The Small Scale Effect on Nonlinear Vibration of Single Layer Graphene Sheets
In the present article, nonlinear vibration analysis of
single layer graphene sheets is presented and the effect of small
length scale is investigated. Using the Hamilton's principle, the three
coupled nonlinear equations of motion are obtained based on the von
Karman geometrical model and Eringen theory of nonlocal
continuum. The solutions of Free nonlinear vibration, based on a one
term mode shape, are found for both simply supported and clamped
graphene sheets. A complete analysis of graphene sheets with
movable as well as immovable in-plane conditions is also carried out.
The results obtained herein are compared with those available in the
literature for classical isotropic rectangular plates and excellent
agreement is seen. Also, the nonlinear effects are presented as
functions of geometric properties and small scale parameter.
[1] H. Chu, G. Herrmann, "Influence of large amplitudes on free flexural
vibrations of rectangular elastic plates", Journal of Applied Mechanics,
23, 1956, pp. 532-540.
[2] N. Yamaki, "Influence of large amplitudes on flexural vibrations of
elastic plates", ZAMM, 41, 1961, pp. 501-540.
[3] G. Singh, K. Raju, G.V. Rao, "Non-linear vibrations of simply
supported rectangular cross-ply plates", Journal of Sound and Vibration,
142, 1990, pp. 213-226.
[4] A.Y.T. Leung, S.G. Mao, "A symplectic Galerkin method for non-linear
vibration of beams and plates", Journal of Sound and Vibration, 183,
1995, pp. 475-491.
[5] M. Amabili, "Nonlinear vibrations of rectangular plates with different
boundary conditions: theory and experiments", Computers and
Structures, 82, 2004, pp. 2587-2605.
[6] M. Haterbouch, R. Benamar, "Geometrically nonlinear free vibrations of
simply supported isotropic thin circular plates", Journal of Sound and
Vibration, 280, 2005, pp. 903-924.
[7] J. Woo, S.A. Meguid, L.S. Ong, "Nonlinear free vibration behavior of
functionally graded plates", Journal of Sound and Vibration, 289, 2006,
pp. 595-611
[8] M. Amabili, S. Farhadi, "Shear deformable versus classical theories for
nonlinear vibrations of rectangular isotropic and laminated composite
plates", Journal of Sound and Vibration, 320, 2009, pp. 649-667.
[9] C.W. Lima, L.H. He, "Size-dependent nonlinear response of thin elastic
films with nano-scale thickness", International Journal of Mechanical
Sciences, 46, 2004, pp. 1715-1726.
[10] S. Kitipornchai, X.Q. He, K.M. Liew, "Continuum model for the
vibration of multilayered graphene sheets", Physical Review B, 72,
075443, 2005, 6 pages.
[11] Y.M. Fu, J.W. Hong, X.Q. Wang, "Analysis of nonlinear vibration for
embedded carbon nanotubes", Journal of Sound and Vibration, 296,
2006, pp. 746-756
[12] S.C. Pradhan, J.K. Phadikar. "Nonlocal elasticity theory for vibration of
nanoplates", Journal of Sound and Vibration, 325, 2009, pp. 206-223.
[13] L.L. Ke, Y. Xiang, J. Yang, S. Kitipornchai, "Nonlinear free vibration of
embedded double-walled carbon nanotubes based on nonlocal
Timoshenko beam theory", Computational Materials Science, 47, 2009,
pp.409-417.
[14] Y.X. Dong, C.W. LIM, "Nonlinear vibrations of nano-beams accounting
for nonlocal effect using a multiple scale method", Science in China
Series E: Technological Sciences, 52, 2009, pp. 617-621.
[15] T. Murmu, S.C. Pradhan, "Thermo-mechanical vibration of a singlewalled
carbon nanotube embedded in an elastic medium based on
nonlocal elasticity theory", Computational Materials Science, 46, 2009,
pp. 854-859.
[16] J. Yang, L.L. Ke, S. Kitipornchai, "Nonlinear free vibration of singlewalled
carbon nanotubes using nonlocal Timoshenko beam theory",
Physica E, 42, 2010, pp. 1727-1735.
[17] M.A. Hawwa, H.M. Al-Qahtani, "Nonlinear oscillations of a doublewalled
carbon nanotube", Computational Materials Science, 48, 2010,
pp. 140-143.
[18] C. Eringen, "On differential equations of nonlocal elasticity and
solutions of screw dislocation and surface waves", Journal of Applied
Physics, 54, 1983, pp. 4703-4710.
[19] I.S. Raju, G.V. Rao, K. Raju, "Effect of longitudinal or inplane
deformation and inertia on the large amplitude flexural vibrations of
slender beam and thin plates", Journal of Sound and Vibration, 49,
1976, pp. 415-422.
[20] A. Beléndez, D.I. Méndez, E. Fernandez, S. Marini, I. Pascual, "An
explicit approximate solution to the Duffing-harmonic oscillator by a
cubication method", Physics Letters A, 373, 2009, pp. 2805-2809.
[21] J.J. Stoker, "Nonlinear vibrations in Mechanical and Electrical Systems",
John Wiley & Sons, Inc., New York, 1950.
[1] H. Chu, G. Herrmann, "Influence of large amplitudes on free flexural
vibrations of rectangular elastic plates", Journal of Applied Mechanics,
23, 1956, pp. 532-540.
[2] N. Yamaki, "Influence of large amplitudes on flexural vibrations of
elastic plates", ZAMM, 41, 1961, pp. 501-540.
[3] G. Singh, K. Raju, G.V. Rao, "Non-linear vibrations of simply
supported rectangular cross-ply plates", Journal of Sound and Vibration,
142, 1990, pp. 213-226.
[4] A.Y.T. Leung, S.G. Mao, "A symplectic Galerkin method for non-linear
vibration of beams and plates", Journal of Sound and Vibration, 183,
1995, pp. 475-491.
[5] M. Amabili, "Nonlinear vibrations of rectangular plates with different
boundary conditions: theory and experiments", Computers and
Structures, 82, 2004, pp. 2587-2605.
[6] M. Haterbouch, R. Benamar, "Geometrically nonlinear free vibrations of
simply supported isotropic thin circular plates", Journal of Sound and
Vibration, 280, 2005, pp. 903-924.
[7] J. Woo, S.A. Meguid, L.S. Ong, "Nonlinear free vibration behavior of
functionally graded plates", Journal of Sound and Vibration, 289, 2006,
pp. 595-611
[8] M. Amabili, S. Farhadi, "Shear deformable versus classical theories for
nonlinear vibrations of rectangular isotropic and laminated composite
plates", Journal of Sound and Vibration, 320, 2009, pp. 649-667.
[9] C.W. Lima, L.H. He, "Size-dependent nonlinear response of thin elastic
films with nano-scale thickness", International Journal of Mechanical
Sciences, 46, 2004, pp. 1715-1726.
[10] S. Kitipornchai, X.Q. He, K.M. Liew, "Continuum model for the
vibration of multilayered graphene sheets", Physical Review B, 72,
075443, 2005, 6 pages.
[11] Y.M. Fu, J.W. Hong, X.Q. Wang, "Analysis of nonlinear vibration for
embedded carbon nanotubes", Journal of Sound and Vibration, 296,
2006, pp. 746-756
[12] S.C. Pradhan, J.K. Phadikar. "Nonlocal elasticity theory for vibration of
nanoplates", Journal of Sound and Vibration, 325, 2009, pp. 206-223.
[13] L.L. Ke, Y. Xiang, J. Yang, S. Kitipornchai, "Nonlinear free vibration of
embedded double-walled carbon nanotubes based on nonlocal
Timoshenko beam theory", Computational Materials Science, 47, 2009,
pp.409-417.
[14] Y.X. Dong, C.W. LIM, "Nonlinear vibrations of nano-beams accounting
for nonlocal effect using a multiple scale method", Science in China
Series E: Technological Sciences, 52, 2009, pp. 617-621.
[15] T. Murmu, S.C. Pradhan, "Thermo-mechanical vibration of a singlewalled
carbon nanotube embedded in an elastic medium based on
nonlocal elasticity theory", Computational Materials Science, 46, 2009,
pp. 854-859.
[16] J. Yang, L.L. Ke, S. Kitipornchai, "Nonlinear free vibration of singlewalled
carbon nanotubes using nonlocal Timoshenko beam theory",
Physica E, 42, 2010, pp. 1727-1735.
[17] M.A. Hawwa, H.M. Al-Qahtani, "Nonlinear oscillations of a doublewalled
carbon nanotube", Computational Materials Science, 48, 2010,
pp. 140-143.
[18] C. Eringen, "On differential equations of nonlocal elasticity and
solutions of screw dislocation and surface waves", Journal of Applied
Physics, 54, 1983, pp. 4703-4710.
[19] I.S. Raju, G.V. Rao, K. Raju, "Effect of longitudinal or inplane
deformation and inertia on the large amplitude flexural vibrations of
slender beam and thin plates", Journal of Sound and Vibration, 49,
1976, pp. 415-422.
[20] A. Beléndez, D.I. Méndez, E. Fernandez, S. Marini, I. Pascual, "An
explicit approximate solution to the Duffing-harmonic oscillator by a
cubication method", Physics Letters A, 373, 2009, pp. 2805-2809.
[21] J.J. Stoker, "Nonlinear vibrations in Mechanical and Electrical Systems",
John Wiley & Sons, Inc., New York, 1950.
@article{"International Journal of Mechanical, Industrial and Aerospace Sciences:62500", author = "E. Jomehzadeh and A.R. Saidi", title = "The Small Scale Effect on Nonlinear Vibration of Single Layer Graphene Sheets", abstract = "In the present article, nonlinear vibration analysis of
single layer graphene sheets is presented and the effect of small
length scale is investigated. Using the Hamilton's principle, the three
coupled nonlinear equations of motion are obtained based on the von
Karman geometrical model and Eringen theory of nonlocal
continuum. The solutions of Free nonlinear vibration, based on a one
term mode shape, are found for both simply supported and clamped
graphene sheets. A complete analysis of graphene sheets with
movable as well as immovable in-plane conditions is also carried out.
The results obtained herein are compared with those available in the
literature for classical isotropic rectangular plates and excellent
agreement is seen. Also, the nonlinear effects are presented as
functions of geometric properties and small scale parameter.", keywords = "Small scale, Nonlinear vibration, Graphene sheet,Nonlocal continuum", volume = "5", number = "6", pages = "1143-5", }
