Nonlinear Wave Modulation in Nanorods Based on Nonlocal Elasticity Theory by Using Multiple-Scale Formalism

Year 2018, , 140 - 158, 04.11.2018

Güler Gaygusuzoğlu

https://doi.org/10.24107/ijeas.422906

Abstract

Many systems in physics,
engineering, and natural sciences are
nonlinear and modeled with nonlinear equations. Wave propagation, as a branch
of nonlinear science, is one of the most widely studied subjects in recent
years. Nonlocal elasticity theory represents a common growing technique used
for conducting the mechanical analysis of microelectromechanical
and nanoelectromechanical systems. In this study, nonlinear wave modulation in
nanorods was examined by means of nonlocal elasticity theory.  The nonlocal constitutive equations of Eringen
were utilized in the formulation, and the nonlinear equation of motion of nanorods was obtained. By applying the multiple scale formalism, the propagation of
weakly nonlinear and strongly dispersive waves was investigated, and the Nonlinear Schrödinger (NLS) equation was
obtained as the evolution equation. A part of spacial solutions of the NLS
equation, i.e. nonlinear plane wave,
solitary wave and phase jump solutions, were presented. In order to investigate
the nonlocal impacts on the NLS equation numerically, whether envelope solitary
wave solutions exist was investigated by utilizing the physical and geometric
features of carbon nanotubes (CNTs).

Keywords

Nanorods, nonlinear wave modulation, nonlocal elasticity theory

References

• [1] Eringen, A. C., Suhubi, E. S., Nonlinear theory of simple micro-elastic solids-I, International Journal of Engineering Science, 2, 189-203, 1964.
• [2] Eringen, A. C., Simple microfluids, International Journal of Engineering Science, 2, 205-217, 1964.
• [3] Eringen, A. C., Theory of micropolar elastisity in Fracture (Edited by H. Liebowitz), Vol. II Academic Press, New York, 1968.
• [4] Kafadar, C. B., Eringen A. C., Micropolar Media-I. The classical theory, International Journal of Engineering Science, 9, 271-305, 1971.
• [5] Eringen, A. C., Nonlocal polar elastic continua, International Journal of Engineering Science, 10, 1-16, 1972.
• [6] Demiray, H., A nonlocal continuum theory for diatomic elastic solids, Int. J. Eng. Sci., 15, 623-644, 1977
• [7] Eringen, A. C., On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics, 54, 4703-4710,1983.
• [8] Toupin, R. A., Elastic materials with coupled stresses, Archive for Rational Mechanics and Analysis, 11, 385, 1962.
• [9] Park, S. K., Gao, X. L., Bernoulli-Euler beam model based on a modified coupled stress theory, Journal of Micromechanics and Microengineering, 16 (11),23055-2359, 2006.
• [10] Ma, H. M., Gao, X. L., Reddy J. N., A microstructure-dependent Timoshenko beam model based on a modified couple stress theory, Journal of the Mechanics and Physics of Solids, 56(12), 3379-3391, 2008.
• [11] Murmu, T., Pradhan, S. C., Small-scale effect on the vibration on the nonuniform nanocantiliver based on nonlocal elasticity theory, Physica E, 41, 1451-1456, 2009.
• [12] Senthilkumar, V., Pradhan, S. C., Pratap, G., Small-scale effect on buckling analysis of carbon nanotube with Timoshenko theory by using differential transform method, Adv. Sci. Lett., 3, 1-7, 2010.
• [13] Rahmani, O., Pedram, O., Analysis and modelling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory, International Journal of Engineering Science, 77, 55-70, 2014.
• [14] Eringen, A. C., Linear theory of nonlocal elasticity and dispersion of plane waves, International Journal of Engineering Science,10, 1-16, 1972.
• [15] Eringen, A. C., Edelen, D. G. B., On nonlocal elasticity, International Journal of Engineering Science, 10, 233-248, 1972.
• [16] Thai, H. T., A nonlocal beam theory for bending, buckling and vibration of nanobeams, International Journal of Engineering Science, 52, 56-64, 2012.
• [17] Aydogdu, M., Axial vibration of the nanorods with the nonlocal continuum rod model, Physica E: Low-dimensional Systems and Nanostructures, 41(5), 861-864, 2009.
• [18] Aydogdu, M., Axial vibration analysis of nanorods (carbon nanotubes) embedded in an elastic medium using nonlocal theory, Mechanics Research Communications, 43, 34-40, 2012.
• [19] Lim, C. W. and Yang, Y., Wave propagation in carbon nanotubes: nonlocal elasticity-induced stiffness and velocity enhancement effects, J. Mech. Mater. Struct., 5, 459-476, 2010.
• [20] Hu, Y. G., Liew, K. M., Wang, Q., He, X. Q., Yakobson, B. I., Nonlocal shell model for elastic wave propagation single- and double-walled carbon nanotubes, J. Mech. Phys. Solids, 56: 3475-3485, 2008.
• [21] Wang, Q., Varadan, V. K., Wave characteristics of carbon nanotubes, Int. J. Solids Struct., 43, 254-265, 2006.
• [22] Narendar, S., Gopalakrishnan, S., Temperature effects on wave propagation in nanoplates, Compos. Part B, 43, 1275-1281, 2012.
• [23] Narendar, S., Gopalakrishnan, S., Nonlocal scale effects on wave propagation in multi-walled carbon nanotubes, Comput. Mater. Sci., 47, 526-538, 2009.
• [24] Aydogdu, M., Longitudinal wave propagation in nanorods using a general nonlocal unimodal rod theory and calibration of nonlocal parameter with lattice dynamics, Int. J. Eng. Sci., 56, 17-28 ,2012.
• [25] Aydogdu, M., Longitudinal wave propagation in multiwalled carbon nanotubes, Composite Structures, 107: 578-584, 2014.
• [26] Wu, X. F., Dzenis, Y. A. ,Wave propagation in nanofibers, J. App. Phys., 100, 124318, 2006.
• [27] Challamel, N., Rakotomanana, L., Marrec, L. L., A dispersive wave equation using nonlocal elasticity, Comptes Rendus Mecanique, 337, 591-595, 2009.
• [28] Narendar, S., Gopalakrishnan, S., Nonlocal scale effects on wave propagation in multi-walled carbon nanotubes, Comput. Mater. Sci., 47, 526-538, 2009.
• [29] Narendar, S., Terahertz wave propagation in uniform nanorods: a nonlocal continuum mechanics formulation including the effect of lateral inertia, Physica E: Low-dimensional Syst. Nanostruct., 43, 1015-1020, 2011.
• [30] Erbay, S., Erbay, H. A., Dost, S., Nonlinear wave modulation in micropolar elastic media-I. Longitudional waves, International Journal of Engineering Science, 29 (7), 845-858, 1991.
• [31] Erbay, H. A., Erbay, S., Nonlinear wave modulation in fluid filled distensible tubes, Acta Mechanica, 104, 201-214, 1994.
• [32] Akgun, G., Demiray, H., Nonlinear wave modulation in a pre-stressed viscoelastic thin tube filled with an inviscid fluid, Int. J. Non-linear Mech., 34, 571-588, 1999.
• [33] Akgun, G., Demiray, H., Modulation of non-linear axial and transverse waves in a fluid-filled thin elastic tube, Int. J. Non-linear Mech., 35, 597-611, 2000.
• [34] Erbay, H. A., Erbay, S., Erkip, A., Unidirectional wave motion in nonlocally and nonlinearly elastic medium: the KdV, BBM and CH equations, Nonlinear Waves, 64, 256-264, 2015.
• [35] Duruk, N., Erbay, H. A., Erkip, A., Blow-up and global existence for a general class of nonlocal nonlinear coupled wave equations, J. Differ. Equations, 250, 1448-1459, 2011.
• [36] Malvern, L. E., Introduction to the Mechanics of a Continuum Medium, Prentice Hall, Englwood Cliffs, New Jersey, 1969.
• [37] Mousavi, S. M., Fariborz, S. J., Free vibration of a rod undergoing finite strain, J. of Physics Conferans Series, 382(1), 2012.
• [38] Fernandes, R., El-Borgi, S., Mousavi, S. M., Reddy, J.N., Mechmoum, A., Nonlinear size-dependent longitudinal vibration of carbon nanotubes embedded in an elastic medium, Phisica E, 88, 18-25, 2017.
• [39] Jeffrey, A., Kawahara, T., Asymptotic Methods in Nonlinear Wave Theory, Pitman, Boston, 1982.
• [40] Lamb Jr., G. L., Mc Laughlin, D. W., in: Bullough, R. K., Coudrey, P. J. (Eds), Aspect of Soliton Physics:in Solitons, Springer, Berlin, 1980.
• [41] Tu, Z-C., Single walled and multiwalled carbon nanotubes viewed as elastic tubes with the effective Young’s moduli dependent on layer number, Physics Rev. B, 65, 233-237, 2002
• [42] Taha, T. R., Ablowitz, M. J., Analytical and numerical aspects of certain nonlinear evolution equations. II Numerical nonlinear Schrödinger equation, J. Comput. Phys., 55, 203-230, 1984
• [43] Wang, H., Dong, K., Men, F., Yan, Y. J., Wang, X., Influences of longitudinal magnetic field on wave propagation in carbon nanotubes embedded in elastic matrix, Appl. Math. Model., 34, 878-889, 2010.