Advances in nano research

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Study on derivation from large-amplitude size dependent internal resonances of homogeneous and FG rod-types

  • Received : 2021.09.12
  • Accepted : 2023.12.11
  • Published : 2024.02.25
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Abstract

Recently, a lot of research has been done on the analysis of axial vibrations of homogeneous and FG nanotubes (nanorods) with various aspects of vibrations that have been fully mentioned in history. However, there is a lack of investigation of the dynamic internal resonances of FG nanotubes (nanorods) between them. This is one of the essential or substantial characteristics of nonlinear vibration systems that have many applications in various fields of engineering (making actuators, sensors, etc.) and medicine (improving the course of diseases such as cancers, etc.). For this reason, in this study, for the first time, the dynamic internal resonances of FG nanorods in the simultaneous presence of large-amplitude size dependent behaviour, inertial and shear effects are investigated for general state in detail. Such theoretical patterns permit as to carry out various numerical experiments, which is the key point in the expansion of advanced nano-devices in different sciences. This research presents an AFG novel nano resonator model based on the axial vibration of the elastic nanorod system in terms of derivation from large-amplitude size dependent internal modals interactions. The Hamilton's Principle is applied to achieve the basic equations in movement and boundary conditions, and a harmonic deferential quadrature method, and a multiple scale solution technique are employed to determine a semi-analytical solution. The interest of the current solution is seen in its specific procedure that useful for deriving general relationships of internal resonances of FG nanorods. The numerical results predicted by the presented formulation are compared with results already published in the literature to indicate the precision and efficiency of the used theory and method. The influences of gradient index, aspect ratio of FG nanorod, mode number, nonlinear effects, and nonlocal effects variations on the mechanical behavior of FG nanorods are examined and discussed in detail. Also, the inertial and shear traces on the formations of internal resonances of FG nanorods are studied, simultaneously. The obtained valid results of this research can be useful and practical as input data of experimental works and construction of devices related to axial vibrations of FG nanorods.

Keywords

References

  1. Akavci, S.S. and Tanrikulu, A.H. (2015), "Static and free vibration analysis of functionally graded plates based on a new quasi-3D and 2D shear deformation theories", Compos. Part B Eng., 83, 203-215. https://doi.org/10.1016/j.compositesb.2015.08.043
  2. Al-shujairi, M. and Mollamahmutoglu, C . (2018), "Dynamic stability of sandwich functionally graded micro-beam based on the nonlocal strain gradient theory with thermal effect", Compos. Struct., 201, 1018-1030. https://doi.org/10.1016/j.compstruct.2018.06.035
  3. Alijani, F. and Amabili, M. (2014), "Effect of thickness deformation on large-amplitude vibrations of functionally graded rectangular plates", Compos. Struct, 113, 89-107. https://doi.org/10.1016/j.compstruct.2014.03.006
  4. Amabili, M. (2008), Nonlinear Vibrations and Stability of Shells and Plates, Cambridge University Press, U.K.
  5. Arefi, M. and Amabili, M. (2021), "A comprehensive electro-magneto-elastic buckling and bending analyses of three-layered doubly curved nanoshell, based on nonlocal three-dimensional theory", Compos. Struct, 257, 113100. https://doi.org/10.1016/j.compstruct.2020.113100
  6. Aydogdu, M. (2009), "Axial vibration of the nanorods with the nonlocal continuum rod model", Physica E, 41(5), 861-864. https://doi.org/10.1016/j.physe.2009.01.007
  7. Aydogdu, M. (2015), "A nonlocal rod model for axial vibration of double-walled carbon nanotubes including axial van der Waals force effects", J. Vib. Cont. 21(16), 3132-3154. https://doi.org/10.1177/1077546313518954
  8. Benatta, M.A., Mechab, I., Tounsi, A. and Bedia, E.A. (2008), "Static analysis of functionally graded short beams including warping and shear deformation effects", Comput. Mater. Sci, 44(2), 765-773. https://doi.org/10.1016/j.commatsci.2008.05.020
  9. Cao, D., Gao, Y., Yao, M. and Zhang, W. (2018), "Free vibration of axially functionally graded beams using the asymptotic development method", Eng. Struct., 173, 442-448. doi.org/10.1016/j.engstruct.2018.06.111
  10. Chakraborty, A., Gopalakrishnan, S. and Reddy, J. N. (2003), "A new beam finite element for the analysis of functionally graded materials", Int. J. Mech. Sci., 45(3), 519-539. https://doi.org/10.1016/S0020-7403(03)00058-4
  11. Chen, D., Yang, J. and Kitipornchai, S. (2016), "Free and forced vibrations of shear deformable functionally graded porous beams", Int. J. Mech. Sci., 108, 14-22. https://doi.org/10.1016/j.ijmecsci.2016.01.025
  12. Civalek, O . (2004), "Application of differential quadrature (DQ) and harmonic differential quadrature (HDQ) for buckling analysis of thin isotropic plates and elastic columns", Eng. Struct, 26(2), 171-186. https://doi.org/10.1016/j.engstruct.2003.09.005
  13. Eringen, A.C. (1972), "Nonlocal polar elastic continua", Int. J. Eng. Sci., 10(1), 1-16. https://doi.org/10.1016/0020-7225(72)90070-5
  14. Eringen, A.C. (1983) "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", J. Appl. Phys., 54(9), 4703-4710. https://doi.org/10.1063/1.332803
  15. Eringen, A.C. and Edelen, D.G.B. (1972), "On nonlocal elasticity", Int. J. Eng. Sci. 10(3), 233-248. https://doi.org/10.1016/0020-7225(72)90039-0
  16. Fariborz, S. (2012), "Free vibration of a rod undergoing finite strain", J. Phys. Conference Series, 382(1).
  17. Faroughi, S. and Goushegir, S.M.H. (2016), "Analysis of axial vibration of non-uniform nanorod using boundary characteristic orthogonal polynomials", Modares Mech. Eng., 16(1), 203-212.
  18. Fernandes, R., El-Borgi, S., Mousavi, S.M., Reddy, J.N. and Mechmoum, A. (2017), "Nonlinear size-dependent longitudinal vibration of carbon nanotubes embedded in an elastic medium", Physica E Low Dimens, 88, 18-25. https://doi.org/10.1016/j.physe.2016.11.007
  19. Gad-el-Hak, M. (1996), "Compliant coatings: a decade of progress", Appl. Mech. Rev, 49(10S), S147-S157. https://doi.org/10.1115/1.3101966
  20. Gheshlaghi, B. and Hasheminejad, S.M. (2011), "Surface effects on nonlinear free vibration of nanobeams", Compos Part B Eng., 42(4), 934-937. https://doi.org/10.1016/j.compositesb.2010.12.026
  21. Guven, U. (2014), "Love-Bishop rod solution based on strain gradient elasticity theory", Comptes Rendus Mecanique, 342(1), 8-16. https://doi.org/10.1016/j.crme.2013.10.011
  22. Hernandez-Acosta, M.A., Martinez-Gutierrez, H., Martinez-Gonzalez, C.L., Torres-SanMiguel, C.R., Trejo-Valdez, M. and Torres-Torres, C. (2018), "Fractional and chaotic electrical signatures exhibited by random carbon nanotube networks", Phys. Scr., 93(12), 125801. https://doi.org/10.1088/1402-4896/aaec46
  23. Hosseini-Hashemi, S., Nahas, I., Fakher, M. and Nazemnezhad, R. (2014), "Surface effects on free vibration of piezoelectric functionally graded nanobeams using nonlocal elasticity", Acta Mech, 225(6), 1555-1564. https://doi.org/10.1007/s00707-013-1014-z
  24. Hsu, J.C., Lee, H.L. and Chang, W.J. (2011), "Longitudinal vibration of cracked nanobeams using nonlocal elasticity theory", Curr. Appl. Phys., 11(6), 1384-1388. https://doi.org/10.1016/j.cap.2011.04.026
  25. Jing, L.L., Ming, P.J., Zhang, W.P., Fu, L.R. and Cao, Y.P. (2016), "Static and free vibration analysis of functionally graded beams by combination Timoshenko theory and finite volume method", Compos. Struct., 138, 192-213. https://doi.org/10.1016/j.compstruct.2015.11.027
  26. Kadoli, R., Akhtar, K. and Ganesan, N. (2008), "Static analysis of functionally graded beams using higher order shear deformation theory", Appl. Math. Modell., 32(12), 2509-2525. https://doi.org/10.1016/j.apm.2007.09.015
  27. Karlicic, D., Cajic, M., Murmu, T. and Adhikari, S. (2015), "Nonlocal longitudinal vibration of viscoelastic coupled double-nanorod systems", Eur. J. Mech A Solids, 49, 183-196. https://doi.org/10.1016/j.euromechsol.2014.07.005
  28. Ke, L.L., Yang, J., Kitipornchai, S. and Xiang, Y. (2009), "Flexural vibration and elastic buckling of a cracked Timoshenko beam made of functionally graded materials", Mech. Adv. Mater. Struct., 16(6), 488-502. https://doi.org/10.1080/15376490902781175
  29. Kiani, K. (2010), "Free longitudinal vibration of tapered nanowires in the context of nonlocal continuum theory via a perturbation technique", Physica E. 43(1), 387-397. https://doi.org/10.1016/j.physe.2010.08.022
  30. Kumar, R., Dutta, S.C. and Panda, S.K. (2016), "Linear and non-linear dynamic instability of functionally graded plate subjected to non-uniform loading", Compos. Struct, 154, 219-230. https://doi.org/10.1016/j.compstruct.2016.07.050
  31. Li, C., Li, S., Yao, L. and Zhu, Z. (2015), "Nonlocal theoretical approaches and atomistic simulations for longitudinal free vibration of nanorods/nanotubes and verification of different nonlocal models", Appl. Math. Model., 39(15), 4570-4585. https://doi.org/10.1016/j.apm.2015.01.013
  32. Li, X.F. (2008), "A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler-Bernoulli beams", J. Sound Vib., 318(4-5), 1210-1229. https://doi.org/10.1016/j.jsv.2008.04.056
  33. Li, X.F., Shen, Z.B. and Lee, K.Y. (2017), "Axial wave propagation and vibration of nonlocal nanorods with radial deformation and inertia", ZAMM J. Appl. Math. Mech., 97(5), 602-616. https://doi.org/10.1002/zamm.201500186
  34. Liu, H., Lv, Z. and Wu, H. (2019), "Nonlinear free vibration of geometrically imperfect functionally graded sandwich nanobeams based on nonlocal strain gradient theory", Compos Struct, 214, 47-61. https://doi.org/10.1016/j.compstruct.2019.01.090
  35. Malekzadeh, P. and Karami, G. (2005), "Polynomial and harmonic differential quadrature methods for free vibration of variable thickness thick skew plates", Eng. Struct, 27(10), 1563-1574. https://doi.org/10.1016/j.engstruct.2005.03.017
  36. Mashat, D.S., Carrera, E., Zenkour, A.M., Al Khateeb, S.A. and Filippi, M. (2014), "Free vibration of FGM layered beams by various theories and finite elements", Compos. Part B Eng., 59, 269-278. https://doi.org/10.1016/j.compositesb.2013.12.008
  37. Mohammadian, M. and Hosseini, S. M. (2022), "A size-dependent differential quadrature element model for vibration analysis of FG CNT reinforced composite microrods based on the higher order Love-Bishop rod model and the nonlocal strain gradient theory", Eng Anal Bound Elem, 138, 235-252. https://doi.org/10.1016/j.enganabound.2022.02.017
  38. Murmu, T. and Adhikari, S. (2010), "Nonlocal effects in the longitudinal vibration of double-nanorod systems", Physica E, 43(1), 415-422. https://doi.org/10.1016/j.physe.2010.08.023
  39. Murmu, T., Adhikari, S. and McCarthy, M.A. (2014), "Axial vibration of embedded nanorods under transverse magnetic field effects via nonlocal elastic continuum theory", J. Comput. Theor. Nanosci., 11(5), 1230-1236. https://doi.org/10.1166/jctn.2014.3487
  40. Narendar, S. and Gopalakrishnan, S. (2011), "Axial wave propagation in coupled nanorod system with nonlocal small-scale effects", Compos Part B Eng., 42(7), 2013-2023. https://doi.org/10.1016/j.compositesb.2011.05.021
  41. Nayfeh, A.H. and Nayfeh, S.A. (1994), "On nonlinear modes of continuous systems", J. Vib. Acoust., 116(1), 129-136. https://doi.org/10.1115/1.2930388
  42. Nazemnezhad, R. and Kamali, K. (2018), "Free axial vibration analysis of axially functionally graded thick nanorods using nonlocal Bishop's theory", Steel. Compos. Struct., 28(6), 749-758. https://doi.org/10.12989/scs.2018.28.6.749
  43. Noroozi, M. and Ghadiri, M. (2021), "Nonlinear vibration and stability analysis of a size-dependent viscoelastic cantilever nanobeam with axial excitation", Proc. Inst. Mech. Eng. C. J. Mech. Eng. Sci., 235(18), 3624-3640. https://doi.org/10.1177/0954406220959104
  44. Pradhan, K.K. and Chakraverty, S. (2013), "Free vibration of Euler and Timoshenko functionally graded beams by Rayleigh- Ritz method", Compos. Part B Eng., 51, 175-184. https://doi.org/10.1016/j.compositesb.2013.02.027
  45. Qing, H. and Wei, L. (2022), "Linear and nonlinear free vibration analysis of functionally graded porous nanobeam using stress-driven nonlocal integral model", Commun. Nonlinear Sci. Numer. Simul., 109, 106300. https://doi.org/10.1016/j.cnsns.2022.106300
  46. Rao, S.S. (2007), Vibration of Continuous Systems, 464, Wiley, New York, U.S.A.
  47. Sankar, B.V. (2001), "An elasticity solution for functionally graded beams", Compos. Sci. Technol, 61(5), 689-696. https://doi.org/10.1016/S0266-3538(01)00007-0
  48. Shakhlavi, S.J. (2023a), "On nonlinear damping effects with nonlinear temperature-dependent properties for an axial thermo-viscoelastic rod", Int. J. Non Linear Mech, 153, 104418. https://doi.org/10.1016/j.ijnonlinmec.2023.104418
  49. Shakhlavi, S.J. (2023b), "Nonlinear nonlocal damping effects under magnetic loads of a ferromagnetic-viscoelastic nanotube exposed to a nonlinear elastic medium with nonlocal viscosity", Commun. Nonlinear Sci. Numer. Simul., 107690. https://doi.org/10.1016/j.cnsns.2023.107690
  50. Shakhlavi, S.J., Hosseini-Hashemi, S. and Nazemnezhad, R. (2020), "Torsional vibrations investigation of nonlinear nonlocal behaviour in terms of functionally graded nanotubes", Int. J. Non Linear Mech, 103513. https://doi.org/10.1016/j.ijnonlinmec.2020.103513
  51. Shakhlavi, S.J., Hosseini-Hashemi, S. and Nazemnezhad, R. (2022a), "Thermal stress effects on size-dependent nonlinear axial vibrations of nanorods exposed to magnetic fields surrounded by nonlinear elastic medium", J. Therm. Stress, 45(2), 139-153. https://doi.org/10.1080/01495739.2021.2003275
  52. Shakhlavi, S.J., Hosseini-Hashemi, S. and Nazemnezhad, R. (2022b), "Nonlinear nano-rod-type analysis of internal resonances and geometrically considering nonlocal and inertial effects in terms of Rayleigh axial vibrations", Eur. Phys. J. Plus, 137(4), 1-20. https://doi.org/10.1140/epjp/s13360-022-02594-x
  53. Shakhlavi, S.J., Nazemnezhad, R. and Hosseini-Hashemi, S. (2020), "On nonlinear torsional vibrations of nanorod", In 28th Annual Conf of Mechanical Engineering, Tehran.
  54. Shakhlavi, S.J., Nazemnezhad, R., Hosseini-Hashemi, S. and Amabili, M. (2021a), "Analysis of nonlinear nonlocal axial free vibrations of gold nanoscale rod", In 29th Annual International Conference of Iranian Association of Mechanical Engineers and 8th International Conference on Thermal Power Plants Industry, Tehran.
  55. Shakhlavi, S.J., Nazemnezhad, R., Hosseini-Hashemi, S. and Amabili, M. (2021b), "On nonlocal nonlinear internal resonances of gold nanoscale rod", In 10th International Conference on Acoustics and Vibration, Tehran.
  56. Simsek, M. (2009), "Static analysis of a functionally graded beam under a uniformly distributed load by Ritz method", Int. J. Eng. Appl. Sci., 1(3), 1-11. https://dergipark.org.tr/en/pub/ijeas/issue/23571/251092 1092
  57. Sina, S.A., Navazi, H.M. and Haddadpour, H. (2009), "An analytical method for free vibration analysis of functionally graded beams", Mater. Des., 30(3), 741-747. https://doi.org/10.1016/j.matdes.2008.05.015
  58. Striz, A.G., Wang, X. and Bert, C.W. (1995), "Harmonic differential quadrature method and applications to analysis of structural components", Acta Mech., 111(1), 85-94. https://doi.org/10.1007/BF01187729
  59. Su, H. and Banerjee, J.R. (2015), "Development of dynamic stiffness method for free vibration of functionally graded Timoshenko beams", Comput. Struct, 147, 107-116. https://doi.org/10.1016/j.compstruc.2014.10.001
  60. Thai, H.T. and VO, T.P. (2012), "Bending and free vibration of functionally graded beams using various higher-order shear deformation beam theories", Int. J. Mech. Sci, 62(1), 57-66. https://doi.org/10.1016/j.ijmecsci.2012.05.014
  61. Trinh, L.C., VO, T.P., Osofero, A.I. and Lee, J. (2016), "Fundamental frequency analysis of functionally graded sandwich beams based on the state space approach", Compos. Struct., 156, 263-275. https://doi.org/10.1016/j.compstruct.2015.11.010
  62. Yadav, A., Amabili, M., Panda, S.K. and Dey, T. (2019), "Non-linear vibration response of functionally graded circular cylindrical shells subjected to thermo-mechanical loading", Compos. Struct, 229, 111430. https://doi.org/10.1016/j.compstruct.2019.111430
  63. Yang, J. and Shen, H. S. (2002), "Vibration characteristics and transient response of shear-deformable functionally graded plates in thermal environments", J. Sound. Vib., 255(3), 579-602. https://doi.org/10.1006/jsvi.2001.4161
  64. Yang, Y., Lam, C.C., Kou, K.P. and IU, V.P. (2014), "Free vibration analysis of the functionally graded sandwich beams by a meshfree boundary-domain integral equation method", Compos. Struct, 117, 32-39. https://doi.org/10.1016/j.compstruct.2014.06.016
  65. Yapanmis, B.E. and Bagdatli, S.M. (2022), "Investigation of the non-linear vibration behaviour and 3: 1 internal resonance of the multi supported nanobeam", Z Naturforsch A, 77(4), 305-321. https://doi.org/10.1515/zna-2021-0300
  66. Yuan, Y., Zhao, K., Zhao, Y. and Kiani, K. (2020), "Nonlocal-integro-vibro analysis of vertically aligned monolayered nonuniform FGM nanorods", Steel. Compos. Struct., 37(5), 551-569. https://doi.org/10.12989/scs.2020.37.5.551
  67. Zhong, Z. and Yu, T. (2007), "Analytical solution of a cantilever functionally graded beam", Compos. Sci. Technol, 67(3-4), 481-488. https://doi.org/10.1016/j.compscitech.2006.08.023
  68. Zhu, X. and Li, L. (2017), "On longitudinal dynamics of nanorods", Int. J. Eng. Sci., 120, 129-145. https://doi.org/10.1016/j.ijengsci.2017.08.003