How to cite this paper
Ebrahimi, F & Nasirzadeh, P. (2015). Small-scale effects on transverse vibrational behavior of single-walled carbon nanotubes with arbitrary boundary conditions.Engineering Solid Mechanics, 3(2), 131-144.
Refrences
Abdel-Halim Hassan, I. H. (2002). On solving some eigenvalue problems by using a differential transformation. Applied Mathematics and Computation, 127(1), 1-22.
Aifantis, E. C. (1984). On the microstructural origin of certain inelastic models. Journal of Engineering Materials and technology, 106(4), 326-330.
Amara, K., Tounsi, A., Mechab, I., & Adda-Bedia, E. A. (2010). Nonlocal elasticity effect on column buckling of multiwalled carbon nanotubes under temperature field. Applied Mathematical Modelling, 34(12), 3933-3942.
Ansari, R., & Sahmani, S. (2012). Small scale effect on vibrational response of single-walled carbon nanotubes with different boundary conditions based on nonlocal beam models. Communications in Nonlinear Science and Numerical Simulation, 17(4), 1965-1979.
Ansari, R., & Ramezannezhad, H. (2011). Nonlocal Timoshenko beam model for the large-amplitude vibrations of embedded multiwalled carbon nanotubes including thermal effects. Physica E: Low-dimensional Systems and Nanostructures, 43(6), 1171-1178.
Ansari, R., Gholami, R., Hosseini, K., & Sahmani, S. (2011). A sixth-order compact finite difference method for vibrational analysis of nanobeams embedded in an elastic medium based on nonlocal beam theory. Mathematical and Computer Modelling, 54(11), 2577-2586.
Baughman, R. H., Zakhidov, A. A., & de Heer, W. A. (2002). Carbon nanotubes--the route toward applications. Science, 297(5582), 787-792.
Chen, C. O. K., & Ju, S. P. (2004). Application of differential transformation to transient advective–dispersive transport equation. Applied Mathematics and Computation, 155(1), 25-38.
Chow, T. L. (2013). Classical mechanics. CRC Press, Boca Raton, Florida, USA.
Eringen, A. C. (1972a). Linear theory of nonlocal elasticity and dispersion of plane waves. International Journal of Engineering Science, 10(5), 425-435.
Eringen, A. C. (1972b). Nonlocal polar elastic continua. International Journal of Engineering Science, 10(1), 1-16.
Eringen, A. C. (1983). On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics, 54(9), 4703-4710.
Ghorbanpourarani, A., Mohammadimehr, M., Arefmanesh, A., & Ghasemi, A. (2010). Transverse vibration of short carbon nanotubes using cylindrical shell and beam models. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 224(3), 745-756.
Iijima, S. (1991). Helical microtubules of graphitic carbon. Nature, 354(6348), 56-58.
Kiani, K., & Mehri, B. (2010). Assessment of nanotube structures under a moving nanoparticle using nonlocal beam theories. Journal of Sound and Vibration, 329(11), 2241-2264.
Kiani, K. (2010). A meshless approach for free transverse vibration of embedded single-walled nanotubes with arbitrary boundary conditions accounting for nonlocal effect. International Journal of Mechanical Sciences, 52(10), 1343-1356.
Kiani, K. (2013). Vibration analysis of elastically restrained double-walled carbon nanotubes on elastic foundation subjected to axial load using nonlocal shear deformable beam theories. International Journal of Mechanical Sciences, 68, 16-34.
Liew, K. M., Hu, Y., & He, X. Q. (2008). Flexural wave propagation in single-walled carbon nanotubes. Journal of Computational and Theoretical Nanoscience, 5(4), 581-586.
Lu, P., Lee, H. P., Lu, C., & Zhang, P. Q. (2006). Dynamic properties of flexural beams using a nonlocal elasticity model. Journal of Applied Physics, 99(7), 073510.
Maranganti, R., & Sharma, P. (2007). Length scales at which classical elasticity breaks down for various materials. Physical review letters, 98(19), 195504.
Mindlin, R. D. (1964). Micro-structure in linear elasticity. Archive for Rational Mechanics and Analysis, 16(1), 51-78.
Peddieson, J., Buchanan, G. R., & McNitt, R. P. (2003). Application of nonlocal continuum models to nanotechnology. International Journal of Engineering Science, 41(3), 305-312.
Samaei, A. T., & Mirsayar, M. M. (2011). Buckling Analysis of Multi-Walled Carbon Nanotubes with Consideration of Small Scale Effects. Journal of Computational and Theoretical Nanoscience, 8(11), 2214-2219.
Torabi, K., & Nafar Dastgerdi, J. (2012). An analytical method for free vibration analysis of Timoshenko beam theory applied to cracked nanobeams using a nonlocal elasticity model. Thin Solid Films, 520(21), 6595-6602.
Wang, Z. G. (2013). Axial Vibration Analysis of Stepped Bar by Differential Transformation Method. Applied Mechanics and Materials, 419, 273-279.
Wang, X., & Cai, H. (2006). Effects of initial stress on non-coaxial resonance of multi-wall carbon nanotubes. Acta materialia, 54(8), 2067-2074.
Wang, C. M., Zhang, Y. Y., & He, X. Q. (2007). Vibration of nonlocal Timoshenko beams. Nanotechnology, 18(10), 105401.
Wang, Q. (2005). Wave propagation in carbon nanotubes via nonlocal continuum mechanics. Journal of Applied Physics, 98(12), 124301.
Wang, Q., & Varadan, V. K. (2006). Vibration of carbon nanotubes studied using nonlocal continuum mechanics. Smart Materials and Structures, 15(2), 659.
Wang, C. M., Reddy, J. N., & Lee, K. H. (Eds.). (2000). Shear deformable beams and plates: Relationships with classical solutions. Elsevier.
Xu, M. (2006). Free transverse vibrations of nano-to-micron scale beams. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, 462(2074), 2977-2995.
Zhang, Y. Q., Liu, G. R., & Wang, J. S. (2004). Small-scale effects on buckling of multiwalled carbon nanotubes under axial compression. Physical review B, 70(20), 205430.
Zhang, Y. Q., Liu, G. R., & Xie, X. Y. (2005). Free transverse vibrations of double-walled carbon nanotubes using a theory of nonlocal elasticity. Physical Review B, 71(19), 195404.
Zhu, H., Wang, J., & Karihaloo, B. (2009). Effects of surface and initial stresses on the bending stiffness of trilayer plates and nanofilms. Journal of Mechanics of Materials and Structures, 4(3), 589-604.
Zhou, J. K. (1986). Differential transformation and its applications for electrical circuits. Huazhong University Press, Wuhan, China.
Aifantis, E. C. (1984). On the microstructural origin of certain inelastic models. Journal of Engineering Materials and technology, 106(4), 326-330.
Amara, K., Tounsi, A., Mechab, I., & Adda-Bedia, E. A. (2010). Nonlocal elasticity effect on column buckling of multiwalled carbon nanotubes under temperature field. Applied Mathematical Modelling, 34(12), 3933-3942.
Ansari, R., & Sahmani, S. (2012). Small scale effect on vibrational response of single-walled carbon nanotubes with different boundary conditions based on nonlocal beam models. Communications in Nonlinear Science and Numerical Simulation, 17(4), 1965-1979.
Ansari, R., & Ramezannezhad, H. (2011). Nonlocal Timoshenko beam model for the large-amplitude vibrations of embedded multiwalled carbon nanotubes including thermal effects. Physica E: Low-dimensional Systems and Nanostructures, 43(6), 1171-1178.
Ansari, R., Gholami, R., Hosseini, K., & Sahmani, S. (2011). A sixth-order compact finite difference method for vibrational analysis of nanobeams embedded in an elastic medium based on nonlocal beam theory. Mathematical and Computer Modelling, 54(11), 2577-2586.
Baughman, R. H., Zakhidov, A. A., & de Heer, W. A. (2002). Carbon nanotubes--the route toward applications. Science, 297(5582), 787-792.
Chen, C. O. K., & Ju, S. P. (2004). Application of differential transformation to transient advective–dispersive transport equation. Applied Mathematics and Computation, 155(1), 25-38.
Chow, T. L. (2013). Classical mechanics. CRC Press, Boca Raton, Florida, USA.
Eringen, A. C. (1972a). Linear theory of nonlocal elasticity and dispersion of plane waves. International Journal of Engineering Science, 10(5), 425-435.
Eringen, A. C. (1972b). Nonlocal polar elastic continua. International Journal of Engineering Science, 10(1), 1-16.
Eringen, A. C. (1983). On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics, 54(9), 4703-4710.
Ghorbanpourarani, A., Mohammadimehr, M., Arefmanesh, A., & Ghasemi, A. (2010). Transverse vibration of short carbon nanotubes using cylindrical shell and beam models. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 224(3), 745-756.
Iijima, S. (1991). Helical microtubules of graphitic carbon. Nature, 354(6348), 56-58.
Kiani, K., & Mehri, B. (2010). Assessment of nanotube structures under a moving nanoparticle using nonlocal beam theories. Journal of Sound and Vibration, 329(11), 2241-2264.
Kiani, K. (2010). A meshless approach for free transverse vibration of embedded single-walled nanotubes with arbitrary boundary conditions accounting for nonlocal effect. International Journal of Mechanical Sciences, 52(10), 1343-1356.
Kiani, K. (2013). Vibration analysis of elastically restrained double-walled carbon nanotubes on elastic foundation subjected to axial load using nonlocal shear deformable beam theories. International Journal of Mechanical Sciences, 68, 16-34.
Liew, K. M., Hu, Y., & He, X. Q. (2008). Flexural wave propagation in single-walled carbon nanotubes. Journal of Computational and Theoretical Nanoscience, 5(4), 581-586.
Lu, P., Lee, H. P., Lu, C., & Zhang, P. Q. (2006). Dynamic properties of flexural beams using a nonlocal elasticity model. Journal of Applied Physics, 99(7), 073510.
Maranganti, R., & Sharma, P. (2007). Length scales at which classical elasticity breaks down for various materials. Physical review letters, 98(19), 195504.
Mindlin, R. D. (1964). Micro-structure in linear elasticity. Archive for Rational Mechanics and Analysis, 16(1), 51-78.
Peddieson, J., Buchanan, G. R., & McNitt, R. P. (2003). Application of nonlocal continuum models to nanotechnology. International Journal of Engineering Science, 41(3), 305-312.
Samaei, A. T., & Mirsayar, M. M. (2011). Buckling Analysis of Multi-Walled Carbon Nanotubes with Consideration of Small Scale Effects. Journal of Computational and Theoretical Nanoscience, 8(11), 2214-2219.
Torabi, K., & Nafar Dastgerdi, J. (2012). An analytical method for free vibration analysis of Timoshenko beam theory applied to cracked nanobeams using a nonlocal elasticity model. Thin Solid Films, 520(21), 6595-6602.
Wang, Z. G. (2013). Axial Vibration Analysis of Stepped Bar by Differential Transformation Method. Applied Mechanics and Materials, 419, 273-279.
Wang, X., & Cai, H. (2006). Effects of initial stress on non-coaxial resonance of multi-wall carbon nanotubes. Acta materialia, 54(8), 2067-2074.
Wang, C. M., Zhang, Y. Y., & He, X. Q. (2007). Vibration of nonlocal Timoshenko beams. Nanotechnology, 18(10), 105401.
Wang, Q. (2005). Wave propagation in carbon nanotubes via nonlocal continuum mechanics. Journal of Applied Physics, 98(12), 124301.
Wang, Q., & Varadan, V. K. (2006). Vibration of carbon nanotubes studied using nonlocal continuum mechanics. Smart Materials and Structures, 15(2), 659.
Wang, C. M., Reddy, J. N., & Lee, K. H. (Eds.). (2000). Shear deformable beams and plates: Relationships with classical solutions. Elsevier.
Xu, M. (2006). Free transverse vibrations of nano-to-micron scale beams. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, 462(2074), 2977-2995.
Zhang, Y. Q., Liu, G. R., & Wang, J. S. (2004). Small-scale effects on buckling of multiwalled carbon nanotubes under axial compression. Physical review B, 70(20), 205430.
Zhang, Y. Q., Liu, G. R., & Xie, X. Y. (2005). Free transverse vibrations of double-walled carbon nanotubes using a theory of nonlocal elasticity. Physical Review B, 71(19), 195404.
Zhu, H., Wang, J., & Karihaloo, B. (2009). Effects of surface and initial stresses on the bending stiffness of trilayer plates and nanofilms. Journal of Mechanics of Materials and Structures, 4(3), 589-604.
Zhou, J. K. (1986). Differential transformation and its applications for electrical circuits. Huazhong University Press, Wuhan, China.