Buckling and Vibration Analysis of Tapered Circular Nano Plate

Document Type: Research Paper

Authors

1 Tarbiat Modares University

2 Department of Mechanical Engineering, Khatam Al Anbia Air Defense University, Tehran, Iran

Abstract

In this paper, buckling and free vibration analysis of a circular tapered nanoplate subjected to in-plane forces were studied. The linear variation of the plate thickness was considered in radial direction. Nonlocal elasticity theory was employed to capture size-dependent effects. The Raleigh-Ritz method and differential transform method were utilized to obtain the frequency equations for simply supported and clamped boundary conditions. To verify the accuracy of the Ritz method, the differential transform method (DTM) was also used to drive the size-dependent natural frequencies of circular nanoplates. Both methods reported good results. The validity of solutions was performed by comparing the present results with those of the literature for both classical plate and nanoplate. The effects of nonlocal parameter, mode number, and taper parameter on the natural frequency were investigated. The results showed that increasing the taper parameter causes increasing of buckling load and natural frequencies, and its effects on the clamped boundary condition is more than the simply support.

Keywords

Main Subjects

[1] Sari, M.S., Al-Kouz, W.G., Vibration analysis of non-uniform orthotropic Kirchhoff plates resting on elastic foundation based on nonlocal elasticity theory, International Journal of Mechanical Sciences, 114, 2016, pp. 1–11.

[2] Sakhaee-Pour, A., Ahmadian, M.T., Vafai, A., Applications of single-layered graphene sheets as mass sensors and atomistic dust detectors, Solid State Communications, 145, 2008, pp. 168–172.

[3] Arash, B., Wang, Q., A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes, Computational Materials Science, 51, 2012, pp. 303-313.

[4] Murmu, T., Pradhan, S.C., Vibration analysis of nano-single-layered graphene sheets embedded in elastic medium based on nonlocal elasticity theory, Journal of Applied Physics, 105, 2009, pp. 64319.

[5] Arash, B., Wang, Q., A Review on the Application of Nonlocal Elastic Models in Modeling of Carbon Nanotubes and Graphenes, Modeling of Carbon Nanotubes, Graphene and their Composites, Springer International Publishing, 2014, pp. 57–82.

[6] Mindlin, R.D., Eshel, N.N., On first strain-gradient theories in linear elasticity, International Journal of Solids and Structures, 4, 1968, pp. 109-124.

[7] Mindlin, R.D., Second gradient of strain and surface-tension in linear elasticity, International Journal of Solids and Structures, 1, 1965, pp. 417–438.

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

[9] Ramezani, S., A micro scale geometrically non-linear Timoshenko beam model based on strain gradient elasticity theory, International Journal of Non-Linear Mechanics, 47, 2012, pp. 863–873.

[10] Alibeigloo, A., Free vibration analysis of nano-plate using three-dimensional theory of elasticity, Acta Mechanica, 222, 2011, pp. 149-159.

[11] Şimşek, M., Dynamic analysis of an embedded microbeam carrying a moving microparticle based on the modified couple stress theory, International Journal of Engineering Science, 48, 2010, pp. 1721–1732.

[12] Sahmani, S., Ansari, R., Gholami, R., Darvizeh, A., Dynamic stability analysis of functionally graded higher-order shear deformable microshells based on the modified couple stress elasticity theory, Composites Part B, 51, 2013, pp. 44-53.

[13] Toupin, R.A., Theories of elasticity with couple-stress, Archive for Rational Mechanics and Analysis, 17(2), 1964, pp. 85–112.

[14] Yang, F., Chong, A.C.M., Lam, D.C.C., Tong, P., Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures, 39, 2002, pp. 2731–2743.

[15] Eringen, A.C., On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics, 54, 1983, pp. 4703–4710.

[16] Peddieson, J., Buchanan, G.R., McNitt, R.P., Application of nonlocal continuum models to nanotechnology, International Journal of Engineering Science, 41, 2003, pp. 305–312.

[17] Lu, P., Lee, H.P., Lu, C., Zhang, P.Q., Application of nonlocal beam models for carbon nanotubes, International Journal of Solids and Structures, 44, 2007, pp. 5289–5300.

[18] Rahmani, O., Pedram, O., Analysis and modeling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory, International Journal of Engineering Science, 77, 2014, pp. 55–70.

[19] Şimşek, M., Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain gradient theory and a novel Hamiltonian approach, International Journal of Engineering Science, 105, 2016, pp. 12–27.

[20] Hosseini-Hashemi, S., Bedroud, M., Nazemnezhad, R., An exact analytical solution for free vibration of functionally graded circular/annular Mindlin nanoplates via nonlocal elasticity, Composite Structures, 103, 2013, pp. 108–118.

[21] Belkorissat, I., Houari, MSA., Tounsi, A., Bedia, E.A.A., Mahmoud, S.R., On vibration properties of functionally graded nano-plate using a new nonlocal refined four variable model, Steel and Composite Structures, 18, 2015, pp. 1063–1081.

[22] Şimşek, M., Yurtcu, H.H., Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory, Composite Structures, 97, 2013, pp. 378–386.

[23] Murmu, T., Pradhan, S.C., 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, 2009, pp. 1232–1239.

[24] Aksencer, T., Aydogdu, M., Levy type solution method for vibration and buckling of nanoplates using nonlocal elasticity theory, Physica E: Low-Dimensional Systems and Nanostructures, 43, 2011, pp. 954-959.

[25] Narendar, S., Buckling analysis of micro-/nano-scale plates based on two-variable refined plate theory incorporating nonlocal scale effects, Composite Structures, 93, 2011, pp. 3093–3103.

[26] Farajpour, A., Mohammadi, M., Shahidi, A.R., Mahzoon, M., Axisymmetric buckling of the circular graphene sheets with the nonlocal continuum plate model, Physica E: Low-Dimensional Systems and Nanostructures, 43, 2011, pp. 1820–1825.

[27] Tornabene, F., Fantuzzi, N., Bacciocchi, M., The local GDQ method for the natural frequencies of doubly-curved shells with variable thickness: A general formulation, Composites Part B, 92, 2016, pp. 265–289.

[28] 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, 94, 2012, pp. 1605–1615.

[29] Farajpour, A., Danesh, M., Mohammadi, M., Buckling analysis of variable thickness nanoplates using nonlocal continuum mechanics, Physica E: Low-Dimensional Systems and Nanostructures, 44, 2011, pp. 719–727.

[30] 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, 39, 2012, pp. 23–27.

[31] Şimşek, M., Nonlocal effects in the free longitudinal vibration of axially functionally graded tapered nanorods, Computational Materials Science, 61, 2012, pp. 257–265.

[32] Efraim, E., Eisenberger, M., Exact vibration analysis of variable thickness thick annular isotropic and FGM plates, Journal of Sound and Vibration, 299, 2007, pp. 720–738.

[33] Zhou, J.K., Differential transformation and its applications for electrical circuits, Huazhong University Press, Wuhan, China, 1986.

[34] Arikoglu, A., Ozkol, I., Vibration analysis of composite sandwich beams with viscoelastic core by using differential transform method, Composite Structures, 92, 2010, pp. 3031–3039.

[35] Mohammadi, M., Farajpour, A., Goodarzi, M., Shehni nezhad pour, H., 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, 2014, pp. 510–520.

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

[37] Behfar, K., Naghdabadi, R., Nanoscale vibrational analysis of a multi-layered graphene sheet embedded in an elastic medium, Composites Science and Technology, 65, 2005, pp. 1159–1164.

[38] Mirzabeigy, A., 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, 27, 2013, pp. 385-394.

[39] 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, 51, 2013, pp. 121–129.

[40] Karimi, M., Shahidi, A.R., Nonlocal, refined plate, and surface effects theories used to analyze free vibration of magnetoelectroelastic nanoplates under thermo-mechanical and shear loadings, Applied Physics A, 123(5), 2017, pp. 304.

[41] Karimi, M., Haddad, H.A., Shahidi, A.R., Combining surface effects and non-local two variable refined plate theories on the shear/biaxial buckling and vibration of silver nanoplates, Micro and Nano Letters, 10, 2015, pp. 276–281.

[42] Karimi, M., Shahidi, A.R., Ziaei-Rad, S, Surface layer and nonlocal parameter effects on the in-phase and out-of-phase natural frequencies of a double-layer piezoelectric nanoplate under thermo-electro-mechanical loadings, Microsystem Technologies, 23(10), 2017, pp. 4903–4915.

[43] Karimi, M., Mirdamadi, H.R, Shahidi, A.R., Positive and negative surface effects on the buckling and vibration of rectangular nanoplates under biaxial and shear in–plane loadings based on nonlocal elasticity theory, Journal of the Brazilian Society of Mechanical Sciences and Engineering, 39, 2017, pp. 1391–1404.

[44] Shokrani, M.H., Karimi, M., Tehrani, M.S., Mirdamadi, H.R., Buckling analysis of double-orthotropic nanoplates embedded in elastic media based on non-local two-variable refined plate theory using the GDQ method, Journal of the Brazilian Society of Mechanical Sciences and Engineering, 38, 2016, pp. 2589–2606.

[45] Karimi, M., Mirdamadi, H.R., Shahidi, A.R., Shear vibration and buckling of double-layer orthotropic nanoplates based on RPT resting on elastic foundations by DQM including surface effects, Microsystem Technologies, 23, 2017, pp. 765–797.

[46] Liu, C., Ke, L.L., Yang, J., Kitipornchai, S., Wang, Y.S., Nonlinear vibration of piezoelectric nanoplates using nonlocal Mindlin plate theory, Mechanics of Advanced Materials and Structures, 2016, doi: 10.1080/15376494.2016.1149648.

[47] Asemi, S.R., Farajpour, A., Asemi, H.R., Mohammadi, M., Influence of initial stress on the vibration of double-piezoelectric-nanoplate systems with various boundary conditions using DQM, Physica E, 63, 2014, pp. 169-179.

[48] Liu, C., Ke, L.L., Wang, Y.S., Yang, J., Kitipornchai, S., Buckling and post-buckling analyses of size-dependent piezoelectric nanoplates, Theoretical and Applied Mechanics Letters, 6(6), 2016, pp. 253-267.

[49] 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, 11(3), 2014, pp. 437-458.

[50] Ke, L.L., Liu, C., Wang, Y.S., Free vibration of nonlocal piezoelectric nanoplates under various boundary conditions, Physica E: Low-dimensional Systems and Nanostructures, 66, 20415, pp. 93-106.

[51] Malekzadeh, P., Farajpour, A., Axisymmetric free and forced vibrations of initially stressed circular nanoplates embedded in an elastic medium, Acta Mechanica, 223, 2012, pp. 2311–2330.

[52] Bedroud, M., Hosseini-Hashemi, S., Nazemnezhad, R., Buckling of circular/annular Mindlin nanoplates via nonlocal elasticity, Acta Mechanica, 224, 2013, pp. 2663–2676.

[53] Anjomshoa, A., Application of Ritz functions in buckling analysis of embedded orthotropic circular and elliptical micro/nano-plates based on nonlocal elasticity theory, Meccanica, 48, 2013, pp. 1337-1353.

[54] Singh, B., Saxena, V., Axisymmetric vibration of a circular plate with double linear variable thickness, Journal of Sound and Vibration, 179, 1995, pp. 879–897.

[55] Liew, K.M., He, X.Q., Kitipornchai, S., Predicting nanovibration of multi-layered graphene sheets embedded in an elastic matrix, Acta Materialia, 54, 2006, pp. 4229–4236.

[56] 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, 45, 2013, pp. 32–42.

[1] Sari, M.S., Al-Kouz, W.G., Vibration analysis of non-uniform orthotropic Kirchhoff plates resting on elastic foundation based on nonlocal elasticity theory, International Journal of Mechanical Sciences, 114, 2016, pp. 1–11.

[2] Sakhaee-Pour, A., Ahmadian, M.T., Vafai, A., Applications of single-layered graphene sheets as mass sensors and atomistic dust detectors, Solid State Communications, 145, 2008, pp. 168–172.

[3] Arash, B., Wang, Q., A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes, Computational Materials Science, 51, 2012, pp. 303-313.

[4] Murmu, T., Pradhan, S.C., Vibration analysis of nano-single-layered graphene sheets embedded in elastic medium based on nonlocal elasticity theory, Journal of Applied Physics, 105, 2009, pp. 64319.

[5] Arash, B., Wang, Q., A Review on the Application of Nonlocal Elastic Models in Modeling of Carbon Nanotubes and Graphenes, Modeling of Carbon Nanotubes, Graphene and their Composites, Springer International Publishing, 2014, pp. 57–82.

[6] Mindlin, R.D., Eshel, N.N., On first strain-gradient theories in linear elasticity, International Journal of Solids and Structures, 4, 1968, pp. 109-124.

[7] Mindlin, R.D., Second gradient of strain and surface-tension in linear elasticity, International Journal of Solids and Structures, 1, 1965, pp. 417–438.

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

[9] Ramezani, S., A micro scale geometrically non-linear Timoshenko beam model based on strain gradient elasticity theory, International Journal of Non-Linear Mechanics, 47, 2012, pp. 863–873.

[10] Alibeigloo, A., Free vibration analysis of nano-plate using three-dimensional theory of elasticity, Acta Mechanica, 222, 2011, pp. 149-159.

[11] Şimşek, M., Dynamic analysis of an embedded microbeam carrying a moving microparticle based on the modified couple stress theory, International Journal of Engineering Science, 48, 2010, pp. 1721–1732.

[12] Sahmani, S., Ansari, R., Gholami, R., Darvizeh, A., Dynamic stability analysis of functionally graded higher-order shear deformable microshells based on the modified couple stress elasticity theory, Composites Part B, 51, 2013, pp. 44-53.

[13] Toupin, R.A., Theories of elasticity with couple-stress, Archive for Rational Mechanics and Analysis, 17(2), 1964, pp. 85–112.

[14] Yang, F., Chong, A.C.M., Lam, D.C.C., Tong, P., Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures, 39, 2002, pp. 2731–2743.

[15] Eringen, A.C., On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics, 54, 1983, pp. 4703–4710.

[16] Peddieson, J., Buchanan, G.R., McNitt, R.P., Application of nonlocal continuum models to nanotechnology, International Journal of Engineering Science, 41, 2003, pp. 305–312.

[17] Lu, P., Lee, H.P., Lu, C., Zhang, P.Q., Application of nonlocal beam models for carbon nanotubes, International Journal of Solids and Structures, 44, 2007, pp. 5289–5300.

[18] Rahmani, O., Pedram, O., Analysis and modeling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory, International Journal of Engineering Science, 77, 2014, pp. 55–70.

[19] Şimşek, M., Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain gradient theory and a novel Hamiltonian approach, International Journal of Engineering Science, 105, 2016, pp. 12–27.

[20] Hosseini-Hashemi, S., Bedroud, M., Nazemnezhad, R., An exact analytical solution for free vibration of functionally graded circular/annular Mindlin nanoplates via nonlocal elasticity, Composite Structures, 103, 2013, pp. 108–118.

[21] Belkorissat, I., Houari, MSA., Tounsi, A., Bedia, E.A.A., Mahmoud, S.R., On vibration properties of functionally graded nano-plate using a new nonlocal refined four variable model, Steel and Composite Structures, 18, 2015, pp. 1063–1081.

[22] Şimşek, M., Yurtcu, H.H., Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory, Composite Structures, 97, 2013, pp. 378–386.

[23] Murmu, T., Pradhan, S.C., 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, 2009, pp. 1232–1239.

[24] Aksencer, T., Aydogdu, M., Levy type solution method for vibration and buckling of nanoplates using nonlocal elasticity theory, Physica E: Low-Dimensional Systems and Nanostructures, 43, 2011, pp. 954-959.

[25] Narendar, S., Buckling analysis of micro-/nano-scale plates based on two-variable refined plate theory incorporating nonlocal scale effects, Composite Structures, 93, 2011, pp. 3093–3103.

[26] Farajpour, A., Mohammadi, M., Shahidi, A.R., Mahzoon, M., Axisymmetric buckling of the circular graphene sheets with the nonlocal continuum plate model, Physica E: Low-Dimensional Systems and Nanostructures, 43, 2011, pp. 1820–1825.

[27] Tornabene, F., Fantuzzi, N., Bacciocchi, M., The local GDQ method for the natural frequencies of doubly-curved shells with variable thickness: A general formulation, Composites Part B, 92, 2016, pp. 265–289.

[28] 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, 94, 2012, pp. 1605–1615.

[29] Farajpour, A., Danesh, M., Mohammadi, M., Buckling analysis of variable thickness nanoplates using nonlocal continuum mechanics, Physica E: Low-Dimensional Systems and Nanostructures, 44, 2011, pp. 719–727.

[30] 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, 39, 2012, pp. 23–27.

[31] Şimşek, M., Nonlocal effects in the free longitudinal vibration of axially functionally graded tapered nanorods, Computational Materials Science, 61, 2012, pp. 257–265.

[32] Efraim, E., Eisenberger, M., Exact vibration analysis of variable thickness thick annular isotropic and FGM plates, Journal of Sound and Vibration, 299, 2007, pp. 720–738.

[33] Zhou, J.K., Differential transformation and its applications for electrical circuits, Huazhong University Press, Wuhan, China, 1986.

[34] Arikoglu, A., Ozkol, I., Vibration analysis of composite sandwich beams with viscoelastic core by using differential transform method, Composite Structures, 92, 2010, pp. 3031–3039.

[35] Mohammadi, M., Farajpour, A., Goodarzi, M., Shehni nezhad pour, H., 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, 2014, pp. 510–520.

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

[37] Behfar, K., Naghdabadi, R., Nanoscale vibrational analysis of a multi-layered graphene sheet embedded in an elastic medium, Composites Science and Technology, 65, 2005, pp. 1159–1164.

[38] Mirzabeigy, A., 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, 27, 2013, pp. 385-394.

[39] 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, 51, 2013, pp. 121–129.

[40] Karimi, M., Shahidi, A.R., Nonlocal, refined plate, and surface effects theories used to analyze free vibration of magnetoelectroelastic nanoplates under thermo-mechanical and shear loadings, Applied Physics A, 123(5), 2017, pp. 304.

[41] Karimi, M., Haddad, H.A., Shahidi, A.R., Combining surface effects and non-local two variable refined plate theories on the shear/biaxial buckling and vibration of silver nanoplates, Micro and Nano Letters, 10, 2015, pp. 276–281.

[42] Karimi, M., Shahidi, A.R., Ziaei-Rad, S, Surface layer and nonlocal parameter effects on the in-phase and out-of-phase natural frequencies of a double-layer piezoelectric nanoplate under thermo-electro-mechanical loadings, Microsystem Technologies, 23(10), 2017, pp. 4903–4915.

[43] Karimi, M., Mirdamadi, H.R, Shahidi, A.R., Positive and negative surface effects on the buckling and vibration of rectangular nanoplates under biaxial and shear in–plane loadings based on nonlocal elasticity theory, Journal of the Brazilian Society of Mechanical Sciences and Engineering, 39, 2017, pp. 1391–1404.

[44] Shokrani, M.H., Karimi, M., Tehrani, M.S., Mirdamadi, H.R., Buckling analysis of double-orthotropic nanoplates embedded in elastic media based on non-local two-variable refined plate theory using the GDQ method, Journal of the Brazilian Society of Mechanical Sciences and Engineering, 38, 2016, pp. 2589–2606.

[45] Karimi, M., Mirdamadi, H.R., Shahidi, A.R., Shear vibration and buckling of double-layer orthotropic nanoplates based on RPT resting on elastic foundations by DQM including surface effects, Microsystem Technologies, 23, 2017, pp. 765–797.

[46] Liu, C., Ke, L.L., Yang, J., Kitipornchai, S., Wang, Y.S., Nonlinear vibration of piezoelectric nanoplates using nonlocal Mindlin plate theory, Mechanics of Advanced Materials and Structures, 2016, doi: 10.1080/15376494.2016.1149648.

[47] Asemi, S.R., Farajpour, A., Asemi, H.R., Mohammadi, M., Influence of initial stress on the vibration of double-piezoelectric-nanoplate systems with various boundary conditions using DQM, Physica E, 63, 2014, pp. 169-179.

[48] Liu, C., Ke, L.L., Wang, Y.S., Yang, J., Kitipornchai, S., Buckling and post-buckling analyses of size-dependent piezoelectric nanoplates, Theoretical and Applied Mechanics Letters, 6(6), 2016, pp. 253-267.

[49] 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, 11(3), 2014, pp. 437-458.

[50] Ke, L.L., Liu, C., Wang, Y.S., Free vibration of nonlocal piezoelectric nanoplates under various boundary conditions, Physica E: Low-dimensional Systems and Nanostructures, 66, 20415, pp. 93-106.

[51] Malekzadeh, P., Farajpour, A., Axisymmetric free and forced vibrations of initially stressed circular nanoplates embedded in an elastic medium, Acta Mechanica, 223, 2012, pp. 2311–2330.

[52] Bedroud, M., Hosseini-Hashemi, S., Nazemnezhad, R., Buckling of circular/annular Mindlin nanoplates via nonlocal elasticity, Acta Mechanica, 224, 2013, pp. 2663–2676.

[53] Anjomshoa, A., Application of Ritz functions in buckling analysis of embedded orthotropic circular and elliptical micro/nano-plates based on nonlocal elasticity theory, Meccanica, 48, 2013, pp. 1337-1353.

[54] Singh, B., Saxena, V., Axisymmetric vibration of a circular plate with double linear variable thickness, Journal of Sound and Vibration, 179, 1995, pp. 879–897.

[55] Liew, K.M., He, X.Q., Kitipornchai, S., Predicting nanovibration of multi-layered graphene sheets embedded in an elastic matrix, Acta Materialia, 54, 2006, pp. 4229–4236.

[56] 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, 45, 2013, pp. 32–42.