Free Axisymmetric Bending Vibration Analysis of two Directional FGM Circular Nano-plate on the Elastic Foundation

Document Type: Research Paper

Authors

Department of Mechanical Engineering, Tarbiat Modares University (TMU), Tehran, Iran

Abstract

In the following paper, free vibration analysis of two directional FGM circular nano-plate on the elastic medium is investigated. The elastic modulus of plate varies in both radial and thickness directions. Eringen’s theory was employed to the analysis of circular nano-plate with variation in material properties. Simultaneous variations of the material properties in the radial and transverse directions are described by a general function. Ritz functions were utilized to obtain the frequency equations for simply supported and clamped boundary. Differential transform method also used to develop a semi-analytical solution the size-dependent natural frequencies of non-homogenous nano-plates. Both methods reported good results. The validity of solutions was performed by comparing present results with themselves and those of the literature for both classical plate and nano-plate. Effect of non-homogeneity on the nonlocal parameter, geometries, boundary conditions and elastic foundation parameters is examined the paper treats some interesting problems, for the first time.                      

Keywords

Sari M.S., Al-Kouz W.G., 2016, Vibration analysis of non-uniform orthotropic Kirchhoff plates resting on elastic foundation based on nonlocal elasticity theory, International Journal of Mechanical Sciences 114: 1-11.
[2] Sakhaee-Pour A., Ahmadian M.T., Vafai A., 2008, Applications of single-layered graphene sheets as mass sensors and atomistic dust detectors, Solid State Communications 145: 168-172.
[3] Arash B., Wang Q., 2012, A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes, Computational Materials Science 51: 303-313.
[4] 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: 64319.
[5] Arash B., Wang Q., 2014, A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes, Modeling of Carbon Nanotubes, Graphene and their Composites 2014: 57-82.
[6] Mindlin R.D., Eshel N.N., 1968, On first strain-gradient theories in linear elasticity, International Journal of Solids and Structures 4: 109-124.
[7] Mindlin R.D.,1965, Second gradient of strain and surface-tension in linear elasticity, International Journal of Solids and Structures 1: 417-438.
[8] 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.
[9] Ramezani S., 2012, A micro scale geometrically non-linear Timoshenko beam model based on strain gradient elasticity theory, International Journal of Non-Linear Mechanics 47: 863-873.
[10] Alibeigloo A., 2011, Free vibration analysis of nano-plate using three-dimensional theory of elasticity, Acta Mechanica 222: 149.
[11] Şimşek M.,2010, Dynamic analysis of an embedded micro-beam carrying a moving micro-particle based on the modified couple stress theory, International Journal of Engineering Science 48: 1721-1732.
[12] Sahmani S., Ansari R., Gholami R., Darvizeh A., 2013, Dynamic stability analysis of functionally graded higher-order shear deformable micro-shells based on the modified couple stress elasticity theory, Composites Part B: Engineering 51: 44-53.
[13] Toupin R.A., 1964, Theories of elasticity with couple-stress, Archive for Rational Mechanics and Analysis 17: 85-112.
[14] 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.
[15] Eringen A.C.,1983, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics 54: 4703-4710.
[16] Peddieson J., Buchanan G.R., McNitt R.P., 2003, Application of nonlocal continuum models to nanotechnology, International Journal of Engineering Science 41: 305-312.
[17] Lu P., Lee H.P., Lu C., Zhang P.Q., 2007, Application of nonlocal beam models for carbon nanotubes, International Journal of Solids and Structures 44: 5289-5300.
[18] Rahmani O., Pedram O., 2014, Analysis and modeling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory, International Journal of Engineering Science 77: 55-70.
[19] Şimşek M., 2016, Nonlinear free vibration of a functionally graded nano-beam using nonlocal strain gradient theory and a novel Hamiltonian approach, International Journal of Engineering Science 105: 12-27.
[20] Hosseini-Hashemi S., Bedroud M., Nazemnezhad R., 2013, An exact analytical solution for free vibration of functionally graded circular/annular Mindlin nano-plate s via nonlocal elasticity, Composite Structures 103: 108-118.
[21] Belkorissat I., Houari M.S.A., Tounsi A., Bedia E.A.A., Mahmoud S.R., 2015, On vibration properties of functionally graded nano-plate using a new nonlocal refined four variable model, Steel and Composite Structures 18: 1063-1081.
[22] Şimşek M., Yurtcu H.H., 2013, Analytical solutions for bending and buckling of functionally graded nano-beams based on the nonlocal Timoshenko beam theory, Composite Structures 97: 378-386.
[23] Murmu T., Pradhan S.C., 2009, Buckling analysis of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity and Timoshenko beam theory and using DQM, Physica E: Low-Dimensional Systems and Nanostructures 41: 1232-1239.
[24] Aksencer T., Aydogdu M., 2011, Levy type solution method for vibration and buckling of nano-plate s using nonlocal elasticity theory, Physica E: Low-Dimensional Systems and Nanostructures 43: 954-959.
[25] Narendar S., 2011, Buckling analysis of micro-/nano-scale plates based on two-variable refined plate theory incorporating nonlocal scale effects, Composite Structures 93: 3093-3103.
[26] 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: Low-Dimensional Systems and Nanostructures 43: 1820-1825.
[27] Tornabene F., Fantuzzi N., Bacciocchi M., 2016, The local GDQ method for the natural frequencies of doubly-curved shells with variable thickness: A general formulation, Composites Part B: Engineering 92: 265-289.
[28] 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.
[29] Farajpour A., Danesh M., Mohammadi M., 2011, Buckling analysis of variable thickness nano-plate s using nonlocal continuum mechanics, Physica E: Low-Dimensional Systems and Nanostructures 44: 719-727.
[30] Danesh M., Farajpour A., Mohammadi M., 2012, Axial vibration analysis of a tapered nano-rod based on nonlocal elasticity theory and differential quadrature method, Mechanics Research Communications 39: 23-27.
[31] Şimşek M., 2012, Nonlocal effects in the free longitudinal vibration of axially functionally graded tapered nano-rods, Computational Materials Science 61: 257-265.
[32] Efraim E., Eisenberger M., 2007, Exact vibration analysis of variable thickness thick annular isotropic and FGM plates, Journal of Sound and Vibration 299: 720-738.
[33] Zhou J.K., 1986, Differential Transformation and its Applications for Electrical Circuits, Huazhong University Press, Wuhan, China.
[34] Arikoglu A., Ozkol I., 2010, Vibration analysis of composite sandwich beams with viscoelastic core by using differential transform method, Composite Structures 92: 3031-3039.
[35] Mohammadi M., Farajpour A., Goodarzi M., Shehni nezhad pour H., 2014, Numerical study of the effect of shear in-plane load on the vibration analysis of graphene sheet embedded in an elastic medium, Computational Materials Science 82: 510-520.
[36] Pradhan S.C., Phadikar J.K., 2009, Small scale effect on vibration of embedded multilayered graphene sheets based on nonlocal continuum models, Physics Letters A 373: 1062-1069.
[37] Behfar K., Naghdabadi R., 2005, Nanoscale vibrational analysis of a multi-layered graphene sheet embedded in an elastic medium, Composites Science and Technology 65: 1159-1164.
[38] Mirzabeigy A., 2013, Semi-analytical approach for free vibration analysis of variable cross-section beams resting on elastic foundation and under axial force, International Journal of Engineering - Transactions C: Aspects 27: 385.
[39] 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: Engineering 51: 121-129.
[40] Alipour M.M., Shariyat M., Shaban M., 2010, A semi-analytical solution for free vibration and modal stress analyses of circular plates resting on two-parameter elastic foundations, Journal of Solid Mechanics 2(1): 63-78.
[41] Shariyat M., Jafari A.A., Alipour M.M., 2013, Investigation of the thickness variability and material heterogeneity effects on free vibration of the viscoelastic circular plates, Acta Mechanica Solida Sinica 26(1): 83-98.
[42] Alipour M.M., Shariyat M., Shaban M., 2010, A semi-analytical solution for free vibration of variable thickness two-directional-functionally graded plates on elastic foundations, International Journal of Mechanics and Materials in Design 6(4): 293-304.
[43] Alipour M.M., Shariyat M., 2010, Stress analysis of two-directional FGM moderately thick constrained circular plates with non-uniform load and substrate stiffness distributions, Journal of Solid Mechanics 2(4): 316-331.
[44] Alipour M.M., Shariyat M., 2011, A power series solution for free vibration of variable thickness Mindlin circular plates with two-directional material heterogeneity and elastic foundations, Journal of Solid Mechanics 3(2): 183-197.
[45] Shariyat M., Alipour M.M., 2013, A power series solution for vibration and complex modal stress analyses of variable thickness viscoelastic two-directional FGM circular plates on elastic foundations, Applied Mathematical Modelling 37(5): 3063-3076.
[46] Alipour M.M., Shariyat M.,2013, Semianalytical solution for buckling analysis of variable thickness two-directional functionally graded circular plates with nonuniform elastic foundations, ASCE Journal of Engineering Mechanics 139(5): 664-676.
[47] Shariyat M., Alipour M.M., 2012, A zigzag theory with local shear correction factors for semi-analytical bending modal analysis of functionally graded viscoelastic circular sandwich plates, Journal of Solid Mechanics 4(1): 84-105.
[48] Neha A., Roshan L., 2015, Buckling and vibration of functionally graded circular plates resting on elastic foundation, Mathematical Analysis and its Applications 2015: 545-555.
[49] Zarei M., Ghalami-Choobari M., Rahimi G.H., Faghani G.R., 2018, Axisymmetric free vibration anlysis of non-uniform circular nano-plate resting on elastic medium, Journal of Solid Mechanics 10(2): 400-415.
[50] Anjomshoa A., 2013, Application of Ritz functions in buckling analysis of embedded orthotropic circular and elliptical micro/nano-plates based on nonlocal elasticity theory, Meccanica 48: 1337-1353.
[51] Singh B., Saxena V., 1995, Axisymmetric vibration of a circular plate with double linear variable thickness, Journal of Sound and Vibration 179: 879-897.
[52] Liew K.M., He X.Q., Kitipornchai S., 2006, Predicting nano-vibration of multi-layered graphene sheets embedded in an elastic matrix, Acta Materialia 54: 4229-4236.
[53] 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.
[54] Shariyat M., Alipour M.M., 2011, Differential transform vibration and modal stress analyses of circular plates made of two-directional functionally graded materials resting on elastic foundations, Archive of Applied Mechanics 81(9): 1289-1306.