Nonlocal effect on the axisymmetric nonlinear vibrational response of nano-disks using variational iteration method

Document Type : Research Paper

Authors

1 Department of Mechanical Engineering, Shahid Chamran University of Ahvaz, Ahvaz, Iran

2 Department of Mechanical Engineering, University of Hormozgan, Bandar Abbas, Iran

Abstract

In this study, the nonlinear free vibration of a nano-disk considering small scale effects has been investigated by using the nonlocal elasticity. To take into account the nonlinear geometric effects, the nonlinear model of von Karman strain has been used while the governing differential equation was extracted according to Hamilton principle. The Galerkin weighted residual method in conjunction with the variational iteration method (VIM) was introduced to solve the governing equations for simply supported and clamped edge boundary conditions. For further comparison, the nonlinear equation was solved using the fourth-order Runge-Kutta method. Very good agreements were observed between the results of both methods, while the former method made the solution much easier. Additionally, it was observed that the ratio of thickness to radius, h/R, plays an important role on the nonlinear frequencies. This effect appears to be minute if the local elasticity theory is adopted. However, results indicated that the nonlocal effect may be ignored provided h/R ratio is very small.

Keywords

[1]           P. Ball, Roll up for the revolution, Nature Publishing Group, 2001.
[2]           R. H. Baughman, A. A. Zakhidov, W. A. De Heer, Carbon nanotubes--the route toward applications, science, Vol. 297, No. 5582, pp. 787-792, 2002.
[3]           B. Bodily, C. Sun, Structural and equivalent continuum properties of single-walled carbon nanotubes, International Journal of Materials and Product Technology, Vol. 18, No. 4-6, pp. 381-397, 2003.
[4]           C. Li, T.-W. Chou, A structural mechanics approach for the analysis of carbon nanotubes, International Journal of Solids and Structures, Vol. 40, No. 10, pp. 2487-2499, 2003.
[5]           C. Li, T.-W. Chou, Single-walled carbon nanotubes as ultrahigh frequency nanomechanical resonators, Physical review B, Vol. 68, No. 7, pp. 073405, 2003.
[6]           S. Pradhan, J. Phadikar, Nonlinear analysis of carbon nanotubes, Proceedings of Fifth International Conference on Smart Materials, Structures and Systems, Indian Institute of Science, Bangalore, pp. 24-26, 2008.
[7]           A. Chong, F. Yang, D. C. Lam, P. Tong, Torsion and bending of micron-scaled structures, Journal of Materials Research, Vol. 16, No. 4, pp. 1052-1058, 2001.
[8]           N. Fleck, G. Muller, M. Ashby, J. Hutchinson, Strain gradient plasticity: theory and experiment, Acta Metallurgica et Materialia, Vol. 42, No. 2, pp. 475-487, 1994.
[9]           Z. E. Hajilak, J. Pourghader, D. Hashemabadi, F. S. Bagh, M. Habibi, H. Safarpour, Multilayer GPLRC composite cylindrical nanoshell using modified strain gradient theory, Mechanics Based Design of Structures and Machines, 2019.
[10]         C. Lim, G. Zhang, J. Reddy, A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation, Journal of the Mechanics and Physics of Solids, Vol. 78, pp. 298-313, 2015.
[11]         A. Babaei, A. Rahmani, Vibration analysis of rotating thermally-stressed gyroscope, based on modified coupled displacement field method, Mechanics based design of structures and machines, pp. 1-10, 2020.
[12]         M. Mohammadimehr, M. Mahmudian-Najafabadi, Bending and free vibration analysis of nonlocal functionally graded nanocomposite timoshenko beam model rreinforced by swbnnt based on modified coupled stress theory, Journal of Nanostructures, Vol. 3, No. 4, pp. 483-492, 2013.
[13]         C. H. Thai, A. Ferreira, H. Nguyen-Xuan, P. Phung-Van, A size dependent meshfree model for functionally graded plates based on the nonlocal strain gradient theory, Composite Structures, Vol. 272, pp. 114169, 2021.
[14]         G. T. Monaco, N. Fantuzzi, F. Fabbrocino, R. Luciano, Hygro-thermal vibrations and buckling of laminated nanoplates via nonlocal strain gradient theory, Composite Structures, Vol. 262, pp. 113337, 2021.
[15]         M. Ebrahimian, A. Imam, M. Najafi, Doublet mechanical analysis of bending of Euler‐Bernoulli and Timoshenko nanobeams, ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 98, No. 9, pp. 1642-1665, 2018.
[16]         M. Ebrahimian, A. Imam, M. Najafi, The effect of chirality on the torsion of nanotubes embedded in an elastic medium using doublet mechanics, Indian Journal of Physics, Vol. 94, No. 1, pp. 31-45, 2020.
[17]         M. Emadi, M. Z. Nejad, S. Ziaee, A. Hadi, Buckling analysis of arbitrary two-directional functionally graded nano-plate based on nonlocal elasticity theory using generalized differential quadrature method, Steel and Composite Structures, Vol. 39, No. 5, pp. 565-581, 2021.
[18]         E. Zarezadeh, V. Hosseini, A. Hadi, Torsional vibration of functionally graded nano-rod under magnetic field supported by a generalized torsional foundation based on nonlocal elasticity theory, Mechanics Based Design of Structures and Machines, Vol. 48, No. 4, pp. 480-495, 2020.
[19]         L. Shen, H.-S. Shen, C.-L. Zhang, Nonlocal plate model for nonlinear vibration of single layer graphene sheets in thermal environments, Computational Materials Science, Vol. 48, No. 3, pp. 680-685, 2010.
[20]         S. Ramezani, A micro scale geometrically non-linear Timoshenko beam model based on strain gradient elasticity theory, International Journal of Non-Linear Mechanics, Vol. 47, No. 8, pp. 863-873, 2012.
[21]         E. Jomehzadeh, A. Saidi, Study of small scale effect on nonlinear vibration of nano-plates, Journal of Computational and Theoretical Nanoscience, Vol. 9, No. 6, pp. 864-871, 2012.
[22]         X. He, J. Wang, B. Liu, K. M. Liew, Analysis of nonlinear forced vibration of multi-layered graphene sheets, Computational Materials Science, Vol. 61, pp. 194-199, 2012.
[23]         S. Ramezani, Nonlinear vibration analysis of micro-plates based on strain gradient elasticity theory, Nonlinear Dynamics, Vol. 73, No. 3, pp. 1399-1421, 2013.
[24]         L. Zhang, Y. Zhang, K. Liew, Modeling of nonlinear vibration of graphene sheets using a meshfree method based on nonlocal elasticity theory, Applied Mathematical Modelling, Vol. 49, pp. 691-704, 2017.
[25]         M. Mohammadi, M. Hosseini, M. Shishesaz, A. Hadi, A. Rastgoo, Primary and secondary resonance analysis of porous functionally graded nanobeam resting on a nonlinear foundation subjected to mechanical and electrical loads, European Journal of Mechanics-A/Solids, Vol. 77, pp. 103793, 2019.
[26]         M. Z. Nejad, A. Hadi, A. Omidvari, A. Rastgoo, Bending analysis of bi-directional functionally graded Euler-Bernoulli nano-beams using integral form of Eringen's non-local elasticity theory, Structural engineering and mechanics: An international journal, Vol. 67, No. 4, pp. 417-425, 2018.
[27]         M. M. Adeli, A. Hadi, M. Hosseini, H. H. Gorgani, Torsional vibration of nano-cone based on nonlocal strain gradient elasticity theory, The European Physical Journal Plus, Vol. 132, No. 9, pp. 393, 2017/09/18, 2017.
[28]         M. Shishesaz, M. Hosseini, K. Naderan Tahan, A. Hadi, Analysis of functionally graded nanodisks under thermoelastic loading based on the strain gradient theory, Acta Mechanica, Vol. 228, No. 12, pp. 4141-4168, 2017/12/01, 2017.
[29]         M. Hosseini, A. Hadi, A. Malekshahi, M. Shishesaz, A review of size-dependent elasticity for nanostructures, Journal of Computational Applied Mechanics, Vol. 49, No. 1, pp. 197-211, 2018.
[30]         M. Mousavi Khoram, M. Hosseini, M. Shishesaz, A concise review of nano-plates, Journal of Computational Applied Mechanics, Vol. 50, No. 2, pp. 420-429, 2019.
[31]         A. Daneshmehr, A. Rajabpoor, A. Hadi, Size dependent free vibration analysis of nanoplates made of functionally graded materials based on nonlocal elasticity theory with high order theories, International Journal of Engineering Science, Vol. 95, pp. 23-35, 2015.
[32]         H. Haghshenas Gorgani, M. Mahdavi Adeli, M. Hosseini, Pull-in behavior of functionally graded micro/nano-beams for MEMS and NEMS switches, Microsystem Technologies, Vol. 25, No. 8, pp. 3165-3173, 2019/08/01, 2019.
[33]         A. Hadi, M. Z. Nejad, A. Rastgoo, M. Hosseini, Buckling analysis of FGM Euler-Bernoulli nano-beams with 3D-varying properties based on consistent couple-stress theory, Steel and Composite Structures, Vol. 26, No. 6, pp. 663-672, 2018.
[34]         A. Barati, M. M. Adeli, A. Hadi, Static torsion of bi-directional functionally graded microtube based on the couple stress theory under magnetic field, International Journal of Applied Mechanics, Vol. 12, No. 02, pp. 2050021, 2020.
[35]         A. Barati, A. Hadi, M. Z. Nejad, R. Noroozi, On vibration of bi-directional functionally graded nanobeams under magnetic field, Mechanics Based Design of Structures and Machines, pp. 1-18, 2020.
[36]         K. Dehshahri, M. Z. Nejad, S. Ziaee, A. Niknejad, A. Hadi, Free vibrations analysis of arbitrary three-dimensionally FGM nanoplates, Advances in nano research, Vol. 8, No. 2, pp. 115-134, 2020.
[37]         M. Moraveji, H. Keshvari, A. Karkhaneh, S. Bonakdar, A. Hadi, N. Haghighipour, The effect of collagen/polycaprolactone fibrous scaffold decorated with graphene nanoplatelet and low-frequency electromagnetic field on neuronal gene expression by stem cells, Advances in nano research, Vol. 10, No. 6, pp. 549-557, 2021.
[38]         M. Najafzadeh, M. M. Adeli, E. Zarezadeh, A. Hadi, Torsional vibration of the porous nanotube with an arbitrary cross-section based on couple stress theory under magnetic field, Mechanics Based Design of Structures and Machines, pp. 1-15, 2020.
[39]         H. Nekounam, R. Dinarvand, R. Khademi, F. Asghari, N. Mahmoodi, H. Arzani, E. Hasanzadeh, A. Hadi, R. Karimi, M. Kamali, Preparation of cationized albumin nanoparticles loaded indirubin by high pressure hemogenizer, bioRxiv, 2021.
[40]         A. Soleimani, K. Dastani, A. Hadi, M. H. Naei, Effect of out-of-plane defects on the postbuckling behavior of graphene sheets based on nonlocal elasticity theory, Steel and Composite Structures, Vol. 30, No. 6, pp. 517-534, 2019.
[41]         M. Shishesaz, M. Hosseini, K. N. Tahan, A. Hadi, Analysis of functionally graded nanodisks under thermoelastic loading based on the strain gradient theory, Acta Mechanica, Vol. 228, No. 12, pp. 4141-4168, 2017.
[42]         M. M. Khoram, M. Hosseini, A. Hadi, M. Shishehsaz, Bending Analysis of Bidirectional FGM Timoshenko Nanobeam Subjected to Mechanical and Magnetic Forces and Resting on Winkler–Pasternak Foundation, International Journal of Applied Mechanics, Vol. 12, No. 08, pp. 2050093, 2020.
[43]         A. Hadi, A. Rastgoo, N. Haghighipour, A. Bolhassani, Numerical modelling of a spheroid living cell membrane under hydrostatic pressure, Journal of Statistical Mechanics: Theory and Experiment, Vol. 2018, No. 8, pp. 083501, 2018.
[44]         R. Noroozi, A. Barati, A. Kazemi, S. Norouzi, A. Hadi, Torsional vibration analysis of bi-directional FG nano-cone with arbitrary cross-section based on nonlocal strain gradient elasticity, Advances in nano research, Vol. 8, No. 1, pp. 13-24, 2020.
[45]         A. Hadi, M. Z. Nejad, M. Hosseini, Vibrations of three-dimensionally graded nanobeams, International Journal of Engineering Science, Vol. 128, pp. 12-23, 2018.
[46]         N.-C. Tsai, J.-S. Liou, C.-C. Lin, T. Li, Analysis and fabrication of reciprocal motors applied for microgyroscopes, Journal of Micro/Nanolithography, MEMS, and MOEMS, Vol. 8, No. 4, pp. 043046, 2009.
[47]         N.-C. Tsai, J.-S. Liou, C.-C. Lin, T. Li, Design of micro-electromagnetic drive on reciprocally rotating disc used for micro-gyroscopes, Sensors and Actuators A: Physical, Vol. 157, No. 1, pp. 68-76, 2010.
[48]         N.-C. Tsai, J.-S. Liou, C.-C. Lin, T. Li, Suppression of dynamic offset of electromagnetic drive module for micro-gyroscope, Mechanical Systems and Signal Processing, Vol. 25, No. 2, pp. 680-693, 2011.
[49]         S. Lee, D. Kim, M. D. Bryant, F. F. Ling, A micro corona motor, Sensors and Actuators A: Physical, Vol. 118, No. 2, pp. 226-232, 2005.
[50]         K. J. Vahala, Optical microcavities, nature, Vol. 424, No. 6950, pp. 839, 2003.
[51]         R. W. Boyd, J. E. Heebner, Sensitive disk resonator photonic biosensor, Applied optics, Vol. 40, No. 31, pp. 5742-5747, 2001.
[52]         A. Dolatabady, N. Granpayeh, V. F. Nezhad, A nanoscale refractive index sensor in two dimensional plasmonic waveguide with nanodisk resonator, Optics Communications, Vol. 300, pp. 265-268, 2013.
[53]         Y. Zhang, S. Tekobo, Y. Tu, Q. Zhou, X. Jin, S. A. Dergunov, E. Pinkhassik, B. Yan, Permission to enter cell by shape: nanodisk vs nanosphere, ACS applied materials & interfaces, Vol. 4, No. 8, pp. 4099-4105, 2012.
[54]         I. Hwang, J. Choi, S. Hong, J.-S. Kim, I.-S. Byun, J. H. Bahng, J.-Y. Koo, S.-O. Kang, B. H. Park, Direct investigation on conducting nanofilaments in single-crystalline Ni/NiO core/shell nanodisk arrays, Applied Physics Letters, Vol. 96, No. 5, pp. 053112, 2010.
[55]         C. Horejs, D. Pum, U. B. Sleytr, H. Peterlik, A. Jungbauer, R. Tscheliessnig, Surface layer protein characterization by small angle x-ray scattering and a fractal mean force concept: from protein structure to nanodisk assemblies, The Journal of chemical physics, Vol. 133, No. 17, pp. 11B602, 2010.
[56]         C. Hägglund, M. Zäch, G. Petersson, B. Kasemo, Electromagnetic coupling of light into a silicon solar cell by nanodisk plasmons, Applied physics letters, Vol. 92, No. 5, pp. 053110, 2008.
[57]         C.-H. Huang, M. Igarashi, S. Horita, M. Takeguchi, Y. Uraoka, T. Fuyuki, I. Yamashita, S. Samukawa, Novel Si nanodisk fabricated by biotemplate and defect-free neutral beam etching for solar cell application, Japanese Journal of Applied Physics, Vol. 49, No. 4S, pp. 04DL16, 2010.
[58]         T. Ito, A. A. Audi, G. P. Dible, Electrochemical characterization of recessed nanodisk-array electrodes prepared from track-etched membranes, Analytical chemistry, Vol. 78, No. 19, pp. 7048-7053, 2006.
[59]         T. Ito, D.-M. Perera, Electrochemical studies of recessed nanodisk-array electrodes prepared from track-etched membranes, in Proceeding of, The Electrochemical Society, pp. 1325-1325.
[60]         L. Luo, H. S. White, Electrogeneration of single nanobubbles at sub-50-nm-radius platinum nanodisk electrodes, Langmuir, Vol. 29, No. 35, pp. 11169-11175, 2013.
[61]         M. K. Chin, D. Y. Chu, S. T. Ho, Estimation of the spontaneous emission factor for microdisk lasers via the approximation of whispering gallery modes, Journal of applied physics, Vol. 75, No. 7, pp. 3302-3307, 1994.
[62]         H. Cao, J. Xu, W. Xiang, Y. Ma, S.-H. Chang, S.-T. Ho, G. Solomon, Optically pumped InAs quantum dot microdisk lasers, Applied Physics Letters, Vol. 76, No. 24, pp. 3519-3521, 2000.
[63]         S. J. Choi, K. Djordjev, S. J. Choi, P. D. Dapkus, Microdisk lasers vertically coupled to output waveguides, IEEE Photonics technology letters, Vol. 15, No. 10, pp. 1330-1332, 2003.
[64]         J. Van Campenhout, P. Rojo-Romeo, P. Regreny, C. Seassal, D. Van Thourhout, S. Verstuyft, L. Di Cioccio, J.-M. Fedeli, C. Lagahe, R. Baets, Electrically pumped InP-based microdisk lasers integrated with a nanophotonic silicon-on-insulator waveguide circuit, Optics express, Vol. 15, No. 11, pp. 6744-6749, 2007.
[65]         S.-H. Kwon, J.-H. Kang, S.-K. Kim, H.-G. Park, Surface plasmonic nanodisk/nanopan lasers, IEEE Journal of Quantum Electronics, Vol. 47, No. 10, pp. 1346-1353, 2011.
[66]         S.-Y. Cho, N. M. Jokerst, A polymer microdisk photonic sensor integrated onto silicon, IEEE photonics technology letters, Vol. 18, No. 20, pp. 2096-2098, 2006.
[67]         Y.-G. Wang, W.-H. Lin, N. Liu, Large amplitude free vibration of size-dependent circular microplates based on the modified couple stress theory, International Journal of Mechanical Sciences, Vol. 71, pp. 51-57, 2013.
[68]         M. Mohammadi, M. Ghayour, A. Farajpour, Free transverse vibration analysis of circular and annular graphene sheets with various boundary conditions using the nonlocal continuum plate model, Composites Part B: Engineering, Vol. 45, No. 1, pp. 32-42, 2013.
[69]         M. Hosseini, M. Shishesaz, K. N. Tahan, A. Hadi, Stress analysis of rotating nano-disks of variable thickness made of functionally graded materials, International Journal of Engineering Science, Vol. 109, pp. 29-53, 2016.
[70]         M. Shishesaz, M. Shariati, A. Yaghootian, A. Alizadeh, Nonlinear Vibration Analysis of Nano-Disks Based on Nonlocal Elasticity Theory Using Homotopy Perturbation Method, International Journal of Applied Mechanics, pp. 1950011, 2019.
[71]         M. Hosseini, M. Shishesaz, A. Hadi, Thermoelastic analysis of rotating functionally graded micro/nanodisks of variable thickness, Thin-Walled Structures, Vol. 134, pp. 508-523, 2019.
[72]         M. Shishesaz, M. Shariati, A. Yaghootian, Nonlocal Elasticity Effect on Linear Vibration of Nano-circular Plate Using Adomian Decomposition Method, Journal of Applied and Computational Mechanics, Vol. 6, No. 1, pp. 63-76, 2020.
[73]         M. Shariati, B. Azizi, M. Hosseini, M. Shishesaz, On the calibration of size parameters related to non-classical continuum theories using molecular dynamics simulations, International Journal of Engineering Science, Vol. 168, pp. 103544, 2021.
[74]         M. Shishesaz, M. Shariati, M. Hosseini, Size effect analysis on Vibrational response of Functionally Graded annular nano plate based on Nonlocal stress-driven method, International Journal of Structural Stability and Dynamics, 2021, In press.
[75]         A. C. Eringen, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of applied physics, Vol. 54, No. 9, pp. 4703-4710, 1983.
[76]         S. S. Rao, 2019, Vibration of continuous systems, Wiley,
[77]         M. Amabili, 2008, Nonlinear vibrations and stability of shells and plates, Cambridge University Press,
[78]         C.-Y. Chia, 1980, Nonlinear analysis of plates, McGraw-Hill International Book Company,
[79]         S. P. Timoshenko, S. Woinowsky-Krieger, 1959, Theory of plates and shells, McGraw-hill,
[80]         W. F. Faris, Nonlinear dynamics of annular and circular plates under thermal and electrical loadings,  Thesis, Virginia Tech, 2003.
[81]         C. Wang, J. N. Reddy, K. Lee, 2000, Shear deformable beams and plates: Relationships with classical solutions, Elsevier,
[82]         J.-H. He, Variational iteration method for autonomous ordinary differential systems, Applied Mathematics and Computation, Vol. 114, No. 2-3, pp. 115-123, 2000.
[83]         J.-H. He, Some asymptotic methods for strongly nonlinear equations, International journal of Modern physics B, Vol. 20, No. 10, pp. 1141-1199, 2006.
[84]         J.-H. He, Variational iteration method—some recent results and new interpretations, Journal of computational and applied mathematics, Vol. 207, No. 1, pp. 3-17, 2007.
[85]         J.-H. He, Variational approach for nonlinear oscillators, Chaos, Solitons & Fractals, Vol. 34, No. 5, pp. 1430-1439, 2007.
[86]         S. Momani, Z. Odibat, Numerical comparison of methods for solving linear differential equations of fractional order, Chaos, Solitons & Fractals, Vol. 31, No. 5, pp. 1248-1255, 2007.
[87]         Ζ. Odibat, S. Momani, Application of variational iteration method to nonlinear differential equations of fractional order, International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 7, No. 1, pp. 27-34, 2006.
[88]         N. Bildik, A. Konuralp, The use of variational iteration method, differential transform method and Adomian decomposition method for solving different types of nonlinear partial differential equations, International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 7, No. 1, pp. 65-70, 2006.
[89]         N. Sweilam, M. Khader, Variational iteration method for one dimensional nonlinear thermoelasticity, Chaos, Solitons & Fractals, Vol. 32, No. 1, pp. 145-149, 2007.
[90]         S. Momani, S. Abuasad, Application of He’s variational iteration method to Helmholtz equation, Chaos, Solitons & Fractals, Vol. 27, No. 5, pp. 1119-1123, 2006.
[91]         A. Soliman, A numerical simulation and explicit solutions of KdV-Burgers’ and Lax’s seventh-order KdV equations, Chaos, Solitons & Fractals, Vol. 29, No. 2, pp. 294-302, 2006.
[92]         M. Inokuti, H. Sekine, T. Mura, General use of the Lagrange multiplier in nonlinear mathematical physics, Variational method in the mechanics of solids, Vol. 33, No. 5, pp. 156-162, 1978.
[93]         J. He, A new approach to establishing generalized variational principles in fluids and CC Liu constraints,  Thesis, Ph. D. Thesis, Shanghai University, 1996 (in Chinese), 1996.
[94]         A.-M. Wazwaz, 2010, Partial differential equations and solitary waves theory, Springer Science & Business Media,
[95]         S. Abbasbandy, Improving Newton–Raphson method for nonlinear equations by modified Adomian decomposition method, Applied Mathematics and Computation, Vol. 145, No. 2-3, pp. 887-893, 2003.
[96]         Z. M. Odibat, A study on the convergence of variational iteration method, Mathematical and Computer Modelling, Vol. 51, No. 9-10, pp. 1181-1192, 2010.
[97]         A. W. Leissa, ’Vibration of Plates’, Office of Technology Utilization, National Aeronautics and Space Administration, Washington, DC, 1969.
Volume 52, Issue 3
September 2021
Pages 507-534
  • Receive Date: 03 July 2020
  • Revise Date: 08 September 2021
  • Accept Date: 16 September 2021
  • First Publish Date: 16 September 2021