Analytical Investigations of Thermal-Magnetic Effects on Nonlinear Vibration of Carbon Nanotube in a Multi-layer Elastic Media using Differential Transformation Method

ABSTRACT

In this work, the differential transformation approach with after-treatment technique is used to evaluate the nonlinear vibration of a single-walled nanotube operating in a multi-layer elastic media in a thermal-magnetic environment. Hamilton’s principle is used to calculate the equation of motion for the nanotube based on nonlocal elasticity theory and the Euler-Bernoulli beam model. The nonlinear vibration model is then broken down into its spatial and temporal components using the Galerkin decomposition technique. Using the differential transformation approach in conjunction with the cosine after-treatment technique, the resulting temporal ordinary differential equation is solved. The effect of nonlocal, elastic media and thermal-magnetic factors on the dynamic behavior of the nanotube is further examined using the established analytical solution. The analytical analysis demonstrates that the frequency ratio rises as the dimensionless amplitude grows as the frequency ratio increases, as do the values of the dimensionless nonlocal, quadratic, and cubic elastic medium stiffness parameters. However, as the temperature change, magnetic force, Winkler, and Pasternak layer stiffness factors increase, the frequency ratio falls. The study, design, and uses of nanotubes in thermal and magnetic settings are greatly aided by the current work.

References

• Iijima. Helical micro tubes of graphitic carbon. Nature 1991; 354:56–58.
• Terrones, F. Banhart, N. Grobert, J. Charlier, C. Terrones, H. Ajayan. Molecular junctions by joining single-walled carbon nanotubes. Phys Rev Lett 2002; 89:07550.
• Nagy, R. Ehlich, L.P. Biro, J. Gjyulai. Y-branching of single walled carbon nanotubes. Appl Phys A Mater 2000; 70:481-3.
• A. Chernozatonskii. Carbon nanotubes connectors and planar jungle gyms. Appl Phys A 1992; 172:173–6.
• M. Liew, C.H. Wong, X.Q. He, M.J. Tan, S.A. Meguid, Nanomechanics of single and multiwalled carbon nanotubes, Physical Review, 2004, B69
• Pantano, M.C. Boyce, D.M. Parks, Mechanics of axial compression of single and multi-wall carbon nanotubes, Journal of Engineering Materials and Technology 126, 2004, 279–284.
• Pantano, D.M. Parks, M.C. Boyce, Mechanics of deformation of single- and multi-wall carbon nanotubes, Journal of the Mechanics and Physics of Solids 52, 2004, 789–821.
• Qian, G.J. Wagner, W.K. Liu, M.F. Yu, R.S. Ruoff, Mechanics of carbon nanotubes, Applied Mechanics Reviews 55, 2002, 495–533.
• P. Salvetat, J.-M. Bonard, N.H. Thomson, A.J. Kulik, L. Forro, W. Benoit, L. Zuppiroli, Mechanical properties of carbon nanotubes, Applied Physics A69, 1999, 255–260.
• Sears, R.C. Batra, Buckling of carbon nanotubes under axial compression, Physical Review, 2006, B73.
• Yoon, C.Q. Ru, A. Mioduchowski, Noncoaxial resonance of an isolated multiwall carbon nanotube, Physical Review, 2002, B66.
• Wang, H. Cai, Effects of initial stress on non-coaxial resonance of multi-wall carbon nanotubes, Acta Materialia 54, 2006, 2067–2074.
• M. Wang, V.B.C. Tan, Y.Y. Zhang, Timoshenko beam model for vibration analysis of multi-walled carbon nanotubes, Journal of Sound and Vibration, 2006, 294, 1060–1072.
• Zhang, G. Liu, X. Han, Transverse vibrations of double-walled carbon nanotubes under compressive axial load, Physics Letters A340, 2005, 258–266.
• Elishakoff, D. Pentaras, Fundamental natural frequencies of double-walled carbon nanotubes, Journal of Sound and Vibration 322, 2009, 652–664.
• Buks, B. Yurke, Mass detection with nonlinear nanomechanical resonator, 2006, Physical Review E74.
• W.C. Postma, I. Kozinsky, A. Husain, M.L. Roukes, Dynamic range of nanotube- and nanowire-based electromechanical systems, 2005, Applied Physics Letters 86.
• M. Fu, J.W. Hong, X.Q. Wang, Analysis of nonlinear vibration for embedded carbon nanotubes, Journal of Sound and Vibration, 296, 2006, 746–756.
• Dequesnes, Z. Tang, N.R. Aluru, Static and dynamic analysis of carbon nanotube-based switches, Transactions of the ASME126, 2004, 230–237.
• M. Ouakad, M.I. Younis, Nonlinear dynamics of electrically actuated carbon nanotube resonators, Journal of Computational and Nonlinear Dynamics, 2010, 5.
• Zamanian, S.E. Khadem, S.N. Mahmoodi, Analysis of non-linear vibrations of a microresonator under piezoelectric and electrostatic actuations, Journal of Mechanical Engineering Science 223, 2009, 329–344.
• Belhadj, A. Boukhalfa, S. Belalia. Carbon Nanotube Structure Vibration Based on Non local Elasticity. Journal of Modern Materials, 2016, 3(1), 9-13. doi:http://dx.doi.org/10.21467/jmm.3.1.9-13.
• M. Abdel-Rahman, A.H. Nayfeh, Secondary resonances of electrically actuated resonant microsensors, Journal of Micromechnics and Microengineering, 2003, 13, 491–501.
• A. Hawwa, H.M. Al-Qahtani, Nonlinear oscillations of a double-walled carbon nanotube, Computational Materials Science48, 2010, 140–143.
• Hajnayeb, S.E. Khadem, Nonlinear vibration and stability analysis of a double-walled carbon nanotube under electrostatic actuation, Journal of Sound and Vibration 331, 2012, 2443–2456.
• Y. Xu, X.N. Guo, C.Q. Ru, Vibration of a double-walled carbon nanotube aroused by nonlinear intertube van der Waals forces, Journal of Applied Physics, 2006, 99.
• Lei, X.W., Natsuki, T., Shi, J.X., Ni, Q.Q., Surface effects on the vibrational frequency of double-walled carbon nanotubes using the nonlocal Timoshenko beam model, Composites Part B 2012; 43:64-69.
• A. Ghorbanpour, M.S. Zarei, S. Amir, M.Z. Khoddami. Nonlinear nonlocal vibration of embedded DWCNT conveying fluid using shell model, Physica B 2013; 410:188-196
• Yoon, C.Q. Ru, A. Mioduchowski, Non-coaxial resonance of an isolated multiwall carbon nanotube, Physical Review B 66 (2002) 233402–233414.
• Yoon, C.Q. Ru, A. Mioduchowski, Vibration of an embedded multiwalled carbon nanotube [J], Composites Science and Technology 63 (2003) 1533–1542.
• Ansari and M. Hemmatnezhad. Nonlinear vibrations of embedded multi-walled carbon nanotubes using a variational approach. Mathematical and Computer Modelling 53(5-6), 2011: 927-938
• Ghorbanpour Arani, H. Rabbani, S. Amir, Z. Khoddami Maraghi, M. Mohammadimehr, E. Haghparast. Analysis of Nonlinear Vibrations for Multi-Walled Carbon Nanotubes Embedded in an Elastic Medium. Journal of Solid Mechanics. 3(3) (2011), 258-270.
• Yoon, C. Q. Ru C, A. Miodochowski. Vibration of an embedded multiwalled carbon nanotubes. Composites Science and Technology 63(2003), 1533-1542.
• M. Wang, V. B. C. Tan, Y. Y. Zhang. Timoshenko beam model for vibration analysis of multi-walled carbon nanotubes. Journal of Sound and Vibration 294(2006), 1060-1072.
• Aydogdu. Vibration of multi-walled carbon nanotubes by generalized shear deformation theory. International Journal of Mechanical Sciences 50(2008), 837–844.
• G. Sobamowo. Nonlinear Vibration Analysis of Single-Walled Carbon Nanotube Conveying Fluid in Slip Boundary Conditions Using Variational Iterative Method. Journal of Applied and Computational Mechanics 2(4) (2016), 208-221.
• G. Sobamowo. Nonlinear Analysis of Flow-induced Vibration in Fluid-conveying Structures using Differential Transformation Method with Cosine-Aftertreatment Technique. Iranian Journal of Mechanical Engineering Transactions of the ISME 18 (1) (2017), 5-42.
• G. Sobamowo. Nonlinear thermal and flow-induced vibration analysis of fluid-conveying carbon nanotube resting on Winkler and Pasternak foundations. Thermal Science and Engineering Progress. 4 (2017), 133-149.
• G. Sobamowo, B. Y. Ogunmola and C. A. Osheku. Thermo-mechanical nonlinear vibration analysis of fluid-conveying structures subjected to different boundary conditions using Galerkin-Newton-Harmonic balancing method. Journal of Applied and Computational Mechanics 3 (1), 60-79, 2017.
• AzamArefi & HassanNahvi(2017)Stability analysis of an embedded single-walled carbon nanotube with small initial curvature based on nonlocal theory,Mechanics of Advanced Materials and Structures,24:11,962-970
• Cigeroglu, H. Samandari. Nonlinear free vibrations of curved double walled carbon nanotubes using differential quadrature method Physica E 64 (2014) 95–105.
• C. Eringen. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves”, Journal of Applied Physics, 54(9) (1983), 4703–4710.
• C. Eringen. Linear theory of nonlocal elasticity and dispersion of plane waves. Inter- national Journal of Engineering Science, 10(5) (1972), 425–435.
• C. Eringen. Nonlocal continuum field theories. Springer, New York 2002.
• C. Eringen and D. G. B. Edelen. On nonlocal elasticity”, International Journal of Engineering Science, 10(3) (1972), 233–248.
• Yang, A. Chong, D. C. C. Lam and P. Tong, “Couple stress based strain gradient theory for elasticity,” Int. J. Solids Struct., 39(10) (2002), 2731–2743.
• Park and X.-L. Gao, “Variational formulation of a modified couple stress theory and its application to a simple shear problem,” Z. Angew. Math. Phys., 59(5) (2008), 904–917.
• Peddieson, G. R. Buchanan and R. P. McNitt, Application of nonlocal continuum models to nanotechnology, Int. J. Eng. Sci., 41(3-5) (2003), 305–312.
• Lu, H. Lee, C. Lu and P. Zhang, Dynamic properties of flexural beams using a nonlocal elasticity model, J. Appl. Phys., 99(7) (2006), 073510.
• Reddy, Nonlocal theories for bending, buckling and vibration of beams, Int. J. Eng. Sci., 45(2-8) (2007), 2–8, 288–307.
• Reddy and S. Pang, Nonlocal continuum theories of beams for the analysis of carbon nanotubes, J. Appl. Phys., 103(2) (2008), 023511.
• W. Lim, “On the truth of nanoscale for nanobeams based on nonlocal elastic stress field theory: Equilibrium, governing equation and static deflection,” Appl. Math. Mech. Engl. Ed., vol. 31(1) (2010), 37–54.
• W. Lim, “Is a nanorod (or nanotube) with a lower Young’s modulus stiffer? Is not Young’s modulus a stiffness indicator ?,” Sci. China Phys. Mech. Astron, 53(4) (2010), 712–724.
• Hosseini and O. Rahmani, “Thermomechanical vibration of curved functionally graded nanobeam based on nonlocal elasticity,” J. Therm. Stresses, 39(10) (2016), 1252–1267.
• Tylikowski, “Instability of thermally induced vibrations of carbon nanotubes via nonlocal elasticity,” J. Therm. Stresses, 35(1–3) (2012), 281–289.
• Ebrahimi and F. Mahmoodi, “Vibration analysis of carbon nanotubes with multiple cracks in thermal environment,” Adv. Nano Res., 6(1) (2018), 57–80.
• Zhang, X. Liu and G. Liu, Thermal effect on transverse vibrations of double-walled carbon nanotubes, Nanotechnology, 18(44) (2007), 445701.
• Murmu and S. Pradhan, Thermo-mechanical vibration of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory, Comput. Mater. Sci., 46(4), (2009) 854–859.
• Karli_ci_c, D. Jovanovi_c, P. Kozi_c and M. Caji_c, “Thermal and magnetic effects on the vibration of a cracked nanobeam embedded in an elastic medium,” J. Mech. Mater. Struct., 10(1) (2015), 43–62, 2015.
• Zarepour and S. A. Hosseini, “A semi analytical method for electro-thermo-mechanical nonlinear vibration analysis of nanobeam resting on the Winkler–Pasternak foundations with general elastic boundary conditions,” Smart Mater. Struct, 25(8) (2016), 085005.
• Ke, Y. Xiang, J. Yang and S. Kitipornchai, “Nonlinear free vibration of embedded double-walled carbon nanotubes based on nonlocal Timoshenko beam theory,” Comput. Mater. Sci., 47(2) (2009), 409–417.
• Togun, “Nonlocal beam theory for nonlinear vibrations of a nanobeam resting on elastic foundation,” Bound Value Probl., vol. 2016 (1), 57.
• Ansari, R. Gholami and M. Darabi, “Nonlinear free vibration of embedded double-walled carbon nanotubes with layerwise boundary conditions,” Acta Mech., 223(12) (2012), 2523–2536.
• S. Ma’en. Superharmonic resonance analysis of nonlocal nano beam subjected to axial thermal and magnetic forces and resting on a nonlinear elastic foundation. Microsyst. Technol., 23(8) (2017), 3319–3330.
• S. Ma’en, “Superharmonic resonance analysis of nonlocal nano beam subjected to axial thermal and magnetic forces and resting on a nonlinear elastic foundation,” Microsyst. Technol., 23(8) (2017), 3319–3330.
• Fallah and M. Aghdam, “Nonlinear free vibration and post-buckling analysis of functionally graded beams on nonlinear elastic foundation,” Eur J. Mech. A/Solids, 30(4) (2011), 571–583, 2011.
• Fallah and M. Aghdam, “Thermo-mechanical buckling and nonlinear free vibration analysis of functionally graded beams on nonlinear elastic foundation,” Compos. B Eng., 43(3) (2012), 1523–1530.
• Murmu and S. Pradhan, “Thermal effects on the stability of embedded carbon nanotubes,” Comput. Mater. Sci., 47(3) (2010), 721–726.
• Simsek. Large amplitude free vibration of nanobeams with various boundary conditions based on the nonlocal elasticity theory. Compos. B: Eng., 56 (2014), 621–628.
• Murmu, S.C. Pradhan, Thermo-mechanical vibration of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory, Comput. Mater. Sci. 46 (2009) 854-859.
• S. Abdullah, S. Hosseini-Hashemi, N. A. Hussein and R. Nazemnezhad. Thermal stress and magnetic effects on nonlinear vibration of nanobeams embedded in nonlinear elastic medium. Journal of Thermal Stresses. 43(10) (2020), 1316-1332.
• Moradi, T. Hayat and A. Alsaedi, Convective-radiative thermal analysis of triangular fins with temperature-dependent thermal conductivity by DTM. Energy Conversion and Management 77 (2014) 70–77

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