, 2013.

[34] M. H. Ghayesh, M. Amabili, H. Farokhi, Nonlinear forced vibrations of a microbeam based on the strain gradient elasticity theory, International Journal Engineering Science, Vol. 63, pp. 52–60, 2013.

[35] Mohammadi M., Farajpour A., Moradi M., Ghayour M., Shear buckling of orthotropic rectangular graphene sheet embedded in an elastic medium in thermal environment, Composites Part B, 2013.

[36] Civalek Ö., Akgöz B., Vibration analysis of micro-scaled sector shaped graphene surrounded by an elastic matrix, Computational Materials Science, Vol. 77, pp. 295-303, 2013.

[37] Murmu T., Pradhan S. C., Vibration analysis of nano-single-layered graphene sheets embedded in elastic medium based on nonlocal elasticity theory, Journal Applied Physics, pp. 105: 064319, 2009.

[38] Pradhan S.C., Phadikar J.K., Small scale effect on vibration of embedded multi layered graphene sheets based on nonlocal continuum models, Physics Letters A, Vol. 373, pp. 1062–1069, 2009.

[39] Wang Y. Z., Li F. M, Kishimoto K., Thermal effects on vibration properties of double-layered nanoplates at small scales, Composite Part B Engineering, Vol. 42, pp. 1311-1317.

[40] Reddy C.D., Rajendran S., Liew K.M., Equilibrium configuration and continuum elastic properties of finite sized graphene, Nanotechnology Vol. 17, pp. 864–870, 2006.

[41] Aksencer T., Aydogdu M., 2011, Levy type solution method for vibration and buckling of nanoplates using nonlocal elasticity theory, Physica Vol. E43, pp. 954 –959, 2011.

[42] Malekzadeh P., Setoodeh A.R., Alibeygi Beni A., small scale effect on the thermal buckling of orthotropic arbitrary straight-sided quadrilateral nanoplates embedded in an elastic medium, Composite Structure Vol. 93, pp. 2083–2089, 2011.

[43] Satish N., Narendar S., Gopalakrishnan S., Thermal vibration analysis of orthotropic nanoplates based on nonlocal continuum mechanics, Physica Vol. E 44, pp. 1950 –1962, 2012.

[44] Prasanna Kumar T.J., Narendar S., Gopalakrishnan S., Thermal vibration analysis of monolayer graphene embedded in elastic medium based on nonlocal continuum mechanics, Composite Structures Vol. 100, pp. 332–342, 2013.

[45] Farajpour A., Mohammadi M., Shahidi A.R., Mahzoon M., Axisymmetric buckling of the circular graphene sheets with the nonlocal continuum plate model, Physica Vol. E 43, pp. 1820 –1825, 2011.

[46] Mohammadi M., Ghayour M., Farajpour A., Free transverse vibration analysis of circular and annular graphene sheets with various boundary conditions using the nonlocal continuum plate model, Composites Part B, Vol. 45, pp. 32-42, 2013.

[47] Bayat M., Rahimi M., Saleem M., Mohazzab A.H., Wudtke I., Talebi H., One-dimensional analysis for magneto thermo mechanical response in a functionally graded annular variable thickness rotating disk, Applied Mathematical Modelling, Vol. 38, pp. 4625-4639, 2014.

[48] Behravan Rad A., Shariyat M., Three-dimensional magneto-elastic analysis of asymmetric variable thickness porous FGM circular plates with non-uniform tractions and Kerr elastic foundations, Composite Structures, Vol. 125, pp. 4625-574, 2015.

[49] Wang X., Dai H.L., Magneto thermodynamic stress and perturbation of magnetic field vector in an orthotropic thermo-elastic cylinder, International Journal of Engineering Science Vol. 42, pp. 539-556, 2004.

[50] Ghorbanpour A., Loghman A., Ahajari A.R., Amir S, semi-analytical solution of magneto-thermo-elastic stresses for functionally graded variable thickness rotating disks, Journal of Mechanical Science and Technology Vol. 24 (10), pp. 2107-2117, 2010.

[51] Nejad MZ, Hadi A, R. A, Buckling analysis of arbitrary two-directional functionally graded Euler-Bernoulli nano-beams ba sed on nonlocal elasticity theory, International Journal of Engineering Science, Vol. 2016 Jun, pp. 1;103:1-0, 2016.

[52] A. Daneshmehr, A. Rajabpoor, A. Hadi, Size dependent free vibration analysis of nanoplates made of functionally graded materials based on nonlocal elasticity theory with high order theories, International Journal of Engineering Science, Vol. 95, pp. 23-35, 10//, 2015.

[53] A. Zargaripoora, A. Daneshmehra, I. I. Hosseinia, A. Rajabpoora, Free vibration analysis of nanoplates made of functionally graded materials based on nonlocal elasticity theory using finite element method, Journal of Computational Applied Mechanics, Vol. 49, 2018.

[54] M. Z. Nejad, Amin Hadi., Non-local analysis of free vibration of bi-directional functionally graded Euler-Bernoulli nano-beams, International Journal of Engineering Science Vol. 105, pp. 1-11, 2016.

[55] M. Hosseini, Mohammad Shishesaz., Khosro Naderan Tahan., Amin Hadi., Stress analysis of rotating nano-disks of variable thickness made of functionally graded materials, International Journal of Engineering Science Vol. 109, pp. 29-53, 2016.

[56] M. Z. Nejad, A. Hadi, Eringen's non-local elasticity theory for bending analysis of bi-directional functionally graded Euler-Bernoulli nano-beams, International Journal of Engineering Science, Vol. 106, pp. 1-9., 2016.

[57] M. Hosseini, H. H. Gorgani, M. Shishesaz, A. Hadi, Size-dependent stress analysis of single-wall carbon nanotube based on strain gradient theory, International Journal of Applied Mechanics, Vol. 9, pp. 1750087, 2017.

[58] Zamani Nejad., M. J. Mohammad, Hadi A., A review of functionally graded thick cylindrical and conical shells, Journal of Computational Applied Mechanics Vol. 48, pp. 357-370, 2017.

[59] M. Shishesaz, M. Hosseini, K. N. Tahan, A. Hadi, Analysis of functionally graded nanodisks under thermoelastic loading based on the strain gradient theory, Acta Mechanica,, Vol. 228(12), pp. 4141-4168, 2017.

[60] D. C. C. Lam, F. Yang, A. C. M. Chong, J. Wang, P. Tong, Experiments and theory in strain gradient elasticity, Journal Mechanic Physics Solids Vol. 51, pp. 1477–1508, 2003.

[61] S. P. Timoshenko, J. N. Goodier, Theory of Elasticity, 3rd edition, McGraw-Hill, 1970.

[62] Farajpour A., Danesh M., Mohammadi M., Buckling analysis of variable thickness nanoplates using nonlocal continuum mechanics, Physica Vol. 44, pp. 719–727, 2011.

[63] Saadatpour M. M, Azhari M., The Galerkin method for static analysis of simply supported plates of general shape, Computers and Structures, Vol. 69, pp. 1-9, 1998.

[64] Shu C., Differential quadrature and its application in engineering, Berlin: Springer, 2000.

[65] Bert CW., Malik M., Differential quadrature method in computational mechanics: a review, Applied Mechanics, Vol. 49, pp. 1-27, 1996.

[66] Malekzadeh P, Setoodeh A. R, Alibeygi Beni A., Small scale effect on the thermal buckling of orthotropic arbitrary straight-sided quadrilateral nanoplates embedded in an elastic medium, Composite Structure, Vol. 93, pp. 2083–9, 2011.

[67] Wang Y. Z., Li F. M., Kishimoto K., Thermal effects on vibration properties of doublelayered nanoplates at small scales, Composites Part B: Engineering, Vol. 42, pp. 1311–1317, 2011.

[68] Zhou ZH., Wong KW., Xu XS., Leung AYT., Natural vibration of circular and annular thin plates by Hamiltonian approach, Journal Sound and Vibration, Vol. 330, pp. 1005-17, 2011.

[69] Leissa A. W., Narita Y., Natural frequencies of simply supported circular plates, Journal Sound Vibration, Vol. 70, pp. 221–9, 1980.

[70] Kim C. S, Dickinson S. M., On the free, transverse vibration of annular and circular, thin, sectorial plates subject to certain complicating effects, Journal Sound and Vibration, Vol. 134, pp. 407–21, 1989.

[71] Qiang L. Y., Jian L., Free vibration analysis of circular and annular sectorial thin plates using curve strip Fourier P-element, Journal Sound and Vibration, Vol. 305, pp. 457–66, 2007.